SOCIALLY MEDIATED METACOGNITION: CREATING COLLABORATIVE ZONES OF PROXIMAL DEVELOPMENT IN SMALL GROUP PROBLEM SOLVING

Transcription

1 MERRILYN GOOS, PETER GALBRAITH and PETER RENSHAW SOCIALLY MEDIATED METACOGNITION: CREATING COLLABORATIVE ZONES OF PROXIMAL DEVELOPMENT IN SMALL GROUP PROBLEM SOLVING ABSTRACT. This paper reports on a three year study of patterns of student-student social interaction that mediated metacognitive activity in senior secondary school mathematics classrooms. Transcripts of small group problem solving were analysed to determine how a collaborative zone of proximal development could be created through interaction between peers of comparable expertise, and to investigate conditions under which such interaction led to successful or unsuccessful problem solving outcomes. Unsuccessful problem solving was characterised by students poor metacognitive decisions exacerbated by lack of critical engagement with each other s thinking, while successful outcomes were favoured if students challenged and discarded unhelpful ideas and actively endorsed useful strategies. In reconceptualising metacognition as a social practice, the study contributes to the growing body of research applying sociocultural theories to understand learning in mathematics classrooms. KEY WORDS: metacognition, social interaction, zone of proximal development 1. INTRODUCTION In a review of progress in mathematical problem solving research in the 25 years to 1994, Lester (1994) lamented that research interest in this area appears to be on the decline, even though there remain many unresolved issues that deserve continued attention. One such issue highlighted by Lester was the role of metacognition in problem solving where metacognition refers to students awareness of their own cognitive processes, and the regulation of these processes in order to achieve a particular goal (Brown, Bransford, Ferrara and Campione, 1983; Flavell, 1976). In mathematical problem solving, regulation of cognition involves such activities as planning an overall course of action, selecting specific strategies, monitoring progress, assessing results, and revising plans and strategies if necessary (Garofalo and Lester, 1985). Although the importance of metacognition is now widely acknowledged, we still lack an adequate theoretical model for explaining the mechanisms of self-monitoring and self-regulation, and understand too little about how metacognition and other aspects of think- Educational Studies in Mathematics 49: , Kluwer Academic Publishers. Printed in the Netherlands.

2 194 MERRILYN GOOS ET AL. ing mathematically cohere to give individuals their mathematical point of view (Schoenfeld, 1992). Also lacking are clear guidelines for teachers on how to foster higher order reasoning and problem solving skills, despite some research evidence that students can be taught to think mathematically. For example, Schoenfeld s work with college students emphasises the benefits of small group problem solving in developing both metacognitive control skills and a sense of what the discipline of mathematics is about. He argues that these group sessions stimulate discussion and argumentation of issues involved in monitoring and evaluating the group s progress, and initiate students into a community of mathematical practice which engenders commitment to the intellectual values of the discipline (Schoenfeld, 1989). Beyond this work, however, research on small group learning in mathematics has yielded few insights into how students think and learn while interacting with peers, since most studies have focused on narrow learning outcomes, such as memorisation of facts or computational skills, rather than learning processes associated with mathematical reasoning (Good, Mulryan and McCaslin, 1992). Consequently, the potential for small group work to develop students mathematical thinking and problem solving abilities has remained largely unexplored, along with related issues concerning the teacher s role in orchestrating students discussion and social interactions. The concerns outlined above are reflected in Schoenfeld s (1999) call for new frameworks, perspectives and methods that bridge the schism between fundamentally cognitive and fundamentally social studies of human thought and action (p. 5). Such a research agenda could lead to practical as well as theoretical advances in understanding the development of mathematical thinking. Issues of practical interest arise from curricular trends that increasingly emphasise problem solving, mathematical reasoning, and communication (Australian Education Council, 1991; National Council of Teachers of Mathematics, 1989, 2000). These trends have begun to influence school mathematics syllabuses; for example, senior secondary mathematics syllabuses introduced in the Australian State of Queensland since 1992 have included communication and justification, and mathematical modelling and problem solving, as assessable objectives for students learning (Board of Senior Secondary School Studies, 1992, 2000). These significant curriculum reforms are intended to engage students more actively in the mathematical enterprise, yet are likely to require a major shift in the classroom practices of many mathematics teachers. How can changes to the social organisation of classrooms be justified when our understanding of the ways in which students mathematical thinking is cultivated by these new forms of classroom interaction is far from com-

