CHAPTER 4

THE ROLE OF PEER INTERACTION IN THE SOCIAL CONSTRUCTION OF MATHEMATICAL KNOWLEDGE

ELLICE FORMAN

Learning Disabilities Program, Northwestern University, 2299 Sheridan Road, Evanston, IL 60208, U.S.A.

Abstract

Observations from a study of collaborative problem solving are used to demonstrate the

processes of informal and implicit teaching or proleptic instruction, that occur in peer work

groups. These observations illustrate how two students with different initial stances toward a

complex task can help each other incorporate new experimentation and reasoning strategies

into their repertoire. The findings from the study indicate that peers can serve as teachers and

students for each other. In addition, the findings show how children can take an active role in

discovering and applying mathematical concepts.

Introduction

In recent years, educational researchers have shown increasing concern about the negative attitudes toward mathematics that are held by many Americans and about the relatively poor achievement in mathematics that is common among American children (Romberg & Carpenter, 1986; Stodolsky, 1985). One complaint about our instructional practices that is frequently heard is that mathematics is not taught the way it is practiced. In other words, students are rarely given the opportunity to discover or to apply mathematical concepts and procedures. Instead, they are expected to absorb a static, fragmented body of mathematical knowledge that is disconnected from other academic disciplines and from every day use (Romberg & Carpenter, 1986).

One instructional strategy which has been recommended as a way of changing students’ motivations, attitudes, and achievement is cooperative learning (Slavin, 1983, 1987; Webb, 1982). In the area of mathematics instruction, Stodolsky and her colleagues (Graybeal & Stodolsky, 1985; Stodolsky, 1985) have argued that mathematics anxiety is a consequence of the way this subject is typically taught. In one study Stodolsky et al.

Address correspondence to: Ellice Forman, Learning Disabilities Program, Northwestern University, 2299 Sheridan Road, Evanston, IL 60208, U.S.A.

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observed 20 fifth-grade mathematics classes and 19 fifth-grade social studies classes. They found that mathematics was taught almost exclusively through recitation or demonstration and lecture, followed by seatwork. In contrast, instruction in social studies showed much more variation, including the use of cooperative work groups. Stodolsky et al. proposed that the predominance of whole class instruction and drill in mathematics communicates the idea that this subject is best learned from experts. Alternatively, the use of cooperative peer work groups in social studies conveys the message that active involvement in generating knowledge through discussion is a valid way to learn concepts in this subject. Finally, they found that children were engaged in different cognitive activities in these two subject areas: the mastery of basic concepts and skills in mathematics versus the application of concepts and the use of higher mental processes in social studies.

The above arguments by educational researchers resemble a position articulated by many social constructivist theories, i.e., that the form of instruction influences the nature of the knowledge learned. This perspective questions whether the current approaches to the teaching of mathematics are capable of instilling enthusiasm for and genuine understanding of this subject. Jean Piaget (1965), for example, argued that cooperative peer relations are crucial for the development of an adult moral code based on norms of reciprocity. He proposed that the morality of constraint which is learned through adult- child relations is incapable of fostering a genuine appreciation for the principles of morality. For Piaget, the morality of constraint needs to be supplemented and eventually replaced by the morality that is socially constructed through working cooperatively with peers. Later on in his writings, Piaget argued that cooperative peer interaction is a causal factor in the development of logical thinking in general (Piaget, 1967).

Piaget’s position on the role of cooperative peer interaction in cognitive development has inspired a small but growing body of research on cooperative learning by developmental psychologists (see Damon, 1984; Tudge & Rogoff, in press, for recent reviews). Unfortunately, Piaget did very little to elaborate his perspective on the cognitive benefits of peer interaction because he was more interested in cognitive development than in cooperative learning. In contrast, another social constructivist, L. S. Vygotsky, was genuinely interested in how instruction in school can foster cognitive development (Vygotsky, 1978,1987). Vygotsky felt that learning and development were necessary and interdependent processes -with learning leading but being constrained by development. This relationship between learning and development is critical to his construct, the zone of proximal development: the difference between a child’s ability to perform a task independently and with assistance. According to Vygotsky, effective instruction must be sensitive to a child’s developmental level. Alternatively, children’s thinking develops because of the kinds of instruction that they receive from others-parents, teachers, more capable peers.

Although Vygotsky made a number of valuable suggestions about instruction, he died too early to elaborate his theory in important ways. Thus, neither Piaget nor Vygotsky provides us with a fully articulated theory to support his assertations about the role of social relations or, more specifically, peer relations in the social construction of knowledge. A third theory, that proposed by Berger and Luckmann (1966), can supply some additional insights into the relationship between social practices and learning.

