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Cooperative Learning in Mathematics: Practice in a Nepalese Classroom

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Article · June 2006

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COOPERATN/E TEARNING IN MATHEMATICS: PRACTICE
IN A NEPATESE CLASSROOM CONTEXT
Tiko Rom Pokhrel*

Background He directly started to teach problem from the

mathematics educator I began to recall textbook and he gave us some ready-made
At; forrnulae to recite. He did not explain even the
those past days ofmy school years and days of
rny teaching careers. Especially I starled to notion of CP(Cost Price), SP(Selling Price),
think on mathentatics teaclring events in the Profit, Loss, etc. Every one was unhappy from
the teacher. When I returned to home I read the
classroom to review those days in the light of
related concepts from book and I tried to solve
theories of mathematics education' The story of
the event revitalized me to think on the the problems given for homework. Next day I
was happy in the class because I had known the
possibility of cooperative learning in place of
problems. Our mathematics teacher entered to
what was practiced in course of teaching
rnathematics in those days. I have stated two the class and asked all thb formulas given.
Some of rny friends told the fortnulae but they
representative events here. When I remembered
the following two events and I got to t'ead the
were in puzzle with the concepts in
of cooperative learning, revive an
I also did not tell
understanding the n.reaning.
benefits
interest to apply this approach to the classroom
them even to read book to get meaningful
understanding, for there was not such culture.
situation of Nepal.
I want to describe two incidents that There was no habit of asking for unknowns to
happened in my life of the two separate the teacher. My friends were not able to
understand the whole chapter Profit and Loss in
porlfolios. One happened when I was a student
the state oftransfer ofskills at application level,
stuclying in grade seven and the next happened
when I was a mathematics teacher in a reputed except solving some problenrs what they have
private school. recited as they thought to be important for
exarninations point of view.
Evenr When I was a Student
I remember the day, when our rnathematics Event When I was u Tcocher
teacher was teachittg mathematics to us in I remember the day, when I was teaching in
grade seven. He was teaching Profit and Loss. a reputed private sclrool. Like rne there was
x Teaching Assistcutt, Scfutol of Edtrcution, Kathmandtt Universitt''
 anothei mathematics teacher teach ing at teacher did with rne. I did the same
secondary level. In those days and to him_ ln
still ihese my later life, I realized that these rvere the
days, there is a belief in teachers ,o
t."p up wrong practices and I attempted to reduce
these
supremacy over the rither in the matter
of practices. My personal otseryations
knowledge richness in mathematics.
When one
of the
present situations get the sarne beliefs
could solve the difficult problem is regarded in
as teachers even today in our schools. When
tlre best teachers and teachers wanted to I
be the nreet students and teachers ofour society, most
best personality in the school showing
the of thern are reporting tlre same facts anJ hare
charismatic power of solving matheriatics
the same practices. I arn thinking it as
problem. Due to the lack of sharing a national
culture, problem because our school cur.riculum
several misunderstanding and unheatthy lras not
given space to cooperative and collaborative
practices are run in the schools even L
today. Iearning and not practiced in the schools
Once I had been a victim of this culture.
One
which \
day, rny colleague came with a problem 11_re!uce such problem of personalizing the
and I difficulties and knowledge, a sort of knowiedge
thought I should dominate him as he did capitalism. Another problem is that most
previously to me. I told him that l.don,t of tiie
know. teaqhers and parents are unaware about
After sometime I went,to the class and instigate what is
cooperative learning, its benefits and
the students to put the problem to the
collefue approaches to apply in classroom teachino
who had asked me to help in solving. ihe
When stakehotdeii ai" not assureJ in ,ir. ,r"
students did as I suggested doing. They Jr
d*id like pedagogy or methods in schools, its
that and he was unable to solve the problem continuity
in and positive effect is nor possible. Similarll,
the class. While I was instigating the students
I educators as well as teachers are not much
had told that the teacher would have not
been directed to action research to make reflection
able to solve the problem when and until
I teach and justifo how a particular approactr,imethod
lrim the way of solving. Some of the students
can be applied in Nepali classroom situation.
told the same thing to him. We used to be lt
is my small effort to carry out an action
opponent to each other in the school.
research regarding the use of cooperative
An lssue of Learning Cutture
learning inschool and teacher ejucation
Now I realized that the above incidents
programme. I
made intervention frame and
implernented in actual classroom practice and
hlRRened intentionally because of the process