3 SOCIALLY MEDIATED METACOGNITION 195 plete? This is an important question that is often neglected by curriculum documents promoting reform in mathematics education. The challenges described above point to the need for further research on mathematical thinking and learning to be conducted in authentic classroom settings. This need was addressed by the study reported here. The aims of the study were to (a) investigate the characteristics of senior secondary students metacognitive activity as they worked together on mathematical tasks, and (b) examine the teacher s role in creating a classroom culture of inquiry which promotes mathematical habits of mind (Goos, 2000). As findings related to the second aim have been detailed elsewhere (see Goos, Galbraith and Renshaw, 1999), this paper deals with only the research question arising from the study s first aim: how is metacognition mediated by collaborative peer interaction? 2. COLLABORATIVE METACOGNITIVE ACTIVITY The theoretical framework for the study drew on sociocultural theories of learning, and related formulations of Vygotsky s notion of the zone of proximal development (ZPD), as providing a coherent rationale for classroom practices that develop students mathematical thinking (see Goos, Galbraith and Renshaw, 1999). In doing so the study focused primarily on the micro social context of classroom interactions, while also considering aspects of the broader macro-context such as teachers and students beliefs about school mathematics learning (Abreu, 2000). Previous research on metacognitive development has used the ZPD to explain how adults can scaffold learners progress from assisted (otherregulated) to independent (self-regulated) performance (e.g., Bruner, 1985; Wertsch, 1985). The scaffolding process is best understood as involving mutual adjustment and appropriation of ideas rather than a simple transfer of information and skills from teacher to learner (Brown et al., 1993; Packer, 1993; Wertsch, 1984). Vygotsky additionally analysed the notion of the ZPD in terms of equal status partnerships, noting that when children played together they acted above their normal level of development and were able to regulate their own and their partners behaviour according to more general social scripts (see Minick, 1987). Applied to educational settings, this view of the ZPD suggests there is learning potential in peer groups where students have incomplete but relatively equal expertise each partner possessing some knowledge and skill but requiring the others contribution in order to make progress. In her research on collaborative problem solving, Forman (Forman, 1989; Forman and McPhail, 1993) has described this interaction between

4 196 MERRILYN GOOS ET AL. peers as creating a bi-directional ZPD in which students were able to coordinate their different perspectives on a problem in order to achieve progress. In a sense, zones of proximal development always have a two way character, since teachers appropriate learners ideas and actions as much as learners appropriate the teacher s guidance. Hence we prefer to use the term collaborative ZPD in our research on small group learning to emphasise this distinction between expert-novice and equal status interactions. Forman s work represents a significant extension of the commonly accepted definition of the ZPD as the distance between learners independent performance and the higher level that can be achieved under the guidance of a more expert partner, such as an adult or more capable peer (Vygotsky, 1978). She claims that this new perspective on collaborative problem solving is compatible with the broad research agenda pursued by Vygotsky and colleagues (Forman and McPhail, 1993, p. 214). Although there is a large body of literature devoted to peer learning (see Good, Mulryan and McCaslin, 1992, for a review), not all forms of peer interaction can be classed as collaborative. For example, Damon and Phelps (1989) distinguish between various approaches to peer education according to the quality of engagement that is fostered. Thus they define peer tutoring as interaction in which students of unequal expertise are brought together so that one may instruct the other, and cooperative learning as an arrangement which allows teams of students to divide a task and master its separate parts. Damon and Phelps reserve the term peer collaboration for the interaction that occurs when students with similar levels of competence share their ideas in order to solve jointly a challenging problem. In this context of supportive communication and assistance, students are encouraged to experiment with new ideas and critically reexamine their own assumptions a form of interaction that seems to hold promise for improving students metacognitive awareness and regulation. Similarly, Granott (1993) maintains that highly collaborative interactions between peers of equal expertise are characterised by shared activity, a common goal, continuous communication, and co-construction of understanding. This view is consistent with the definition of collaboration offered by Teasley and Roschelle (1993), as a coordinated, synchronous activity that is the result of a continued attempt to construct and maintain a shared conception of the problem (p. 235). From the above the distinguishing feature of peer collaboration can be defined as mutuality a reciprocal process of exploring each other s reasoning and viewpoints in order to construct a shared understanding of the task. Producing mutually acceptable solution methods and interpretations thus entails reciprocal interaction, which would require students to

5 SOCIALLY MEDIATED METACOGNITION 197 propose and defend their own ideas, and to ask their peers to clarify and justify any ideas they do not understand. Because this kind of reasoned dialogue involves comparing one s own ideas with those of another person, collaborative interaction need not be based purely upon agreement and cooperation, but may also include disagreement and conflict. Thus the process of co-constructing understanding is more complex than simply reaching consensus on an agreed answer (Kruger, 1993). The literature that deals specifically with improving metacognitive strategy use via collaborative peer interaction is limited, and has produced conflicting results. For example, Goos (Goos, 1994; Goos and Galbraith, 1996) noted that student-student interactions could either help or hinder metacognitive decision making during paired problem solving, depending on students flexibility in sharing metacognitive roles such as idea generator, calculation checker, and procedural assessor. Further evidence that peer interaction does not always deliver metacognitive benefits comes from a study by Stacey (1992), who found that problem solving performance of Grade 9 students diminished when they worked in groups. Videotape analysis revealed that although groups were able to propose a range of strategies, the correct solution method was frequently overlooked due to lack of checking and evaluation procedures. Nevertheless, Artzt and Armour-Thomas (1992) have argued that the small group format for mathematical problem solving does seem to promote spontaneous verbalisation, which allows students to offer their ideas for critical examination. Their cognitive-metacognitive framework has been designed to analyse communication between students during small group problem solving (see also Curcio and Artzt, 1998). Individual students problem solving statements and behaviours are coded at one minute intervals and recorded on a chart to represent the group s progress, an analysis which reveals patterns of metacognitive activity associated with different group interaction patterns. For example, members of one group of seventh grade students that was unable to solve the assigned problem worked essentially as individuals, with little monitored or regulated exploration; while another more successful group was nevertheless dominated by one student who did most of the monitoring while others watched and listened. While Artzt and Armour-Thomas (1992) have made a major contribution to the study of metacognitive processes in small group settings, their analysis method in tracking individual contributions to problem solving within the group does not distinguish between a student s monitoring or regulation of their own thinking and that which is directed at a partner s thinking. Consequently, it fails to capture the reciprocal nature of metacognitive activity which is implied by the notion of a collaborative

