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Building a Better Mathematics Classroom

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 Hridaya Kant Dewan
hardy.dewan@gmail.com

Building a Better Mathematics Classroom

Abstract
This Paper focuses on the holistic teaching-learning process for Mathematics
as embedded in the current thinking on Mathematics education which
argues for the child building her own understanding through the process of
engagement with concepts in different ways including having opportunities
to collectively construct definitions and examine and modify them, developing
formulas or/and algorithms, proofs and solving new problems, building
patterns and generalisation, creating and setting problems and puzzles,
talking and expressing mathematical concepts in written form, etc.

1. Introduction 2. Nature of the Subject and its

Whenever we think of the methodology Manifestation
of teaching a subject some basic aspects The nature of the subject and the way
have to be kept in mind. These include: it manifests itself and constructs and
A. The nature of the subject and the accepts new knowledge informs the
way it manifests itself and constructs way classrooms must be constructed.
and accepts new knowledge; Mathematics is considered different
B. The feelings and attitude of the from other disciplines; less driven
learners, teachers, the system and by context, more abstract and in
the community to the subject and comparison to other disciplines more
how they view its importance and hierarchical implying each idea is
its learnability; intricately linked to other mathematical
ideas and objects. This would suggest a
C. The understanding about how linear class-room content and process.
humans, particularly children, However, the recent discourse on
learn; Mathematics education elaborates the
D. The background of children and above and this brings out nuances that
teachers along with the kind of qualify above statements. While these
schools and the education system nuances are stated in many different
that exists; ways, we choose to follow one that is
E. The purpose for which we want to also reflected in the position paper on
educate children, in this case in Mathematics NCERT 2005. It points
Mathematics; out:
F. The curriculum and the content of – Mathematics learning is about
Mathematics for the classes children knowing, finding and creating
are in and the resources that can be patterns, relationships and
used. The class-room depends on discovering other properties among
the way these are understood. It is mathematical objects.
not possible to discuss all of them – Mathematics is present in what we
in this but we would discuss the do and is best learnt through that.
essential aspects of most of them. – Knowing Mathematics means being
Each of these have been discussed able to create and use ideas to form
briefly to lay down the contours new ideas for yourself. It is about
and their implications on the class- leading learners to find solutions to
rooms.

6
 Voices of Teachers and Teacher Educators

unfamiliar problems and to create have been used as models for ideas.
new problems. Validity of mathematical ideas is not
All these add to what Mathematics is established through multiple repeated
and what is worth learning and teaching observations to conclude the veracity
in it. These would be embedded in the of the statement. They have to be
chosen content areas like numbers, logically deduced from axioms and
shapes, angles, data, functions, prior known results. Mathematics is
equations and their relationships about generalising rules and being able
to each other and to life, etc. This to see and understand them and its
decade old formulation is far from ideas emerge from precise presentation
being operational in the classrooms. of the underlying concepts and their
The common notion of Mathematics consistent use to build a hierarchically
continues to be limited to jugglery with structured understanding.
numbers and different combinations 3. Attitudes to Mathematics
of operations on them. The other areas
are added as definitions and algorithms “The feelings and attitude of the
(standard and non-standard short- learners, teachers, the system and the
cuts). The attempt is to acquire enough community to the subject and how they
bits and pieces to go through the exam view its importance and its learn ability
without it all coming together to form a also affects classrooms.” The present
picture. In the absence of recognition day perceptions of Mathematics
of even the need to differentiate as it is to be taught and learnt are
a mathematical statement from a characterized by some fundamental
non-mathematical statement or to underlying commonalities:
understand what it means to generalise – Mathematics is difficult and meant
and to prove them. The students do not for a few intelligent children.
get a feel for the process of accepting – Mathematics is not about
mathematical knowledge as valid. understanding concepts but
The other difficulty is that the abilities to do.
NCF like statements are nuanced and – Mathematics is not useful to learn
has ideas about Mathematics that as it does not aid our day to day
seem to suggest opposite ways. One life except in the utilitarian sense of
argument that can be made is that calculations and data.
Mathematics is fully a part of our life
– Mathematical knowledge and all
and has meaning only when completely
other knowledge is learnt bit by bit
embedded in that and hence as a
through the process of simplification
utilitarian area of knowledge. There is
and explanation. These are what
a consequent over-focus on concrete
the child must reproduce exactly in
materials and models. The above are
the way they are given to her.
however nuanced and have to be read
together with other statements of the – Mathematics definitions, shortcuts
NCF. While Mathematics does relate and solutions to problems are to be
to our lives and is always manifest in explained and given to children and
it, yet the form that it is seen in daily that they would then memorise and
life has to be generalized and the remember them.
underlying Mathematics abstracted – Repeated practice of the same
from the experiences. It must be algorithm on the same kind of
delinked from any particular context problems can lead to conceptual
and from physical materials that may understanding.

