PEDAGOGY OF MATHEMATICS

Prepared by
Dr. K.R. SELVAKUMAR
ASST. PROFESSOR,
DD & CE – B.Ed.
Manonmaniam Sundaranar University,
Tirunelveli.
Unit I CONCEPT, NEED, OBJECTIVES AND SCOPE
1.0 Introduction
1.1 Meaning, characteristics and definition of Mathematics
1.2 Mathematics as a science of measurement and quantification
1.3 Mathematics and Its Relationship with Other Disciplines
1.4 Characteristics of a Good Mathematics Teacher
1.5 Contributions of Some Indian Mathematicians to the Development of
Mathematics.
1.6 Let Us Sum Up
Unit II AIMS AND OBJECTIVES OF TEACHING MATHEMATICS
2.1 The need and significance of teaching Mathematics
2.2 Aims - Practical, social, disciplinary and cultural
2.3 Instructional Objectives
2.4 General Instructional Objectives (G.I.Os) and behavioral
2.5 Specific Learning Outcomes (S.L.Os)
2.6 Relating to the cognitive, affective and psychomotor domains based on
Bloom's Taxonomy of Educational Objectives.
Unit III LESSON PLANNING AND ITS USES
3.1 Micro teaching
- Origin, need, procedure, cycle of operation and uses
- Skill emphasis
- Explaining, questioning
- Probing and Fluency in questioning, using black board, reinforcement,
Stimulus variation, introduction, Closure
3.2 Link Lesson
3.3 Macro teaching
– Lesson plan, Unit plan & Year plan
– Herbartian steps
- Format of a typical lesson plan
– G.I.O‘s & S.I.O‘s, teaching aids
– Motivation, presentation, application, recapitulation and assignment
Unit IV METHODS AND TEACHING AIDS
4.1 Inductive, deductive, analytic, synthetic, heuristic, project, problem solving
and laboratory methods of teaching mathematics
4.2 Activity Based Learning (ABL)
4.3 Active Learning Method (ALM)
4.4 Applications of ABL and ALM
- Format of a typical lesson plan based on ALM
4.5 Introduction: Evocation, Recall, Survey
- Understanding: Concept, Teacher and Individual Solving Problems
- Group Work, Presentation
- Evaluation: Reinforcement, Homework, Remedial measures
4.6 Computer assisted instruction
- E-learning, mobile learning
4.7 Importance of teaching aids
- Projected and non-projected aids
- Improvised aids: Paper folding and paper cutting etc.,
4.8 Criteria for selection of appropriate teaching aids
4.9 Use of mass media in teaching mathematics
4.10 Field trip as a teaching technique
4.11 Characteristics of a good mathematics text book.
Unit V EVALUATION AND ANALYSIS OF TEST SCORES
5.1 Different types of tests in Mathematics, achievement, diagnostic, prognostic
5.2 criterion and norm referenced evaluation
- Construction of achievement test
- Continuous and comprehensive evaluation
- Formative and summative assessment
- Grading pattern
5.3 Statistical measures
- Mean, median, mode, range, average deviation, quartile deviation,
Standard deviation
- Graphical representation of data Bar diagram, Pie diagram, Histogram, Frequency
Polygon, Frequency Curve and Ogive curve.
- rank correlation
Unit I CONCEPT, NEED, OBJECTIVES AND SCOPE
1.0 Introduction
1.1 Meaning, characteristics and definition of Mathematics
1.2 Mathematics as a science of measurement and quantification
1.3 Mathematics and Its Relationship with Other Disciplines
1.4 Characteristics of a Good Mathematics Teacher
1.5 Contributions of Some Indian Mathematicians to the Development of
Mathematics.
1.6 Let Us Sum Up
1.0 Introduction

Mathematics is the creation of human minds. Because mathematics is made by persons and exists
only in the minds it must be made or remade in the mind of each person who learns it. In this
mathematics can only be learnt by being created. Mathematics is not just about number and space.
The traditional definitions, meaning and the modern thinking about the nature of mathematics such as
the language of mathematics and symbolism, logical structure, precision and abstraction are considered
in this unit as you, the student teacher, will have to be familiar with the definitions and the nature of
mathematics to become a good and efficient teacher of mathematics.
1.1. MEANING AND DEFINITION OF MATHEMATICS
We all know that mathematics is defined in simple terms as the science of quantity,
measurement and spatial relationships. It is a systematized, organized and exact branch of sciences. It
deals with quantitative facts, relationships as well as with problems involving space and form. The
dictionary meaning of mathematics is that it is either the science of number and space or the science
of measurement, quantity and magnitude-also the logical study of shape, arrangement and quantity.
Benjamin Peirce expressed it more explicitly as "Mathematics is the science which draws
necessary conclusions."According to Comte, Mathematics is the science of indirect measurement.
Distance between planets, diameter of an atom and rate of plant growth are some of the indirect
measurements made.
For Smith "Mathematics is the rock upon which art and science of the world rests". According
to Gauss, "Mathematics is the queen of sciences and Arithmetic is the queen of mathematics. Bacon
says "Mathematics is the gate way and key to all sciences" and Kant feels that "Mathematics is the
indispensable instrument of all physical researches."
Mathematics can be viewed as one of the great humanities because it is a method of expressing,
explaining and communicating man's total behavior: it reigns as a queen of all sciences in its clear,
rigorous and logical structure and in doing so, it serves as an ideal arid goal for perfection of other
sciences. Moreover, it is also considered as the servant of all sciences, meaning thereby, it provides
service to all other sciences. At the time of difficulties or hurdles they face for exact explanation and
precise direction. Then mathematics has its utilitarian aspect because of its numerous applications. It
is an instrument to train the mind.
1.1 NATURE OF MATHEMATICS: ITS CHARACTERISTICS
From various definitions and descriptions about mathematics it is easy to know and understand
the characteristics features of mathematics. They are: abstractions, language and symbolism, logical
structure and precision. We shall discuss these characteristics in detail now.
1.1.1. Logical Sequence
Mathematics is much more than counting, computing, drawing figures, using symbols with
mysterious code. It is a way of thinking, a way of reasoning. Sometimes mathematics involves
experimentation and observation but most of the cases it concerns with deductive reasoning.
Mathematics is not just adding of figures all day long, which can best be done by machines. It
involves mainly logical reasoning.
By reasoning in mathematics it can be proved that if something is true then something else must
be true. For example
A proposition as "If a quadrilateral is a square, then it is a parallelogram" can be written as
for any quadrilateral x, if x is a square, then x is a parallelogram; which is true?
By logical reasoning it is possible to explore the various ways in which a problem can be
tackled or solved. Sometimes one can show by reasoning that the problem has no solution.
For example
"Place 36 marbles in 9 boxes such that each box must contain odd number of marbles" has no
solution.
1.1.2 Structure
The inter related concepts and nature are becoming the structure of mathematics. Earlier,
mathematics was used to count and measure, then it becomes the science of structure. In counting,
there is a sequence viz number theories, subsets numbers, counting numbers, Natural numbers, Whole
numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers and Complex Numbers. This
is called the structure of Mathematics.
When the numbers undergo in the fundamental operations by using +, -, X, 4 and some basic
rules, it will get some principles and properties.
For e.g.: 3 4 = 4x3 [called reversible property]
These so many mathematical structures are found like Algebraic structures and topological
structures in mathematics.
There is a closer relationship among the concepts and theorems in mathematics. In this
structure, first elements are formulated and then operations and properties are created according to
the elements.
In other words, structures are used to create a new branch in mathematic. For e.g.:
Correlating the Algebra and Geometry, We get "Analytical Geometry".
1.1.3. Precision
Mathematics has its own precision, that is why its called an exact science. Mathematically
obtained result are either accepted or rejected, either correct or incorrect there is no double stand.
Even in the case of approximation, the result of mathematics can be had to any degree of accuracy
required, Measures at mathematics are precise to within certain specified units as 1/10 of 1cm, 500
Rs, 50 paise etc. Numbers are precise to units, tenths, hundredths etc. A measure or a number is
precise within certain percent of error or a certain number of significant digits. Effective measures
are in terms of errors involved. Such errors are either positive or negative. The precision of a measure
is evaluated in terms of apparent error. The accuracy of a measure is evaluated in terms of the
relative error or percent of error made.
Pupils should possess the sense of precision in the act of measurement made. Developing a
sense of precision helps to discipline pupils mind. Pupils acquire the sense of precision when they
observe critically, perceive relationship, tabulate data, analyse results and arrive at correct & precise
inferences or conclusions. There is a need to develop skills in using measures and estimating
quantities and magnitudes without measuring and to gain appreciation of the role of measurement. In
modern life by introducing sound courses of mathematics in school stages.
1.1.4 Abstractness
Mathematics is mostly theoretical and difficult to understand. Mathematics is characterized
by geometric and formalized attributes or otherwise by non-representational qualities without
reference to specific circumstances on practical experience. It has no reference to material objects or
specific examples. Mathematics is not concrete but sometimes the abstract mathematical concepts can
be explained through concretization and. practical activities. For example, the concept of 'number' in
mathematics is abstract since it does not refer to any specific quantity; for instance the number five
does not refer to live objects [marbles/books, students ... ].The symbols [5, V,] represent quantitive
measure of five. Five simply means the fiveness of five the 'representative ness' of five objects;
which is an abstract concept of five.
'Fiveness' of five objects thus, representative ness of five/5/V is a cardinal concept of number
which is an 'abstract concept' which refers to fiveness of five objects. 'The concept of 'fiveness' of
five objects can be compared with of 'whiteness' of paper or wall, egg or idly, which is an abstract
quality. The obvious illustrations are to appreciate the fiveness of five students, five pens ... etc.
Similarly to understand the division into seven equal parts of Rs.14,35cms or 56 ones is the mental
functioning of identifying mathematical relationships between the elements of a set or any other. All
the numbers and operations such as 5,6... and union [u] and intersection [n] are abstractions; percent,
length, volumes are abstractions; sum, difference, product, average are facts that concern elements or
aspects which may appear with countless different concrete surroundings. Concretization helps to
develop abstract thinking while learning mathematics. When concrete experiences are not possible,
the mathematical concepts, principles well understood by students serve as concrete foundation for
further abstract level of mathematic learning.
1.1.5. Symbolism
Mathematics is a language. It is a technical way of expressing a certain range of ideas in a
form which is convenient for applying reasoning processes. The language of mathematics is
applicable to a limited range of ideas with comparative degree. It has its own precision, conciseness an
accuracy which are absent from any other language.
A language of mathematics is much more than the word, symbols and the grammar it contains.
The way all these components are put together provides a complete logical reasoning and thinking
with which the concept principles and rules of mathematics are associated. Pupils learning
mathematics should be able to distinguish the uses of symbols; because they represent basically three
different kinds of meanings and understanding; namely
1. The elements or mathematical object: they are "numbers represented by the symbols [numerals];
e.g.5,12; sets such as A, B, or geometrical objects such as lines 1, m which are used in
 mathematical sentences like 5< 12; ACB = 0,1/m etc.
2. The relationship between elements: The symbols like >, =, < etc. represent the verbs of
mathematics. These express the particular way in which the elements are related to each other.
3. The operation on elements: The symbol; +,x, u, n ...represent operations performed on elements
such as addition, multiplication, union and intersection respectively and brackets or parenthesis
perform the function of punctuation in the language of mathematics as could be seen from the
different answers for
81 ^ [27^ 3] =9 and [81 4- 27] H- 3 = 1
Thus, it is essential to insert the appropriate mathematical punctuation the bracket or parenthesis.
1.2. MATHEMATICS AS A SCIENCE OF MEASUREMENT AND
QUANTIFICATION
We all know that mathematics is defined in simple terms as the science of quantity,
measurement and spatial relationships. It is a systematized, organized and exact branch of sciences. It
deals with quantitative facts, relationships as well as with problems involving space and form. The
dictionary meaning of mathematics is that it is either the science of number and space or the science
of measurement, quantity and magnitude also the logical study of shape, arrangement and quantity.
1.3. MATHEMATICS AND ITS RELATIONSHIP WITH OTHER DISCIPLINES
Leaving aside Mathematics, languages, physical education and work experiences all the other
subjects taught in the high school classes can be classified into two groups i.e. subjects related to arts
group in which History, Geography, Civics, Economics can be included and the subjects related to science
group in which we can place Physics, Chemistry, Astronomy, Botany, Zoology etc. Here the question
may arise: in which group Art or Science should Mathematics be placed. The answer is difficult
because Mathematics has its connection and roots in the Science as firmly as it has with the Arts
subjects. It nourishes and in turn gets nourished from Science as well as Art. That is why it is termed
as Science of all Sciences and Art of all Arts.
Whatever may be the form of the society it has an educational structure for realizing certain aims
and objectives. Different subjects of the curriculum help in the realisation of these set goals.
Although the courses are different yet they have common goals. This commonness draws them
nearer and in this way learning in .a particular subject affects the learning of other subjects and vice
versa. But here Mathematics plays a more specific role in helping the learning of other subjects. It has
direct or indirect relationship with almost all the subjects. Let us begin with the relationship of
Mathematics with Science.
Relationship of Mathematics with other Disciplines
1.3.1. Mathematics and Physics
If we take physics we see that its study requires the knowledge of Mathematics at every point.
All the physical laws, laws of motion, laws of lever and pulleys, laws of refraction and reflection, laws of
magnetization, laws of electric current, movement of the earth and planets and laws of quantum
energy can only be understood and applied with the help of the understanding of Mathematics. The
need of the numerical calculations in dealing with the problems in Physics clearly reveals the value of
Mathematics in learning Physics. The lenses and other equipments used in microscope, telescope,
photographic camera movie can only be made useful and workable with the help of intensity, power
and arrangement decided by the basic principles of Mathematics. In this way what we study in
'Physics' can only be studied effectively with the proper use of Mathematics.
1.3.2. Mathematics and Chemistry
Study in chemistry is also helped by the knowledge of Mathematics. The compositions and
properties of the different elements in Chemistry can only be understood properly with Mathematics.
For example the type of composition, no matter whether volumetric or graviometric is decided by the
laws of ratios and proportions governed by Mathematics. The study of compounds, mixtures, laws of
chemical combination and the study of molecular or atomic structures, chemical names of formula and
chemical equations all are based on the laws of Mathematics. In the preparation of different gases
and chemical products like bleaching powder, salts, acids, medicines and other day to day use
products, we need exact measurements in terms of weight, ratios and other calculations. In this way it is
no exaggeration in saying that Mathematics is used in the study of chemistry right from petty
chemical reaction to the preparation of chemical fuels of modern rockets and bombs.
1.3.3 Mathematics and Botany-Zoology
In all the experiments and studies of Botany and Zoology, we take the help of Mathematics.
The cellular construction of animals and vegetables, heredity, process of reproduction, balanced diet
and similar other topics need the knowledge of Mathematics. In any organism if we try to study the
anatomical structure and pattern of definite growth and development, we have to take the help of the
subject Mathematics. The graphs and statistical concepts used in these branches can also reveal the
need of Mathematics or Geometry.

1.3.4. Mathematics and Astronomy

Astronomy in one sense is totally based on the learning of Mathematics. The complicated
and intensive study connected with the movements of planets and satellites, their relative attraction
and distances and study of their orbits can only be possible with the knowledge of Mathematics. For an
ordinary individual it would be a great wonder to know that eclipses, tides, and the rising and the
setting of planets and starts happen at a fixed day and time but for a student of Astronomy it is a usual
phenomenon conducted through the rules of Mathematics. In this way Mathematics does not only
help in the understanding of Astronomy but also renders the astronomers a reliable help in the
realization of their dreams to walk straight on the distant planets like Venus and Mars.
1.3.5. Mathematics and Medical Science
In medicines the diagnosis as well as remedial treatment is based on the knowledge of
Mathematics. Temperature, blood pressure, deficiency and excess of the minerals and other
substances, the pressure or absence of undesirable substances and parasites in the blood, urine and
stool tests can only be detected and correctly measured with the adequate knowledge of Mathematics.
In preparation of the doses of medicines one has to take it to account the mechanism of measurement
which is not possible without Mathematics. Can we imagine that particular ingredient of the medicine
may prove most fatal or injurious to a person if its ratio or quantity is increased or decreased a little.
Both nurses and compounders will feel handicapped in the proper look after of the patients and
preparation of the proper doses of the mixtures and medicines if they happen to be ignorant of
Mathematics
1.3.6. Mathematics and Engineering
Mathematics is the base of all the Engineering, Surveying and measurements which help the
Science of Engineering to construct large bridges, plan the net work of canals and dams, extend
railway lines across the wide forests and lofty mountains, control the floods and establish the heavy
industry. Wise engineers with the help of the knowledge of applied mathematics at command, are
always in a position to serve the society and country in any front of Engineering - mechanical,
electrical, civil etc.
In this way we can realise that Mathematics is quite indispensable in learning Science Subjects.
Actually the relationship between Mathematics and Science is just like the relationship between the body
and its soul. Body [Science] has no meaning without its soul [Mathematics]. Soul may have its existence
without body. But in true sense, the existence of the soul will prove fruitful only when it carries body
along with it. In the same way Mathematics or Science cannot bring any fruitful result if they are not
integrated and used in combination. Whatever we see in the modern world of Science and technology
has its root in the progress and improvement of Mathematics. That is why Bacon has said that
"Mathematics is gateway and key of Sciences". A student of Science should therefore, try to learn
Mathematics for gaining adequate success in the field of sciences.
1.3.7 Mathematics and History
History is nothing but the systematic study of the past events which requires the knowledge of
Mathematics for its exact description and interpretation. A historical record, however, important it
may be, is meaningless if it does not carry the adequate concept of time and this concept can only be
understood with the knowledge of Mathematics. History will merely tell a student that Alexander the
Great invaded India in the year 326 B.C. where Mathematics would help him to realise the total time
passed to the occurrence of the said event in relation to the present, running year. In the same way the
extent, organisation and duration of the different empires of the past can only be understood and
appreciated with the knowledge of Mathematics. A student of History who has to acquire the
knowledge about time line, dynasty and historical maps can be helped by the knowledge of Mathematics
at his command.
1.3.8 Mathematics and Geography
Geography is nothing but a scientific and mathematical description of our earth in its universe.
We can quote so many examples from the subject Geography that its proper learning is only possible
with the adequate knowledge of Mathematics. The dimension and magnitude of the earth, the
formation of days and nights, change of seasons, lunar and solar eclipses, tides, currents, movement of
winds, falling of rain, factors influencing the climate of a region etc. are so many learning areas of
Geography which needs the knowledge of Mathematics. A student of Geography is required to possess
sufficient knowledge of drawing and understanding the maps. Study of Mathematics may certainly help
him in this task. He can locate and describe the position of a particular place in the world map. He can
also understand and calculate the local, standard and international time with the help of the
knowledge of Mathematics.
1.3.9 Mathematics and Economics
In learning Economics also the students use the language and knowledge of Mathematics. The
production, sale and purchase and distribution of commodities can only be regularised and maintained
by the help of Mathematics. The rate of exchange of different currencies belonging to different
nations are always decided by Mathematics. Whether we talk of export and import or value of
currency and international business relationship of a country or we talk of the internal economic
structure in terms of the budget preparation, planning and tax-collection of a country. We have to
take help from the terms used in the study of Economics like de-valuation of the currency, long term
saving schemes; Labour and capital relationship can be understood with the help of Mathematics. The
whole commercial system, Banking and Insurance etc. gets its nourishment from Mathematics. In any
economic planning one has to collect different types of statistics. These statistics can only be collected,
maintained and interpreted with the help of the knowledge of Statistics—an offshoot of Mathematics.
The language of Economics in terms of different types of tables and graphs is nothing but the language
spoken by Mathematics. It needs its own dictionary in terms of the knowledge of averages, ratio and
proportion, formulae, discount, interests and the necessary geometrical as well as statistical concepts.
1.3.10. Mathematics and Psychology
Today in the field of Psychology, statistical methods are occupying very important place and
therefore the study of Mathematics has become a necessary for gaining adequacy in Psychology. In this
connection Herbert Spencer has rightly said that "It is not only possible but necessary that Mathematics
should be applied to Psychology". When a student of Psychology wants to study the human behaviour
with adequate precision and objectivity he has to take the help of Statistics. Sometimes he has to
calculate the measure of central tendency in terms of mean, median or mode and at the other time he has
to compute coefficient of correlation, standard deviation and other Statistics for deriving valid and reliable
inferences. In all these tasks one has to take the help of Mathematics and hence Mathematics is
indispensable in any study concerning Psychology.
1.3.11. Mathematics and Languages
The key of learning any language—regional, national, or international, lies in its grammar. In
grammar we apply some set rules and principles governed by Mathematics. Where to apply comma,
full stop and put question mark is decided as exactly as we decide to write 10 after 9. Nouns,
pronouns, verbs, adjectives and other forms of sentence are used at their appropriate set places. Not
only the prose but the poetry also follows the rules of Mathematics. The aesthetic value of the poetry
can only be enjoyed if it is decorated with the regularity, symmetry and precision of Mathematics.
Mathematics has its own language in the form of numbers, signs, symbols, formulae and equations. This
language of Mathematics is borrowed by languages for bringing objectivity, precision, exactness and
accuracy in communication and expression of ideas and thoughts.
1.3.12. Mathematics and Drawing
Drawing and Mathematics go side by side as far as acquisition of the knowledge in any of these
areas is concerned. In drawing any picture, portrait, model or design one needs the knowledge of
Mathematics in terms of geometrical forms and skills as well as arithmetical measurement and
computation. In the same way the topics of area, menstruation as well as geometrical propositions and
constructions in Mathematics need the drawing skills for their proper learning. That is why in any
scheme of learning Mathematics or Drawing integration of these two subjects is properly cared for.
1.3.13. Mathematics and Music and Dance
The sound systematized and organized on some set Mathematical principles is known as Musical
sound. All the musical instruments—harmonium, violin, guitar, sitar, flute etc. follow the laws of
mathematics. Which particular 'Raga' has what type of rhythmical order, the ascending order of tone
and pitch all need the knowledge of Mathematics. In the art of dance also the movements of the limbs
and postures require mathematical rhythm and tuning.
1.3.14. Mathematics and Physical Education
In Physical Education one has to learn about Physiology and Hygiene. The knowledge about the
body systems requires mathematical calculations. Similarly to know about temperature and blood
pressure of the body, pulse rate, balanced diet for the different kinds of individuals, the student has to
take the help of Mathematics. In games and physical activities also, the learning of essential skill requires
an adequate knowledge of Mathematics because everywhere one has to use weighing, counting and
measuring as standard setting and evaluating devices.
1.3.15. Mathematics and Work Experience Activities
For integrating Education with work, to develop a positive attitude towards manual work and
increasing productivity a subject work-experience is being included in the Secondary School Curriculum.
The students are asked to choose one or two areas such as vegetable growing, soap making, some
useful direct experiences. Gaining experiences in such areas also requires the knowledge of
Mathematics. For example soap or Ink making requires a set procedure based on measurement
governed by Mathematics. In floriculture or vegetable growing also one cannot do without
Mathematics. Right from surveying, measuring and distribution of land to the ploughing, manuring,
watering, taking care of plants and selling of the produced goods one needs the knowledge of
Mathematics. Similarly clay modeling, woodwork, metal work, cane-work and leather work all require
the knowledge of Mathematics for purchasing raw material, processing finished products and disposing
them properly by getting adequate economic and educational gains. In a nutshell in all the areas of
work experiences wherever the task of computation and measurement is asked for, the knowledge of
Mathematics is indispensable.
As teachers of mathematics we should encourage our students to solve the problems by logical
reasoning or otherwise and be ready to help them if they have difficulties, individual help and general
class discussion if necessary would be fruitful.
It becomes crystal clear from the above discussion that Mathematics occupies a key place in
the school curriculum. It gives language to the languages, artistic touch and beauty to the arts,
scientific essences to the sciences and movement to the work experiences activities of the students. It
appears that Mathematics is a life blood of all the activities going inside a school. If we take different
subjects and activities as different pearls, Mathematics may be compared with that golden chain
which fixes firmly many pieces of different pearls [other subjects of the curriculum] to form a
beautiful necklace. In a necklace where the chain is essential for every piece of the pearls, the pearls
are also very much essential for magnifying the importance and usability of the chain.
1.4. CHARACTERISTICS OF A GOOD MATHEMATICS TEACHER
♦ A good mathematics teacher should have an extensive knowledge and love of
mathematics.
♦ A good mathematics teacher should have good content knowledge of Mathematics.
♦ He/she needs to have a profound understanding of basic mathematics and to be -able to
perceive connections between different concepts and fields.
♦ A good mathematics teacher needs to understand pupil's thinking in order to able to arrange
meaningful learning situations.
♦ A good mathematics teacher needs to be able to use different strategies to
promote pupil's conceptual understanding.
♦ A good mathematics teacher needs additional pedagogical knowledge the ability to arrange
successful learning situations, knowledge of the context of teaching and knowledge of the
goal of education.
♦ A good mathematics teacher's beliefs and conceptions should be as many-sided as possible
and be based on a constructivistic view of teaching and learning. The view of mathematics
viz., knowledge, beliefs, concepts, attitudes and emotions and about on self as a learner, as a
teacher of mathematics.
 ♦ A good mathematics teacher should know:
o What is Mathematics?
o How is Mathematics taught?
o How is mathematics learned?
♦ A good mathematics teacher should posses an endless amount of patience because there as
many different ways that students actually learn mathematics. And they learn at many
different speeds.
♦ A good mathematics teacher should understand Piaget's theory on how
youngsters create logic and number concepts is time well spend for mathematics
teacher.
♦ A good mathematics teacher should have strong classroom management
skills.
♦ A good mathematics teacher should be able to explain mathematics well.
♦ A good mathematics teacher should have the ability of making mathematics relevant and
incorporating real-world examples.
♦ A good mathematics teacher should have a good rapport with pupils.
♦ A good mathematics teacher should focus on concepts as well as procedures.
♦ A good mathematics teacher should help students when they are struck by showing them
how to do problems.
♦ Good mathematics teacher needs the ability to do quick error analysis, and must
be able to concisely articulate what a student is doing wrong, so they can fix it.
1.5. Contributions of Some Indian Mathematicians to the development of Mathematics.
1.5.1. Aryabhata and his contributions
In the history of Indian mathematics, Aryabhata is a very respectable name. Very often it is
being repeated. Thereby it is difficult to ascertain how many mathematicians of this name had been in
ancient India. However, it is quite certain that there were at least two Aryabhatas. One of them was born
in 476 A.D. at Kusumapura, the city of flowers [Patliputra] near present city of Patna in Bihar and wrote
the book Aryabhatiya, is known as "Aryabhata first". The other mathematician bearing the same name
who wrote the book "Maha Aryn Siddhant" in 950 A.D. is known as Aryabhata Second. The period of
Aryabhata first or the elder Aryabhata has been the golden period of Indian mathematics. Let us try to
know something about the life and work of this great mathematician.
Scholars differ about the birth place of the elder Aryabhata. He worked and lived at Kusumpur
but it does not prove that it was his birth place. Some writers claim that he belonged to Kerala and in
support of this claim they assert that the Calendar System invented by Aryabhata is still prevalent there.
Whatever may be his birth place, it is quite certain that he rose to fame while working at Kusumpur.
Here at the age of 23 years he wrote Aryabhatiya the first Indian astronomical text to contain a section
devoted entirely to basic mathematics.
Aryabhatiya in fact, is a small publication of 121 shloks [verses].
This work of Aryabhata shows his greatness, originality and creativity in the field of
mathematics by bringing into light some of his following contributions :-
1. Aryabhata invented a notation system consisting of alphabet numerals. Digits are denoted by alphabet
numerals in this system. Devnagri script contains Varga letters [Consonants] and Avarga letters
[Vowels]. Digits from 1 to 25 are denoted by the first 25 varga letters.
2. Although earlier to Aryabhata, the method of extracting square root was evolved by Jain
mathematicians, yet Aryabhata is known for giving its simple and clear explanation. He writes as :-
"One should always divide the avarga by twice the [square] root of the [preceding] varga. After
subtracting the square [of the quotient] from the varga, the quotient will be the square root to the
next place."
3. Aryabhata put more appropriate uses of the decimal system.
4. He gave almost all the formulae for knowing area of different figures like area of a square,
rectangle, triangle, rhombus circle, and volumes of sphere and cone etc. He also tried to point out
the construction of different geometrical figures—triangle, quadrilateral, circle. It shows how much
interested he was in the practical geometry.
5. Not only in arithmetic but in algebra also. Aryabhata contributed a lot. In Aryabhatiya he has given
the method
of addition, subtraction, division and multiplication of simple and compound algebraic
6. He tried to give a rule for summing an arithmetic series after the pth term.
"The desired number of terms minus one, halved plus the number of terms which precedes,
multiplied by the common difference between the terms, plus the first term, is the middle term. This
multiplied by the number of terms desired is the sum of the desired number of terms or the sum of the
first and last terms is multiplied by half the number of terms."
[where a and 1 are the first and the last terms of the progression, d the common difference
between terms and n being the number of terms extending the[p+ l]th to the [p+n]th terms in
arithmetical progression.
7. One can imagine the intelligence and work of Aryabhata know that he tried to solve indeterminate
linear equations like ax ± by = c by the method of the continued fractions which is substantially the
same as the method in use today.
8. The identities like the following are found in Aryabhatiya for the first time in the history of
mathematics.
I2 + 22 + ......... + n2= l/6[n [n +l][2n +1]]
l3+23 + .............. + n3= [1+2+...... +n]2
In this way 'Aryabhata was much ahead of his time. He dared to begin a new chapter in the
development of mathematics and astronomy through his valuable researches and contributions. Surely
he has an irremovable place in the history of mathematics.
1.5.2. Bramgupta and his contributions
Brahmgupta is known as one of the most distinguished, and prominent mathematician of the
ancient India. He was born in 598 A.D. in Sindh province [now in Pakistan]. His father's name was
Vishnugupta. According to Smith he later on began to live and work, in the great astronomical centre
Ujjain [now in Madhya Pradesh]. At this place he wrote valuable books on Astronomy and-
Mathematics. Among them few like Brahm-Sphuta Siddhanta, Khand Sadhak and Dhyan
Grahopdesh are quite popular.
'Brahm-Sphuta-Siddhanta, which was written by Brahmgupta when he was only 30 years old,
contains 21 chapters. The most useful chapters are 12th and 13th which are named as Ganitadhya and
Kutakhadhyaka. Ganitadhya deals with arithmetic and geometry whereas Kutakhadhyaka deals with
algebraic problems. The other remaining chapters deal mostly with astronomical knowledge. This
book Brahm Sphuta-Siddhanta helped Arabs to acquaint with the Indian Astronomy. Khalifa
Abbasial Mansoor [712-775 A.D.] called astronomer Kanka of Ujjain for explaining Brahm-Sphutat-
Siddhanta to the astronomers of his country and made the book translated into Arabic language.
Brahmgupta contributed a lot to almost all the branches of mathematics. The following
information may reveal this fact
Contribution in Arithmetic
1. The method of squaring first occurs in Brahm-Sphuta-Siddhanta. Concisely it has been
written as :-
"Combining the product twice the digit in the less [lowest] place into the several others [digits] with
its [i.e. of the digit in the lowest place] square [repeatedly] gives the square."
He also gave method of cubing and extracting square roots as well cube roots. About square root
he writes, "The pada [root] of a kriti [square] is that of which it is the square."
2. He also explained clearly the operations of addition, subtraction, multiplication and division
with different types of fractions. For example for the multiplication of fractions he writes:
"The product of the numerators divided by the product of the denominators is the [result of]
multiplication of two or more fractions."
3. The method of inversion called 'vilomgati' [working backwards] also for the first time
was completely explained by Brahmgupta. He writes:
"Beginning from the end, make the multiplier divisor, the divisor multiplier; [make] addition, subtraction
and subtraction, addition, [make] square, square root and square root, square ; this gives the
required quantity."
4. He contributed a lot in understanding the concept of zero. He defines it as a - a = 0. In.
addition to this he also explained operations of addition, subtraction, multiplication and
division with zero but he fell into the error of assuming that "cipher [zero] divided by
cipher [zero] is cipher [zero]."
Contributions in Algebra
He, for the first time, treated algebra as a separate branch by dealing it separately in
Kutakhadyaka. His main contributions in this branch are as follows:
1. Brahmgupta's work on indeterminate equations displays his greatest
power. Aryabhata had indicated a method of arriving at a solution of the indeterminate equation
of the first degree. Brahmgupta went ahead by giving a complete integral solution of the equation
ax +by -[a. band c being integral] and by his elaborate treatment of the indeterminate equation
ax2+l=y2

2. He was the first Indian writer who applied algebra to astronomy of any great extent. As
illustration the following problem may be cited
3. "A bamboo 18 cubits high was broken by the wind, its top touched the ground 6 cubits from the
foot. Tell the length of the segments of the bamboo."
1.5.3. Bhaskaracharya and his contributions
The history of Indian mathematics has been illuminated twice by the name Bhaskaracharya.
Bhaskaracharya first, was born a few years after the Aryabhatta first. He was a good student as well
as critic of mathematics and astronomy. The other Bhashkaracharya, known as Bhaskar II was born
many centuries after the first. He has been the most powerful and creative mathematician India ever
produced. In the following pages we would try to focus our discussion on this Bhaskaracharya.
Bhaskaracharya served as head of the astronomical observatory at Ujjain where Brahmgupta
had served in a similar capacity some five hundred years ago. According to his own book "Siddahanta-
Shiromani" he was born in 1114 A.D. in the village Bijjada Bida [at present situated in Bijapur-
Mysore State] near Shahyadri mountain. He came of an old learned family of some nobility and title,
possessing an established tradition of scholarship. The name of his father [as well as Guru teacher]
was Maheshwar. The book Siddhanta Shiromani was written in 1150 A.D. at the age of 36 years.
Four chapters of this book have survived. They are Lilavati, Vijaganit, Goladhyaya and Grahganit. Why
did he name the first chapter as Lilavati, is disputable. Some say it is because of the name of his daughter
Lilavati, the others consider it a mere style of writing prevalent in those days. Whatever may be the
reason for the naming, Lilavati is a valuable treatise, having 278 verses.
In addition to Siddhant Shiromani, Bhaskaracharya, wrote other valuable books like Karan
Kotuhal, Samiaya Siddhant Shiromani. Goladhayay Rasguna, Surya Siddhant. The original texts of
these works are however, missing.Through his valuable writings, Bhashkaracharya contributed a lot
in the field of mathematics and astronomy. Some of his important contributions are mentioned below.
1. Bhaskar for the first time brought the idea of infinity while dividing a number by zero. He writes,
"The fraction of which the denominator is cipher is termed as infinite quantity. In this quantity there
is no alteration, though it may be inserted or extracted ; as no change takes place in the infinite
and immutable God at the period of destruction or creation of worlds, although numerous orders
of being are absorbed or put forth."
2. He also contributed much in the field of mensuration. He gave many important formulae for the
computation of the area and volumes of different figures like the following :-
Area of a sphere = 4 ' area of a circle
Volume of a sphere = area of a sphere 'l/6of its diameter
3. Bhaskaracharya also dealt with cubic equations and Biquadratic equations in his writings.
The following types of problems are available in the 2nd chapter of Siddhant Shiromani.
x4 + 12x = 6 x2 + 35 [Biquadraticequation]
x3 - 2x2 - 400x = 9999 [Cubic equation]
It reveals that he was quite ahead in this field.
4. Bhaskara's greatest strength lay in his ability to handle problems which lead to
indeterminate equations. An attempt is here made to arrange the various problems in
the order in which they would appear in a modern treatise.
[i] ax+c=by
[ii] ax+by+cz = d
[iii] ax+by+d = xy
 Equation of the types considered above were fairly common among earlier writers. But Bhaskara
advanced well beyond his predecessors in his approach to indeterminate equation of the second
degree like the equation ax2+ 1 = y2, the so called Pell's equation.
5. Bhaskaracharya is known for his poetic presentation of the complicated and abstract
problems of mathematics.
6. Bhaskaracharya was much ahead of his time in so many aspects. In Goladhayaya he
gives the following evidence in support of the roundness of the earth.

7. "The hundredth part of the circumference of a circle seems, to be a straight line. Our
earth is a huge sphere, we can see only a small fraction of it; therefore it appears flat.
8. He also had the knowledge of gravitational power long before Newton. He named it
as Dharnikatmak Shakti He explains it as below :-
"All things appear to fall on the earth simply because they are attracted by it on account of its
Dharnikatmak Shakti [gravitational power].
In this way what has been contributed by Bhashkaracharya is unaccountable. In every sense he
was a celebrated astronomer and mathematician. He shines like sun [Bhaskar] in the world of
mathematics.
1.5.4. Shrinivasa Ramanujan and his Contributions
One day a primary school teacher of the third form was telling to his students, "If three, fruits
are divided among three persons, each would get one. Even if 1,000 fruits are divided among 1,000
persons, each would get one." From this he tried to generalize that any number divided by itself was
unity. This made a child of that class jump and ask, 'Is zero divided by zero also unity ? If no fruits
are divided among nobody, will each get one?"
Do you imagine who was this little child? He was Shrinivasa Ramanujan, the wonderful
young Indian mathematician of the 20th century who was so intelligent that as student of class III of
a primary school he successfully worked out the properties of the Arithmetical, Geometrical and
Harmonic progression and up to class IV he almost solved all the problems of the Loney's
Trigonometry meant for the degree classes.
This extra-ordinary mathematician of his time was born in a poor Brahmin family on 22nd
December 1887 at Erode in Tanjore district of Madras State. His father Srinivasa Ayyangar was an
accountant at a cloth merchant at KumbkaKonam. While his mother, Komalammal, was the daughter
of a petty official [jamin] in the district Munsif's court at Erode.Ramanujan got much of his earlier
education in the town high school at Kumbka Konam. He always stood first in his class and got
scholarship. He was very much popular for his interest and extra-ordinary abilities in mathematics
He was so bright that he was declared child mathematician, at the age of 12 by his teachers. He used
to entertain his friends with theorems and formulae, with the recitation of complete list of Sanskrit roots
and with repeating the value of pi and the square root of two to any number of decimal places. In the
year 1903, when he was 15 and in the sixth form at school, a friend of his gave him a book "Carr's
Synopsis of Pure and Applied Mathematics" from the library of the local government college. It was
this book that awakened and stimulated his genius. He verified many of the results in the book and
discovered many new results of his own. Besides engaging in this original work he did not miss his
regular studies and as a result gained a place in the first class in the matriculation examination of the
University-of Madras held in December 1903. This enabled him to secure Subramaniam Scholarship
and join the F.A. [First examination in arts] class in the Government College, Kumba Konam.
Owing to weakness in English, for he gave no thought to anything but mathematics, he failed in his
next examination and lost his scholarship. He then left Kumbka Konam, fast for Vizagapatam [Andhra
Pradesh] and then for Madras. He resumed his studies, completed his second year course in the
Pachiappa College in 1906. Unluckily he got ill at the time of examination and therefore appeared for
the university examination in December 1907 but at this time also he got failed, and then determined
not to try again.
For the next few years he continued his independent work in mathematics. In 1909 he was married to
Janaki and it became necessary for him to find a job of permanent nature. In the course of his search
for work, he was got introduced to a true lover of mathematics, Diwan Bahadur Ramchandra Rao. For
some months he was supported by Shri Ramachandra Rao. Then he accepted his appointment as a
clerk in the office of the Madras Port Trust. While working as a clerk he never slackened his interest
in mathematics. He made his one of the works published in the Journal of the Indian Mathematical
Society in 1911 at the age of 23. He wrote a long article on "Some properties of Bernoulli's numbers'
in the same year. In 1912 he contributed two more notes to the same journal and also several
questions for solution. Meanwhile he began correspondence with Professor G.H. Hardy, a leading
mathematician of his time. To his first letter he attached 120 theorems of his own creation.
At last in May of 1913, as the result of the help of many friends, Ramanujan was relieved of
his clerical post and was given a special scholarship. Hardy made efforts to bring Ramanujan to
Cambridge and helped him to learn modern mathematics so as to acquaint him with all the up-to-date
development in the field of mathematics. In 1916 he got honorary B.A. degree of the University of
Cambridge. About making Ramanujan to learn at ambridge. Hardy writes "It was impossible to ask such
a man to submit to systematic instruction, to try to learn mathematics from the beginning once more I
had to try to teach him and in a Measure succeeded though obviously learnt from him much more than
he learnt from me".
In the spring of 1917, Ramanujan first appeared to be unwell. He went to nursing home at
Cambridge in the early summer and was never out of bed for any length of time again. For brief
period he resumed some active work, stimulated perhaps by his election to the Royal Society and
Trinity Fellowship. Due to Tuberculosis he left for India and died in Chelpet, Madras on account of
this disease on April 26, 1920 at the age of thirty three.
In his short span of life he contributed so significantly in the field of mathematics that many
eminent authorities accepted him "quite the most extraordinary mathematician of his age." Some of his
achievements are enumerated below
1. Divergent Series: His first investigation in this direction was sent to Professor Hardy in form of 120
theorems in the year 1913. Commenting on the merit of these theorems, Hardy wrote, "I had
never seen anything the least like them before. A single look at them is enough to show that they
could only be written down by a mathematician of the highest class"
2. Hyper geometric series and continued fractions: He was unquestionably one of the greatest
masters in this field. Commenting on this Hardy writes, "It was his insight into algebraic formulae,
transformation of infinite series and so forth that was most amazing. On this side, most certainly I
have never met his equal and I can compare him only with
Euler and Jacobi"
.... '.
3. Definite Integrals: He produced quite a number of results in this field in the form of general
 formulae. These are all included in his three quarterly reports to the University of Madras.
4. Eliptic Functions: He tried to handle eliptic functions profusely commenting on his ability in this
direction Hardy writes, 'Ramanujan shows at his very best in the parts of the Theory of Elliptic
Functions allied to the Theory of Partitions."
5. Partition Functions: Before Ramanujan very little was known about the arithmetical properties
of a partition function P [n] where n is odd or even. Ramanujan was the first, and till his time, the
only mathematician to discover any such properties. For the first time in 1917, Hardy and
Ramanujan, jointly examined, the question of how large the number of partitions of n is, when n
itself is large. They gave the answer in the form of an asymptotic series and also estimated the
error involved in taking a definite number of terms only.

6. Fractional Differentiation: The insight and generalizations derived in this field by Ramanujan
are quite wonderful. He gave a meaning to Eulerian Second Integral for all values of n negative,
positive and fractional.
In fact Ramanujan had a very original and intuitive approach to numbers. It appears that every
integer was one of his personal friends. In the simplest array of digits he detected wonderful
properties, congruence's, symmetries and relationships which had escaped the notice of even the
outstanding gifted theoreticians. Hardy recalls a meeting of Mr. Little-wood with Ramanujan. He
had taken a taxi cab no. 1729 and remarked that the number seemed to him rather a dull one. No,
Ramanujan replied. "It is a very interesting number, it is the smallest number expressible as a sum
of two cubes in two different ways."
[1729=l3+23=93+103]
In this way one can judge the merit and competency of Ramanujan as a first rate
mathematician. Although he had not got enough opportunity for college education in the subject
mathematics and most of his time was spent either in struggling for the means of livelihood or
fighting his ill health he contributed a lot in the field of mathematics by showing his profound and
invincible originality. He was a mathematician whom only first class mathematicians can follow and
it is not surprising, therefore that he attracted little attention outside his profession. But his work has left
a memorable imprint on mathematical thoughts.
1.5.5. Euler and his Contributions
Leonhard Euler
The 18th-century Swiss mathematician Leonhard Euler [1707-1783] is among the most
prolific and successful mathematicians in the history of the field. His seminal work had a profound
impact in numerous areas of mathematics and he is widely creed for introducing and popularizing
modern notation and terminology, particularly in analysis.
Mathematical Notation
Euler introduced much of the mathematical notation in use today, such as the notation f[x] to
describe a function and the modern notation for the trigonometric functions. He was the first to use the
letter 'e' for the base of the natural logarithm, now also known as Euler's number. The use of the Greek
letter <3 to denote the ratio of a circle's circumference to its diameter was also popularized by Euler
[although it did not originate with him]. He is also creed for inventing the notation i to denote -
Complex Analysis

Euler made important contributions to complex analysis. He discovered the scientific notation. He
discovered what is now known as Euler's formula, that for any real number q, the complex
exponential function satisfies
e = cosG + z s i n G
This has been called "the most remarkable formula in mathematics " by Richard Feynman.
Euler's identity is a special case of this:
e™+l = 0
This identity is particularly remarkable as it involves e, 3, i, 1, and 0, arguably the five most
important constants in mathematics.
Analysis
The development of calculus was at the forefront of 18th century mathematical research, and
the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field.
Understanding the infinite was naturally the major focus of Euler's research. While some of Euler's
proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for
many great advances. First of all, Euler introduced the concept of a function, and introduced the use of
the exponential function and logarithms in analytic proofs.
Euler frequently used the logarithmic function as a tool in analysis problems, and discovered
new ways by which they could be used. He discovered ways to express various logarithmic functions in
terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly
expanding the scope where logarithms could be applied in mathematics. Most researchers in the field
long held the view that log[x] = log[ - x] for any positive real x since by using the additivity property of
logarithms 21og[ - x] = log[[ - x]2] = log[x2] = 21og[x]. In a 1747 letter to Jean Le Rond d'Alembert,
Euler defined the natural logarithm of -1 as id a pure imaginary.
In addition, Euler elaborated the theory of higher transcendental functions by introducing the
gamma function and introduced a new method for solving quartic equations. He also found a way to
calculate integrals with complex limits, foreshadowing the development of complex analysis. Euler
invented the calculus of variations including its most well-known result, the Euler-Lagrange equation.
Euler also pioneered the use of analytic methods to solve number theory problems. In doing so,
he united two disparate branches of mathematics and introduced a new field of study, analytic number
theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series,
hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he
proved the infinitude of primes using the divergence of the harmonic series, and used analytic methods
to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to
the development of the prime number theorem.
Number Theory
Euler's great interest in number theory can be traced to the influence of his friend in the St.
Peterburg Academy, Christian Goldbach. A lot of his early work on number theory was based on the
works of Pierre de Fermat, and developed some of Fermat's ideas.
One focus of Euler's work was to link the nature of prime distribution with ideas in analysis. He
proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the
connection between Riemann zeta function and prime numbers, known as the Euler product formula
for the Riemann zeta function.
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two
squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the
totient function 6[n] which assigns to a positive integer n the number of positive integers less than n
and coprime to n. Using properties of this function he was able to generalize Fermat's little theorem to
what would become known as Euler's theorem. He further contributed significantly to the understanding
of perfect numbers, which had fascinated mathematicians since Euclid. Euler made progress toward the
prime number theorem and conjectured the law of quadratic reciprocity. The two concepts are
regarded as the fundamental theorems of number theory, and his ideas paved the way for Carl
Friedrich Gauss.

Applied Mathematics
Some of Eider's greatest successes were in applying analytic methods to real world problems,
describing numerous applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler
numbers, e and 5 constants, continued fractions and integrals. He integrated Leibniz's differential
calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus
to physical problems. In particular, he made great strides in improving numerical approximation of
integrals, inventing what are now known as the Euler approximations. The most notable of these
approximations are Euler method and the Euler-Maclaurin formula. He also facilitated the use of
differential equations, in particular introducing the Euler-Mascheroni constant:
One of Euler's more unusual interests was the application of mathematical ideas in music. In
1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually integrate music theory as part
of mathematics. This part of his work, however did not receive wide attention and was once
described as too mathematical for musicians and too musical for mathematicians.
Works
The works which Euler published separately are:
♦ Dissertation on the physics of sound
Mechanica, sive motus scientia analytice; expasita
♦ Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes
♦ Additamentum II [English translation]
Theoria motuum planetarum et cometarum
♦ Beantwortung, &c. or Answers to Different Questions respecting Comets
♦ Thoughts on the Elements of Bodies
♦ Introduction to the analysis of the infinites - ■

1.5.6. Euclid of Alexandria

Known for "The Elements' Background
Euclid of Alexandria lived in 365 - 300 BC [approximately]. Very little is known about
Euclid's life except that he taught in Alexandria, Egypt. He may have become educated at Plato's
Academy in Athens, or possibly from some of Plato's students. Basically, all of the rules we use in
Geometry today are based on the writings of Euclid, specifically 'The Elements'. The Elements
includes the following Volumes:
Volumes 1-6: Plane Geometry
Volumes 7-9: Number Theory
Volume 10: Eudoxus' Theory of Irrational Numbers
Volumes 11-13: Solid Geometry
The first ion of the Elements was actually printed in 1482 in a very logical, coherent framework.
More than one thousand ions have been printed throughout the decades. Schools only stopped using the
Elements in the early 1900s, some were still using it in the early 1980's, however, the theories
continue to be those that we use today.
Euclid's book the Elements also contains the beginnings of number theory. The Euclidean
algorithm which is often referred to as Euclid's algorithm is used to determine the greatest common
divisor [gcd] of two integers. It is one of the oldest algorithms known, and was included in Euclid's
Elements. Euclid's algorithm does not require factoring.

Euclid and His Contributions

He is famous for his treatise on geometry: The Elements. The Elements makes Euclid one of
if not the most famous mathematics teacher. The knowledge in the Elements has been the foundation
for teachers of mathematics for over 2000 years!
Geometry Tutorials like these wouldn't be possible without the work of Euclid.
Famous Quote: "There is no royal road to geometry."
Little is known about the life of Euclid. Although he is often referred to as 'Euclid of
Alexandria', this city was not the one of his birth. We don't know exactly when he was born either,
though it is known that he was in Alexandria during the reign of Ptolemy I [323-283 BC].Euclid was a
Greek and before arriving in Alexandria, it is likely that he studied with Plato at his Academy in
Athens.
Alexandria was a port on the northern Merranean coast of Egypt, and there Ptolemy I
established an institute which was known as The Museum. We sometimes refer to it today as The
University of Alexandria. Euclid was a professor of mathematics at The Museum and established
the Alexandrian School of Mathematics.
Euclid is best remembered for the mathematical textbook he wrote - in fact 13 books
altogether - that are collectively known as The Elements. The Elements contains 465 propositions
altogether. Some of the propositions and proofs were Euclid's own work, but the book also recorded
many results from his mathematical predecessors. Euclid arranged them logically into a single
coherent book and used a system of mathematical proofs based on axioms.
The Elements is often thought of as a book on geometry, but it also contains number theory and
some elementary algebra. Euclid also wrote at least ten other books, five of which have survived.
No copy of the original Elements is in existence today. Modern ions are based on a revision
by Theon of Alexandria almost 700 years later. Nevertheless, The Elements has proved to be the
most widely studied book in the world with the exception of The Bible. The Elements was used as a
textbook in schools until the last century, and is still the basis of Euclidean geometry taught in
modern schools.
As the most influential book on mathematics ever written, the importance of The Elements
 cannot be overestimated.