6 198 MERRILYN GOOS ET AL. zone of proximal development established through peer interaction. The purpose of this paper, then, is to identify mechanisms of peer interaction that mediate collaborative metacognitive activity in problem solving tasks. 3. THE STUDY The research study that is the subject of this paper was conducted in senior secondary school classrooms (i.e. Years 11 and 12) over a three year period from Five teachers and their mathematics classes, all in different schools, contributed to the study. The schools were located in or near a large city in the Australian State of Queensland. As the emphasis was on interpreting learning in complex social settings rather than experimental manipulation and control of variables, research methods were consistent with a naturalistic inquiry approach (Lincoln and Guba, 1985), and included long term participant observation of classrooms (supplemented by audio and video recording), interviews with students and teachers, and survey questionnaires. Complementary perspectives provided by questionnaire and observational data revealed that in one classroom, more so than others, students seemed to be developing positive metacognitive dispositions and a preference for learning through interaction with peers. Consequently, this classroom was selected for intensive analysis to investigate how the teacher and students together created a culture of mathematical inquiry, and to explore the anatomy of collaborative metacognitive activity (see Goos, Galbraith and Renshaw, 1999, for an analysis of how such a learning culture was established within this classroom). Target students within this classroom were chosen for observation on the basis of their metacognitive sophistication and preference for working collaboratively with peers, as judged from preliminary observations and questionnaire responses (refer to Goos, 1995 and Goos, 1999a for questionnaire details). One lesson was observed each week, from late 1994 until the end of 1996, and target students were videotaped and audiotaped as they worked together on tasks set by the teacher as part of their regular mathematics program Data coding and analysis Selected portions of audio and videotapes were transcribed and the resulting verbal protocols parsed into Reading, Understanding, Analysis, Exploration, Planning, Implementation, and Verification episodes, following the methods pioneered by Schoenfeld (1992) and adapted by Artzt and Armour-Thomas (1992). However, this paper is concerned with a second,

7 SOCIALLY MEDIATED METACOGNITION 199 more detailed, level of analysis that focused on the conversational turns of all speakers (referred to here as Moves). Moves in the protocol were first coded to identify their metacognitive function. A coding scheme developed in an earlier study (Goos and Galbraith, 1996) was used to identify metacognitive acts where new information was recognised or an assessment of particular aspects of the solution was made. The first type of metacognitive act, New Idea, occurred when potentially useful information came to light or an alternative approach was suggested. The second type involved making an Assessment of the execution or appropriateness of a strategy, the accuracy or sense of a result, or of one s knowledge or understanding. Conversational Moves were then coded a second time to identify their contribution to the collaborative structure of the interaction, indicated by the transactive quality of the dialogue as defined by Kruger (1993). Transactive reasoning is characterised by clarification, elaboration, justification, and critique of one s own or one s partner s reasoning. Three types of transacts were coded: spontaneously produced transactive statements and questions, and passive responses to transactive questions. The orientation of each transact was also noted: operations on one s partner s ideas were labelled other-oriented, while reasoning directed at one s own ideas was coded as self-oriented. This procedure produced six transact codes: (three types) (two orientations). Transactive coding of peer discussion is consistent with Vygotskian approaches to studying links between collaborative interaction and cognitive change (Teasley, 1997); nevertheless this approach does not do justice to the reciprocal nature of collaboration. The scheme was therefore modified by grouping the codes to produce an operational definition of collaboration, as follows: Self-disclosure Self-oriented statements and responses that clarify, elaborate, evaluate, or justify one s own thinking. Feedback Request Self-oriented questions that invite a partner to critique one s own thinking. Other-monitoring Other-oriented statements, questions and responses that represent an attempt to understand a partner s thinking. As the aim of the study was to identify social interaction processes that mediated collaborative metacognitive activity, our initial analysis examined relationships between conversational Moves coded as having both metacognitive function and transactive structure. To investigate differences between successful and unsuccessful collaborative problem solving sessions, we then looked beyond single utterances that were coded as being simultaneously metacognitive and transactive, and analysed transactive discussion of metacognitive decisions that extended beyond the specific Move in which the New Idea or Assessment was offered. The principles underlying