7
Voices of Teachers and Teacher Educators

– In total therefore, Mathematics mapping and projecting and so on. It is

is a subject in which algorithms, not only adults who learn and use all
facts and in fact even solutions to these but also children.
problems have to be remembered These are not however, learnt or even
and reproduced. used in the formal way as in school and
These perceptions have led to a later Mathematics. Mathematics learnt
classroom methodology that believes and used in everyday life is often not
that practice makes a learner perfect. An useable in all situations and contexts
often quoted couplet about the constant that formal Mathematics considers
rubbing of a rope against a stone for similar. For example a newspaper
a long time resulting in a mark on the vendor can track the papers sold and
stone is used as an illustration of the money collected; a vegetable seller or a
maxim. We need to examine what this paan seller can do all accounts needed
view means, especially with reference for maintaining the shop, but would
to our ideas and understanding of perhaps find it difficult to do the same
how children learn. It also leads to type of mathematical processes when
strategies that are stratified as most placed in another shop, even with the
children considered not destined for price list.
Mathematics be only given short cuts It is not that the Mathematics learnt
to memorize and attention be given to in daily life is not at all generalised and
those who are learning and hence also abstracted. The situations presented
appear keener. are varied. The forms of generalisation,
The above described commonalities the way of reaching answers, the way
run counter to not just the nature of communicating, etc are all however,
and purpose of Mathematics, but also different from formal Mathematics.
principles of pedagogy, articulated The important point is that some
in the NCFs, particularly in NCF Mathematics is known to all children
2005. NCF 2005’s view, aligned with and each child not only acquires that,
learnings from across the world, but develops her own strategies to deal
argues for conceptual and procedural with the Mathematics that she needs.
understanding arrived through solving In this sense Mathematics has
making new problems, alongwith an important role in our life and our
formulation of own definitions and experience of it helps us learn more of
arguments. All this has ramifications it too. This also tells us that human
for the role of the teacher, the teaching- children have the ability to learn to
learning process and the nature of generalise, abstract, visualise and deal
resources . with quantities.

The knowledge of children prior to 4. Learning Formal Mathematics

school: A robust system to help all The above discussion suggests that
students learn Mathematics is helped Mathematics when acquired in a
by recognition of the extent to which generalised and somewhat formal
Mathematics penetrates our lives. way would develop abstract thinking,
Any approximate or exact task of logical reasoning and imagination.
quantification, requires enumeration, It would enrich life and provide new
estimation, comparison, scaling, use dimensions to thinking. The struggle
of the operations, conceptual ability to to learn abstract principles would
deal with space and spatial relations develop the power to formulate and
including transformations, visualising, understand arguments and the

