Analysing power relationships in collaborative groups in mathematics

Mary Barnes, University of Melbourne
 

Abstract
This paper describes the development of a framework for analysing power relations in small groups of students working on collaborative activities, and is based on an approach to power derived from the work of Michel Foucault. Student-student interactions in two classrooms were observed and videotaped. Key features that emerged were techniques used to control the flow of the discourse in the group and behaviours which influenced the mathematical knowledge constructed. Other factors included gesture and the use of resources.

Introduction

In recent years, collaborative learning has been widely recommended as a strategy to enhance mathematics learning for all students (e.g., NCTM, 1989) and especially girls (Cordeau, 1995; Jacobs, 1994; Solar, 1995). As part of a study of collaborative learning, I am developing a framework for the analysis of power relationships among students working on mathematical tasks, in small groups, with shared goals. This paper reports work-in-progress on this project.

My study has as its main focus studentsí experiences of collaborative learning and the ways in which gender impacts on, and is affected by, these experiences. In particular, I am investigating patterns of interaction among students working collaboratively. In addition I aim to discover how they perceive themselves as learners of mathematics. I chose to focus on senior students around the stage when they make key course choices affecting their post-school options, and their future relationship to mathematics. These decisions may be mediated by the studentsí evolving constructions of themselves as learners of mathematics.

Feminist theory and my own experience both suggest that a study exploring gender effects in collaborative learning needs to take account of the exercise of power within groups, and the potential of this to influence learning outcomes. A framework for analysing power relationships could also be useful in studies involving class and/or race where power and status differences may be salient.


Power and knowledge

What is power and how can it be investigated?

My starting point is an understanding of the nature of power, and an approach to analysing power relations, proposed by Michel Foucault. The questions most often asked about power deal with its nature and sources, (the "What?" and the "Why?" of power). Foucault, on the other hand, chose to ask about the "How?" of power. The shift in focus, from theorising about the sources of power to asking how it is exercised, opens up the possibility of empirical investigation. Questions of the nature and sources of power are not ignored. On the contrary, empirical evidence of the exercise of power may provide insights into these questions and so help us understand how power functions in different contexts.

Foucault claimed that power in modern society is not a commodity, which some possess and others do not. Rather, it is a structure of relationships, jointly constructed, which shapes peopleís actions. "Power exists only when put into action." (Foucault, 1982 p. 219). Furthermore, the effects of power are not all negative: "it induces pleasure, forms knowledge, produces discourse. It needs to be considered as a productive network which runs through the whole social body, much more than as a negative instance whose function is repression." (Foucault, 1980 p. 119). This is particularly important in studying classrooms, where the formation or construction of knowledge is the object of the enterprise.

Power relationships and the construction of knowledge

Systematic observation is needed to clarify the operation of power in pedagogy (Gore, 1997). It is particularly important in a study of collaborative learning, which involves a shift in traditional classroom power relationships. By relinquishing some control over classroom interactions, the teacher shares power with the students. I claim that the exercise of power among students working together on a mathematical activity can influence the construction of knowledge by the groupóboth the personal understanding of mathematics constructed by each individual, and the knowledge which is "taken-as-shared" within the group.

The extent of a studentís influence on a groupís discussions has the potential to affect their self-perceptions of mathematical competence and of ownership of the mathematics constructed; and also how their capabilities are perceived by others. Thus the exercise of power within small groups is potentially important in a study of how students construct themselves as learners of mathematics.

Studies of teacher-student power

A Foucauldian view of classroom power sees it as a relationship between the participants, claiming that there can be no power relations without the possibility of resistance. Manke (1997) adopted an interactive conception of power, taking into account actions of students as well as teachers. But her focus was on the struggle for power between teacher and students, and strategies which teachers adopt to achieve their objectives in the classroom. In a current project, Gore (1997) is using categories derived from Foucaultís work to analyse the practice of power in a variety of educational settings.

