Small- group students’ involvement in a mathematical activity and its relation
to their beliefs about cooperation
Petros Chaviaris & Sonia Kafoussi
University of the Aegean, Greece
chaviaris@rhodes.aegean.gr , kafoussi@rhodes.aegean.gr
Abstract: This study investigates the way by which different groups of students participate in a mathematical
activity, in order to establish common solutions. The groups of students were formed in accordance to their beliefs
concerning the cooperation in the class of mathematics as well as their conceptual level in this subject. We are going
to analyse some representative dialogues of these groups’ work.
Résumé: Cette étude recherche la façon de laquelle des groupes différents des élèves participent dans une activité
mathématique, afin qu’établir des solutions communes. Les groupes des élèves ont été formes par rapport à leurs
vues concernant la coopération dans la classe des mathématiques ainsi que leur niveau de connaissance dans ce sujet.
On va analyser des dialogues représentatifs du travail des groupes.
Introduction
As it is accepted nowadays, mathematical learning is characterized as both an individual and
a collective process. Mathematical meaning can be considered as a product of social interactions
in the class of mathematics, where all the members negotiate meanings by continually modifying
their interpretations (Cobb & Bauersfeld, 1995- van Oers, 2000). Moreover, “the participant of
the interaction monitors his action in accordance with what he assumes to be the other
participants’ background understandings and expectations” (Voigt, 1994, p. 176). That is,
students’ individual beliefs about their own role, the role of the other members of the classroom
as well as the nature of mathematics influence their mathematical behavior. Therefore, the study
of potentiality that a mathematical activity gains through the cooperation between pupils presents
a particular interest. The understanding of how a mathematical activity is productive for different
small groups of learners is useful for the mathematics’ teachers, especially for the design of a
cooperative teaching.
A lot of researches have focused on the relation between the composition of the group and
the productive work. Their basic conclusion is that the most productive is the work in the group
that is consisted of students who don’t have a big divergence in their conceptual level in
mathematics (Webb,1989-Cobb,1991-Good et al.,1992). Furthermore, Murlyan(1989) has
mentioned the connection that may exist between interlocutors’ beliefs about mathematics and
especially about cooperation and the behaviors that they present during their involvement in
mathematical activities. Nevertheless, the influence of students’ beliefs about cooperation in
mathematics in the development of constructive dialogues during the working groups needs
further investigation.
The purpose of this study is to investigate the relation between the way that students in small
groups behave when they are engaged in a mathematical activity and their beliefs about
cooperation in the class of mathematics in primary school. More specifically, this paper is
focused on the way that students with similar or different beliefs about cooperation in
mathematics, as well as with similar or different mathematical conceptual levels, behave when
they are engaged in a mathematical activity for establishing common mathematical solutions.
This study constitutes the initial phase of a wider research program that aims at investigating the
way by which students develop their cooperation in mathematics while they are discussing about
problems of cooperation and they are establishing targets for its improvement.
Method
The collection of the information was based on the tape-recorded dialogues of six
representative pairs of students in a fifth grade class of a public school of Athens, who were not
systematically involved in cooperative teaching mathematics before the research program.
Firstly, the students were asked to complete a questionnaire concerning their beliefs about
cooperation in the class of mathematics. More specifically, they answered the following
questions:
1.The children of a fifth grade class were asked about how they would like the lesson of mathematics to be done.
Some of the pupils gave the following answers:
A. I would like to
B. I would like
C. I would like one of
D. I would like to
have some silence
the teacher to help my classmates, who is
solve the problems
in the class, so that I me more, so that I good at mathematics, to quickly so I could
could solve the
would solve the
help me solve more
help someone who
problems on my
problems more
quickly a problem I
finds them difficult.
own.
quickly.
find difficult.
With which of the previous answers do you agree the most and why?
E. I would like me
and my classmates
to discuss about
and solve all the
problems together.
2. Do your classmates help you in mathematics in class? If they do, write down how they help you. If they don’t then
write down why they don’t help you.