8
 Voices of Teachers and Teacher Educators

capacity to see interrelations among widening the manner of organising

concepts. This enriched understanding and analysing experiences. Helping
would also help us deal with abstract learners find mathematical ideas and
ideas in other subjects and in our find instances of where mathematical
lives. Supporting us to understand and ideas may be useful and what they
make better patterns, maps, appreciate can tell us. We would like them to
area, volume and similarities between be able to create new ways to solve
shapes and implications of their sizes. problems and develop the ability to find
As Mathematics includes many aspects solutions to new problems. In order to
of our life and our environment, the be able to recognise situations where
symbiosis, inter dependence and inter- mathematical ideas may be useful,
relationship between learning and we must be able to convert them into
using Mathematics in life needs to be mathematical descriptions and present
emphasised. them as mathematical expressions
We also need to ensure that even or statements that can be solved and
though Mathematics deals with a lot interpreted.
of symbols, abstractions, logic, spatial The background of children and
perceptions, generalisations, patterns teachers, kind of schools and the
and rules, it must not appear as education system that exist for
difficult and meaningless to children. learning, and choice of materials and
A classroom must give children a tasks: While the principle of universal
feeling that they are doing something education and equity in the nature and
meaningful that relates to reality
quality of education demands that all
around particularly at the primary
learn Mathematics and the evidence of
school stage without getting trapped
engaging with Mathematics in daily life
in utilitarianism. We must embed
shows that all can learn it.
Mathematics and problems in it, for
The current common belief and
children to do through meaningful
enjoyable situations and space for attitude about is that Mathematics can
childrens’ creativity and allow for be understood only by those few who are
multiple strategies thereby valuing bright and ‘intellectually’ well endowed.
children’s articulation and logical Only they can understand and for the
formulations, even though not fully rest therefore it has to be memorisation
aligned. The classroom process must of solutions, or short cuts or algorithms
allow and demand that children create and formulas. NCF 2005 recognises the
tasks, questions and problems for the abstract nature of mathematical ideas
classroom discussion. The NCF 2005 and the distance some children may
would suggest that the way forward have from some of these, but considers
is towards Mathematics class-rooms these as constraints to be overcome
where children have opportunities to through concern and respect for the
explore mathematical ideas and models child, her knowledge, language and
not necessarily nor primarily concrete. culture. In all pre-NCF 2005 documents
The purpose of Mathematics (except perhaps the NCF 2000, where it
learning thus becomes developing the was not so clear) one common feature
ability to explore mathematical entities was the impression that Mathematics
and add to what is known. A growing is not easy to learn and that many
ability extending beyond the classroom children will fail to learn it. The kind
to help the learner mathematise of ideas about the resources and the
her experiences. Using concepts to methods to help children learn were
perceive the world differently and also different.

9
Voices of Teachers and Teacher Educators

5. The curriculum and the content is open to change with challenges and
of Mathematics:While there is an new situations.
agreement that Mathematics must Constructivism does require that
be learnt by all, but what to teach in Mathematics classrooms consider
Mathematics remains contentious. learners as naturally exploratory, keen
The emphasis in Nai Talim and in the to learn and act. They need tasks to
Kothari Commission was different stretch their minds and challenge
as being core in one and useful for logical abilities, through discussions,
Science and engineering in the other. planning, strategizing and implementing
them. The Mathematics classroom
National policy on education 1986
should not desire blind application
focussed on this and ability to use
of not understood algorithm or one
Mathematics in daily life. The NCF
way of solving a problem but suggest
2000 underlined the importance of many alternative algorithms and
the utilitarian purposes for life of expect learners to also find new ones.
Mathematics. With the scope largely Problems with scope for many different
around numbers and their use in correct solutions must be included to
market, in mensuration or other develop nuanced understanding of
areas in life. concepts. Class room must involve all
The NCF 2005 with an emphasis children and give space to do things at
on abstraction, use of logical forms, their own pace and in their own ways.
grasping, discovering, creating as Besides, children need opportunities to
well as appreciating patterns and solve problems, reflect on solutions and
new ideas brought in new focus on examine the logical arguments provided
mathematisation giving a dialectic to evaluate them and find loopholes.
relationship to Mathematics to its Learning Mathematics is not about
daily use and opened space to discover remembering solutions or methods
Mathematics. Focus also shifted but knowing how to solve problems.
to developing concepts and new Problem solving provides opportunities
algorithms and learners’ own ways of to think rationally, understand and
solving problems. create methods motivates students
Since the NCF 2005, there is to become active participants and
talk without clarification of making not passive receivers. This can help
the Mathematics class-rooms learners abstract, generalise, formulate
constructivist. Constructivism in and prove statements based on logic.
classrooms cannot mean that children In learning to abstract children would
rediscover all knowledge or form also need some concrete materials,
curriculum. The school Mathematics experiences and known contexts
has to be suitable and useful for the scaffold to help them. In Mathematics
stage of the children. Constructivism we need to separate verification from
can shape the manner and pace of proof and explanation from exploration
transaction within overall goals and and recognising that.
expectations. Besides classrooms Mathematics not only helps in day-
are not about individual children to-day situations, but also develops
but about collective teacher assisted logical reasoning, abstract thinking and
learning. Constructivism here means imagination. Enriched understanding
for each child space to think, formulate developed through it helps us deal
ideas, descriptions and definitions. A with abstract ideas in other subjects
conceptual structure for the child that as well. These two dimensions have