The goal of this article is to use social constructivist theory to illustrate the educational value of cooperative work groups. First, some basic concepts from the theories of Berger and Luckmann and of Vygotsky will be presented. Second, the application of these

Peer Interaction, Problem-Solving, and Cognition 51

concepts to the study of cooperative learning will be discussed. Finally, data from a study of peer collaboration in children and adolescents will be used to illustrate how peer relations can function in the social construction of mathematical knowledge.

Social Constructivist Theory

A complete discussion of the theoretical perspective of Berger and Luckmann (1966) is beyond the scope of this article. However, some of their important constructs will be outlined below. Berger and Luckmann argued that there are three components to the social construction of reality: externalization, objectivation, and internalization. All three components are necessary to their theory and together they explain how social institutions, technologies and knowledge are created, maintained, legitimated, and transmitted through social interaction. They proposed that knowledge begins as a natural by-product of the externalization of human activity. As people try to interact over time with each other, an implicit mutual understanding develops between them. Soon, however, this tacit knowledge becomes objectified in explicit concepts and rules to which language and other sign systems can refer. The final step in the process occurs when this knowledge needs to be internalized by people who were not part of its creation. In order to communicate the objectified understanding and to legitimate the social group that created it, knowledge becomes reified, fragmented, and rationally ordered. In this way, the originally informal, fluid, and implicit understanding between people in a face-to-face situation becomes formal, static, explicit, and impersonal knowledge which can be culturally transmitted.

In summary, Berger and Luckmann argued that there is a correspondence between two different types of interactional processes and forms of knowledge. In face-to-face communication, implicit mutual understandings are established. In cultural transmission, an objectified body of knowledge is taught by experts to novices. However, the causal connections between face-to-face communication and cultural transmission are reciprocal. In other words, communication depends upon the existence of cultural knowledge systems (e.g., a shared linguistic code) while cultural transmission arises through a process of face-to-face communication. Thus, both of the two prototypical teaching-learning processes, informal communication and formal cultural transmission, may be present, but in different proportions in any educational situation.

Vygotsky also discussed different kinds of teaching-learning processes. Although Vygotsky’s theory of instruction in the zone of proximal development is often equated with “scaffolding” (Wood, Bruner, & Ross, 1976), there are subtle differences between these two constructs. Wood et al. (1976) defined scaffolding as a “process that enables a child or novice to solve a problem, carry out a task, or achieve a goal which would be beyond his unassisted efforts” (p. 90). More recently, investigators have emphasized the adjustable, temporary, and interactive nature of scaffolded instruction (Palincsar, 1986). Stone and Wertsch (Stone, 1985; Stone & Wertsch, 1984) have argued that the original definition of scaffolding did not emphasize the role of dialogue in instruction which they felt was central to Vygotsky’s notion of the zone of proximal development (cf. Palincsar ,1986). Thus, they have adopted the term “proleptic instruction” to characterize teaching and learning within the zone of proximal development. They used the Greek term “prolepsis” to refer to a means of communication in which the interpretation of a message requires an understanding of the speaker’s presuppositions. “Since the speaker’s presuppositions are

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left unstated, the listener must construct them for himself in order to understand fully the intended message. The process of construction makes the message more alive for the listeners” (Stone, 1985, p. 135).

Stone and Wertsch have argued that, for Vygotsky, instruction is more than just didactic teaching via explicit demonstration and explanation. They suggested that an effective form of teaching, proleptic instruction, requires the learner to construct his or her own interpretation of the teacher’s often implicit instructional messages. The important difference between explicit demonstration and explanation on the one hand and proleptic instruction on the other is the degree to which the instructor’s message presupposes background knowledge on the part of the student. Effective demonstration and explanation presuppose little background knowledge. i.e., they are muximnlly explicit. In contrast, effective proleptic instruction presupposes much more background knowledge, i.e., it is minimally explicit. The background knowledge required for proleptic instruction comes from the prior knowledge that the instructor and the student bring to the task and the shared understanding or intersubjectivity that is developed over time in that situation between those two specific people.

Combining both of the social constructivist theories discussed above, one can hypothesize that instruction consists of a mixture of prolepsis and explicit demonstration/ explanation. When prolepsis dominates, then the knowledge that is conveyed would be primarily informal, tluid, and implicit. When explicit demonstration or explanatioi~ dominates, then the cultural transmission of an objectified body of knowledge would result. It follows that the assessment of learning would need to differ in contexts that vary in their degree of implicit versus explicit instruction. The acquisition of an objectified and reified body of explicit knowledge could be assessed using traditional techniques such as achievement tests. However, since the knowledge that is internalized during proleptic instructi~~n is tacit, it is unlikely that traditional means of assessment will be able to measure changes in this type of understanding.