made reflections on the possibility, challenges
of socialization and sharing culture of learnins
in school. The cause of that sort o"f
and risks of using cooperative learning in
Nepalese context.
uncooperative understanding and
individualized culture is not a day's grown
ups, Intervention Frames and Work
it is the part of socialization process ii courssof
I wanted to develop two frames of applying
learning both in schools as well as at home.
cooperative learning in Nepalese classioo,nl
During student life, we used to be more one for post graduate level ofstudents who
competitive rather that cooperative in course of are
the teacher of mathematics and another is
Iearning mathematics. That belief continued for
school level students. I went through an article
even in our professional lives too. We were
not 'Cooperative Learning in Mathematics Teacher
given the value of group learning, group Education' by Alice F. Artzt (Vol. 92, No.l
discussion, and culture of sharing. We weie oJanuary 1999, in the site: www.nctm.org
not
made realized that we can learn many things ) and
by
3lo1h"l article 'Cooperative Learniig in
discussing within a group of fi.iends. In the Mathematics' by Roza Leikin ancl Zasli,sky
above mentioned event, I applied the principle
(Vol.92, No.3 oMarch 1999, in the site:
"tit for tat" that what anothe, matirematlcs www.nctm.org ) and realized the difficulties
of
 applying things discussed about the use of 3. Complete the record sheet. Be sure to divide
cooperative learning approach in mathematics the work so that if possible, each person in
teaching. With implelnentation of planned the group solves the same number of
activities, I drew reflections and presented here. equations.
Here in this article, I am trying to find the Students worked together on this activity for
difficulties of applying these things into thirty minutes in the respective-groups and each
practice in Nepalese situation. group submitted one assignment sheet with
agreed-on solutions. They were also asked to
lntervehtion Plan.for Posl Graduate Students' Class respond the following questions as home
In the first article, I found that cooperative assignrnent.
learning as activities of grouping, naming the l. Describe how your group worked on the
groups, problem assignment, students w-orking activ ity.
in group, students' presentation and reflection. 2. Describe the nature of the participation in
The gr,ouping was done as students entered the the group. Did people explained rhings to
classroorn; they were asked to sit in a specific one another? What accounted for this level
group of three or four. The group formation was and degree of participation?
also on the basis of student's previous requests i. Describe how the cooperative-leaming
to work with certain people and also ensured activity was designed to create mutual
diversity with respect to gender, academic and individual accountabi lity.
physical ability, personal characteristics, and 4. Describe how the cooperative-learning
ethnic background of the students. In the activity was ,designed to create mutual
process of naming the groups, the students were interdependence.
free to name their groups themselves. They 5. Describe how the grouB formations
named as 'The Wizards', 'The Supreme affected the interactions within your
Mathematicians', 'Bash', 'The Happy group.
Calculators', 'The Solutions', 'lnsight and New I applied this frame to the students of
York's Finest'. Then the stuldents were given the teacher education in mathematics. When the
fol lowing mathematical problem : same activity was designed to my students of
The numbers 2, 3 and 5 are substituted at Graduate Level in Education (mathematics), I
random for a, b, and c in the equation ax + b : divided them into three groups consisting 2 or 3
c where a*b*c. in each. The same activities and instnrctions as
l. What is the probability that the solution is stated above were provided to them. They
negative? worked for thirty minutes and each group
2. What is the probability rhat the absolure submitted one assignment sheet. I had assigned
value of the solution is 1? them the same homework.
3. What is the probability that the soiution is a
proper fraction? InterventionPl an on School Students
4. lf c is not 5, what is the probability that the Another interesting article of Leikin and
- solution is negative? Zaslavsky about cooperative learning for
And the groups were instructed as follows: school students. The cooperative learning was
l. Engage in a group discussion for managed carefully. In this approach of
understanding of the problem and select cooperative leaming setting was made with
possible strategies for solution. exchange of experience and jigsaw methods
2. Work in group to write down all the possible that made the students mutually and positively
substitutions of the numbers 2,3, and 5 for dependent :on one another, mathematical
a, b, and c in rhe given equation. Use the computations increased and ninety percent
record sheet to record vour result. expressed positive attitudes towards this
method of leaming. Being:irnpreSsed from the with school children.'t devtiloped the problem
positive aspects of 'learning mathematics, I set (see figure l-a) to teach ar.u of
thoughtthat I ntlself should apply this method parallelograrn and rhombus in class lX.

