Sue Helme and David ClarkeUniversity of Melbourne

Identifying Cognitive Engagement in theMathematics Classroom

This paper reports an analysis of videotape and interview data from four Year 8mathematics lessons from the perspective of student cognitive engagement. Thestudy extends our understanding of cognitive engagement by locating empiricalevidence for its occurrence within the classroom. On the basis of the data we haveexamined, it appears that cognitive engagement can be consistently recognised byspecific linguistic and behavioural indicators and is promoted by particular aspectsof the classroom situation, the task, and the individual.

2001, Vol. 13, No.2, 133-153Mathematics Education Research Journal

Interviewer: So can you tell me what made it, what was good about it, whatworked for you?

Student: Uh, I think, urn, it was just that we all really put in together, we reallyput our minds to it, thought about it.

In this paper, we undertake an analysis of Videotape and interview data fromfour mathematics lessons from the perspective of student engagement. Our focuson the nature, role, and significance of engagement stems from our concern thatthe literature on learning has not provided a satisfactory empirical demonstrationof the role played by engagement in the learning process. Our primary purposewas to identify indicators of cognitive engagement in a mathematics classroom, asan initial step in exploring the relationship between cognitive engagement andlearning.

The term engagement usually refers to the extent to which a student is activelyinvolved with the content of a learning activity, where active involvement suggeststhat the person acts to maintain or extend their contact with the object in order toincrease their knowledge of it (Ainley, 2001). The quality or level of thisinvolvement is generally believed to have a profound effect on learning outcomes,in that students who "really put their minds to it" are much more likely to learnsuccessfully than students whose engagement with the subject matter is low.

The term engagement has been used by researchers to encompass both themotivational and cognitive aspects of the construct, encompassing students'initiation of action, effort, and persistence on academic tasks, as well as theirambient emotional states during learning activities (Skinner, Wellborn & Connell,1990; Skinner & Belmont, 1993). There seems to be general agreement in theliterature about this definition. In a recent review of the literature, Stipek (1996)describes actively engaged students as approaching challenging tasks eagerly,exerting intense effort using active (that is, deliberate) problem-solying strategies,and persisting in the face of difficulty.

Various studies have found that motivational and cognitive components ofstudent engagement do not operate in isolation from each other, but rather supportand complement one another in a synergistic manner. Pintrich (1989) explored the

134 Helme & Clarke

interactive relationships between students' motivation and cognition and foundthat students who were more mastery and challenge oriented used more cognitivestrategies and engaged in more metacognitive activities than students who wereless intrinsically oriented. He concluded that motivation and cognition wereinterdependent and proposed a multivariate contextual model of student learningwhich stressed the importance of the pattern of relationships among the variouscognitive, metacognitive, and motivational components of learning, particularlyemphasising "the dynamic interplay between motivation and cognition" (Pintrich,1989, p. 153).

The interdependence of cognition and motivation is a key feature of notions ofself-regulated lean~ing (Corno & Mandinach, 1983; Zimmerman, 1990).Zimmerman described such learners as "metacognitively, motivationally, andbehaviourally active participants in their own learning" who "set goals, organize, .self-monitor and self-evaluate at various points during the process of acquisition"(p. 4). According to Zimmerman, self-regulated learners have a greater sense ofcontrol over their learning processes and outcomes than their more passivecounterparts. Skinner et al. (1990) explored the notion ot perceived control in moredetail. They found that children's perceived control influenced academic outcomesby promoting or undermining the effort children exerted in performing cognitivetasks.

A number of studies have attempted to clarify the relationship between themotivational, cognitive, and behavioural aspects of engagement. Fullarton (1996)'studied students' belief structures over the primary - secondary school transitionwith the aim of identifying students whose belief structures put them at risk byundermining their engagement with the subject. She identified an underlyingconstruct of control and confidence, and found that a substantial proportion of thestudents showed a decline in both their perceived control and their engagement (asrated by teachers) over the transition from primary to secondary school.

The concept of flow, which has been theorised and researched in manydifferent settings (Csikszentmihalyi & Csikszentmihalyi, 1988; Csikszentmihalyi &Rathunde, 1992), also underscores the dynamic interplay between cognition andmotivation. The notion of flow refers to an individual's experience of an activity inwhich they are intensely involved. The flow experience is usually associated with ahigh degree of control, enjoyment and cognitive engagement with activities whichare just within the limits of an individual's capacity, and a dynamic balancebetween the challenge of the task and the individual's skill level.

Ainley (1993) examined the interaction between motivation, learningstrategies, and achievement. Three general styles of engagement with learning(deep, achieving, or surface) were identified from questionnaire data andcombined with information on examination preparation strategies andachievement outcomes. Students with a stronger orientation to performance andmastery goals were more likely to use transformational, rather than reproductive,learning strategies, highlighting the association between students' goals and thetype of learning strategies they adopted. The study lends support to the notion that·"characteristics that the individual learner brings to the learning context shape andcombine with the learner's construction of the task and its demands, to influencethe strategies adopted and the learning outcomes" (p. 395). These associations

suggest possible indicators by which engagement might be recognised withinclassroom videotape and interview data such as that collected in this study.