Stone and Wertsch have assessed knowledge acquisition in proleptic instruction by observing changes in the explicitness of the teacher’s messages and by evaluating the degree to which the student’s task performance is self- or other-regulated, They assume that the teacher’s implicit definition of the task setting, e.g., the task structure, goals, and procedures, needs to be eventually understood by the learner. Initially, however, the student’s view of the task is quite different from the teacher%. The student brings much less background knowledge to the problem-solving situation and is likely to view the task more narrowly than the instructor. After working on the task with varying degrees of other- regulation, the learner is eventually able to do more and more of the task procedures by him or herself or with minimal and often indirect other-regulation. When the student is able to perform the task (e.g., complete a puzzle or read words) with little or no other- regulation, it can be presumed that the instructor and student share more of a common perspective on the task goals and procedures. Thus, according to Stone and Wertsch, the success of proieptie instruction needs to be assessed dynamicaIly, through changes in the division of labor in a problem-solving task and through corresponding changes in the explicitness of communication between the teacher and student.

Cooperative Peer Interaction and Instruction

The notion of proleptic instruction would seem to apply not just to adult-child

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instruction but to peer instruction as well. In fact, proleptic instruction may be the preferred instructional format in cooperative work groups because peers are likely to be less skilled than adults in providing explicit explanations and demonstrations (Ellis & Rogoff, 1986).

Cooperative peer instruction differs from adult-child instruction in at least two ways. First, adult~hild relationships represent an unequal distribution of power and knowledge while peer relationships exhibit a more equal distribution. When one person in a relationship has greater power and/or knowledge the relationship tends to be defined by complementarity of interaction: one teaches while the other learns. When the distribution of power and/or knowledge is more equal, the relationship tends to be defined by reciprocity of interaction: each may take turns at instructing the other (Hinde & Stevenson-Hinde, 1987). This does not mean that adults cannot learn from children or that children cannot serve as peer tutors. Nevertheless, peer and adult-child relationships typically differ in their degree of complementarity and reciprocity. Therefore, one essential difference between adult-child and cooperative learning situations is the fact that the role of teacher and student may be assumed at different times by different people during cooperative peer instruction (Stodolsky, 1984).

The second difference between adult~hild instruction and cooperative learning is the degree to which children can assume the responsibility for defining task goals and strategies. In adult-child instruction, it is assumed that the adult takes primary responsibility for task definition. In contrast, Rommetveit (1985) has argued that a symmetric control of communication is likely in situations where power and knowledge are more equally distributed. He proposed that people enter a communicative situation with their own private stance toward a task. In the process of attempting to coordinate their activities with others, a temporarily shared social world is co-constructed. Part of this social world is a set of intersubjective task definitions that are established, negotiated, and modified by all the participants. Thus, children in a cooperative peer work group can learn to share the responsibility for establishing a common understanding of task goals and strategies.

If one assumes that some form of proleptic instruction may occur in cooperative work groups, then it follows that a detailed analysis of the dynamics of interpersonal interaction and communication could yield insights into the cognitive benefits of peer cooperation. In particular one would need to see if children can create a bi-directional zone of proximal development for each other by assuming, at different times, the role of teacher or student. Also, one would want to understand how a shared task definition is established, negotiated, and modified. This kind of dynamic assessment of learning within the zone of proximal development may provide a valuable supplement to the static, decontextualized assessments of explicit learning that one gets from more traditional measures of achievement or cognitive development.

The remainder of this paper will be used to illustrate the educational potential of cooperative peer instruction by describing, in detail, how two girls collaborated on a complex problem-solving task. These two girls were selected from a sample of 50 fourth- and seventh-graders who were involved in a study of collaborative problem solving. In this study, students were seen in both individual pretest and posttest sessions and were paired with a familiar classmate for two dyadic sessions. Three different types of analyses were done from the videotapes of the experimental sessions: (1) a quantitative analysis of reasoning during the individual pretest and posttest sessions; (2) a qualitative analysis of

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experimentation and situation definition during the pretest sessions; and (3) a quantitative and qualitative analysis of the division of labor, reasoning, experimentation and situation definition during the two dyadic sessions. The data will be used in two ways. First, the quantitative pretest-posttest change scores will be presented to show what a more traditional, static, and relatively decontextualized assessment can provide as evidence of cognitive growth. Second, qualitative data from the subjects’ pretest performance will be contrasted with both quantitative and qualitative data from the two dyadic sessions to illustrate what a dynamic assessment of learning within the zone of proximal development can show.