Figure l(a)
Problem Cartl
Part-I

Study the' following 'activity and do [rngth

activities in your group. We know that area of
rectangle is length x breadth. Breadth
Or, briefly we write A:l x b. ;
l

Here, we are going to find' arca of a

parallelogram. Take a piece of parallelogram
and cut it as shown in the figure. (Each group of
students gets a paper model parallelogram and a
sc issors)

Cut the parallelogram along line vertically

straight. (The cutting line can be any where in
the parallelogram even in breadth too)

in the figure.

Repositioning the' pieces we obtaiir the

following rfigure.
What figure do you get? Can you find its
area?
(This is a rectangle and its area is base x
height)
What is the area of parallelogram?
(Area of the parallelogram
Area of the rectangle so formed
: Base x Height.
Part- II

Again study the following activities and

learn. Take another piece of parallelogram and
cut it as shown in the figure. (Each group of
students gets a paper rnodel parallelogram and a
scissors)

Cut the parallelogram along vertically

through vertex as shown

Get the two pieces as qhqwn

Move'and position the triangle to form the

figure
Then, : ' 'i..
Area of the parallelogram
' : Area of rectangle so formed
Length x Breadth
Base x Height[ In parallelogram]

Part-III

'To formulate the area of parallelogram we

proceed as follows:
Given: Parallelogram ABCD
To find: Area of Parallelogram ABCD

Construction:
a) Draw DE perpendicular to AB.
b) Draw CF perpendicular to AB produced.
 We first show AADE: A BCF
i) AD : BC I how?]
ii) tDAE: tCBF how?l
iii) ZAED = /.CFB how?l
.'. AADE ABCF how?l
=
:'. AADE: ABCF how?l

Now from figure,

Area of pararrerogram ABCD
r ^_rffiIdiflSs8
= Length x Breadth
:CDXDE
Base x Height
.'. Area of the parallelogram is base x height.

Similar problem card was designed to find the area of triangle. Here it is not given.

I was convinced for management of groups A jigsaw is a cooperative learning

in any number of students in class, with an idea structure in which material to be learned
that if the number of students is odd we can is divided into separate components.
make it even as explained by taking two Groups of students are assigned
average students as one workgroup, I arfr very responsibility ./br each component and
rnuch interested with this style of cooperative learn together how to teach that
learning as exchange of experience in two component. 'lhen, teunts with one
phases of groups namely the group of experts individual is responsible for each
and tle new reshuffled groups of students for component, conte together to teach eqch
exchange of experiences. To apply this method other the entire set .of material. FirSt,
iq teaching mathematics, I selected one section students work together ,to learn how to
of class IX (Sec B) in a school and I decided to best teach the waterial far which they
teach area of parallelogram and rhombus to are responsible. Second, students
them in cooperative learning setting. When I interact in their final teams to teach
entered into the classroom as mathernatics each other what they haye learned.
teacher, I found there were 10 sets ofbenches
and desks in rows and columns with 40 students There were I6 persons taken as number of
sitting with four students in each bench. The set students and they were divided into four groups
of benches and desks were placed in such a way each consisting of4 as described in the article.
that there were two eolumnS and each column Particularly when there are .16, or 36 or 64
consisting 5 sets of benclres and desks. There students, this rnethod can be done very easily.
were 40 students in the class. In the beginning, Nothing is mentioned in the article if the
I was puzzled how many groups should I form? numbers of students vary. But there is a hint
How many to be in a group? How to manage for that
lhose students in cooperative learning activity? the number of students within a group
The rnethod I wanted to apply was jigsaw. This should equal to the number of cards
was an exchange of knowledge method sharing within the learning unit. Each Student
some characteristics with jigsaw method as has gained expertise in his or her own
indicated by Buck and Wage (2005) card.... If the number of students is not
 evenly divisible into groups, a teacher discuss within the group. Each member of the
can let a student who is at a low group should be an expert. This is a chance of
achievement level work with a middle- gaining expertise within their group.
level achiever all the time. This After this, again I have to manage students
pair then works as one student in the for the group ofexchange of knowledge. To do
learning arrangement. They solve the this as in my reading there was necessary of
problem individually and explain making equal nurnber of sets and students in
different cards to their partner in turn. each group. Also this sorl of row setting does
not allow as in the case of round table setting. I
I decided to clivide tlte group of studenls as have to fix them for their partners. I decided
they were sitting in a bench. Heterogeneity of that a student and another student just behind
each group was maintained by replacing himiher would be a pair. This was done for each
students from one seat to another with the help first four sets of groups of students of each
of mathematics teacher. Then I announced that column. Those students who were sitting in last
the students sitting in a bench would form a benches of both rows were asked to sit
group. There were 4 members in each group. alternatively so that they got with a partner of
There were only two sets of problem. (Problem d ifferent sets (fi gure I -b). Mathematics teacher