Power relations have not been an explicit focus of most studies of interaction in mathematics classrooms, including those dealing with gender issues (Koehler, 1990; Leder, 1990), but some of the findings of these studies suggest the exercise of power by male students. Leder, for example, found fairly consistent differences in teachersí interactions with male and female students, and noted "the pervasiveness of malesí domination of teacher attention" (Leder, 1990 p. 165). Jungwirth (1991) found gender-related modifications of "typical" teacher-student interaction patterns, and argued that their effect was the interactive constitution of boysí mathematical competence and of girlsí mathematical incompetence. Interactions between students were not analysed in these studies.

Power relations within collaborative groups

Forgasz (1995), studying groups working together in two Year 7 classrooms, observed disruptive behaviour by boys, occasional abusive behaviour by boys towards girls, and work-avoidance tactics by boys who left the girls in their group to do most of the work for which all group members would receive credit. These could all be interpreted as ways of exercising power. Forgasz, however, did not explicitly address power issues, choosing to focus instead on autonomous learning behaviours and attributions for success and failure.

 
Procedures

The present study used naturalistic inquiry methods, in order to disturb normal classroom processes as little as possible. Case studies were conducted in two government high schools in large Australian cities, involving Year 11 classes working on elementary calculus. The teachers of both classes used collaborative learning methods, but they implemented them in very different ways. In one class, the teacher first introduced and explained the topic, and then the class worked on a variety of short collaborative activities in which they applied the ideas they had learned. I call this collaborative practice. In the other class, groups worked, without prior instruction, on carefully chosen open-ended problems, and in the process developed the new mathematics they needed. At intervals, groups reported progress to the whole class, so that ideas and methods could be discussed and shared. I describe this as collaborative inquiry.

Data included videotapes of lessons, field notes from classroom observations, and copies of student worksheets. I prepared "rich" transcripts of the videotapes, including descriptions of actions, gestures, facial expressions or voice intonations which I judged relevant. To validate these judgements, a colleague was asked to view a sample of the taped lessons and comment on the information included in, or omitted from, the transcripts.


Indicators of the exercise of power

My aim was to develop a set of criteria for identifying the exercise of power which could be applied to rich transcripts by someone with no information about the gender, class or ethnicity of the participants. This would make it possible for a colleague working from transcripts alone to verify the reliability of my analysis, and so provide a safeguard against any unintentional bias on my part.

I sought a framework for analysis grounded in the data, which could be applied to groups using either collaborative inquiry or collaborative practice. I began by studying and reflecting on the transcripts of the lessons, and re-viewing the videotapes, trying to gain a feeling for the power relations involved. I was able to classify significant influences under two main headings: control of the flow of discourse, and influence over the construction of knowledge. Use of resources, body language and voice emphasis were also used, but in subsidiary ways.

Control of the flow of discourse

A student can control the discourse in a group by influencing the topic to be discussed, including the timing of transitions from one topic to another. A study of the resolution of uncertainty (Clarke & Helme 1997) provided a useful approach to this. Clarke and Helme proposed that a mathematics lesson can be divided into episodes, each defined by a consistent purpose such as the solving of a particular problem. An episode is made up of one or more negotiative events involving identifying a sub-goal, and attempting to resolve it. These are usually initiated by an expression of uncertainty such as the asking of a question.

I found that the transcripts I was analysing did not divide neatly into sequences of negotiative events, each satisfactorily resolved before the group moved on to the next. Discussions were often inconclusive, or interrupted by off-task talk. Nevertheless, negotiative events appear to be a key unit for analysis, because transitions from one to the next mark the progress of a groupís work on an activity. A student who enacts closure of a negotiative event by initiating a new one is exercising considerable control over the discourse.