8 200 MERRILYN GOOS ET AL. this dual coding approach are similar to those that guided Cobo and Fortuny s (2000) recent analysis of students social interactions while solving area problems. However, the research we present here differs significantly from this work in that our focus was specifically on the metacognitive function of students dialogue in natural classroom settings over an extended period of time, rather than a short term, out-of-class experimental context. 4. SUCCESSFUL COLLABORATION Three transcripts were selected for detailed analysis to illustrate common features of collaborative metacognitive activity. The transcripts were drawn from different years of the study (1994, 1995, 1996), and involve three different groups of students within the research classroom. All record students interactions in the early stages of learning a new mathematical concept or process (compound interest, projectile motion, and Hooke s Law), so that the tasks on which they worked were unfamiliar, challenging, and hence genuine problems. The tasks were therefore likely to require metacognitive control of problem solving actions, and to elicit collaborative interaction. Separate analyses of the metacognitive and transactive nature of the dialogues were undertaken for each transcript. Before considering the overall findings, an abbreviated analysis of one of the transcripts is presented below to illustrate the analysis methods. The following conventions were adopted in transcribing students interactions: (a) conversational Moves are numbered sequentially; (b) non-verbal information from the videotape is included in parentheses; (c) the symbol [...] indicates that part of the transcript has been omitted; and (d) annotations indicating metacognitive acts and transacts are recorded in italics The Cannon problem This problem comes from a Year 11 lesson on projectile motion. The teacher began the lesson by drawing a diagram on the blackboard showing one of the students in the class about to be fired from a cannon, at an angle of 30 degrees to the horizontal, with muzzle velocity 30 metres per second. He then drew a brick wall in the path of the projectile, at a distance of 3 metres from the point of projection, and placed a 1 metre high landing trampoline beyond the wall to break the fall of the human cannon ball. Students were to find the height at which a hole should be cut in the wall to allow passage of the projectile, and how far from the wall the landing pad should be placed. As this was the first time the class had attempted

9 SOCIALLY MEDIATED METACOGNITION 201 Figure 1. The cannon problem, with solution to part (a). a problem involving projection at an angle, this task represented a genuine problem for which they had not yet been shown a standard solution method. The teacher s diagram, and a model solution to the problem of finding the height of the hole in the wall, are presented in Figure 1. The target students in this lesson, Alex and Dylan, took 56 Moves and 10 minutes to solve the first part of the problem, concerning the height of the hole in the wall. Before video recording started, they had calculated the time of flight and range as 3 seconds and metres respectively. Dylan then started to use this information in a proportional reasoning strategy which, if carried to its conclusion, would have allowed him to find the time for the projectile to reach the wall as follows: distance to wall range = time to wall time of flight

10 202 MERRILYN GOOS ET AL. 3m 77.94m = = time to wall 3s time to wall 3s time to wall = s = 0.115s Alex immediately challenged Dylan s strategy for finding the time to the wall, and persisted throughout with his claim that there must be another way. Instead he tried to reason out the answer by first finding the time to travel one metre horizontally, as: 1m 30 cos 30 m/s = s = 0.038s This result corresponded to the distance fraction, (distance to wall)/(range) = 0.038, calculated by Dylan. However, later events suggest that, instead of finishing this calculation, Dylan had interpreted the interim result of as the time to reach the wall, and used it to calculate the height of the hole in the wall as s = ut at2 =(30sin30 ) ( 10) (0.038)2 2 = 0.563m which he rejected as an unreasonable answer. Alex continued to insist that you shouldn t need (the range), and to search for the simplest way of finding the time to reach the wall. Dylan, however, showed little interest in investigating Alex s working, urging his partner to just rearrange the formulas without worrying about the underlying reasoning. Unmoved by Dylan s impulsiveness, Alex steadfastly maintained his concentration by verbalising and monitoring his thinking, as the following excerpt shows: 27. Alex: So it takes one second to do twenty-five point nine eight, so you want to do three, so how many three s go into that... or do you do it the other way around? That s right you do...(pause) Hang on, what am I doing wrong? Can someone tell me? (Assessment strategy execution; Self-oriented transactive question, requesting critique)

11 SOCIALLY MEDIATED METACOGNITION Dylan: Just rearrange the formulas, right? 29. Alex: No, I want to do it this way! (Dylan throws up his hands. Alex taps him on the shoulder to get his attention again.) (Indistinct, trying to justify to himself what he has done) So it s...ok, so do you get...three, um Dylan: Three s the displacement. 31. Alex: So if it s three you want to divide that by...(uses calculator) What have I done? Oh OK, yeah (Assessment strategy execution)... Twenty-six t plus zero...i m being stupid... it takes...(long pause as he works) What am I dividing by? Nine point six six now... I ve just got to think... eight point six six Dylan: What are you doing? Simply, it takes twenty-five metres to (Inaudible). 33. Alex: Point...one one. (announcing answer) I get, point one one. Didn t you get point one one? 34. Dylan: (Showing what he has done) Divide that by ten. (New Idea) And there s, there s, that s, point one of a second. 35. Alex: Why did you divide by ten? (Assessment strategy appropriateness; Other-oriented transactive question, seeking justification of Dylan s answer) 36. Dylan: To get point one of a second. That s, that s its (inaudible) Otherwise it s point 01 of a second. (Self-oriented response, justification of answer) 37. Alex: Why d you do that? (Other-oriented transactive question, seeking further justification) 38. Dylan: That s velocity...that s displacement...(alex shows disagreement) It s easier to do it that way! Whatever you figure out off this is going to (inaudible) velocity. 39. Alex: I know, I know, but I can t 40. Dylan: (frustrated) Just use v equals s over t, but you replace s with... oh. What have I done? (Assessment strategy execution) Displacement over... new displacement over old displacement...thus...(uses calculator) Alex carefully reasoned his way through the calculations concerning the horizontal motion of the projectile, beginning by asking himself how long it would take to travel 3 metres if it moved metres in one second (Move 27). After some confusion over whether to divide by 3 or 3 by 25.98, he turned to the equation and, with a = 0, found t =0.11 seconds. Alex s query at this point (Move 33) prompted Dylan to try to rationalise his own working (Move 34). What Dylan has done here is not