10
 Voices of Teachers and Teacher Educators

implications for the resources to be It also must be recognised that the

used and the way of using them. nature, specific purpose and use of the
materials evolve as we move to different
6. Ideas on Resources in the
stages of learning Mathematics.
Mathematics Classroom
This movement may not be entirely
The idea of materials in Mathematics linear, but as we go towards building
classrooms has become fashionable. of mathematical ideas the nature of
While, the nature and structure of what may be constituted as a concrete
classrooms, remains unchanged, experience changes and slowly the
terms such as ‘engaging materials’ and requirement and limitation of using
activities are widely used. The need concrete experiences to describe
to interrogate the nature of materials mathematical ideas starts becoming
is more for Mathematics as it is not evident. Materials and concrete
empirical or experimental. Empirical contexts are pegs to create temporary
proofs like showing by measurements models and to help visualise and
that the sum of the angles of a few manipulate mathematical objects in
triangle is 180 degree do not constitute the form of their concrete models till
a proof. Mathematics being abstract learners can do without them and deal
expects us to separate objects of with mathematical concepts without
Mathematics from their context. For mediation.
example, numbers are independent As Mathematics develops
of objects and the unit-ten-hundred hierarchically the earlier learnt abstract
system and operations on it not limited ideas become concrete models for
to bundles and sticks, place holder further abstraction and formalisation.
cards or any other models. Base 10 system becomes the concrete
Resources for Mathematics model for generalised representation
teaching must align with the purpose system in any base. Numbers and
of Mathematics teaching, nature of operations on them generalise to
Mathematics, assumptions about the algebra. Geometrical ideas generalise
learner and teaching learning process. from objects to nets to representations
Printed materials and other concrete as faces and edges on two dimensional
objects are examples of materials but, surface and their co-ordinates through
their use is different in Mathematics. point, line, plane, etc. The 3 D system,
The other important point is that physically visualisable, moves to a
almost everything around us can be n-dimensional system that has to be
used as a concrete material to support mentally visualised. We now look at
learning of Mathematics. Stones, some aspects of Mathematics and
flowers, leaves, water, etc can help resources for it and different levels.
quantification. Some objects can aid in a) Teaching Mathematics in primary
visualisation when placed in different classes
positions at different angles, ways and
We know that conversations around
then trying to anticipate how they would
matematical concepts with opportunity
appear when changed. The materials
for the learner to express and get
presenting models as scaffold, help
feedback is useful for development of
build foundations of mathematical
thought and conceptual structures.
ideas. Obviously, the use of these has to
In addition concrete experiences
be linked to the textbooks, worksheets,
and memories can form the basis of
etc.
generalization and abstraction.