One example of a contribution of Euclid to mathematics is the Euclidean algorithm for finding
the greatest common divisor of two given positive integers. This is found in Book VII of The
Elements.This is how the algorithm works:

1. Divide the larger number by the smaller number and find the remainder.
2. Divide the divisor [the smaller of the two original integers] by the remainder from step 1 and find the
next remainder.
3. Repeat step 2 until the division is exact. The final divisor gives the greatest common divisor of
the original two integers.
As an example, find the greatest common divisor of 444 and i 1512.
1. 1512 -r 444 = 3, remainder 180
2. 444 -r 180 =2, remainder 84 3.180 4-84 =
2, remainder 12
84 4- 12 = 7, exactly
Now the division is exact, the final divisor [which is 12] gives the greatest common divisor of
444 and 1512.
It is not meant to be a full history of Euclid. Neither are the other articles I've written in the
series 100 Great Mathematicians 'full histories'. If you want a full history, buy Euclid's biography!
1.5.7. Pythagoras and his contributions
PYTHAGORAS
Pythagoras was an ancient Greek mathematician and philosopher who was one of the most
influential men in all of history. Pythagoras is one of the most famous mathematicians. Even though he
was a mathematician, his contributions help all sorts of fields of study, including math, science, music
and astronomy. Most of hiscontributions, and those of his followers, the Pythagoreans, are known and
used throughout the world. Almost everyone knows his theorem that the sum of the squares of the
lengths of the two shorter sides of a right triangle is equal to the square of the length hypotenuse.
Pythagoras was born on the Greek Island of Samos, Greece around 582 B.C. and he died
around 475 B.C. He travelled around the known world visiting Babylonia and Egypt along with seeing
other parts of Greece. Early on in his life several other philosophers, such as Anaximander majorly
influenced him. Pythagoras was a patron of the ancient Greek Olympic games, even though he
criticized them. He died at some point around the turn of the century in Metapontion
Although he made many important contributions to math we do not know much about him.
We do not have anything that he wrote because he was the leader of a secret society where they didn't
write down what they did. All that we have about Pythagoras are a few biographies from a long time
ago.
Some accounts state that Pythagoras went to Egypt in about 535 B.C. to learn more about
mathematics and astronomy. In Egypt Pythagoras visited many of the temples and talked to the
priests, but he was not allowed in any of the temples except for one. In 525 B.C. the king of Persia
invaded Egypt and Polycrates sent 40 ships to help the Persians invade Egypt. During the invasion
Pythagoras was captured and taken to Babylon. After Polycrates and the king of Persia died
Pythagoras went back to Samos, but nobody knows how he was freed.
Pythagoras studied pure mathematics. He didn't try to solve mathematical problems. Instead, he
was interested in the concept of numbers and geometric figures. He was also interested in properties
of numbers; such as, odd and even numbers. He is best known for his famous geometry theorem
known as the Pythagorean Theorem.
The Pythagorean Theorem is that if you have a right triangle the two sides that form the right
angle are equal to the other side when each number is squared [multiplied by itself].
The equation for the Pythagorean Theorem is:
Sidel2 + Side 22 = Side 32
For example, if you have a right triangle with sides of three, four, and five, you square three
which is nine, you square four which is sixteen, and you square five which is twenty-five. Then you
add nine and sixteen and you get twenty-five which is what the other side is squared.

This proves that the Pythagorean Theorem works!

You can use the Pythagorean Theorem in everyday life. It can be used on a baseball field to
determine distances. Since the infield is a square with right angles you can use the Pythagorean
Theorem to determine the distance from home plate on a baseball field to second base if you know
the distance from first to second base.
Pythagoras also experimented with music. He found a relationship between music and math.
Pythagoras found the relationship one day when he was walking past a blacksmith shop and heard the
hammers hit the anvil. When Pythagoras heard the hammers hit the anvil he noticed that each hammer
made a different sound according to the weight of the hammer. So, the weight of the hammer [i.e.
number] decides the sound the hammer will make.
He started a religious and philosophical order known as the Pythagoreans and most of his work
was completed between himself and his followers. His society was founded in Croton, in southern
Italy.
The society was made up of two groups, the pupils and the learned, or teachers. These
people set up the world's first organized schools.
They were a very strict order and had many restrictions, such as no meat or beans, and the new
members couldn't even speak until they had listened to the teaching of their master for 5 years.
Even after the first 5 years the pupil's work was too be left anonymous, the discoveries made were
either cred to the master or to the school itself.
The Pythagoreans believed that everything in the universe revolved around mathematics,
such as music and the magicks. Every number had a soul and a specific meaning and value to the
universe.
You of course know of my a2+b2 = c2, but Pythagoras created this method at first just to prove
that a triangle was a right triangle. Pythagoras also discovered a formula to find out how many degrees
there are in a polygon. Pythagoras came up with [n-2]180°= the number of degrees in a polygon,
where n represents the number of sides in the polygon. For example, a triangle has three sides, 3-
2=1,1x180=180, which is the total sum of all the inner angles of a triangle. Along with that
Pythagoras found out that the sum of all the outer angles of a polygon is always equal to three
hundred sixty
Pythagoras did a lot of work with proportions in other fields. Pythagoras looked at and showed
the difference in pitch in ratio to the length of string plucked. It was not as Pythagoras expected.
Pythagoras found out that half way along the string is not half the pitch. Pythagoras also looked to
the stars and saw that the further away a planet is from where it orbited the longer it would take to
go around the sun.
Through this, Pythagoras discovered a lot of different things, many being very useful.

1.5.8 Gauss and his contributions

Johann Carl Friedrich Gauss

[Latin: Carolus Fridericus Gauss] [30 April 1777 - 23 February 1855] was a German
mathematician and scientist who contributed significantly to many fields, including number theory,
statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.
Sometimes referred to as the Princeps mathematicoruml [Latin, "the Prince of
Mathematicians" or "the foremost of mathematicians"] and "greatest mathematician since antiquity,"
Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of
history's most influential mathematicians. He referred to mathematics as "the queen of sciences."
Carl Friedrich Gauss was born on April 30, 1777 in Braunschweig, in the duchy of
Braunschweig-Wolfenbiittel, now part of Lower Saxony, Germany, as the son of poor workingclass
parents. Indeed, his mother was illiterate and never recorded the date of his birth, remembering only
that he had been born on a Wednesday, eight days before the Feast of the Ascension, which itself
occurs 40 days after Easter. Gauss would later solve this puzzle for his birth date in the context of
finding the date of Easter, deriving methods to compute the date in both past and future years. He
was christened and confirmed in a church near the school he attended as a child.
Gauss was a child prodigy. There are many anecdotes pertaining to his precocity while a
toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He
completed Disquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though it was not
published until 1801. This work was fundamental in consolidating number theory as a discipline
and has shaped the field to the present day.
Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig, who sent
him to the Collegium Carolinum [now Technische Universitat Braunschweig], which he attended
from 1792 to 1795, and to the University of Gottingen from 1795 to 1798. While in university, Gauss
independently rediscovered several important theorems;citation needed his breakthrough occurred in
1796 when he was able to show that any regular polygon with a number of sides which is a Fermat
prime [and, consequently, those polygons with any number of sides which is the product of distinct
Fermat primes and a power of 2] can be constructed by compass and straightedge. This was a major
discovery in an important field of mathematics; construction problems had occupied mathematicians
since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics
instead of philology as a career. Gauss was so pleased by this result that he requested that a regular
heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult
construction would essentially look like a circle.
The year 1796 was most productive for both Gauss and number theory. He discovered a
construction of the heptadecagon on March 30. He invented modular arithmetic, greatly simplifying
manipulations in number theory. He became the first to prove the quadratic reciprocity law on 8
April. This remarkably general law allows mathematicians to determine the solvability of any quadratic
equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good
understanding of how the prime numbers are distributed among the integers. Gauss also discovered
that every positive integer is representable as a sum of at most three triangular numbers on 10 July
and then jotted down in his diary the famous words, "Heureka! num = A + A + A." On October 1
he published a result on the number of solutions of polynomials with coefficients in finite fields,
which ultimately led to the Weil conjectures 150 years later.

Gauss's method involved determining a conic section in space, given one focus [the sun] and the
conic's intersection with three given lines [lines of sight from the earth, which is itself moving on an
ellipse, to the planet] and given the time it takes the planet to traverse the arcs determined by these lines
[from which the lengths of the arcs can be calculated by Kepler's Second Law]. This problem leads to
an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought
is then separated from the remaining six based on physical conditions. In this work Gauss used
comprehensive approximation methods which he created for that purpose.
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never
published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from
the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-
contradictory. Research on these geometries led to, among other things, Einstein's theory of general
relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with
whom Gauss had sworn "brotherhood and the banner of truth" as a student had tried in vain for many
years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, Janos
Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it,
Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of
the work... coincides almost exactly with my own mations which have occupied my mind for the past
thirty or thirty-five years."
Gauss's presumed method was to realize that pairwise addition of terms from opposite ends
ofthelist yielded identical intermediate sums: 1 + 100= 101,2 + 99 = 101,3 + 98 = 101, and so on, for a
total sum of 50 x 101 = 5050.

:■
1.6 LET US SUM UP
Mathematics is a core subject at school level and has got number of applications in daily life.
Thus the teachers are expected to develop understanding about the nature of mathematics and the
factors that would enable to treat mathematics as a discipline. Being the first unit of the block, we
have discussed the concept and nature of Mathematics in detail. As children validate mathematical
knowledge they make use of various processes such as developing hypothesis, formulating
conjectures and generalisation, proving mathematical arguments and statements, etc. These are the
key processes used in mathematical reasoning and validation of mathematical knowledge. In later
portion of this unit, the various processes involved in mathematical reasoning have been extensively
discussed citing examples pertaining to them. The unit ends with the discussion on the ways of
developing skills of creativity among children.
Unit II AIMS AND OBJECTIVES OF TEACHING MATHEMATICS
2.1 The need and significance of teaching Mathematics
2.2 Aims - Practical, social, disciplinary and cultural
2.3 Instructional Objectives
2.4 General Instructional Objectives (G.I.Os) and behavioral
2.5 Specific Learning Outcomes (S.L.Os)
2.6 Relating to the cognitive, affective and psychomotor domains based on
Bloom's Taxonomy of Educational Objectives.
■,

2.1. THE NEED AND SIGNIFICANCE OF TEACHING MATHEMATICS

Mathematics is being as the mother of all sciences. If a student wants to function effectively
according to the Global changes, he/she must understand the mathematics and be able to use
mathematics in their daily life both personal and professional. The need and significance of teaching
mathematics are as follows:
♦ To enable the students to get good mathematical background with the knowledge of concepts
and theories.
♦ Make the students to apply mathematical concepts and theorems in new situations.
♦ To create the ability of transfer the mathematical type of thinking and reasoning to daily life
situations.
♦ Make the students to understand the law of nature.
♦ Enable the students to understand the culture and development of civilization.
♦ Teaching mathematics gives sufficient mathematical skills to meet the demands of daily life.
♦ If provides better understanding of the world.
♦ It provides good deal of self-reliance, self-confidence, and open-mindedness.
♦ It promotes the skill of problem solving among the students.
♦ It helps to learn the other subjects which are expecting mathematical operations, particular
science.
2.2.1. AIMS
The goals of education may be usefully divided into the utilitarian, social, disciplinary and cultural
goals; so also the aims of teaching mathematics. Mathematics is taught in schools because it is useful,
disciplines the mind and is beautiful. Every student will have to pick up mathematical knowledge and
skills since it cater to the individual, societal and national needs.
While starting to teach a particular subject it is essential to know why we are going to teach
that subject. Until we have clear-cut aims of teaching a subject, we would not be able to proceed on
the right track Aimlessness makes the work uninteresting and results in the wastage of time, energy
and other material resources both on the part of the teacher and the taught. Therefore, we must have
some definite aims of teaching a subject before starting its actual teaching.
 Now the question arises what would be the aims of teaching Mathematics in our schools. The
answer needs the knowledge of all the advantages that can be drawn by the teaching of Mathematics
as we know that aims and values are interrelated and interdependent things. One aims at a thing because
one values it or by aiming at a thing one would be able to realise its values. Therefore, aims help in the
realisation of values or drawing of advantages while the knowledge of the advantages or values of a
subject helps in setting the aims to get all the essential advantages.
This makes us to conclude that the knowledge of the advantages or, values of teaching
Mathematics may help us a lot in setting the aims of teaching Mathematics in our schools. In other
words it may help us in realising what we should expect from our Mathematics teaching.
The teaching of Mathematics should essentially help the students in acquiring essential
Mathematical knowledge, skills, interests and attitudes for the following purposes.

2.2.2 Practical Aim

To enable the student, to make use of the learning in Mathematics in their day to day life. To
have clear ideas of number and a comprehension of the way the number is applied to measure of all
varieties, but most particularly to those physical concepts he meets with most frequently, length,
volume, weight, area, temperature, speed and acceleration. Able to apply his knowledge of
mathematics to a wide range of problems that continually occur in his every day life. To understand
the concepts of ratio and scale drawing and read, interpret graphs diagrams and tables especially
those relating to statistical evidences. Able to use correctly, accurately and with understanding the
four fundamental operations of addition, substraction, multiplication and division as applied to both
number measurement and check both his own and other peoples calculations by appropriate
approximation.

2.2.3 Social Aim

The social goals are to make the students understand how mathematics methods namely
scientific, intuitive, deductive and inventive are used to investigate, interpret and to make decisions in
human affairs and also how it contributes to his understanding of natural phenomena. The scientific
method is that in which one seeks to discover order, pattern and relations not only in the sets or
numbers in series, quantities or measures but in the natural world as well. Teachers should arrange for
activities and situations to make it possible for students to discover or rediscover the relations for
themselves. The intuitive method is one by which advances are made step by step by a flash of insight,
and a sudden illumination of a concept brings understanding of a difficult problem situation. The results
of ones insight or intuition are linked logically and thus instruction is closely related to deductive
reasoning, the other method inventive method emphasizes the need for providing opportunities to
students to make their own investigations and find solutions. Besides the mathematical value of
discussion, co-operation, learning the importance of organization and corporate endeavour.
2.2.4 Disciplinary Aim
To develop their intellectual powers and disciplining their minds. According to Locke,
"Mathematics is a way to settle in the mind a habit of reasoning." It trains or disciplines the mind. Due
to its very nature, it possesses a real disciplinary value. It is exact, true and to the point knowledge, and
therefore creates a discipline in the mind. Its truths are definite and exact. The learner has to argue
the correctness or incorrectness of a statement. It taught in the right sense, it develops reasoning and
thinking powers more and demands less from memory. The student come to realize that thinking
makes him a successful student of all the subjects. Its study results in the development of power rather
than the acquisition of knowledge and knowledge also comes as a natural consequence or by-product.
Reasoning in mathematics possesses certain characteristics which are suitable for the training of the
learners mind. If properly emphasized and streamlined, these characteristics are likely to develop the
corresponding habits in the learner. Here ensues a discussion of those characteristics and their
influence.
i. Characteristic of simplicity: There is a vast scope for simple reasoning in this subject. It teaches
that definite facts are always expressed in a simple language and definite facts are always easily
understandable. So if you want to be understood, you must express yourself in a definite or simple
way. More over, one can easily follow a gradation going from simple to complex. The teacher
advances by degrees to harder and harder portions. The procedure when practiced for a pretty
long time becomes a habit.
ii. Characteristic of accuracy: Without accuracy there is no chance of progress and credit in
mathematics. Accurate reasoning, thinking and judgment are essential for its study. It is in the
nature of this subject that it cannot be learnt through vagueness of thought and argument. In other
subjects, it may sometimes be possible for the student to hide his ignorance by beating about the
bush, but such tricks never play in mathematics. Accuracy, exactness and precision compose the
beauty of mathematics. The student learns the value and appreciation of accuracy and adopts it as
a principle of life. He learns to influence and command others by his accuracy.
iii. Characteristic of certainty of results: There is no place for subjectivity and proposal equation in
mathematics. The answer is either right or wrong. Subjectivity of difference of opinion between the
teacher and the taught. The student can verify his result by reverser process. It is possible for the
child to remove his difficulties by self-effort and to be sure of the removal. The success of
personal effort is a source of pleasure for him. He develops faith in self-effort which is the secret
of success in life. He inculcates the habit of being certain about his achievement.
iv. Characteristic of originality: Most work in mathematics demands original thinking.
Reproduction and cramming of ideas of others is not very much appreciated. In other subjects, of
course, ideas of others occupy a prominent place and have to be grasped by the student. Therefore
he can safely depend on memory in other subjects; but without original thinking and intelligent
reasoning there cannot be satisfactory progress in mathematics. When he has a new or a different
mathematical problem, it is only his originality which keeps him going. The discovery or
establishment of a new formula is also his original work. This practice in originality enables the
child to face new problems and situations with confidence in his future career.
v. Characteristic of similarity to the reasoning of life: Clear and exact thinking is as important in
daily life as in mathematical study. Before starting with the solution of a problem, the student has
to grasp the whole meaning. Similarly in daily life, while understanding a task, one must have a
firm grip on the situation. This habit of thinking will get transferred to the problems of daily life
also.
vi. Characteristic of verification of results: Results can be easily verified. As already
pointed out, this gives a sense of achievement, confidence and pleasure. This verification
of results is also likely to inculcate the habit of self-criticism and self evaluation. After
making any attempt in life, the child would like to satisfy himself about its success or
failure.
2.2.5 Cultural Aim
To make them understand the contribution of Mathematics in the development of culture and
civilization. As regards the cultural goals the aim is to enable the student to see the part mathematics
has played in the culture of the past and continues to play in the culture of the world or today. The
students will have to be encouraged to preserve, promote and transmit the mathematical knowledge
further to the younger generation. They have to appreciate the roles of mathematics in our modern
culture and civilization through science and technology and various cultural arts like drawing, design
making, painting, poetry, music, sculpture and architecture. Hence the aim is to help the student to
explore the creative fields such as art, architecture and music apart from making them aware of the
strength and virtues of the culture they have inherited.
Some other aims are,
Aesthetical and Recreational Aim:
To develop their aesthetic abilities, meet their varying interests and help them in the task of
utilisation of their leisure time.
Moral Aim:
To help them in imbibing essential moral virtues.
Vocational Aim:
To prepare for the future vocation or occupation.
Pre-preparational Aim:
To help in the study of other subjects and future learning in Mathematics.
Inter-disciplinary Aim:
To give them insight to recognise relationships between different branches and topics of
Mathematics.
Self-learning Aim:
To help them in becoming self-dependent for mastering new topics and problems of
Mathematics.
If the instructional process is to be effective, all three activities must be oriented to certain
common objectives. Objectives should be started for each course, unit and topic. They are the mental
skills that pupils should develop as a result of teaching. Objectives direct the pupils as to what he/she is
expected to do, what should be the minimum level of acceptance for his/her performance and under
what conditions it will be achieved.
2. 3 Instructional Objectives
♦ Define what a student be able to do after a period of instruction.
♦ Help the teacher plan appropriate learning experiences.
♦ Help the examiner plan a relevant test to actually measure these outcomes.
♦ Provide direction for the instructor and it clearly conveys his instructional intent to others.
♦ Provide the frame of reference for decisions about selection and gradation and organization of
subject matter, the mode of instruction and techniques of evaluation.
♦ Till the students, what is expected of him after a period of instruction and these by enable him
to use his study time more efficiently.
 ♦ Till the student how he should be able to use the material from the syllabus and in what ways
he is expected to display his mental skills and abilities.
The interrelationship among Objectives, Learning Experiences and Evaluation Techniques.
Objectives

Learning Experience ^\ y\ Evaluation Techniques

Relationship Between Aims and Objectives

Education is given for achieving certain ends and goals. The various subjects of the school
curriculum are the different means to achieve these ends. The aim is to achieve the goals or broader
purposes of education. Thus by the term aims of teaching Mathematics we should mean the goals,
targets or broader purposes that may be served by the teaching of Mathematics, in the general scheme
of Education. Aims are like ideals. They need a long term planning. Their realisation becomes a
difficult task for a subject teacher. Therefore, they are divided into some definite, functional and
workable units named as objectives. Objectives of teaching Mathematics are therefore, those short
term immediate goals or purposes that may be achieved within the specified classroom resources by a
subject teacher. They help in bringing appropriate behavioural changes in the learning for the ultimate
reahsation of the aims of teaching Mathematics.
In this way the aims of teaching Mathematics are usually broken into some specified objectives
to provide definite learning experiences for bringing desirable behavioural changes. A teacher thus may
have certain clear-cut well defined objectives before him at the time of teaching a particular topic.
Objectives of Teaching Mathematics at the entire school stage
The objectives of teaching Mathematics may be classified as under
a] Knowledge and understanding objectives.
b] Skill objectives.
c] Application objectives.
d] Attitude objectives.
e] Appreciation and Interest objectives.
For making the objectives unambiguous and attainable they are always expressed in the
behavioural terms [testable behaviours]. What the student is expected to achieve is clearly known by
the teacher while teaching a particular topic. We will try to keep this thing in view while describing the
different objectives of teaching Mathematics. A.
a) Knowledge and understanding objectives:
 Through Mathematics a student acquires the knowledge and understanding of the language of
 Mathematics
[in terms of symbols, formulae, figures, diagrams, technical terms and definitions].
 The various Mathematical concepts [number concept, concept of units and measurement and
concept of direction etc.].
 Mathematical ideas, facts, principles, processes and relationships.
 The development of the subject and contribution of Mathematicians.
 The inter-relationship among different topics and branches of Mathematics.
 The basic nature of the subject Mathematics.
B. Skill Objectives:
Mathematics helps a student in the following ways
1. He learns and develops essential skill in the use and understanding of Mathematical language.
2. He develops speed, precision, brevity, accuracy and neatness in the computation and calculation
work.
3. He learns and develops the technique of problem solving.
4. He develops the ability to estimate and check results.
5. He develops the ability to perform calculations orally or mentally.
6. He develops the ability to think correctly, to draw inferences and to generalize.
7. He develops the ability to use Mathematical apparatuses and tools skillfully.
8. He develops the essential skill in drawing geometrical figure.
9. He develops essential skill in surveying, or measuring and weighing processes.
10. He develops essential skill in drawing, reading and interpretation of graphs and statistical tables.
11. He develops skill in the use of Mathematical tables and ready reckoners.
C. Application Objectives
Mathematics helps a student in applying above knowledge and skills in the following ways
1. He is able to solve the problems of Mathematics independently.
2. He makes use of mathematical concepts and processes in everyday life.
3. He develops ability to analyse, to draw inferences and generalise the collected evidences and data.
4. He develops the ability to make use of Mathematics learning in the learning of other subjects
and equips himself for higher mathematical studies.
5. He can think and express precisely, exactly and systematically by making proper use of
Mathematical language.
D. Attitude Objectives
Mathematics helps in the development of correct attitudes which may be stated as below:
1. The student tries to analyse the problem.
2. He develops the habit of systematic thinking and objective reasoning.

3. He develops heuristic attitude and tries to discover the facts or solve the problems with his own
independent efforts.
4. He tries to collect enough valid evidences for drawing inference, conclusion or generalization.
5. He recognizes the adequacy or inadequacy of given data in relation to the problem.
6. He tries to verify his results.
7. He understands and appreciates logical, critical and independent thinking in others.
8. He tries to express his opinions precisely, systematically and logically without any biases and
prejudices.
9. He develops personal qualities e.g. regularity, punctuality, honesty, neatness and truthfulness.
10. He de velops proper self-confidence for solving the mathematical problems.
11. He develops mathematical perspective and outlook for observing the realm of nature and social
world.
12. He shows originality and creativity.
E. Appreciation and interest Objectives:
The student is helped in the acquisition of appreciations and interests in the following way:
1. He appreciates the role of Mathematics in everyday life.
2. He appreciates the role of Mathematics in understanding his environment.
3. He appreciates Mathematics as the science of all sciences and art of all arts.
4. He appreciates the contribution of Mathematics in the development of culture and civilization.
5. He appreciates the aesthetic value of Mathematics by observing symmetry, similarity, order and
arrangement in Mathematical facts, principles and processes.
6. He appreciates the contribution of Mathematicians in the development of subject and civilization.
7. He appreciates the recreational value of the subject Mathematics and learns to utilise his leisure
time properly.
8. He appreciates the vocational value of the subject Mathematics.
9. He appreciates the role of Mathematical language, graphs and tables in giving precision, exactness
and accuracy to the expression.

10. He appreciates the power of computation.

11. He develops interest for the learning of the subject Mathematics.
12. He feels relaxed and entertained by Mathematical recreations and amusements.
13. He takes active interest in the activities of Mathematics club.
14. He takes interest in independent library reading, working on Mathematical projects and doing
practical work in Mathematics laboratory.
Objectives of Teaching Mathematics at sub-stages-Elementary and Secondary
What we want to achieve through Mathematic teaching at the entire school stages is achieved
slowly and slowly through the well defined objectives, planned learning experiences and methods at the
different stages.
In the 10 + 2 pattern of educational structure these stages are as follows
1. Primary or Elementary Stage [first four or five years].
2. Secondary Stage [from 5th or 6th year to 10th year of schooling].
Since it is still undecided whether the last two years should be spent in the school or college we would
like to think for the objectives of teaching Mathematics at the elementary and secondary stage only.
Objectives of Teaching Mathematics at the Secondary Stage
Objectives of teaching Mathematics at this stage are more or less the same as listed earlier for
the entire school stage. However, the objectives of teaching Mathematics at the elementary stage
[being elementary in nature] may be specified as follows :-
Objectives of Teaching Mathematics at the Elementary Stage
Objectives of teaching Mathematics at this stage may be specified in terms of the following
expected learning outcomes of student behaviour:
A. Knowledge and Understanding Objectives:
The student develops knowledge and understanding for the following:
1. Concepts like number concept, concept of units of measurement, concept of size and
shape, concept of fractions, concept of direction and distance, concept of grouping and
sub grouping.
2. Mathematical facts and processes like the place values of numbers, the meaning and
significance of zero, the four fundamental operations, L.C.M. and H.C.F., Percentage,
Unitary Method, Simple Interest, Profit and Loss and Mensuration.
3. Arithmetical terms and symbols like digits and numbers, symbols for fundamental operations,
fractions, percentage etc.
4. Relationship between different topics of arithmetic.
B. Skill Objectives:
The students develops the following skills :-
1. He develops ability in reading, writing and counting of numbers.
2. He develops skill in four fundamental operations dealing with integral numbers and fractions.
3. He develops a reasonable speed, accuracy and neatness in the computation of oral as well as
written work in Mathematics.
4. He develops the technique of solving problems involving elementary Mathematical processes
and simple calculations
5. He develops skill in the use of Multiplication Tables.
6. He develops proficiency in making quantitative estimate of size and distance.
C. Application Objectives:
The student is able to apply the above knowledge and skills as follows:
1. He is able to solve elementary mathematical problems independently [orally as well as in writing].
2. He makes use of elementary mathematical concepts and processes in every day life.
D. Attitude Objectives:
1. The student develops proper self-confidence for solving elementary mathematical problems.
2. For the solution of a problem he tries to read it carefully, analyses, collects all the known evidences
and then draw proper inferences.
3. He develops habit of regularity, neatness, truthfulness and honesty.
E. Appreciation and Interest Objectives:
1. He develops interest for the learning of the subject Mathematics.
2. He appreciates the contribution of Mathematicians and gets inspiration from their work.
3. He appreciates the power of computational skills.
4. He appreciates and takes interest in using his learning of Mathematics in solving daily life
problems.
5. He appreciates the recreational value of the subject Mathematics and learns to utilise his leisure
time properly.
2.4 & 2.5 THE GENERAL INSTRUCTIONAL OBJECTIVES [GIO'S] AND SPECIFIC
INSTRUCTIONAL OBJECTIVES[SIO'S]
An objective of instruction is an instructional objective, a statement of what the learner is
likely to be after successfully completing the specified learning task.
General instructional objectives are stated using terms as knows, understands, applies, appreciates,
develops etc. the student understands the base two system of numbers [This is not an observable
behaviour as understanding by the student cannot be observed by the teacher]. The GIO begins with
verbs such as knows, develops, appreciates etc which are not action verbs.
Specific instructional objectives[SIO's] are the learning outcomes or terminal student behaviour
which are observable; they are action verbs such as defines, states, compares, distinguishes,
identifies, illustrates, classifies, computes, converts, checks, derives, extracts, groups, solves, tabulates,
constructs, verifies, interprets, discriminates etc. The student converts a number in base ten into
number in base two system [This is an observable behaviour because converting from one form to the
other can be observed by the teacher]. The SIO are the expected learning outcomes after the
instruction. The SIO begins with action verbs such as recalls, compares, identifies measures etc. which
indicate terminal behaviours that are definite and observable.
The GIO and SIO when stated clearly give direction for action to bring about the desired
changes in the students behaviour enabling them to attain the goals or aims.
Writing Objectives in Behavioural Terms
The major weakness about the taxonomies of objectives given above lies in the fact that they do
not state objectives in terms of terminal behaviour i.e. what the learner should be able to do at the end
of teaching. Specification of objectives in a task of teaching and learning may prove more effective
and purposeful if they are translated into behavioural language.
The structure of the educational or instructional objectives mainly consists of two parts,
namely,
[i] the modification part and [ii] the content part.
The modification part represents the behavioural changes that are designed to occur in the
behaviour of the learner through the related instruction or learning experiences.
The content part refers to the syllabus in particular and to the curriculum in general to be
covered by the related instruction.
Therefore, the writing of an objective in behavioural terms is always done in related to the
following three things
i. The nature of the objective i.e. knowledge, application etc.
ii. The area or domain of the behaviour i.e. cognitive, affective etc.
iii. The specific content areas in which behavioural changes are planned to be brought about i.e.
Fundamental rights, Means of irrigation, Sources of heat, etc.
A list of Associated Action Verbs for the Cognitive Domain Objectives [Based on Bloom's
Taxonomy]
1. Knowledge: Define, List, Label, Measure, Name, Recall, Recognise, Reproduce, Select, State,
Write, Underline, etc.
2. Comprehension: Change, Classify, Distinguish, Explain, Formulate, Identify, illustrate, Indicate,
Interpret, Justify, Judge, Name, Represent, Select, Summarize, Transform, Translate, etc.
3. Application: Assess, Change, Choose, Conduct, Construct, Compute, Demonstrate, Discover,
Explain, Establish, Find, Generate, Illustrate, Modify, Predict, Perform, Select, Solve, Use, etc.
4. Analysis: Analyse, Associate, Compare, Conclude Contrast, Criticize, Differentiate, Identify,
Justify, Point out, Resolve, Select, Separate, etc.
5. Synthesis: Argue, Conclude, Combine, Derive, Discuss, Generalize, Integrate,Organise, Precise,
Prove, Relate, Restate, Select, Summarize, Synthesize, etc.
6. Evaluation: Associate, Choose, Compare, Criticize, Conclude, Defend, Determine, Evaluate,
Judge, Identify, Recognize, Relate, Select, Summarize, Support, Verify, etc.

A list of Action Verbs for Affective domainObjectives [Based on Bloom's Taxonomy]

1. Receiving: Ask, Accept, Attend, Beware, Catch, Discover, Experiment, Identify, Favour, Follow,
Observe, Prefer, Perceive, Receive, Select, etc.
2. Responding: Answer, Assist, Complete, Derive, Discuss, Develop, Help, List, Label, Name,
Obey, Present, Practise, Record, Select, State, Write, etc.
3. Valuing: Accept, Attain, Complete, Choose, Decide, Demonstrate, Discriminate, Develop, Increase,
Indicate, Influence, Participate, Prefer, Recognise, etc.
4. Organising: Add, Associate, Change, Compare, Complete, Coordinate, Correlate, Determine,
Find, Form, Generalize, Integrate, Judge, Project, Prepare, Relate, Select, synthesize, Organise,
etc.
5. Characterising: Accept, Change, Characterize, Decide, Discriminate, Demonstrate, Develop,
Experiment, Face, Identify, Judge, Prove, Revise, Serve, Solve, Verify, etc.
A list of Action Verbs for Psychomotor domain Objectives
1. Imitation: Chooses, describes, detects, differentiates, distinguishes, identifies, isolates, relates,
selects, separates
2. Manipulation: begins, displays explains, moves, proceeds, reacts, responds, shows, starts,
volunteers
3. Precision: measures, sketches, mends, mixes
4. Articulation: Assembles, builds, displays, organizes
5. Naturalization: composes, constructs, originates
Instructional Objectives and their relationships to General Aims and Objectives of Teaching
Mathematics
At the time of imparting instruction i. e. teaching-learning of a particular lesson, unit or subunit
of the subject mathematics, teacher has to place before him some definite and very specific objectives
for being attained within a specified classroom period and resources in hand. Through these so specific
classroom teaching-learning objectives, known as instructional objectives, a teacher tries to bring
desired changes in the behavior of his pupils. In this way, the term instructional objectives in relation to
the teaching of mathematics may be defined as a group of statements formulated by the teacher for
describing what the pupils are expected to do or will be able to do once the process of classroom
Instruction is over. In fact instructional outcomes is the teaching-learning product in the form of
behavioral changes in the pupils that a teacher expects as a result of his instruction related with a
particular lesson, unit or sub-unit of the subject. Instructional objectives are thus nothing but descriptions
of the pupil's terminal behavior expected out of the on going classroom instruction.In comparison to
general aims and objectives of teaching mathematics, instructional objectives are quite narrow and
specific. They are definite, tangible, precise and functional. They are predetermined and are always
formulated in such a way that their attainment becomes quite practicable through the usual classroom
teaching within the stipulated period of fixed duration. They are the desired learning or teaching outcomes
and are always stated in terms of expected pupil's behaviour or desired behavioural changes. They are,
therefore, may be termed as teaching-learning objective or behavioural objectives. The main purpose
of these objectives is to provide statements of skills, concepts or the behaviour, learners are expected
to demonstrate after going through particular instruction. Objectives of teaching mathematics falls
midway between goals or aims of teaching mathematics and instructional objectives. They are more
specific and definite than the general aims or goals but less specific and much wider than the
classroom instructional objectives. Their attainment is quite possible within the educational structure
and means.

Classroom instructional objectives may be submerged in the reservoir of general objectives

of teaching mathematics at a particular school stage which in turn are further submerged in the
ocean of general aims and goals of teaching mathematics.
2.6 BLOOM'S TAXONOMY OF INSTRUCTIONAL OBJECTIVES
"Taxonomy" means a system of classification and in this sense a taxonomy like Bloom's Taxonomy
presents a system of classification of the objectives in the similar way as Dewey's Decimal system
tends to classify a number of books in a library.
The taxonomy, of educational and instructional objectives has been worked out on the
assumption that the teaching-learning process may be conceived as an attempt to change the behaviour
of the pupils with respect to some subject matter or learning experiences Behaviour is divided into
three domains—Cognitive [knowing], affective [feeling] and psychomotor [doing]. The taxonomy
of educational and instructional objectives have also been considered to be belonging to these three
domains.
The taxonomy related to cognitive' domain has been presented by Bloom [Bloom, et. al, 1956], the
second related to affective domain by Krathwohl, Bloom and Mosia [Krath Wohi,' et al, 1964] and the
third related to psychomotor domain by Harrow' [Harrow, 1972] and Simpson [1966].
Let us have a brief description of these taxonomies.
Taxonomy of objectives in the Cognitive Domain
Bloom and his associates have classified the objectives related to cognitive domain into six
categories arranged from the lowest to the highest level of functioning as described below.
1. Knowledge

a. Knowledge of specifics
b. Knowledge of terminology.
c. Knowledge of specific facts.
d. Knowledge of ways and means of dealing with specifics.
e. Knowledge of conventions.
f. Knowledge of trends and sequences.
g. Knowledge of classifications and categories.

h. Knowledge of criteria.
i. Knowledge of methodology.
j'. Knowledge of universals and abstractions in a field.
k. Know 1 edge of principles and generalizations.
l. Knowledge of theories and structures.

2. Comprehension
a] Translation
b] Interpretation
c] Extrapolation

3. Application
4. Analysis

a] Analysis of elements
b] Analysis of relationships
c] Analysis of organisational principles
5. Synthesis

[a] Production of unique communication

[b] Production of a plan or a proposed set of operations
[c] Derivation of a set of abstract relations
6. Evaluation

a] Judgement in terms of internal evidence

b] Judgement in terms of external criteria.
Let us try to elaborate the above taxonomy of objectives of cognitive domain given by Bloom
1. Knowledge. It represents the lowest level of the objectives belonging to the cognitive
domain and primarily aims for the acquisition of the knowledge concerning.
[i] Specific facts, terminology, methods and processes and
[ii] Generalized principles, theories and structures.
The knowledge objectives mainly call for the recall and recognition level of one's memory and
therefore their evaluation is primarily made through a simple recall or multiple choice type questions.
2. Comprehension. Comprehension is based upon the knowledge. If there is no knowledge,
there will be no comprehension. On the ladder of the acquisition of cognitive abilities its
level is little higher than the knowledge. Specifically, it means the basic understanding of the
facts, ideas, methods, processes, principles or theories, etc. As a result, what is communicated
to a learner, he may

i. translate or summarize the communicated knowledge in his own words

ii. interpret i.e. cite examples, discriminate, classify, verify or generalize and

iii. extrapolate i.e. estimate or understand the use of knowledge and extend it to other subjects
and fields.
3. Application. The knowledge is useful only when it is possible to make it employed. The application
of an idea, principle or theory may be made possible only .when it is grasped and understood properly,
Therefore, the category of application automatically involves both the earlier categories i.e.
knowledge and comprehension. Under this objective the learner is required to acquire the ability to
make use of the abstract or generalized ideas, principles in the particular and concrete situations.
4. Analysis. Analysis refers to an understanding at higher level. It is a complex cognitive process that
involves knowledge, comprehension as well as application of an idea, fact, principle or theory.
Through the realization of these objectives the learner is expected to acquire the necessary skill in
drawing inferences, discriminating, making choices and selection and separating apart the different
components or elements of a concept, object or principle.
5. Synthesis. The objectives belonging to this category aim to help the learner to acquire necessary
ability to combine the different elements or components of an idea, object, concept, or principle as to
produce an integrated picture [i.e] figure of wholeness. As a result he may be expected to propagate or
present a theory or principle by combining different approaches, ideas or view points. He may arrive at
something new or originate some novel thing or idea after synthesizing all what is known to him
earlier. In this war, it calls for the creativity aspect of the cognitive abilities and therefore may be
considered definitely a higher level of learning involving knowledge, comprehension, application as
well as analysis.
6. Evaluation. This category of objectives aims to develop in the learner the ability to make proper
value judgement about what has been acquired by him in the form of knowledge, understanding,
application, analysis and synthesis. It involves all the five categories described earlier. As a result the
learner is expected to take proper decision about the quantitative and qualitative value of a particular
idea, object, principle or theory. He may arrive at an appropriate decision about the matter and methods by
making use of all the cognitive abilities acquired through the earlier categories of cognitive
objectives.
Taxonomy of objectives in the Affective Domain
Taxonomy of Objectives in the Affective Domain arranged from lowest to highest
level of functioning.
1. Receiving [Attending]
[a] Awareness
[b] Willingness to receive
[c] Controlled or selected attention
2. Responding
[a] Acquiescence in responding
[b] Willingness to
respond
3. Valuing
[a] Acceptance of a value
[b] Preference for a value
[c] Commitment
4. Organization
[a] Conceptualization of a

[b] Organisation of a value system

5. Characterization by a value or value Complex

[a] Generalized set
[b] Characterisation
Let us try to elaborate further the above classification for its clarity and understanding as below.
1. Receiving. It represents the initial category for the objectives belonging to
affective domain. For the inculcation of certain interests, attitudes, values or ideas it is essential
that learner should be made to receive or attend the desired ideas, events or objects. This
category points out towards this necessity and takes into consideration three types of following
sequential activities :-
[i] Firstly, the learner is sensitized or made aware about the existence of certain stimuli.
[ii] Then the desired intention or willingness for receiving or attending the stimuli is created in the
learner.
[iii] Lastly, the efforts are made for the control of the attention of the learner. He may be trained to
pay selective attention and sustain it for a desired period.
2. Responding. It represents the second level of the objectives for the categories
belonging to affective domain. Once a learner receives or attends to a particular idea, event
or thing he must be made to respond to it as actively as possible. The responses here do not confine
itself in just paying attention or arousal of a simple intention or desire of getting a thing, as in the
first category of receiving but manifest, themselves in the active behaviour like obeying, answering,
reading, discussing, recording, writing and reacting to a stimulus, etc.
3. Valuing. When one attends as well as responds to a particular thing, idea or event he is naturally
drifted towards taking value judgement about that thing, idea or event. Therefore, this category of
valuing depends upon both the former categories i. e. receiving and responding. Here the learner is
expected to imbibe a definite value pattern towards different ideas, events, and objects. In practice
the objectives belonging to this category are usually concerned with the development of typical
value patterns, attitudes, etc.
4. Organising. This category of objectives concern with the construction of relatively enduring
value structure in the learner by organising and synthesizing the different value patterns imbibed by
him from time to time. Ultimately this category of objective leads the learner to form a set value
structure or philosophy of life.
5. Characterizing by a value or value complex. It is the highest level category of the objectives
belonging to the affective domain. Up to this stage, the learner is able to imbibe all the essential
affective behaviour i.e. various interests, attitudes, values, value complex or value patterns, a
permanent set value structure and therefore, all the earlier categories are automatically involved in
the objectives of this category. At this stage, the learner is destined to imbibe typical characteristics
of his individual character i.e. life style of his own- In fact it is the end point or ultimate goal of the
process of education.
Taxonomy of objectives in the Psychomotor Domain

Taxonomy of Psychomotor Objectives [R.H. Dave] *

1. Imitation
[a] Impulsion
[b] Over t repetition
2. Manipulation
[a] Following direction [bj Fixation
3. Precision
[a] Reproduction
[b] Control

4. Articulation
[a] Sequence
[b] Harmony
5. Naturalization
[a] Automatism
[b] Internalization
Let us have a necessary explanation for the above steps
Imitation For the learning of a psychomotor activity i.e. drawing or surveying skill in
mathematics, the task begins with the imitation of observed acts. The child observes the
demonstrated behaviour related to drawing of a line, angle or circle etc. He feels an inner push or an
impulse [by having an inner rehearsal of the psychomotor activities] to imitate action It is followed
by the overt repetition [imitation] of the demonstrated behaviour.
Manipulation. This second category of psychomotor objectives emphasizes manipulation on
the part of the learner for the acquisition of skills by following directions, performing selected action
and fixation of performance through necessary practice.
 Precision. In the third category of psychomotor objectives learner is able to perform skilled
acts or motor activities with a desired level of precision [accuracy, exactness and right proportion]
and as such may be said to reach a higher level of refinement in reproducing a given act or skilled
task.
Articulation. It is the fourth category in the hierarchy of learning the psychomotor activities
or skills. At this stage learner becomes capable of coordinating a series of acts by establishing
appropriate sequence and accomplishing harmony or internal consistency among different acts.
Naturalization. It is the highest stage reached in terms of the development or proficiency
acquired in the learning of a skill or psychomotor act. One can now perform a single act or a series
of articulated acts with a greater refinement, ease and convenience as automatic and naturally as
possible.
Unit III LESSON PLANNING AND ITS USES
3.1 Micro teaching
- Origin, need, procedure, cycle of operation and uses
- Skill emphasis
- Explaining, questioning
- Probing and Fluency in questioning, using black board, reinforcement,
Stimulus variation, introduction, Closure
3.2 Link Lesson
3.3 Macro teaching
– Lesson plan, Unit plan & Year plan
– Herbartian steps
- Format of a typical lesson plan
– G.I.O‘s & S.I.O‘s, teaching aids
– Motivation, presentation, application, recapitulation and assignment

3.1 MICRO TEACHING: ORIGIN

Micro teaching was first introduced at the Stanford University, USA in 1963. The Stanford
Teacher Education staff members sought to identify, isolate and build training programmes for critical
teaching skills. Micro teaching, has since then been refined and applied not only in teacher training but
also in business, Nursing and army research in India and other developing countries has shown that
conventional micro teaching methods help to improve teaching competence.
Micro teaching is now considered not only as a constructive teacher training teaching technique
but also as a versatile research tool which dramatically theologises investing of certain teaching skills
and learning variables. Teaching constitutes a number of verbal; and non-verbal acts. A set of related
behaviours or teaching acts aiming at specific objectives and performed with an intention to facilitate
pupils learning can be called a teaching skills. All these technical skills which go to make good teaching
can be defined, observed, measured, controlled by means of practice. Micro-teaching concentrates on
specific teaching behaviours and provides opportunity for practicing teaching under controlled
conditions. Micro teaching is a method of teacher training and not a method of teaching in class room.
It simplifies the complex teaching process so that the student teacher can cope with it.
Need for Micro Teaching:

The heart of any educational process is teaching which includes training, instruction and development of
cognitive processes and abilities. Experts have commented that the quality of a nation is judged by the
quality of its education, which is in turn, is decided by the type of teachers it has. The teacher in a
classroom uses several techniques and procedures too bring about effective learning on the part of
his students. These activities include introducing, demonstrating, explaining or questioning and the
teacher could also use nonverbal behaviours such as smiling, gesturing and nodding. These activities
form what are called teaching skills. In order to achieve objectives in all three domains the teacher has
to acquire all teaching skills and use them appropriately. Hence it is necessary that a student teacher is
introduced to a quite range of teaching skills. Micro teaching allows the student teacher to practice
any skill independently and integrate it with other skills in familiar environment.
Meaning and Definitions of Micro teaching:
The teacher education programme is made up of two parts: the theoretical course and the
practical course with the main focus on practice teaching. The student teachers are provided with a
relatively strong theoretical base and are exposed to a couple of demonstration and observation lessons:
then they have to actually practical or put into action what they have seen and heard, in a real
classroom situation. Here he learns the art and craft of teaching by trial and error. To rectify this
micro teaching approach was introduced. Micro teaching is a procedure in which a student teacher
practices teaching with a reduced number of pupils in a reduced period of time with emphasis on a
narrow and specific teaching skill. Microteaching is a scaled down teaching act, scaled down in terms
of class size and class time.
D.W. Allen defines "Micro teaching is a scaled down teaching encounter in class size and
time".
Allen and Eve defined micro teaching as a system of controlled practice that makes it possible
to concentrate on specific teaching skills and to practise teaching under controlled conditions.
David B.Young defines "Micro teaching as a device which provides the novice and experienced
teacher alike, new opportunities to improve teaching".
Clift and others defined it as a teacher training procedure which reduces the teaching situation to
simpler ad more controlled encounter by limiting the practice teaching to a specific skill and reducing the
teaching time and class size.
Characteristic features of Micro teaching:
♦ Micro teaching is an experiment in the field of teacher education which has been incorporated in
the practice teaching schedule.

♦ It is a student teacher skill training technique and not a teaching technique or method like lecture -
demonstration or inductive-deductive methods.
♦ Micro teaching is micro or minature in the sense that it scales down the complexities or real
teaching with provisions of

• Practicing one skill at a time

• Reducing the class size to 5-10 pupils.
• Reducing the duration of lesson to 5-10 minutes.
• Limiting the content to a single concept.

♦ Micro teaching advocates the choice and practice of one skill at a time.
♦ Feed back is provided immediately after the completion of the lesson.
♦ It is an analytic approach to training.
♦ Micro teaching is a highly individualized training device permitting imposition of a high degree of
control in practicing a particular skill.
Components of Micro Teaching
A. T u
v Teacher trainee
♦ Students [5-10]
♦ Observers [2]
♦ Supervisor [1]
Micro teaching procedure
In a micro teaching procedure the student teacher is involved in a scaled down teaching
situation-in terms of class size, class time and teaching tasks. The tasks may include practicing and
mastering of a specific teaching skill such as explaining, questioning, introducing, mastering of specific
teaching strategies, flexibility; use of instructional materials and class room management.
The short lesson are taken by the student teachers. The pupiis who attend the lesson are asked
to fill in the rating questionnaires and evaluate the specific aspects of the lesson. The supervisor also
records and suggests remedial measures. If asked to reteach, the student teacher replans his lesson
and immediately reteaches the lesson to the group of peer members which is observed, recorded and
feedback provided. This cycle continues until he masters the specific skill. The micro teaching cycle
is represented below.
Micro Teaching Cycle of operation
Steps of Micro-teaching
The Micro-teaching programme involves the following steps :
Step I
Particular skill to be practised is explained to the teacher trainees in terms of the purpose and
components of the skill with suitable examples.
Step II
The teacher trainer gives the demonstration of the skill in Micro-teaching in simulated conditions
to the teacher trainees.
Step III
The teacher trainee plans a short lesson plan on the basis of the demonstrated skill for his/her
practice.
Step IV
The teacher trainee teaches the lesson to a small group of pupils. His/Her lesson is supervised
by the supervisor and peers.
Step V
On the basis of the observation of a lesson, the supervisor gives feedback to the teacher
trainee. The supervisor reinforces the instances of effective use of the skill and draws attention of
the teacher trainee to the points where he could not do well.
Step VI
In the light of the feed-back given by the supervisor, the teacher trainee replans the lesson plan
in order to use the skill in more effective manner in the second trial.
Step VII
The revised lesson is taught to another comparable group of pupils.
Step VIII
The supervisor observes the re-teach lesson and gives re-feed back to the teacher trainee with
convincing arguments and reasons.
Step IX
The 'teach - re-teach' cycle may be repeated several times till adequate mastery level is
achieved.
Uses of Micro Teaching
♦ It is an effective device for modifying the behaviour of teachers under training.
♦ It is highly individualized type of teacher training.
♦ It is useful for pre-service and in-service teacher training. The teachers can improve their
competency of teaching.
♦ Feed back being quick, there is scope for early remedy of drawbacks and hence over all
improvement in teaching sis possible.
♦ It provides a lot of scope for research work especially of experimental type.
♦ It helps in developing useful type of curriculum.
♦ Usually class room teaching is a complex and complicated type of activity, but micro teaching
simplifies it so as to make it suitable for the beginner teachers.
♦ It helps in acquiring various types of skills which ultimately forms the basis of successful teaching.
♦ It develops a lot of confidence in the teachers.
♦ It helps in sorting out problems related to class room teaching, proper solutions can be thought of.
♦ Close supervision is possible.
♦ The objectives of micro lesson are clearly given in behavioural terms.
♦ It caters to the individual differences as opportunities are provided for re-planning and re-
teaching till a skill is mastered.