12 204 MERRILYN GOOS ET AL. clear. However, Alex s persistence in seeking justification seemed to alert Dylan to a flaw in his reasoning (Move 40). In the next segment of the transcript the students simultaneously evaluated and amended their previous solution attempts: 43. Alex: Yeah. I think it s point one one! So why did I get...? with the other thing it should have been like that. (Assessment strategy execution; Self-oriented transactive question, requesting critique) 44. Dylan: No, because that was displacement over displacement. (New Idea; Self-oriented response, clarification of own strategy) 45. Alex: What s wrong with that? It s just a fraction of the complete thing. (Assessment strategy appropriateness; Otheroriented transactive statement, critique of New Idea proposed in Move 44) 46. Dylan: If it s a fraction you can find point 038 of twenty-five point nine eight. (New Idea; Self-oriented transactive statement, justification of New Idea proposed in Move 44) 47. Alex: (pause) No. (pause) You find... that of one point five (an error, corrected in Move 49). (Assessment strategy appropriateness; Other-oriented transactive statement, critique of New Idea proposed in Move 46) 48. Dylan: (to himself) Fifteen...times...point one...this is going to be more Alex: (Pause) Oh we just didn tfinish the other thing! If you times it by one point five it d be bigger. (Assessment strategy execution) (Pause)Oh sorry,you times it by three. (Assessment strategy execution) (Pause, both using calculators) 50. Dylan: (sounding pleased) That s a better answer. (Assessment sense of result) One point seven... (muttering his calculations) 51. Alex: Ah yes, yes, we just didn t finish it. 52. Dylan: So what did we do? (Other-oriented transactive question, seeking clarification of metacognitive Assessments in Move 49) 53. Alex: We just found the percentage! But we forgot to times it by the time of flight. (Self-oriented transactive response, providing clarification of metacognitive Assessments in Move 49) Fools! Glad I picked it up, eh? Otherwise we would have been wrong. As well as being stupid and...we would have been wrong. (Assessment sense of result) The boys clarified their initial proportional reasoning strategy (Moves 43 to 45), and Alex corrected Dylan s erroneous use of the distance fraction

13 SOCIALLY MEDIATED METACOGNITION 205 (Move 47). This brought about the realisation, on Alex s part, that they had simply forgotten to finish their initial calculations with this strategy by finding time to wall = (Move 49). At this point, Dylan was happy to complete the calculation of the distance to the wall (Moves 48 and 50), and only then did he ask for an explanation of their error (Move 52). Alex s response to this query shows his concern for monitoring his work to prevent calculation errors, and for assessing his results for both accuracy and sense (Move 53). Metacognitive function and transactive structure of the dialogue The metacognitive function and transactive structure of the dialogue were analysed by coding the transcript to record the number and type of metacognitive acts and transacts produced by each speaker, and in total. Because speakers may have unequal opportunities to contribute to the discussion, these coding categories were counted as both frequencies and proportions. For each speaker, proportions were calculated as the number of metacognitive acts or transacts they produced divided by that person s number of Moves. For the whole dialogue, proportions were calculated as the total number of metacognitive acts or transacts divided by the total number of Moves in the protocol. Twenty of the 56 Moves in the dialogue (a proportion of 0.36) were coded as having metacognitive function. Analysis of individual contributions shows that Alex made fourteen Assessments (mainly of strategy execution and appropriateness) to Dylan s two, which identifies Alex as the driving force in locating and correcting errors, and in evaluating the usefulness of the strategies employed. The second coding pass identified fifteen of the 56 Moves (a proportion of 0.27) as transactive. These transacts were then grouped to reflect three elements of collaboration identified previously: Self-disclosure (selforiented statements and responses), Feedback Request (self-oriented questions) and Other-monitoring (other-oriented statements, questions and responses). This analysis indicated that the dialogue was genuinely collaborative, in that it contained a balance of Self-disclosure (proportion = 0.11), Feedback Requests (0.04) and Other-monitoring (0.13). However, when each student s contributions are examined separately, it becomes clear that the interaction was characterised by Alex s desire to understand Dylan s thinking, since Alex s transacts chiefly involved Other-monitoring (proportion = 0.21) while Dylan s transacts were predominantly Self-disclosure (proportion = 0.18).

14 206 MERRILYN GOOS ET AL. TABLE I Moves double coded as metacognitive acts and transacts successful collaboration Metacognitive function Transactive structure (frequencies) Self- Feedback Otherdisclosure request monitoring New Idea Assessment strategy Assessment result 3 Assessment understanding 6 Total Collaborative metacognitive activity The double coding criterion As the central purpose of this analysis was to discover how student-student interaction mediated metacognitive activity, conversational Moves were identified which had both a metacognitive function (New Idea or Assessment) and transactive structure (self- or other-oriented statement, question, or response). The metacognitive function of students collaborative dialogue across all three problem solving protocols illustrating successful collaboration is summarised by Table I and the comments that follow. These results indicate that joint metacognitive activity was characterised in several ways. First, students clarified, elaborated, and justified their New Ideas for the benefit of a partner (Self-disclosure). For example, in the Cannon transcript, Dylan justified his proportional reasoning strategy by claiming: 46. Dylan: If it s a fraction you can find point 038 of twenty-five point nine eight. (New Idea; Self-oriented transactive statement, justification of New Idea proposed in Move 44) Second, students sought feedback on the New Ideas they proposed, and also asked their peers for help in finding errors by inviting critique of strategies and results (Feedback Request). This is illustrated in the way that Alex invited an evaluation of his strategy execution: 27. Alex: [...] Hang on, what am I doing wrong? Can someone tell me? (Assessment strategy execution; Self-oriented transactive question, requesting critique)