Having positive beliefs about cooperation, were considered those students who agreed mostly
with the fifth option and some of those who agreed with the fourth option but declared receptive
to help from their classmates (e.g. they help me, they ask me where I have difficulties in, they
explain the problem). Then, the students were separated in pairs according to their beliefs on the
previous topic as well as to their mathematical possibilities, which were determined by the
classroom teacher’s comments (high, average and low). Each group has to be engaged with the
same mathematical activity: Six children want to have a fair share of 10 chocolates. How much
chocolate does each child take?
The transcripts of the dialogues provided the data for the analysis of the students’
mathematical behavior. The analysis of the conversation between the children was based on the
interactivity analysis that Kieran & Sfard have developed (Kieran, 2001) allowing us to detect
critical moments concerning the students’ discourse1. Furthermore, students’ phrases were
analyzed with regard to the types of talk as they were described in Edwards’ work (2002).
Specifically, the characterization of every phrase was done according to the dialogue’s
organization, the type of help, the way of explanation and justification, the negative behavior (off
task work, silence, anger etc).
Results
In order to reveal the different ways by which the mathematical activity was treated by the
different pairs of students, with positive or negative beliefs about cooperation in mathematics, we
are going to present three illustrative examples of the dialogues between the students. The first
example refers to the difficulties experienced by children with different beliefs about
cooperation. The second one illustrates the characteristics of a productive cooperation between
students with positive beliefs about cooperation in mathematics. Finally, the third one points out
obstacles that might make the cooperation between students with positive beliefs about it
difficult.
First example: Giorgis is a student who has declared that he wishes to work on his own in
mathematics and that he usually doesn’t accept help from the others, unless the problem is too
difficult. On the contrary, Eni has declared that she would wish to discuss about the problems
with her classmates and that they would help each other. Concerning their conceptual level in
mathematics, their teacher had characterized both students as “average”.
Giorgis:[1] Let’s do this: 6 children will take 10 chocolates; since the children are less than the chocolates,
chocolates will be left.
Eni:[2] 4 chocolates will be left.
Giorgis:[3a] That’s right, 4 chocolates left that nobody has eaten…
[3b] They’ll cut 3 of them in 6 pieces that is each one in 2 pieces and the last chocolate they’ll cut it in 6 pieces and
each one will take 1 piece. This way, they will share all of them.[3c] What do you say?
1
According to their analysis, every utterance of the conversation is interpreted to be either reactive (responding to a
previous utterance) or proactive (response inviting) and is represented on a flowchart by an arrow upward or
downward correspondingly (cf. appendix).
Eni: [4] I didn’t understand what you said.
Giorgis: [5a] The children are 6, each one will take a chocolate and 4 chocolates will be left. [5b] all right so far?
Eni: [6] Fine.
Giorgis: [7a] They will divide the 3 chocolates in 2 pieces each, and so they will make 6 pieces and each child will
take 1 piece. They’ll cut the 4th chocolate in 6 pieces and each child will take a small piece, so this way they will all
have eaten the same amount of chocolate.[7b] Fine! We found it.[7c] All right?
Eni: ...
Giorgis: [8a] Did you understand it the way I solved it? [8b] You didn’t understand it! It was easy.
Eni: ...
Giorgis: [9]Eni, what’s your opinion?
Eni:(she looks embarrassed, she is bent over her notebook, avoiding to look at Giorgis and the researcher.)
Researcher: Eni you can ask Giorgis anything you want, if you disagree with the way he solved the problem, if you
want him to explain to you something more…
Eni: (hesitating)…I don’t have anything to ask.
Researcher: Giorgis, do you want to try to persuade Eni about your solution?
Giorgis: I don’t know. Whether Eni will be persuaded if my solution is correct or not, that depends on her, it doesn’t
depend on me. This is mathematics; I am not inside Eni (he points at his head).
