GOOSM97.264

Self-Directed and Peer-Assisted Thinking in a Secondary Mathematics Classroom

Merrilyn Goos

The University of Queensland

Paper presented at the

Annual Conference of the Australian Association for Research in

Education

Brisbane, 30 November-4 December 1997

Self-Directed and Peer-Assisted Thinking in a Secondary Mathematics Classroom

Merrilyn Goos

The University of Queensland

Abstract. This paper examines the metacognitive monitoring and

regulatory strategies used both individually and collaboratively by

senior secondary school mathematics students. Analysis of interviews

and videotaped lesson transcripts revealed that metacognitive activity

was organised around routine monitoring of progress and recognition of,

and response to, various types of impasse presented by challenging

problems. The findings help to extend existing episode-based models of

mathematical thinking, and shed light on the social processes of peer

collaboration through which students monitor and extend each other's

thinking.

The last decade has seen the emergence of an international movement

calling for reform in the teaching and learning of mathematics. In both

the United States and Australia, for example, new curriculum and policy

documents place increased emphasis on problem solving, communication

and mathematical reasoning, and endorse greater use of small group work

and peer interaction as a means of encouraging students to become

self-directed learners (Australian Education Council, 1991; National

Council for Teachers of Mathematics, 1989). Such significant curriculum

reforms require a sound research base if they are to be effectively

implemented. However, our theoretical understanding of problem solving

processes, and how students' problem solving abilities are cultivated

by these new forms of classroom interaction, is far from complete.

In a recent review of progress in mathematical problem solving research

over the past 25 years, 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 mathematical

problem solving where metacognition refers to what students know about

their own thought processes, and how they monitor and regulate their

thinking while working on mathematical tasks. Although the importance

of metacognition is now widely acknowledged, we still lack an adequate

theoretical model for explaining the mechanisms of individual

self-monitoring and self-regulation (Schoenfeld, 1992), and, despite

increasing research interest in the social and cultural aspects of

mathematics learning (e.g. Brown et al., 1993), we have barely begun to

examine the possibilities for collaborative metacognitive activity that

may occur when students work in small groups.

This paper reports on a study which investigated metacognitive activity

in secondary school mathematics classrooms, and connections between

self-directed (individual) and peer-assisted (collaborative) monitoring

and regulation. Two research questions are addressed:

1. What monitoring and regulatory processes do individual students use

while working on classroom mathematics tasks?

2. How are these metacognitive processes elicited and supported during

collaborative student-student interaction?

Mathematical Problem Solving and Metacognition

Frameworks for analysing task-oriented mathematical thinking typically

identify phases or episodes representing distinctive kinds of problem

solving behaviour. For example, Schoenfeld (1985) developed a procedure

for parsing verbal protocols into five types of episodes: Reading,

Analysis, Exploration, Planning/Implementation, and Verification. A

recent study by Artzt and Armour-Thomas (1992) has modified and

expanded Schoenfeld's model in order to delineate the roles of

cognitive and metacognitive processes in small group problem solving.

The modifications adopted in the present study separate Schoenfeld's

Planning/Implementation episode into two distinct categories, and

include an additional episode type, Understanding the problem, which

overlaps somewhat with Schoenfeld's Reading and Analysis episodes.

These modifications may be applied equally well to individual and

collaborative problem solving. The ideal characteristics of each

episode are described in the far left column of Figure 1.

In addition to identifying and categorising episodes of problem solving

behaviour, the analysis frameworks of both Schoenfeld (1985) and Artzt

and Armour-Thomas (1992) acknowledge the central role of metacognitive

monitoring and regulation in keeping the solution process on track, for

example, by noting that the solution status or one's general progress

should be monitored and plans modified if necessary. However, a

deficiency in both frameworks is the lack of detail in describing the

types of monitoring and regulatory activities that would be appropriate

and expected in each episode. The suggested scope of these

metacognitive activities is detailed in Figure 1, in the columns headed

Monitoring and Regulation.

The model offered in Figure 1 also makes a distinction between two

kinds of monitoring. First, the routine assessment of activity during

each episode (for example, assessing one's understanding of the

problem, assessing execution of the strategy) confirms that problem

solving is on track. On the other hand, this routine monitoring may

alert the student to specific difficulties and signal the need for a

pause or some backtracking while remedial action is taken. Possible

warning signals, or metacognitive 'red flags', associated with this

second kind of monitoring are shown in Figure 1 as shaded boxes.