15 SOCIALLY MEDIATED METACOGNITION 207 Finally, students made an effort to understand their partners thinking by offering critiques of their strategies, elaborating on and monitoring their understanding of partners ideas, or requesting explanations (Other-monitoring). Hence we see Alex assessing Dylan s ideas: 44. Dylan: No, because that was displacement over displacement. (New Idea; Self-oriented response, clarification of own strategy) 45. Alex: What s wrong with that? It s just a fraction of the complete thing. (Assessment strategy appropriateness; Otheroriented transactive statement, critique of New Idea proposed in Move 44) and asking him to justify his approach: 35. Alex: Why did you divide by ten? (Assessment strategy appropriateness; Other-oriented transactive question, seeking justification of Dylan s answer) Table I shows that, for the three transcripts (a total of 351 Moves), 26 of the metacognitive-transacts were self-oriented (Self-disclosure or Feedback Request) and 26 were other-oriented (Other-monitoring). Thus, when the students interacted with each other, their monitoring activity was directed at both their own thinking and the ideas of their peers. In other words, the evidence presented here suggests that collaborative metacognitive activity proceeds through offering one s thoughts to others for inspection, and acting as a critic of one s partner s thinking. 5. UNSUCCESSFUL COLLABORATION This microscopic analysis of problem solving transcripts provides a lens through which to view the peer interaction processes that establish a collaborative zone of proximal development. Nevertheless, since previous research has demonstrated that collaboration does not guarantee successful problem solving outcomes (e.g. Stacey, 1992), it is necessary also to examine situations in which collaboration was metacognitively fruitless and to identify reasons for this lack of success. A further three transcripts of students dialogue were selected as before to illustrate unsuccessful collaboration, with the analysis showing how poor metacognitive decisions contributed to problem solving failure. The problem solving tasks dealt with finding the area of the Koch snowflake, analysing the motion of a body on an inclined plane, and combinatorics. Again, before drawing any conclusions about the nature and causes of metacognitive failure, an analysis of one transcript is provided below to

16 208 MERRILYN GOOS ET AL. Figure 2. Combinations problems. facilitate comparisons between successful and unsuccessful collaborative problem solving The Combinations problem This videotape transcript comes from another Year 11 lesson in the early stages of a unit of work on combinatorics. Knowing that he would be unavoidably absent for this lesson, the teacher had set a series of problems which would give the students their first opportunity to apply their newly gained knowledge of combinations. These problems were contained in a teacher-prepared handout that also included explanations and worked examples, and served as the students sole text for the topic. The target students are Dylan and Alex, who featured in the Cannon transcript presented earlier, and Sean and Rhys. Figure 2 shows the first problem on which they worked (Question 19 in the problem set), together with model solutions. After reading the stem to Question 19 and making the observation that there were 52 C 5 hands in total, Dylan immediately recognised that this would give too large a number for part (a) of the question, which imposed the constraint of having at least three aces in the hand. Nevertheless, all three boys used the n C r buttons on their calculators to gain a feel for the problem and discovered that 52 C 5 does indeed represent a very large number of hands ( ). Although they had identified the relevant information in the problem, the students struggled to formulate a strategy for taking account of the specified selection of at least three aces. 13. Alex: How do you do it with three aces? (No response) Maybe we have to work out the probability of aces or something. (New Idea)

17 SOCIALLY MEDIATED METACOGNITION Sean: Well that s... four out of fifty-two. That s one out of thirteen chances you ve got an ace. (Other-oriented transactive statement, elaboration of New Idea proposed in Move 13) 15. Alex: (doubtfully) Yeah, but how do you work out these three aces? (Other-oriented transactive question, seeking further clarification) 16. Dylan: No, you ve got five cards, so it s only fifty-two, ah...fiftytwo C (New Idea) 17. Alex: Ohh! Do C two, that s how many won t have (New Idea) 18. Dylan: Yeah, and you got to have (Other-oriented transactive statement, elaboration of New Idea proposed in Move 17) 19. Alex: (simultaneously) a certain three cards. 20. Dylan: That certain three cards, will be your aces. (Self-oriented transactive statement, providing further elaboration) Alex and Dylan proposed that 52 C 2 might represent the number of five card hands without three aces (Moves 17 20), foreshadowing an approach based on mutually exclusive operations and the addition principle. (Note that they still had not come to grips with the at least condition.) Despite their initial enthusiasm for this strategy, it soon became apparent that the boys had no way of knowing whether or not they were on the right track: 24. Dylan: Aarrh I don t know if I m doing it right or I m doing it wrong! (Assessment strategy appropriateness) Before long, the boys abandoned part (a) of the problem and acknowledged that they were stuck on Question 19 as a whole. However, they were not yet willing to give up completely. 40. Dylan: So how do you do it? 41. Sean: If we had an answer an answer sheet (New Idea) 42. Dylan: Yeah, we could figure it out. (New Idea) 43. Alex: You could always think about it without the C rule. And go like, OK, for hearts you ve got, however many choices, and, the next choice you ve got however many choices, the next choice you ve got...(new Idea) 44. Dylan: You got a quar ter...(hesitation, draws out this word) A quarter of fifty. (New Idea) 45. Alex: (Not listening to Dylan) Think about it the long way. Hey, is there an example somewhere? (New Idea) (Checks quickly through handout, overlooks Example 12.) Here the boys considered two potentially useful strategies for dealing with impasses such as the one they faced working backwards from the answer