Illustrations from a Study of Collaborative Problem Solving

The two girls, “Miriam” and “Diana”, worked on the same complex problem-solving task for a total of four sessions each (two individual sessions and two dyadic sessions). The two girls were in seventh grade when they were tested and declared themselves to be best friends. They were both 13 years, 3 months of age and of average intelligence (Slosson IQ scores of 91 and 96, respectively). Both girls were from white, middle-class families who lived in a suburban neighborhood located in the Chicago metropolitan area. The problem- solving task and the procedures employed in the study are described below.

The Problem-Solving Task

The study employed a transformational geometry task which involved the projection of shadows of two-dimensional geometric shapes onto a screen. Different shapes were employed within and across problem-solving sessions. Each geometric shape contained a hole in the center which allowed it to be screwed onto a metal stand. The stand was attached to a wooden track. On one end of this track was a wooden easel which supported a large pad of paper and on the other end was a tensor lamp. The stand was located between the pad of paper and the light. The shape could be moved on the stand in a number of specific directions: (1) closer to or further from the source of light; (2) about the center screw; (3) around a line parallel to the floor (horizontal axis rotations); (4) around a line perpendicular to the floor (vertical axis rotations); and (5) all possible combinations of the above movements.

Each geometric shape was presented with several shadow choice cards (5” X 8” pieces of cardboard on which grey geometric shapes were pasted). Each shadow choice card was numbered in the lower right-hand corner. For example, during both the pretest and posttest sessions, an equilateral triangle was attached to the stand. The subjects were given eight shadow choice cards: a larger equilateral triangle, three isosceles triangles, a line, a trapezoid, a right triangle and a scalene triangle.

During each of the dyadic sessions, subjects worked with two shapes and six shadow choices per shape. One shape was presented during the first half of these sessions and it was replaced with the second shape during the second half of the session. The shapes used during the dyadic sessions were: a square, two different isosceles triangles, and a right triangle. The shadow choices included lines and a variety of three- and four-sided figures.

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Procedures

All of the testing was done by a female examiner in a spare room in the children’s school. The examiner followed a semi-structured script throughout all the problem-solving sessions. No attempt was made to explain or demonstrate relevant geometric concepts, e.g., symmetry. The examiner referred to all geometric shapes by their number, not by name. That is, a large equilateral triangle was labeled “shadow choice No. 2” and was referred to by that label. In addition, subjects were not told if their answers were right. When subjects raised procedural or evaluative questions about the task, e.g., “Does the shadow have to look exactly like the shadow choice‘?“. they were told that that was a decision they should make on their own.

During the dyadic sessions. the examiner was instructed to address her questions to both girls and to determine whether the answer that was provided was acceptable to both of them. If there was a disagreement about an answer. the girls were asked to try to reach an agreement.

Each problem-solving session (pretest, posttest, and dyadic 1 and 2) followed the same format. The problem-solving sessions were divided into three phases per shape: prediction; experimentation; and evaluation. During the prediction phase, the subjects were shown a new geometric shape and were given 6-X shadow choice cards. Subjects were asked by the examiner to decide, with the light off, which of the shadow choices could be projected from that particular shape and to justify their decisions. During the experimentation phase. the examiner turned on the light and asked the subjects to determine which of the shadow choices could actually be projected from the given shape. Finally, during the evaluation phase, the light was turned off and the examiner asked the subjects to state and justify their conclusions about each shadow choice.

Dutu Coding and Analysis

The videotapes of each problem-solving session served as the primary data base. Verbal transcriptions were made from the audiotrack of each videotape and were edited. Coding activities employed the videotapes, the edited transcripts, and examiner’s notes of subjects’ predictions and conclusions.

The pretest, posttest, and dyadic sessions were coded in order to assess changes in each subject’s ability to make accurate prediction and conclusion decisions about the projection of shadows and her ability to justify these decisions using appropriate transformational reasoning. The dyadic sessions were also coded to describe how the two subjects divided up the problem-solving duties: which girl produced predictions or conclusions in response to the examiner’s questions and which projected shadows or monitored the shadows. These quantitative indices of individual and dyadic activity were supplemented by a qualitative description of each subject’s problem-solving goals and experimentation strategies within and across sessions.