11
Voices of Teachers and Teacher Educators

These are useful whenever new ideas to solve problems. It must point out
are introduced but, are critical for the that many alternative adapted working
primary stage. They change according algorithms and strategies exist and
to the concept and the maturity of the problems can have many different
learners and in primary classes would correct solutions and their analysis.
be informed by the struggle of children The classroom must involve all children
to read and use the textbook. One point and give them space.
in that is the language may differ from The obvious question is, can this
the common languages of learners, be possible? The elements that are
but there are difficulties with the way required have been mentioned above.
arguments are formulated as well. This To put theses together, the teacher
implies an important role for language needs to have scope for choosing what
to help children learn concepts and she needs to do and is capable of and
more Mathematics. then ask how she would proceed to fulfil
In the absence of prior, familiar, above suggestions. Some suggestions
easily usable and available abstract as examples are: Divide grade 2
conceptual structures with the child, children into groups of 6-8 with one or
concrete models are critical for the two dice and a pile of stones for each
primary classes. The recognition group. Each child throws the dice and
of Mathematics as a discipline picks up stones on each turn to see who
emerging from some basic axioms has more after 5 turns. They slowly
and assumptions based on logical abstract that two numbers together
procedures requires concrete produce a third and relate to these and
experiences and learners engaging with other numbers as abstract entities.
classification, matching, counting, etc. Similarly fractional numbers
For this they may use their bodies and children can have many relevant
parts, stones, leaves, any other object examples from daily experience using
in the classroom, games, scores, etc. unequal and equal parts of a whole.
as means. Dice with a Ludo or a snake They can list and consider these. For
and ladder board gives many such example, including concrete models
experiential opportunities. Pictures where the fractional number is
can be used for various comparing and more than one whole (e.g. 3/2) and
matching tasks, identify groups that recognising that the nature of whole is
have more objects than the others. important.
Each of these examples has a different Experiences of spatial relations,
degree of concreteness. However, for transformations, symmetry, congruence,
Mathematics 5 chairs represent 5 as patterns, measurements through
well as 5 stones can. Five, is just the opportunities to play with shapes,
name of an idea relating to certain etc. build foundations for further
other ideas in a specific way. Concrete development of ideas. The tasks for this
representations need to use different can include building shapes, arranging
models for the varied situations and them, observing and anticipating
nuances of the concepts. transformed forms, using themselves
Promotion of thinking, exploring and surroundings as data sources for
answers, their comprehension and organisation and presentation of simple
analysis would not expect or follow from data using for example counters. In
a blind application of un understood all this we know some children start
algorithm and should encourage enjoying the play with abstract entities
children to find many different ways sooner than others. Also different

12
 Voices of Teachers and Teacher Educators

children enjoy play with different the characteristics of Mathematics.

abstractions though all eventually The concepts no longer be imbued
began to use logic and understand with materials and tasks that are
Mathematics. also different. The empirical and
b) Teaching Mathematics in Upper measurement aspect giving way to
Primary classes logic and proof. Concept building now
would require more dialogue and at
Upper primary Mathematics is linked
most consciously temporary modeling.
to experience, but moves further
The problem formulation and solving
towards abstraction. Children yet
must expand here. Not remembering
need context and/or models linked to
solutions or methods, but knowing how
their experience to find meaning but
to solve problems, thinking rationally
eventually must work just with ideas.
to create methods as well as processes.
This challenge is to engage each child
This motivates and makes active
through context and move her from this
participants to construct knowledge
dependence. So while the child should
rather than being passive receivers.
be able to identify principles useable
Problem solving requires students to
in a context, she should not be limited
select or design possible solutions and
to contexts. At this stage they may be
revise or redesign the steps, if required.
asked to build their own models and
These are thus essential parts of the
use them as supports.
Mathematics classroom program.
This stage links the more concrete
and direct experience linked with 7. Mathematics Lab and Beyond
Mathematics in primary classes to
a) In talking about resources and
formal, less experience dependent
materials there is a lot of talk about
abstract secondary Mathematics. This
Mathematics laboratory. It is important
stage must acknowledge that many
to better understand the purpose of
people after school would take different
materials in learning Mathematics and
occupations, most not requiring
the notion of lab so that we use the
formulae and algorithms. What all
possibilities in an appropriate manner.
do require is mathematisation of
We know learning requires experiences
understanding, a deep understanding
related to the concepts being learnt,
and appreciation of Mathematics to
but Mathematics deals with ideas that
sharpen analysis and maintenance of
are eventually with abstract ideas. For
logical thread in thought.
example, numbers are not related to
Ideas like negative numbers,
the objects that are used to represent
generalised fractional numbers,
them, a function not related to the
rational numbers, letter numbers,
curve that depicts it, a triangle that has
ideas like point, line, etc. introduced
points of zero dimension and lines of
and developed in upper primary are
zero thickness can only be visualized in
not easy to model. The representations
the mind, etc.
and examples used to introduce these
With this and the recognition
can be confusing and are inaccurate
that Mathematics relationships are
unless dealt with care and with
not empirically provable or verifiable
changing examples with including
means the the purpose and scope of
warnings about limitation and inherent
Mathematics labs need to be sharpened.
inappropriateness of models.
To illustrate, no amount of measuring
Through upper primary to
angles of quadrilaterals can convince
secondary classes, these models
anyone that the sums of 4 interior
have to be discarded giving way to