'Red flags' may be of three types: lack of progress, error detection,

and anomalous result. Recognising lack of progress during a fruitless

exploration episode should lead students back to analysis of the

problem in order to reassess the appropriateness of the chosen strategy

and to decide whether to persist, salvage whatever information is

useful, or abandon the strategy altogether. In the latter case it is

likely that students will need to reassess their understanding of the

problem, and search for new information or a new strategy. Error

detection during an implementation episode should prompt checking and

correction of calculations carried out so far. Finally, if attempts to

verify the solution reveal that the answer does not satisfy the problem

conditions, or does not make sense, then this anomalous result should

trigger a calculation check (assess execution of strategy), followed,

if necessary, by a reassessment of the strategy.

Collaborative Problem Solving

The analytical framework used by Artzt and Armour-Thomas (1992)

revealed the type and extent of metacognitive behaviour for individual

students working together in small groups, but it was not designed to

capture the interactive nature of the groups' monitoring and regulation

of their collective mathematical activity. Little is known about how

collaboration between peers of comparable expertise might mediate

metacognitive strategy use, and the few studies which have been

reported have produced conflicting results (e.g. Hart, 1993; Stacey,

1992). One source of difficulty may lie in the lack of attention given

to the quality of students' interaction, since instructing students to

work together does not guarantee that they will collaborate. For

example, Damon and Phelps (1989) reserve the term peer collaboration

for the interaction that occurs when students with similar levels of

competence share their ideas in order to jointly solve a challenging

problem. Similarly, Teasley and Roschelle (1993, p. 235) described

collaboration as 'a coordinated, synchronous activity that is the

result of a continued attempt to construct and maintain a shared

conception of the problem'. However, collaborative work can also

include periods of individual activity when participants withdraw to

grapple with difficult or partially-formed ideas, and turn-taking is

not responsive to their partner. Neither is the students' overlap of

meaning always complete or certain, producing periods of conflict

during which mutual understanding has to be renegotiated.

Thus, although collaboration may involve both cooperation and conflict,

its distinguishing feature mutuality the 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 propose and defend their own ideas, and to

ask their peers to clarify and justify any ideas they do not

understand.

The study reported in this paper focussed on the metacognitive

monitoring and regulatory processes reported by individual students and

observed during collaborative work in senior secondary school

mathematics classrooms.

Figure 1. An episode-based model of metacognitive activity during problem solving

(Available from the author: see end of paper for details)

Method

The data for the present study come from a larger project, the purpose

of which was to investigate both individual and collaborative

metacognitive monitoring and regulation in senior secondary school

classrooms, patterns of teacher-student and student-student interaction

associated with metacognitive activity, and assumptions about teaching

and learning mathematics underlying teachers' and students' actions.

The project involved up to eight secondary school teachers and their

classes in five schools over a period of three years. Methods used to

investigate the metacognitive strategies students employed while

working on mathematics tasks included questionnaires (Years 1-3 of the

study), individual interviews (Year 3 only), and classroom observation

supplemented by audiotaping and videotaping (Years 1-3). This paper

draws on interview and audio/videotape data gathered from one

independent and one government school (labelled School A and School B

respectively) in Brisbane's northern suburbs.

Target students was chosen for interview and observation on the basis

of their metacognitive sophistication and preference for working

collaboratively with peers, as judged from preliminary observation and

responses to questionnaires probing their beliefs about mathematics and

metacognitive self-knowledge (see Goos, 1995 for details of

questionnaires). Individual interviews were conducted with seven

students at School A and eight at School B. The semi-structured

interview script is shown in Figure 2. The interview was designed to

probe and follow up some of the issues raised in the open ended

metacognitive self-knowledge questionnaire. Questioning generally

followed the order indicated; however, since the course and substance

of interviews were determined by students' responses, some variation in

the sequence of questioning occurred. All interviews were audiotaped

and transcribed.

One lesson each week was observed in participating classrooms, and

groups of target students were audiotaped or videotaped as they worked

and discussed their ideas together in class. The tapes were reviewed to

identify instances of collaborative metacognitive activity, and these

portions were transcribed for later analysis. Only transcripts from

School A are discussed in this paper.

Individual Interview

1. What aspects of maths do you find most difficult?

What do you do to overcome these difficulties?

What strategies seem to help?