Pretest and Posttest Sessions

Two of the three task phases were coded during the pretest and posttest sessions: the prediction and evaluation phases. These were the periods in which the examiner asked for

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decisions about each shadow choice and explanations for those decisions. The prediction and evaluation phases of the pretest and posttest sessions were examined for two indices of reasoning: the accuracy of each subject’s prediction or conclusion decisions and the adequacy of her explanation. The accuracy and adequacy scores were independent. That is, a subject could make an accurate decision but base it on an inadequate explanation. Conversely, a subject could make an inaccurate decision but justify her decision by referring to an appropriate transformation. In these cases, subjects were given credit for the part of their answer that was correct.

The accuracy of a prediction/conclusion was determined by whether or not a subject’s decision about a particular shadow choice was correct. For example. a subject who predicted/concluded that it is impossible to make a four-sided shadow from a triangular shape was ,judged to bc making an accurate decision about that particular shadow choice. Each correct decision received a score of I which resulted in a maximum possible score of X (1 point for each shadow choice).

The adequacy of a subject’s explanation was determined by the degree to which she was able to explain verbally or to demonstrate nonverbally the movements of a shape which wcrc necessary for producing a particular shadow choice. For example, if a subject was

able to explain or demonstrate that one can make a shadow of a large equilateral triangle by moving a small equilateral triangle shape closer to the light, then she would get full credit (a score of 1) for providing an adequate explanation for that shadow choice. Other shadow choices, however, required different kinds of explanations. In order to show how to make a shadow which resembled an isoscelcs triangle from an equilateral triangle shape, a subject would have to mention or demonstrate that the shape would need to be rotated about an axis parallel to the floor (horizontal axis) or about an axis perpendicular to the floor (vertical axis). One pretest and posttest shadow choice was impossible to make and two shadow choices required two scparatc kinds of rotations (e.g.. vertical urrd horizontal axis rotations). The maximum number of correct transformations that could be mentioned for the pretest and posttcst shape was equal to nine.

All three phases of the task were coded for the two dyadic sessions. The accuracy and adequacy of their predictions and explanations were coded. In addition, the contribution of each subject to their collective problem-solving activities was coded in order to determine the division of labor in the task. For the prediction and evaluation phases, the subjects who answered the examiner’s questions about each shadow choice were noted. During the experimentation phase, the subjects who manipulated the shape or the stand vthile attempting to produce a particular shadow choice were also noted. In addition, the subjects who evaluated the accuracy of the shadows produced and gave feedback about them (c.g., “No, that’s upside down” or “Too big”) were recorded.

During the two dyadic sessions, two shapes were presented with six shadow choices per shape. Thus, each subject could provide a maximum of six predictions and six conclusions and could attempt to produce a maximum of six shadows and to provide feedback about the accuracy of a maximum of six shadows per shape. Frequently, however, subjects took turns giving predictions and conclusions or producing and monitoring shadows. Across the two dyadic sessions, there were multiple opportunities for subjects to perform each of these problem-solving roles. For example, one subject could provide all the predictions

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and another all the conclusions for a particular shape. Then, on the next shape, they could exchange roles. Alternatively, one subject could produce all the shadows in both sessions.

In the following two sections, Miriam and Diana’s ability to benefit from collaborating on the transformational geometry task will be discussed. First, evidence from each girl’s individual pretest and posttest sessions will be outlined. Second, a description of how their problem-solving activity changed from the pretest to dyadic sessions and within the dyadic sessions will be presented. Previous research by Stone and Wertsch (Stone, 1985; Stone & Wertsch, 1984) and by Forman and Cazden (1985) demonstrated the value of assessing cognitive growth dynamically through documenting changes over time in interpersonal and cognitive activity. Therefore, we felt that it was important to provide information about the process of peer instruction over time instead of summarizing the data across the sessions.

Results From a Traditional Assessment of Pre-post Change

During their individual pretest sessions, the reasoning of the two girls did differ in subtle ways (see Table 4.1). They were equally successful at making accurate predictions, but were not equally successful at drawing accurate conclusions from their experiments. Diana increased the number of correct decisions that she made during the evaluation phase while Miriam did not.

Table 4.1 Summary of Pretest and Posttest Performance

Number of correct Number of correct decisions” explanation$

Pretest Posttest Pretest Posttest

Miriam Predictions Conclusions

6 6 1 4 6 5 3 3

Diana Predictions Conclusions

6 6 3 5 7 7 6 4

“Maximum correct decisions = 8. h Maximum correct explanations = 9.

The adequacy of the reasoning used by both girls to justify their predictions conclusions during their individual pretest sessions is also displayed in Table

and 4.1.