13
Voices of Teachers and Teacher Educators

angles would always be 360 degree. Therefore, in the secondary classes,

No model constructed would be free of Mathematics lab can only provide
experimental blemish to show exactly opportunity to help children concretely
360 degree. But this does not lead us to visualise some of the ideas to which
conclude that such figures do not have they have not been exposed earlier.
interior angles with sum equal to 360. Over emphasis on materials and
So what is the purpose of materials in expecting their use to prove statements
Mathematics learning? This example can be extremely misleading and
and question is true for all such uses of become a barrier for an appropriate
Mathematics lab including the oft used understanding of Mathematics. The
and quoted verification/demonstration tasks in the so called “Mathematics
of Pythagorous theorem activity room” therefore must help
(b) This does not mean that there is no children explore ideas and start dealing
use for materials in Mathematics. They with them in more abstract forms.
are for many purposes and stages not It has been pointed out above that
just useful, but essential in helping the drawing of any geometrical shape
learners deal with abstract ideas is a model and a representation. A
initially and concretely visualise them. circle is not the line as drawn, but
Materials in Mathematics learning while the locus of points equidistant from
initially helping children experience the a point in a two dimensional plane.
abstract ideas concretely have to be The plot of a function itself, seems far
withdrawn eventually making the child from a concrete reality, but is actually
constructed them in the mind and a representation of the more abstract
move away from concrete examples. relationship. The plot displays how the
For example starting from an angle or function behaves and shows its form.
a ring as a model of a circular shape to (c) One example of a resource with
a circle drawn on paper, we go through significant possibilities is Geogebra.
different stages of concreteness in This is different from the usual as it
the depiction of the idea of a circle, allows creative exploration of graphs
which can be only imagined as a shape and curves as models of functions and
bounded by a line of zero width. When use geometrical diagrams to represent
we draw a chord and find the angle and explore relationships. For example
subtended at the centre we are dealing marking equal sides and angles,
with representations of lines and seeing symmetry, transformations
angles. Representing a general circle by and congruence, even constructions of
a diagram is crucial to understanding shapes, etc. can be explored through
of the proof of statements about chord Geogebra. It does not demonstrate but
and other properties of a circle. In these allows the user to model what she wants
we are not taking circles with specific to explore. Its effective use becomes
lengths of radius and length of chords, possible only when the user recognises
but the generalized abstraction relation. the abstract nature of Mathematics
In primary classes we encourage the and uses modelling through Geogebra
use of a lot of concrete materials and recognise patterns and reach
this usage must drastically reduce generalisations. Another example is of
through upper primary to secondary using coins, dice or coloured balls etc.
classes. to set up an experimental distribution
Learners may use these of outcomes helps build understanding
representations, but not see them as of chance, independence of events,
being mathematical objects themselves. probability, etc.