Their dialogue can be divided in two phases:
During the first phase (ut. 1-7b), most of Giorgis’ phrases are proactions which are connected
to Eni’s corresponding answers (reactions) (cf. appendix). Giorgis is presenting his solution to
Eni (1+1/2+1/6), while at the same time he cares about his interlocutor’s opinion (ut. 3c,5b) and
Eni shows interest in Giorgis’ solution (ut. 2). Nevertheless, we could remark that he doesn’t
justify the steps he has taken to be led to the solution, not even when Eni doesn’t understand it.
He seems to consider that the only way to help Eni is to repeat his solution at a slower pace
(ut.5a-7c) and not to present it in a different way. This is confirmed by his allegation that the
understanding of the solution is Eni’s own problem and that he is not responsible for that.
During the second phase (ut. 8-9) the conversation was led to failure. A critical moment for
this failure was Giorgis’ phrase: “Did you understand it the way I solved it? You didn’t
understand it! It was easy”. Although it seems that he keeps caring about continuing his
cooperation with Eni (ut. 9), the way he expresses his invitation revokes his intention. The
emphasis he puts on the fact that the solution was his, his surprise that Eni didn’t understand it
and the characterization of the solution as “easy” created negative feelings to Eni concerning this
specific cooperation. Eni seems sad and stressed and doesn’t correspond to Giorgis’ proactions.
Giorgis’ negative beliefs about cooperation seem to have a negative influence on the
evolution of their meeting. Such a point of view, according to which the understanding of
mathematics is an individual procedure and is not produced during the sharing of ideas, leads not
only to unsuccessful cooperation but can also create negative emotional reactions about
mathematics.
Second example: Constantina and Giannis are both students with positive beliefs about
cooperation and their teacher had characterized both of them as “very good” in mathematics.
WHAT IS SAID
Constantina:[1] May be the division?
Giannis:[2] First of all, the chocolates cannot be divided because the children are 6. We
must consider the fractions. …(silence for 1 minute)
Constantina:[3] What are you doing?
Giannis:[4] I’m trying to find a number that will give me 40, but it doesn’t work out.
Constantina: [5] Why 40;
Giannis: [6] 100/10 all of the chocolates, 60/10 the children are sharing, from one
chocolate each. But they cannot share the 40/10, because 6.7=42. We need 2/10 more.
Constantina: [7] I don’t understand you. I’ m not very concentrated.
Giannis: [8] One way or the other, it doesn’t work out.
…(silence for 30 seconds)
Constantina: [9a] If we cut in the middle each one of the 4 chocolates that are left, …[9b]
8 pieces, 6 children,[9c] it doesn’t work out.
Giannis: [10a] If we cut the 8 pieces in the middle as well, …[10b] 2.8=16, it still doesn’t
work out[10c] 6.2=12, and there are still 4 pieces left.
Constantina:[11a] If they cut each chocolate in 3 pieces, [11b] they will have 12
pieces…[11c] each one will take 2(pieces)
Giannis: [12] So they will take 1 chocolate and 2…
Constantina:[13] and 2/3.
Giannis: [14] That’s right, 1 chocolate and 2/3 of a chocolate.
Constantina:[15] I think that’s it.
Giannis: [16] Should we check it again?
Constantina:[17] Yes.
Giannis: [18a] 6 times by 1 2/3 …6 and 12/3…6 plus 4…10. [18b] It’s correct.
Constantina: [19]That’s fine, we found it.
WHAT IS DONE
He is writing
42:6=7
[16]He is writing
vertically
1 2/3
Χ 6
6 + 12/3 =
6+4=10
The previous dialogue can be divided in three phases. During the first phase (ut.1-8) Giannis
is trying to solve the problem on his own and Constantina is constantly challenging him to
explain his solution to her. Their cooperation is then interrupted when Giannis realizes that his
solution is not effective (ut 8). He considers that ineffective solutions cannot form a subject of
negotiation. During the second phase (ut. 9-15), the children seem to have a productive
cooperation. Their dialogue presents a regularity that relates to the way by which they are sharing
their thoughts. More specifically one of them appears to appropriate and extend the other one’s
thought by realizing the corresponding actions: conjecture- application- evaluation. (ut. 9a,b,c10a,b,c-11a,b,c). During the third phase they agree to check the validity of their solution.