2. What do you do when you get stuck on a problem?

3. What is the best way you have found to learn and understand maths?

What kinds of things do you do to learn and understand maths while

4. (a) When you're working on a problem, how can you tell whether you're

(b) How can you tell if you've made a mistake?

(c) How can you tell whether you've solved the problem correctly?

5. If you and a friend got different answers to the same problem, what

6. Is it possible to get the right answer to a maths problem and still

Results

Research Question 1: What monitoring and regulatory processes do

individual students use while working on classroom mathematics tasks?

Analysis of students' interview responses was guided by the

episode-based model of problem solving shown in Figure 1. From the

interview script (Figure 2), Questions 2, 4 and 5 were identified as

those probing metacognitive self-monitoring and self-regulation

strategies that might be used while working on a mathematical task:

specifically, during Implementation (Questions 4a, 4b), Exploration

(Question 2) and Verification (Questions 4c, 5) episodes. Responses to

only these questions are included in the following analysis. Note that

the interview did not include questions specifically probing

metacognitive activity during the initial Reading, Understanding,

Analysis and Planning episodes of problem solving. However, students'

responses to the question 'What do you do when you get stuck on a

problem?' (Interview Question 2) indicate that this lack of progress

can trigger a reassessment of their task-specific knowledge and

understanding of the problem monitoring activities which are associated

with these initial episodes (see Figure 1). Thus the interview deals

either directly or indirectly with all of the episodes typically found

during problem solving.

The interview implicitly investigated students' ability to recognise

and act on metacognitive 'red flags' (see Figure 1). The analysis began

by identifying the 'red flags' targeted by each interview question:

Interview Question 'Red Flag'

2. What do you do when you get stuck on a problem? Lack of progress

4. (a) When you're working on a problem, how can you Lack of progress

tell whether you're doing it the right way? How do you Error detection

decide whether to change your approach? Anomalous result

4. (b) How can you tell if you've made a mistake (while Error detection

working on a problem)?

4. (c) How can you tell whether you've solved the Anomalous result

problem correctly?

5. If you and a friend got different answers to the Anomalous result

same problem, what would you do?

Students' responses to each interview question were then collected

together and grouped into categories corresponding to the monitoring

and regulatory strategies identified in the episode-based framework of

Figure 1.

Since each 'red flag' was targeted by more than one interview question,

the next step in the analysis procedure drew together all strategies

triggered by each of the three 'red flags'. The results are presented

below, together with a sample of actual responses to exemplify the

range of monitoring and regulation strategies mentioned by students.

The proportion of students (n=15) who reported each type of strategy is

also noted.

'Red Flag': Lack of progress

¥ Assess progress towards goal (Monitoring), 60%

Leading nowhere, going round in circles.

No pattern forming.

Realise you're not heading in the right direction, run out of patience

¥ Assess strategy appropriateness (Monitoring), 13%

See if I'm doing it the right way first.

Go back to the beginning and check my reasoning.

¥ Change strategy (Regulation), 67%

Try different approaches, start afresh.

If I'm attacking it one way, change directions.

If I know any other ways, try them.

¥ Assess knowledge and understanding of problem (Monitoring), 20%

Reread the question, make sure I've understood it properly.

Make sure I haven't missed anything the question gives.

¥ Identify new information, reinterpret problem (Regulation), 53%

Jot down all information given, try to piece together to find pattern.

Look for underlying similarities to something you've done before.

Think about what we've learned, see what could apply to that.

Additional regulatory strategies triggered by lack of progress included

individual persistence (47%), seeking help from the teacher (33%), or

collaborating with peers (40%).

'Red Flag': Error detection

¥ Assess strategy execution (Monitoring), Correct calculation errors

(Regulation), 33%

Redo every calculation a few times.

Pause at critical stages in the problem to check what you've done

'Red Flag': Anomalous result

¥ Assess result for accuracy and sense (Monitoring), 93%

Check it another way, or work backwards.

Does it make sense.

Ask someone else what they got.

¥ Assess strategy execution (Monitoring), Correct calculation errors

(Regulation), 60%

Check my own working for errors, then check each other's working.

Explain our working to each other.

Decide whose working matches what we've been taught.

¥ Assess strategy appropriateness (Monitoring), Change strategy

(Regulation), 53%

Compare approaches and decide which makes most sense.

Criticise my own work, and my friends'.

Try it together and see where the two ways branch off.