Adequacy of reasoning refers to the appropriate use of transformational criteria in their justifications. Although both girls showed an increase in the use of transformational reasoning during their pretest sessions, Diana used more transformational reasoning than did Miriam. That is, Diana was more likely to justify her predictions and conclusions by showing the examiner how one would be able to produce a given shadow choice by moving the shape in the appropriate ways on the stand. Diana was able to supply three correct transformations out of the nine possible in her predictions and six out of nine in her conclusions. In contrast, Miriam was more likely to justify her decisions by comparing the

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static properties of the shadow choice and shape (e.g., “This shadow choice has the same number of sides as the shape so I think you can make it.“). Miriam was able to provide only one correct transformation in her predictions and only three in her conclusions. Despite these differences, both girls used a mixture of static and transformational justifications during their pretest sessions.

The initial differences between the two girls in their pretest sessions were maintained in their posttest sessions (see Table 4.1). Diana continued to make a greater number of correct decisions about shadow choices during the evaluation phase of this session than did Miriam. Diana also used more transformational reasoning in her predictions and conclusions (five out of nine and four out of nine, respectively) than did Miriam (four out of nine and three out of nine, respectively).

What is most striking, however, about their posttest performance is the relative absence of pretest to posttest improvement shown by either girl. When the accuracy of each girl’s prediction and conclusion decisions was compared, it appeared that Diana made no gains from pretest to posttest while Miriam showed some regression. When the adequacy of their reasoning from pretest to posttcst was compared. it seemed that Miriam’s predictions improved but her conclusions did not. For Diana, improvement was noted in the adequacy of her predictions but regression was found in the adequacy of her conclusions.

The quantitative analysis of the individual pretest and posttest sessions did not seem to indicate many cognitive gains in either subject as a result of working collaboratively with a peer. Both girls seemed to perform on the posttest in a fashion that was similar to their pretest performance. Although Diana was more adept at using transformational reasoning than was Miriam, there is little evidence that Miriam benefitted from her exposure to Diana’s thinking. Even more disturbing is the possibility that Diana declined in her use of transformational reasoning from the pretest to posttest sessions. What this static, decontextuahzed assessment of cognitive growth cannot reveal is the interpersonal and cognitive dynamics of their collaboration. Therefore, the above analysis will be supplemented with a description of the interpersonal and cognitive processes that occurred from the two pretests through the second dyadic session.

Results From a Dynamic Assessment of Learning

In order to evaluate learning within the zone of proximal development one needs to know how a child functions with and without assistance from others. In addition, as Stone and Wertsch have shown, changes in the division of labor within an instructional context can also provide information about a child’s ability to learn from proleptic instruction. Information from each girl’s pretest performance was used to assess her initial stance toward the task as well as her level of independent functioning. Then, this information was compared with data from the two dyadic sessions to illustrate changes in task approach and performance within the collaborative context.

Reasoning Performance in the Pretest and Dyadic Sessions

Miriam made an average of 75% correct decisions during both the prediction and evaluation phases of her pretest session while Diana made an average of 82% correct

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decisions (see Table 4.1). During their two dyadic sessions they made an average of 90% correct decisions (88% during session 1 and 92% during session 2).

Miriam provided an average of 22% adequate justifications during these same two phases of her pretest while Diana supplied an average of 50% adequate justifications during her pretest. During their two dyadic sessions they provided an average of 47% adequate justifications (43% during session 1 and 51% during session 2).

Therefore, the accuracy of their decisions increased when they were able to collaborate. In contrast, their use of transformational explanations was equal to the independent performance of the one girl, Diana, who showed the better pretest performance on this variable.

Pretest Differences in Task Definition

An informal, qualitative analysis of the experimentation phase of the two girls’ pretest sessions revealed some important differences in their task goals and problem-attack strategies. As one might expect from their differential use of transformational reasoning in their predictions and conclusions, Diana showed a greater ability than Miriam to anticipate which movements of the shape were likely to produce a specific shadow. Thus, Diana’s strategy for producing shadows was more efficient and less reliant upon trial and error. In contrast, Miriam paid more attention to comparing the shadows she produced with each shadow choice. Unlike Diana, Miriam asked the examiner several times whether the shadow and shadow choice should be the same size or oriented in the same direction.

Taking into account each girl’s reasoning and problem-attack strategies during their pretests, one can outline their differences in initial stance towards the problem-solving task. Diana appeared to view the task as an opportunity to explore her ideas about light rays and their effect upon different kinds of shapes. For her, the task seemed to represent a science experiment such as one might encounter in a physics laboratory. Miriam, however. appeared to be more interested in analyzing and comparing two-dimensional geometric shapes. For her, the task was an opportunity to learn, informally, about the essential similarities and differences between two shapes (e.g., these two triangles both have “two long sides and a short side”).