The transcripts revealed the following ways in which students act to control or influence flow of group discourse: initiating a negotiative event; initiating off-task talk; and rejecting or ignoring off-task talk (by continuing the negotiative event, or initiating a new one). Examples of these are given below:

Explanation of symbols used in the examples:
Ė> the significant turn in an excerpt 

(Ö) an indecipherable utterance 

(by) the best guess for an indistinct utterance 

why emphatic speech

[ ] observations from tape or field notes 

= "latching", i.e., no perceptible gap between speakers, usually experienced as an interruption.

        Initiating a negotiative event.

The person who initiates a negotiative event is attempting to take control of the discussion. Initiation is frequently, but not always, signalled by a discourse marker like "Okay", "So", "Now", "Well" or "Right". In both examples below, the initiation was followed by several turns of discussion of the question.

Example 1:

Ė>D: Well, letís think of things we can do, like what?

Example 2:

Ė>D: Okay, this graph tells us Ö ó no it doesnít tell us ó what does it tell us?

        Initiating off-task talk

Changing the subject is also a way to control the discussion, but as the examples below show, the effect depends on the context and frequency. Occasional off-task talk can provide a break after a period of intense engagement, but the sustained or repeated introduction of irrelevant topics disrupts the group effort.

Example 3:

    B: Mm yeah. I think we plus it together.

Ė>L: Oh, shit. [Looks up and smiles to T (sitting opposite).] Iím so tired.

    T: Go to sleep then.

This was the beginning of a series of 13 off-task turns, terminated only when L herself brought the group back on task by initiating a new negotiative event. Example 4:

    P: How about 2.23, I mean 2.25?

Ė>G: [laughing] I canít be bothered. Itís too hot. Itís too hot!

This exchange occurred when the group had been engaged on the problem for some time and had effectively solved it. Immediately afterwards, they returned to work and completed the solution. Thus Gís aside did not disrupt the group effort, and so did not function as an exercise of power.

        Rejecting or ignoring off-task talk

Example 5:

    L: Iím so tired, man.

Ė>B: Okay, whatís the rest of the question?

Example 6:

    P: I reckon that=

    G:                      =What?

    P: We should all go to the beach.

Ė>G: Okay, I need paper. Okay, so the basic formula. What did we have?

In both examples, one student controlled the groupís discussion by ignoring the distraction, and in a business-like manner drawing attention back to the topic.

Terminating a negotiative event may also seem to be a powerful move. But it is never clear that a negotiative event has ended until the next event, or discussion of another topic, is established. Until then, although a group may appear to have reached agreement, one member can always have a change of mind and resume negotiation. The sequence of events is thus interactively constituted by all participants. One member can only control the proceedings if the others allow it.

Construction of knowledge

The sequence of topics discussed tells only part of the story. In a mathematics lesson, it is the mathematical ideas that are important, so we need to look at the influence of different students on the knowledge constructed or negotiated. For this, we must pay attention to individual turns within negotiative events. A study of the transcripts revealed that the following types of moves could be significant: introducing a new idea or making a suggestion about solving the problem; rejecting an idea or suggestion; endorsing an idea or suggestion; asking for an explanation or justification; giving an explanation or justification; correcting or questioning an error; and assigning tasks to the group.

        Introducing a new idea, or making a suggestion about solving the problem.

Example 7:

Ė>A: Could I suggest that Ö we choose one variable to work it around, and then work from the lowest to the highest one, using integers in      the table, in that way we get a really good pattern, you know, that we can see.

Example 8:

Ė>L: How do you do that?

    B: Sub one in as x.

Following example 7, Aís group adopted and used his idea. After example 8, B was challenged about his suggestion, expressed uncertainty, resorted to looking up notes, and the pairís work on the problem ground to a halt. I suggest that these different responses derive from differences in the studentsí sources of authority for their ideas, and differences in the type of problem. Aís authority came from himself, and he was able to give a reason for his suggestion. Bís authority derived from remembering a procedure, which he was later unable to explain. The depth and open-endedness of the problem Aís group were working on gave opportunities for the use of original ideas. The more routine question Bís group were tackling did not provide such opportunities.