The analysis of students' interview responses summarised above has

identified all the appropriate metacognitive monitoring and regulation

strategies proposed in the episode-based model of mathematical thinking

of Figure 1 (although no single student mentioned every type of

strategy). It is possible to map students' reported strategies onto the

model, as shown in Figure 3. Monitoring and regulation pathways

triggered by each of the 'red flags' are plotted on the model. Path 1

describes the sequence triggered by lack of progress: the

appropriateness of the strategy is reconsidered and a decision may be

made to modify it; and the problem may need to be reinterpreted and new

information identified. Path 2 begins with error detection, followed by

assessment of strategy execution and correction of calculation errors.

Path 3, followed when an anomalous result is recognised, involves

eliminating any errors in the execution of the strategy, and trying a

new approach if the original strategy was responsible for the incorrect

answer.

Here it is worth recalling that 'red flags' will not be recognised

without routine monitoring to check whether problem solving is on

track. If students are to avoid persisting with an inappropriate

strategy, detect and correct errors, and reject nonsensical answers,

then continuous assessment of progress, strategy execution, and results

is essential.

Research Question 2: How are metacognitive processes elicited and

supported during collaborative student-student interaction?

Three transcripts from a senior secondary mathematics classroom in

School A were selected for analysis to illustrate common features of

collaborative metacognitive activity. Each transcript was drawn from a

different year of the study, and each was obtained from a different

group of students as they worked on problems set by the teacher in the

normal course of a lesson. Students were not specifically told to work

together, but were free to choose whether, and how, to interact with

their peers. Since a full description of the analysis of each

transcript is beyond the scope of this paper, only a summary of the

findings is presented here. The three problems involved:

¥ Spreadsheet calculations of compound interest (one male and two female

¥ Hooke's Law (three male Year 12 students, second year of the project),

¥ Projectile motion (two male Year 11 students, third year of the

The transcripts were first parsed into macroscopic episodes which

represented the distinctive kinds of problem solving behaviour shown in

Figure 1.

Figure 3. Monitoring and regulation pathways identified from student

interviews

(Available from the author: see end of paper for details)

A second, microscopic, level of analysis focussed on the metacognitive

function and collaborative structure of the conversational turns of all

speakers. A coding scheme developed in an earlier study (Goos &

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 proposed. The second type of metacognitive act

involved making an Assessment of a strategy (execution or

appropriateness), a result (accuracy or sense), or of one's knowledge

or understanding.

Conversational turns were then coded a second time to identify

transactive dialogue, defined as discussion in which an individual's

reasoning operates on a partner's reasoning, or significantly clarifies

his or her own reasoning (Kruger, 1993). Following Kruger, three types

of transacts were coded: transactive statements and questions, and

responses to transactive questions. The orientation of each transact

was also noted, with operations on one's partner's ideas being labelled

other-oriented, and reasoning directed at one's own ideas coded as

self-oriented. This procedure produced six transact codes: (three

types) x (two orientations). Kruger's coding scheme was then extended

to highlight the reciprocal nature of collaborative interactions. This

was done by grouping the codes to produce an operational definition of

collaboration possessing the following three elements:

Self-disclosure Self-oriented statements and responses that clarify,

elaborate, evaluate, or justify one's own thinking. (Here is what I think.)

Feedback request Self-oriented questions that invite a partner to

critique one's own thinking. (What do you think about my idea?)

Other-monitoring Other-oriented statements, questions and responses that

represent an attempt to understand a partner's thinking. (Is this what

you mean? Here is what I think of your idea.)

A balance of Self-disclosure, Feedback Requests and Other-monitoring,

indicating mutual engagement with each other's reasoning, would confirm

that students' interaction was collaborative.

Metacognitive Function of Students' Dialogue

In the Hooke's Law problem, routine monitoring of progress prompted a

disagreement over the meaning of the problem's conditions, and the

students initiated Understanding and Analysis episodes in which they

clarified their interpretation of the problem and chose an appropriate

solution strategy. The other two transcripts involve monitoring and

regulation triggered by the recognition of metacognitive 'red flags'.

For the projectile motion problem, careful monitoring of strategy

execution during Implementation and Verification episodes resulted in

error detection and correction. In the compound interest problem, an

anomalous result recognised during a Verification episode triggered a

series of Analysis and Verification episodes, involving assessments of

strategy appropriateness and of the accuracy and sense of the results

these strategies produced.