14
 Voices of Teachers and Teacher Educators

The Mathematics lab needs tasks embedded in their own language, culture
that make learners explore ideas. The and their daily activities, and make
word itself denotes exploration and the classroom inclusive, participative,
curious thinking not fixed and correct exploratory with simultaneous focus
explanations. Lab therefore has to be on conceptualisation, formulation and
multi-dimensional allowing learners articulation of ideas.
to explore ideas and to add to their Secondly for the textbooks to be
library of experiences. The focus of used in the spirit intended appropriate
the lab must be aligned to objectives guidance, support and enabling ambi-
of Mathematics teaching, not trapped ence for the teachers and the learners
in explanation and telling syndrome has to be available. It should be able
with the recognition that material are to struggle with the notion that Math-
ematics classroom and text book are
temporarily scaffolded to form ideas.
needed for examples of solved problems
All this as a part of the classroom with methods, techniques, short cuts
process and not a separate visit to an and memory devices with guidance on
exotic location called mathematical lab how to use and replicate them. The
or something like that. textbook would be organised assuming
The idea of the Mathematics lab, that since Mathematics is hierarchi-
therefore, has to be in conjunction with cally organised, learning would be or-
the nature of mathematical ideas and ganised similarly and once a topic has
questions of what materials, till when, been covered, it can be revised by doing
for what and how they should be used problems similar to the ones done ear-
need to be considered. lier.
In the alternative perspective of
(d)Text Books as a resource in Mathematics textbook must help the
Mathematics learner engage with mathematical
For organised transaction of knowledge ideas in different ways and experience
along a certain syllabus, including the nuances. The materials should help
content, abilities and perhaps elaborate and interlink her concepts
dispositions as well, textbooks are embedding them in her language
a necessary evil. The text books in and experience. Hierarchical nature
Mathematics are reputed to be dreary requires spiralling not linear sequence
and unattractive as they are full of for concepts to become internalised.
numbers, letter numbers, abstract Learners must come back to the ideas
geometrical shapes, unusual brackets explored on multiple occasions in
and symbols interspersed by terse different contexts and in alignment
sentences. Their standard format with different concepts.
suggests that learning mathematical The following note to the teacher
ideas is about seeing examples and from the textbook of the NCERT
then following them to do similar things exemplifies not just the way the book is
without conversation or dialogue. intended to be used, but also the way it
Context and experience is a post has been organised;
concept development application or a “We have tried to link chapters with
mere entry point. each other and to use the concepts learnt
The textbooks of Mathematics in the initial chapters to the ideas in the
have to act at a major resource for the subsequent chapters. We hope that you
teachers to create engaging processes will use this as an opportunity to revise
for learners. It has to indicate how to these concepts in a spiraling way so
bring and use the experiences of the that children are helped to appreciate
children, their mathematical ideas the entire conceptual structure of

15
Voices of Teachers and Teacher Educators

Mathematics. Please give more time to started changing recently particularly

ideas of negative number, fractions, after the NCF 2005, both at NCERT and
variables and other ideas that are new in some states however, many remain
for children. Many of these are the basis in the earlier framework.
for further learning of Mathematics. Some of the books also have
For children to learn Mathematics, be illustrations that show children engaged
confident in it and understand the in doing Mathematics differently. They
foundational ideas, they need to develop are shown using resources, chatting
and discussing with each other,
their own framework of concepts.
workout, exploring, imagining and
This would require a classroom where
visualising.
children are discussing ideas, looking
The way the textbook is to be
for solutions of problems, setting new
used and the nature of the classroom
problems and finding not only their own
comes out well from this excerpt form
ways of solving problems, but also their
the NCERT book “There are many
own definitions language they can use
situations provided in the book where
and understand. These definitions yet
children will be verifying principles or
need to be as general and complete as
patterns and would also be trying to
the standard definition.”
find out exceptions to these. So while
The indication is explicit on
on the one hand children would be
spiralling, on the need for greater time,
expected to observe patterns and make
multiple contexts and nuances for
generalisations, they would also be
internalising mathematical ideas and
required to identify and find exceptions
the way the classroom conversation
to the generalisations, extend patterns to
architecture should be organised as
well as the expectations from what new situations and check their validity.
children would achieve in this process. This is an essential part of the ideas of
As Mathematics in school moves Mathematics learning and therefore, if
across grades its structure and you can find other places where such
organisation must be such that the exercises can be created for students it
same ideas do not occur concentrically. would be useful.” The key points here
The textbooks and other materials being that children are expected to
must require Mathematics teacher explore, think and work out the answers
to think and reflect on her class- to the problems. It expects teacher
room experiences and not move and childrens to create exercises and
mechanically. Such textbooks also call suggests that the teachers should look
for reasonably long and well-structured for more places for problems that could
orientation program. They should pool be located, created or found.
new ideas and develop new activities to
supplement the textbook. (8) Summing Up:
The language used and the nature Mathematics subject in secondary
and extent of illustrations in it help classes includes elaborating and
reduce terseness of the textbook, consolidating the conceptual edifice, to
making it comprehendible for the make logical and organised arguments,
learner. The flow of the book must precisely and concisely formulate ideas,
aid the learner to pause, reflect and
to perceive rules and generalization
engage with it. It must expect the
and found mechanisms to prove them.
learner to articulate ideas, concepts,
explanations, generalisations, Go beyond numbers to understanding
definitions and attempt to prove or abstract number systems, their
disprove mathematical statements. The properties and general rules about
nature of the Mathematics books have them and similarly in other areas.