Card).The alternating set was given to each again helped in this setting. The students then
group for discussion. That is starting from left were instructed to teach turn by turn what they
column, first bench students' group got Set A had learned in the previous setting. Two
and second bench students' group got Set B and students were teaching to his/her partner turn
so on. I was there to help thern as their by turn. At the end of these activities, I gave
requirements. The students were instructed to them a question "how have you leamed in a
study the materials individually first and group and taught to partner?

Figure I (b)
Classroom Setting During Cooperative Learning

Group of Experts

Group A: A, A, A.r A, Group J: JrI"J3I

Group B: Br B, B. 84 Croup I: [l 12 13 11

Group C: Cr C, c, C, Group H: HI H2 H., rt

Group D: Dr D, D, D, Croup G: Gr G2 G.q

Group E: E, E, E-, E4 Group F: Fr F, qF4

 Group of Exchange of Knowledge

Ar4ArAl Jr I, J., I

Br B, 83 84 Il I, I3 14

cr c, c. c4 Hr H2 H. H4

Dr D2 D. D4 Gr G, G. G4

Reflection over the Methods Practiced in students, the hope was that each student
Two Different Groups of Students would solve two equations... ". Next
I collected the reflections frorn the students reflection was shown to my students was
of post graduate after a week and from the "... each member took on a role and then
students ofgrade IX the next day. I analyzed the they all worked together. As one of the
students' reflections over the methods of students explained- because we all had very
teaching under which they learned things in different abilities in our group, we each took
previous day. The reflection of each group of on a different role. Gopal was stronger in his
students is discussed separately. confidence of his mathernatics skill. and he
was explaining the problem. I felt I was
Sttrdents ol Post Graduate asking the questions. Rita was the checkeq
L From the article, I found that'despite making sure everything was OK.... There
detailed directions, students often do things was nice balance. After reading similar
their own ways, which sometimes work out other reflections they speak that we don't
better'. This is also observed with my have habit to work in team cooperatively.
students and I found that only one member We don'i share the work but we talk tnore.
was working for the solution and other were 2. I compare 'the structure of a cooperative-
giving oral suggestions. But the work was Iearning activity has an impact on the
not divided among themselves to tlreir participation of the group members'that my
members. Some students I found were students reflected me that though the
passive workers but active observers. I number was smaller we all could not
offered time to share the reflections the participate actively. One did and other we
students had made over the learn ing knew it by looking it. Cooperative learning
activities. They shared in group. After their activity that was to lead to mutual
reflection I gave them to read the reflection- interdependence and individual
"The problem was designed so that the work is least applicable in my
accountability
could easily be divided among the students. students.
That is it was necessary to solve six J. The reflection about the following three
equations to find the sample space. Since ideas was not mentioned by my students.
each group usually consisted of three That may be the cause that they hurry in
 writing and do not remember all the they have and developed. They develop the
activities of the class. habit of listening to others. In course of
a. Imposing a time restraint on group work learning, students develop an understanding of
can have both positive and negative shared living habit in the community.
effects.
b. The nature and level of difficulty of a Conclusion
problem have an impact on tlre degree I found that cooperative learning is useful