Although each transcript illustrates different functions of

metacognitive monitoring, taken together they also reveal that

consistent monitoring patterns accompany the distinctive problem

solving behaviours represented as episodes. Table 1 shows the numbers

and functions of metacognitive acts recorded in all three transcripts,

grouped according to episode type. (No Reading episodes were observed,

possibly because the students had read the problems before the camera

and microphone were positioned.)

The main patterns which emerge can be summarised as follows:

1. Understanding episodes were monitored by students assessing their

understanding of the problem (four Assessments), and it was often

necessary to identify new information or reinterpret the problem in

order to make further progress (eight New Ideas).

2. Analysis episodes featured assessments of strategy appropriateness

(twelve Assessments) as well as understanding (six Assessments), and

New Ideas (seventeen) were proposed to help reformulate the problem.

Table 1. Number and Function of Metacognitive Acts in Each Type of

Problem Solving Episode

Metacognitive Act Understanding Analysis Implementation Verification Total

New Idea 8 17 1 6 32

Assessment

¥ knowledge - - - 1 1

¥ understanding 4 6 2 1 13

¥ strategyÐappropriateness - 12 6 7 25

¥ strategyÐexecution - - 10 3 13

¥ resultÐaccuracy - - - 2 2

¥ resultÐsense - - - 4 4

Total Metacognitive Acts 12 35 19 24 90

3. Students monitored Implementation episodes by assessing strategy

execution (ten Assessments) and appropriateness (six Assessments). (New

Ideas and Assessments of understanding were also recorded in small

numbers, but these monitoring activities were observed across all

episode types, and did not contribute significantly to metacognitive

activity during Implementation.)

4. Because Verification episodes are intended to review the entire

solution process, almost all metacognitive functions were represented

in the transcript analyses. Results were assessed for accuracy and

sense (six Assessments in total), the execution of strategies was

checked (three Assessments), and the appropriateness of the solution

method evaluated (seven Assessments). If the solution could not be

successfully verified, further task-related knowledge (six New Ideas)

had to be identified and work on the task continued.

The monitoring strategies students used during collaborative problem

solving are consistent with those reported by individual students,

illustrated in the episode-based model of Figure 3. However, the

analysis has demonstrated that multiple strategies were employed to

monitor each episode, rather than the single strategies suggested by

the model. As students worked through the stages of solving a problem,

from Understanding the problem to Verification of the solution, they

needed to call on an ever increasing range of metacognitive strategies

to keep track of progress and deal with obstacles (see Table 1). It may

be useful to think of this process as backtracking, so that episode

types and their corresponding monitoring and regulatory strategies from

the earlier stages of problem solving are accessible at later stages

when difficulties arise.

Collaborative Structure of Students' Dialogue

Although the teacher did not impose a formal group work structure, the

transcripts show that students did choose to work together in class,

particularly when they faced difficulties. In the Hooke's Law problem,

it was a disagreement which prompted collaborative debate, while

students' initial uncertainty over constructing a solution to the

unfamiliar projectile motion problem led them to work together. For the

compound interest problem, one student consulted with others in order

to deal with an obstacle in the form of an answer which did not make

sense.

The collaborative quality of students' interaction was measured by

coding conversational turns to identify transacts produced by each

speaker. Because speakers had unequal opportunities to contribute to

the discussion, the six transact codes (self- and other-oriented

statements, questions and responses) were counted as both frequencies

and proportions. The proportion of transacts in each whole dialogue was

calculated as the total number of transacts divided by the total number

of conversational turns. As described earlier, the transact codes were

then grouped to reflect three elements of collaboration:

Self-disclosure (self-oriented statements and responses), Feedback

Request (self-oriented questions) and Other-monitoring (other-oriented

statements, questions and responses). The proportions of grouped

transacts for all three transcripts are displayed in Figure 4.

Figure 4. Distribution of transact groupings by transcript

(Available from the author: see end of paper for details)

The balance of Self-disclosure, Feedback Requests and Other-Monitoring

transacts found in each dialogue suggests that the interaction of each

group of students was consistently collaborative. However, when

individual students' contributions were examined they did not always

display the same balance. There were times when individual students

were mainly occupied with explaining their ideas to a peer

(predominantly Self-Disclosure), or with trying to understand a

partner's ideas (predominantly Other-Monitoring). These different

findings are not contradictory instead, they highlight the various

roles individual students may take in order to initiate and maintain

collaborative interaction. It seems possible that students' roles in

contributing to collaborative dialogue might change according to the

circumstances in which they find themselves.