♦ It reduces time and energy as there is no room for trial and error.

♦ Modification of teacher behaviour and learning of specific tasks are the main outcomes
of micro teaching.
Merits of Micro-teaching
♦ It helps to develop and master important teaching skills.
♦ It helps to accomplish specific teacher competencies.
♦ It caters the need of individual differences in the teacher training.
♦ It is more effective in modifying teacher behavior.
♦ It is an individualized training technique.
♦ It employs real teaching situation for developing skills.
♦ It reduces the complexity of teaching process as it is a scaled down teaching.
♦ It helps to get deeper knowledge regarding the art of teaching.
Phases of Micro-teaching
Micro-teaching has the following three phases.
Phase - I : Pre-active phase
Phase - II : Interactive phase
Phase - III : Post-active phase
Phase - I : Pre-active phase
Phase I emphasises on the understanding of the teaching skill that is to be leant by the teacher
trainee. It envisages the followings,
1] Orientation to micro-teaching
2] Discussion of teaching skills with their components and teaching behaviours.
3] Presentation of model demonstration lesson by the teacher educator.
4] Observation of the model lesson and criticism by the teacher trainees.
Phase - II : Interactive phase
Objective of interactive phase is to enable the teacher - trainee to practice the teaching skill
following the microteaching cycle like,
1] Preparation of micro-lesson plan for the selected teaching skill.
2] Creating microteaching setting
3] Practice of teaching skill
4] Feedback 5] Re-planning 6]
Re-teaching
7] Repetition of the microteaching cycle.
Phase - III : Post-active phase
Objective of Post-active phase is to enable the teacher - trainee to irrigate the teaching skill in
real or normal classroom situation. Integration may be defined as the process of selection, organization
and utilization of different teaching skills to form an effective pattern for realising instructional
objectives in a teaching - learning process.
Principles of Micro-teaching
♦ Micro-teaching is based on the premise that teaching can be analysed into various component
behaviours called teaching skills.
♦ The teaching skills can be defined, practised, observed, controlled, measured and evaluated.
♦ Micro-teaching technique seems to be based on skinner's operant conditioning i.e. reinforcing
an operant response increases the possibility its recurrence of a response. This principle is
fundamental to the feedback session.
♦ Skinner's theory of shaping in acquiring new patterns of behaviour seems to have been,
applied to 'teach-feedback - reteach' pattern in the micro-teaching.
 ♦ The steps involved in behaviour modification are,
Stating the behaviour in operational terms o Fixing criteria for operational terms. o Pre-treatment
stage involving measuring entry behaviour. o Giving actual treatment for behaviour modification. o
Obtaining post-treatment measures.
3.1.1 MICRO-TEACHING SKILLS (SKILL EMPHASIS)
Micro-teaching skills according to 'Allen and Ryan' are as follows:
1. Skill stimulus variation
2. Skill of set Induction
3. Skill of closure
4. Teacher silence and non-verbal cues

5 Skill of Reinforcing pupil participation

6. Skill of fluency in questioning
7. Skill of Probing questioning Skill of Higher
questions Skill of Divergent questions

10. Skill of Recognizing and attending behaviour

11. Skill of Illustrating and use of examples
12. Skill of Lecturing
13. Skill of planned repetition
14. Skill of completeness of communication.
Micro-teaching skills according tot he P.K. Parsi are as follows:

1. Skill of Writing Instructional Objectives

2. Skill of Introducing a lesson
3. Skill of Fluency in questioning
4. Skill of Probing questioning
5. Skill of Explaining
6. Skill of Illustrating with examples
7. Skill of Stimulus variation
8. Skill of Silence and non-verbal cues
9. Skill of Reinforcement
10. Skill of Increasing Pupil participation
11. Skill of Using black board
12. Skill of Achieving Closure
13. Skill of recognizing attending behaviour.
Micro-teaching skills identified by the NCERT in its publication of 'Core Teaching Skills' are as
follows,
1. Skill of Writing Instructional Objectives.
2. Skill of Organizing the content
 3. Skill of Creating set for Introducing the lesson
4. Skill of Introducing a lesson
5. Skill of Structuring classroom questions

6. Skill of Question delivery and its distribution.

7. Skill of Response Management
8. Skill of Explaining
9. Skill of Illustrating with examples
10. Skill of Using teaching aids
11. Skill of Stimulus variation
12. Skill of pacing o the lesson
13. Skill of Promoting pupil participation
14. Skill of use of Blackboard
15. Skill of Achieving closure of the lesson
16. Skill of Giving Assignments
17. Skill of Evaluating the pupil's progress
18. Skill of Diagnosing pupil learning difficulties and taken remedial measures
19. Skill of Management of the class.
Skill of Explaining
A teacher is said to be explaining when he is describing How, Why, What concepts, events,
actions and etc., It can be described as an activity to bring about an understanding in the learner about
a concept, principle etc.
Explaining bridges the gap in understanding the new knowledge by relating it to the past
experience. Explaining depends upon the type of the past experience, the type of the new knowledge
and the type of the relationship between them.
Explanations can be made more effective by using simple and clear language for clarity,
examples and illustration materials for better understanding and appropriate link words for relating the
concepts.
Components of Skill of Explaining
 Beginning Statement
 Link words
 Concluding Statement
 Questions to test pupils understanding
 Correct answers getting from the questions

Undesirable behaviours in the skills of Explaining

1) Making Irrelevant Statements:
Statements which are not related sand do not contribute to the understanding of the concept
being explanied are irrelevant for explanation.
2) Lack of continuity:
Lack of continuity in terms of logical sequence, relationship with previous statement, references
to earlier experiences and so on have to be avoided.
4.2.1.3. Model micro lesson plan [Skill of Explaining]
Name of the teacher - trainee : xxxx
Skill : Explanation
Topic : Mensuration
Teachers : In the last class we have desired the formula to find the area of a right -angled
triangle. What is the formula? [beginning statement]
Student : 1/2 ab [Where a, b are the sides]
Teacher
B C
What kind of triangle is ABC? Why? [Beginning Statement] ABC is a
Students
equilateral triangle because all the sides are equal. A

Teacher
B D C
In AABC, BC is the base, AD is perpendicular to BC.
Therefore, [Link Word] AD is the attitude.
Teacher
What king of triangles are AABD and AACD? Why?
[Questions to test pupils understanding]
AADB and AACD are right angled triangle because ZADB=ZADC = 90°
Student
[Correct answers getting from the questions asked]
Teacher : So [Link word] in a triangle, among the three angles any one has 90°, it is said to be a
right - angle triangle [Concluding statement].
Observations sheet
Components Very Good Average Poor Very
good poor
Beginning Statement
Link words
Concluding Statement
Questions to test
pupils understanding
Correct answers
getting from the
questions
Skill of Questioning
Questions are the most important tool of thinking, reasoning, learning and teaching. Questions
are used at every stage of teaching.
i.e. pre-active, interactive and post-active stages.
Objectives of Questioning
1. Finding out previous knowledge
2. Revising the topic
3. Stimulating though process
4. Encouraging discussion
5. Getting students participation
Questioning Approaches
1] Irrigating questions:
A list of questions is put to a particular student or a group of student. After which it is rotated in
the class from one student to another.
2] Delivery
In this type of approach all participants have to consider the questions. Students are given time
to think about the answers after teaches calls individual students and asks them to give answers.
3] Acknowledging the answers
Sometimes teacher put questions to acknowledge the answer of the previous question and
reinforce their positive response.
Types of questions
1] Open questions
Questions for which there is no a single definition answer.
2] Closed questions
Questions which have only one exact correct answer.
3] Rhetorical questions
Questions for which there are no easy answers. Those are highes order thinking questions.
Components of skill of Questioning:
1]Structure
2]Process
3] Product
Skill of probing and Fluency in questioning
Probing questions are those which help the students to think in depth about the various
aspects of the problem. By asking such questions again, the teacher makes the public more
thoughtful. Teacher enables the students to understand the subject deeply.
Components of skill of probing and fluency in questioning
1. Prompting
2. Seeking further information
3. Refocusing
4. Redirecting Questions
5. Increasing Critical Awareness

Undesirable behaviours in the skill of Probing and Fluency in questioning

♦ Very often or often questions and answer could make the students inattentive.
♦ Reframing the questions could result in confusion.
 ♦ Suggestive questions should be avoided.
♦ Questions requiring the student to respond by 'yes or no' are leading questions and do not
stimulate the students to think.

Observation sheet
Components Very Good Average Poor Very
good! poor

Skill of using Black Board

Blackboard is an important visual aid used by teachers for effective teaching. A mathematics
teacher makes extensive use of blackboard for daily classroom teaching for working out problems,
deriving formulae, proving theorems, drawing figures, constructing geometrical figure and etc.
Purpose of Black board
1] Effective visual aid
2] Provides clarity in understanding concepts 3] Draws
attention of students at relevant points 4] Presents holistic
picture of the content.
Components of skill of using Black board
1] Legibility of hand writing
2] Neatness
3] Appropriateness
4] Orgnization
Observation sheet

Components Very Good Average Poor Very

good poor
Legjbility of hand
writing
Neatness
Appropriateness
Organization

Skill of reinforcement
Reinforcement is strengthening the connection between a stimulus and a response.
Types of reinforcements
i] Positive reinforcements ii] Negative reinforcements.
i] Positive reinforcements •
Positive reinforcements provide pleasant experience or a feeling of satisfaction which contributes
towards strengthening of desirable responses.
ii] Negative reinforcements.
Negative reinforcements result in unpleasant experiences, which help in weakening the
occurrences of undesirable responses.
The skill of reinforcement increases the student involvement towards learning. The skill is
used, when the teacher reinforces correct responses with a smile, when the teacher praises a good
response or encouraged a slow learner. The use of more and more positive reinforces maximize
students involvement towards learning rather than the use of negative reinforces.
Components of skill of reinforcement
1] Positive Verbal reinforcement
2] Positive non-verbal reinforcement
3] Negative Verbal reinforcement
4] Negative non-verbal reinforcement
5] Contact reinforcement
6] Proximity reinforcement
7] Activity reinforcement
8] Token reinforcement
9] Inappropriate use of reinforcement
10] Denial reinforcement

Components Description
Positive Verbal reinforcement Repeating students answer, Good, very good,
excellent, right.
Positive non- Smiling, nodding the head, moving towards the
verbal student when correct response, mm-hmm etc.
Negative Verbal Nonsense, Very bad, No, worst etc. ,
reinforcement
Negative non- Anger face reaction, Negative head movements
verbal etc
Contact reinforcement Patting the back, hand shaking, putting hands on
the students head.
Proximity reinforcement Going nearer to the pupils and making them
more involved and interesting in learning
Activity reinforcement Giving a task, project, home work, assignment,
Token reinforcement etc.
Awarding marks, grader, etc.
Inappropriate use When the teacher does not encourage the student
of reinforcement with respect to quality of his response. Teacher
uses same type of comment for every response.
Denial reinforcement The teacher does not give reinforcement when
the situation is demanding encouragement.
Undesirable components of skill of reinforcement
1] Negative verbal reinforcement
2] Negative non- verbal reinforcement
3 Inappropriate use of reinforcement
4. Denial reinforcement
Observation Tally sheet
Components Tally Total
1. Desirable Components
1] Positive verbal reinforcement
2] Positive non-verbal reinforcement
3 ] Contact reinforcement
4] Proximity reinforcement
5] Activity reinforcement
6] Token reinforcement
II. Undesirable components
1 ] Negative verbal reinforcement
2] Negative non-verbal reinforcement
3] Inappropriate use of reinforcement
4] Denial reinforcement V

Skill o stimulus variation

If the classroom environment becomes monotonous, then it put a negative impact on the teaching
and learning process. It is therefore, essential to make the classroom environment challenging and
interesting such that the teaching - learning process becomes lively, interesting, pleasant and a though
provoking experience. This process of brining variation in the overall interactive environment of the
class with the help of stimuli change is called stimulus variation.
It is very important for a teacher to secure and sustain student's attention. For this purpose, the
teacher uses some pester, body movements, verbal statements, etc. All these behaviours are related to
stimulus variation. The skill of stimulus variation can be defined as deliberate change in the attention
drawing behaviours of the teacher in order to secure and sustain student's attention towards the
lesson.
Components of the skill of stimulus variation
1] Movement
2] Gesture
3] Change in interaction style
4] Change in speech pattern
5] Oral - visual switching
6] Pupil activity
7] Focusing

Components Description
] ] Movement Teacher should not be static while teaching. He/she should more
around the teacher's table. So that both the teacher and students
be active.
2] Gusture The Body language and the facial expression of the teacher
should be pleasant, relevant to explain the concepts and make
teaching alive experience.
3] Change in The classroom interaction pattern must be changed constantly to
interaction style make the class lively and everybody participate in the learning
process. Change in interaction style may be in the following four
types:
1] Teacher to whole class
2] Teacher to group
3] Teacher to student
4] Student to student
4] Change in speech Teacher should very his/her speech pattern depending on the
pattern relevance o the concept and to break monotony of the class. It also
5] Oral - visual switching helps in gaining and maintaining students attention and in
reflecting the importance of the concept being expressed. Speech
pattern may be in the following
ViZ,■■ : : ■ ■ . ■
1J Pausing 2] Low pitch 3J High pitch

According to the topic, it is necessary to shift sensory channels of

students. Oral-visual switching may be in the following ways:
1 ] Verbal to visual
2J Verbal to verbal - visual
3] Visual to verbal
4] Visual to verbal - visual

6] Pupil activity In the activity based learning, the par f the •dents.

7J Focusing It implies drawing the attention of t! els a

---------------------------- _ particular point which the teacher wishes to emphasize, Focusing
may be verbal focusing, gestural f , and verbal-gestural
focusing.

Components Very Good Average Poor Very

good poor
1] Teacher's Movement
2] Teacher's Gesture
3] Change in interaction style
4] Change in speech pattern
5] Oral-visual switching
6] Pupil activity
7] Focusing

Skill of introduction
Introduction skill is the skill required to begin the teaching-learning process on a good note.
The objectives of the skill of introduction are as follows:
 ♦ Get students attention and their readiness for learning
♦ Arouse student's motivation
♦ Clearly indicate the learning experience to be provided
♦ Suggest ways and means of the approaching activity to be done
♦ Review provisions experiences and knowledge and makes its link to the present content.
Component of the skill of introduction
1 j Gaining Attention
2] Use of pervious knowledge
3] Use of Appropriate Device

Components Description
1] Gaining Attention ♦ By using voice, gesture and eye contact
♦ By using audio - visual aids.
♦J* Changing the pattern of teacher - student interaction
2] Use of pervious ♦ Previous knowledge refers to the student's level of knowledge
achievement before instruction begins. Use of previous
knowledge is a
must, because it helps to establish integration between the pre-
existing
knowledge of the learner and the new knowledge that the teacher
3] Use of ♦ In order to motivate the learner, the teacher should make use of
Appropriate Device appropriate devices or techniques while introducing a lesson. Eg:
Dramatization, models, audio-visual aids.

Skill of closure
It is an important skill to achieve closure at the end of the lesson
Components of the skill
1] Consolidation of major points
2] Application of the present knowledge in a new situation
3] Linking the part knowledge with the present knowledge
4] Linking the present knowledge with the future learning
3.2. LINK LESSON
In micro-teaching technique, teaching skills are practiced one by one separately. At a time, only
one skill can be practiced. While practiced one skill, the use of that particular skill is maximized and other
related skills may also occur inking indirect-role. Skills practiced in isolation have no meaning unless
they are integrated in teaching.
Hence after attaining mastery in various skills, opportunity should be given to the teacher
trainees to teach in real situation integrating the skills mastered already. So separate training
programme is necessary for this purpose. This programme is called link practice.
Line practice is a bridge between micro-teaching and macro-teaching where micro-teaching
skills are effectively integrated and transferred.
These is a big contrast between micro-teaching and macro-teaching. In micro-teaching, there
is a scaled down process in terms of class room size, skills, scope of the lesson, time etc. Macro-
teaching is practiced under stimulated conditions. In macro-teaching in addition to the existence of
macro elements, there are also class room management problems. In link lesson the trainee are given
chance of teaching real students.
There are many method for link lesson. On the method sis that after practicing three sub skills
separately, the trainee may combine all the three sub skills in a lesson of 10 minutes. He then practices
another three sub skills separately and links them. He then combines all the six sub skills in a single
lesson of 15 minutes. And so on till the entire sub skills are combined in a macro lesson of 40
minutes and teaching a full class.
Link lesson sessions are arranged with about 20 students for about the normal class period i.e. 20
minutes. The trainee prepares a series of eight short lessons on single unit and teaches each lesson for
20 minutes using appropriate skills particular to the content. The number of lessons used in link
practice is flexible but selected topic should be adequately covered. The teaching skills namely 'Set
Induction' and 'Closure' cannot be practiced in micro-teaching session in isolation. So in link lesson,
the trainees include these skills also. At the end of each lesson the trainee should get feed back about
the lesson.
3.3 MACRO - TEACHING
The word 'Macro' comes from Greek work 'Makros which means 'Long / Large'. Macro-
teaching occurs when a teacher provides instruction to the entire class at one time for an extended
period of time, usually longer than 40 minutes. Macro-teaching is after done in lecture format and
may be used to introduce a new concept, or to practice a new concept. Macro-teaching allows a
teacher to introduce new information to everyone at once. While that's sometime an advantage, it can
be a drawback if several students are performing below grade level and aren't yet ready academically
to learn the new concepts. At the same time, macro-teaching can give a teacher an idea of what
subjects or concepts she/he needs to spend more time on, as well as who in her / her class needs
additional help. Planning lessons at the macro level helps a teacher stay on track so she's able to meet
her/ his goals and cover the entire curriculum before the school year ends.
LESSON PLANNING AND ITS USES
The purpose is to help the teachers to decide about the type of lesson most appropriate in various
circumstances and plan the strategies and materials accordingly. This lesson also highlights the
importance of lesson plan and unit plan, its components and its uses. A good programme of instruction
will contain some lessons given to the whole class, others involving groups of students and still others to
cater to the individual needs and ability of the students. If teachers concentrate on their daily lessons
they really would be drawing out the best in the students helping them to develop, progress and
achievement.

Lesson Plan
Carter V. Good defines a lesson plan as "a teaching outline of the important points of a lesson
arranged in order in which they are to be presented. It may include objectives, points to be asked,
references to materials, assignments etc."
Bossing defines, "A lesson plan is an organized statement of general and specific goals
together with the specific means by which these goals are to be attained by the learner under the
guidance of the teacher on a given day."
Truly it is commented, "Lesson plan is teacher's mental and emotional visualization of class
room activities."
Well planned lessons give confidence to the teacher and that is why more emphasis is laid on the
daily lesson plan. This ensures progress and continuity. The teacher should answer a comprehensive list
of questions based on three basic questions what? Why? and How?
1. What are the goals and objectives I want the students to attain?
2. What am I going to teach and what experiences am I going to provide for the students? and why is
it important?
3. How best could I introduce the topic?
4. How do I present and develop the lesson?
5. What teaching aids will I need?
6. What are the various activities that would be appropriate for effective learning?
7. How will I get the evidence of the student, understanding?
8. How do I provide review?
9. How do I evaluate student's attainments and the effectiveness of my teaching?
At the end of the lesson every teacher should ask himself, Are the objectives realized? If not, why?
What points from this lesson I should bear in mind in preparing my next lesson? Could I use some of the
features in this lesson for future reference?
Functions of a Lesson Plan
The daily lesson plan specifies the area of classroom activity. It serves as a check on the
possible wastage in time and energy of both the teachers and the taught. It gives an opportunity to
think and be imaginative. It helps the teachers not to deviate from the right direction unnecessarily.
Principles of Lesson Planning:
1. Formulation of objectives.

2. Deciding and selecting the content to be covered.

3. Collecting and selecting suitable information.
4. Presenting of the subject matter in an organized, systematic and effective manner.
5. Participation on the part of the learner as a co-share in the educative process.
6. Assessing the outcomes of the lesson and the achievement of the objectives.
Daily Lesson Planning
While yearly planning is made for the year or session and unit planning is made for the
teaching-learning of a particular unit drawn from the prescribed syllabus of the year, the daily lesson
planning as the name suggest is the planning made for the instructional work carried out by the
teacher on a day to day basis. However, the term daily lesson planning is not much in use. It has been
replaced by the term lesson planning for conveying the same meaning. Let us try to understand the
meaning and purposes of lesson planning more clearly through the following discussion.
What is Lesson Planning?
The duties of a teacher demand from actual classroom, teaching. He has to teach daily one or
more subjects to one or more classes in his school. While trying to perform his duties regarding the
classroom teaching, a teacher has to pass through the following phases
1. Pre-active phase of the teaching.
2. Inter-active phase of the teaching.
3. Post-active phase of the teaching.
Where what does a teacher do in the classroom with regard to his actual classroom teaching
along with his students are covered in the interactive phase, the theoretical activities performed at the
cognitive level by the teacher before the actual classroom teaching are known to be related with the
pre-active phase of teaching.
In simple words lesson planning means the planning of a daily lesson related with a particular
unit of a subject to be covered by the teacher in a specific school period for the realization of some
stipulated instructional objectives. It is a sort of theoretical chalking out of the details of the journey
which a teacher is going to perform practically in the classroom along with his students.
Now the work of chalking out the details of such journey or preparation on the part of a teacher
for executing the task of actual classroom teaching may be done either at the cognitive level or
preferably in the written form by writing a lesson plan.
In this planning a teacher of mathematics may thus have to pay considerations to the
following essential aspects
♦ Broader, goals or objectives of the subject mathematics.
♦ Setting and defining of the classroom objectives related with the present unit or topic of the subject
mathematics.
♦ Organisation of the relevant subject matter to be covered in the given lesson for the realization of
the set objectives.
♦ The decision about the method of presentation of the subject matter, teaching strategies, classroom
interaction and management.
♦ Appropriate provision for evaluation and feedback.
How to plan Lesson for Teaching Mathematics?
A teacher has to plan for the teaching-learning of a small subunit or portion of the prescribed
syllabus of his subject going to be covered in the fixed duration of the classroom period. It is a quite
routine affair for him to be done on the all working days in a scheduled period of the school time
table.
The educationists and researchers in the field of pedagogy have suggested from time to time
appropriate guidelines for the planning of these daily lessons. However, the schedule suggested by the
renowned educationist J.F. Herbart in the shape of his famous six steps has remained quite popular for
the lesson planning in almost all the subjects of the school curriculum.
The six steps suggested by him for the lesson planning are as below:
Components of a lesson plan
i. Preparation:
The teacher should first of all prepare the students to get new knowledge. He should excite
the students in such a way that they feel the need of learning new things by means of stories or through
charts, models and pictures or from student's pervious experiences.
ii. Presentation:
The subject matter should be presented in simple and familiar way. It should be done with
the active participation of students. Students should not be passive listeners. Questions asked should
be relevant, according to the mental level of students and evenly distributed. Aids should be used and
black board summary should be developed side by side.
iii. Comparison and Association:
Whatever students learn, they compare it with similar set of examples. iv.
Generalization:
It should be done with the participation of students.
v. Application:
Whatever the students learn should be applied to new situation and unfamiliar facts so that
there is transfer of training and knowledge gained becomes permanent.
vi. Recapitulation:
This is the last but not in any way the least important step. There the teacher knows what the
students have learnt, where they stand and whether the teacher himself is successful in his aims or not.
The educationists like H.C. Morrison and B.S. Bloom have tried to forward their own
schemes and guidelines for the purpose of planning the daily lessons, in the name of unit approach and
evaluation approach respectively. In our country also some attempts have been made to suggest
suitable scheme and guidelines for the planning of the lesson.
Influenced by the above
1. Identifying Data:
It is the beginning step of the lesson planning. Here the following things are written by the pupil
teacher in his notebook and on the black board while initiating to deliver his lesson in the class.
I. Name of the pupil teacher or his roll number.
II. Class and section to be taught.
III. Name of the subject [Mathematics is to be written here]
IV. Topic [on which lesson has been prepared]
It should be written just after making announcement of the aim or purpose of the lesson e.g. at
the introductory stage when the teacher has made announcement that "today we will study about
simple interest." then he may write name of the topic as "simple Interest".
V. Date on which lesson in being taught.
VI. Duration of the period for which lesson has been prepared. Generally it is 35 or 40 minutes.
 2. Aid Material:
Here the type of aid material needed for the classroom teaching-learning of the lesson in hand is
mentioned. Note that it is only written in the lesson planning notebook No mention of such aid material
either verbally or by writing of the board is ever made by the pupil teacher in the class. Usually the
following types of aid material may be mentioned by the pupil teacher in his notebook.
1. Backboard, chalk, duster, pointer etc. [the type of aid material essentially needed for every
type of mathematics lesson].
2. Concrete material or real objects.
3. Graphical material such as charts, pictures and graph or models.
4. Slides, films, transparencies or computer presentation etc.
3. Instructional objectives:
These are also written in the note book and never mentioned to the students either orally or by
writing on the black board in the class. The framing and writing of these class room instructional
objectives is a serious task which needs essential knowledge about the taxonomies of instructional
objectives related to all the domains of human behaviour and then the methodologies of writing these
objectives in behavioural terms. Pupil teachers are advised to go through the relevant chapter of this
text for acquiring essential knowledge and skill for framing and writing instructional objectives of the
different types of lessons taught by them.
4. Previous Knowledge:
The pupil teacher here mentions in his note book about the type of knowledge, skill and
experiences etc. needed on the part of the students of his class for the teaching and learning of the
present lesson. These things are also written in the notebook and never mentioned to the students.
5. Testing of previous knowledge and Introduction:
Under this step attempts are made to test the previous knowledge of the students [assumed
under the previous step] through some well framed questions or asking the pupils to do certain types of
activities, or making them to solve certain problems or putting them in some specific problematic
situation so that they may feel an urgent necessity of studying the present lesson [by being enabled to
find its solution through the previously acquired knowledge and skills]. In this way while trying to explore
the previous knowledge of the pupils related to the lesson in hand, the teacher tries to make them arrive
at the necessity of studying the very

lesson. All the questions and activities related to this step are also meant for the writing on the
notebook, and also their help is actively sought for the task of testing the previous knowledge of the students
and, making them eager to study the present lesson.
6. Announcement of the Aim:
Under this step, the necessary announcement regarding the topic to be taught is made by the
teacher to the students of the class e.g. today students, we will know about the sum of three angles
of a triangle or let us know how the sum of three angles of a triangle is equal to two right angles etc.
After making such announcement, the pupil teacher should write the name of the topic to be taught
on the blackboard.
7. Presentation:
This step is related to the planning of the real teaching-learning task performed in the class.
What type of teaching-learning activities will be performed by the teacher and pupils in the class for
the realization of the stipulated behavioural objectives are properly mentioned under this step. In the
beginning, the contents to be covered or the learning experiences to be given are mentioned as
briefly as possible. After them the activities performed by the teacher and pupils are mentioned under
two separate columns named as teacher's activities and pupil's activities. This step really represents
the heart of the lesson planning. Here all types of decisions regarding the type of learning experiences
provided to the students and the methods, strategies, techniques and tactics used for providing these
experiences are properly taken. The questions asked, the responses managed, the mutual interactions
among the students and teacher held, the teaching-learning experiences shared, the aid material and
other resources to be utilized all are properly planned for the better realization of the stipulated
instructional objectives. The planning about all such things find proper mentioning either in term of
teacher activities or pupils activities.
8. Generalization:
Many of the mathematics lessons are concerned with the establishment of certain rules,
principles formulae etc. Usually the problem solving task related to almost all the topics and
branches in mathematics needs proper generalized way of proper solving. As a result in this step,
attempts are made by the teacher with the help of questioning device and the illustrations and
examples given to help the students in arriving at some or the other appropriate conclusions and
generalization.
9. Practice or Drill work:
It is a quite important step for the planning of mathematics lessons and is related with the
post-teaching stage. In the lesson planning of other subjects it is known by the term

recapitulation. What is being taught by the teacher and learnedly the pupils is well practiced
through the activities performed in this step. The mathematical skills like computational,
measurement, surveying, drawing, graphical presentation, problem solving etc. can be properly
developed through the practice or drill work, this step is also being utilized for the evaluation of the
students learning and acquision of the things taught to them.
10. Homework:

done by them at their homes. It is also aimed for the fixation of the knowledge, understanding
and skills acquired by the students regarding the topic taught to them in the class. Usually it may be
assigned in the following forms
i. They may be asked to memorize some facts, principles, proof of the theorems and
prepositions, ways of solving the problems, formulae and their application etc.
ii. They may be asked to solve some problems either, given in their textbooks or presented to him by
the teacher in the class orally or through charts and transparencies.
iii. They may be asked to perform certain practical activities for-making use of the knowledge
and skills acquired by them through the classroom instruction e.g. measurement of the total floor,
 wall and roof areas of the rooms and other constructions of their own homes and computation of
the
iv. Expenses incurred for the white-washing, flooring and paints etc.
In this way an appropriate lesson plan can be properly prepared by a mathematics teacher by
following the above mentioned steps for the planning of the lessons in mathematics. However, it should
be always kept in mind that a prepared lesson plan howsoever good it may, be, can merely provide
some guidelines for showing the path to be followed in the classroom instructional activities. Its
implementation rests with the teacher. He has to take right decisions at the right time for making use
of these guidelines. These are meant for good teaching and better learning outcomes. These are not
meant for following blindly. A teacher should take them as a means for attaining the desired teaching-
learning objectives and not as an end in themselves. Therefore, it is always appropriate and advisable
to make some or the other changes in the prepared lesson plan while making its use in the classroom
instructional work depending upon the situations and circumstances prevailed at the time of teaching
and learning.
How should the task of lesson planning be performed by the pupil teachers for the teaching
of mathematics may further be well understood through some of the following illustrative lesson
plans.

Planning is a must for the successful execution of a task or a project. It not only caters to the
proper realization of the aim or purposes of doing that tack but also helps in proper utilization of the
time and energy on the part of human and material resources. The same is equally true for the process
of teaching-learning. The teachers who plan their work properly prove quite effective in their teaching
task. It is why, a mathematics teacher should also concentrate on a wise planning of his teaching and
instructional work carried along with his students during the whole session He may have three types of
schemes for such planning named as below:
♦ Yearly Planning
♦ Unit Planning
♦ Daily Lesson Planning
Yearly Planning in Mathematics
In such type of the scheme of planning a teacher of mathematics tries to take a complete view of
what he has to do in the whole session regarding the instructional work of a particular mathematics class.
In this way by a yearly plan we mean the sessional programme, that has to be chalked, out by the
teacher in his subject of teaching in the shape of teaching-learning activities to be carried out with his
students. For chalking out such a programme a teacher has to take care of the things like below:
1. The total number of working days available for the teaching-learning of the subject during the
year.
2. The total number of periods or time available for the teaching-learning of subject during the year.
3. The nature and scope of the subject in relation to the number of topics included in the syllabus, the
contents covered in these topics, the type of learning experiences to be provided to the students
and objectives of teaching-learning to be realized etc.
4. The means and material available for the teaching-learning of the prescribed syllabus.
Unit Planning
This yearly plan [what is to be done during the whole session] is then further subdivided into
monthly, weekly and daily plans for the purpose of proper implementation. There is still another way for
its division and implementation which is known by the term unit planning. Let us try to get
acquainted with the term unit planning.
What is Unit Planning?
In it simple meaning unit planning stands for the planning of the instructional work of the session
by dividing the prescribed syllabus into some well defined and meaningful units.
For more clarity let us first define the term unit.
The term unit has been variously defined as under Carter V. Good. "Unit may be described as
an organisation of various activities, experiences and types of learning around a central problem or
purpose, developed cooperatively by a group of pupils under teacher-leadership." [1959].
H.C. Morrison. "A unit consists of a comprehensive series of related and meaningful activities so
developed as to achieve pupil purposes, provide significant educational experiences and result in
appropriate behavioural changes. [1961].
The analysis of the above two definitions may reveal that
♦ The contents of a unit are always woven or organised around a central problem or purpose.
♦ Students may cooperate the teacher in the formulation of units.
♦ A unit consists of well integrated meaningful wholes capable of providing useful learning experiences
to the students for achieving the desired teaching- learning objectives.
♦ The subject matter or content of a unit represents continuity and comprehensiveness in
conveying a proper sense or understanding of a particular problem, theme or knowledge area
related to a curricular subject.
♦ A unit represents a wholesome and complete sub-division of the contents of a syllabus quite useful
and meaningful in providing rich educational experiences.
In this way by the term unit we may understand one of the complete and meaningful sub
divisions of prescribed course of a subject, centered around a single problem or purpose capable of
helping in the realization of the desired teaching- learning of the subject. After grasping the meaning
of the term unit now we can attempt for knowing about the nature and meaning of the term unit
planning. As pointed out already the syllabus for its proper coverage in a session is divided into some
complete and meaningful sub-divisions known as units. Now to think about a proper way for the
teaching-learning of the subject matter or learning experiences contained in these units by keeping
an eye over the proper realization of the teaching- learning objectives of the subject mathematics is
known by the term unit planning in mathematics. In this way the term unit planning may be defined
as a scheme or plan chalked out for the teaching-learning of a particular unit, [a meaningful and
complete subdivision of the learning experiences to be given in the whole session] mentioning the
ways and means of imparting learning experiences related to that unit in view of the proper realization of its
teaching-learning objectives.
Units Formation in the Subject Mathematics
The task of unit planning in Mathematics starts with the formation of proper units out of the
subject matter and learning experiences to be imparted to the students during the whole session. For
the formation of units a teacher may proceed as under.
1. He may accept the various topics as mentioned in the prescribed syllabus as different units for
the task of his unit planning.
2. He may combine the different alike topics of the syllabus for the formulation of units i.e.
Area of all types of geometrical figures, volume of all types of bodies and figures or even
more broader like mensuration or Graphs.
3. He may formulate the units on the basis of similarities or symmetry observed in the structure,
principles and system of mathematics like Number System, Decimal System, Rational and
Irrational numbers, system of integers, Measurement systems, Congruency, etc.
4. He may organise the subject matter into units on the basis of mathematical formulae and
principles or similarities found in the solutions of mathematical problems like Algebraic
formulae, Simple equations, Simultaneous equations, Quadratic equations, Trigonometric
formulae and functions etc.
For the purpose of dividing the contents of the syllabus, into various units, a teacher of
mathematics should try to take note of the following things.
i. The total days and working hours [classroom periods and other extra time] available for the teaching
of Mathematics in a particular grade or class.
ii. The completeness and meaningfulness of the units formulated in terms of some special purpose or
objectives achieved.
iii. Suitability in terms of the age interest, needs and abilities of the learners.
iv. Suitability in terms of the resources and teaching-learning conditions available for the teaching-
learning of the units.
v. The proper division of the whole syllabus [contents and learning experiences] in view of the total
time and resources available.
vi. Being in perfect tune with the realization of the teaching-learning objectives of mathematics.
vii. The proper integration and correlation of the subject matter and learning experiences available
within the units themselves.

viii. The proper correlation, coordination and integration among the different units formulated out of the
prescribed syllabus for the needed continuity and convenience.

How To Proceed Further In Unit Planning?

After organising the subject matter and learning experiences into some complete and meaningful
subdivisions i.e. units, the further work of planning may be undertaken in the following ways.
1. A unit in hand should be further divided into some suitable sub-units or parts. A sub-unit or part,
as far as possible, should contain that much of the subject matter or learning experiences as
could be covered within the available classroom period of 35 or 40 minutes.
2. Objectives related to the teaching-learning of the unit should be predetermined and properly
framed by expressing them into behavioural terms for making clarity about the types of
behavioural changes expected from the students after going through the unit.
3. Proper decisions should be taken about the methods and technique used, and audiovisual aids
and other material utilized and teaching-learning experiences given for the realization of the set
teaching-learning objectives.
4. Decisions should also be taken about the type of interactions among the teacher and students and
the relative roles played by them in performing the various activities during the teaching-
learning process.
5. Proper decision should also be taken for the evaluation of the teaching- learning of the unit
covered. For this purpose it is always better to prepare before hand the desired unit test. The
time and resources needed for the administration of unit test should also be well decided.
Let us now illustrate, the procedure of unit planning in mathematics by an example. An
Example of Unit Planning
1. Subject: Mathematics
2. Name of the Unit: Various types of measures and their measurement
3. Name of the sub-units:
a. Length and its measurement
b. Mutual conversion of the bigger and smaller units of length.
c. Weight and its measurement
d. Mutual conversion of the bigger and smaller units of weight
e. Temperature and its measurement
f. Mutual conversion of the measurement of temperature from one scale to another
g. Time and its measurement
h. Mutual conversion of the bigger and smaller units of time
i. Our currency and its units of measurement.
j. Mutual conversion of the bigger and smaller units of currency.
4. Teaching-Learning Objectives of the unit:
a. The following types of behavioural changes are expected from the students as
a result of teaching-learning of the above unit.
b. Students tell what they mean by measurements related to length, weight,
temperature, time and currency.
c. Students tell about the utilities of these measures and their measurement in day
to day life.
d. Students tell about the units bigger as well as smaller, employed for the
measurement of these measures.
e. Students, demonstrate the skill of mutual conversion of the bigger and smaller
units of the measurement of these measures.
5. Decision about the methods and aid material:
For the attainment of objectives through the teaching-learning of the contents related to various
sub-units, the required pre-planning for the selection of appropriate methods and aid material etc.
6. Evaluation Work:
At the end of the teaching learning work related to various sub units resulting outcomes will be
evaluated through a unit test. This test will help in, determining the extent to which stipulated teaching
learning objectives have been realized. The follow up work in the form of remedial education or
necessary practice and drill work will be done iii the subsequent period.
On the similar lines a teacher of mathematics may plan for the teaching learning of other units
in the subject mathematics related to syllabai of more higher classes. For doing so let us assume 'for
one's setting for the planning of the unit named as "mensuration or Areas and volumes of different
figures." For the planning of this unit he may then advance by subdividing the unit into the following
sub-units each to be covered in a single classroom period.

Importance and Advantages of Unit Planning

1. The syllabus in terms of content and learning experiences to be covered in the whole session is
suitably divided into units in view of the time available for the teaching of mathematics.
2. Units represent the unified and integrated wholes of the meaningful and purposeful content
material and learning experiences.
3. Unit planning lays proper stress on the formulation of teaching learning objectives of the unit in the
behavioral terms.
4. In unit planning, a teacher is well informed about the type of methods and strategies used, aid
material and resources utilized for the teaching learning of the various sub units.
5. Unit planning helps in the proper organization and sy stematization of the teaching learning process. A
teacher has a prior knowledge of the task and'activities to be executed for the teaching and
learning of the unit and sub units and such knowledge makes him mentally and professionally
prepared for the fulfillment of his obligations as a teacher.
6. The planning of the activities and resources beforehand and division of the content material into
complete and meaningful units makes the task of teaching and learning quite interesting and
absorbing leaving no scope for the problem of indiscipline in the class.
7. The administration of the pre-prepared unit, test in view of the stipulated teaching-learning
objectives helps in the proper evaluation of the teaching- learning task.

8. Unit planning has a proper provi sion for the diagnosis of the learning difficulties of the students
and subsequent remedial instruction.
9. Unit planning has a proper provision for the review, recapituation, practice and drill work related to
the contents and learning experiences relative to the sub-units.
10. Unit planning paves the way for a proper and appropriate daily lesson planning. A teacher feels quite
at home in the task of his daily lesson planning in view of the blue print provided by unit
planning.
Demerits and limitations of unit planning.
Unit planning is discredited on account of its following shortcomings and limitations.
1. The division of the contents of the syllabus into meaningful and complete units and sub-
units is not an easy task. The improper and faulty formulation of units and sub-units may
create hurdles in the path of teachers and students for the proper teaching and learning of the
subject.
2. Mathematics is a well organised, systematic and sequenced subject. The certain types of
information, pre-knowledge and skills are required for the learning of one or the other topics.
However, while organising the syllabus into some or the other units, we have to set aside the
logical and sequential development of the subject. It creates too much difficulty for the teacher as
well as students of mathematics in their respective tasks of teaching and learning.
3. The unit planning puts restrictions on the freedom of teachers. The predetermined objectives,
learning experiences, methods and resources, method of evaluations etc. leave little scope for the
originality and creativity of the teacher needed in the changed situations and circumstances faced
in the classroom environment.
4. The teaching-learning process becomes too much time bound through the adoption of unit
planning. Mathematics is a skilled and applied subject that needs a lot of exercises and activities for
the practice and drill work. Limitations and restriction of time provided by the unit planning thus
may prove a big obstacle for the teaching and learning of this subject.
5. Unit planning may make the teacher too much conscious for the implementation of planned
scheme as the planning becomes an end instead of remaining means for the realization of the
teaching-learning objectives.
6. The students and teachers have no ways other than to follow the guidelines provided by the unit
planning. It makes the teaching and learning as planning centered rather than students centered.
Their needs, interests, abilities and capacities are thus may be unnecessarily sacrificed for the
sake of following the path laid down by unit planning.
7. The task of unit planning needs expertization as well as labour and time spent on the part of a
teacher. Overloaded with teaching and other functionary duties teacher takes too little interest in
the proper planning of the units. Absence of needed skill for unit planning, as well as lack of interest
and enthusiasm for doing such work, thus may prove a big handicap in the successful realization
of the purposes of unit planning.
Shortcoming and limitations listed above may leave an impression that unit planning is not at all
beneficial for the subject like mathematics. However, this conclusion is not justified. It is only one
side of the picture. A close analysis may reveal that the defects and problems do not lie in the
purposes, processes and outcomes of the unit planning. Planning for any task is always aimed for the
proper execution of that task. Unit planning in this way
is always aimed first for the proper division of the syllabus into suitable units and then thinking
proper ways and means for the reahzation of the stipulated teaching-learning objectives through the
meaningful instructions related to these units. The limitations and shortcomings if any thus lie in the
system of implementation and not in the planning. Therefore, there should not be any hitch or doubt in
the adoption of unit planning for the teaching-learning of mathematics.
Distinction between Unit Planning and Daily Lesson Planning
Unit planning and Daily lesson planning can be distinguished from each other on some of the
following grounds.
1. Unit planning is meant for the division, organisation and planning of the prescribed syllabus
being covered in the whole session while daily lesson planning helps in the organisation of
teaching- learning in terms of a lesson delivered during a classroom period.
2. The scope of unit planning is much wider than the scope of daily lesson planning.
3. The unit planning may be done for the teaching-learning carried out in a classroom period or its
duration can be extended to several days but in the case of lesson planning its duration is strictly
limited to a single day task i.e. fixed duration of a classroom period of 35 to 40 minutes.
4. The unit planning carries the objectives of the teaching-learning to be done for the whole unit
comprising of the various sub-units. These objectives may have quite wider coverage in
comparision to the objectives of daily lesson planning strictly limited to the expected behavioural
changes brought out in the limited time of the classroom period.
5. A unit planning may give birth to a number of daily lesson planning depending upon the number
of sub-units carved out of a unit in hand. Sometimes a subunit may need its handling through
two or more days lesson planning depending upon the nature of instructional work needed to be
carried out.
HERBATIAN STEPS
John Fedrik Herbat advocated the following lesson plan steps.
1] Preparation
2] Presentation
3] Association or comparison
4] Generalisation
5] Application
6] Recapitulation
1] Preparation
It pertains to preparing and motivating the children to learn the new topic. The mind of the child
should be prepared to receive new knowledge. The preparation should not only set the atmosphere for
learning, but it should also arrest attention of the students and focus it on the new topic. The
preparation may involve.
♦ Testing the previous knowledge relevant for learning the new topic.
♦ Integrating the previous knowledge with the new lesson to be learned.
♦ Capturing the attention and maintaining interest using audio-visual aids, story telling, etc.,
Arousing the curiosity of the students by creating a problematic situation or posing an
intriguing question.
♦ Announcing the aim of the lesson.
2] Presentation
Actual teaching takes place at presentation stage. Students acquire new knowledge and ideas.
For an effective learning out come, the teacher should ensure active student participation by
providing a number of learning activities. Teacher can make use of audiovisual aids and Teaching -
learning materials to make the learning interesting, effective and meaningful. The teacher should
stimulate the mental facilities of the students by asking questions.
 3] Association or comparison
The new knowledge becomes more meaningful when it is compared, contrasted, associated
and integrated with already existing knowledge. This step is particularly significant in subjects like
mathematics where the students have to learn definition, establish principles or generalizations or
formulae deducted from already learnt concept, postulates, theorems and axioms.
4] Generalisation
Students are required to establish formula, law, rule and etc., in mathematics. This is possible
presenting particular examples and requiring the students to observe and compare for co
5] Application
In mathematics the students have to apply the rules, formulae or generalizations that they have
learnt in order to solve problems. By application the validity of the generahzation is tested and verified.
6] Recapitulation
Recapitulation can be done through the following,
♦ asking questions on the contents of the lesson.
♦ giving a short objective type test.
♦ asking the students to arrange the steps in solving problem in its logical sequence.
MODEL LESSON PLAN
Name of the teacher : Subject: Mathematics Date:
School : Topic : Menstruation Time:
Standard : IX Sub Topic: Sector of a circle Period:
Strength
General Instructional objective: The Pupil
1. acquires knowledge about the method of finding the area of the section of a circle.
2. develops understanding of the method of finding the area of the section of a circle.
3. applies the knowledge to solve some problems using the formula.
4. develops skill in a) drawing free hadn sketch of the section of a circle b) computation
5. develops interest by dicussing the given problem.
6. develops scientific attitude towards the sector of a circle and appreciation by making the sector of
circles with different measures.
Specific Instructional objectives: The Pupil
1. recalls central angle and formulas for length of arc of the sector of a circle.
2. recognizes the method of finding the area of the sector of a circle.
3. identifies the shape of circle.
4. translates the verbal form of central angle to symbolic form.
5. suggests alternate formula for finding the area of the sector of a circle.
6. analyses the given problem
7. establishes the required relationship
8. generalizes the area of the sector of a circle with length of arc '/' and radius V.
9. does written calculations systematically.
10. discussess the means to solve the problem.
 mmon elements or pattern. Thus, students are led to frame a general law or principle.

11. considers the given problem in all its aspects.

12. makes models with the help of paper cuttings.
Teaching Aids:
Colour Papers for cutting sector of circles with different measures.

Specifications Content analysis Learning Experiences Evaluation

The pupil It is circle in shape What is the shape of this
identifies picture?
Ring, ball, bangle etc. Can you give some examples of
things that are circle in shape?

Area of circle = ror What is the formula to find the

area of a circle whose radius
is 'r?
recalls Part of a circle within In the circle what is sector?
two radii and an arc is
sector.
recognizes To find the area of a Let us find the area of a
sector of a circle with sector of a circle whose
length 1 units and length of arc T units and
radius r units radius 'r' units |
analyses Length of the arc i. What are the measures What are the
radius of the circle r, needed to find the area of the different types
and central angle. sector of a circle? of angles
Makes models ZACB is the central In the paper cutting of circle : :■
translates angle with 'c' as centre draw to
draws neatlv arcs at two different points
;
and denotes it as 'A' and 'B\
Now how can you write the
central angle in the form of
\ symbol?
How can you represent the
above circle in a diagram?
recognizes ACB is the sector of a What can you say about the What is the
circle sector of a circle? value of Jt?
D -, What is the area of a sector of i ■ .............
A = ----- 7tr a circle? *
Whose centra! angle is 'D'
and radius 'r' Units
D is the c ;ngle What is'D'?
Specifications Content analysis Learning Experiences | Evaluation
D
suggests 2r How can you rewrite the
A = ---- xitr — formula in some other form?
360 2
 D
recalls , 2 What is the formula to find
360 length of arc of the sector of a
circle?
lr
recognizes A Now what is the formula for
A=— finding the area of sector in
2 terms of 1?
generalizes Yes Whether the formula for If radius is 5cm, and
finding the area of sector in the length of the arc
terms of T and 'r' is true for all is 157t What 'D' is?
measures of circle?

Let us solve a problem. Find the

length of arc and radius if the
area of sector a circle is 24071
and whose central angle is 150

-analyses Given. A=204jt What is given in the What is the formula

D=150 problem? to find 1?