18 210 MERRILYN GOOS ET AL. (Moves 41 and 42), and looking at a similar problem (Move 45). Unfortunately, they were unable to take advantage of either strategy, since the teacher-prepared handout did not provide answers to the problems, and they overlooked a worked example in the text that might have provided some clues. While Alex continued to hunt for a helpful example in the text, Dylan moved on to Question 19 (b), and began hesitantly to reason out a strategy which would lead him to the correct answer (Move 44). There is evidence here that Dylan was beginning to develop a general understanding of how the choices of cards can be constrained. In the case of Question 19 (b), if a five card hand is to contain three hearts, then the hearts are selected from only one suit (a quarter of fifty-two cards), not the full pack. 54. Dylan: (writes) This is thirteen out of fifty-two... is... hearts. So what would you go? Would you go, thirteen... C... (inaudible). (New Idea) 55. Sean: So are you still trying to work out something for? 56. Alex: No, I m just going to leave that for now. And wait until he comes up with 57. Sean: Leave Question 19 altogether? 58. Alex: Yeah, I don t know how to do it. (Assessment understanding) Dylan now became absorbed with completing Question 19 (b), and he worked in silence while his friends considered their next move. Eventually his persistence was rewarded, but his triumphant announcement that he had worked out how to attack Question 19 (b) (Move 63) was not acknowledged by Alex and Sean, who were busily working on Question Dylan: (to himself) So should we go...? I know, I ve figured it out! I ve figured it out! (Assessment Understanding) (Pause) Multiply that by...what s the (inaudible)? It s thirteen take fifty-two. (New Idea) 64. Rhys: Thirteen take fifty-two? (Assessment strategy execution) 65. Dylan: Sorry! Fifty-two take thirteen. Thirty-nine, yeah. (Quietly, to himself) Thirty-nine C two. 66. Alex: (Reading Question 20) How many committees of five...? 67. Dylan: (to himself, using calculator) Two hundred and eighty-six times...seven hundred and forty-one! (Sounds surprised) 68. Rhys: Is that for (a) or (b)?

19 SOCIALLY MEDIATED METACOGNITION Dylan: That s for (b)! I think (a) s wrong actually, but anyway...(assessment accuracy of result) (Long pause, writing. Goes on to Question 19c.) C... one... C four... is thirteen times... eight thousand two hundred, no, eighty-two thousand two hundred...(long pause, writing. Responds to inaudible question, from student off camera.) Well we don t have any answers, so we don t even know if we re right. (Assessment accuracy of result) (Continues working) Thirteen Ctwo...(now doing Question 19c) Although Dylan did not verbalise all his working, it is clear that he was pursuing the correct approach to solving parts (b) and (c) (see Moves 67 and 69, and Figure 2). Nevertheless, the lesson ended with all students still at a loss to know whether they had found the correct way to approach these problems. Metacognitive function and transactive structure of the dialogue Of the 69 Moves in the transcript, 22 were coded as having a metacognitive function (proportion = 0.32). Dylan and Alex were the main contributors, sharing ten of the eleven New Ideas (six for Dylan, four for Alex) and all eleven Assessments (seven for Dylan, four for Alex). However, this quantitative representation does not tell the full story, since it obscures an important quality of the students metacognitive activity while working on the combinations problems their inability to make valid judgments about their strategies and answers. In particular, Dylan was unsure of the appropriateness of the strategy he implemented for Question 19 (a) (Move 24), and he expressed his doubts as to the accuracy of his answers (Move 69). Only nine Moves were coded as transactive, a proportion (0.13) considerably lower than found in transcripts of successful collaboration. No Feedback Requests were identified, and no transacts were classified as justifying a speaker s or a peer s ideas. The significance of these results is examined in the next part of the paper. 6. WHAT MAKES THE DIFFERENCE BETWEEN SUCCESSFUL AND UNSUCCESSFUL COLLABORATION? What are the metacognitive and transactive characteristics of students interactions that distinguish successful from unsuccessful problem solving activity? Let us begin to examine this question by calculating proportions of success and failure transcripts coded as having metacognitive function and/or transactive structure. Figure 3 provides a summary of the rel-