Changes in Performance During the Dyadic Sessions

These differences between the performances of the two girls on their individual pretests played a role in their joint performance during their two dyadic sessions. Four shapes were presented during these two sessions. Each shape was accompanied by six shadow choices. The distribution of responsibility for providing explanations and for producing shadows is displayed in Table 4.2.

From Table 4.2 one can see that Miriam provided at least 50% of the explanations for their mutual predictions about each of the four shapes. Also, it is apparent that Miriam’s involvement with giving justifications for their predictions increased over time while Diana’s involvement fluctuated. One sees a different distribution of responsibility for justifying conclusions than we saw for predictions. The two girls tended to take turns explaining their conclusions or to rely exclusively upon Diana to explain their conclusions. Therefore, both girls were actively involved in supplying justifications for their mutual

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Table 4.2

Division of Labor During the Two Dyadic Sessions

Session I Session 2 Shape 1 Shape 2 Shape 3 Shape 4

Justified Predictions”

Miriam Diana

Justified C‘onclusions”

Miriam Diana

Produced Shadows’

Miriam

Diana

Monitored Shadows’

Miriam Diana

,’ Maximum numhcr of predictions = 6. ” Maximum number of conclusions = 6. ’ Maximum number of experiments = 6

predictions and conclusions but Miriam took slightly more responsibility for their predictions while Diana took more responsibility for their conclusions.

What Table 4.2 does not show is the extent to which each subject’s justifications reflected their different stances toward the task that were seen in the pretest sessions. During the initial dyadic session, Miriam tended to make verbal comparisons between the configurations of the shadow choice and the shape while Diana tended to demonstrate nonverbally how the shape needs to be moved in order to produce a particular shadow choice. In addition, Diana produced several explicit gestures which demonstrated her conception of light rays spreading out from a source, being blocked by the two- dimensional shape, and resulting in a particular shadow configuration. Over time. however, each subject’s explanations incorporated characteristics of her partner’s stance. Miriam, for example, made some verbal and nonverbal references to the effect of the light hitting the shape. By the second dyadic session, both subjects tended to make verbal comparisons between the shadow choice and the shape while demonstrating nonverbally how the shape must be moved.

Table 4.2 also shows the role that each subject played during the experimentation phases of the dyadic sessions. Diana took primary responsibility for producing shadows by manipulating the shape or the stand. However, over time, Miriam became increasingly involved in that activity as well. In fact, by the beginning of the second dyadic session, Miriam was more involved in producing shadows than Diana. By the end of the second dyadic session, both girls worked closely together in producing virtually every shadow they made.

Finally. Table 4.2 shows that Miriam took an early and active role in monitoring the

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shadows that were produced during the first dyadic session. That is, she was quite involved in noticing how the shadows and the shadow choices differed and in pointing out those differences to her partner. Over time, Diana’s tendency to monitor their experimentation increased while Miriam’s involvement first decreased then increased as well. By the final shape in the second dyadic session, both girls were actively participating in the monitoring process.

The division of labor that we saw occurring for these two girls during all phases of their first dyadic session reflected, to some degree, their different initial stances toward the task during their pretests. Diana’s scientific stance may have encouraged her to assume primary responsibility for producing shadows. It also may have resulted in her use of nonverbal demonstrations rather than verbal explanations during this first dyadic session. In contrast, Miriam continued to show an interest in analyzing and comparing static geometric shapes. She became involved in monitoring the shadows that Diana produced and comparing them with the shadow choices. Also, she tended to use verbal explanations which focused on static comparisons between the shape and shadow choice.

Over time, however, both girls took over more and more responsibility for the role that had been originally assumed by the other. This does not mean that they merely switched roles. Instead, they became more involved in both roles-often switching back and forth between roles during the production of a single shadow. In addition, they combined the type of explanation preferred by their partner with their own style of explanation. This resulted in a collaborative problem-solving style which incorporated both stances toward the task: an experimental perspective and a geometric analysis perspective.

Summary and Conclusions

Earlier in this paper the notion of proleptic instruction was invoked in order to describe a particular type of teaching-learning interaction. It was argued that proleptic instruction may play a prominent role in cooperative peer work groups. One principle difference between adult-child and peer proleptic instruction being that the roles of teacher and student may be assumed at different times by different people during peer interaction. The second difference being that task goals and strategies in cooperative work groups may be mutually constructed.