Length of arc and What is to be determined?

radius.
!50 2 Substituting the given details in
2407t = ---- X7T the area of the sector of a
circle, the length of the arc and
360 radium can be found
D
does not written _2407tx360 Now find the length of the A 2
calculations 150JI = arc? A ------- rtr
systematically V576 = 24cm 360
It is true.
Establishes D
relationship L = ----- 27tr
360
= — 2rcx24 360
=24

Review
1. What is central angle?
2. How will you find the arc length and area of the sector of a circle?
3. What is the area of the sector of a circle?
Assignment
1. The length of arc and radium is 87t cm and 10 cm. Find the area of the sector of a circle.
2. Find the length of arc if the central angle of the sector of a circle is 135 and radius 8JI cm
General Instructional objectives [G.,I.O's]
i] Knowledge ii] Understanding iii] Application iv] Skill v] Appreciation vi] Interest
vii] Scientific attitude
GIO's Writing:
♦ The pupil acquired the knowledge of the construction of a triangle.
♦ Pupil acquires knowledge about the derivation of the formula for area of a circle.
 ♦ The pupils acquire knowledge about the derivation of the formula for the perimeter of
rectangle.
♦ Pupil acquires knowledge of the expansion of (a+b)2
♦ Pupil gains on understanding of the construction of a trapizium.
♦ Pupil develops an understanding about the derivation of the formula for the area of semi-circle.
♦ Pupil develops understanding about the derivation of the formula for the area of acquire -
angled triangle.
♦ Pupil gains understanding of the derivation of the expansion of the identity (a+b)2
♦ Pupil applies their knowledge and understanding in new situations.
♦ Pupil applies their knowledge and understanding about the perimeter of rectangle in an
unfamiliar situation.
♦ Pupil applies their knowledge and understanding of the mathematical formula in new
situation.
♦ Pupil applies their knowledge and understanding of the expansion of identities with numerals.
♦ Pupil gains skill in construction of circle.
♦ pupil develops skill constructing mathematical figures.
♦ Pupil develops skill of computation by the drawn geometrical figures
Specific Instructional objectives [S.I.O's]
S.I.O's under knowledge:
♦ Pupil recalls
♦ Pupil names the different figures
♦ Pupil recognises the given values
S.I.O's under understanding:
♦ Pupil identifies the measurements given for construction geometrical figures

♦ Pupil indicates the different parts *t*

♦ Pupil evaluates
♦ Pupil differentiates
♦ Pupil gives examples

♦ Pupil substitutes the values

♦ Pupil explains
♦ Pupil compares
♦> Pupil generalises the different concepts
♦ Pupil formulates
♦ Pupil concludes
S.I.O's under Application:
♦ Pupil analyses the problem
♦ Pupil select the suitable method
♦ Pupil judges the sufficiency of the measurements
♦ Pupil computes
♦ Pupil analyses the problem
♦♦♦ Pupil verifies the result
 S.I.O's under skill:
♦ Pupil takes measurements with speed and accuracy
♦ Pupil uses the geometrical instruments
♦ Pupil selects the appropriate geometrical instruments
♦ Pupil draws the figures accurately and neatly

♦ Pupil does the calculation

S.I.O's under Appreciates:

♦ Pupil appreciates that the mathematical formula which helps in daily life.
♦ Pupil appreciates the symmetry of mathematical regular figures.
S.I.O's under Interest:
♦ Pupil does the puzzles

♦ Pupil catches the tricks and shortcuts in mathematics

♦ Pupil reads and writes articles in mathematics
S.I.O's under scientific attitude:
♦ Pupil accepts only after logically proved concepts
♦ Pupil cross checks the results
♦ Pupils develops the habit of logical thinking
♦ Pupils accepts their errors.
Teaching aids
Teaching aids are an important means of making mathematics teaching and learning most
effective and interesting. Teacher of mathematics needs to explore the full potential of teaching aids
for more effective mathematics teaching. The use of teaching aids help in breaking the monotony of
the classroom, teaching - learning process as most of the learning takes place through multi-sensory
experiences like listening, watching, touching and feeling.
Eg: Cut-outs, Models, Objects, Magnetic board, Charts, Overhead projector, Improvised aids, etc.
Motivation
♦ Teacher motivates the pupils by testing their previous knowledge by using the teaching aids.
♦ Motivation can be done through questions, stories, songs, riddle, teaching aids, previous knowledge
and etc.,
♦ Motivation should be done psychologically.
♦ Motivation is the stimulating factor towards learning of mathematics.
Presentation
♦ While presentation teacher writes the topic on the black board by inter-relating the teaching
aids and the discussion done on motivation.
♦ Presentation is the important step because, pupils are mentally clean and fresh through motivation
so that they can be ready for new concepts.
Application
♦ Teacher writes or give directions and asks the pupils to do on their own.
 ♦ In this step, pupils are getting opportunity to apply their knowledge and understanding in an
unfamiliar or new situations.
Recapitulation
♦ Teacher suits few questions to test the comprehension of the pupils.
♦ Recapitulation can be done through summarization also by the teacher.
Assignment
♦ In assignment step, teacher gives simple and innovative follow-up activities related to the day's
topic.
♦ Assignments should be given in order to stimulate pupils creativity and interest.
Unit IV METHODS AND TEACHING AIDS
4.1 Inductive, deductive, analytic, synthetic, heuristic, project, problem solving
and laboratory methods of teaching mathematics
4.2 Activity Based Learning (ABL)
4.3 Active Learning Method (ALM)
4.4 Applications of ABL and ALM
- Format of a typical lesson plan based on ALM
4.5 Introduction: Evocation, Recall, Survey
- Understanding: Concept, Teacher and Individual Solving Problems
- Group Work, Presentation
- Evaluation: Reinforcement, Homework, Remedial measures
4.6 Computer assisted instruction
- E-learning, mobile learning
4.7 Importance of teaching aids
- Projected and non-projected aids
- Improvised aids: Paper folding and paper cutting etc.,
4.8 Criteria for selection of appropriate teaching aids
4.9 Use of mass media in teaching mathematics
4.10 Field trip as a teaching technique
4.11 Characteristics of a good mathematics text book.

We are concerned with the methods of teaching. "How to impart its knowledge? How to enable the
child to learn it' are the questions to be answered in this discussion. Different methods of teaching
have been proposed or propounded by different educational thinkers or schools of thought in
education. The knowledge of procedures of the methods will broaden the outlook of the teacher. The
choice for him is not to be made narrow. It would be then left for him to decide form his wide
information, which of the methods to use and when. Senses are gateways of knowledge. Sensory
experiences form the foundation for any intellectual activity. These experiences could be affected
by means of number of teaching aids. They provide for a variety of methods. While using teaching
aids the teacher should be clear about the importance of teaching aids. This unit helps the teachers to
decide about the type of lesson most appropriate in various circumstances and plan the strategies and
materials accordingly. This unit also focuses on cooperative learning, individualized instruction
which represents one of the effective innovations in teaching learning process. Computer assisted
instruction and Dalton plan is discussed. Seminar, group discussion, team teaching and guided
discovery are also discussed. After deciding 'why to teach' and 'what to teach', it is proper to think
about 'how to teach'. In which way the subject matter and learning experiences, to be imparted,
should be given to the pupils so that the set aims and objectives may be properly realised? Keeping
this very thing before him, a subject teacheradopts some special ways or devices for imparting the
desired theoretical or practical experiences to his students. These ways or devices are known as
methods of teaching.
Success in teaching depends mainly on two factors
i. Mastery over the subject matter, and
ii. Skill in teaching.
The experiences gained about the subject in schools and colleges help in acquiring the former
but for the latter a person has to undergo some specialized training so that he may be acquainted with all
the modern and techniques of teaching a subject. For the benefit of those who are going to devote
themselves for the cause of Mathematics teaching, some of the important methods are discussed
below.
4.1. INDUCTIVE METHOD
You may still have in your memory the way of proving binomial expansion like [a +b]n etc. It
was named as method of induction—the way of proving any universal truth by showing that if it is
true of any particular case, it is true of the next case in the same serial order. Inductive method takes
into account the process of induction. While adopting this method students are required not to accept
the already discovered formula without knowing how it has been established. They are helped in its
discovery by adopting inductive reasoning. In inductive reasoning one proceeds from particular to
general, from concentrate facts to abstract rules and. from the special examples to the general
formula. The results are always generalized by studying particular concrete cases and examples. If
one rule applies to a particular case and is equally applicable to different similar cases, it is accepted
as a generalised rule or formula. How this is done may be understood through the following
examples.

Inductive Reasoning in our daily life

1. A small child meets the accidental death of his father. Some time later he comes across the
death of his playing mate and a year after his neighbor uncle passes away. These are special
and particular cases of death but they may lead the child to conclude that "Man is mortal".
2. A child observes the rising of the sun and getting of darkness after the setting of the sun. He
observes at a particular day in the beginning when he becomes able to observe the nature's
mystery, on the next day, the same thing happens. He collects these particular instances of
rising and setting of the sun in his memory and after some time he himself feels that "The sun
rises every day and also sets everyday"
3. A child eats a green apple and feels its sour taste. Again on any day he takes another green apple
and experiences the same sour taste. These few examples are, enough to make him conclude that
"green apples are sour" and hence afterwards whenever he is offered a green apple he at once
refuses to eat it.
The method of acquiring knowledge as mentioned above, is known as inductive method. It is
nothing but the learning from direct experiences. Here conclusions are based on the repetition of a
particular kind of experience at so many times. Scientists and inventors usually adopt this procedure
in order to derive some general law by performing a number of experiments. For example 'the law
that matter has weight' has been discovered only after weighing so many types of matter at so many
places.
Teaching Mathematics with the help of Inductive Method
1. Knowledge of the sum of three angles of a triangle:
Children in a class maybe asked to construct a few triangles of any size or shape. Then they may
be asked to measure and sum up the angles in each case. They may find that the sum is the same in
all cases. It may lead to conclude that the sum of the three angles of a triangle is equal to two right
angles.
1. Establishing the Formula:
(a + b)2 = a2 + b2 + 2ab. Students may be asked to find out the square value in each of the
cases like (a.+ b)2, (x + y)2, (m + n)2, etc. by the simple method of multiplication. After doing these
different multiplications, they may be helped in generalizing that (1st term + 2nd term)2 = (1st
term)2 + (2nd term)2+2 (1st term) x (2nd term).

DEDUCTIVE METHOD
In deductive method one follow deductive reasoning which is just opposite to the Inductive
reasoning as may be seen through the following examples.
1. When the child comes across a case of death he may be told on enquiry that on one day or
the other one has to depart. He may have verification of this established fact after coming
across some other cases of death in his life.
2. The child may be told that he should never eat the green apples because they are sour.
Afterwards he may verify this fact by tasting some green apples.
3. In this way deductive reasoning begins with the deduced results or generalised conclusions.
Therefore in deductive method, one proceeds from general to particular, from abstract rules to
the concrete cases and from the general formula to the special example. In this method the
students are told to accept a generalized truth or pre- constructed formula as a well
established truth and then asked to apply it in solving so many particular relevant problems.
How deductive method is used in the classroom for teaching mathematics can be understood
through the following examples:
1. The teacher may announce that today he is going to teach simple interest. He will
the give the relevant formula i.e.,
S.I=PNR/100
2. For acquainting the students with its application he may also solve a few problems. Then, he
may ask the students to solve similar problems directly with the help of the given formula.
3. The teacher may tell the students that the sum of the three angles of a triangle is equal to two
right angles. Afterwards students may be asked to verify this established fact by measuring the
angles of the different triangles.
4. Student may be told about the formula of the area of a rectangles i.e. A = Breadth x Length
and then be asked to apply it in finding the areas of different rectangles.
Which Methods—Inductive Or Deductive—Should Be Used
The above discussion may lead us to conclude that inductive method is a method of
establishing formula or deriving generalised results from the particular examples, direct experiences
or experiments. It is a method of discovery whereas deductive method is a method of applying the
deduced results. Here, although the students do not get opportunity for original thinking and.
discovering, yet it is very much useful in saving their time and labour. In this way both the methods
have their own advantages and disadvantages. Therefore an attempt has been made to combine both
these methods in such a way as to derive maximum advantages by doing away with the possible
defects of both the methods.. This method in combination—is named as Inducto-deductive method.
In adopting this combined method, beginning is made with the inductive, method. The students are
made to discover truth or establish a formula with the help of inductive method.
Later on, deductive method is used for the following:
a. to provide opportunity for the application of the generalised formula.
b. to memorize the formula through drill and practice work.
c. to develop skill, speed and accuracy in the process of computation.
In this way Inducto-deductive method requires the combined use of inductive and deductive
reasoning.

ANALYTIC METHOD
Analytic and Synthetic methods are two such methods which in spite of their opposing nature
are used in combination. Let us try to study these methods one by one.
What is analytic method?
Etymologically the word 'analytic' has been derived from the word 'analysis' which means to
take apart or to separate the things that are together. In other words, analysis is a process of breaking
a thing into its smaller parts. By analysing a problem, thus, we mean to break the problem into
simpler elements or unfold its hidden aspects in such a way that its solution may appear quite
obvious. In this method, one moves from unknown to known by adopting the process of analysis.
The beginning is always made from the conclusion or what is to be proved and then by operating it
analytically the unknown is ultimately linked with the known.
SYNTHETIC METHOD
What is synthetic method?
In contrast to analytic method, synthetic method takes into consideration the process of
synthesis. In synthesis, the smaller constituents or parts of a thing are combined or put together so as
to give something new. Synthetic method leads us from known to unknown as the known bits of
information are synthesized for reaching the unknown. What is already given or known is arranged
in such a way that the synthesized structure may lead us to the desired results or conclusion. Here
the start is always made from the hypothesis and not with the conclusion as in analytic method.

How can you further simplify it?

acd + 2b2d= bc2 + 2b2d will be true
If acd = bc2 [canceling common from both sides]
or if ad=bc [doj [Canceling off common from both sides] or acd + 2b2d = bc2 + 2b2d [by
cross multiplication] Q. 2. How can you further simplify it? acd + 2b2d= bc2 + 2b2d will be true If
acd = bc2 [canceling common from both sides]
or if ad=bc [do]
How would you write this relation in some other form?
 Exp. Ans. This will be true if a/b=c/d [which is known and true]
Conclusion
In this way a close analysis may reveal that both of these methods have advantages as well
as disadvantages. Therefore, it is unwise to advocate one method by completely discarding the
other. The need is to synthesize these methods. In the beginning we should try to use analytic
method for making the students discover proofs and solutions and then afterwards synthetic method
should be used for the presentation of the discovered proofs and solutions As far as the
understanding of something is concerned; it should be acquired with the help of analytic method
but for fixing and retaining what has been understood, the synthetic method should be brought into
use. In this way analysis and synthesis both should be used in combination. Analysis must take lead
and necessarily be followed by synthesis for realising the better results in the teaching of
Mathematics.
HEURISTIC METHOD
Etymologically, the term "heuristic' has been derived from the Latin word 'heurisco' which."
means T have found out.' The method as such was originated by Professor N.E. Armstrong. He was
of the view that the child must be made to discover things himself. It is of no use to acquaint the
students with facts, rather they should be made to investigate or discover the facts. It is not the
knowledge but the way or method of enquiry or investigation that should be aimed in the teaching
and learning process.
Keeping in view the above ideas he propagated the use of heuristic method. Elaborating the
meaning of this method h; writes, "Heuristic method is the method of teaching which, places the
students as far as possible in the attitude of a discoverer".
A close analysis of this definition suggests that heuristic method is not any specific or
separate method in itself. A broader definition of this method may be evolved as below.
"Any method, which is opposed to 'be dogmatic method of teaching, which can make the
students learn or acquire knowledge independently by exercising their thinking and reasoning
powers and which can foster the habit of self-activity and self dependence can be called as heuristic
method."
In this way, in true sense heuristic method aims to develop a particular type of attitude,
named as 'heuristic attitude' among the students. It seeks to bring a change in the nature of learner.
He must not remain as a mere passive listener or receiver of the knowledge. From a passive recipient,
he must be changed into an active independent enquirer, investigator and researcher. Instead of telling
the things he must get opportunity of discovering them. In this way a!i the good new methods like
Inducto-deductive method. Analytic-synthetic method, laboratory method, can take the form and
shape of heuristic method, provided if the development of heuristic attitude may be properly aimed
in the teaching-learning process.
How to use heuristic method?
While teaching through heuristic method teacher puts .a definite problem before the students.
Every student is asked to solve the problem or to discuss and investigate the ways and means to
solve the problem theoretically or practically as he desires. The main thing is that the students as far
as possible have to remain independent thinkers and discoverers. The teacher has to remain very
cautious in providing necessary guidance and help to the students. He has to look for the
maintenance of heuristic attitude. The students are to be made active independent enquirers and
discoverers of the knowledge rather than mere passive recipients. For this purpose the teacher asks
such thought provoking questions that may lead the students to independent thinking, reasoning and
striving for the discovery of the solution of the problem: The teacher in this way only provides a
suitable direction to the students to proceed independently for the solution of their problem.
Let us try to illustrate the procedure adopted by solving a. problem on profit and loss.
Problem: Ram Singh bought a cow for Rs. 500/- later on at his own need he sold it to Jailal for Rs.
400/- Try to find out his loss percentage.
Procedure: The students will be asked to study this problem carefully. Then, the teacher will lead
them to the solution of the problem through the suitable heuristic questions as illustrated below:
1. What have you to find out in this problem?
A: Loss percentage
2. How will you find out the loss percentage?
A: First of all net loss will be calculated. It will lead us to compute loss percentage.
3. Try to find out the net loss in this problem
A: Students will discover that net loss to Ram Singh is of Rs. 100/-
4. How will you compute loss percentage now?
A: Students, on the basis of their previous knowledge know that profit or loss always incurs on the
cost price. Hence they can readily respond that this lOORs. Loss has incurred on Rs.500/- They
also know that percentage is always calculated on 100 and hence they can very well proceed to
solve the problem.
In this way the students may be persuaded to solve such more questions and then they may be
helped to generalize the following formulae related to computation of loss or gain percentage.
Loss percentage = [Net loss x 100]/ Cost Price
Profit percentage = [Net profit x 100]/ Cost Price
MERITS OF HEURISTIC METHOD
1. Psychological method:
Heuristic method is a psychological method in the sense that it is based on the psychological
principles. Knowledge is not thrown upon the child, rather he is made to discover the things himself
with his own pace. In this method the needs, interests and motives of the child are fully taken into
account and hence heuristic method takes the students on the right path of learning through a
healthy psychological wave.
2. Development of Scientific attitude:
Heuristic method helps in the development of scientific attitude among children. In this
method a great emphasis is laid over the development of reasoning and thinking powers. The students
are made independent enquirers and discoverers of the knowledge. It makes them critical and
scientific minded. It helps them to be original and creative in their outlook and thinking. In this way
heuristic method helps in the inculcation of scientific attitude among children.
3. Emphasis on activity:
Child does not remain passive recipient of the knowledge in this method, rather he becomes
quite active for making independent efforts to discover or re-disorder the knowledge

Moreover, this method demands quite alertness and presence of mind on the part of the student for
responding to the heuristic questions asked by the teacher. In this way heuristic method encourages
activity on the part of students.
4. Learning through independent efforts:
This method encourages independent learning. The child learns all what he wants to learn
through his own efforts. It ensures his mastery over the subject. Instead of spoon feeding the real
understanding of the subject matter is acquired. Knowledge thus gained becomes real, lasting and
useful.
5. Teacher-pupil Contacts:
Heuristic method gives enough opportunity of developing teacher-pupil intimacy. Teacher
has to remain in living touch with the students for giving necessary guidance at the appropriate time.
He has to put heuristic questions for loading the students to think and discover the things by their
own efforts. For this purpose he has to study every student carefully and know their interests,
aptitudes, abilities and limitations. Students, on the other hand, for 'their independent work have to
seek guidance and necessary help from the teachers. In this way the success of this method depends
upon the nearness of the pupils and the teacher which leads to the establishment of their harmonious
relationships.
6. Helpful in discipline:
Heuristic method, as said earlier, needs a lot of mental alertness on the part of students. They
are busy in exercising their reasoning and thinking powers and making their independent efforts.
They are entrusted with the great responsibility of self-learning and discovering the things and thus
have no empty mind for making mischief.
7. No problem of assigning home-task:
In heuristic method the students have to make independent efforts. They remain engaged in
the assigned responsibilities even in the class-rooms. This leaves no extra scope for assigning home
work and its correction etc. Therefore, no usual procedure of assigning homework is adopted in
heuristic method and teacher as well as student get a sort of relief.
DEMERITS OF HEURISTIC METHOD
1. Too much expectation from the students:
This method expects much from the students in demanding that they should discover the things
with their own independent efforts. To discover or to explore a new field is not a joke. It needs a lot
of hard work, patience, deep concentration, sound reasoning and thinking powers and creative
abilities. Every child is not supposed to possess such qualities.
2. Too much expectation from the teachers:
Heuristic method also expects too much from the teachers. They have to frame appropriate
heuristic questions for stimulating thinking in the students and provide them individual guidance
and help at the proper time. Every child of the class is to be properly studied by them in terms of his
basic potentialities and interests. In a class of sixty to seventy students this type of individual
attention and guidance becomes almost impossible for the teachers with their limited resources and
average abilities.
3. Obstacle in the progress:
Heuristic method requires that the students should discover or rediscover the things. In this way
much of the time is wasted in discovering the already discovered facts. This actually creates
obstacles in the progress. Human civilization has stood on the legs of imitation and invention. With
the absence of one of the legs imitation, the progress of learning would be surely hampered.
4. Difficulty in covering the syllabus:
Heuristic method is in fact a slow and time consuming method as the students have to go
ahead with the self-discovery. It creates difficulties in covering the lengthy syllabus in time.
5. Possibility of erroneous conclusions:
When the students discover the things by themselves, it cannot be safely said that they would
always be on right lines. Due to lack of experience and proper abilities, they often draw erroneous
conclusions or inferences out of their observations and independent work which in fact proves very
harmful to them.
6. Lack of proper aid material:
The successful use of heuristic method requires adequate laboratory and library facilities. There is
lack of such facilities in our schools. Moreover textbooks written on heuristic lines are not
available. Lack of such facilities creates potential difficulties in the use of heuristic method.
Conclusion
The merits and limitations described above reveal that in spite of being a good method
heuristic, method suffers from so many practical difficulties and handicaps. Too much time is
wasted in learning a thing and hence joy of discovery, in spite of its greater value, loses its
significance. The knowledge gained, although real and useful, remains quite inadequate and
insufficient. Moreover the conditions prevailing in our schools do not favour use of this
method and hence it is not possible to use heuristic method in its pure form. The spirit behind the
method is quite constructive and useful and hence it should be preserved. The students should not
be told everything. They should be encouraged for independent thinking and making conscious
efforts to learn by themselves. But on the other hand, children should not be forced to discover
everything and left unguided and undirected. The real thing is that we should try to avoid extremes
and concentrate on the inculcation of heuristic and scientific attitude among the children.
PROJECT METHOD
Project method is the outcome of the pragmatism-ideas propagated by Sir John Dewey. "What is
to be taught should have a direct relationship with the actual happenings in life," this central idea
forms the basis of project method. The principle of correlation has been given a very practical shape
through this method as it tries to impart education of all the subjects in an integrated way by
correlating them with the real life activities. In order to understand this method let us try to think
over the meaning of the term 'project'. It has been defined by the different educationists as under:
According to Stevenson "A project is a problematic act carried to completion in its most
natural setting."
According to Kilpatrik "A project is a whole-hearted purposeful activity proceeding in a social
environment."
Ballard says, "A project is a bit of real life that has been imported into school."
These definitions clearly reveal the following characteristics of a project:-
1. A project is an act related to actual life activities.
2. It is that activity which is undertaken to solve an emerging or felt problem or to realise some
useful and purposeful objectives.
3. It is always completed in a social environment and natural setting.
4. It is such act which is most interesting and absorbing.
How to work with Project Method?
Project method gives opportunities to teach various subjects of the school curriculum through
working on a suitable well selected project. Essentially the following six steps are involved while
working on a project:
1. Providing a situation
2. Choosing and purposing

3. Planning of the project

4. Executing the project
5. Evaluation of the project
6. Recording of the project
1. Providing A Situation:
In this step, a situation is provided to the students to think over in choosing some project to
work on. They may be confronted with a problem while studying in the classroom, participating in
co-curricular activities and going on excursion etc. It may force them to think about for choosing
some project.
2. Choosing and purposing:
In this second step, students try to choose a definite and appropriate project keeping in view
of the resources in hand and the nature of the problem faced in the first step. They are properly
guided by their teachers in this selection task. Then the aims and, objectives of choosing such a
project are properly discussed through group participation.
3. Planning of the project:
The project chosen is again discussed in terms of laying down a plan and procedure for the
execution of the project.
4. Executing the project:
In this step, students are engaged in the execution of the project in a natural way without
involving any artificiality. They play their roles according to their abilities and capacities with a true
social and cooperative spirit.
5. Evaluation of the project:
In this step, the work done on the project is evaluated from time to time. The line of action
and mode of execution may be modified on the results of such evaluation.
6. Recording of the project:
This step is concerned with the task of recording. How the project was chosen, planned, and
executed, what type of difficulties were faced and how they were solved? How far the project work
achieved the desired aims and objectives? All these things; noted down by the students, are
properly recorded for the future guidance. The working on a project, while going through the steps
given above, requires an extensive knowledge of the various subjects. In a project, therefore,
emphasis is laid over in the integration of various areas of knowledge.
A project as defined earlier, is bit of real life and therefore, knowledge of all the areas and aspects is
being used in the implementation of a project without caring for the name and nature of a subject. The
essence of teaching by the project method lies in the proverb "necessity is the mother of invention".
The knowledge of various subjects is given at the time, then and there, when its need is strongly felt
while working on a project. Therefore, project method proves to be a way of incidental teaching. Like
other subjects mathematics is also taught through such incidental teaching. While working on a project
when and where the need of a particular type of mathematical knowledge is felt it is given then and
there irrespective of its sequence simplicity or difficulty. This is how mathematics is taught through this
project method.
Below we illustrate this type of teaching through an example.
Project :-
Laying out flower garden in the school compound.
One day students go on a picnic to some nearby garden. They happen to see beautiful flowers
growing everywhere in the garden. They wish that such flowers should also be there in their school.
They convey their feelings to their teacher in charge. Now the problem how to have such flowers in
the school is discussed in all aspects. Some students propose to purchase some of the flower pots from
the garden. Some say that the services of a gardener should be taken to plant the flower garden in the
school. In the end one of the proposals that the students should themselves lay out a flower garden in
the school compound is accepted.
Now the prospect of the execution of the project is discussed in a very democratic way. The
work to be done is planned well in advance. The students in the group are entrusted with various tasks
of the project according to their abilities, interests and capacities. Some engage themselves in studying
literature about the growing of flower plants. Some take responsibility of taking necessary guidance
from the expert gardeners. Some go to bring seeds and nursery plants, some try to look after the
cultivation, manuring of the soil, watering and safety of the plants and others try to keep account of all
the expenditure and income involved in the, project. In doing all these activities concerning the
project, students get enough opportunities for the incidental teaching of various subjects. In the
following lines we would try to think over the opportunities for teaching mathematics through the
project in discussion. The activities like the survey of the land to be used for laying out flower garden,
division of land into small square, rectangular, triangular or circular pieces, laying out the hedge or
wire for safety purpose provide opportunities for the teaching of the topics related to area and
mensuration. Number sense can be developed through counting of flowers and plants. Four
fundamental rules and multiplication table may be learnt through counting of plants in rows or number
of flowers in the plants. Besides this, the accounts concerning all types of purchases, sales, income and
expenditure may provide enough opportunities for the learning of these topics. The activities like
division of work, maintenance of accounts and various records provide opportunities for the learning
of the topics time and work, profit or loss, average, partnership, simple and compound interest etc. In
arranging the plants into some organised form, to make way for watering and putting hedges and wires
many of the topics concerning geometry may be taught.
In this way the project laying out a flower garden offers so many opportunities for the learning
and making practical use of so many principles and facts of mathematics. Besides learning
mathematics, it provides: enough opportunities for the learning of the other school subjects like
history, geography, Science, language and crafts and thereby present a clear picture of the integration
of the knowledge in a very natural, simple and practical way.
MERITS OF PROJECT METHOD
1. Psychological method:
Project method is based on psychological principles. The innate tendencies, interests and
aptitudes of the students are best utilised in this method. The instincts of curiosity, creativeness and
hoarding also get their satisfaction in project method. In fact it is a play way method of teaching where
a useful interesting and purposeful activity becomes the centre of teaching-learning process. This
method also suits the modern theories of learning and hence it can be termed as a psychological
method.
2. Democratic way of learning:
Liberty, equality and fraternity are some of the cardinal principles which form the basis of
project method. Right from the selection of the project till its execution students are provided
sufficient freedom for thinking, decision making and going ahead in their assigned tasks. All of these
students cooperate in a common project according to their tastes, temperaments, abilities and
capacities. In this way this method teaches a good lesson of democratic living.
3. Development of social virtues:
So many virtues essential for good citizenship like self confidence, tolerance, patience, self-
dependence, self-respect, sense of responsibility, duty boundness, resourcefulness, mutual love and
cooperation etc. are inculcated through project method.

4. Practical method:
This method of the maxims of teaching like 'learning by doing' and 'learning by living'. Therefore,
what is to be learnt in project method is learnt by doing that in a very practical way. In project method
the problems concerning actual life activities are undertaken. Therefore, it provides sufficient training lo the
students to use their learning in their practical life.
5. Dignity of labour:
Project method emphasizes dignity of labour. Students irrespective of their caste, creed and
social status join their hands in 'doing mental and manual labour in the execution of the project. A little
apathy or slackness on their part towards doing their duties may bring disappointment to them.
Therefore, they learn value of the work through this method.
6. Correlated teaching:
Project method presents an ideal picture of the correlated teaching. Teaching of mathematics
while having integration with other subjects of the school curriculum can be properly linked with the
work experience areas and day to day life activities. The project itself belongs to the very realm of
physical and social world and has a direct link with the commonly felt problem of the student
Moreover, it is completed in its most natural setting and thereby knowledge is imparted as a unified
whole. In this way project method provides good opportunity for, the correlated teaching.
7. The end of various educational problems:
Project method presents incidental way of teaching. Therefore, there is no need of time table
construction in this method. The problem of indiscipline is also automatically solved as the students are
completely absorbed in their tasks. Moreover, no home-task is assigned in this method because the work
in, hand is completed in a co-operative way in this method. This method also provides relief to the
authorities as they have not to arrange big classrooms, furniture's and other costly aid material. Here the
actual life situations and setting in the physical and social environment may become real platform for the
dissemination of knowledge. In this way so many educational problems are automatically solved in
teaching through project method.
DEMERITS AND LIMITATIONS OF PROJECT METHOD
1. Difficulty on the part of teacher:
Project method provides many challenges to the teachers in charge right from the selection of
the project till its execution. Every teacher is not equipped with such enthusiasm, abilities and leadership
essential for working with such a method. It expects that teacher must be a walking encyclopedia
having an all round knowledge of every subject along with its practical application in. day to day life. No
teacher is supposed to have a theoretical and practical thorough command over all the subject of the
school curriculum and hence this method suffers from serious handicaps in terms of its practical
application.
2. Uneconomical method:
Project method is an uneconomical method in the sense that the time, labour and amount spent
in this method is quite larger than the return received. Students, irrespective of the guidance given to
them, waste so much energy, time and money due to their lack of experience and immaturity.
3. Not suitable for teaching Mathematics:
No organised and systematic teaching is possible in the project method as it provides incidental
teaching. But Mathematics needs well organised and systematic teaching. It is a sequence subject where
all pieces and units a knowledge require an integrated and systematic approach but in project method such
provisions cannot be availed. For example let us take the project - Running a Cooperative shop in the
School. Now think how can all of the students in a class learn principles and facts of mathematics
through such project. It is difficult to teach 60 or 70 students of a class to make them learn thousands of odd
combinations of the multiplication tables. They cannot even learn all the facts of profit and loss through the
role playing of shopkeepers and customers.
4. Moreover, it is not the end of mathematics:
There are so many branches, topics and aspects of mathematics that may be hardly covered
through such projects. Really speaking if we analyse the fruits of project method we can know that
project method is not a method of learning the new facts and principles but it is a way of utilising or
learning the- application of already discovered learned facts. But here also conies a difficulty that the
drill and practice work which is the backbone of mathematics teaching can also no be done properly
through the teaching by this method. Therefore, project method provides serious handicaps in learning
mathematics.
5. Difficulty in covering the syllabus:
Project method puts obstacles in terms of the coverage of lengthy school syllabus. A particular
project for example laying out a flower garden takes so many months in its completion. Through
projects hardly a part of the syllabus can be covered and therefore, it does not suit the present day
classroom teaching.
6. Not suitable to the present day school conditions:
In India, our schools can neither afford sufficient money nor provide appropriate personnel for
teaching with project method. For using this method, suitable text-books are also not available.
Schools are overcrowded and the educational structure is examination oriented. No provision has been
made for the project method in our examination system. Therefore, it seems quite impracticable to
make use of the project method in our schools. Conclusion
In this way project method, irrespective of having so many good points on its credit side, suffers
from so many handicaps and limitations. It is clear that in no case project method may be taken as a
reliable method of teaching mathematics. Incidental teaching always needs a support from the
organised classroom teaching. The present classroom teaching, therefore, cannot be replaced by
project method teaching. On the other hand it is also true that a mathematics teacher should have a
practical and natural base for his teaching. Mathematics must be taught in the way it is utilised in our
practical life. Here lies the need of working with some useful and productive activity covered by the
name of a project and a wise teacher should utilise it for the learning of real and useful mathematics.
PROBLEM SOLVING METHOD
Students while studying mathematics have to make constant efforts for finding the solution of
the assigned problems. In their day to day life also they are confronted with so many problems requiring
appropriate solution. Problem solving is thus an essential skill which needs to be learned by everyone for
his adequate adjustment to his environment. It would be better if this skill could be learnt during the school
days. Mathematics is full of the opportunities for learning this skill through a wide treasurer of problems,
the solutions of which students are made to find out. In this way, problem solving as a method of
teaching-learning becomes an urgent necessity in the subject mathematics. Let us try to know about the
mechanism and use of this method in the subject mathematics.
Meaning of the term - "Problem Solving"
Usually the term "problem" is used to describe a situation when one is faced with some unknown
and asked to find about its identity or it's confined to a place, the existence of which is unknown to him
and he feels necessity of coining out. Similarly a student while studying mathematics may be asked
some question or is confronted with a mathematical problem given in his textbook the answer or
solution of which is unknown to him. At this juncture, the process in which he is bound to be engaged
for finding out the answer or solution of the confronted problem or situation may be named as problem
solving. In this way the term problem solving stands for "What one does when he does not know what
needs to be done." Such problem solving activity is very much essential in the subject like mathematics. The
students can very well develop the desired problem solving ability in them by solving various problems
in mathematics.
Defining Problem Solving Method
Problem solving method as a method of teaching represents a method which provides opportunity
to the pupil for analysing and solving a problem faced by him on the basis of the previous stock of his
knowledge enriched with the present means available to him, quite independently by following some
systematic and scientific steps and arriving at some basic conclusions or results to be utilised in future
for the solution of the similar problems in the identical situations.
The Process Adopted
In problem solving method a systematic and orderly process is adopted for carrying out the
teaching-learning process. The process begins with the felt difficulty or problem. The student is then
made to think out all the possible solutions of the confronted problem on the basis of what does he know.
Inability of finding out the solution with the help of his previous knowledge and experiences makes
him to engage in serious exploration with the help of self-study, mutual discussion and independent
practical work. He tries to test one by one the possible alternatives and solutions of his problem and then
by his continuous efforts get success in finding out the best possible solution of his problem. The
practicability and validity of this solution may be further verified on the basis of its applicability and
reliability in the solution of similar problems in other identical situations.
Stages or Steps in Problem Sohing Method
The following systematic steps or stages arc usually employed in following, the problem solving
method.
1. Confrontation with the problem:
In this step or stage students are made to face a problematic situation. It may occur spontaneously
or is created by the teacher deliberately, As far as teaching and learning of mathematics is concerned,
there is no scarcity of such a situation. Enough problems in the form of exercises and assignments are
already there in their prescribed textbooks. They may be given problems for the drill and homework
by the teachers after the classroom instructions. They may come across with various mathematical
problems while studying in the library or doing self study in their homes. The day to day happenings in
their surroundings may create many situations requiring mathematical solution. For example, a child may
become

eager to know about the computation of the cost of labour for the white washing or painting work
done at his home and it may generate likewise problems related to the surveying and measurement of
the areas of different figures.
2. Proper understanding of the problem:
Problem should be well analysed and understood before attempting for its solution. What is its
nature, what is there to be found out, how is it different from the problems solved in the past, the
questions like these need to be properly answered for knowing about the problem in depth. One can,
think about the probable solution of a given problem only when he is fully aware about the nature,
magnitude and direction of the problem. Therefore, the students should devote enough time for
understanding and analysing the given problem. The following things may prove helpful to them in this
task.
[i] They must carefully study and think over the problem with the desired concentration,
patience and peace of mind.
fii] The problem should be analysed properly as to know what it is and how one can proceed to
solve it step by step.
[iii] The problem maybe divided into some small meaningful parts and where needed it should be
summarised through numbers, words or geometrical figures for its proper analysis and
understanding.
3. Search for the probable solutions of the problem;
After proper understanding and careful analysis of the given problem serious attempts should be
made for finding out the solution of the problem. At the first instance, where possible help should be
taken from the previous experiences helpful in solving the problem in hand. In case, problem is new and
can't be solved on the basis of previous knowledge, skills and experiences, then alternate attempts
should be made. One can have required study in the library, perform practical work in the workshop and
laboratory, do surveying, weighing, measuring or any other useful activity, knock the doors of related
sources and agency for gathering relevant and useful information, and take help and guidance from the
concerned persons including subject teachers and experts. In this way multi-dimensional efforts should be
made in search of some possible solutions of the given problem. Where needed trial and error method
can also be adopted for listing out these probable solutions.
4. Solution and testing of appropriate solution:
Out of the possible tentative solutions or hypotheses, the attempts are now made to search out the
best. For this purpose all the possible solutions are taken one by one, discussed and weighed in terms of
their validity and practicability. The most relevant one is then subjected to proper testing. The selection of
the most relevant solution out of the so many solutions is made quite cautiously. In general, the following
considerations are kept in mind while making such decision.

[i] The selected solution must be able to present a valid as well as reliable solution of the
problem.
[ii] The solution should be in tune with the pre-established facts and principles.
[iii] The situations and negative examples that may cast doubt over the validity of the selected
solution should also be kept in mind.
5. Utilization of the accepted solution:
A particular solution after proper testing is accepted as a best possible way of finding out the
solution of the problem in hand. Now further attempts are made to apply it in the solution of the similar
other problems. In case it helps, it may be accepted as a valid and reliable conclusion or solution for being
applied in the solution of the problem of a particular nature in a particular situation. If not, further
attempts are made to search again for some more reliable and valid solution helpful in solving the
problems in hand. In doing s, it is also kept in mind that the solution must work in the real life situations
so that the child may learn to find out the solution of the problem not merely on theoretical grounds but
also on the sound practical footings.
Importance and Utility of Problem Solving Method
Problem solving carries the following advantages in the teaching of mathematics.
1. Mathematics teaching primarily aims to make the students able to solve mathematical problems
with their own independent efforts. Problem solving method helps much in the attainment of this
objective.
2. In the study of mathematics students are constantly required to solve different types of problems,
learn different methods for the solution of the problems, acquire new information, knowledge, and
skills and make use of the acquired knowledge and skills in the practical and applied situations of the
life. All such requirements can be easily met by adopting problem solving method for the teaching
and learning of mathematics.
3. This method provides valuable opportunity for the development of mental and cognitive abilities of
the students. In fact it has no parallel in the task of developing the power of reasoning, thinking
analysing, synthesizing, generalizing and making conclusions among the students.

4. This method helps the students in learning the most systematic and scientific method of problem
solving.
5. The development of problem solving ability makes the student quite self reliant and self confident in
solving any type of problems related to curricular or non-curricular areas.
6. The use of problem solving method proves a good source of the internal motivation to the students.
The solution of the problem found out by the independent efforts makes a student further motivated
in the task of learning mathematics.
7. This method is helpful in the development of harmonious relationship between the teacher and taught.
8. It is a psychological sound method on account of its being child centered and problem oriented. The
children get completely absorbed as they accept voluntarily the challenge of solving the problem with
their own independent efforts. What they learn, is learnt with their own efforts and the satisfaction
they get after finding out the solution of their problem makes them more enthusiastic and energetic
in further learning and solving of the problems.
9. This method is specially helpful in getting rid of many teaching learning problems like the problems of
indiscipline, assigning the home-work, etc.
10. This method proves quite helpful in making the study of mathematics more useful and practicable in
the day to day life. The students learn to utilize the acquired facts and procedure for solving their
day to day life problems.
Demerits and Limitations of Problem-Solving Methods
Problem solving method is said to suffer from the following demerits and limitations:
1. Problem solving method is best suited for the teaching of solution of mathematical problems.
However, there lie so many things in the syllabai of Mathematics apart from the teaching of the
solution of problems. Knowledge, understanding and skills related to these topics may not be
successfully given with the use of problem-solving method. In this way, this method has a partial
applicability in dealing with the mathematics contents.
2. Problem solving method requires independent efforts on the part of students to find out the solution
of the problems. It demands from them to be trained in the scientific procedure, scientific thinking and
problem solving. Every student is not to be expected to possess such abilities and consequently to
carry on all the students of the class for the adoption of this method poses a great problem.

. The task of thinking about the possible tentative solution or hypothesis is a quite challenging one. The
students are more often tempted to pick up wrong hypotheses or follow the wrong path of solving
problem and are thus found to waste time and energy in the useless and irrelevant attempts.

LABORATORY METHOD
Laboratory method of teaching mathematics is that method in which we try to make the students
learn mathematics by doing experiments and laboratory work in the mathematics room or laboratory on
the same lines as they learn sciences by performing experiments in the science rooms or laboratories.
This method involves the maxims of teaching like 'learning by doing', 'learning by observation',
and 'concrete to abstract' etc. It provides a practical base to our inductive reasoning. The present day
teaching is criticized vehemently on the ground that it provides a bundle of theoretical knowledge
without any practical ground. Laboratory method is quite competent to check this evil. It can help in
learning mathematics in the way it is used in our day to day life. The theory and practice may proceed
side by side in this method and hence it may make the teaching and learning process as interesting,
useful and lively as possible.
The laboratory method aims to arouse teachers to a belief, not only theoretical but practical and
effective as well that mathematical dishes must be made appetizing and palatable. They are to be accepted
with pleasure and digested with ease.
J.W.A.Young.
In this way, laboratory method has too much to contribute in mathematics teaching. For its use it
needs a well equipped laboratory and a laboratory conscious, skilled mathematics teacher. Emphasizing the
need of laboratory in this method Young has remarked that "a room specially filled with drawing
instruments, suitable tables and desks, good blackboards and the apparatus necessary to perform the
experiment of the course is really essential for the best success of the laboratory method".
What is to be kept in the Mathematics Laboratory?
What is to be kept in the mathematics laboratory should be known to the teachers of mathematics.
1. To teach counting and four fundamental rules teacher may make use of laboratory method by taking
help of concrete materials. How many sticks or stones does a student have? The total number of
concrete items possessed by these students may be known through actual counting. Similarly
subtraction, multiplication, division etc, can also be taught with the help of concrete objects, charts
and models etc.
2. Knowledge of multiplication tables may also be given through concrete objects. Let us take
multiplication table of 5. Students may be asked to make several bundles of slicks in a group of five
each by typing them with the help of rubber band. Let it be arranged in such a way that the first line
has only one bundle, the second two bundles, third three bundles and so on up to 10th line. Let the
bundles now be untied and sticks be counted by the pupils. They will find that by combining 5 at 2
times one gets 10, combining 7 at 3 times 15 and so on and so forth. In this way students may be
properly acquainted with the multiplication table of 5.
3. In geometry much can be experimented and practiced with the help of laboratory method The drawing
of straight lines and angles, dividing them into required parts and construction of triangles, quadrilaterals
etc. all demands the use of laboratory work and can only be learnt through experimenting with
geometrical instruments.
4. For determining the ratio between the circumference and the diameter of a circle, laboratory
method will involve the following steps:
i. Students may be asked to draw circles with a diameter of 8n cm on their cardboard pieces and ask
them to cut out such circular figures.
ii. Ask them to measure the circumference of these circular figures either by
a. Measuring the length of the thread tied around the figure or
b. Measuring the distance traveled by the circular figure on rolling it down on
a piece of paper while it makes one complete revolution.
iii. Ask them to compare the measured circumference with the diameter.
iv. Ask them to repeat the same above experiment with the circular pieces of varying diameter and
 note the deduced results.
v. Now help the students to think inductively and establish the fact that
Area of circle/diameter=22/7=5
5. Laboratory method is easily applicable for the calculation of the area of the figures like
rectangle, square, triangle and circle etc. as may be seen from the following examples:
a. Area of rectangle
a. Ask the students to take a rectangular piece of a cardboard and measure its
length and breadth.
b. Ask them to divide the length ad breadth into parts

c. Ask them to measure the area of any one these square pieces.
d. The above experiment may then be repeated with the help of other rectangular
cardboard pieces of different dimensions and the student may be persuaded to
conclude inductively that
Area of a rectangle =length x breadth.
ii. Area of a triangle
i. Ask the pupils to draw a figure of a triangle on a piece of thick paper and cut it out with the
scissors.
ii. Now ask them to find out its weight with the help of a physical balance.
iii. Ask tern to cut a unit area figure from this triangular piece.
iv. Now ask them to determine the weight of this small square piece with the help of a spring
balance.
v. Then, the students may be helped to determine the area of the triangle by comparing the
weight of a unit area with the weight of the whole triangular piece.
MERITS OF THE LABORATORY METHOD
1. Psychological method:
Laboratory method has a psychological base. It is a child centered method in which the basic
interests and natural instincts of the child are given due weightage. Every child is active by nature and
has a spontaneous desire to create or manipulate something or the other. Laboratory method suits such
psychological requirements of the children and thus proves an appropriate means of learning
mathematics.
2. Follows maxima of teaching:
Children learn very easily by learning by doing, proceeding from concrete to abstract and from
known to unknown etc.
3. Scientific method:
Laboratory method is so much systematized and organized method that it may be safely labelled as a
scientific method. The entire steps essential for a scientific method are followed in this method. The
student, after analyzing the problem solves it through experimentation and after necessary verification
arrives at a definite conclusion. In this way laboratory method provides the way for systematic enquiry
and investigation. New facts may be
explored through this method or the discovered facts may be further testified in similar other practical
situations.
1. Clarity and fixation of knowledge:
Senses are the gate way of knowledge. Here the child gets the opportunity of receiving
knowledge through his so many senses-eyes, ears and hands. Moreover, the knowledge is earned
through ones own attempt. Genuine interest is also developed through practical work. All these things
make learning easy, lasting and enduring.
2. Making mathematics useful and practicable:
Laboratory method helps in getting practical and useful information rather than bookish one. What a
child learns he learns through doing and hence he finds himself quite capable of using the acquired
knowledge in his life.
3. Development of some good virtues:
Laboratory method helps in imbibing some good virtues among the students. Day to day
experimental and practical work teaches them dignity of labour and inculcates love for truth and
honesty. It also makes them self-reliant and helps them in building their self-confidence.
4. Suitable for individual's capacity and ability:
This method provides sufficient freedom to work according to ones capacity and abilities. The
gifted child need not wait for the dullard for his independent experimentation. On the other hand average
or even sub-average children can work at their own pace and finish their work at their convenience
without disturbing the progress of the gifted children.
5. Intimate contacts between teacher and taught:
Laboratory method provides opportunities to bring teacher and the taught closer and closer.
Practical work demands individual attention from the teacher. The teacher has to keep watch on the
working of students and render appropriate guidance at the appropriate time. Students need to seek
help form the teacher now and then in this laboratory work. This makes essential to both of them to draw
nearer and nearer. Consequently, the relationship becomes intimate which helps in smoothening of the
teaching learning process.
6. Problem of indiscipline solved:

The students are always busy in doing practical work. Their mind is engaged either in the observation of
phenomena or outcome of the results. Also the work done suits their interests, urges and capacities.
Therefore, there are very remote chances of the emergence of the indiscipline problem in a class taught
by laboratory method. Development of co-operative feelings and social outlook:
In this method the students get opportunity of working in a group them in the inculcation of
group spirit and we-feeling. They develop a social outlook and learn to work and get long with others.
DEMERITS AND LIMITATIONS
1] Partial usability:
Only those topics which can be experimented upon can be taught through this method. Therefore,
laboratory method has a partial applicability.
2] Not too useful in mental development:
Laboratory work is useful in acquainting the students with the facts or learning of some useful skills.
But it does httle help in providing opportunity for the proper mental development of the students.
3] Too expensive:
It is a very expensive method on account of the following reasons; a.
Maintenance of laboratory:
This method requires a well equipped and properly maintained mathematics laboratory. To
keep a separate mathematics laboratory in a school is not a joke. Therefore, the present circumstances
do not allow us to work with the laboratory method as it will involve huge expenditure.
. Need of more stall:
In comparison to usual lecture or text book method, laboratory method needs sufficient strength of
the staff as there is need of paying individual attention for the practical work done by the students. For
the maintenance of laboratory, laboratory staff is also needed. The enrolment of the extra staff would
definitely bring extra financial burden. Moreover, the mathematics teachers need to be trained on the lines
of laboratory method failing which this method cannot be used satisfactorily.
4] Too much expectation from the pupils:
In laboratory method it is expected that students must work independently. They should
discover and verify the facts like a scientist or mathematician. But how can we expect from every
pupil to work as scientists. It is hoping against hope. Immatures cannot be expected to investigate or
discover the things independently.
5] Difficulty on the part of teachers:
Laboratory method needs individual attention to be paid to every pupil. All types of students work
with their own pace. It is very difficult for the teacher to take care of all types of students and give them
guidance at the appropriate time.
6] Wastage of time:
The process of analysis, experimenting, getting results or verifying - is a very slow process.
7] Non-availability of appropriate text-books:
Text book written on the lines of laboratory method is not available. 8] Non-
practicable in the higher classes:
Not possible to teach all the concepts in the higher classes in a concrete way.
4.2 ACTIVITY BASED LEARNING [ABL]
It is one of new teaching learning method. Activity joyful learning approach is a strategy of
teaching learning aims at securing maximal participation of students in the teaching learning process. Almost
all the children love fun and show interests towards play. Therefore ABL method follows the principles
of learning by playing, learning by doing, learning by enjoying and learning by problem solving. This
approach required the involvement of multi-sensory organs of the child in the teaching learning process.
Much of the researches have been conducted on different teaching strategies relating to joyful learning
activities and the results were found to be positive. This approach is unique to attract out of school
children's to schools. The teachers who are involved in implementing this method have developed
activities for each learning unit, which facilitated readiness for learning, instruction, reinforcement and
evaluation. ABL transforms classrooms into hubs of activities and meaningful learning.
Activity-based learning or ABL describes a range of pedagogical approaches to teaching. Its
core premises include the requirement that learning should be based on doing some hands-on
experiments and activities. The idea of activity-based learning is rooted in the common notion that
children are active learners rather than passive recipients of information. If child is provided the
opportunity to explore by their own and provided an optimum learning environment then the learning
becomes joyful and long-lasting.
Characteristics of Activity-based Learning
The key feature of the ABL method is that it uses child-friendly educational aids to foster self-learning
and allows a child to study according to his/her aptitude and skill. Under the system, the curriculum is
divided into small units, each a group of Self Learning Materials [SLM] comprising attractively designed
study cards for English, Tamil, maths, science and Social Science. When a child finishes a group of
cards, he completes one "milestone". Activities in each milestone include games, rhymes, drawing, and
songs to teach a letter or a word, form a sentence, do maths and science, or understand a concept. The
child takes up an Exam Card only after completing all the milestones in a subject. If a child is absent
one day, he/she continues from where he/she left unlike in the old system where the children had to learn
on their own what they missed out on.
Activity-based Learning
The "Joyful Learning" experiment of the mid-nineties had started as an effort to provide special
schools for children who had been freed from being bonded labour. Methods and materials, which were
devised to help the children catch up on the lost years of childhood, seemed both appropriate and
attractive to all children.
The Montessori system has proved to be a tremendous enrichment to ABL. The materials now
available in ABL are colourful, easy to handle, hardy and meticulously developed and enable children to
understand place value [units, tens, hundreds] and the basic mathematical processes.
By bringing the blackboard from the teacher's eye level to the child's, and by increasing the
blackboard space, two more learning aids have been created: a specific space for each child to write and
a large space to read each others' exercises. Every child can proudly own a part of that blackboard.
The learning materials are not only systematically stacked on the shelves, but they are colour-
coded, for each class level. Also logos of animal and insect forms are used for different aspects of the
curriculum. When the child completes one set, there is a card for Self Evaluation. This can be
administered by oneself or with the assistance of another child.
In building in the opportunity of recall of learnt material at each stage, evaluation has become part
of the process. For the children, there is no failure and therefore, there is no fear of failure. In the
conventional school system, so many children drop out of school because they fail! The need for an
examination at the end of the school year is made redundant in this system. So easily, has the asura
called "Annual Exam" been vanquished!
If we pause for a second, to think of how children are generally given ranks for their performance
in school subjects and how ranking becomes a subtle way of indicating the "value" of a child, we have
a sense of liberation from ranking here. No child is "better then" or "worse than" another. The teacher
keeps an eye on the levels attained by every child and

sometimes helps by pairing an advanced learner with a slower one, for specific exercises. This kind of
peer teaching works well.
It must be noted that the entire system allows for diversity and differential rates of progress. The
Achievement Chart clearly shows the positions of the children in each area. Thus the teacher is enabled
to track every learner's progress. Monitoring of progress by the teacher is subtly combined with the
child's freedom to select the pace of learning.
The ruthlessness of ranking and peer competition is further reduced by mixing the age groups
and classes. In a room of 40 children, there could be ten each from Classes 1, 2, 3, and 4. This vertical
grouping has several advantages. It recreates a family model, where the older child automatically
becomes a guide and helper for the younger one. It encourages cooperation between children, rather than
competition among them. We are told that a multi-grade classroom is a problem which many rural schools
confront. ABL is a simple solution to that complicated issue. The system absorbs different age groups and
different ability levels within the same age group.
Taking the daily attendance is a ritual in most schools, with the teacher calling out the names and
the pupils responding. In the ABL method, this process is made child-friendly. There is an Attendance
Card for each child, to be filled up everyday by the child. Children love the sense of trust that this
procedure implies. When they assemble in the morning, one student from each class level in the room
distributes the Attendance cards and collects the filled up ones. The entire process is orderly. It puts the
responsibility for marking attendance on the child and not on the teacher. We have all heard about
teachers losing their voices because of their shouting and screaming, to keep the children quiet. In the
schools with ABL, there are no apparent discipline problems. The structured learning materials have
their own logic, which supports the children's involvement in reading, writing and calculating. Children
find that they can learn at any speed, without being taunted by classmates or scolded by the teacher.
Also, there is no scrambling for adult attention. Discipline is intrinsic to the material and internalized by
the children.
The text book is not the only source of knowledge, just as the teacher is not the sole authority. The
text book is integrated into the materials. For instance, one of the steps of the ladder contains an
instruction to read a specified page of the text book. Clearly, when a child goes step by step on the ladder,
his steady progress gives him the skills to read the connected page in the text book. Of course, if he needs
help, he could ask the teacher when he is in the teacher-assisted group, or just go across to where she is
sitting. Students appear to have no fear of being reprimanded by the teacher. The conventional distance
has been bridged here.
Generally, one of the constant problems of schooling is absenteeism. For example, if a child is sick
for a week, he cannot follow the lesson when he gets back. He has the feeling of running a race he can
never win. However, ABL has a simple strategy to take care of missed classes. The mastering of a skill
is not a collective exercise. The child's work is individual. Therefore, he goes to the points on the
ladders, where he left off and starts learning from there.
In rural areas, harvest time is when children are needed on the farm. Their short-term absence
from school is no longer a problem. Time away from school can be made up. Fairs and festivals can be
enjoyed without their seriously disrupting a child's learning activities.
Repetition of a lesson acts as reinforcement. That is accepted pedagogy. But instead of sing-song
chanting of tables or whatever, the child in ABL writes on the blackboard first, his notebook next and
finally in the workbook. Since he writes the same material three times, the pattern [be it spelling or
grammar], gets well established. Whatever the lesson [names of animals, masculine and feminine nouns
or singular and plural words], the strengthening of the connections by repetition, is certainly achieved.
With so many materials directly accessed by the pupils, one would worry about the displacement
of the learning cards. However, the pupils have understood that any disorder in the stacking of material is
a problem. Each time the materials are returned to the shelf, they are checked out by the child who used
it, sometimes helped by another child. It is like the practice of putting their school bags and their
footwear neatly on the verandah, which has become second nature to the children. Order and structure in
the materials seem to result in systematic habits in the children.
Gender equality seems to have been achieved rather effortlessly. Girls and boys sit at the low
desks or on reed mats together and share their work, without any awkwardness. This is particularly
important in a culture, where the girl child needs to struggle for equal rights in home and school.
Inclusion is the word that defines an equitable education system, where all children are together in
the same school. Children, with disability are admitted into the school and can be seen wearing calipers
and participating in the activities. They are fully part of the class and their peers accept them as friends.
Once the ABL system has been mastered by the teacher and the pupils, the burden on the teacher
is reduced. Even though the teacher needs a period of un-learning and re-learning, when moving from
the conventional system to the ABL, the end result is very satisfying. She is justifiably proud of her
mastering the administration of the new system and of the children's achievements Furthermore, up to
Class IV, there is no homework. This reduces the teacher's work considerably and frees the young
children to continue learning a variety of things from the family and community and from Nature.
Knowledge can be garnered from many sources. In this system, the time table is in units of half days. If
Mathematics is on the curriculum, the children will be involved with the materials from the beginning of
the school day until lunch time. This is done so that the children can concentrate on one subject without
the intrusive bell breaking up their lesson. In the conventional school, this would have been tedious for
the child, but in the ABL, there is a lot of movement and activity, exchange of ideas and group work.
There is no question of boredom.