        Rejecting an idea or suggestion

Example 9:

[R and N are working on a task involving matching functions and derivatives]

    R: So, which one is this?

    N: Negative six. [Moves a card forward, but does not put it in place]

Ė>R: Negative six. [Looks at the card suggested] No, it should have only x.

This sequence is similar to the initiation-response-feedback (I-R-F) pattern common in teacher-led classroom dialogue (Stubbs, 1983). Although R did not at that point know the answer to the question, he echoed the "teacher" role by first initiating, and then evaluating Nís response. Throughout the whole time that the pair worked on this task, R continued to exercise control in this way. Example 10:

    M: Itís half. Because, like, it seems like (pause) you see how here, this is half of the base. [pointing to the model they have made].

Ė>D: Is that, just a coincidence?

Following this remark, nothing more was said about Mís suggestion for 10 minutes, while the group fruitlessly pursued other approaches to the problem. Thus Dís incorrect rejection had more influence than Mís correct suggestion. Example 11:

      I: Hey, what if you graph it?

Ė>G: Yes, but then, this will give us the exact volume.

    V: Oh no, the graph gives the exact volume.

Ė>G: Yeah well, if someone else wants to draw a graph Ö

G seems to have had her mind fixed on the trial-and-error method she had been using with success, and so rejected the suggestion of drawing a graph. Although she did suggest that the others could draw a graph if they wanted, the fact that none of them attempted to do so illustrates her dominant influence in the group.

        Endorsing an idea or suggestion

Example 12:

    M: I just thought itís got to be half of x, so it will fold up.

    D: It goóhang on, is this x? [studying the model]
Ė>    Youíre right, it is. [Looks towards M and nods.]

This sequence took place 10 minutes after example 10 above. It constituted a significant breakthrough for the group concerned, changing the track of their work on the problem. The key to the breakthrough, however, was not Mís suggestion, which had been made earlier without effect, but Dís endorsement of it. This underlines Dís power and Mís lack of power within the group. Example 13:

    R: What do you think this one is?

    N: [Points to a card] This?

Ė>R: [Moves into place the card N was indicating.]

This again has the form of an I-R-F sequence. Nís suggestion was tentative, expressed as a question rather than a statement. R endorsed it wordlessly by moving the card into place, and the pair moved on to the next stage of the task.

        Asking for an explanation or justification

Asking a question can be productive, or counter-productive.

Example 14:

    D: Because we canít have three variables in an equation.
Ė>A: Why canít you?

Aís influence here was productive. His question helped the group to focus on the next step in solving the problemóusing substitution to reduce the number of variables and obtain a function of one variable which they could differentiate. Example 15:

Ė>P: Thatís your base? Is that your base? Is that going to be your base?

    G: Yes.

Ė>P: And thatís going to be a side there?

    G: Those are the sides.

Three members of this group had worked together on their problem, while P pursued his own ideas independently, and fruitlessly. When the three had completed the first stage of the problem, P began asking G to explain what they had done. He persisted with questions like these, interrupting other remarks and requiring each step to be clarified, halting the groupís progress until eventually the solution had been explained to him three times. Thus P exercised power obstructively, first by non-participation and then by persistent questioning. Example 16:

Ė>L: Substitute x is one yeah. Wouldnít that still be the same thing?

    B: I donít know. I think Iíll go check my notes.

This exchange happened soon after example 8. B had seemed confident, but had a poor understanding of the concepts. Lís question undermined his confidence. He turned to a reliable source of authority, while progress on the problem halted.

    Giving an explanation or justification

Explanations and justifications are powerful if the rest of the group find them convincing, whether or not they would be regarded as correct by a trained mathematician. Gestures and manner of speech can help to direct the attention of the group to the explanation, but it appears that the status and power of the speaker are crucial. An argument presented by a student of lower status may not be found convincing, or may not even be attended to, as was seen in example 10.