Collaborative Metacognitive Activity

Metacognitive activity which was collaborative in nature was described

by examining conversational turns double coded as having both

metacognitive function and transactive structure. The numbers and types

of metacognitive transacts for all three transcripts are shown in Table

2.

These results indicate that joint metacognitive activity was

characterised by:

¥ students clarifying, elaborating and justifying their New Ideas for

the benefit of a partner (Self-disclosure);

¥ students asking their peers for help in finding errors by inviting

critique of strategies and results; and students seeking feedback on

the New Ideas they proposed (Feedback Request);

¥ students making an effort to understand their partners' thinking by

offering critiques of their strategies, and also by elaborating on and

monitoring their understanding of partners' ideas (Other-monitoring).

For the three transcripts in total, 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.

Table 2. Conversational Turns Coded as Metacognitive Acts and Transacts

Transactive Structure (Frequencies)

Metacognitive Function Self-Disclosure Feedback Request Other-Monitoring

New Idea 14 2 4

AssessmentÐstrategy 2 5 16

AssessmentÐresult Ð 3 Ð

AssessmentÐunderstanding Ð Ð 6

Total 16 10 26

Not all of the students' metacognitive activity was jointly transacted.

Monitoring sometimes took the form of self-directed 'thinking aloud'

which did not seek acknowledgment or response from a partner. On other

occasions students did seek a response to their New Ideas and

Assessments, but were ignored by their partners. There were also times

of conflict and disagreement, when Assessments simply rebutted a

partner's strategy and New Ideas merely reasserted the speaker's

position. This finding is not surprising, since collaboration does not

preclude periods of individual reflection, nor is it necessarily based

on immediate or continuing consensus (Kruger, 1993; Teasley &

Roschelle, 1993).

Discussion

Empirical evidence from students' interview responses and videotaped

classroom interactions has been presented to support a theoretical

framework which extends existing models of the ideal cognitive and

metacognitive characteristics of mathematical problem solving

behaviour. Most interview questions asked students to report on the

strategies they used while working individually on problems; however,

some evidence of collaborative strategies was also obtained, for

example, in seeking the help of peers to locate errors, evaluate

strategies, and deal with obstacles to progress. Analyses of student

interviews and of transcripts of collaborative problem solving were

guided by the episode-based framework developed here. As well as

demonstrating that the same monitoring and regulatory processes

reported by individuals are used when students work with peers, the

analysis also suggested mechanisms through which peer interaction

mediates metacognitive activity.

The three problem solving transcripts record spontaneous

student-student interaction during regular classroom lessons.

Collaboration was initiated by the students, without being imposed by

the teacher or researcher for organisational or experimental purposes.

It is therefore worth asking why the students chose to collaborate in

these specific instances. The answer is found in the various types of

impasse presented by these problems a disagreement, uncertainty, or an

obstacle where progress was either halted or slowed and the students

turned to their peers to work their way through the difficulty by

monitoring their own and each other's thinking. It is important to

recognise that collaborative interaction stimulated by challenging

problems may feature periods of individual activity or conflict as well

as agreement and cooperation. Consensus may only be reached after

students have debated each other's perspectives and retired from the

discussion to reconsider their own.

The mathematics education reform movement has identified new goals for

school mathematics and recommended changes in teaching and learning

practices which call for increased attention to be given to small group

work and student communication of mathematical ideas. This paper has

suggested that reforms which favour such peer interaction have the

potential to develop the metacognitive aspects of students'

mathematical thinking. Individual metacognition is organised around

routine self-monitoring and the recognition of, and response to, 'red

flags' which warn that problem solving has gone astray. Collaborative

metacognitive activity involves the same processes, but proceeds

through offering one's thoughts to others for inspection, and acting as

a critic of one's partner's thinking. If students are encouraged to

engage with each other's reasoning in this way, the resulting

discussion makes visible the processes that individuals could

appropriate to monitor and regulate their own thinking, and become

self-directed learners.

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Hillsdale, NJ: Erlbaum._

Merrilyn Goos

Graduate School of Education

The University of Queensland 4072

Ph (07) 3365 6492

Fax (07) 3365 7199

email M.Goos@mailbox.uq.edu.au