In the beginning, many parents were skeptical about classes where children of all ages were mixed. It did
not look like a school to them. Could the pupils learn well, when they seemed to be enjoying it so much?
The parents were invited to come and sit in the classroom. Gradually, as they watched their child's
reading skills and general knowledge develop, most parents were convinced that the system worked.
The tangible achievements impressed the parents. Many PTA meetings were also held to explain the
new methods to the parents. There was Open House on Saturdays, when parents could find out how
the system worked. They were willing to suspend their disbelief then, but it was the child's obvious self
confidence and self esteem, as well as demonstrable abilities in the formal skills that won the case.
Parents soon became allies of the teaching staff.
The key element in the story of the Silent Revolution is the competence of teachers and
supervisors in Tamil Nadu, who were exposed to the new system. Some of them, who were trained at
Rishi Valley, became the core group. Others were trained by the core group. It was the translation and
adaptation of the learning materials that consumed their energy and invigorated them, at the same
time. For nearly six months, when a hundred teachers worked on developing materials, working after
school hours, from 4.00 pm to 8.00 pm every evening. Their involvement in the process of material
 development was total. [[Not surprisingly, their sense of ownership of the method enabled them to work for
long hours and strengthened their allegiance to it.
The ABL system has some senior teachers who are experts in handling teacher education and
are treated like wise elders. But there are also young trainers, selected to do the training for their
enthusiasm and communication skills. A teacher handling Class I or II could have demonstrated a
tremendous grasp of the principles and procedures of the new system and be asked]] to conduct the
entire training for a rural school. One of the byproducts is a breaking down of hierarchy and a dynamic
interchange of personnel, materials and methods.
The ladder is not a mechanical structure with equidistant steps. For each specific academic
achievement, different ways of learning are built in. There are several ways of reinforcing learning,
while making it enjoyable: song, game, reading, writing and finally evaluating. For the children the
most exciting aspect is that they can learn actively and have a sense of fulfillment. The teacher knows the
exact level of the child's achievement and can take remedial measures for a child who has slowed down.
There are a few blank steps in every ladder. These are intended for any new area the teacher
may wish to include. There is scope for the teachers to be creative. They keep adding songs that the
children sing together and stories for the shadow puppet shows. They use familiar material from their
own environment.
Apart from all the Corporation schools in the city of Chennai, there are ten schools in every rural
block in Tamil Nadu, functioning on this method. They serve as demonstration schools for the entire
block. It is expected that the ABL's obvious success in making children competent is the best tool for
advocacy.
There is a commonly held idea that children need to be motivated by the teacher or parent, to
study. This is not always the case. Children are naturally highly motivated to know and to learn. Most
school procedures dampen the enthusiasm of children and suppress their intrinsic motivation. When
school methods and materials are devised to be attractive and easy to use, as in the ABL, the inherent
motivation of the children is sustained. Learning to learn comes from wanting to learn.
The learning achievements of the children are resoundingly convincing. Indeed, they provide a
strong case for the ABL method to be extended to all State-run schools in Tamil Nadu.
Analysis of ABL
One can examine the ABL method and materials through the following five lenses:
a] Clarity of lessons
b] Classroom environment
c] Children's involvement in process
d] Teacher's role
e] Scope for creativity
Clarity of Lessons
Clarity of the lesson is probably the ABL method's most valued asset. The Learning Ladders
provide structure as they are planned in a systematic way. The child knows what

must be done next. Each unit of information or process is broken up in such a way that clarity of the
lesson is ensured. The method is particularly effective in the fundamentals of Mathematics, as many
children said that it was their best subject. However, there is a need for a review of the language
material.
The criteria for the selection of the vocabulary is not too the same in ABL as in the REC of Rishi
Valley. There they had built up the set of words to be taught, by listening to the conversations of children
around them. For the Tamil kit, the teachers selected letters that are easy to write, made small words out
of them and then gradually increased the number of words on the same criterion. For making the ABL
English language kit, a different rule has been used. The alphabet is not taught directly, of course, but
five words [which begin with each letter of the alphabet] are introduced at a gradual pace. This method
has resulted in the addition of words, which are not directly relevant to their everyday experience.
Classroom Environment
Those who have seen the documentary film on the scheme will vouch for the pleasant relaxed, yet
disciplined climate of the classroom. There is order in the stacking of materials and in all the procedures
that the children follow. The body language of the children shows their enthusiasm. The closing of the
physical and psychological distance between teacher and child reflects a very satisfactory feature of the
system.
Children's Involvement in the Learning process
There is absolutely no doubt that the children are truly engaged in the act of learning, though there
could be degrees of difference among them. During the several hours of observation, one rarely came
across a child who was not pursuing an academic task or a related task. Watching the children move
into the classes after Assembly was a heartening sight. There was an eagerness in their step and a sense
of purpose in their deportment. One is left in no doubt that a feeling of mastery is the best reinforcement
for the development of competence. It seems to work far better than external symbols of recognition like
'stars' and 'medals'.
The Teacher's Role
The teacher has a very important role in this system, though it is not obvious to a casual visitor.
She has to learn the entire ABL system and work effectively with it. She has to exercise a quiet
authority, without becoming authoritarian. An egalitarian attitude may require some un-learning and re-
learning for teachers, but when they see it as part of the new culture of education, they are quick to
accept it and practice it. They are also able, in this system, to spend some time on children who are
slow.
The research team felt, however, that there should be some time allotted to the teacher's voice. For
instance, she could read a story or explain a scientific principle. For such an activity, as for sports, it
might be necessary to group the children by age. The advantages of the mixed age group have been
demonstrated here, but there is also a value for being with peers of one's own age. Some suggestions on
doing this will follow in the last section of the report. While teacher domination is not desirable, teacher
participation is advisable for at least 30 - 45 minutes a day.
Scope for Creativity
As we noticed there are some blank slots in all the ladders, for the teacher to fill up. This gives her
an opportunity for bringing in new material or for including a locally relevant theme. Clearly, there is
here, recognition that knowledge is not a pre-determined set of facts. Changing perspectives, new
information, the opinions of students and teachers, views of others in the community - all these can and
do constitute knowledge. That there is a provision to introduce a new item for study is to be highly
commended. However, a new item is not necessarily a creative addition to the curriculum. Some special
monitoring of the items filled in the blank slots would be recommended. The child's understanding of
open-endedness to new perceptions may not get enough emphasis, when the materials are presented as
an end in themselves. Other ways of allowing children to be creative must be consciously introduced.
Suggestions for Enrichment of the ABL Method
1] India is a country with tremendous diversity in every aspect. When one has a generalization
about any fact in India, an exception to it will crop up immediately. In the school curriculum, the
experiencing of the vastness of the cultural spectrum must find some place. The 'empty slots' must seek
to bring originality and variety. Towards this end, the training of teachers should be strengthened.
2] Music, as it seems to be taught, is a collective effort by children to sing rhymes and songs at the
top of their voices. In this activity, there is no sign that much has changed from an earlier era. This needs
to be modified and moderated. Children can learn to sing softly, to sing in tune and to take turns to sing.
One does not get the idea that they understand what they are singing. There is a sense of enjoyment, of
course and that is good, but a feeling of competent singing will be a value addition.
3] Flexibility is allowed in pace of learning and this is a boon. A certain level of flexibility must be
available for the occasional re-grouping of children. The practice of forcing children to compete and
ranking them according to their performance is shunned by most enlightened educators. And the ABL is
quite child-friendly in this respect. Here it is important to see that
having children of the same age together in an activity does not necessarily entail competition. Also it is
possible to introduce a small element of competition without hurting anyone, a strategy which has been
tried with success. Children of the same age are divided into two or three groups. The quiz question or
alternatively, the athletic task is given to the group. Every child must have one chance, but can get help
from others in the group.
4] The shadow puppet stories are good. They are simple enough for all children to know the
entire dialogue by heart, as we observe from watching a performance. There is scope for introducing
other themes for shadow puppets and also other styles of puppetry and dramatization. Hand puppets, glove
puppets, finger puppets and a host of other kinds of play materials will bring joy to the children. Drama
enables them to cultivate the imagination and enhances their ability to speak clearly and articulately, to
express feelings and to convey messages directly and indirectly. Expanding the scope and variety of
theatre-based activity is strongly recommended.
5] Every school should have a Dictionary in Tamil and one in English. Children should be taught
"dictionary skills". Knowing the order of the alphabets is certainly the first step. Likewise, an
Encyclopaedia in one of the languages would be a tremendous asset for their learning. In the ABL, it is
not clear what a child, who has completed the ladders, can do with his time. In other words, there must be
access to other kinds and higher levels of knowledge. The information ceiling must be raised to provide
room at the top.
6] Story books for reading in class and out of class must be provided in large numbers. This should
be treated as a priority.
7] The Rishi Valley rural schools, which provided the template for the ABL schools, had one very
important part of education i.e. being sensitive to the environment and conserving water, growing plants
and creating a green space around the school. That aspect has been totally neglected in the city schools
which we visited. Just outside the school room, there was rubble and dying grass. No attention had been
given by anyone in the system, to keeping it clean or attractive. Since manual labour of any kind is
totally absent in the set of school activities, it might be a matter to take up after the first rains.
8] Many of the formal sports, which would be ideal for young children, require space and
equipment. And lack of funds may be cited as the reason for their conspicuous absence. But athletics can
be introduced at very little cost. A good sand pit and a few metres of rope can take care of High Jump
and Long Jump. As for running, one needs only some safe space, preferably adjoining the school.
9] This system is better than any other which one would come upon in India, to handle the problem
of understaffed schools. The inadequate number of teachers in our rural schools is a constant problem. On
one hand, there are thousands of trained teachers waiting to get employment and on the other, there are
a number of Primary schools which are short of two or even three teachers. The ABL can be used with
advantage, but its success in the long run, will be determined by the children's access to a teacher in the
classroom
10] Many a time, we open the newspaper and read about an accident at an level crossing. This, in a
country, where the youth are facing unemployment and even their right to 100 days of work a year is the
end result of a long struggle by activists. Why cannot we have a match between those who need a job and
the obvious vacancy? The question one would address to the Railway Minister is quite similar to the
question that we would pose to the Education Minister.
11] The educational scene in Tamil Nadu has many positive ratings to its credit. ABL must build on
the strengths. There are many achievements to be proud of, but one cannot afford to be complacent.
There must be an annual review of the materials, the methods and the learning processes to ensure
success and to reach even higher levels. This educational initiative could well be a forerunner for a
positive change in educational standards across the country. We are now at the threshold of a silent
revolution.
The Process Of ABL Approach
♦ Competencies are split into different parts/units and converted into different activities.
♦ Each part/unit is called a milestone.
♦ In each subject, the relevant milestones are clustered and linked as chain and this chain of
milestones is called LADDER.
♦ Each milestones has different steps of learning process and each step of learning process is
represented by logo.
♦ Milestones are arranged in a logical sequence from simple to complex and also activities in each
milestone.
♦ To enable the children to organise in groups, group cards are used.
♦ Evaluation is inbuilt in the system. Separate cards/activities are used for this purpose.
♦ Each child is provided with workbook/worksheet for further reinforcement activities.
♦ Children's progresses are recorded through annual assessment chart.
♦ Each milestone has different type of activities such as introduction, reinforcement, practice,
evaluation, remedial and enrichment activities represented by different logos.
Benefits of ABL Approach
♦ Children learn on their pace.
♦ Provision of more time for self directed learning and teacher directed learning is reduced considerably.

♦ Group learning, mutual learning and self-learning are promoted. ♦♦♦ Teachers
teaching time is judiciously distributed among children.
♦ Teachers address only needy children.
♦♦♦ Children's participation in every step is ensured in the process of learning.
♦ Evaluation is inbuilt in the system it is done without the child knowing it.
♦ Rote learning is discouraged and almost no scope for wrote learning.
♦ Periodical absence of child from school is properly addressed.
♦ Classroom transaction is based on child's needs and interests.
♦ Freedom to child in learning as he chooses his activity.
♦ Multigrade and multilevel in learning is effectively addressed.
♦ No child can move to the next higher step of learning unless attains the previous one.
♦ Sense of achievement boosts child's confidence and morale.
♦ Attractive cards and activity create interest among children.
♦ Scope for child's development in creative and communicative skills.
♦ Children will have a feel of security as they sit in rounds in the groups.
♦ Children are allowed to move in the classroom as they choose their activity.
♦ Moreover the distance between the teacher and the child is largely reduced and the
teacher acts as a facilitator rather than teacher.
Yet another silent revolution is in innovative education.
I hear; I forget,
I see; I remember,
I do; I understand.
The education system improvements depends upon the teachers attitude and their interest
towards the system, learners, welfare and parents Involvement. These three are the main factors of the
educational system. The ABL schema is one of the hallmarking
schemes in the field of education. It paves the way for learning by doing. It includes play way method,
demonstration, cooperative learning. It motivates the students of primary school to get involved in the
learning activities by themselves. It is quiet an innovative one form the conventional method.
ABL method is very effective and attractive method. Students like this method very much and
also they are involved and learn well through this method. The method is newly-introduced, so to see
the true colour of it, some more capsule period is required.
Active Learning Exercises
Bonwell and Eison [1991] suggested learners work in pairs, discuss materials while role-playing,
debate, engage in case study, take part in cooperative learning, or produce short written exercises, etc. The
argument is when should active learning exercises be used during instruction. While it makes some sense
to use these techniques as a "follow up" exercise or as application of known principles, it may not make
sense to use them to introduce material. Proponents argue that these exercises may be used to create a
context of material, but this context may be confusing to those with no prior knowledge. The degree of
instructor guidance students need while being "active" may vary according to the task and its place in a
teaching unit. Examples of "active learning" activities include:
♦ A class discussion may be held in person or in an online environment. Discussions can be
conducted with any class size, although it is typically more effective in smaller group settings.
This environment allows for instructor guidance of the learning experience. Discussion requires
the learners to think critically on the subject matter and use logic to evaluate their and others'
positions.
♦ A think-pair-share activity is when learners take a minute to ponder the previous lesson, later to
discuss it with one or more of their peers, finally to share it with the class as part of a formal
discussion. It is during this formal discussion that the instructor should clarify misconceptions.
However students need a background in the subject matter to converse in a meaningful way.
Therefore a "think-pair-share" exercise is useful in situations where learners can identify and
relate what they already know to others. So preparation is key. Prepare learners with sound
instruction before expecting them to discuss it on their own.

♦ A learning cell is an effective way for a pair of students to study and learn together. The learning
cell was developed by Marcel Goldschmid of the Swiss Federal Institute of Technology in
Lausanne [Goldschmid, 1971]. A learning cell is aprocess of learning where two students alternate
asking and answering questions on commonly read materials. To prepare for the assignment, the
students will read the assignment and write down questions that they have about the reading. At
the next class meeting, the teacher will randomly put the students in pairs. The process begins by
designating one student from each group to begin by asking one of their questions to the other.
Once the two students discuss the question. The other student will ask a question and they will
alternate accordingly. During this time, the teacher is going around the class from group to group
giving feedback and answering questions. This system is also referred to as a student dyad.
♦ A short written exercise that is often used is the "one minute paper." This is a good way to review
materials and provide feedback. However a "one minute paper" does not take one minute and for
students to concisely summarize it is suggested[who?] that they have at least 10 minutes to work
on this exercise.
♦ A collaborative learning group is a successful way to learn different material for different classes. It is
where you assign students in groups of 3-6 people and they are given an assignment or task to
work on together. This assignment could be either to answer a question to present to the entire
class or a project. Make sure that the students in the group choose a leader and a note-taker to
keep them on track with the process. This is a good example of active learning because it causes
the students to review the work that is being required at an earlier time to participate. [McKinney,
Kathleen. [2010]. Active Learning. Normal, IL. Center for Teaching, Learning & Technology.]
♦ A student debate is an active way for students to learn because they allow students the chance to
take a position and gather information to support their view and explain it to others. These debates
not only give the student a chance to participate in a fun activity but it also lets them gain some
experience with giving a verbal presentation. [McKinney, Kathleen. [2010]. Active Learning.
Normal, IL. Center for Teaching, Learning & Technology.]
♦ A reaction to a video is also an example of active learning because most students love to watch
movies. The video helps the student to understand what they are learning at the time in an
alternative presentation mode. Make sure that the video relates to the topic that they are studying
at the moment. Try to include a few questions before you start the video so they will pay more
attention and notice where to focus at during the video. After the video is complete divide the
students either into groups or pairs so that they may discuss what they learned and write a review
or reaction to the movie.

[McKinney, Kathleen. [2010]. Active Learning. Normal, IL. Center for Teaching, Learning &
Technology.]
♦ A class game is also considered an energetic way to learn because it not only helps the students to
review the course material before a big exam but it helps them to enjoy learning about a topic.
Different games such as jeopardy and crossword puzzles always seem to get the students minds
going. [McKinney, Kathleen. [2010]. Active Learning. Normal, IL. Center for Teaching, Learning
& Technology.]
While practice is useful to reinforce learning, problem solving is not always suggested. Sweller
[1988] found solving problems can even have negative influence on learning, instead he suggests that
learners should study worked examples, because this is a more efficient method of schema acquisition.
So instructors are cautioned to give learners some basic or initial instruction first, perhaps to be
followed up with an activity based upon the above methods.
Active learning method: Learning by teaching
An efficient instructional strategy that mixes guidance with active learning is "Learning by
teaching" [Martin 1985, Martin/Oebel 2007]. This strategy allows students to teach the new content to
each other. Of course they must be accurately guided by instructors. This methodology was introduced
during the early 1980s, especially in Germany, and is now well-established in all levels of the German
educational system. "Learning by teaching" is integration of behaviorism and cognitivism and offers a
coherent framework for theory and practice
4.3 ACTIVE LEARNING METHODOLOGY
As a parallel initiative, Active Learning Methodology has been introduced in VI, VII and VIII
classes in all Government and Aided Schools. The classroom process in ALM, inter aha, includes self-
study, group study, mind mapping, presentation and discussion by the children with the teacher playing the
role of a facilitator. The exercise of mind-mapping of the concepts through self-study provides a lot of
scope for kindling and inculcating creativity among the children. As is the case with ABL in primary
classes, the classroom processes at upper primary level also have undergone a complete change with
children exhibiting unlimited curiosity, interest and enthusiasm for learning. The classroom space has
been transformed into a comfortable zone for the children. All children in upper primary classes [Std 6-
8] in Tamilnadu underwent a dramatic refreshing learning process, empowering them to break into
knowledge systems effectively. It is probably the most rapid transformation of schooling ever attempted.
SS A was intensely searching for a pedagogy that would continue and sustain
the focus on the learner, at upper primary level also. Similar to the focus of Activity Based Learning for
primary classes, it could spot a pedagogy practiced at The School, KFI, Adyar, and Chennai. This was
termed Active Learning Methodology [ALM]. The main thrust of ALM is to support the sure footed
emergence of the Life Long Learner, through active engagement of the student in constructing
knowledge. It emphasizes the importance of the engagement of the learner with the sources of
knowledge. Students are not seen as were recipients of information from the teacher.
The scope of ALM
The aim of ALM is empowerment of the learner in such a way that he or she is confident and
able to function in many contexts. In the middle school years [classes 6-8], such learning can be
blended into the curriculum of any school easily. It includes:
♦ Learning to affirm oneself and one's learning style.
♦!♦ Learning to be healthy and safe - Biology curriculum enrichment.
♦ Learning to think skillfully, recognize and deal with one' s feelings and be resourceful in a variety
of situations -Units on Learning for Life
♦ Learning to live in social systems - living and working together with other people, good
citizenship skills and being able to participate in debate and discussion.
♦ Learning to live in and interact with a physical environment - finding environmentally viable
responses in terms of lifestyle and choices.
♦ Developing leadership and personality among all children in classes 6 to 8.
Above all, Active Learning Skills will help students negotiate the world of knowledge with
competence and enthusiasm, confident of their own abilities and opening widening newer avenues to
learning.

Enrichment Activities
♦ Supply of DVDs depicting ALM model classes
♦ Training and supply of Audio and Video CDs to promote the basic skills
♦ Supply of modules to facilitate teachers for interesting and challenging homework, which extend the
classroom processes to home.
♦♦♦ Conducting workshops to promote hands-on learning experiences and experiments.
Simplified Active Learning Methodology
V Std. - Pilot Study ABL is adopted for I to IV standards and ALM is search of pedagogy for Class V
that would sustain the interest of the learner and at the same time prepare them for the active learning
methodology at upper primary level. The implementation of Simplified Active Learning Methodology in
Class V sustains and enhances skills already acquired by the child in ABL. Every child is actively
involved in the process of self study, pair study, observation, logical thinking, questioning, small group
discussion, large group discussion and linking life situation to class room learning. The special feature of
SALM is that the students are able to evaluate themselves [Self Evaluation] which makes the class not
only very lively but also ensures the child, where he/she is in the learning process. Above all this,
SALM serves as a preparatory mode for the children to cope up with ALM at the upper primary level
and also acts as a bridge between ABL and ALM.
Approach of SALM
SALM has its own unique integrated approach, in which the child becomes very,
♦ Multi-sensorial
 ♦ Creative
♦ Functional Learning Steps in SALM

1. Introduction or Evocation
2. Understanding
a. Guided reading
b. Individual reading
c. Underlining unfamiliar words
d. Discussion in peer group, small group, large group or with the teacher to find out the
meanings.
3. Mind map [guided]
4. Consolidation [focused summary]
5. Reinforcement and Enrichment Activities
a. Framing questions
b. Small group discussion
c. Home work
d. Project
6. Presentation
7. Writing
8. Evaluation

In Mathematics, the brainstorming activities to kindle the thinking skill are noteworthy to be
mentioned here. A single query which results in different answers are discussed in small
groups/Thereby, the conceptual knowledge of a particular concept is linked with the life situations
through the thinking skill.
Scope of SALM
The skills developed by adopting and practising SALM, which bring the leadership quality
among the children, are listed below:
♦ Learning to learn
♦ Reading to learn and understand
♦ Learning to think and question
♦ Learning to relate with life situations
♦ Learning to live in society - working together with others and participate in group discussions.
It is no doubt, that SALM is so reviving and empowering that the child is an active learner
throughout the learning process
Active Learning Method [ALM]
Active learning methodologies require that the student must find opportunities to meaningful
talk and listen, write, read and reflect on the content, ideas, issues ad concerns of an academic
subject[Meyers & Jones, 1993].
Bonwell and Eison[1991] state that some merits of active learning are:
♦ Students are involved in more than listening.
♦ Less emphasis is placed on transmitting information and
♦ Greater emphasis on developing student's skills.
♦ Students are involved in higher order thinking [analysis, synthesis, and evaluation].
♦ Students are engaged in activities [e.g., reading, discussing, and writing].
♦ Greater emphasis is placed on student's exploration of their own attitudes and values.
Active learning shifts the focus from the teacher to the student and from delivery of subject
content by teacher to active engagement with the material by the student. Through appropriate inputs
form the teacher, students learn and practice how to apprehend knowledge and use them meaningfully.

♦ The educator strives to create a learning environment in which the student can learn to restructure the
new information and their prior knowledge into new knowledge into new knowledge about the
content and to practice using it.
♦ Students are assumed to be an intelligent participant in knowledge creation who can lookup
definitions before and after class independently.
♦ Students can develop skills in constructing and using knowledge with the educator's guidance,
alone and also with others in small and large groups.
♦ The educator may explain concepts, principles and methods.
♦ Visual aids, demonstrations, etc., integrated into class presentations.

♦ Students have the opportunity to remember upto50% of the content of each class session.
♦ Students care deeply about their own education.
♦ Students learn to monitor and discuss their own learning.
♦ Students collaborate with other students to discover and construct a framework of knowledge that
can be applied to new situations.
Scope of ALM
The aim of ALM is empowerment of the learner in such a way that he or she is confident and
able to function in many contexts.
♦ Learning to affirm oneself and ones learning style -ALM classroom
♦ Learning to be healthy and safe
♦ Learning to think skillfully, recognize and deal with ones feelings and be resourceful in a variety of
situations.
♦♦♦ Learning to live in social systems-living and working together with other people, good citizenship
skills, being able to participate in the debate of our times.
♦♦♦ Learning to live in and interact with a physical environment-living environmentally viable response in
terms of lifestyle and choices.
Advantages of The ALM
♦♦♦ Active engagement on the child's part
♦ Provides a template for learning and learning to learn ♦♦♦ Te child is
not subjected to endless passivity
♦ Applicable in large classrooms and schools with few teachers

♦ Requires no special aids or special equipment

♦ Children can be resources for each other through paired and group activity
♦ The teacher can devote some time to children who need special help
♦ Allow the child to check her/his work against the teachers and thus save the teacher endless
corrections while ensuring accuracy in Childs learning.
♦ Works at child friendly and realistic assessment formats
The beauty of the process is its simplicity. Allows room for all children's voices to be heard
through discussions and presentations.

4.5 COMPUTER ASSISTED INSTRUCTION [CAI]

Computer Assisted Instruction [CAI] is an advanced technology in the teaching learning
process of mathematics. CAI is an improved version of programmed learning. Nowadays computers
are integral and inseparable part of education process. CAI, the computer provides instruction directly
to the learner and allow them to inneract with it though the lessons programmed in the system. The
computers put questions and expects the students to respond. Computers provide instant feedback to
the leaner on the basis of his response. Computer is, therefore act as a teacher to the student in this
respect. There are many kind of exercises and Practices provide by computer in teaching various
aspects in mathematics.
♦ CAI provides self-learning
♦ Learners proceed according to the own interest and peace of learn i ng.
♦ Computers provide unbiared instant feedback.

E-LEARNING
♦ E-learning comprises all forms of electronically supported learning and teaching.
♦ The information and communication systems, whether networked or not, serve as specific
media to implement the learning process.
♦ The term will still most likely be utilized to reference out-of-classroom and in-classroom
educational experiences via technology, even as advances continue in regard to devices and
curriculum.

M-LEARNING
♦ The term M-Learning is known as "mobile learning 1'.
♦ It related to e-learning and distance education
♦ It is distinct in its focus on learning across contexts and learning with mobile devices.

M- E-
Learning Learning
The term covers: learning with E-learning is essentially the
portable technologies including computer and network-enabled
but not limited to handheld transfer of skills and knowledge.
computers, MP3 players,
E-learning applications and
notebooks and mobile phones.
processes include Web-based
M-learning focuses on the learning; computer-based learning,
mobility of the learner, virtual classroom opportunities and
interacting with portable digital collaboration.
technologies, and learning that
Content is delivered via the Internet,
reflects a focus on how society
intranet/extranet, audio or video
and its institutions can
tape, satellite TV, and CD-ROM. It
accommodate and support an
can be self-paced or instructor-led
increasingly mobile population.
and includes media in the form of
M-learning is convenient in that text, image, animation, streaming
it is accessible from virtually video and audio
anywhere.
Abbreviations like CBT (Computer-
M-Learning, like other forms of Based Training), IBT (Internet-
E-learning, is also collaborative; Based Training) or WBT (Web-
sharing is almost instantaneous Based Training) have been used as
among everyone using the same synonyms to e-learning.
content, which leads to the
Today one can still find these terms
reception of instant feedback and
being used, along with variations of
tips,
e-learning such as e-learning, e-
M-Learning also brings strong learnins, and e-learning.
portability by replacing books
and notes with small RAMs,
filled with tailored learning
contents. In addition, it is simple
to utilize mobile learning for a
more effective and entertaining
experience.
4.7 IMPORTANCE OF TEACHING AIDS - PROJECTED AND NON-PROJECTED AIDS-
IMPROVISED AIDS [PAPER FOLDING AND PAPER CUTTING]-ITS SPECIFIC USES IN
TEACHING MATEMATICS
A teacher has an inherent desire that his teaching should be as effective as possible. What he teaches
should be clearly understood, grasped and fixed in the minds of his students. In order to realise his objective,
the teacher makes use of different types of aid materials just as charts, models, concrete objects, apparatuses,
instruments and other resources. All such material and resources which help the teacher of mathematics in the
realisation of his objective of effective teaching can be termed as aids in. teaching of mathematics. These aids
are also named as audio-visual aids in the sense that they call upon the auditory and visual senses of students.
The aids like radio, tape recorder which help the individuals to learn through listening, are called audio aids.
The aids like filmstrip, projector, epidiascope, newspaper, magic lantern and black-board which help in
learning through watching are called visual aids. Some aids like cinema, television, where one learns through
listening as well as watching are known as audio visual aids.
Need and Importance of A.V. Aids in Teaching Mathematics
The use of audio-visual aids in teaching of mathematics may be supported on the following grounds:
1. Clarity of the subject:
+Audio-visual aids help in clarifying the various abstract concepts of mathematics instead of struggling
hard only with the theoretical talks, if the teacher takes the help of some aid material he can make the' subject
more clear and meaningful to his students. For example the simple facts of addition like 7+5 = 12 can only be
taught effectively if the children are given opportunity to count seven and five concrete objects first separately
and then in combination.
2. To make the subject interesting:
Audio-visual aids help in creating and maintaining interest in the learning of Mathematics. The
subject no longer remains as boring, dull and unreal one.
3. Based on maxims of teaching:
The use of audio-visual aids facilitate to the teacher to follow the important maxims of teaching like,
'simple to complex', 'concrete to abstract', 'known to unknown' and 'learning by doing' etc.
4. Psychological value:
Use of audio-visual aids has some psychological advantage also. Children always like to manipulate or
observe the new things. Once they are attracted towards an object or activity, their attention can be easily
captured and desired interest in the learning can be safely maintained. The satisfaction of various interests and
innate tendencies through audiovisual aids thus helps much in the task of learning.
5. Fixing up the knowledge:
The knowledge gained needs to be fixed in the minds of the students. It needs a lasting impression in
their minds which can be easily engraved through audio-visual aids.
6. Saving of time and energy:
Much of the time and energy of both the teachers and the taught may be saved on account of the use of
audio-visual aids as most of the abstract concepts may be easily clarified and understood through their use.
7. Use of maximum senses:
Senses are said to be the gateway of knowledge. Audio-visual aids help in the maximum utilisation of
sense organs and there by facilitate the gaining of knowledge by the students.
8. Meeting the individual differences requirements:
There are wide individual differences among children: Some are ear minded, some can be helped
through visual demonstration while others learn better through doing. The use of various types of audio-visual
aids helps in meeting the retirements of different types of pupils.
9. Encouraging activity:
Teaching learning process becomes quite stimulating and active through audio-visual aids. Here passive
listening does not help in the realisation of the objectives of teaching mathematics. Use of audio-visual aids
helps in converting the passive environment of the class-rooms into a living one.
10. Development of scientific attitude:
Use of audio-visual aids helps in cultivating scientific attitude among students. Instead of agreeing to the
listened facts, they resort to observe or use them practically with the help of audio-visual aids and ultimately
adopt the habit of generalization through actual observations and experiments.
VARIOUS TYPES OF AIDS USED IN TEACHING OF MATHEMATICS
Mostly, the types of aids given ahead are used in the teaching of Mathematics.
1. Weighing and measuring instruments:
Various types of weighing and measuring instruments help in acquainting the students with different scales
and units of weight and measurement. The teacher should try to take help of such instruments like tape
measures, balances and weights, graduated cylinders etc. for gaining practical knowledge of the mathematical
facts.
2. Drawing instruments:
Drawing instruments help much in learning mathematical facts and skills. Specially in geometry and
mensuration the use of these instruments is a must. Therefore, every student of mathematics should be
asked to keep a geometry box containing essential geometrical instruments and the teacher should try to take
help of the wooden instruments for his demonstration work on the black-board.
3. Real objects:
As an aid real objects are said to be the most useful and effective means of providing direct experiences to
the students. The list of such objects may consist of objects like beads, ball frames, coins, seeds, sticks, pebbles,
coloured balls or solids, pencils, the material used and produced in various work- experiences areas etc. Various
topics and concepts concerning four fundamental rules, average, percentage, fraction, profit and loss etc. may be
successfully taught through these objects. The real life situations may also be exploited as an aid in the
teaching of mathematics. Classroom may work as a real object for teaching area of the four walls. Similarly,
black-board, classroom tables, play-grounds and gardening plots etc. may prove very helpful in teaching area
and other facts of mensuration.
4. Models:
Models are the copies of the real objects. When for some reasons or the other it is not possible or
advisable to use the real objects, models prove very useful and effective means of educating the students. As
far as possible a model should be least expensive and be made by the students themselves. Models can be
successfully used to acquaint the students with the shape and forms of different numerals and geometrical
figures. For this purpose even the square, round or rectangular process of cardboard or thick paper may
serve as models. To give practice in writing numbers, numerals and digits engraved on the wooden or stone
pieces may prove useful models. Also for teaching topics like, area of four walls, cross roads and other squares,
rectangular and circular figures models may be made out of thick paper and cardboard.
Models may also serve the best purpose in teaching various concepts and facts related to geometrical
theorems and exercises. For example to acquaint the students with the fact that 'sum of the three angles of a
triangle is equal to two right angles'.
5. Pictures and Charts:
In case where it is not possible to have an appropriate model or use real objects, pictures and charts
prove very useful aid in teaching Mathematics. For example to teach
fractions-simple, compound and decimal, fractional parts [rectangular or circular] may be drawn on the charts.
In teaching profit and loss, unitary method, percentage, interest and work and time the help of charts may be
taken to work out principles and formulae. Pictures may also be used to make the students understand the basic
things about the problems. In the problems related to area, volume and mensuration, charts may be used for
analysing the problems: In geometry, the use of charts may be made in showing figures concerning the proof
of the theorem, or proposition and helping the students in the construction of various geometrical figures and
diagrams. In algebra, the charts may be effectively used in teaching of directed numbers, four fundamental rules
and problems based on equations. The general formulae may also be demonstrated through charts. The pictures
and chart prove a helping hand to the teachers as they save their time, and energy otherwise wasted in
drawing figures and diagrams on the black-board. In addition to this, they may prove constant source of
inspiration and means of imparting self-education to the students. For this purpose the following types of
charts and pictures may be hung in the classroom
i. Charts concerning geometrical figures and shapes.
ii. Charts depicting different principles and formulae.
iii. Charts concerning various units of weights and measures.
iv. Pictures concerning great mathematicians.
v. Pictures related to the history of mathematics.
vi. Pictures and charts showing use of mathematics in day to day life.
The following points should be kept in mind for the effective use of the pictures and charts :-
a. There should not be too many things or facts demonstrated through a simple chart. It
should concentrate on a single definite purpose.
b. Charts should be coloured and attractive.
c. Charts should have a proper size. The drawing and writing must be so distinctive and
clear that all the pupils may get benefitted through its use.
d. No irrelevant thing or theme not connected with the topic should be demonstrated
through a chart.
e. As far as possible the charts and pictures should be got prepared by the students.
6. Black-Board:

As an aid in teaching of mathematics, black-board is so much effective that it is usually termed

as the right hand of a mathematics teacher. The secret of the popularity of the black-board lies in the
fact that one can write or remove what has been written on it at his own will without involving any
significant expenditure. The teacher writes and explains his writing while writing on' the black board. In
this way students get both the benefit of observing and listening at a time. The use of black-board is
indispensable for all the branches and topics of Mathematics. It begins with the first lesson of mathematics
and then goes up to highest learning in the subject. In all tasks like drawing of diagrams and figures,
giving definitions, writing the language of the problem and their solutions, drawing generalizations and
giving principles and formulae, having practice and drill work and assigning home-task, black-board
proves a good helping hand. The black-board with graph lines may be successfully used for thawing all
types of geometrical figures, plotting graphs, presenting statistical data, solving problems on areas and
volume. In short, what a teacher wants to communicate to his students may be succesfully done through
this readily available aid and it helps in quick understanding as well as fixing up of the knowledge of
the subject. But for getting the desired advantages, the teacher must know its proper use. The following
may serve a useful purpose in this direction :-
i. The black-board should be properly cleaned before making its use; it should be got polished or
painted from time to time.
ii. The teacher should have sufficient practice for writing, sketching and drawing legibly on the black-
board.
iii. The black-board writing should be in straight lines. It should be visible to all the students in
the class.
iv. Teacher should try to speak what he is writing on the black board.
v. He must be very careful for not writing anything wrong and inappropriate on the
black-board.
 vi. He must learn how to face the black-board while writing on it. He should have a watchful eye over
his
students while keeping himself busy on the black-board.
vii. The problems solved or the work done on the black-board should follow a logical as well as
psychological
sequence.
viii. Teacher should always make use of the standard terminology and symbols on the, black-board so
that
the students may not get confused. ix. The chalk used should be of a superior quality. It should
be
properly pressed while writing.
x. Coloured chalks should also be used for emphasizing particular facts and making the diagrams and
sketches more attractive and meaningful.
xi. The students should also get proper opportunity for solving the problems and drawing the diagrams
on the
black-board.
7. Magic Lantern:
This instrument of science had proved very useful for teaching mathematics. It helps the teacher to
demonstrate different types of figures, diagrams, pictures related to various topics of mathematics through
the slides. For getting better results, the teacher may also give explanation of the things demonstrated on the
magic lantern. The demonstration may further be followed by group discussion for the clarification of the
various issues on the topic.
8. Epidiascope:
This instrument is used for enlarging and then demonstrating the contents, figures and diagrams of the
printed or handwritten pages. It has shown its value in teaching of mathematics too, specially at the time when
the teacher feels difficulty in drawing or giving things on the blackboard.

Pretetten lens

9. Film-Strip Projection:
 In a film strip, 15 to 20 slides concerning useful topics are photographed on a 35 or 16m.m. films.
These film strips are then projected on the screen through a projector. The teacher may demonstrate the
pictures for any period of time irrespective of speed as the situation demands. These film strips give
altogether a new colour and attraction to different ideas in mathematics. The film strips may be easily
obtained from the market or borrowed from the Central Library, Department of NCERT and SCERT and
some leading foreign embassies.
10. Cinema:
Cinema, the popular, means of entertainment may be successfully used for the teaching of
various principles, definitions and facts of mathematics. The life history of mathematicians, their
discoveries and the historical landmarks of the development of mathematics may also be successfully
demonstrated through cinema films. The films may be used effectively for teaching demonstrative
geometry. The students may learn how to use various geometrical instruments, draw different types of
figures and diagrams, survey and measure the different dimensions areas and volumes and use graphs.
Though films students may be acquainted with the use of mathematics in solving day to day life problems
used in different occupations and field of actions. In this way the use of cinema films may prove, quite
effective, stimulating and useful for the teaching and learning of mathematics. Cinema is such an aid that
calls on both the visual and auditory senses. The students at the same time may listen as well as observe the
facts and therefore it provides to them a greater stimulating and motivating value for learning something
new. It can help not only the students but also the teachers in acquainting them with the growing
knowledge, and methods of teaching being adopted in other, countries. In a school hail or big room, the
teacher may demonstrate films through some good projector. The films for this purpose may be borrowed
from state or central education departments and libraries. The NCERT and similar state level bodies as
well as some foreign embassies may also help in this direction.
11. Radio and Television:
Radio and television both have established their due credit in the field of education. Almost all the
important centres of A.I.R. broadcast programmes concerning education. For the programmes on
mathematics eduoatioi4 either the regular classes on topics of mathematics are being held or the important
discussions and speeches concerning principles and laws of mathematics, life history and contributions of
mathematicians, historical development of the knowledge of mathematics, the application of mathematics
in practical life are broadcast. Highly experienced teachers, teacher educators, mathematicians and
research scholars take
part in such programmes. The Radio as a means of communication takes their voices to the millions of
students and teachers listening to their programmes. Television has far greater advantages as it not only
conveys the voices but the pictures and actual scenes also. The students sitting far, away from the TV
stations may be benefited through the telecasting programmes almost in the same way as it is
happening just before their eyes. A teacher of mathematics should try to take advantage of such learning
opportunities by making himself and his students fully conversant with such programmes.
12. Newspapers:
Newspapers may be used as an effective aid for teaching and learning of mathematics. They
help in correlating teaching of mathematics with day to day- happenings of life. The statistics given in the
newspaper in the form of weather charts, the prices of various commodities, budgets of state and
central government, interest rates of various private and government agencies, stock and shares etc. all
provide good means for making the teaching of mathematics interesting, useful and purposeful. The cutting
of the newspapers may thus be employed to help the students in learning the practical application of
mathematics in day to day life.
13. Running cooperative store:
A cooperative store running in a school may also be utilized for mathematics education. The students
may learn practically the various principles and facts of mathematics regarding profit and loss, four
fundamental rules, unitary method, percentage, weighing and measuring etc. by becoming shopkeepers or
customers in such a store.
14. Mathematical games and riddles:
Mathematical games and riddles besides playing their recreational roles, may be effectively utilised
for learning, practising and using various principles and facts of mathematics. There are so many games which
may be successfully utilized for mathematics education. In such games and riddles, competitions may be
usually organised by dividing the students into groups. It will provide learning opportunities while
playing.
15. Visits and excursions:
Visits and excursions do play an effective role in learning mathematics by providing . direct
experiences. The students get opportunity of learning mathematics in the manner it is being used in
practical life. For visits and excursions they may be taken to different places like workshops, industries,
mills, and telegraph offices, station, market, agriculture fields, forest, some picnic spots and visiting
places. During these visits students may face so many problems requiring the knowledge of mathematics
for their solutions. The teacher may utilize
such situations for giving the necessary knowledge of mathematics or sometimes students may get
opportunities for applying the already gained knowledge.
Conclusion about the Aids used
In this way it may be seen that there is no dearth of aids for making the learning of mathematics
easy, interesting and useful. A resourceful teacher may choose the appropriate aids suiting to his needs, time
and occasion. It is also true that financial resources put obstacle in the way of taking advantages from aids.
But this is not the end of the story one should not get disappointed. If a teacher has the courage and will he
may get these aids prepared with the help of his students- through indigenous and cheap material. In this
direction the services of the agencies like education departments of central and state governments, teachers
training institutions, NCERT and SCERT, and extension services centers may also be received for
collecting the essential and useful teaching aids.
LOW COST IMPROVISED TEACHING AIDS
As emphasized earlier mathematics is a type of subject the need of which is felt well for
performing the day to day activities of our life. Not only this a child from his tender age observes the use
and application of mathematics in the existence and activities of all the animal and inanimate objects
surrounding him. In fact nature itself is a big source and a treasurer of aid material for learning the facts
and principles of mathematics. The children may thus be helped to gather valuable direct and indirect
experiences for the learning of mathematics from their local environmental surroundings. In their local
setup comprising of their homes, community, physical and social environment, they may get a lot of
opportunity to practise and learn so many valuable concepts regarding the teaching learning of mathematics.
There is a lot of cheap and sometimes waste material available in children's local environment that can be
successfully utilised for the improvisation of valuable aid material for the teaching of mathematics.
1. Concrete material and objects:
A wide variety of collection of different types of concrete material like beads, seeds, balls, sticks,
match boxes, pebbles, different types of corns etc. may be made with the help of students. All that
material may prove quite helpful in the learning of counting, four fundamental rules [addition,
subtraction, multiplication, division], multiplication tables etc.
2. Improvising an abacus:
An Abacus containing a number of wood, metal or even thermocol beads in several wires can be
easily improvised for teaching the students the facts of counting, four fundamental rules, place value system,
etc.
3. Place value pockets may also be improvised:
The required boxes for this purpose can be made with the help of thermocol, thick paper or
wooden etc. and the system be made so operated as help in the learning of the concept of place value.
4. Preparation of models:
By using the easily available low cost or waste material various types of models may be
improvised. Let its have different types of such models representing the shape of various geometrical
figures like rectangle, square, parallelogram, trapezium, triangle, circle, ellipse, cone, cylinder, pyramid,
sphere etc. We can have models made of clay, match box sticks, thermocol, wood, wax etc. to
demonstrate the various geometrical figures, their properties and operations. Area of the cross roads,
Area of the four walls of a room, circumference and area of a circle, circumference and area of
rectangle, triangle and so many other things related to the learning of mensuration may be easily taught
through the use of the models made of the locally available low cost material. The waste paper, card
boards, thick papers, wooden boxes etc. available in the packages of the household goods purchased from
market can be effectively used for making that model with a simple use of pins, nails, scissors, hammers,
threads, ropes etc. easily available in the houses and chooi workshop. For illustration purpose let me point
out a working model for helping the students to understand geometrical concept and the related theorem
"sum of the three angles of a triangle is equal to two right angles. It has been already mentioned in the
present chapter. We may utilise the wood available from the carton boxes or the boxes received from
shopkeepers during the purchases of fruits and other household articles for the improvisation of this model.
Match box sticks and rubber bands may also be utilized for the construction of such model.
5. Preparation of Charts and pictures:
With the help of the, chart paper and drawing material easily available in the market, the low cost
visual aid material may be easily prepared for the teaching learning of almost all concepts related to all
the branches of mathematics. You may hang these charts in the mathematics classrooms, laboratory or
library for the grasping of the essential mathematical concepts with no extra efforts. At the time of class
room teaching these charts and pictures may be successfully used for the clarity of the needed
mathematical concepts, processes and operations.
A teacher of mathematics in this way may utilize the waste or last cost material available in the
local surroundings for the preparation and improvising valuable teaching-learning aids for bringing
efficiency and effectiveness to the ongoing teaching Learning process. He may even utilize the real
surroundings as an aid to his teaching. For example while teaching about the area of the four walls of a
room, he can have the class room as a living concrete model. In teaching the concept of mensuration, then
he may utilise the sports ground, school, garden and neighbouring plots as a living models for necessary
surveying and measurement The scrap book prepared by the students through the news paper cuttings of
the data of mathematical interest thus may also be used as one of the useful aid material for ac
acquainting the students with the facts and principles of mathematics.
Paper folding and Paper cutting etc.
Paper folding encourages students toward content and invites then to reflect upon the meaning of
proof. Significant benefit of paper folding technique is its accessibility to students and the effective
benefits of learning.
Paper folding and paper cutting activities are selected because it is simple and interesting hands
on activity that students can build their learning experience. The process incorporates both mental and
physical involvement in the learning process. It is especially appealing to the concrete learner who need
to visualize and sense to learn.
4.8 CRITCRIA FOR SELECTION OF APPROPRIATE TEACHING AIDS:
Teaching and learning material, whether purchased or donated, should be selected and acceded in
was which ensure they:
♦ are directly related to a preschool's or school's curriculum policy and program, based on the
department's framework of standards and accountability, and include, where relevant, support for
the recreational needs of children and students
♦ support an inclusive curriculum, thus helping children and students to gain an awareness of our
pluralistic society and the importance of respectful relations with others
♦ encourage understanding of the many important contributions made to our common Australian
heritage by men, women, Aboriginal and Torres Strait Islander peoples, people from diverse cultural
and linguistic groups, people with disabilities and minority groups
♦ motivate children, students and educators to examine their own attitudes and behaviour and to
comprehend their duties, responsibilities, rights and privileges as citizens in our society
♦ are relevant for the age of the children or students for whom they are selected and for their
emotional, intellectual, social and cultural development. This includes the
assurance that children and students will not be exposed to offensive materials; that is, materials
which describe, depict, express or otherwise deal with matters of nudity, sexual activity, sex, drug
misuse or addiction, crime, cruelty, violence or revolting or abhorrent phenomena in a manner that a
reasonable adult would generally regard as unsuitable for minors of the age of the relevant
children and students
♦ provide opportunities for children and students to find, use, evaluate and present information and
to develop the critical capacities to make discerning choicer so that they are prepared for
exercising their freedom of access, with discrimination, as informed and skilled adults
♦ represent a range of views on all issues.
4.9 USE OF MASS MEDIA IN TEACHING MATHEMATICS
Let's take quick trip through the main types of media outlet. There are others webpages. CD-
ROMs, DVDs, blogs, podcasting, whatever.
Magazines
Popular science magazines have the advantage that their readership is self-selected for an interest in
science. Surveys have shown that mathematics is very popular among such readers. Each magazine has its
own level, and its own criteria for what will appeal. Scientific American is justly famous for the
'mathematical games' columns originated by the peerless Martin Gardner, which unfortunately no longer
run. In the UK there are New Scientist and Focus, which regularly feature mathematical items ranging from
primarily testing to su doku. If you are thinking of writing an article for such a magazine, it is always better
to consult the editors as soon as you have a reasonably well formulated plan. They will be able to advise
you on the best approach, and will know whether your topic 1640 Ian Stewart has already been covered
by the magazine - a problem that can sink an otherwise marvellous idea. Expect the editors and subeditors
to rewrite your material, sometimes heavily. They will generally consult you about the changes, and you
can argue your case if you disagree, but you must be prepared to compromise. Despite this editorial input, the
article will usually go out under your name alone. There is no way round this: that's how things are in
journalism.
Newspapers
Few newspapers run regular features on mathematics, bar the odd puzzle column, but most
'quality' newspapers will run articles on something topical if it appeals to them. Be prepared to write 400
words on the Fields medallists with a four-hour deadline, though, if you aspire to appearing in the national
news.