Third example: The previous dialogue constitutes a representative example of a productive
cooperation between students with positive beliefs about it. Nevertheless, there were cases of
groups with similar characteristics whose cooperation was not equally productive. Anastasia and
Stefanos had both positive beliefs about cooperation and their teacher had characterized both
students as «very good» in mathematics, but their dialogue had a different evolution than the
previous one2. At first sight the two students seemed to communicate and to cooperate
effectively. The proactive and the reactive utterances were equally divided between the two
interlocutors. At the beginning, the conversation developed around the question of whether the
operation of the division (10:6) could give a solution. The wrong interpretation that both students
gave to the relation between the rest and the dividend led them to confusion. Then they tried to
use an informal strategy (to cut the 4 rest chocolates in pieces) but they didn’t trust it so they
searched again for the security of formal division. This interchange between formal and informal
strategies that the students were using to solve the problem restrained the evolution of the
cooperation, because of their restricted beliefs about the nature of mathematical activity (I apply
rules and operations). Both students laughed when Anastasia suggested they cut the chocolates in
pieces, as if that was a suggestion that wasn’t related to any mathematical activity. This reaction
demonstrates the restriction of their cooperation into negative beliefs about mathematics.
Concluding remarks
The results of the previous research indicate the following:

The students’ beliefs about cooperation in mathematics (positive- negative) influence its
development.

The students’ competence to present their solutions in different ways, so that these can be
understood by their interlocutors, facilitates the cooperation.

The interlocutors’ belief that ineffective solutions cannot form a subject of negotiation holds
the cooperation back.

The beliefs about the nature of mathematical activity influence the cooperation.

The quality of students’ mathematical solutions doesn’t seem to obstruct the cooperation
between students with similar possibilities.
The wider investigation of more cases of cooperation in mathematics will facilitate the
further analysis of the previous remarks. Furthermore, the study of the evolution of cooperation
in different groups of students for a longer period of time presents a particular interest.
References
2
The dialogue and the interactivity flowchart are not presented due to luck of space.
Cobb, P. & Bauersfeld H.:1995, Introduction: “The coordination of psychological and sociological
perspectives in Mathematics Education”, in P. Cobb and H. Bauersfeld (eds.), The Emergence of
mathematical Meaning: Interaction in Classroom Cultures, NJ: LEA,.pp. 1-16.
Cobb, P., Wood, T. and Yackel, E.:1991, “A constructivist approach to second grade mathematics”, in E.
von Glasersfeld (ed.), Constructivism in mathematics education, Dordrecht, Netherlands:Kluwer,
pp.157-176.
Confrey, J. :1991, “Learning to listen: A student’s Understanding of Powers of ten”, in E. von
Glasersfeld (ed.), Radical Constructivism in Mathematics Education, Kluwer Academic Pubishers,
pp.111-138.
Good, T., Mulryan, C. and McCaslin, M.:1991, “Grouping for instruction in mathematics: Research on
small-group processes”, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and
Learning, New York: Macmillan, pp. 165-196.
Kieran, C.:2001, “The mathematical discourse of 13-old partnered problem solving and its relation to the
mathematics that emerges”, Educational Studies in Mathematics, 46,187-228.
Van Oers, B.:2001, “Educational Forms of initiation in mathematical culture”, Educational Studies in
Mathematics, 46, 59-85.
Voigt, J.:1994, “Negotiation of mathematical meaning and learning mathematics”, in P. Cobb (ed.),
Learning Mathematics: Constructivist and Interactionist theories of Mathematical Development,
Kluwer Academic Publishers, pp. 171-298..
Appendix The interactivity flowchart of the 1st group
Giorgis
1
2
3a
3b
3c
4
5a
5b
6
7a
7b
7c
8a
8b
9
Eni