20 212 MERRILYN GOOS ET AL. Figure 3. Proportions of transcripts coded as having metacognitive function and/or transactive structure. evant coding of the six transcripts that were selected to illustrate problem solving success (Cannon, Compound Interest, Hooke s Law) and failure (Koch Area, Forces, Combinations). To enable valid comparisons to be made between transcripts of varying lengths, metacognitive acts and transacts are recorded as proportions of the total number of conversational Moves in the relevant transcript. For example, eight of the 56 Moves in the Cannon transcript were double coded (metacognitive and transactive), so 8 56 gives a proportion of Averages for the two groups of transcripts categorised as success and failure are calculated as the proportions of the total number of Moves in the three transcripts belonging in that group. Thus, 52 out of a total 351 Moves in the three transcripts illustrating successful collaborative metacognitive activity were double coded, so that the mean proportion of the dialogue coded as having both metacognitive function and transactive structure for this group of transcripts is 0.15 (52 351). The total metacognitive and transactive proportions of the success and failure groups of transcripts have been calculated in a similar fashion and also recorded in Figure 3. For transcripts illustrating successful collaborative metacognitive activity, proportions of the dialogue coded as metacognitive or transactive were 0.27 and 0.26 respectively, while the correspond-

21 SOCIALLY MEDIATED METACOGNITION 213 ing proportions for transcripts of unsuccessful metacognitive activity were 0.29 (metacognitive) and 0.17 (transactive). This analysis shows that there was very little difference in the metacognitive proportions illustrating successful (0.27) and unsuccessful (0.29) collaboration. However, a different picture emerges from the summary of transact proportions: first, there was a lower incidence of transactive discussion in unsuccessful problem solving sessions (proportion = 0.17, compared to 0.26 in successful sessions); and second, a large part of this discrepancy was accounted for by the difference in the proportions of non-metacognitive transacts (proportion = 0.05 in unsuccessful problem solving sessions, compared to 0.11 in successful sessions). It appears, then, that success is characterised not only by utterances which are simultaneously metacognitive and transactive (the double-coding criterion applied previously), but also by interactions involving purely transactive discussion. This finding suggests that the discussion around, and generated by, individual metacognitive acts is crucial to the success of the mathematical enterprise. All transcripts were therefore re-examined to identify metacognitive acts that either led to or were prompted by a transactive statement, question, or response. Moves so identified were thus connected to at least one transact, and were labelled metacognitive nodes. If the node was connected to more than one transact, then a transactive cluster was said to have formed around the node. Figures 4 and 5 provide a visual representation of these nodes and clusters for the Cannon and Combinations transcripts respectively. Numbers refer to Moves in the transcript coded as either metacognitive acts or transacts, while letters identify the speakers. Symbols have been used to distinguish the different types of metacognitive acts defined in section 3.1. Clusters of transacts have been enclosed in boxes. Arrows trace the connections between metacognitive nodes and non-metacognitive transacts. The important features to note are: 1. Circles superimposed on symbols represent metacognitive transacts (i.e. double coded Moves previously highlighted as identifying collaborative metacognitive activity); 2. Arrows connecting metacognitive nodes with transacts, and particularly with transactive clusters, pinpoint instances of extended discussion of metacognitive ideas and assessments, and thus highlight the role of non-metacognitive transacts. Visual representations similar to those in Figures 4 and 5 were constructed for each transcript. The numbers and proportions of nodes and clusters for

22 214 MERRILYN GOOS ET AL. Figure 4. Metacognitive nodes and transactive clusters Cannon problem. successful and unsuccessful problem solving transcripts are recorded in Table II. Successful collaboration was found to feature roughly twice the proportion of metacognitive nodes (0.07 for success and 0.04 for failure), and three times the proportion of transactive clusters (0.03 for success and 0.01 for failure), as unsuccessful collaboration. In other words, transactive discussion of metacognitive New Ideas and Assessments appears to be a significant factor in successful collaborative problem solving. The following section examines the impact of these nodes and clusters (or

23 SOCIALLY MEDIATED METACOGNITION 215 Figure 5. Metacognitive nodes and transactive clusters Combinations problem.

24 216 MERRILYN GOOS ET AL. TABLE II Frequencies and proportions of metacognitive nodes and transactive clusters for successful and unsuccessful collaborative metacognitive activity Frequencies (proportions) Success transcripts Failure transcripts Metacognitive nodes 26 (0.07) 11 (0.04) Transactive clusters 10 (0.03) 3 (0.01) Total Moves their absence) on students ability to solve the Cannon and Combinations problems Metacognitive success The Cannon problem While there are only two transactive clusters in the map of the Cannon transcript (Figure 4), both were important in revealing errors in the students working (see section 4.1). The first cluster (Moves 36 and 37) originates from Alex s request to Dylan that he explain how he arrived at a time of 0.1 second for the projectile to reach the wall (Move 35). Dylan s explanation (Move 36) was not deemed satisfactory so Alex persisted, asking Why d you do that? (Move 37). His query forced Dylan to articulate further, and in so doing to discover a flaw in his reasoning (Move 40) Oh. What have I done? The second cluster, around the node represented by Move 49, was preceded by a significant exchange in which every conversational turn was double coded as having metacognitive function and transactive structure (Moves 43 to 47). Here the students paused to evaluate their previous solution attempts, while Alex tried to reconcile the answer he had obtained using the conventional projectile motion approach (0.11 seconds to reach the wall) with the proportional reasoning strategy which Dylan had unsuccessfully pursued. This exchange led to the next metacognitive node, where Alex identified the cause of the discrepancy (Move 49) as We just didn t finish the other thing, and the accompanying transactive cluster. Dylan s request for clarification (Move 52) prompted Alex to explain in more detail what was required to finish the proportional reasoning approach, and to identify the specific error which had prevented them from arriving at a sensible answer via this method, we forgot to times it by the time of flight (Move 53).