Books

Books, of course, occupy the other end of the deadline spectrum, typically taking a year or so to
write and another year to appear in print. They really deserve an article in then-own right, and I won't say
a lot about them here, except in Section 7 below. Sometimes expediency demands a quicker production
schedule. I once wrote a book in 10 weeks. It was short, mind you: 40,000 words. The quality
presumably did not suffer because it was short-listed for the science book prize. If you want to write
semi-professionally, you will need an agent to negotiate contracts. At that level, book writing is much like
getting a research grant. Instead of ploughing ahead with the book, you write a proposal and go for a
contract with a specified advance on royalties.
Radio
Radio is my favourite medium for popularising mathematics. This is paradoxical, because radio
seems, to have all of the disadvantages [such as no pictures] and none of the advantages [such as being
able to write things down and leave them in full view while you discuss them] of other media. However,
it has two huge advantages: attention-span and imagination. Radio listeners [to some types of programme]
are used to following a discussion for 30 minutes or longer, and they are used to encountering unfamiliar
terminology. And radio has the best pictures, because each viewer constructs a mental image that suits
them. On radio you can say 'imagine a seven-dimensional analogue of a sphere" and they will. It may not
be a good image, but they'll be happy anyway. Say the same on TV and the producer will insist that you
build one in the studio for the viewers to see. TV removes choice: what you get is what they choose to
show you. On radio, what you see is what you choose to imagine.
Television
Television is far from ideal as a medium for disseminating science, and seems to be becoming
worse. As evidence: every year the Association of British Science Writers presents awards for science
journalism in seven categories in 2005 no award was made in the television category, and [Acker] the judges
stated: Mathematics, the media, and the public 1641. To say the quality of entrants was disappointing is
an understatement. We were presented with 'science' programmes with virtually no science in them. Some
were appalling in their failure to get across any facts or understanding. Whenever there was the possibility
of unpicking a little, highly relevant, science, or research methodology, the programmes ran away to non-
science territory as fast as possible, missing the whole point of the SLOP, as far as we were concerned. I
still vividly recall a TV science programme which informed viewers that Doppler radar uses sound waves to
observe the speed of air in a tornado. No, it uses electromagnetic waves - the word 'radar' provides a
subtle clue here. Sound waves come into the tale because that's where Doppler noticed them. The
reasons why television is far from ideal as a medium for disseminating are equally disappointing. It is
not the medium as such that is responsible - although it does discourage attention-spans longer than
microseconds. The responsibility largely rests with the officials who commission television programmes,
and the companies who make them. Television changed dramatically in the 1980s, especially in the UK.
Previously, most programmes were made "in-house' by producers and technicians with established track
records and experience. Within a very short period, nearly all programming was subcontracted out to
small companies [many of them set up by those same producers and technicians] on a contract-by-
contract basis. This saved television companies the expense of pensions schemes for their employees
[since they now had none] and protected them against their legal responsibilities as employers [ditto].
But as time passed, contracts were increasingly awarded solely on the basis of cost. A new company
would get the commission to make a programme, even if they had no experience in the area, merely
because they were cheaper. Very quickly, most of the companies that knew how to make good science
programmes were ousted by new kids on the block whose main qualifications were degrees in media
studies and, the decisive factor, cheapness. Any lessons previously learned about how to present science
on television were lost, and had to be re-learned, over and over again, by a system dedicated to the
perpetual reinvention of the wheel. There is still some good IV science, but nowhere near as much as
there ought to be given the proliferation of satellite and cable channels. The good news here is that TV
is once again wide open as a medium for popular science, especially now that there arc hundreds of
channels desperate for content. But we will have to fight all the old battles again.

4.10 FIELD TRIP AS A TEACHING TECHNIQUE

♦ A field trip is defined as any teaching and learning excursion outside of the
classroom.
♦ There are two types of field trips - Physical and Virtual.
Physical Field Trip Examples
♦ school playground, school board outdoor education centres, provincial parks,
protective wetlands, science centres, museums, zoos, grocery stores, fire stations,
veterinary clinics, agricultural operations, natural resource operations
Why Field Trips?
♦ To make a connection between reality and theory - hands-on
♦ Can be used as an introduction to a unit or a culminating actively.
♦ To provide an authentic learning experience
♦ Exciting, children get to meet and interact with others
♦ They can experience all five senses, see, touch, feel, smell, taste
 ♦ Children remember the field trips because they learn using different methodology How to Plan
and Run a Successful Physical Field Trip
♦ Planned and effectively organized
o Check for school/board policy on field trips o Children to
supervisor ratios o Transportation procedures o Fund raising
♦ Plan with children as much as possible
♦ Involve school principal and vice-principal
♦ Ensure field trip compliments the curriculum by meeting specific expectations

♦ Ensure students have necessary background knowledge prior to field trip, if introduction to
field trip provide essential preparatory information in order to prepare students for the
experience
♦ Plan post-trip activities that build on the knowledge gained in partaking in the field trip [eg.
reports, displays, photos, graphs].
♦ Prepare a checklist to ensure that all tasks are completed [e.g. booking facilities and
transportation, parental notifications, medical forms, supervision, safety precautions,
emergency information] and have the school administrator sign the checklist once completed.
♦ Be sure to visit the site ahead of time, in order to plan for safety, resources and resource
personnel, facility.
♦ Plan on route activities to enrich their experience during the field trip.
♦ Provide parents with rationalization for the field trip and trip itinerary.
Guidelines for Safety and Behaviour
♦ There are many potential liability situations that can occur on a field trip, it is your ultimate
responsibility to ensure that the following safety guidelines are meet concerning safety and
behaviour while outside the classroom.
♦ Set behavioural expectations for the field trip and describe and discuss them with the
children prior to departure.

■ ■..,..■■■.'-. ■
♦ Have children create their own code of behavi our with teacher involvement and veto power.
♦ If junior students are mature enough to be responsible and accountable for their own behaviour,
have them sign a written code of conduct; therefore, creating a behavioural contract.
♦ Introduce the idea of team work to enable students to live to the written code of conduct.
♦ Describe the consequences for not behaving properly prior to embarking on the trip.
♦ Provide parents with behavioural expectations and ask them to ensure that the children know
and understand the code of conduct and the consequences.
♦ Ensure that the student/supervision ratio meets board/school standards.
♦ Eliminate all the safety concerns identified in the school/board policy.
♦ Use board approved transportation.
 ♦ Create passenger manifest and file with appropriate school personnel. Also, take along
passenger manifest to check that everyone is accounted for.
♦ Implement a buddy with students as an additional safety precaution.
♦ Ensure that safety gear and first aid equipment are readily available and in plain view.

Tips and Variations for Field Trips

Before the Trip, Teachers Should:
♦ Visit the site to find connections to curricula, assess potential problems, and plan how the
students could best use their time.
♦ Give as much context as possible bso that the students will understand what they see. Teachers
might consider having the students do something like a journal or a K/ W/L chart in which they list
questions they have, expectations for their visit, or plans for ways to use what they will see.
♦ Create a trip sheet like Stanlee Brimberg's that prompts students to draw, write responses, answer
questions, or find items for a "scavenger hunt" of the location. This sheet, however, should not be
so directive that the students can't see and respond to the site in their own ways.
♦ Set standards of etiquette and respectful behavior.

During the Trip, Teachers Should:

♦ Build in opportunities for students to view the site or work alone, in pairs, or in small groups. On a
trip to a museum, for example, the students could be asked an open-ended question like, "Find a
work that represents our theme or time period and sketch it. In class we will share our choices
and discuss why we chose them." The students could also choose one aspect or part of the site to
explore.
♦ Consider giving some students disposable cameras, small tape recorders, or mandates to record
specific information. When the class is back at school, they can compile a complete picture.
After the Trip:
♦J* Allow the students to synthesize their experience creatively. For example, they might create trip
brochures for other classes or the school library. They might create children's books about a
theme from the field trip. Or they might present their experience orally to another class or grade.
Benefits of Field Trips
♦♦♦ Field trips bring classroom study alive for students and help them remember and relate to what
they have learned. They provide rich resources that can rarely be approximated in the classroom.
They also help connect school to the world.
♦ Field trips provide new cultural contexts for literature and provoke questions.
♦ Field trips stimulate and focus class work by helping students synthesize information.
4.11 QUALITIES OF GOOD MATHEMATICS TEXT BOOK
A] Introduction
The mathematics textbook is an important source for learning mathematics and it plays a key role
in effective teaching and learning. A textbook should stimulate reflective thinking and develop problem-
solving ability among students. The textbooks should present real learning situations, which are challenging
and interesting for the students and should not render itself as a means of rote learning.
Text books and teachers' guides occupy a unique place in the teaching learning process. Text
book are an indispensable part of primary and secondary education. The text book is a teaching instrument.
It is not only a source of information, but a course of study, a set of unit plans and learning guide. It helps
to revise and reinforce the language material already taught. In the absence of any other instructional
material, the text book becomes a potent tool in the hand of a teacher to teach the skill of a language and
the more so of a foreign language.

B] Qualities of a Mathematics Textbook

The qualities of a good textbook in mathematics can be broadly classified under the following
heads
1] Physical features
2] Author
3] Content
4] Organization and presentations
5] Language
6] Exercise and illustration
7] General
1. Physical features:
♦ Paper: the paper used in the textbook should be of superior quality
■
♦ Binding: it should have quality strong and durable binding
♦ Printing: it should have quality printing, bold font and easily readable font.
♦ Size: bulky and thick. It should be handy
♦ Cover: it should have an appealing and attractive cover page.
2. Author:
♦ Qualified author should write it
♦ Experienced teacher should write it
♦ Competent teachers should write it
♦ It should be written by committee of experts constituted by the state government
♦ For the authors, certain minimum academic and professional qualifications may be prescribed.
3. Content
♦ It should be chiId centered
♦ The subject matter should be arranged from simple to complex and concrete to abstracts.
♦ The subject matter should create interest in the pupil.
♦ It should be obj ective oriented
♦ It should be written according to prescribed syllabus
♦ It should satisfy the demands of examination
 ♦ The answers given at the end of each section should be correct
♦ It should include the recent developments in the mathematics relating to the content dealt with.
♦ Oral mathematics should fine its due place in the textbook.
4. Organization and presentation
♦ It should provide for individual differences.
♦ There should be sufficient provision for revision, practice and review.
♦ It should stimulate the initiative and originality of the students
♦ It should offer suggestion to improve study habits.
♦ It should facilitate the use of analytic, synthetic, inductive, deductive, problem solving and heuristic
approaches to teaching.
♦ Content should be organize in a psychological consideration
♦ Content should be organize in a logical way
♦ It should suggesting project work, fieldwork and laboratory work.
5. Language
♦ The language used in the textbook should be simple and easily understandable and within the
grasp of the pupils
♦ The style and vocabulary used should be suitable to the age group of student for whom the book
is written.
♦ The term and symbols used must be those, which are popular and internationally accepted
♦ It should be written in lucid, simple, precise and scientific language.
6. Exercise and Illustrations:
♦ The illustrations should be accurate
♦ The illustrations should be clear and appropriate
♦♦♦ It should contain some difficult problems
♦ It should contain exercises to challenge the mathematically gifted students.
♦ There should be well-graded exercises given at the end of every topic.
♦ The exercise should develop thinking and reasoning power of the pupils.
7. General:
♦ At the end of book there should be tables and appendices.
♦ The textbook should be of latest edition with necessary modifications
♦ The book should be moderate price and readily available in the marker.
C] Conclusion
Every teacher of mathematics uses a textbook. An average teacher uses it as "his stock in hand"
but a good teacher uses it as "a helper". The textbook that is considered "as store house of basics
information" can facilitate a teacher to do wonder in his subject:
The textbook should not be used as the only source of instructional material. It should be used as an
aid in teaching. The following are the qualities of a good math textbook:
I] A mathematics textbook should be written in accordance with the aims and ob
jectives of teaching the subject in that particular class.
2] It should be well illustrated.
3] There should be diagram and figures wherever needed.
4] The textbook should be written in simple and understandable language.
5] It should be free from mistakes.
6] It should be written within the grasp of the children.
7] It should provide sufficient materials to motivate the students to solve problems.
8] The students should get adequate opportunity of learning through initiative and independent
efforts.
9] The problems should relate to the real life needs and physical & social environments of the
learners.
10] It should foster the right attitude towards self-study and self-reliance among pupils and it should
be done by promoting project works, field works and laboratory works.
II] It should promote the use of analytic, synthetic, inductive-deductive, problem-
solving and heuristic approaches to teaching.
12] The content should be up-to-date.
13] The exercises should aim of all level of students. It should be challenging for
intelligent students and should give opportunity for average & below average
students also.

14] It should satisfy individual difference in students and should meet the varying abilities,
interest and attitudes.
15] It should promote logical and psychological arrangement of contents.
16] It should provide for practice, revision and satisfy the demands of examination.
17] Also the textbook should be appealing and should have the necessary external qualities i.e., its
get-up, paper and printing, etc., should be good.
EXERCISE
1. Which is the best method of teaching mathematics according to your option? Support your preference
with arguments.
2. Illustrate and discuss the inductive-deductive methods of teaching mathematics.
3. Heuristic spirit is the golden rule which a teacher of mathematics should never forget, whatever
method of teaching he may adopt?
4. How will you employ the project method for the teaching of mathematics?
5. What do you understand by problem method? What is its scope in the teaching of mathematics?
How will you employ it?
6. What do you understand by the term "teaching aids"? Discuss their need and importance.
7. Discuss the role of teaching aids in the teaching of mathematics of the school stage.
8. Give your views regarding the development of low cost improvised teaching aids in the subject
mathematics.
9. Write the advantages of lecture method.
10. Write the disadvantages of lecture method.
11. Write the purpose of demonstration method.
12. Write the criteria for a good demonstration method.
Unit V EVALUATION AND ANALYSIS OF TEST SCORES
5.1 Different types of tests in Mathematics, achievement, diagnostic, prognostic
5.2 criterion and norm referenced evaluation
- Construction of achievement test
- Continuous and comprehensive evaluation
- Formative and summative assessment
- Grading pattern
5.3 Statistical measures
- Mean, median, mode, range, average deviation, quartile deviation,
Standard deviation
- rank correlation
- Graphical representation of data
- Bar diagram, Pie diagram, Histogram, Frequency Polygon, Frequency
Curve and Ogive curve.

INTRODUCTION
measurement is an important feature of our daily life. From birth to death, almost every aspect of
our life is touched by measurements in its natural form. Measurement enters into all branches of
science. Measurement is an effective in physical sciences, biological sciences, social sciences as in
applied sciences. For that reason measurement is indispensable in the domain of education.
What is Measurement?
Measurement of any kind is determining how much or how little, how great or how small, how
much more than or how much less than. In the words of James. M. Brad field "Measurement is the
process of assigning symbols to dimensions to phenomena inorder to characterize the status of a
phenomena as precisely as possible". However measurement in education is more complex than
measurement in physical situation. Educational measurement involves the mental processes of the individual
which are not visible and which are interpreted in terms of the behaviour of the individual in certain
situations.
What is Examination?
An educational examination may be defined as the assessment of a persons performance, when
confronted with a series of questions, problems or tasks set in order to ascertain the amount of knowledge
that he has acquired and the extent he is able to utilize or the quality and effectiveness of the skills he has
developed. The examination is intended only to focus at the tangible and easily measurable objectives of
education like knowledgeand skill. As an increasing emphasis was given in educational philosophy to
other goals of educational effort satisfaction mounted with the existing system of examinations. Attempts
were accordingly made to release examinations and to replace by a wider and more encompassing
concept of evaluation.
Evaluation:
evaluation in general is an act or a process that allows one to make a judgment about the desirability
or value of a measure, evaluation in educational situation is thus a relatively new term introduced to
designate a more comprehensive concept of measurement than is implied in conventional tests and
examinations, the emphasis in evaluation being upon broad personality changes and more objectives of an
educational programme and therefore include not only subject matter achievements, nit also attitudes, ideals,
ways of thinking, work habits and personal and social adaptability. Thus evaluation is not just a testing
programme. Tests are but one of the many different techniques that may contribute to the total
evaluation programme. Evaluation is any systematic continuous process of determining
♦ the extent to which specified educational obj ectives previously identified and defined are attained.
♦ the effectiveness of the learning experiences provided in the class room.
♦ how well the goals of education have been accomplished.
Thus evaluation is integrated with the whole task of education and its purpose is to improve
instruction and not merely to measure its achievement. In its highest form evaluation brings out the factors
that are inherent in student's growth such as proper attitudes, habits, manipulative skills, appreciations and
understanding in addition to the conventional acquisition ofknowledge.
evaluation is the process of finding out the extent to which the desired changes in behaviour have
taken place in the student. It differs form the concept of measurement in the sense that evaluation is more
comprehensive. Measurement consists of rules for assigning number to attributes or characteristics of
behaviour where as the evaluation aims at providing detailed and comprehensive meaning and
interpretation to the bahavioural attributes of a learner. It expresses quantitative as well as qualitative
description of learners' performance. Te purpose of evaluation is different at various stages of instruction.
Prior to beginning of instruction, the assessment of the learners' present achievement should serve the
basis for selecting and formulating instructional objectives and then for planning appropriate learning
experiences. The evaluation also helps the teacher to know how effective the instruction has been in
helping learners to master the "instructional objectives"

Effective evaluation of student's achievement with respect to accepted and planned objectives of
instruction is considered an indispensable aspect of good teaching. Teachers use various evaluation
procedures i.e., tests (oral and written), practical, assignments, observation, interview etc., for assessing
and monitoring the progress of the students achievement in scholastic and co -scholastic areas. These
evaluation procedures and techniques have become an integral part of the instructional process and
influence students in many ways. One of the functions that evaluation serves is to enable students to
determine how well they are learning and achieving. When students are aware of the learning progress, their
performances will be superior to what it would have been without such knowledge. The purpose of
evaluation
♦ To provide information for grading, reporting to parents and promoting students.
♦ To evaluate the effectiveness of a single teaching method or to appraise the relative worth of
several methods.
♦ To motivate the students.
♦ To select students.
 ♦ To evaluate the entire educational institution and to show how several of its aspects could be
improved.
♦ To collect information for effective educational and vocational counseling.
5.1 TYPES OF TESTS IN MATHEMATICS:
♦ Achievement test
♦ Diagnostic Test
♦ Prognostic test
♦ Criterion test
♦ Norm referenced test
Achievement test:
The term achievement is often understood in terms of pupil's scores on a certain school test. If,
for instance, a student is tested in two school subjects, say Language and Arithmetic and in one subject he
gets 70% marks while in the other 60% marks, it is understood that his achievement in language in which he
gets 70% marks is better that that in arithmetic in which he gets 60%. This is a loose way of
understanding the concept of achievement. More intelligently understood, achievement means ones learning
attainments, accomplishments, proficiencies, etc., achievement is directly related to pupils growth and
development in educational situations where learning and teaching are intended to go on.
The concept of achievement involves the interaction of three factors, namely, aptitude for learning,
readiness for learning and opportunity for learning. Besides these factors, the concept involves health and
physical fitness, motivation and special aptitude, emotional balances and unbalances. Achievement in
education, precisely speaking, implies ones knowledge, understanding or skills in a specified subject or a
group of subjects.
Achievement test constitute an important tool in school evaluation programme. It is necessary for
the teacher to know how far the pupils have attained in a particular subject-area. Pupils differ in their
attainments. In the school evaluation programme, various forms of achievement tests are used to measure
the extent of learning of the pupils.
"Any test that measures the attainments or accomplishments of an individual after a period of
training or learning is called an achievement." - N.M. Dowinie.
"An achievement or proficiency test is used to ascertain what and how much has been learnt or
how well a task can be performed, the focus is on evaluation of the past without reference to the future,
except for the implicit assumption that acquired skills and knowledge will be useful in their own right in
the future."- Super.
"Achievement tests are useful aids for diagnosing a students specific learning needs, for identifying
his relative strengths and weaknesses , for studying his progress and for predicting his success in a
particular curriculum".- Waters.
Of all the different types of examinations, achievement tests are used most frequently. Achievement
tests differ from intelligence or aptitude tests in that - former measures the quality and quantity of
learning attained in a subject of study or group of subjects after a certain period of instruction, the later
measure pupil's innate capacity for attainment or accomplishment independent of any learning. These
tests predict performance in a certain subject or group of subjects.
Functions of achievement tests
The major functions of achievement tests are:
♦ To provide basis for promotion to the next grade.
♦ To find out at the beginning of a year where each student stands in the various academic areas.
♦ In many schools, a certain grade or class has a number of sections. Achievement tests help in
determining the placement of a student in a particular section.
♦ A teacher can use achievement test to see for himself how effectively he is doing, what is
getting across pupils and what is not.

♦ To motivate students before a new assignment is taken up.

♦ Achievement tests expose pupils difficulties which the teacher can help them solve.
♦ To report to the parents the place of a student in a particular section according to the
achievement scores.
♦ To diagnose a students specific learning needs, relative strengths and weaknesses.
♦ To predict future progress and for predicting his success in a particularly curriculum.
♦ To reflect teachers effectiveness.
Advantages to the teachers and administrators:
♦ Through achievement tests, the teacher can know the general range of abilities of students in
the class.
♦ The teacher can select appropriate materials of instruction.
♦ The teacher can determine and diagnose the strengths and weaknesses of students in various
subjects.
♦ The teacher can find out gifted and backward children.
♦ Tests help to discover backward children who need help and plan for remedial instruction for
such students.
♦ Tests help to select talented pupils for providing enhanced curriculum.
♦ Through test, teachers can select students for the award of special merits or scholarships.
♦ Tests help to evaluate the extent to which the objectives of education are being achieved.
♦ They help to discover the type of learning experiences that will achieve those objectives with
the best possible results.
Types of achievement tests
Achievement tests are of two main types:
♦ Teacher-made achievement tests and
♦ Standardized achievement tests.
Teacher made achievement tests may further be divided into two categories.
1. Written or paper and pencil test
2. Oral tests
Written tests can still further be classified as: (i) Long essay type (ii) Short answer type (iii)
Objective type
Teacher-made achievement tests
Oral tests
In the past, classroom teachers relied very heavily on the oral work of pupil's in order to arrive an
estimate of the extent to which they measured the work of his course. The value of the oral examination
is quite apparent. Oral tests are the oldest form of achievement tests. These tests are mostly used in lower
classes. But, even in higher classes, oral tests or examinations are used. The viva voce as used in
graduate and post-graduate classes is nothing but oral examination. An advantage of this test is that a
large number of areas can be covered and knowledge of the student can be assessed. Another advantage
is that both the examiner and the examinee sit face to face with each other and the examiner can give a
proper turn to the test, as the situation demands. More over, it is less time consuming. However, the
chief limitations of oral tests are:
i. It is difficult to test each pupil on the basis of total curriculum
ii. It is difficult to ask same questions from every pupil.
iii. The examiner does not have any written proof regarding the pupil's attainment.
iv. Much depends upon the examiners personal choice.
Short Answer Type Test
Depending upon the length of the answer we can have short answer and long answer questions. A
question which can be answered in less than tour steps may be called a short answer question. For
example the following questions will be treated as short answer ones.
1. Prove that log(l+2+3) = logl+log2+log3
2. For what value of K will 25x2 +70x+K be a perfect square
3. In a right angle triangle ABC, AB=4cm, AC=3cm, what is the measure of the hypotenuse BC?
This form of questions can be easily related to objectives and can be made more stimulating for
the pupils. The responses will be more specific than the long answer type.

Objective Type Questions

The essay type questions require the formulation of an extended verbal answer to the question.
Objective type tests on the other hand consists of questions to which the pupil responds by the selection of
one or more of several given alternative, by giving or filling in a word of a phrase or by some other device
which does not call for an extensive written response.
FORMS OF OBJECTIVE TYPE TEST ITEMS
Simple recall tests
In this type of test in which the pupil is required to recall a response from his past experience to a
direct question, a specific direction or a stimulus word or phrase. It completely eliminates the elements of
guessing.
Examples
1. How many apples can I buy for rupees 50 if each apple cost Rs.10? --------
2. What is he formulae for the circumference of a circle? --------
Completion Type (Supply Type)
A competition test comprises a series of sentences in which certain important words or phrases have
been omitted and blanks are supplied for the pupils to fill in. There may be more than one blank and each
may be taken to count one point. This test, if carefully prepared has a wide applicability; it is likely to
measure rote memory. The scoring of competition test is a laborious job, as the responses are scattered all
over the page.
The competition item requires he pupil to complete the sentence by filling in the word or words that have
been omitted or if directs him to respond to a question by writing the answer in the blank space provided.
Because the pupil needs to decide upon his answer and then write it out. Test composed of competition
items takes longer time to administer them than other forms of objective tests.
Examples:
1. The degree of the polynomial 2x2 y+3x+2 is ---------------
What is the degree of the polynomial? ------------------
2. The formula for the circumference of a circle is -----------------
3. Area of a square is----------
4. Perimeter of a rectangle is --------
The competition item, however offers a natural form of questioning. It can be used readily with
material calling for specific information. These items are particularly useful for using mathematics and
science where the results of complex reasoning processes can be represented by or few symbols or
numbers.
Suggestions for Writing Completion Type Items
♦ If possible, use a direct question, rather than the complicated declarative sentences.
♦ The blank must call for a single specific response of the question can be answered by a unique
word, number or symbol.
♦ Avoid using statements lifted directly out of the book, since this tends to over emphasize rote
learning.
♦♦♦ In computational problem specify the units in which the answer is to be given and also the degree
of precision expected.
♦ Avoid indefinite statements.
♦♦♦ Avoid lifting statements straight form the text book. This brings into prominence rote memory.
♦ Blanks should be of uniform length.
♦ Statements should be so chosen or worded that there is only one correct response to be supplied in
the blank.
♦ Avoid grammatical clues to the answer expected.
For example
1. Give the value of 5 correct to 3 decimal places.
2. What is the area of the cloth in square meters needed for a conical tent of height 4m, radius of whose
base is 175 cm?
Alternate Response or True-false Items.
An alternate response test is made up of items each of which admits only two possible responses.
The usual form is the familiar true-false, right-wrong, correct-incorrect, yes-no test. It is adapted to the
testing of simple facts ideas and concepts. Scoring is easy and objective. True -false item requires the pupil
to express his judgment of a given statement by indicating true or false, yes or no, correct or incorrect,
right or wrong or some similar response.
True false item should only be used when a simple statement is either completely true or
completely false. Since only a small percentage of important items in most areas of
learning meet this criterion, the number of true false items which can be asked is limited. True false
item is a special type of multiple choice question having only two choices.
Example
1. The quantity denoted by 11,800 can also be written as 11x103 (T/F)
2. If the base angles of a triangle are equal, the sides opposite to them are also equal (T/F)
In Constructing True-false items should be taken to avoid Several Common Pit Falls
1. Avoid broad generalizations
2. Use true or false item form only for statement which are either absolutely true or false, there
should no exception.
Example
One of the co-ordinates in any point on x axis is zero (T/F) Here the exception
of origin may create confusion. A better way of asking the question would have been
At least one of the co-ordinates of any point on x-axis is zero.
3. Avoid giving clues in the statements
Words like all, always, never, only, often some times, generally, may etc., will give clues in answering
them.
Example
An angle may be equal to its complement.
Here the word "may" is likely to give a clue to a clever student.
The question could better be written as
There exists an angle which is equal to its complement (T/F)
4. The use of negative statements in true-false items should preferably be avoided.
Example
If the base angles of a triangle are equal, the sides opposite to them cannot be unequal.
It could be simply worded as
If the base angles of a triangle are equal, the sides opposite to them are also equal.
5. Every statement involved in a true-false item should be self contained.
Example
The radius of a circle whose area is 75 would be greater than 5.
The question seems to be making very bold assumptions about the units of measurement involved in it.
The ambiguity could be removed by wording it as follows.
The radius of a circle whose area is 75 sq. cm would be greater than 5 cm. (T/F)
6. Avoid double statements each true-false items should test a single concept.
7. Avoid lifting up statements from the text books.
8. Have the number of true statements equal to the number of false statements.
9. Omit specific determiners that are likely to be associated with a true or false statement.
10. Avoid ambiguous statements or statements which may be correct by one interpretation and wrong by
another interpretation.
11. Avoid statements in complex structure or figurative language.
12. Require the simplest method of response.
Multiple Choice Items
The multiple choice item consists of a stem, which presents a problems situation and several
alternatives, which provide possible solutions to the problem. The stem, may be a question, or an
incomplete statements. The alternative includes the correct answer and several plausible wrong answers
called distracters. Their function is to distract those students who are uncertain of the answer. This test is
made up of a number of items each which carries two or more responses out of which only one is
correct or definitely better.
The following items illustrate the use of both the question form and the incomplete statement
form of multiple choice items.
Example
1. Which one of the following is an example of a quadratic expression?
a. X+2
b. X+Y=4
c. X2+2x+3=0
2. An example of a quadratic expression is
a. Y-t-0
b.X+Y=4
c.X2+2X+3=0
d.YH4Y+6
3. An example of a quadratic expression is
a.X+2 b.X+Y=4 c. X2+2X+3=0 d.Y2+4Y+6
4. Correlation lies between
a. 0&+1 b. -1 &+1 c. -1&0 d. -2 &+2

The alternative in the above examples certainly only one correct answer and the distracters are
clearly incorrect. Multiple choice items typically include either four or five choices. The larger number
will of course reduce the student's chances of obtaining the correct answer by guessing.
Rules for Constructing Multiple Choice Items
♦ Design each item to measure an important learning outcome.
♦ Present a single clearly formulated problem in the stem of the item.
♦ State the stem of the item in simple, clear language.
♦ Put as much of the wording as possible in the stem of the item.
♦ Avoid repeating the same material over again in each of the alternatives.
For example
Hexagon is defined as
a. Polygon with four sides
b. Polygon with five sides
c. Polygon with six sides
d. Polygon with seven sides
This could more elegantly be put as a hexagon is a polygon with number
of sides equal to
a. 4 b. 5 c. 6 d. 7
♦ State the stem of the item in positive form whenever possible.
♦ Emphasis negative wording whenever it is used in the stem of an item.
♦ When negative wording is used in the stem of an item, it should be emphasized by
underlining or capital letters or by being placed near the end of the statement.
Example
All of the following are quadrilaterals except
a. Rectangle
b. Square
c. Parallelogram
d. hexagon
♦ Make sure that the intended answer is correct and clearly best.
♦ Make all alternative grammatically consistent with the stem of the item and parallel in form.
♦ $he correct answer is usually carefully phrased so that it is grammatically consistent
with the stem. A general step that can be taken to prevent grammatical inconsistency is
to avoid using the article 'a' or 'an' at the end of the stem.
Example
In the expression 3x, x would be called as
a. Root
 b. Base
c. Radical
d. Exponent
Here the right answer is the only alternative that correctly follows the article 'an' One could
easily modify the question as In the expression 3x, x would be called
a. a root
b. a base
c. a radical
d. an exponent
Avoid verbal clues which might enable students to select the correct answer or to eliminate an
incorrect alternative.
Some of the verbal clues commonly found in items are
a. Similarly of wording in both the stem and the correct answer.
b. Stating the correct answer in text book language.

c. Stating the correct answer in greater detail.

d. Including two responses that have the same meaning.
♦ Make the distracters plausible and attractive.
♦ Vary the relative length of the correct answer to eliminate length as a clue.
♦ Avoid use of the alternative 'all the above' and use 'none of the above with extreme caution.
♦ Vary the position of the correct answer in a random manner.
Matching type items
A matching type typically consists of two columns, each tern in the first column to be paired with a
word or phrase in the second column upon some basis suggested. Usually, the number of items in the two
columns is exactly the same. In the words of Lindquist, "The matching exercise is particularly well
adapted to testing in who, what, why and when types of situations, or for naming and identifying
abilities." In fact, matching comprises an economical form of having a number of multiple-choice items
in the same question.
The matching type is simply a modification of the multiple choice form. Instead of testing the
possible responses underneath each individual stem, a series of stems, called premises is tested in one
column and the responses are tested in another.
Rules for Constructing Matching Items
♦ Include only homogeneous material in each matching item.
As far as premises and responses are concerned they should be homogeneous with in a single item.
They should be related to the same concept more or less. If for example they deal partly with algebra
and quadrilateral, the students will get a number of clues in the item and it will serve no purpose.
♦ Keep the lists of items short and put the brief responses on the right.
The list of premises and responses should be relatively small. It is difficult to provide long lists and
to maintain homogeneity at the same time.
♦ Use a larger or smaller number of responses than premises and permit the responses to
 be used more than once.
The two columns should seldom be the same length so as to provide perfect one-to-one matching.
Preferably the number of responses should exceed the number of premises by two or three. This is
done to control the chances of guessing and having the test answer by the elimination itself. He can
attain the same effect by permitting some of the responses to be used more than once in the same
time.
♦ Specify in direction the basis for matching and indicate that each response may be used once, more
than once or not at all.
The basis on which the matching is to be done should always be made clear, if it is not obvious. This
can be done by making the directions clear and specific and also by providing appropriate column
headings.

Example
Consider the following matching question which does not follow some of the rules stated above.
Directions: Match column I with Column II by writing appropriate numbers in the blank spaces.

COLUMN 1 COLUMN II
1 Isosceles triangle A Diagonals neither bisect nor are at right angles
2 ■are B is bisect but are not at right angles
3 Rhombus C Dia>.;. e at right angles but only one is
bisected by other
4 Quadrilateral D Diagonals bisect at right angles but i ;
5 Parallelogram E All the four sides and for angle s are equal
6 Kite F Two sides arc equal
■:: Equilateral triangle G All t ies are equal
■'/.:
One can easily see a number of defects in the items. At the out set, the directions are two
sketchy to clarify the basis of matching to the students. The number of premises and responses is the same
to provide one-to-one matching. The homogeneity is test by mixing together the properties of triangle
and quadrilateral.

Limitations of Matching Tests

♦ It is not well adapted to the measurement of understanding as distinguished form mere memory.
♦ With the exception of the true-false test, the matching test is form most likely to include irrelevant
clues to the correct response.
♦♦♦ Unless skillfully made, it is time-consuming for the pupils.
How to improve Matching Test Items:
♦ Include only homogeneous or related materials in each matching type exercise. Dissimilar
matches such as persons and events, dates and events, terms and definitions should not
 be mixed together.
♦ Indicate clearly the basis for the matching. ♦♦♦ Avoid
making the test too easy.
♦ Be sure that exercises do not indicate clues to matching pairs.

♦ The complete matching exercise should appear on the same sheet and be not carried over to the
next pages.
♦ The matching test should contain at least five and not more than fifteen items.
♦ The items in the response column should be arranged in a systematic order.
General considerations for Writing Objective Type Tests
♦ Rules governing good language expression should be observed.
♦ Difficult words should be avoided.
♦ Text book wording should be avoided.
♦ Ambiguities should be avoided.
♦ Items having obvious answers should not be used.
♦ Clues and suggestions as far as possible should be avoided.
♦ Items that can be answered by intelligence only should not be included. ♦♦♦
Quantitative words rather than qualitative one should be used.
♦ Catch-words should not be employed.
♦ Items should not be inter-related.
♦ Response positions should preferably be aligned.
Advantages of Objective Type Tests
♦ Objective type tests have the major advantage of extensive sampling. A test which
contains a hundred or more items, will adequately sample pupil achievement for many
purposes.
♦♦♦ Objective test items are so stated that usually the answers to them are brief and only one correct
answer is possible for a certain item. There should be no disagreement between different persons
scoring the items at the same item of different examinees.
♦♦♦ Generally, objective tests can be scored on the basis of the key. This involves less time. Moreover,
the pupil can record his response definitely and briefly in minimum of
time.
■
♦ Since the pupil is to give correct response which is correct for all times, the pupil cannot bluff the
examiner as may happen in case of essay-type tests in which sometimes it is not possible for the
examiner to read the whole answer.
♦ In essay-type tests, sometimes, even good students cannot do well because their writing speed fails
them. Objective-type tests eliminate the fact that one pupil can write more material than another in
the same length of time.
Possible Limitations of Objective Type Tests
♦ Objective tests measure only factual knowledge. They fail to measure intellectual skills, such as
ability to interpret, ability to analyse critically and ability to solve problems.
♦ Objective type test items are often ambiguous, particularly for the better students, and
therefore penalize them.
♦ Objective type tests have a negative effect on teaching, since they encourage the student to learn
small bits of knowledge rather than broad understandings and since they discourage writing
efforts on the part of the students.
♦ It needs careful thought and enough time to write objective type tests. All teachers cannot prepare
highly objective items.
♦ Objective tests are quite valuable when they are available for class room use in printed or in
mimeographed form. But this would mean considerable cost which most schools cannot afford.
♦ Objective type tests encourage the student to engage in guessing. Once this habit is deeply
engrained, it will be hard to change and this may very well yield serious consequences.
Construction of Achievement Test
A test can be made an effective instrument of evaluating the achievement of objectives, the content and
the learning activities. We should not evaluate only the content; we have to 3evaluate the total behaviour
of the pupils. For this, the achievement of the objectives of all the three domains (cognitive, affective and
psychomotor) has to be evaluated. Tests can be conducted at different times during a course.
They are:
♦ At the end of teaching a daily lesson;
♦ At the end of teaching a unit;
♦ At the end of the term;
♦ At the end of the year or curriculum.
The time factor for testing depends upon the nature of the objectives to be tested. Knowledge
objective can be suitably tested either in the course of teaching or at the end of teaching a lesson. The
objective of understanding or comprehension requires a comparatively longer period for its testing.
Objectives of application and skill require a sufficiently longer period for their testing. The long ranged
objectives, such as interest, attitude and appreciation take a lot of time for their development. These
objectives can well be tested at the end of the year or when the curriculum has been completed.
Teachers should exercise their discretion in testing the objectives, keeping in mind the nature of
objectives in relation to the time factor. They should change the weightage of the objectives to be tested
at different times. Testing, if properly done, leads to good learning. Testing should be pre-planned,
systematic and scientific. Planning of lessons will give the teachers a full idea of how much weightage is
to be given to the content, the objectives, the form of questions, etc., while planning the tests.
The preparation of a good test is a systematic process having well defined stages. The important

steps envisaged in the preparation are as follows:

1. Planning of the test
2. Writing the test items
 3. Reviewing and editing
4. Arranging the items
5. Providing directions
6. Preparing the scoring key and marking scheme
7. Administering and scoring the test
1. Planning of the test:
Test planning encompasses all of the varied operations tat go into producing the test; but it must also
involve careful attention to item difficulty, to type of items to directions to the examiner.
a. Preparation of design:
Designing is the first and most important step in the construction. It is at this stage that we plan to
build in the test important qualities-Validity, Reliability, objectivity and practicability. In order to
accomplish that, the test constructor has to take a number of decisions regardingthe selection of
objectives, content, form of questions, the difficulty level of test items and the weight ages to be allotted
to the objectives to the comment and the forms of questions.
b. Identification of the objectives and allotting weightage to the objectives:
Identification of the instructional objectives and stating them in terms of specific observable
behaviors. After the objectives are identified and stated the test maker has to decide the relative weight
in the test. The important fundamental principal to be observed here is that the test should reflect the
actual emphasis being given to various mental process enduring instruction. There cannot be any cut and
dried formula for assigning weights to various objectives.

Table showing the weightage allotted to the objectives:

S.No. Objectives Marks Percentage

1. KNOWLEDGE Alloted
58 20
2. UNDERSTANDIN 10 32
3. G APPLICATION 2 40
4. SKILL 8
TOTAL 25 100
------------
c. Selection of the content and allotting weightage to the content:
It becomes very necessary to decide the weights to be given to different parts of it. As the whole
syllabus cannot be covered through any single test a convenient number of units can be selected for
testing. When this is done a decision about the weights to be given to those units has to be taken so as to
represent the actual emphasis on them in instruction. In assigning relative weights to units a number of
factors will have to be taken into account. How important is the unit in the total learning experience? How
much time was devoted to it during instruction? Although there are a number of such considerations, the
easiest method is to decide the weightages on the time required to teach various units.
Example
Table indicating the weightage given to three units namely polynomials, functions and
quadrilaterals.

S.No Unit Marks given Percentage

. 1. Polynomials 10 40
7 Functions 8 32
3. Quadrilaterals 7 28
Total 25 100

e. selection of the form of question and giving weightage to the questions:

Decide about the form of questions to be used, the number of questions to be chosen and the
relative weightage to be given to each form.
Example
The weightages to different forms of questions could be as follows.

S.No. Fonn Murks given Percentage

1. Long b9 24
*> answer(L.A) 10 36
3.
Short 40
Total 25 100

f. distribution of difficulty level:

A decision also has to be taken concerning the distribution of difficulty level. The distribution of
difficulty level in a test will depend upon the purpose of the test as also on the group of students for whom it
is designed. To get optional discrimination through a test most of its question should be of average
difficulty level. A few questions here and there may be easy and difficult.
Example

S.No. D ; level Percent;

i difficult 15
z. questions 70
3. average 15
Total 100

g. preparation of Blue print:

Preparation of the Blue print refers to the final stage of the planning of a test. It is a three
dimensional chart showing the weightage given to the objective, content and the form of questions in terms
of marks. It is also called as a table of specifications as it related outcomes to the content and indicates
the relative weight given to each of the various areas. The blue print helps to improve the content validity
of teacher made tests. It defines as clearly as possible the scope and emphasis of the test; it relates
objectives to the content; it provides greater assurance that the test will measure learning outcomes and
course content in a balance manner. The major and final responsibility will be that of the teacher, who the
decision maker is. To be of the utmost benefit, it should be prepared well in advance. It would thus
assist the teacher in organizing his teaching material, serve as a monitoring agent, and help keep the
teacher form straying off his instructional track.
Example
Objectives Knowledge „.__.. ----- --- Application Skill Total
■
Content L.A S.A 0 L.A S.A 0 L.A 1!A- 0 L.A S.A 0
Polynomial 1(2) 3(1) KD 3(1) 1(1) 10
Functions 1(2) 3(1) 3(1) 1(2) 8
Quadrilateral KD 4(1) 2(1) 7
Total 8 10 2 25
W*i
-
The number inside the bracket indicates the number of questions and the number outside the bracket
indicates marks allotted to each questions.
Scheme of options:(choices)
There are two types of options: They are external options and
internal options
In case of external options, there are various methods.
♦ The students are asked to attempt any six out of the given ten questions. This is over all options.
♦ The students are asked to attempt any tow from the first section, any four from the second section and any
fiver from the third section. This is section wise options.
♦ The students are asked to answer either one question or the other alternate question. This is question wise
options.
In case of internal options, the option is given within a question for example; write an essay on any two of
the following.
Options should not be given, because the purpose of a unit test is to have complete evaluation of the
achievement of objectives of the content and the learning experiences. This purpose is not served if options are
given. It is extremely difficult to set two alternative questions or two alternative groups of questions, which
measure the achievement of the same objective, the same content or the same learning experiences. The scheme
of options may develop some wrong practices among students. They may omit some content. The scheme of
options in a test paper influences even the teachers training.
Sections in the question paper
Depending on the type of question, the question paper can be divided into sections. One section
can include essay type questions, another section short answer type and the third section objective
types. Separate marks for each section should be mentioned. For each section, separate instruction also
has to be given.
2. Writing the test items:
Write the test items according to the table of specifications. He should take up each cell of the blue
print and draft an item taking care of the various dimensions the objective, the content and the form as laid
down in the blue print.
While teaching a lesson, the teacher can prepare questions also. Continuously this work has to
be done. When the teacher asks questions in the class room or when the teacher conducts a test, the
students respond to them. From the responses of the students multiple choice items can be coined.
Depending on the concept, the teacher teaches in the class, different types of questions can be
prepared in the sub concepts. The same item can be presented in different forms and kept ready to be
used at any time. So question banks should be prepared by the teachers of the same subjects. Newly
prepared questions can be added to the existing questions in the question bank every day, every month
and every year.
If an item bank is ready and if a blue print is ready, the only job that remains to be done by the
teacher is to select test items form the item bank in accordance with the design of the blue print. A lot of
effort is needed to coin better test items. It calls for a mastery of the content and practice in wording
test item. But once they are ready, editing a question paper will become easier.
3. Grouping of test items
While grouping test items, the following points are to be considered.
♦ All the obj ective type items should be grouped in one sections, while the short answer in the next
section and essay type items should be grouped in another sections.
♦ In the section of objective type items, the items having the same format, examples yes-no type, true-
false type, matching type, multiple choice type should be grouped together.
♦ Items in each section should be arranged in order of their difficulty, as far as possible.
Sections in a question paper
Generally, objective type items are grouped under section A and short answer type questions
under section B and essay type questions under section C.

4. Reviewing and editing:

The pool of items for a particular test after being set aside for a time can be reviewed by the help of
experts. The following are to be considered:
1. Does each item measure an important learning outcome included in the table of specification?
2. Is each items appropriate for the particular learning outcome to be measured?
3. Does each item present a clearly formulated test?
4. Is the item stated in simple, clear language?
5. Is the difficulty of the item appropriate for the students to be tested?
 6. Does each item fit into one of the cells of the blue print?

5. Arranging the items:

The items should be arranged so that all items of the same type are grouped together. The items
should be arranged in the order of increasing difficulty. It may be desirable to group together items
which measure to same learning outcome or the same subject matter content.
6. Providing directions:
The direction should be simple and concise and yet contain information concerning each of the
following:
Purpose of the test, time allotted to complete the test, how to record the answers, whether to guess when
in doubt about the answer and marks allotted for each question. General instructions may be given at
the beginning of a question paper.
e.g: This paper consists of A, B and C sections.
♦ Answer any two from section A: answer any four form section B and all questions are compulsory
in section C.
♦♦♦ Specific instruction related to each section may be give at the beginning of each section.
e.g: Answer toe section A should be given on the question paper itself.
e.g: Answer to each question under section BN should not exceed one page.
7. Preparing the scoring key and marking scheme:
It becomes necessary to prepare some other important accessories in the form of scoring key for
objective type questions and marking scheme for supply type questions.
Scoring key refers to the prepared list of answers to a given set of objective type questions. The
examiner simply compares the answers given by the students with these in the scoring key and thus
arrives at the marks to be awarded to the students.
A marking scheme is essential in the case of short answer and essay type items: the following are
the important items in a scheme.
♦♦♦ Points or steps expected in the answer.
♦ Description of each point or step expected in the answer.
♦ The weightage to each of these points or steps.
Advantages

♦ A marking scheme helps the examiner to bring about a uniformity of standard in assessing and
there by increases the objectivity of the est.
♦ Many examiners may be judiciously involved in assessing answer books. It will be of help to
them.
A scoring key is essential in the case of objective test items. In the scoring key, the item wise
correct response in terms of tits number is to be mentioned.
Marking scheme in mathematics will be desirable not only to analyse the solution into important
stages and to distribute marks over them but each stage my be looked upon from the point of view of the
method involved as also from expected accuracy. The marks for each stage therefore may be divided
into two components; marks for the method and those for accuracy.
■
REVIEWING THE QUESTION PAPER
The question paper is to be reviewed because this will enable the paper setter to remove any
flaws in the question paper.
■
a. Question wise analysis
b. Critical evaluation of the test
c. Item analysis
a. Question wise analysis
This is done in order to know the strength and weakness of a question paper more thoroughly;
totally the question paper with the blue print; to determine the content validity of a test and to give
satisfaction to the paper setter that the paper exactly agrees with the design.