Example 17:

    D: Ö weíve got three variables, thatís what I donít=

    A:                                                                   =We donít have three variables=

Ė>D:                                                                                                                 =Oh, because we can do x over two.        Look what we can do! [with great excitement]

Here, Dís excited voice and emphatic speech commanded the groupís attention for the detailed explanation which followed, but the commanding role which he had taken in the group from the start of their work may have played a key role.

    Correcting or questioning an error

Example 18:

Ė>R: Minus n minus 1, which is minus three, oh yeah.

R recalled the differentiation rule for powers of x (although expressed in a rather confused way), realised they had misapplied it to a negative power, and made the correction, without reference to his partner. In so doing he retained control. Example 19:

Ė>I: How come you multiplied here?

    V: Because [pause] I donít know.

When V found that he could not answer the question, he reassessed, and later amended, what he had done. Thus Iís question was challenging and productive. Example 20:

Ė>L: Should it be eight x squared from four x and two x?

    B: Nuh.

    L: Oh nah, yeh, yeh yeh yeh.

In this case, L correctly pointed out an elementary algebraic mistake that B had made, but acquiesced immediately when B rejected her correction. She appears to mistrust her own correct reasoning. For her, Bís rejection carried more power.

    Assigning tasks to group members

A group may sometimes decide to share out parts of the work among members. The student who makes this decision, and the one who allocates the tasks (not necessarily the same) exercise poweróif the rest of the group accept their direction. But, as example 11 shows, such suggestions may not be followed.

Other factors

Other factors seen to be important included gestures, body language, and use of resources. A gesture such as pointing to a model, diagram or graph can draw attention to what one is saying, so can be an exercise of power. But gestures may also help to explain oneís ideas or clarify oneís thoughts, and so can support the construction of knowledge without necessarily exercising power.

Observation revealed that resources such as worksheets, textbooks, models and calculators are used in a variety of ways to support both the control of discourse and the construction of knowledge. If there is a single copy of a worksheet or other resource, the student who has it is at an advantage in controlling the discourse. Conversely, passing a worksheet to another student can be a way of handing over control. Occasionally, one student hands over a worksheet to another but then dictates what to write, in this way maintaining control.


Discussion

The examples demonstrate that power is interactively constituted: the influence of an utterance cannot be determined until its reception by the rest of the group is known. The most important indicators of the exercise of power seem to be the initiation of a new topic, either a negotiative event or off-task talk; and the endorsement, rejection or challenge of a statement. Gestures and the use of resources can act to intensify or moderate the effects of an utterance.

The objective of a negotiative event is the resolution of uncertainty. The key to this may be contained in an idea suggested by one group member, but what really counts is its reception by the rest of the group. No matter how good the idea, it will not advance the groupís endeavour if it is rejected, so a successful rejection move is powerful. Being ignored by the rest of the group is a form of rejection, and signifies the individualís lack of power within the group.

Similarly, endorsement of a suggestion resulting in its adoption by the group, is clearly an exercise of power, whereas uncritical acquiescence is not. A good idea may be accepted because of the status of its originator, without all group members understanding its significance. Or a misleading idea may be accepted, and cause a time-wasting digression, or failure to complete the task successfully. In such cases, the originator of the idea exercises power. Discrimination between weak and powerful acceptance moves needs careful interpretation. The manner of saying and doing things can be as important as what is said or done.

Asking for an explanation or justification can be an important and powerful move, but again this depends on the manner of asking, whether it is interpreted as a challenge or a threat, or simply a request for help or clarification. Finally, giving an explanation or justification can be powerful, but only if it convinces the hearers. It will be important to take into account the nature of the authority to which the respondent appeals, and which the group find convincing. Do they, for example, rely on an external authority like the teacher, a textbook, or an established formula or rule, or on the internal authority of a rational argument.

 
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