and quality of the discottrse that occur for mathematical learning activities as it
within the small group. developed students' interest in .working
c. Although students are nrembers of the together. lt can develop the positive attitude to
same group, they rnay have different work in group. The learning environment offers
perceptions of how well they worked all members of the group an equal opportunity
together and the solutions at which they to interact and each member feel the
arrived. responsibility of learning. I hope that if the
cooperative learning approaches are frequently
School Students used then such problerns as I faced as a student
After reading the work of the students, I and as a teacher will also be reduced. Besides
noticed that they learn many things working in the benefits of cooperative learning, there are
group. One student, Rohit writes, "l used to be risks of no learning in students. When students
pvzzled with area of parallelogram that area of do not participate in learning activities actively
parallelogram : Length x Breadth. But now we and depends upon the others, this would
all are clear that the area of parallelograrn : damage the learners. Other risk is it could take
Length x Height. We enjoyed with tlre paper more time even to learn a simple mathematical
work and we learned ourselves in group and we content. So, teachers should be aware on risk
teach each other. I am feeling that if we work in management.
such group we can solve most of the problems
of our mathematics book.l' Next boy, Prakash References
writes, "l learned it from my friends because I Buck, J.R. & Wage, K.E.(2005). Active and
have to teach to nty partner later and I knew it Cooperative Learning in Signal Processing
very easily by the help of iny friends". Anil Courses. IEEE Signal Processing
describes that their fruitful talking in the class Magazine, 22(2), 76-81. In
and understanding made tl.rem teach and learn http:/www.foundationcoal ition.orglcol labor
the topic and this made hirn to listen to his ativelearning.html
f,riends. Artzt, A.F.(1999). Cooperative Learning in
From above description, I came to know the Mathematics Teacher Education. Nol. 92,
effect of cooperative learning in students that No. l). ln http://www.nctm.org
students learn more in group work than Leikin, R. and Zasavsky, O. ( 1999)'
individual work. The activities to be undertaken Cooperative Learning in Mathematics.(Vol'
in Group of Exchange of knowledge made them 92, No. 3). ln http://www.nctm.org
realized a sott of compulsion to learn in the
Group ofExperts, for they have to teach to the
partner each other. By the exchange of lv,
,/A\\
experience students can share the ideas that
 Mathematics Education Forurn
Year 10 June 2006 Volume ll lssue 2A

Contents
Topics Authors Pages

qrqT-(frtq ii

1. Geneology of Mattrcmatics:
Pure Verses Applied Dr. Kanhaiya Jha I

2. Mathematics and lndustries:

A Pospective Dr, R.P. Ghimire 8

3. Anxiety at the Developmental Stage of

Learning Mathematics Dandapani Gautam t0
4. An Outlook on Different Levels of
Discourse in School Geometry Dr. Min Bahadur Shrestha t2

5. Concept Mappings Krishna Prasad Pokhrel T9

6. Cooperative Learning in Mathematics:

Practice in a Nepalese Classroom.Context Tika Ram Pokhrel 2t
7. Dice in the Classroom Dwarika N. Amgain 30

8. Trar ifl-grdd fqo.reqT rI:( (Gauss) a1

ftril{ q-cD-Rc'I 3il-dTd 36

9. qrE{rfutr f€rdq ildqr wEqrd

(Proportion) i{ren"r w.gr qGI-IIT;T +s 4l
10. qril{eD'rrwflzFl ffiq-6F lzlzFRI rFT

,idr iurfrtil66
(Some Strategies DeveloPing the

Skills of Mental Calclrlation) fra5fi S'd-ft 45

tt. rriurd ftT&Tr. qffircs 51

12. List of Life Members of MEC 54

 4

6 2 I 5 genelal
concept
gensd
concepl

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qz-
?o (q, iq

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C.0U $6R'm EMATICSEDUCffiION

KATHMANDU, NEPAL
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