Each question is analysed into objective, specification, content type or form of questions,
estimated difficulty level, approximate time required and marks allotted. These details are generally
collected in a tabular form.
b. Critical evaluation of the test
Before printing, the question paper should be subjected to a critical revision by subject experts
who may be able to discover its shortcomings. The following points should be considered while
evaluation of the test.
♦ The directions to students should be as clear, unambiguous, complete and concise as possible.
Nothing should be left for guess work.
♦ There should be an adequate coverage of all the topics, specifications and learning activities.
♦ There should not be any items that test only trivial points.
♦ All the criteria that are necessary, while preparing the obj ective type test items, should be fulfilled.
♦ Similar types of items should be placed together to speed up scoring ad evaluation and enable
students to take full advantage of the mind set imposed by a particular item format.
♦ The test items should be graded, that is, arranged in order of their estimated difficulty level.
♦ Classroom tests are power tests. The time allotted should be such that at least 95 percent of the
students in the class should be able to completer the test within the stipulated period of time.
♦ A marking scheme and scoring key should be kept ready.
a. Item analysis
After an achievement test has been constructed and administered in the classroom, the scores
obtained by the students maybe taken on their face value and recorded to suggest which students are good,
which average and which stand low in merit on the combined list. One may leave the results as such,
without bothering about the suitability and stability of the various test items included in the classroom
testing programme of pupils achievement. But, on the contrary, it is bound to pay rich dividends if the
relevance of each test item is obtained, that is, if the relative difficult and discriminating power of the test
item is ascertained.
The process of determining the relative difficulty and discriminating power of the test item is
known as item analysis.

Item analysis is an analysis of responses made to teacher made tests by pupils in the class, an
inspection of the percentage of students who correctly answered each item, the process which reveals
areas in which instruction was especially good or especially poor. The process of item analysis is best
applied to objective type test items; it can also be applied to essay type tests, provided they are scored by
the analytical method, which is a difficult process.
Functions of Item Analysis
♦ Item analysis provides valuable information concerning student performance.
♦ Item analysis provides the effectiveness of teachers.
♦ It also describes about the test-item characteristics.
♦ The primary purpose of the item analysis is to improve the quality of the test item for further
administration.
♦ The second purpose of considerable importance is to study the strengths and weaknesses in the
academic achievement of the students who have responded to the test item.
♦ Another advantage of the items analysis is that by studying the nature of the incorrect responses, the
teacher is able to gain a superior view of the relative position of each student with regard to the
material being tested.
♦ The immediate purposes of an item analysis are to determine the difficulty and discrimination of
each item.
♦ When an item analysis is performed on a test, one is almost certain to gain additional important
insight into the examinees thinking, understanding and test taking behaviour.
The process of item analysis should improve an instructors skills in test construction beyond that
possible otherwise.
Item Difficulty and Discrimination Power
The two important characteristics desired for a test item are its difficulty level and
discriminating power.
Most test experts believe that satisfactory evidence concerning these characterists can be
obtained by considering the performance of only a portion of the group, namely, those who performed very
well on the total test(the high group) and those who performed very poorly(the low group). For statistical
computation, the high group comprises the upper 27% of the total group and the low group, the lower
27%. The computations of difficulty and discrimination indices are based on these two fractions of the
total group.
Item Difficulty
The difficult level of a test item means how well the test item distinguishes between students who
know more and the students who know less.
In the words of Prof. Ahmann, "By test item difficulty is meant the percent of students who
correctly answer to a given test item."
 The information regarding the test item difficulty can be of great help to the teachers if his test is to
"yield scores on the basis of which students are to be ranked in terms of their achievement". Such
information also provides some indication of "the extent to which the item is doing its job."

The following formula is used to determine the difficulty level of a test item: V »,

where p is the item difficulty index

Nr is the total number of students in the combined high and low group who give correct answer to the
test item.
Nt is the total number of students in the combined high and low groups.
The difficulty level index may range form 0 to 1.
Administering and scoring the test
This is the last step. The main thing is to make certain that all students know exactly what do and
then to provide them with the most favorable conditions to do it. After the administration of the test, the
scoring can be done with the help of the scoring key and marking scheme.
Standardized Achievement Tests
A standardized test is a test which comprises of carefully selected items, having been given to a
number of samples or groups under standard conditions and for which norms have been established after
careful evaluation. It is a specially designed test keeping in view the instructional objectives. It is the
product of the joint efforts of a number of persons, including test experts. It is produced by some test
publishing agency. The standardized tests have high validity, reliability and discriminating power.
Merits of Standardised Tests
♦ The standardized tests are accompanied by carefully established norms. Norms provide basis for
comparing the achievement of pupils in the same age and grade. Pupil's achievement may also be
compared with national norms to get a more realistic picture of the achievement of pupils.
♦ Standardized tests are constructed after carefully defining instructional objectives. They possess
greater objectivity.
♦ Test experts devote considerable time in their planning, try out and evaluation. Thus these tests are
superior in all technical aspects.
♦ Since items of standardized tests are well written, these tests possess high degree of validity and
reliability.
Comparison between standardized and Teacher made Achievement Test

Characteristics Teacher Made Standardized Achievement

Achievement Tests Test
Sampling of content Sampling and content are Content is determined by experts
both determined by after extensive investigations of
classroom teacher. existing syllabi, text books and
programmes; sampling of content
is one systematically

Construction Constructed in a hurried and Constructed after carefully

haphazard manner; often defined instructional objectives;
there is no blue prints, no involves blue print, item tryouts,
item tryouts or item analysis item analysis and revision.
and revision.
rrms Only local classroom norms Besides local, national, school,
are available. district norms are available.

^ministration and No uniform directions in Specific instructions standardize

'ring this regard are available. administration and scoring
procedures.

poses and Use Best suited for measuring Best suited for measuring broader
particular objectives set by curriculum objectives and for
teacher and for intra class inter class, inter school and
comparison. national level comparison.

Diagnostic Test
The Meaning and Purpose of Diagnostic Testing
Literally diagnostic testing stands for the testing and evaluation programme carried out for the
diagnosis of something. In case one has some problem regarding his physical or mental health he is subj
ected to one or the other land of testing, (blood test, urine test, stress test and blood pressure, ECG, X-ray
checking etc) for diagnosing the nature as well as roots or causes of his ailments. The results of all such
diagnoses then form part of the remedial or treatments programmes for helping the individual from
getting rid of that physical and mental health problem. Similar is the case with the diagnostic testing
and evaluation programmes carried out in the field of education. Educational efforts are aimed to bring
desirable behavioural changes for an all round development in the personality of an educand.
Most standardized and teacher made achievement test are designed to give and indication of
how far the student has progressed towards the accomplishment of specific objectives measured by the
test. These objectives, however are grouped into broad categories. They cover a broad area and result
in a total score which reflects over all achievement in the area tested. Thus the teachers can say that a
pupil is doing well in arithmetic or poorly in arithmetic, but they do not know why nor do they know
what are the concepts causing difficulty. It will identify students who are having relative difficulty in an
area but it will not identify the causes of the difficulty.
Such achievement tests serve a useful function, but inorder to help the student with a disability,
the teacher will need to analyse the nature of the difficulty and the causes for the trouble. There are
tests which have been devised of to provide information about the specific nature of pupils difficulties in
given subject areas. These tests are called diagnostic tests.
Thus diagnostic test measure some what narrower aspects of achievement than do survey tests.
Diagnostic test yield measures of highly related abilities underlying achievement in a subject. They are
designed to identify particular strength and weakness on the part of the individual child and within
reasonable limits to reveal the underlying causes.
The diagnostic test attempt to break a complex still like computation into related parts such as
addition, substraction, multiplication and division and to provide separate measures of these subskills.
Such measures can help the teacher to locate the sources of difficulty form which constructive action
can be taken. Hence we are going to diagnoise and find remedies for learning mathematics.
However many times these efforts may not yield the satisfactory results or the individual
students may not be duly benefited through such efforts resulting into one or the

other kinds of behavioural problems or educationally failure. We are alarmed when we find student turning
into a problem child or observe him lagging behind in his studies related to one or more subjects of the
school curriculum. Here comes the need of diagnosing his behaviour and state of educational progress in
one or the other subjects of the school curriculum.
Going in this way, the need of diagnostic testing arises in the subject mathematics specifically at
the time when a particular student exhibits some or the other signs and symptoms of his failure or
difficulties with regard to the learning of the subject mathematics. Why is one subjected to repeated
failure in the subject mathematics? Why is he feeling difficulty in learning a particular concept or skill
in one or the other branches and topics of mathematics? Why is he not attending the classes in the
subject mathematics? Why does he create fuss or problems in the mathematics class? Why does, he
hate the teacher of mathematics and his subject? The list may be quite exhaustive with regard to such
day to day or occasional problems faced by a teacher of mathematics with his one or more students.
Surely like a physician or psychiatrist, here he has to resort to the methods of diagnosing the
learning and behavioural difficulties of his students for chalking out some remedial programmes aiming
to help them in getting rid of their difficulties and problems.
Looking in this way diagnosting testing in mathematics may be defined as a testing or evaluation
programme carried out by a mathematics teacher for diagnosing the nature and extent of the learning
difficulties and behavioural problems of an individual or group of students along with the inherent causes
by chalking out suitable remedial programmes aimed to help them in getting rid of their difficulties and
problems.
What is Diagnosis?
Diagnosis has been borrowed from the medical profession where it implies identification of
disease by means of patients symptoms. The word diagnosis is used more or less in the same sense in
education. The only difference perhaps is that in medical diagnosis it is physical or an organic break
down that is examined, while in educational diagnosis it is the failure of the process of education or
learning that is located and attended to be remedied. We may say that educational diagnosis is the
determination of the nature of learning difficulties and deficiencies. Of course it cannot stop only at the
identification of weaknesses in learning but has to go a litter deeper to locate their causes and also
suggest remedies for getting rid of them.
Characteristics of Diagnosis
♦ Diagnosis must be made in connection with worthy objectives of a good educational programme.
s
♦ It should be obj ective, reliable and valid.
♦ It should be as specific as the desired outcomes permit and as the possibility of
localization of symptoms allows with in the limitations of practicability.
♦ It should yield results that would be comparable over a period of time and
between groups of students.
♦ It should be sufficiently precise to note progress during small units of time.
♦ It should be comprehensible.
♦ It should be appropriate to the educational programme.
♦ The person making the diagnosis must understand the educational
programme and be familiar with the fundamental problems of children.
Need for Diagnosis in Teaching
1. To identify and gain information about the learning difficulties of individual
students in a specific concept or content.
2. To provide a specific remediation or remedial instructional measures to help
the students overcome or correct the learning difficulties.
3. To individualize instruction and enhance mastery learning.

Purposes of Diagnosis
The following are the purpose served by diagnostic tests:
♦ To point out inadequacies in specific skills.
♦ To locate areas in which individual instruction is required.
♦ To furnish continuous information inorder that learning activities may be
most productive of desirable outcomes.
♦ To serve as a basis for improving instructional methods, instructional
materials and learning procedures.
LEVELS OF DIAGNOSIS

Classification:
It is the process of sorting out students into groups particularly of under achievers and low
achievers in the context of educational diagnosis. The reference point for each student should rightly be
his own expected achievement and we have to sort out the students with regard to their levels of expected
achievement. If they have not reached their levels they will be the ones who are in need of remediation.
However, if have crossed their level, we may plan some enrichment programme help them improve their
achievement further.
Finding the nature of difficulties:
In this level of diagnosis we have to pinpoint the specific areas where they experience of difficulties.
Achievement test, unit test etc, can be used for this purpose as they cover as many learning point as
possible. Do the item analysis. The analysis helps to locating the weaknesses of the students as revealed
by the test. If most of the students do poorly on a particular learning point, we get an indication that
something is wrong with the instruction relating to the point. But this type of analysis gives no ideas
regarding the causation of these errors which will have to be sought by the other means.
Aetiology:
This is the most difficult stage in diagnosis. The main difficulty of this stage lies in the fact that
test appraise only the products of learning and not the process of learning. They may establish where
the break down in learning has taken place but can seldom tell anything about the causes of it. These
causes are generally varied and complex in nature. We have to seek for them in different areas some of
which need not be connected in anyway with classroom, instruction or school. Broadly speaking under
achievement may be due to factors within the student or environment factors outside the control of the
students or a combination of the two. Most of the causes internal to the student may however be located
in the areas of scholastic aptitude, retardation of basic skills, study habits, physical factors and emotional
factors. Interviews, check lists, rating scales, questionnaire etc can be used to sort out the areas.
Remediation:
When the basis of pupils difficulty is understood, we come to the stage of applying remedial
measures. There is however, no set pattern and no cut and dried formulate for remediation. In some
cases it may be a simple matter of review and reteaching. In other, and extensive effort of improving
motivation, correct emotional difficulties and over come deficiencies in work study skills maybe
required. The hared fact is that there are no patent remedies in educational practice for two students
having the same learning difficulty may

have suffered it because of different causes and may have to be tackled differently. Moreover since each
subject has the own genius and personality, remedial programmme have to be planned accordingly.
Despite the different methods and techniques needed in remediation, there are certain guiding principles
that apply to all subject areas and provide a frame work in which the teacher can operate. Remediation
should be accompanied by strong motivational programme. Remediation should be individualized in term
of the psychology of learning. There should be continuous evaluation giving the pupil a knowledge of
results. Remediation programme may not always need a separate time allocation for them. But they will
always mean some extra work for both the teachers and the affected students.
Prevention:
Prevention is better than cure in education as else where. A programme of diagnostic testing should
help an imaginative teacher in getting an insight into the types of possible errors that are likely to occur
in learning their possible causes and the ways of preventing them in future. Thus educational diagnosis
does not and should not end at remedial measures, but also should become a means for improving
instruction modifying its curriculum and also for retaining instructional materials. The real importance is
rather in the prevention of its reappearance elsewhere under similar conditions. Any weakness
identified should afford the basis for decisions calculated to reduce the probability of their recurrence in
the future.
Remedial Teaching
 Remedial teaching employs a greater variety of instructional and learning correctives to overcome
the constraints experienced by the students during learning a particular concept or topic. It assumes that
quality of instruction could best be defined in terms of:
i. The clarity and appropriateness of the instructional cues for each student
ii. The amount of active participation in and practice of learning allowed to each student and
iii. The amount and variety of reinforcements and feedbacks available to each student to
complete his unit oflearning.
The teachers may use various instruction/teaching correctiv4es. Viz., small group problem
session, individual tutoring, alternative learning materials including textbooks, workbooks, audio visual
techniques, academic games and puzzles and re-teaching.
Small group problem sessions involving three to four students with very different learning
difficulties work best with elementary school children. While teaching these children we may observe that
they lack both independence and perseverance, which are required to complete a learning task by
themselves. As a remedial measure, small group problem sessions involved three to four students
provide a specific block of time when students are formerly constrained to attempt to complete their
learning tasks.
Individual tutoring is also an effective learning corrective remedial measure, which can be used for
correcting individual learning problems of students. It can be done either by the teacher himself or by
someone other than the teacher. It is advisable that the tutoring should be done by someone other than
the teacher who can bring new perspectives to students in the teaching learning process.
Alternative learning materials and textbooks already available or easily acquired by the school are
also useful remedial materials. You may adopt a textbook or some instructional/ learning material for course
use, which is not of consistently high quality in explaining particular points and processes. You can use
alternative learning material/textbooks to fill in these gaps and provide better explanation of difficult
learning spots.
Workbooks are most useful remedial tools especially for students who have great difficulty in
grouping ideas and processes from a highly verbal type of instruction. Workbooks provide drill and the
specific problem solving practice, the students need for learning.
Audio visual materials, viz., film strips, motion pictures, classroom demonstrations, charts and
illustrations can be used in the case of students who have difficulty in grasping material presented in a
verbal abstract instructional mode.
By diagnosing we may identify the learning deficiencies and hard spots. After this diagnosis, it
would be advisable to re teach the entire class.
All these remedial correctives are added to supplement and not to replace the original instruction and
instructional materials used in the class. They may be viewed as helping measures to be used for a
student at those particular hard points where his original instruction was not of optimal quality.
The main focus of diagnostic testing and remedial teaching is to build into the instruction the feedback
devices so as to point out deficiencies in student learning of a given instructional unit and to suggest the
alternative instructional correctives required to overcome them. Re-teaching of selected aspects of an
instructional unit, small group co-operative study sessions, individual tutoring, and use o various
alternative instructional materials and methods may prove effective and useful instructional correctives
by teachers for the students. These correctives attempt to provide each student with instructional cues, the
learning participation-practice, and the reinforcements, which are best, suited to his characteristics and
needs. With the help of diagnostic testing followed by effective remedial correction procedures. We can
transform classroom instruction of any initial quality into instruction of optimal quality for each student.
Distinguishing Diagnostic Tests from the Acheivement Tests
A diagnostic test is primarily carried out for knowing about the nature and extent of the learning
difficulties and weaknesses of a student or group of students in a particular learning area, subject, topic
or concept. Here all efforts are concentrated on the search for the areas of weaknesses (or strength in the
case of above average) and error analysis (for knowing the probable causes of weaknesses) resulting into
one or the other form of remedial teaching to help the needy students in getting rid of their weaknesses and
learning difficulties.
The search can be better made if the area and scope of this search is kept small and extensive as
well as vigorous efforts are made for such search. Similar is the case with diagnosis in Mathematics.
Here the task of diagnosis should be broken into simpler and smaller curriculum units and students
should be asked a sufficient number of questions, preferably short answer on very short answer type
including completion type objective questions for going into the depth of their areas of difficulty and
weaknesses.
With the help of what has been above said about the diagnostic and achievement tests, we can
be able to distinguish them from each other in the manner given below:
1. The achievement tests while focusing on knowing about the level of the learner's achievement may
only provide a hint or clue about the learning difficulty or weakness of a student, the diagnostic
testing goes, deeper in concluding about the nature, extent as well as reason for such learning
difficulty and weakness.
2. Sampling of questions in an achievement test need detail and not to be too exhaustive to cover each
and every minute points with regards to the coverage of the contents. In diagnosis on the other hand,
we have to go quite in depth. Therefore, here the coverage of the subject matter is more detailed and
exhaustive, though based on a smaller area than an achievement test. Diagnostic tests, have
therefore, to be much longer than the achievement test to make necessary subtests sufficiently
reliable.
3. Although the results of an achievement lest may provide a signal or warning about the poor
performance of a student yet these are unable to provide necessary base for the planning of a
remedial programme. Diagnostic testing on the other hand necessarily leads to the formulation of a
well thought remedial programme aiming to help a particular learner from getting rid of his learning
difficulties and weaknesses.
4. The performance of the individual student in a diagnostic test is subjected to a careful analysis with
regard to the common as well as specific error made by him for concluding about the exact nature,
extent and causes of his learning difficulties and weaknesses in a particular learning area. The
results of achievement tests are not subjected to such error analysis as these are primarily utilised for
measuring the levels of the achievement or performance of the students with regard to their gain in
learning during a specified period.
5. In diagnostic testing we require more detailed and exhaustive content analysis i.e. breaking of the unit
into multiple learning points and then arranging these points in a hierachical order without breaking
their continuity. We can't break this continuity in case of the diagnostic test as the test in itself will
become faulty by becoming unable to untouch the weaknesses of those students who are weak and
 deficient in respect of one or the other learning points omitted in the test. Achievement test on the
other band, do not require such type of compulsion for arranging the testing learning experiences
into some hierarchical order and emphasize upon each and every learning points by maintaining
their continuity in a hierarchical order.
A. Planning.
An appropriate thorough planning is very much essential for the construction of a diagnostic test.
Usually it may involve the following considerations:
(i) Identifying the areas of weakness or learning difficulties
First of all the need of constructing a diagnostic test should be properly identified. It may be based
on the findings of the achievement test, classroom drill and practice Work, homework and assignments,
classroom behaviour of the students etc. The performance and behaviour of the students of the class (or
a particular student) during such evaluation encounters may provide a clue or evidence of some or the
other types of weaknesses and learning difficulties 'suffered by the students (or a particular student) in
one or the other learning areas of mathematics.
(ii) Isolating a unit, sub-unit or concept for diagnosing in depth
Suppose, section B of the class IX (or one or the other particular students) has demonstrated a
quite poor performance in the subject mathematics. What are the different weak areas.
Certainly, it needs a careful analysis of the results of performance tests and academic encounter with
the students: Let us further assume that this analysis has led us to conclude that particular student or class
is lagging behind or feeling difficulty in one or some particular units of the IX class mathematics
curriculum. Now these units may work as a base for the construction of an achievement test. However
since diagnosis is to be carried out in quite detail and depth, therefore it is not advisable to take all these
units or even a full unit for the construction of diagnostic test. Let us therefore try to take only one unit,
or a sub-unit and single concept for the construction of a diagnostic test. In this way several sub-unit or
single concept diagnostic tests may be constructed and then joined and combined into composite
diagnostic test aiming to locate the learning difficulties and weaknesses of the students in the whole unit
or units of the total prescribed curriculum in mathematics.
(iii) Content Analysis
The Contents of the sub unit or a single concept should then be further analysed to determine:
♦ the pre-requisite behaviour i.e. the previous knowledge, skills etc. needed for the learning of
a particular sub-unit or a single concept.
♦ the expected behaviour demonstrated by the learner after learning the contents related to that
unit or a single concept.
(iv) Deciding about the nature of the items of the test
A proper decision should then be taken about the nature and numbers of the items of the
diagnostic test since it should be as exhaustive, detailed and lengthy as possible, hence there must be more
weightage to the short and very short answer type questions in comparison to the large and essay type
questions. It is preferable to use completion type items in the test. However, for diagnosing the actual
practical difficulties, of the students in solving the mathematical problems, a sufficient number of
problems related to the know how of the knowledge, skills and application of the mathematical
concepts should necessarily be included in diagnostic test of mathematics.
(v) Taking decision about necessary administrative measures
It is better to take decision regarding various administrative functioning of the test before sitting
for its construction like, the time limit, direction for the proper administration of the test.
B. Construction of the Diagnostic test
In view of the things planned at the planning stage now attempts should be made to select
appropriate items for being included in the proposed diagnostic test. This selection is mainly focused on
the following three aspects.
1. The nature of the contents of the sub-unit or single concept.
2. The pre-requisites behaviour in terms of the previous knowledge, skills needed for learning
the sub-unit or concept.
3. The expected behaviour i.e. knowledge, skills and application etc. acquired by the students
after studying the contents of the sub-unit or single concept.
For the illustration purpose let us choose one of the sub-units of major unit mensuration of the subject
mathematics namely "area of a circular figure" for the construction of a sample Diagnostic test.
Diagnostic Testing and Remedial Teaching in Mathematics
It can be easily concluded from the above discussion that diagnostic testing and remedial
teaching are inter-related and complementary to each other. Each is based on and results into the other.
One resort to diagnostic testing for searching the appropriate remedial instructions. However, the diagnosis
not followed by the remedial treatment is useless. Similarly remedial task not based on the diagnosis of the
nature and extent of the weaknesses may prove not only the wastage of the resources but can also prove
dangerous to the well being of the affected persons. It is therefore, essential that diagnostic testing in
mathematics should necessarily be followed by the suitable remedial 'teaching. In fact neither diagnostic
testing nor remedial teaching should ever be considered in isolation. They should form a part of a cycle
known as Diagnostic testing and Remedial teaching cycle which may be considered to involve the
following processes for its complete execution.
1. Diagnostic testing for knowing the child's weaknesses and learning difficulties in mathematics.
2. Hypothesizing the probable causes for these weaknesses and difficulties.
3. Applying remedial teaching for removing these weaknesses and difficulties.
4. Evaluating the outcomes of the remedial teaching
5. Continuing to repeat the above four processes to achieve desired success in removing the diagnosed
difficulties and weaknesses
A close analysis of the above mentioned Diagnostic testing and Remedial teaching cycle may at
once reveal that here the cycle starts With the application of its first step, Diagnostic testing—testing
undertaken for diagnosing the weaknesses or learning difficulties of an affected student (who has been
identified so as a result of achievement tests teachers observations, etc.) Remember here, the diagnostic
testing or need of administrating diagnostic tests and other diagnosis measures arises only when something
wrong and serious is reported about the progress of the child in the learning of one or the other subjects
By applying the diagnostic testing the efforts are made to know the nature and extent of ones
weaknesses and difficulties in the learning of that particular subject say mathematics
Once the weaknesses and difficulties regarding the learning of a particular concept, knowledge
and skill area etc are identified, efforts are then made to list out the probable causes responsible for
these weaknesses and difficulties.
The listing of such probable causes then may be made a base for building a remediation program. The
affects or outcomes of such remedial program or remedial teaching are then evaluated to see whether or
not we have got desired success in getting rid of the child
from his identified weaknesses or difficulties in the learning of mathematics. In case we don't get the
desired success indicating the ineffectiveness of the remedial teaching, then the diagnostic-remedial
cycle must be re-initiated and the processes of diagnosis, hypothesizing remediation evaluation etc must be
brought into operation again for the attainment of desired success in helping the child to get rid of his
learning difficulties or weaknesses in one or the other areas of mathematics, With all what has been
discussed so far, one can assume that diagnostic testing and the subsequent remedial teaching is
exclusively meant for he students who suffer from weaknesses or learning difficulties in one or the areas
of learning a subject or those who are under achievers, slow learners backward, learning disabled or
suffer from the learning or behavioural problems with regard to the teaching-learning of a subj ect.
However it is not suffice as the term diagnostic testing and remedial teaching can be equally
applied in the case of students who are gifted, creative and meritorious. Diagnosis helps in diagnosing
the weaknesses as well as strengths, therefore diagnostic testing may properly carried out for knowing
what is appreciable, original, creative and above average among the geneous creative and meritorious
students.
Based on such diagnosis, the efforts should be made to harmness on their potentialities and
maximise their strength for helping them further to grow and develope in the better way.
Therefore, it is no harm in carrying out the diagnostic testing and remedial teaching programmes
for the gifted, creative or meritorious. Rather it will be a boon for them in one or the other ways.
However, as you know necessity is the mother of invention. We try only when we feel the urgency of
trying, similarly here, we resort to diagnostic testing and remedials teaching only when the necessity of
doing such work is urgently felt. Since this urgency in mostly felt in the case of those who are weak or
feel difficulty in learning, the diagnostic testing and remedial teaching programmes in mathematics are
then usually carried out only with the students who are identified with one or the other types of learning
difficulties and weaknesses in the study of the subject mathematics. In the pages to follow thus we will
be concentrating our efforts for evolving diagnostic testing and remedial teaching strategy only for such
children.
REMEDIAL TEACHING IN MATHEMATICS Meaning and
Purpose
The term remedial teaching as the name suggests stands for the teaching or instructional work carried
out to provide remedial measures for helping the students (or individual student,) in getting rid of their
common or specific weaknesses or learning difficulties diagnosed through diagnostic testing or some
other measures carried out for such diagnosis.
Diagnosis thus provides a solid base for hypothesizing the general and specific causes underlying the
weaknesses or learning difficulties of the students of a class group (or a particular students). It is thus
true that as the diagnosis so is the remedy for the removal of the difficulty Let us think over the possible
remedial measures in the light of this very assumption.
1. In case the class as a whole is demonstrating a particular type of weakness or
learning difficulty then the treatment should be followed by a common remedial programme.
For this purpose, special classes may be organised or special methods and techniques may
be employed for making the class understand properly the concepts or skills etc. related
to the area of weakness.
2. In case the weakness or learning difficulties diagnosed are of specific nature
applicable to the individual students. Then the treatment should also be individual specific.
Mathematics is a quite sequential subject. Here the study of a topic is dependent upon the
previously learnt topics. In case the student is lagging behind in the learning of the pre
requisites essential for the study of the topic in hand, then the treatment in terms of the
remedial teaching must necessarily be given for making up the previous deficiencies in
learning. The child should be helped individually for this prupose by making use of the methods,
techniques and teaching-learning situations best suited to the individuality as well as nature
of the content.
In case the learning difficulty felt at present is not the product of the deficiencies in terms of prerequisites
or previous learning, then its roots must necessarily lie in the present set up of the teaching learning. The
results of the diagnostic testing and adequate analysis of the errors committed by the individual student
then may bring on the surface some or the other probable causes for the diagnosed weakness and learning
difficulty of the individual students. In the light of such diagnosis a proper remedial teaching programme
should be taken into hand.
Any remedial teaching programme includes three major steps:
♦ Diagnose a learning difficulty.
♦ Remedy the learning difficulty with a prescription,
♦ Encourage and support the learner all the while.
Remedial Teaching Programmes.
Let us now initiate he task of such individualized remedial teaching programme based on the diagonised
difficulties and error analysis of the responses of the students.

(i)Let us take the case of an elementary class student. He commits the errors in multiplication of
numbers in the way as.4 x l = 5 o r 5 x l = 6 , etc. Error analysis may here reveal that he is lacking in
terms of the understanding of the basic concept of multiplicative identity n x 1= n. Now the remedial
teaching programme in this case may, be proceeded as below:
(a) Use concrete aid material representing a single obj ect repeated 4 and 5 times or a single bundle of the
items containing 4 and 5 objects making the students conclude that 1 x4 or 4 x 1 = 4 and 1 < 5 or 5 x 1
= 5.
(b) Making the students practise as below
4x =4 5x =5 4x1 = 5x1 =
x 1 =4
x1=5
(ii) Let us take the case of another students who has been diagnosed to feel difficulty in multiplying a (b
+ c) or (a + b) (c + a). From such diagnosis it is natural to conclude about the ignorance of the student for
the distributive properly of multiplication over addition [(for any three number a, b and c,
a (b+ c) = ab + ac]. Let us further assume that diagnosis in this case also reveals his ability to
identify this property with whole numbers, through his written response as 5x47 =5 x (40 + 7)
= (5 x 40) + (5 x 7) = 200 + 35 = 235

It may now lead us to conclude that, in the present case the student is feeling difficulty either
on account of the weakness in the programme of his algebra instruction or due to his own inability to
transfer the knowledge and skill acquired for using distributive property with whole numbers to
algebraic expression.
Thinking on this line a remedial teaching in this case is to be aimed to help the student to
generalize the distributive property of multiplication over addition from whole numbers to algebraic
expressions. Accordingly the remedial instructional work may be planned through the following
comparison table to observe the step by step analogical connection between operations with whole
numbers and algebraic expressions
(i) 5x47 (i)a(b + c)
(ii) 5 x (40 + 7)
(ii) a x (b + c)
(iii) (5 x 40) +
7) (iii) (a x b )+(a x c)
(iv) (5
(200) + (35)
(iv) ab + ac
(v) 235
(v) sum of ab and ac.
The knowledge of the general applicability of the distributive property of multiplication over addition,
then may be further strengthened by helping the students to leam about (a + b)(c + d)=ac + ad+ be + bd,
through the step by step comparison of the process with the multiplication of whole numbers like 35 x 47
[expression (30 + 5) and(40 +7)].
(iii) While responding to the item 2 of the part p of the diagnostic test on "The area of a circular
figure." let us assume that a student has committed the error in the recollection of the formula. His
answer was 27tr instead of 7ir2. Subsequent error analysis, then may give rise to the following hypothesis
for his weakness or underlying difficulty in providing correct answer:
(a) The concepts of the circumference and area of a circle are not clear to him.
(b) He has not been able to recollect the correct formula as the same was learned through
cramming i.e. memory level teaching-learning instead of understanding level teaching
learning.
 In view of the above probabilities, the desired remedial instructional work in this case may be
proceeded as under.
(i) The concept of the length of the circumference and area of a circle will be made
clear to the students with the help of concrete examples and laboratory experiments. He will be
presented with a circular disk or any solid figure of circular shape. A circle with a
given radius may also be drawn on a white paper. The student may be asked to put a dot or mark on the
circular edge of this figure and then, taking it a starting point he may be asked to roll up a piece of thread
around the circular shaped solid and then finally measure its length being termed as the length of the
circumference of the circle.
Here O is the center of the circle drawn with the length OP as its radius starting from P, the
length travelled all along the circular path passing through Q, R and S (taking the whole round-starting
from and coming back to P) is known as the length of the circumference of the circle computed by the
formulae / = 2nr.
Area of circle denotes the measurement of the total space available in the circular figure. For
the clarity of the concept about the area of a circle some concrete illustrations should be provided. How
much cloth or sun mica is needed to cover the top of a circular table? How much space in available in a
circular plot? How much coloured paper is needed to cover a circular cardboard figure etc.
The total covered space within the circular boundary of the given circular figure is termed as the
area of this figure.

(ii) The students should be made to learn about the establishment of the formula area of a circle = 7tr2
through the use of laboratory method. It has already been mentioned somewhere earlier in the text. Here the
given circular figure drawn on a piece of paper is gradually converted into a parallelogram or a rectangle
(approximately) to let the students calculate its area as length x breadth. The length of this converted
figure measures as equivalent to the half of the circumference of the circular figure i.e. nr and breadth is
measured as equivalent to the radius of the circle 'r' giving the area of the converted rectangular figure
(or the circular figure in original) as jrr x r = 7ir2. In this way the students are made to learn all about the
establishment of the area of a circle, through their own doing and understanding. Such understanding level
instruction, then may help the student to get the formulae remembered well or recollect soon (if forgotten) by
going through the process of its establishment.
 We have tried to mention about some of the remedial measures meant for removing the
weaknesses and learning difficulties of the children caused mainly on account of the lapses in the
instructional process or learning inability or deficiencies of the students. The roots of the behaviour
problems needs to be rectified thus here lies in the cognitive domain of the students. However, there may
be so many cases where for the felt learning difficulties or demonstrated weaknesses the physical as
well as emotional reasons maybe found more or even sold)/ responsible for a particular child or group
of children. The situation like this may be pointed out as under.
1. The child may feel difficulty in learning and progressing well in the subject mathematics
on account of his poor health, ailments or physical and mental handicapness.
2. A particular child or the group as a whole may suffer academically on account of the poor
physical and learning environment as well as facilities available to him/them is the school
as well as in their homes.
3. The child may feel difficulty in the path of his learning and may demonstrate serious
backwardness, deficiencies and weaknesses in terms of the learning of the subject
mathematics 'purely on account of the emotional reasons. A simple rebuff from the teacher
may cause a child to become completely disabled, intellectually and emotionally. The
vicious circle of misunderstanding once started between the teachers and taught may
develop into a bigger problem of first hating the teacher and than the subject eventually
terming a so capable child into a problem or backward child in mathematics. Sometimes
such behavioural problems are the product of the over expectations, mishandling and over
anxiousness shown by the parents about their children.
There are some cases where the parents were demanding high achievement in mathematics
from their child on account of getting him admitted in a very competitive course. They were loosing their
patience and getting extremely tense when he did not meet their standards. Eventually to save his
face, the child developed an escaping device in the shape of becoming physically ill at the time of
testing and examination.
Now in such a case the remedial programme in terms of tight mathematical instructions
making the child to work more and more can cause more emotional block to his learning of
mathematics. Therefore, here the treatment needs psychological help more than the academic caring.
The same happened to this child. The parents were told to behave in a normal way not to be too
worried about his mathematics study. The demands for high achievement in mathematics were
lessened.
The efforts were made in the school and home to love and accept him with no strings (such as
accepting or loving him only when he achieved high in mathematics) attached. Test and examination
phobia was also removed to bring positive and favourable changes in the environment. He was allowed
to team up with his close friend in test situation. In this way with in due course of time, applying
behavioural therapy, the child became normal and began to develop interest in the more serious
learning of the subject mathematics.
In this way the corrective measures for the removal of the learning difficulties and weaknesses
of the students in the subject mathematics must involve all the aspects and domains of the children
behaviour-cognitive, conative and affective (assuming that cause of their problem and deficiency may
be cognitive, physical and emotional in nature). Many times, the problems, deficiencies and weakness
do not arise on account of a single factor.
 There may be a multiple number of causes, academic, physical or emotional in character
responsible for one or the other learning difficulties and weaknesses of the students in a particular area or
learning aspects of the subject Mathematics. Therefore, a global strategy involving all the possible
corrective measures in terms of improved instructional strategy, modifying teaching, learning
environment, adjusting the teaching learning according to their physical and mental abilities as well as
emotional requirements and seeking guidance from the clinical experts should be adopted for helping the
needy students in the task of being get rid of the diagnosed learning difficulties and weaknesses in the
subject mathematics.
While planning diagnostic evaluation, the teacher should bear in mind the following points:
♦ The diagnostic tests should be prepared in accordance with the specific tasks regarding a
particular subject or area of a subject.
♦ There should be large number of items or questions covering various aspects of the relevant
subject matter for which the test is designed.
♦ The questions have to be easily administrable.
♦ Scoring of the items should be easier and quicker.
♦ Variety of questions like very short answer type and multiple choice type should be included.
♦ Different kinds of tests may be used to suit the students with different abilities.
While administering the tests, the teachers should keep in mind the following points:
♦ The teachers should make their students understand that the tests are to help
them to ^improve their learning.
♦ The tests should be administered in a relaxed environment.
♦ Students should not consult each other while taking the test.
♦ The teacher may ensure that the students attempt all the questions.
♦ Time schedule should not be enforced strictly.
Uses of Diagnostic Tests
♦ Diagnostic tests serve as guides to the attainments of students.
♦ They serve as guides to locate difficulties of students i.e., isolating difficulties of
students individually.
♦ They help in dividing students into groups for remedial teaching or special coaching
Diagnostic Tests for Reading and Arithmetic
Diagnostic tests are designed to analyse individual's performance and provide information on
the causes of difficulty. Such tests are generally administered in reading and arithmetic.
Diagnostic tests in reading range form group tests to intensive clinical programmes for
individual case study. IOWA silent reading tests are short, but widely used group test. The common
functions covered by both the batteries are:
♦ Rate of reading
♦ Vocabulary
♦ Sentence comprehension
♦ Paragraph comprehension
Diagnostic Test in Mathematical Skills
The Compass diagnostic tests in arithmetic comprise twenty tests covering different arithmetic
operations.
The diagnostic tests for fundamental process in arithmetic prepared by Buswell and John are
individual tests. Each problem in this test is to be solved orally by the subject. When the subject is
engaged in the process of solving the problem, his method can be observed. Errors as well as
undesirable work habits are recorded on a check list.
A Sample Diagnostic Test in Mathematics
Name of the unit— Mensuration
Name of the sub-unit Area of a circular figure
Pre-Requisites.
Students are expected to have previous knowledge regarding the following :
1. They are well acquainted with the circular figures in terms of naming and recognizing
their parts like centre, circumference, diameter, radius etc.
2. They are familiar with the concept of area.
3. They are familiar with the concept of circumference.
4. They can recall the formula for computing the length of the circumference of a circle.
5. They are familiar with the concept of n and are able to tell its value as 22/ 7
6. They are able to compute the length of the circumference of a circle of a given radius r
Expected Learning Outcomes
1. They will be acquainted with the concept of the area of a circular figure.
2. They will be able to recall the formula for the computation of the area of a circular figure.
3. Given radius, they will be able to compute the area of a circular figure with utmost
accuracy in their respective sq. units.
4. Given area of a circular figure, they will be able to tell about the length of the radius of this
circular figure.
5. Given two circular figures of the different diameters or radius, they will be able to tell about
the largeness or smallness of these areas.
6. They will be able to apply the knowledge and skills acquired in the computation of the
area of a circle in solving day-to-day life problems.
Let us now try to frame the items of the proposed diagnostic tests in view of above cited pre-
requisites and expected learning outcomes.
Note. Answer all the question given below at your own. Although there is no time limit yet try
to finish your work as early as possible preferably in an hour or so.
1. In the centre of the floor of a big hall there is a circular spot of the radius measuring 3.5
meters. It is to be covered by the gracious Italian tiles costing 1000/ per sq. metre. Tell
what one of the following you have to compute for the payment of the cost incurred in
installing these tiles.
(a) Circumference of the circular spot
(b) Length and breadth of the circular spot..
(c) Area of the circular spot.
(d) Volume of the circular spot.

2. Write down the formula for the computation of the area of a circular figure.
3. Compute the area of the circular spot (with a radius of 3.5 meters) in the problem 1 mentioned
above.
4. Compute the cost involved in flooring the circular spot (with a radius of 3.5 meters) with the tiles in
the problems 1 mentioned above.
5. There are two pieces of agricultural land. One is rectangular measuring 80 meter in length and 60
meter in breadth. The other one is circular in shape with a measured radius of 42 meters. Tell which
one of them is bigger in size.
6. A person wants to cover 616 sq. meter area of his farmhouse with a circular shaped spot. Tell
what will be the radius of this circular shaped spot.
7. Confirm the constancy of the value of 6 with the given data : Area of the
circle = 1386 sq. meter
Radius of the circle = 21 meter
Along with the writing of the items for the diagnostic test, attempts should be made to prepare a
scoring key and model answers for helping in its proper interpretation.
B. Administration and interpretation of the diagnostic test.
The constructed diagnostic test should now be administered to the class or individual student for
knowing about the weaknesses and learning difficulties pertaining to the learning of the sub unit or a
single concept. The necessary directions related to the proper administration of the test should here be
clearly explained or demonstrated to the students. When they have finished their task, the answer
sheets along with the test paper should be carefully collected for the analysis and interpretation of
their responses. For this purpose these may be scored on the basis of the scoring key and model
answers suggested in the constructed test.
The interpretation about the weaknesses and strength of the students related to their
performances in the learning of a sub-unit or single concept then be made on the basis of their scores
on the total test items, parts of the test or individual items. A wise step for knowing in depth the
nature of learning difficulties and weaknesses of the student lies in carrying out error analysis of the
responses given by the group of students or an individual student.
Error Analysis
The term Error analysis as the name suggests stands for the analysis of the error committed by
the students in providing responses to the items of the administered diagnostic test for the proper
diagnosis of their weaknesses and learning difficulties with the purpose of framing a suitable remedial
programme for the removal of these diagnosed weaknesses and learning difficulties.
Error analysis of the diagnostic test in four fundamental rules

S.No. The error Diagnosed weakness (as

committed a result of the error analysis)
1, 45 The student is lacking in the mastery of basic addition
+32 combinaions.
76
9 67 The student is lacking in the mastery of basic subtraction
-43 combinations.
23
3. 23 The student is lacking in the mastery of basic
multiplication combination.
x 3
66
4. 42 -T- 6 = 6 The student is lacking in the mastery of basic division
combination.
.■ o* 63 The student is in the habit of rewriting numerals without
■ +35 computing.
88
6. 567 The student multiplies by only one digit out of the two
x 14 present in the given number.
2268
7. 90 The student ignored remainder because: i. he has not .
8)721 completed subtraction ii. he is not feeling the need for
720
further computation
iii. he does not know what to do with "1" if subtraction
occurs, so does not compute further.

In this way, through careful study and analysis of the error committed by the students in
responding to the various items of a diagnostic test one can reach to some or the other most
convincing reasons (in terms of the learner's difficulty and weakness) for the errors committed.
Such type of error analysis and interpretation of the responses of the students in terms of
their weaknesses and learning difficultis then may lead, a mathematics teacher to think about the
possible remedial measures with the sole objective of helping the students to get rid of their weaknesses
and learning difficulties. In this way, the findings and results

of the diagnostic tests (coupled with some other types of surveys, results of achievement tests,
administration of interest and aptitude tests, attitude scales, case history etc.) may help him to
proceed on the path of remedial teaching.
Difference between Achievement and Diagnostic Test
Achievement Tests Diagnostic Test

The achievement test tells the teacher how much Diagnostic tesr t. •. how much content still
content the individual has learnt. he has to learn to master a specific content.

It never tells how he still has to learn in a specific Objective type questions are included.
content. But it evaluates whether not the
particular instructional objective is mastered by
the student. All types of questions like
essay, short answer and objective types are
included.
Achievement tests are meant for the average Diagnostic tests are normally meant for
students. the below average students.

Achievement tests are used to evaluate the Diagnostic tests are used to identify the
achievement level of the students. difficulties and weaknesses of the
students.
Achievement tests are used to compare Diagnostic tests lead to remedial teaching
achievement of students and also for grading. or special coaching.

These relate to the entire unit covered. These concentrate on difficult areas.

These at various occasions determine the These are not for evaluating the
effectiveness of the teacher in the teaching efficiency of the teaching or the system.
learning process.

Prognostic Test
The first of these functions is that of prognosis function. Any test tells about some differences
among peoples performance at this movement. All decisions involve prediction.

When psychological test is mentioned, so called I.Q. test (intelligence test) administered to students in
school to predict their academic performance comes to mind. The measurement provides the extent of a
variable which has the specific purpose of predicting future behaviour.
Prognosis means foretelling, prediction and forecasting. The prognostic tests are designed to
predict the student's ability or readiness to undertake the study of a school subject successfully.
Uses of Prognostic Test
The test scores gained through prognostic testing indicate not only students present level of
achievement, but they also tell or indicate about the possibility of future achievement. Prognostic tests tell
the teacher as to what extent his students can derive advantage for further learning experiences. The
prognostic testing will indicate whether a student has acquired the mastery of the subject or not if that
student wants to take up a higher course in the subject he studied.
 Several tests have been developed in India for prognosis purpose. These tests are used
generally with children shortly after their entry into the first grade. The objective at this stage is to give
the school as accurate an indication as possible of the child's ability to progress in reading. It may be
remembered that the reading readiness test is given only to predict ability to profit form reading
instruction in the near future. It is not used to forecast ultimate level of learning.
A large number of reading readiness prognostic tests have been developed. Following are the
main categories of these tests.
♦ Ability to read letters or word tests
♦ Oral vocabulary tests
♦ Rhyming or matching sounds tests
♦ Visual matching of figures, letters or word tests.
5.2 CRITERION TEST AND NORM REFERENCED TEST
In order to interpret the marks, marks are referenced to something outside the test. To forms of
marks referencing schemes are generally used. They are
1. Criterion test
2. Norm referenced test
Criterion test
It is meant to measure the achievement of an examinee on a certain domain to find out his level
of achievement in that domain. It has little to do with the achievement level of other examinees. It
relates a student's score on an achievement test to a domain knowledge rather than to another students
score.
Its main objective is to measure student's achievement of curriculum based skills. It is prepared
for a particular grade or course level. It has balanced representation of goals and objectives. It is used
to evaluate the curriculum plan instruction progress and group students' interaction. It can be
administered before and after instruction.
♦ It stresses what examinees can do and what they cannot do.
♦ It focuses on a delimited domain of learning tasks with a relatively large number of items
measuring in each specific task.
♦ It contains easy and difficult items.
♦ A student is tested after each unit for mastery of objectives.
♦ A student is allowed to proceed to the new material if mastery is obtained.
♦ A student is allowed to go to next unit along with the whole class.
♦ A student is presented with the new materials of the next unit.
♦ A student is tested for the new material and assigned marks.
Norm referenced test
A Norm-referenced test is designed to measure individual differences in achievement, intelligence,
interest, attitudes or personality. Percentile Ranks, Grade-equivalently scores and standard scores are
the examples of norm referenced test. Norm,-referenced test if primarily used for comparing
achievement of an examine to that of a large representative group of examinees at the same grade
level. The representative group is known as the 'Norm Group'. Norm group may be made up of
examinees at the local level, district level, state level or national level.
Gilbert Sax
A norm referenced test is designed to measure the growth in a student's attainment and to
compare his level of attainment with the levels reached by other students and norm group.
Bormuth
Norm referenced test is prepared for a particular grade level. It is administered after
instruction. It can be used for forming homogeneous or heterogeneous class groups. It classifies
achievement as above average or below average for a given grade.
♦ It stresses discrimination among individuals.
♦ It covers a large domain of learning task with just a few items measuring each
specific task.
♦ It contains items of average difficulty.
♦ A student is tested after each unit of the new material presented.

♦♦♦ A student is assigned the marks or grades to indicate his■ performance.

♦ A student is given remedial instruction if the material presented is not mastered.
♦ A student is tested again after remedial work, to check for mastery of the material.
♦ A student is tested for mastery of the already learned unit.
comparison of norm referenced test and criterion referenced test
♦ A norm referenced test typically attempts to measure a more general category of
competencies.
♦ A Criterion referenced test typically focuses on a more specific domain of examinee
behaviours.
♦ A 100 item on referenced test may be needed to cover the entire range of learners reading
comprehension skills as against five separate twenty item criterion referenced test
focusing only on five well defined skills within the overall realm of reading
comprehension.
CONTINUOUS AND COMPREHENSIVE EVALUATION [CCE]
Philosophical Basis :
The primary purpose of education is the manifestation of perfection already in man and woman
[Swami Vivekananda]; purpose of education is all round development of the child / individual. The
Report of the International Commission on Education for 21st Century to UNESCO referred to four
planes of living of human individuals namely; physical, intellectual, mental and spiritual. Thus, all round
development as the stated purpose of education implies optimization of hidden potential of every child in
the physical, intellectual, mental and spiritual planes. The CBSE in 2010 initiated for the first time an
effort to translate the lofty goal of all round development into practice by introducing CCE - scheme in
schools.
Globalisation in every sphere of society has important implications for education. We are
witnessing increasing commercialization of education. We need to be vigilant about the pressures to
commodity schools and the application of market-related concepts to schools and school quality. The
increasingly competitive environment into which schools are being drawn and the aspirations of
parents place a tremendous burden of stress and anxiety on children, to the detriment of their personal
growth and development and thus hamper the joy oflearning.