Observations from a study of collaborative problem-solving were used to demonstrate how proleptic instruction could be applied to peer work groups. Those observations illustrated how two subjects of comparable intellectual ability but with somewhat different stances toward a problem-solving task can help each other incorporate new problem- attack and reasoning strategies into their repertoire. This outcome occurred despite the fact that explicit instruction between the two subjects was rarely used. Instead, we saw an instance of the communicative situation described by Rommetveit where each subject began with her private interpretation of the task and then modified this interpretation in the process of creating a temporarily shared social world. In addition, we saw how these girls were able to create a bi-directional zone of proximal development where the roles of teacher and student were assumed at different times by different people.

These findings were revealed through a dynamic assessment of changes in goal definition, experimentation, and reasoning within and across problem-solving sessions.

68 NOREEN M. WEBB

The findings of the dynamic assessment, however, appeared to contradict the lack of pretest to posttest change. Why were the changes that we saw during the dyadic sessions not apparent during the posttest? The answer to that question may be found in the distinction made earlier in this paper between the explicit knowledge which is taught through didactic instruction and the implicit knowledge which is taught through proleptic instruction. It was proposed that traditional, decontextualized tests would be capable of assessing change in explicit knowledge but would not be accurate measures of change in implicit knowledge.

The inadequacy of a decontextualized approach to evaluating the success of peer collaboration in this study was apparent when the two posttest sessions were examined. First, neither Diana nor Miriam seemed to be motivated to perform at her highest level during her posttest session. This change in motivation seemed to be a direct result of the success of their collaboration. In the previous two sessions, they had been encouraged to work as a collaborative team. As we have seen, they achieved that goal over the course of those two sessions. Thus. one of the principle incentives for working on the task was missing during the posttest. Second, the traditional assessment of performance on the posttest focused on the use of transformational reasoning. Diana’s performance on this variable appeared to decline because she had incorporated Miriam’s style of explanation. In other words, a traditional analysis of cognitive growth which employed pretest to posttest change scores, was insensitive to the qualitative shift in stance toward the task that Diana had learned from working with Miriam.

Although we claim that Miriam and Diana’s collaborative relationship was not unique, it does represent a particular kind of collaboration. Our observations as well as those of many others confirm the fact that collaborative relationships differ widely in their degree of mutuality and in their potential for enhancing cognitive growth (Bcarison. Magzamen. & Filardo, 1986; Forman Rc Cazden, 1985). One advantage of the collaborative relationship described above is the degree to which each subject respected the pcrspectivc of the other. This kind of mutual respect may be a necessary prerequisite for the willingness of each to accept and incorporate the other’s task approach (Sullivan, 1953).

In other collaborative relationships where power and knowledge are not equally distributed, an asymmetric division of labor may be seen. Some of these situations may result in a form of peer tutoring if the more powerful is also the more knowledgeable. For some dyads, however, the more powerful is not the more knowledgeable. In these situations, the educational potential of collaborative learning is greatly diminished. It is important to keep in mind these different types of collaborative relationships when one is arguing for the value of peer work groups.

Thus, we have seen that exposure to peer collaboration can result in an expanded view of a transformational geometry task. The subjects that we discussed above were not only learning from each other, they were also actively involved in increasing their knowledge of geometry and in discovering how the shadows of geometric shapes change when their distance from a source of light is systematically varied. This enthusiastic involvement in the process of generating and applying geometric knowledge seems to be what educational researcher feel is missing from mathematics instruction at the present time. Since this type of learning is largely implicit, it needs to be assessed by examining changes over time in subjects’ ability to assume responsibility for different problem-solving roles and to co- construct a shared understanding of task goals and strategies. As we become more knowledgeable about the factors that serve to maximize the cognitive benefits of peer

Peer Interaction, Problem-Solving, and Cognition 69

collaboration, we may be in a better position to help educators use peer work groups to supplement their traditional instructional practices in mathematics.

Acknowledgements-This research was supported by grants from the Spencer Foundation, the Northwestern University Research Grants Committee, and the Bowman Lingual Fund. I would like to thank Jean McPhail, Gavriel Salomon, Addison Stone, Jonathan Tudge, Noreen Webb, Israel Weinzweig, and Meredith Williams for their comments on an earlier version of this paper. I would also like to thank the faculty and students of the Winnetka and Roselle, Illinois Public School districts for their participation in this study.

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Biography

Ellice Forman is an Assistant Professor in the Learning Disabilities Program, Northwestern University. She received her Ed.D. from the Harvard Graduate School of Education. Her research focuses on both developmental and individual differences in normally-achieving and learning disabled children. She has studied children’s reasoning and problem-solving strategies, social support, self-esteem, and peer relations.