16
 Voices of Teachers and Teacher Educators

Gradually a broad and tenuous international thinking, the NCF 2005

agreement on the universal purpose has enlarged the scope of this with
and scope of concepts for Mathematics focus on mathematization to attempt
in secondary classes has been arrived enrichment of the scope of thought
at. Spelt out in the NCF 2005, it is and visualisation. The secondary
however, yet to reach the classrooms. school Mathematics, therefore, on
With multiple formulations of its the one hand, needs to focus on
implications and approaches for the consolidation of the conceptual
the materials and the classroom edifice initiated in the classes 6 to 8,
architecture and processes there is no but also take it forward to help child
consensus on strategy to be adopted. On explore wider connection and deeper
deeper analysis some of the differences understanding. The logical formulation
in strategy seem to emerge from basic and the arguments included in each
purpose and perspective differences. step along with the precision of
The unfortunate linking of a child presentation is of value to engage with
engaging meaningful program to a the world in more forceful manner.
confused terminology of child-centered To summarise, a much
or constructivist program has led to a larger number of students are now
feeling that a Mathematics or a school attempting Mathematics as a part of
program could be evolved based on what their secondary program as we push
children want to do on any particular towards universalisation of secondary
day with an overdose of materials education. The purposes of teaching
and physical activity. In a pragmatic Mathematics, the pedagogy for it and
formulation of meaningful school the materials for secondary classes
Mathematics program constructivism has somewhat evolved over the last
would imply the child space to think decade or so, but there is a need to
in the classroom, formulate her ideas, put all this in some framework and
her descriptions and definitions with much more thinking and clarifying is
an attitude and a conceptual structure needed. All this has also brought forth
that is open to changes when presented the need for context and resources in
with new kinds of challenges and the secondary classrooms which were
situations. earlier devoid of these. The materials
We know that for many children in the class-room would not only be
class X is a means to study further, but an aid to scaffold introduction, but
in the context of Indian education, the also of engagement with concepts.
secondary classes are the final year of Like in the upper primary classes and
general education and after this students in fact now much more than that, the
would go in to different roles. A complete students need to be asked to create
general education requires a rounded contexts and resources and present
up Mathematics understanding and them rather than being given materials
capabilities (not mere skills) that are to manipulate as would be likely in the
needed by all citizens. In line with the primary and occasionally in the upper
primary classrooms.
References
1. National Curriculum Framework – 2005, National Council of Educational Research and Training,
New Delhi
2. National Focus group- Position Paper on Teaching of Mathematics (2006), National Council of
Educational Research and Training, New Delhi
3. Mathematics, Textbooks of Mathematics for classes VI, (2006), (pp ix & x) National Council of
Educational Research and Training, New Delhi

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