The aims of education simultaneously reflect the current needs and aspirations of a society, its
lasting values, concerns as well as broad human ideals. At any given time and place, they can be
called the contemporary and contextual articulation of universal human values.
The Frame Work
An understanding of learners, educational aims, the nature of knowledge, and the nature of the
school as a social space can help us arrive at principles to guide classroom practices. Conceptual
development is thus a continuous process of deepening and enriching connections and acquiring new
layers of meaning. Alongside is the development of theories that children have about the natural and
social worlds, including themselves in relation to others, which provide them with explanations for
why things are the way they are, the relationships between causes and effects, and the bases for
decisions and acting. Attitudes, emotions and values are thus an integral part of cognitive development,
and are linked to the development of language, mental representations, concepts and reasoning.
As children's metacognitive capabilities develop, they become more aware of their own beliefs
and capable of regulating their own learning.
Accordingly, National Curriculum Framework - 2005 [NCF-05] proposing Examination
Reforms has stated - "Indeed, Boards should consider, as a long-term measure, making the Class X
examination optional, thus permitting students continuing in the same school [and who do not need a
Board certificate] to take an internal school examination instead".
As a sequel to the above, the Position Paper on 'Examination Reforms' by NCERT 2006, says,
"Indeed, it is our view that the tenth grade exam be made optional forthwith. Tenth-graders who
intend continuing in the eleventh grade at the same school and do not need the Board certificate for
any immediate purpose, should be free to take a school-conducted exam instead of the Board exam."
Obviously, the efforts of CBSE to provide a leadership and pioneering role in implementing
CCE is a major breakthrough which attempts to elevate the status of the schools as equal partners of
the Board in assessing the attainment levels of learners.
Place of Evaluation in the Curriculum
A curriculum is what constitutes a total teaching-learning program composed of overall aims,
syllabus, materials, methods and assessment. In short it provides a framework of knowledge and
capabilities, seen as appropriate to a particular level. The syllabus provides a statement of purpose,
means and standards against which one can check the effectiveness of the program and the progress
made by the learners. Evaluation not only measures the progress and achievement of the learners but
also the effectiveness of the teaching materials and methods used for transaction. Hence evaluation
should be viewed as a component of curriculum with the twin purpose of effective delivery and further
improvement in the teaching-learning process.
What is Continuous and Comprehensive Evaluation?
Continuous and Comprehensive Evaluation [CCE] refers to a system of school-based
assessment of students that covers all aspects of students' development.
It is a developmental process of assessment which emphasizes on two fold objectives. Continuity
in evaluation and assessment of broad based learning and behavioural outcomes.
 In this scheme the term 'continuous' is meant to emphasise that evaluation of identified aspects of
students' 'growth and development' is a continuous process rather than an event, built into the total
teaching-learning process and spread over the entire span of academic session.
The second term 'comprehensive' means that the scheme attempts to cover both the
Scholastic and the Co-Scholastic aspects of students' growth and development. Since abilities,
attitudes and aptitudes can manifest themselves in forms other than the written word, the term refers
to application of a variety of tools and techniques and aims at assessing a learner's development in
higher order thinking skills such as analyzing, evaluating and creating. Assessment during the course
of studies or formative assessment must be based on a variety of evidences and lead to diagnosis of
learning gaps and their remediation.
The scheme is thus a curricular initiative, attempting to shift emphasis from memorizing to holistic
learning. It aims at creating citizens possessing sound values, appropriate skills and desirable qualities
besides academic excellence. It is hoped that this will equip the learners to meet the challenges of life
with confidence and success. It is the task of school based co-scholastic assessment to focus on holistic
development that will lead to lifelong learning.

The objectives of the CCE scheme are

♦ To help develop cognitive, psychomotor and affective skills.

♦ To lay emphasis on thought process and de-emphasise memorization.
♦ To make evaluation an integral part of teaching-learning process.
♦ To use evaluation for improvement of students' achievement and teaching- learning strategies on
the basis of regular diagnosis followed by remedial measures.
♦ To use evaluation as a quality control devise to raise standards of performance.
♦ To determine social utility, desirability or effectiveness of a programme and take appropriate
decisions about the learner, the process of learning and the learning environment.
♦ To make the process of teaching and learning a learner-centered activity.
Features of Continuous and Comprehensive Evaluation
♦ The 'continuous' aspect of CCE takes care of the 'continual' and 'periodicity' aspect of
evaluation.
♦ Continual means assessment of students in the beginning of instructions [placement evaluation]
and assessment during the instructional process [formative evaluation], done informally using
multiple techniques of evaluation.
♦ Periodicity means the assessment of performance done at the end of a unit/term [summative].
♦ The 'comprehensive' component of CCE takes care of assessment of the all round development
of the child's personality. It includes assessment of Scholastic as well as Co-Scholastic aspects
of the pupil's growth.
♦ Scholastic aspects include curricular areas or subject specific areas, whereas Co-Scholastic
aspects include Life Skills, Co-Curricular activities, attitudes and values.
♦ Assessment in Scholastic areas is done informally and formally using multiple techniques of
evaluation continually and periodically. The diagnostic evaluation takes place at the end of
 unit/term test. The causes of poor performance in some units are diagnosed using diagnostic tests.
These are followed up with appropriate interventions and remedy measures.
♦ Assessment in Co-Scholastic areas is done using multiple techniques on the basis of identified
criteria; where assessment in Life Skills is done on the basis of Indicators.
FORMATIVE AND SUMMATIVE ASSESSMENTS
Formative Assessment [FA] is a tool used by the teacher to continuously monitor a student's
progress in a non-threatening, supportive environment. It involves regular descriptive feedback, a chance
for the student to reflect on the performance, take advice and improve upon it. If used effectively, it
can improve student performance tremendously, while raising the self-esteem of the child and reducing
the work load of the teacher.
Salient features of Formative Assessment:
♦ diagnostic and remedial.
♦ makes the provision for effective feedback.
♦ provides the platform for the active involvement of students in their own learning.
♦ enables teachers to adjust teaching to take account of the results of assessment.
♦ recognizes the profound influence assessment has on the motivation and self-esteem of students,
both of which are crucial and influences learning.
♦ recognizes the need for students to be able to assess themselves and understand how to
improve.
♦ builds on students' prior knowledge and experience in designing what is taught.
♦ incorporates varied learning styles into deciding how and what to teach.
♦ encourages students to understand the criteria that will be used to judge their work.
♦ offers an opportunity to students to improve their work after feedback.
♦ helps students to support their peers, and expect to be supported by them.
Formative Assessment is thus carried out during a course of instruction to provide continuous
feedback to both the teachers and the learners, to take decisions regarding appropriate modifications
in the transactional procedures and learning activities.
It involves students' being an essential part of assessment. They must be involved in a range of
activities right from designing criteria to self assessment or peer assessment. Summative Assessment
[SA] is carried out at the end of a course of learning. It measures or 'sums-up' how much a student has
learned from the course. It is usually a graded test, i.e., it is marked according to a scale or set of
grades.
Assessment that is predominantly of summative nature will not by itself be able to illustrate a
valid measure of the growth and development of the child. It, at best, certifies the level of achievement
only at a given point of time. The paper-pencil tests are basically a onetime mode of assessment; and to
exclusively rely on it to decide about the development of a child is not only unfair but also unscientific.
Overemphasis on examination marks focusing on only scholastic aspects makes children assume that
assessment is different from learning, resulting in the 'learn and forget' syndrome. Besides encouraging
unhealthy competition, the overemphasis on Summative Assessment system also produces enormous
stress and anxiety among learners. It is this that has led to the emergence of the concept of Continuous
and Comprehensive School-Based Evaluation.
Formative Assessment [b]
The marking and grading of FA[b] activities would be based on the assessment of written work.
A few types of questions /tests for FA [b] arc suggested as under:
Right/Wrong, True-false, Yes/No. Matching Type, Matrix Type, Multiple Choice Questions,
Choose the correct answer, Fill in the blanks, Completing Railway reservations forms and bank
challans etc., Sequencing questions, Dictation, Very Short Answers, Short Answers, Filling Summary
Formats/ Graphic Organizers, Riddles, Puzzles and Competency-related Drawing. For differently-abled
children, or children who have been recently mainstreamed, other appropriate modes of assessment
may be evolved and used.
Recording FA [b] Activities
As soon as the FA [b] tests are completed in each term, learners are to be facilitated to record the
completion of FA [b] in the Student Learning Activity Record [Annexure -1].
The teachers need to meticulously evaluate the answer scripts of learners and award marks to
them according to the correctness of the answer. The mark secured by each student for each subject
in all the four tests is to be recorded in the Teacher Assessment Record [Annexure - II].
Out of a minimum of four tests conducted for learners, two tests that reflect the best
performance may be chosen for marking and grading.
The corresponding grade for the marks secured by each student is to be entered in the Student
Cumulative Record. If the mark secured by the students is 5 or less than 5, they need to be given
special attention and remedial activities

Guidance for summative Assessment

♦ The focus of marking and grading will be on written work.
♦ This will be a normative pen-and-paper Examination given at the end of each them.
♦ The Summative Assessment will be based on the prescribed blueprint.
♦ The questions for summative assessment may be asked not only from the exercise given at the
end of lesson but teachers can also ask questions related to higher order thinking skills.
GRADING PATTERN
Table - 1
Formative Summative Total
Assessment Assessment FA + SA
FA SA 40+60=60
40 60
Marks Grade Marks Grade Marks Grade
37-40 Al 55-60 Al 91-100 Al
'33-36 A2 49-54 A2 81-90 A2
29-32 Bl 43-48 Bl 71-80 Bl
25-28 B2 37-42 B2 61-70 B2
21-24 CI 31-36 CI 51-60 CI
17-20 C2 25-30 C2 41-50 C2
13-16 D 19-24 D 33-40 D
9-12 El 13-18 El 21-32 El
8& Below E2 12 & E2 20 & E2
Below Below

Table - TI
Grade Grade Explanation for the Grade
Point
A 5 Excellent [If all the five indicators are exhibited]
B 4 Very Good [If any four indicators are exhibited]
C 3 Good [If any three indicators are exhibited
D 2 Satisfactory [If any two indicators are exhibited]
E 1 To be Strengthened [If any one indicators are
exhibited]
Table - III
Average
Grade Point Grade
4,5-5 A
3.5-4.4 B
2.5-3.4 C
1.5-2.4 D
1.0-1.4 E

5.3 STATISTICAL MEASURES

As much as teaching and testing go together, statistics and testing go hand in hand. Through tests, we
can find out
(1) how much pupils learn from our teaching, i.e the achievement of our pupils, and
(2) whether our teaching is effective or not.
This can be done by treating and processing the test results.
Statistics, as we know, is the study of the collecting and analyzing of data to see what
inference and conclusions can be drawn.
Statistics can be defined as the legal data, which describe, summarize, generalize the condition
of a country or a school or a department or a student. It is essential and indispensable in important
fields of specialization in education.
After a test has been given, it is essential to arrange the scores in some order and do some
calculations to interpret the results. The basic knowledge of statistics is essential to enable us to
process these test scores.
The aim of teaching statistics in this course is to enable the students to
(1) construct test systematically,
 (2) compile and analyze data, and
(3) interpret the results scientifically.
Frequency Distribution
The score by itself is meaningless. They are only useful, if order can be put into the collected
scores. It is the statistical methods that put orders into data.
One of the most fundamental techniques for putting order into a disarray of data is the
frequency distribution. Basically, it is a systematic procedure for arranging individuals from least to
most in relation to some quantifiable characteristics.
Frequency distributions are constructed primarily for two reasons. First, they put the data into
order so that visual analysis can be made of the results of the measurements, and secondly, they
provide a convenient structure for simple computations.
Frequency distributions mean the way the scores within a group are spread out. From the
frequency distribution, we can know
(1) the number of scores above and below the Mean.
(2) whether me scores are all bunched up around the Mean or spread about quite far away
form the Mean towards either end of the range.
The two types of Frequency distribution tables are as follows:
(1) Frequency distribution table where actual scores are used as groups, and
(2) Frequency distribution table where ordered data are grouped within the range of intervals.
Frequency distribution table can be set up by
(1) finding the range,
(2) dividing the difference by 15 or 10
(3) writing down the class intervals, and
(4) taking frequency counts.
An example using English test scores of 25 students will illustrate the procedure for setting up
the two types of frequency distribution table.
English test scores of 25 students

75 72 77 68 71

80 78 63 87 72

67 70 66 70 72

82 83 76 76 70

62 65 84 68 62
Table 1 A Frequency Distribution of English Test Scores of 25 Students

Scores Number of Scores Number of

X Student X Student
(frequency) (frequency)

87 1 74 0

86 0 73 0

85 0 72 3

84 1 71 1

83 1 70 3

82 1 69 0

81 0 68 2

80 1 67 1

79 0 66 1

78 1 65 1

77 1 64 0

76 2 63 1

75 1 62 2
Table 2 A Second Frequency Distribution of English Test Scores of 25 Students

Class Interval Frequency

X f

85-87 1

82-84 3

79-81 1

76-78 4

73-75 1

70-72 7

67-69 3

64-66 2

61-63 3

Total 25

From the above frequency distribution, we can calculate the followings:

(1) The Range
Range = Highest Score – Lowest Score
From Table 1, Range = 87 – 62 = 25
From Table 2, Range = 86 – 62 = 24
(2) Exact Limits of The Class Interval (85-87)
Exact Lower Limit = 84.5
Exact Upper Limit = 87.5
(3) Size of class interval (CI)
CI = U – L
CI for the interval (85-87) = 87.5 – 84.5 = 3
( U  L)
(4) Mid point of the class interval x =
2
(84.5  87.5)
Mid point of the class interval (85-87) = = 86
2
 Range
(5) CI =
No. of Class Intervals
(i) If range is 60, and number of class intervals is 12, then CI equals 5.
(ii) If range is 80, and CI is 5, then the number of class interval equals 16
(iii) If the number of class intervals is 10 and CI is 3, then range equals 30

Exercises
(1) Find the exact limits, mid point and CI of the interval (56-60).
(2) Given the following IQ data, construct a frequency distribution according to prescribed
procedures.
144, 116, 97, 111, 112, 85, 132, 128, 123, 106, 80, 93, 118, 113, 104, 121, 101, 117, 138, 122,
118, 112, 109, 114, 105, 125, 129, 133, 103, 92.
(3) Using the given data, construct two frequency distribution table assuming CI = 3, and CI = 5.
115, 108, 102, 106, 103, 97, 104, 105, 110, 97, 101, 103, 92, 92, 105, 104, 106, 93, 103, 95,
104, 98, 106, 91, 102, 99, 103, 96, 99, 102.

Graphic Representation of Data

Graphs are used in the practical handling of real set of data. It is also used as visual models in
thinking about statistical problems.
The graphic representation of education of educational data takes various forms. They are:
(1) Pie Chart
(2) Bar Graph
(3) Pyramid
(4) Histogram
(5) Frequency Polygon
(6) Cumulative Frequency Curve
(7) Cumulative Percentage Curve or Ogive
 (1) Pie Chart
The pie chart is a convenient way of indicating the various components of a whole. It is
useful when one wishes of picture proportions of the total in a striking way.

Pie Chart showing the Socio-economic Classes in

a Primary School
10%

Lower-middle
15%
Upper-middle
Lower
40%
Upper
35%

Fig 1 Pie Chart showing the socio-economic classes in a Primary School

(2) Bar Graph

Bar graphs are commonly used to represent the size of measures, scores, percentages etc.
It is often used in Education and Psychology to compare the relative amounts of some traits (height.
intelligence) possessed by two or more groups.

Bar Graph Showing the Socio-economic

Classes in a Primary School

50%
Percentage

40%
30%
20%
10%
0%
Lower- Upper- Lower Upper
middle middle
Class

Fig 2 Bar graph showing the socio-economic classes in a Primary School

 (3) Pyramid
Pyramids are used for the purpose of comparing the number of students enrolled by levels,
by grades and by sex etc.
The data represented by the pyramid are very helpful for educational administrators and
planners.

High Girl Boy

Middle Girl Boy

Primary Girl Boy

Fig 3 Pyramid showing the general feature of enrollment in a High School.

(4) Histogram
One way of presenting frequency distribution in a graphic form is a histogram. A
histogram is a graph plotted by exact limits and their respective frequencies. In a histogram, the
scores are assumed to be spread uniformly over the entire interval.
It shows how the scores in the group are distributed-whether they are piled up at the low
or high end of the scale or are evenly distributed over the scales.

f
8

0 Score
60.5 63.5 66.5 69.5 72.5 75.5 78.5 81.5 84.5 87.5
Fig 4 Histogram showing distribution of English test scores of 25 students.
(5) Frequency Polygon
Sometimes we wish to reflect the continuous nature of our data. By placing a point at the
center of the top of each bar (class interval) and connecting these points and lines, we will get a
frequency polygon. In a frequency polygon, we assume that all cases in each interval are concentrated
at the mid-point of the interval. Thus, a frequency polygon is a graph plotted by actual scores or mid
points and their respective frequencies. It is likely to be more useful in comparing two or more graphs
plotted on the same axes.
 Frequency Polygon
8
7
6
Frequency

5
4
3
2
1
0
62 65 68 71 74 77 80 83 86
Scores
Fig 5 Frequency Polygon showing distribution of English Test Scores or 25 students.
Frequency distribution curve can generally be classified into two types, depending on the
shape of the curve.
(1) Symmetrical Curve
(2) Skewed Curve
- Positively Skewed Curve
- Negatively Skewed Curve

(1) Symmetrical Curve

A curve is symmetrical when one half of the curve is a mirror image of the other half.

Fig 6 An example of a symmetrical curve

(2) Skewed Curve
A curve is skewed it is not symmetrical.

(i)Positively Skewed Curve

If the massing of the scores is at the left end of the curve with the tail extending to right
end, then the curve is positively skewed.

Fig 7 An example of positively skewed curve

(ii)Negatively Skewed Curve

If the massing of the scores is at the right end of the curve with tail extending to the left
end, then the curve is negatively skewed.

Fig 8 An example of negatively skewed curve

 (6)Cumulative Frequency Curve
The cumulative frequency is used to determine the number of cases falling above or below
particular values. In cumulative frequency curve, we plot points above the top of the exact limits of
the interval.

X f F F%

85-87 1 25 100

82-84 3 24 96

79-81 1 21 84

76-78 4 20 80

73-75 1 16 64

70-72 7 15 60

67-69 3 8 32

64-66 2 5 20

61-63 3 3 12

30

25

20

15

10

0 60.5 63.5 66.5 69.5 72.5 75.5 78.5 81.5 84.5

Score
Fig 9 Cumulative frequency curve showing distribution of English test scores of 25 students
 (7) Cumulative Percentage Curve or Ogive
We may determine the cumulative percentage frequencies by converting raw frequencies
to percentage. A cumulative percentage or Ogive can be obtained by graphing these frequencies.
The advantage of this type of graph is that, from it, we can read off directly the
percentage of observations less than any specified values. Percentiles and percentile rank can be
determined quickly an fairly accurately from this Ogive.
120

100

80

60

40

20

0 60.5 63.5 66.5 69.5 72.5 75.5 78.5 81.5 84.5

Score
Fig 10 Cumulative Percentage Curve or Ogive showing distribution of English test scores of
25 students
Exercises
(1) Draw a (i) Pie Chart (ii) Bar Chart from the given percentages of students attending
school in a township.
Primary Level 45%
Secondary Level 35%
High School Level 15%
Colleges Level 5%
 (2) Plot a (i) Histogram (ii) Frequency Polygon (iii) Cumulative Frequency Curve (iv)
Cumulative Percentage Curve or Ogive, from the given distribution of spelling test
scores.

X f

46-50 4

41-45 6

36-40 10

31-35 18

26-30 6

21-25 3

16-20 1

Total 50

Measures of Central Tendency

In the previous chapters, we are concerned with the organization of collections of numbers in
the form a frequency distribution and how such frequency distribution could be presented in tabular
or graphic form. Now we will proceed to a consideration of how a collection of numbers may be
described in a single value.
Measures of central tendency is a value around which a distribution tends to center. It gives us
a concise description of the typical performance of the group as a whole. It enables us to compare the
performance of two or more groups. The indicators of central tendency are the mean, the median and
the mode.

The Mean
The mean is the arithmetic average. Of the three statistics, it is the most stable from sample to
sample.
The mean is simply the sum to series of measures divided by the number of measures. It is
represented by the Symbol X.
The formula for mean is:

X = X
N

where X = mean
X = individual score
 N = total number of measures
= summation
Example
(1) Find the mean of the following scores.
19, 20, 26, 18, 21, 8, 20, 24, 15, 18, 20

X = 209 N = 11

X = X =
209
= 19
N 11
X = 19
The Median
By simple definition, the median is the mid-point in a set of ranked scores.
The median is not influenced by how far extreme scores may range in a given distribution
because the range of these scores does not change the point that divides the distribution into two
equal-sized groups.

Examples
(1) What is the median of the following set of scores?
22, 28, 19, 25, 8, 23, 20
First, arrange the scores in order of magnitude.
28, 25, 23, 22, 20, 19, 8
Here the middle score is 22.
 The median of this set of scores is 22. There is only one score in the center when working
with odd number of scores.
When even number of scores are given, the arithmetic mean of two middle scores would
be the median.
(2) What is the median of the following set of scores?
20, 19, 22, 17, 15, 25, 12, 18
First arrange the scores in order of magnitude.
25, 22, 20, 19, 18, 17, 15, 12
18  19
 Median = = 18.5
2
The Mode
The mode is simply the most frequently occurring score in a distribution. It is less frequently
used measure of central tendency. It is used in preference to either the median or the mean, when a
measure of the most characteristic of a group is desired.
Example
(1) Find the mode of the given set of scores.
2, 3, 3, 2, 4, 5, 2
Here the most frequently occurring score 2 is the mode.
Mode = 2
(2) Find the mode of the given set of scores.
3, 4, 5, 6, 7, 8
Since all the frequency of each score is equal, there is no mode.
Also if the given scores are 2, 2, 3, 3, 4, 4, 5, 5, there is no mode since the frequency
of each score is equal.
(3) Find the mode of the given set of scores.
4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9
Since the most frequently occurring scores 6 and 7 are consecutive,
67
Mode = = 6.5
2
(4) Find the mode of the given set of scores.
4, 4, 4, 5, 5, 6, 6, 7, 7, 7
Here, there are two modes namely 4 and 7.
Calculation of Measures of Central Tendency from a Frequency Distribution Table
Calculation of the Mean
Example (1) Find the mean using the given frequency distribution table.
Table 1 : Table showing mathematics test scores of 50 students.

X f fX

10 2 20

9 4 36

8 9 72

7 15 105

6 8 48

5 7 35

4 5 20

Total 50 336
 Here  fX = 336, f = 50

Using the formula X =

 fX
f
336
X = = 6.72
50
 Mean = 6.72
Steps
1. Multiply each score by its respective frequency.
2. Add these products.
3. Divide the sum of the products by the sum of frequencies.

Example (2) Find the mean using the given frequency distribution table.
Table 2 Table showing English test scores of 50 students.

X f X fx

65-69 2 67 134

60-64 10 62 620

55-59 11 57 627

50-54 12 52 624

45-49 9 47 423

40-44 5 42 210

35-39 1 37 37

Total 50 2675

Here  fX = 2675,  f = 50
Using the formula X =
 fX
f
2675
X =
50
= 53.5
Mean = 53.5
Steps
1. Calculate the mid-point of all intervals.
2. Multiply each mid-point by the corresponding frequencies.
3. Sum the product of mid-point by frequencies.
4. Divide this by sum of frequencies, i.e., (N)

Calculation of the Mean by the short method

Example (1) Find the mean from the following distribution table.

X f d fd

10 2 3 6
9 4 2 8
8 9 1 9
7 15 0 0
6 8 -1 -8
5 7 -2 -14
4 5 -3 -15

Total 50 -14

The formula for using the shorts method is

X = AM +  fd x CI
f
where, AM = Assumed Mean
X  AM
d=
CI

From the table

AM =7
fd = -14
f = 50
CI =1
  14
 X =7 x1
50
= 7 – 0.28
= 6.72
Example (2) Find the mean from the following distribution table.

X f d fd

65-69 2 3 6
60-64 10 2 20
55-59 11 1 11
50-54 12 0 0
45-49 9 -1 -9
40-44 5 -2 -10
35-39 1 -3 -3

Total 50 15

Here AM = 52
fd = 15
f = 50
CI =5

X = AM +  fd x CI
f
15
X = 52  x5
50
= 52 + 1.5
= 53.5

Combined Mean
Sometimes we have means or several different samples and would like to know the mean for the
total combined group.
The formula for computing a combined mean is
N1X1  N 2 X 2  .......  N K X K
X =
N1  N 2  ......  N K
Where X = mean or combined groups
N1, N2, ….. , NK = numbers of cases in sample 1, 2 and k
 X1 , X 2 , ...., X K = means of sample 1, 2, K

Example
The mean IQs for there sections of eight standard students are given.
Find the combined means for the eighth standard.

X N

Section 1 101 30

Section 2 107 35

Section 3 95 26

Using the formula

N1X1  N 2 X 2  .......  N K X K
Combined mean X =
N1  N 2  ......  N K
30 (101)  35 (107)  26 (95)
=
30  35  26
= 101.6

Calculation of the Median

Example : Calculate the median for the following distribution table.

X f F

65-69 2 50
60-64 10 48
55-59 11 38
50-54 12 27
45-49 9 15
40-44 5 6
35-39 1 1

Total 50

In calculating the Median, we must first set up a cumulative frequency column. The formula for
calculating the Median is
 N
F
Mdn = L 2 x CI
f
where, L = exact lower limit of the interval containing the median
F = cumulative frequency below the class containing the median
CI = class interval
N 50
In this example = = 25, L = 49.5 , F = 15, f = 12
2 2
N
F
 Mdn = L 2 x CI
f
25 15
= 49.5  x5
12
= 49.5 + 4.166
= 53.666
 Mdn = 53.67

Calculation of the Mode

Example : Calculate the mode for the following distribution table.

X f

65-69 2
60-64 10
55-59 11
50-54 12
45-49 9
40-44 5
35-39 1

Total 50

The formula for calculating the mode is

1
M0 = L  x CI
1   2
 where M0 = Mode
L = exact lower limit containing mode
1 = f0 – f-1
2 = f0 – f+1
CI = Class Interval
In this example
L = 49.5
1 = 12 – 9 = 3
2 = 12 – 11 = 1
CI = 5
3
M0 = 49.5  x5
31
15
= 49.5 
4
= 49.5 + 3.75
= 53.25

Comparison of the Mean, Median and Mode

If the frequency distribution is Symmetrical, the mean, median and mode coincide.
If the frequency distribution is skewed, these three measures do not coincide. (Fig 1)
If the mean is greater than the median, then the distribution is positively skewed. (Fig 2)
If the mean is less than the median, then the distribution is negatively skewed. (Fig 3)
Thus just by knowing the values of mean, median and mode, the shape of the distribution can
be easily calculated.

Mean = Median = Mode

Fig (1) A Symmetrical Curve

 Mdn Mean

Fig (2) A Positively Skewed Curve

Mdn Mean

Fig (3) A Negatively Skewed Curve

Exercises
1. Compute means, medians and modes for the following data.
(a) 3, 8, 18, 36, 54
(b) 11, 20, 19, 29, 29, 45
(c) 3, 3, 4, 6, 6, 6, 7, 7, 7, 9
2. Calculate the mean, median, and mode for the following distribution and trace the shape of it.

X f

85-89 4
80-84 5
75-79 5
70-74 20
65-69 18
60-64 10
55-59 13
50-54 10
45-49 4
40-44 6
35-39 5

Total 100

Measures of Variability
Statisticians have taken advantage of two outstanding features of most frequency distributions:
(1) The tendency of the data, to cluster around some value lying between the smallest and
largest data.
(2) The tendency of the data to be dispersed around this central value.
The first feature is central tendency and the second feature is dispersion or variability.
Although measures of central tendency are very useful statistics for describing a set of data,
they do not tell us enough. Two sets of data, which are very different, can have identical means, or
medians. As an example, consider the following sets of data.

Set A 59 59 59 60 61 61 61

Set B 30 40 50 60 70 80 90

The mean of both sets of score is 60 and the median of both is 60, but set A is very different
from set B. In set A, the scores are very close together and clustered around the mean. In set B, the
scores are much more spread out; that is, there is much more variation or variability in set B. Thus,
there is a need for a measure that indicates how spreads out the scores are and how much variability
there is.
 Measures that indicate the amount of scatter in a distribution scores are referred to as
measures of variability. In other words, measures which reflect the way in which data are spread in
either direction from the center of a distribution are called measures of variability or dispersion or
spread ness or scatter.
Five measures have been devised to indicate the variability or dispersion within a set of data.
They are :
(1) The range
(2) The quartile deviation
(3) The mean deviation
(4) The standard deviation
(5) The variance

The Range
The range is defined either as the difference between the highest score and the lowest score (R
= H – L) or as the difference Plus one (R = H – L + 1), the latter being more accurate. For example
the range for the scores 59, 59, 59, 60, 61, 61, 61 is 3 where as the range for the scores 30, 40, 50, 60,
70, 80, 90 is 61. Thus, if the range is small, the scores are close together whereas the range is large,
the scores are more spread out. Like the mode, the range not a very stable measure of variability and
its chief advantage is that it gives a quick, rough estimate of variability.

The Quartile Deviation

The quartile deviation (also referred as the semi-inter quartile range) is one-half of the
difference between the upper quartile and the lower quartile in a distribution. The upper quartile is
the 75th percentile that point below which are 75% of the scores, the lower quartile. Correspondingly,
is the 25th percentile, that point below which are 25% of the scores. By subtracting the lower quartile
from the upper quartile then dividing the result by two, we get a measure of variability.
Q3  Q1
QD =
2
As an example, if there are 60 scores, Q1 is the point below which are 15 of the scores (15 =
25% of 60), and Q3 is the point below which are 45 of the score (45 = 75% of 60). If the quartile
deviation is small, the scores are together whereas if the quartile deviation is large, the scores are
more spread out. The quartile deviation is a more stable measure of variability than the range and is
appropriate whenever the median is appropriate. Calculation of the quartile deviation involves a
process very similar to that used to calculate the median, which just happens to be the second quartile
Q2.

The Mean Deviation

Unlike the range and the quartile deviation, the mean deviation takes into account every score
in the distribution. The difference, or deviation, or each score from the mean of the distribution is
determined without regard to the direction of the difference. These deviations are added and the sum
is divided by the number of scores. The result is the mean of the deviations of the scores from their
mean.
The mean deviation does not have a wide range of application, nor it is very useful for making
comparisons among sets of data. The main reason is that the absolute values of the deviations of
scores the mean are used in the computation. In other words, we ignore the signs of deviations.

The Standard Deviation

The standard deviation is appropriate when the data represent an interval or ratio scale and is
by far the most frequently used index of variability. Like the mean, the measure of central tendency
which is its counterpart, the standard deviation is the most stable measure of variability and takes into
account each and every score. In fact, the first step in calculating the standard deviation involves
finding out how far each score is from the mean, that is, subtracting the mean from each score. If we
square each difference, add up all the squares, and divide by the number of scores, then we have a
measure of variability called variance. The square root of that measure is called the standard
deviation. A small standard deviation indicates that scores are close together and a large standard
deviation indicates that the scores are more spread out.
If you know the mean and the standard deviation of a set of scores, you have a pretty good
picture of what the distribution looks alike. If the distribution is relatively normal, then the mean plus
3 standard deviations and the mean minus 3 standard deviations encompasses just about all the scores
over 99% of them.
For example, suppose that for a set of scores the mean ( X ) is calculated to be 60 and the
standard deviation (SD) to be 1. In this case, the mean plus three standard deviations, X + 3SD is
equal to 60 + 3(1) = 60 + 3 = 63. The mean minus three standard deviations, X - 3SD is equal to 60
– 3 = 57. The scores fall between 57 and 63. This makes the scores to be close together or not to be
very spread out.
As another example, suppose that for another set of scores the mean (X) is again calculated to
be 60, but this time the standard deviation (SD) is calculated to be 5. In this case, the mean plus three
standard deviations and the mean minus three standard deviations are 75 and 45 respectively. Thus,
almost all the scores between 45 and 75. This almost makes sense since a larger standard deviation
indicates that the scores are more spread out.
The Variance
The variance is nothing but the square of the standard deviation. Thus, in order to determine
the variance of a distribution, each of the deviation scores is squared. These squared deviations are
then added together and divided by the number of scores in the distribution. The mean of sum of
these square deviations is called the variance of the distribution.
If the variance is small, the scores are close together, if the variance is large, the scores are
more spread out. Both variance and standard deviation are most widely used statistical indices of
variability and are fitted into further statistical analyses.

Calculation of Measures of Variability from Ungrouped Data.

Example (1) Find range, quartile deviation, standard deviation and variance of the following data.
10, 15, 13, 8, 10, 10, 15, 13, 11, 7, 4, 4
(1) Calculation of range.
Range = Highest Score Lowest Score
= 15 - 4 = 11
(2) Calculation of quartile deviation

N X f F
F
Q1 = L 4 x CI 15 2 12
f
12 13 2 10
2
= 6. 5  4 x 1 = 6.5 + 1 = 7.5
1 11 1 8
3N
F 10 3 7
Q3 = L 4 x CI
f 8 1 4
3 x 12
8 7 1 3
= 12.5  4 x 1 = 12.5 + 0.5 = 13
2 4 2 2
Q  Q1 13  7.5 5.5
QD = 3 = = = 2.75
2 2 2 Total 12

(3) Calculation of mean deviation

X = X =
120
= 10
X X- X X X (X- X )2
N 12
10 0 0 0
 XX 34
MD = = = 2.83
N 12 15 5 5 25

13 3 3 9
(4) Calculation of standard deviation
8 -2 2 4
SD =  (X  X) 2

10 0 0 0
N
10 0 0 0
154
=
12 15 5 5 25

= 12.83 13 3 3 9
= 3.58 11 1 1 1
 7 -3 3 9

4 -6 6 36

4 -6 6 36

Total 34 154

(5) Calculation of variance

Variance =
 (X  X) 2 =
154
= 12.83
N 12
Note : (1) Variance = SD2 (2) SD = Square root of Variance

Calculation of Measures of Variability from Grouped Data

Example (2) Find range, quartile deviation, standard deviation and variance of the following
frequency distribution table.

X f

70-74 5

65-69 5

60-64 8

55-59 10

50-54 20

45-49 14

40-44 12

35-39 16

30-34 10

Total 100
For calculating such measures, we have to construct the given table adding with another four
columns. (F, d, fd, fd2) as follows:

X f F d fd fd2

70-74 5 100 4 20 80

65-69 5 95 3 15 45

60-64 8 90 2 16 32

55-59 10 82 1 10 10

50-54 20 72 0 0 0

45-49 14 52 -1 -14 14

40-44 12 38 -2 -24 48

35-39 16 26 -3 -48 144

30-34 10 10 -4 -40 160

Total 100 533

(1) Calculation of range

Range = Highest Score – Lowest Score = 72 – 32 = 40

(2) Calculation of quartile deviation

N
F
4 25  10
QI =L+ x CI = 34.5 + x5 = 39.18
f 16
3N
F
4 75 72
Q3 =L + x CI = 54.5 + x 5 = 56
f 10
Q3  Q1 56  39.18
Q = = = 8.41
2 2

(3) Calculation of Standard Deviation

2
 fd 2    fd 
2
533   65 
SD = CI x =5x  
 f   f  100  100 
 = 5 x 2.21 = 11.05

(4) Calculation of Variance

Variance = (SD)2 = (11.05)2 = 122.1025

Exercises
1. The data below represent raw scores on a test taken by a class of 15 students.

32 36 38 39 42

44 45 46 45 45

48 48 50 51 51

(a) Find the range of distribution.

(b) Compute mean deviation and standard deviation by using the ungrouped data method.
(c) Arrange the scores into frequency distribution with intervals three points wide (i.e. CI = 3)
beginning with 30-32. Compute again the standard deviation and quartile deviation of the
given distribution.

 X  X 
2
2. If = 144, N = 16, find the variance and standard deviation.

Percentile and Percentile Rank

The Nature of Percentile and Percentile Rank
A numerical value which summarizes the responses actually made on a test by an individual is
called a raw score. Raw score alone is not enough to interpret a student's performance. Knowing a
student's raw score merely is about how well he did on the test and it also means that how much a
student about how well he did on the test and it also means that how much a student made his/her
actual performance without comparing to either a criterion score or the performance of others who
took the same test. Clearly there is a need for methods of transforming raw scores into values which
facilitate the interpretation of scores on both an individual and group basis. Then it becomes the
measures of relative position more meaningful in interpretation of scores and in comparison of
student's performance as well.
Measures of relative position indicate where a score is in relation to all other scores in the
distribution. In other words, measures of relative position permit a teacher to express how well an
individual has performed as compared to all other individuals in same group who have been given the
same test. Raw scores that have been transformed systematically into equivalent values which
indicate relative position are referred to as derived scores. The most common types of derived scores
are percentile and percentile ranks.
A percentile is a point which cuts off a given percentage of a distribution.
There are 99 percentiles that a distribution is divided into 100 equal parts. The first percentile
symbolized as P1 is the point below which are one percent of the scores. Similarly, P50 is the point
below which 50% of the scores and P90 is the point below which are 90% of the scores.
 A percentile rank (PR) indicates the percentage of scores that lie below a given value in the
total distribution.
Thus, if a score of 48 corresponds to a percentile rank of 80 (symbolized PR80), this means
that 80% of the score are below the score 48. In other words, if a student had a percentile rank of 98,
this would mean that he/she did as well or better than 98% of the members of some group which took
the same test.
The median of a set of scores corresponds to the 50th percentile which makes sense since the
median is the middle point and therefore the point below which are 50% of the scores. Similarly, the
first quartile (Q1) corresponds to the 25th percentile and second quartile (Q2) corresponds to the 50th
percentile and the third quartile (Q3) corresponds to the 75th percentile. thus, percentile ranks
basically allow us to determine how well an individual did in relative terms, as compared to others
who took the same test. If percentile ranks are given for a number of subtests, they also provide a
rough mean if comparing an individual's relative performance in a number of different areas.
One point to keep in mind when interpreting percentile ranks is that they are ordinal, not
interval measures. Therefore, we do not have equal intervals between percentile points. An increase
of a given number of percentile points corresponds to a different number of ranks score points
depending upon where they are in the distribution. For example, the difference between the 45th and
50th percentile does not represent the same increase in raw scores as the difference between the 90th
and 95th percentiles.
Uses of Percentile and Percentile Rank
The percentile rank is useful to compare not only the performance of two different tests. It can
be used to set up tables that list the percentile equivalents of each raw score. Moreover, it can also to
draw a profile of the student's performance (relative to a reference group) on a several different tasks.
Percentiles are also useful in determining the cutoff score. Cutoff scores are never
predetermined in a relative system. Percentile ranks must be set up first according to the quota. After
that percentiles must be computed to find the raw scores equivalent to the percentile ranks which have
already been set up. For example, 20% of the candidates who sit for a competition examination is
supposed to be selected. It means that 80% will be rejected. Thus, P80 must be computed to determine
a raw score, below which 80% of total candidates lie. This calculated raw scores can be regarded as
the cutoff score, because students who receive below this raw score will be rejected. It is obvious that
percentile and percentile rank are interdependent.

The Computation of Percentiles and Percentile Ranks

The method of calculating percentiles is essentially the same as that employed in finding the
median and quartiles. So, it can be calculated by using the following formula.
PxN
F
Pp = L + 100 x CI
f
Where, P = Percentage of the distribution wanted.
L = exact lower limit of class interval upon which Pp lies.
PxN
= Part of N to be counted off in order to reach Pp.
100
 F = Sum of all scores upon intervals below L. This is, the cumulative
frequency below the class containing Pp.
CI = Length of the class interval.
N = number of cases.

Calculation of Percentiles from Ungrouped Data

Example (1)
Find P70 of English test scores of ten students given below.
70, 60, 55, 55, 50, 46, 46, 46, 40, 35
In order to make the computation easier, it is necessary to set up a frequency
distribution table using the given data shown in the table.

X f F

70 1 10

60 1 9

55 2 8

50 1 6

46 3 5

40 1 2

35 1 1

Total 10

The formula for P70 can be written as:

70 N
F
P70 = L + 100 x CI
f
70 N 70  10
 = = 7
100 100
Therefore from the table, it can be observed that class interval in which P70 falls is 5.5.
L = 54.5, F = 6, f = 2, CI = 1
76
P70 = 54.5 + x1
2
 P70 = 54.5 + 0.5
P70 = 55

Calculation of Percentile from Grouped Data

Example (2)
Find P70 by using the following frequency distribution table. The table is showing the
Mathematics test scores of 100 students of Standard Five.

X f

70-74 5
65-69 8
60-64 7
55-59 10
50-54 30
45-49 15
40-44 18
35-39 4
30-34 3

Total 100

Before applying the formula, it is necessary to reconstruct the table with its cumulative
frequencies as follows.

X f F

70-74 5 100
65-69 8 95
60-64 7 87
55-59 10 80
50-54 30 70
45-49 15 40
40-44 18 25
35-39 4 7
30-34 3 3

Total 100
 70 N  F
P70 = L + 100 x CI
f
70 N 70 x 100
= = 70
100 100
L = 49.5, , F = 40, f = 30, CI = 5
70  40
P70 = 49.5 + x5
30
 P70 = 49.5 + 5
P70 = 54.5

Calculation of Percentiles Ranks from Ungrouped Data

The formula for calculating percentile rank is
No. of cases below the score + .5(No. of cases at the score)
PR = x 100
N
Example (3) Compute PR of the student with the test score of 50 in the distribution, 30, 32, 33, 38,
40, 48, 50, 58, 70, 75
By the problem,
Number of cases below the score 50 = 6
Number of cases at the score 50 = 1
Total number of students in the distribution = 10
6  0.5 1
PR = x 100
10
6.5
= x 100
10
= 65%
Therefore, percentile rank of score 50 is 65%

The Computation of Percentile rank from grouped data

If percentile ranks have to be computed from a frequency distribution, where the
scores are grouped within the range of 2 intervals, we have to use the following formula.
  f  100
PR = ( S  L )  F 
 CI  N
where PR = percentile rank
S = the given score
L = the lower limit of the class interval in which the given score lies
f = frequency of the class interval in which the given score lies
F = cumulative frequency below the class containing the given score
CI = class interval

Example (4) Find PR of the score 67 in the following frequency distribution.

X f

70-74 5

65-69 8

60-64 7

55-59 10

50-84 30

45-49 15

40-44 18

35-39 4

30-34 3

Total 100
 The given table is reconstructed as follows and L, f, F and CI are calculated from that table.

X f F

70-74 5 100

65-69 8 95

60-64 7 87

55-59 10 80

50-54 30 70

45-49 15 40

40-44 18 25

35-39 4 7

30-34 3 3

Total 100

Score 67 falls on the interval (65-69)

L = 64.5, f = 8, F = 87, CI = 5, S = 67, N = 100
 f  100
PR = (S  L)  F
 CI  N
 8  100
= (67  64.5)  87 
 5  100
 8 
= 2.5   87  1
 5 
= [4 + 87] x 1 = 91%
Therefore, percentile rank of score 67 is 91%.

Exercise
(1) By using the data given in example (4), find P91.
(2) By using the data given in example (4), find how many percentages of students who
passed the examination when their
cutoff score is decided as 40 marks.
(3) By observing the given data also, find cutoff score so that 90% of students should pass
the examination.
Correlation
So far we have been concerned only with a single variable and its frequency distribution. Now
in this chapter, our concern will be on two variables or even more.
Many variables or events in nature are related to each other. As the sun rises, the day warms
up, as children becomes matured, they think more completely and the persons bright in one area tend
to be bright in certain areas too. Such relationships are called correlations. This relationship of one
variable to another is known as correlation.
The correlation coefficient is a statistical measure that describes the degree of relationship
between two or more variables. All correlation coefficients lie between – 1.00 and +1.00 passing
through 0.00. One might more easily visualize possible coefficients as existing on a line as indicated
below.

-1.00 0.00 +1.00

The sign of coefficient expresses the direction of the relationship, and the size of coefficient
expresses the degree of relationship.
A scatter diagram is one of the clearest ways of showing the meaning of the correlation
graphically. By using scatter diagrams the student can get the indication of the magnitudes of the
correlation coefficient and also an indication of the signs of relationship.

Classification of Correlation
Correlation can be classified into five categories as follows:
(1) Perfect positive correlation.
(2) Positive correlation
(3) Perfect negative correlation
(4) Negative correlation
(5) Zero correlation

(1) Perfect positive correlation

As X increases a given amount, Y increases in a proportionate amount. The relationship is
therefore a perfect positive one.
X: 20, 18, 16, 14, 12, 10, 8, 6, 4, 2
Y: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1& was To
 12
10 +
+
8 +
+
Y 6 +
+
4 +
+
2 +
+
0
0 10 20 30
X
Fig (1)
When the pairs of these variables are plotted, the points will fall along a straight line. It runs
upward from left to right as shown in Fig (1). The correlation coefficient r is equal to 1.00.

(2) Positive correlation

But if all pairs of variables do not progress together unit for unit, their relationship is positive
but less than perfect.
X (Age): 43, 48, 56, 61, 67, 70
Y (Pressure): 128, 120, 135, 143, 141, 152

Y
160
150 +

140 + +
+
130 +
120 +
110
100 X
30 40 50 60 70 80
Fig (2)

Thus the plotted points fall near but not directly on the straight line as shown in Fig (2). The
correlation coefficient would be less than 1.00.
Some example of positively correlated variables are intelligence and school success, high
rainfall and high humidity.

(3) Perfect Negative Correlation

For every increase on the X, Y decreases a constant number of units, the relationship therefore
is a perfect negative one. When the pairs of the variables are plotted, the points will fall along a
straight line. It runs downward from left to right as shown in Fig (3). The correlation coefficient r
equals to – 1.00.
X: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Y: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Y
12
10 +
+
8 +
+
6 +
+
+
4 +
+
2 +
0 X
0 5 10 15 20 25
Fig (3)

(4) Negative Correlation

As X increases a unit, Y has a decreasing trend but not in a proportionate amount. The
relationship therefore is negative but not perfect.
When the pairs of the variables are plotted, it will not fall along a straight line, as shown in
Fig (4). The correlation coefficient r will be a negative number between 0.00 and – 1.00.
Some examples of negative correlation are fatigue and test performance, decrease in teaching
quality and increase in number of failures.
X (No. of Absence) 6, 2, 15, 9, 12, 5, 8
Y (Final Grade) 82, 86, 43, 74, 58, 90, 78
 Y
100
90 +
+
80 +
+
70 +
60 +
50
40 +
30
20
10
0 X
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Fig (4)
(5) Zero Correlation
If an increase or decrease in one variable tells us nothing about the likely condition of the
other variable, there is no relationship, and we may say that there is zero correlation.
When the pairs of the variables are plotted, we will have no definite direction as shown in Fig
(5). The correlation r will be zero.
An example of zero correlation is adult height and intelligence.

Y
20
18 +
16
14
12 + +
10 +
8
6 ++ +
+
4 +
2
0 X
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Fig (5)
The Strength of Correlation Coefficient
When statisticians refer to high and low correlation, they are usually using the absolute scale
of 1.00 to 0.00 as their guide. 'High' or 'Strong' correlation coefficients therefore, should not
necessarily be thought as 'important' nor should 'low' or 'weak' correlation coefficients necessarily be
considered 'unimportant'.
Generally correlation coefficient r equal to or greater than 0.70 can be regarded as high,
between 0.40 and 0.70 as fair and equal to or less than 0.40 as low correlation.
But it should be noted that for certain purposes r of 0.50 might be considered satisfactory
whereas in other situations, an r of 0.90 or high would be required.
If two test forms are to be considered equivalent, we would expect a very high positive
correlative between scores on the two forms, i.e. a correlation of somewhat about 90. Yet in other
educational situations, for example, in relating academic achievement to a predictor test of
achievement, such as IQ test, we are often satisfied with an r of between 0.40 and 0.50. An r of 0.70
in such situation would be exceptional indeed.

The Computation of the Pearson Product Moment Correlation Coefficient

Computing a Correlation Coefficient from Raw Scores

Formula:

 XY   N
X Y
r =
  X 2    Y 2 
 X  N   Y  N 
2 2
  

Examples
(1) Given the following scores on an arithmetic test and corresponding scores on a reading
comprehension test, correlate the two set of scores.

X Y X2 Y2 XY
Arithmetic Reading

3 6 9 36 18

2 4 4 16 8

4 4 16 16 16

6 7 36 49 42

5 5 25 25 25

1 3 1 9 3

 X = 21  Y = 29  X2  91  Y2  151  XY 112
 21 x 29
112 
r = 6
 (21) 2   (29) 2 
91   151  
 6  6 
609
112 
= 6
 441  841
91  6  151  6 

112  101.5
=
91  73.5151  140.16
10.5
=
17.5  10.84
10.5 10.5
= = = 0.76
189.7 13.77

Computer r using the following information.

 X2 = 8.5 X = 83  XY = 788

 Y2 = 766 Y = 82 N = 10

 XY   N
X Y
r =
  X  2   Y 2   Y  2 
 X  
2
 
 N  N 
83 x 82
788 
= 10
 (83) 2   (82) 2 
835   766  
 10   10 

788  680.6
=
835  688.9766  672.4
107.4
=
146.1  688.9
107.4
=
13674.96
 107.4
=
116.94
= 0.918
= 0.92

Computing a Correlation Coefficient from Standard Sores or z Scores.

Formula: rxy =
 Z x Yy
N
XX YY
where ZX = , ZY =
x y

N = number of pairs of X, Y scores.

Example Zx Zy ZxZy
1.4
Given the following pairs of z scores representing seven 1.0 1.40
1.0 1.4 1.40
children's position on the IQ scale an on a reading test,.7what is the
.7 .49
reading comprehension? .0 .0 .00
-.6 -1.2 .72
IQ scale = x -1.3 -.6 .78
Reading Comprehension = y -1.2 -1.3 1.56
 Z x Z y =6.35

rxy =
 Zx Zy
N
6.35
=
7
= 0.91
Exercises
(1) Ten individuals made the following scores on two methods (X and Y) of spelling test.
Compute r between their scores.

X Y

14 13

12 15

10 8

9 10

8 7

7 5

6 4

3 2

2 4

2 5

(2) By using the following information, calculate r.

 X2 = 393  X = 57  XY = 431
 Y2 = 514  Y = 62 N = 10

(3) Calculate r from the given pairs of Z scores.

Zx Zy

1.50 0.50

2.00 1.00

0.90 -0.30

-0.35 2.00

-3.00 -0.25

0.50 0.10

0.30 0.20
 EXERCISE

1. What are the limitations of objective type test items?

2. List the steps involved in construction of an achievement test
3. Explain item analysis
4. What do you mean by diagnosis?
5. List the essential for the construction of a diagnostic test
6. Explain the concept of diagnostic testing. How is it different from achievement testing? Illustrate the
difference with the help of an example.
7. Describe the role of diagnostic testing in continuous and comprehensive evaluation.
8. What if Remedial teaching? Describe various techniques/measures, which can be used in remedial
teaching.
9. Distinguish between the achievement test and diagnostic test in their functions and preparation.
10. What is meant by a diagnostic test? How do you prepare it?
11. What are the advantages of objective type test items?
12. Explain what is diagnosis?
13. List the levels of diagnosis
14. Describe the characteristics of diagnosis
15. Explain what do you mean by remediation?
16. What do you mean by educational diagnosis?
17. List the levels of diagnosis.
18. What is Norm referenced tests?
19. Give an account of criterion referenced test.
20. Compare Norm referenced and criterion referenced tests.
21. How will you prepare a frequency distribution table?
22. What are the main uses of a frequency distribution in statistical analysis?