Cognitive EngagementOur research into engagement has focused on what we have called cognitive

engagement, in part to distinguish active mental involvement from the motivationaland emotional aspects of engagement. The term cognitive engagement appears tohave sufficient currency in the literature however for some authors to see no needto define the construct, either theoretically or operationally. For example, Pintrichand De Groot (1990) made repeated reference to cognitive engagement in a paperdealing with motivation and self-regulated learning, without definition. Nolen(1995) similarly employed the construct in relation to self-efficacy,· but withoutdefinition.

Researchers generally agree, though, that cognitive engagement involves thethinking that students do while engaged in academic learning tasks. For example,Meece, Blumenfeld, and Hoyle (1988) used the term active cognitive engagement torefer to "students' reported use of metacognitive and self-regulation strategies" (p.515). Similarly, Blumenfeld, Mergendoller, and Puro (1992) defined cognitiveengagement as the use of "thinking, metacognitive, and self-regulatory strategiesto approach learning thoughtfully" (p. 207).

Corno and Mandinach (1983) proposed a model of cognitive engagementwhich distinguished four hypothetical forms of cognitive engagement: self­regulated learning (where students' cognitive processing is driven by higher-orderor metacognitive components); task focus (where students use task-specificplanning and self-monitoring for tasks where information transformation ratherthan acquisition is required), resource management (in which students garner helpfrom external sources); and recipience (in which students respond passively withlittle mental investment, often to instruction which has short circuited their self­regulatory cognitive processes). Although self-regulated learning is the highestform of cognitive engagement in this model, Corno and Mandinach (1983)emphasised the need for students to adapt their forms of engagement to varyingtask situations.

A number of studies have explored the relationship between the motivationaland cognitive aspects of engagement. Meece et al. (1988) investigated therelationship between student goal orientations and cognitive engagement. Studentcognitive engagement was assessed using questionnaire items that tappedcognitive strategies and dimensions of self-regulated learning (active engagement) aswell as help-seeking and effort-avoiding strategies (superficial engagement).Students who placed greater emphasis on task-mastery goals reported more activecognitive engagement than students oriented toward gaining social recognition.

Blumenfeld et al. (1992) explored the relationship between motivation andcognitive engagement, and found that high levels of motivation are not necessarilyassociated with high levels of cognitive engagement. Student cognitiveengagement was assessed using the same instrument as that used by Meece et al.(1988). The study comprised a comparison of two science classes taught by twodifferent teachers. Although both classes reported high levels of motivation, there

Identifying Cognitive Engagement in the Mathematics Classroom 135

136 Helme & Clarke

were significant differences between the classes in levels of reported cognitiveengagement. The different levels of cognitive engagement were associated withdistinctly different teaching strategies, suggesting that teaching practices have an

.. important role in shaping cognitive engagement. Although teacher practices weremonitored via classroom observation, there was unfortunately no such detailedobservation and analysis of students' actual classroom behaviour. Nevertheless,the findings of these and other studies highlight the importance of motivationaland instructional factors in explaining variations, in students' levels of cognitiveengagement in the classroom, and by implication, their learning outcomes.

We define cognitive engagement as the deliberate task-specific thinking that astudent undertakes while participating in a classroom activity. Our consistentapproach in the Classroom Learning Project has been to avoid sole reliance onstudent self-reports of cognitive behaviour and to seek evidence of constructs suchas cognitive engagement within videotapes of classroom behaviour. From thisevidence, we hope to construct a viable model of classroom learning, in whichstudent cognitive engagement is usefully situated.

Cognitive Engagement and LearningStudents need to have both the will and the skill to be successful learners. It is

the experience of teachers that students who are motivated to learn and who thinkcarefully about what they are learning develop deeper understanding of thematerial being covered. There is also a growing body of research indicating theimportant role that self-regulated learning strategies (as well as key motivationalfactors) play in students' academic achievement (Pintrich, 1989; Pintrich & DeGroot, 1990; Zimmerman, 1990).

Swing, Stoiber, and Peterson (1988) undertook a large and complex studywhich, among other things, explored the effects of increasing the quality ofcognitive engagement in classroom tasks. The study compared two classroom­based interventions on students' mathematics achievement: increasing students'academic learning time versus instruction in using specific cognitive strategies. Thestudy did not directly study students' actual skill use in the classroom, but usedvideo-stimulated recall to facilitate students' reporting of these strategies, andpresented a ·problem to students in interview which was solved using a thinkaloud protocol. As would be expected, students in the thinking skills groupreported significantly greater use of specific cognitive strategies than students inthe learning time group. The achievement results were equivocal: In high andmedium ability classes, thinking skills instruction led to better high-level andconceptual achievement than did instruction aimed at promoting academiclearning time, but this effect was not observed for lower ability classes. In contrast,within a given class, lower ability students appeared to benefit more from thethinking skills treatment than did higher ability students. A detailed comparison ofthe verbal protocols of two lower ability students illustrated the possibleimpact ofthe thinking skills training: The success achieved by the student in the thinkingskills group appeared to be facilitated by the use of specific cognitive strategies.Further qualitative analysis may have provided more insight into the impact ofparticular cognitive strategies on mathematics achievement.

Identifying Cognitive Engagement in the Mathematics Classroom 137

Shortcomings in defining and operationalising student engagement limit theusefulness of the results of many studies in this area. Skinner et al. (1990), forexample, found that scores on a student engagement scale completed by teacherscorrelated significantly with grades and other measures of student achievement.However, information on the actual behaviours associated with engagement wasnot collected.

Studies employing the process-product approach, such as the work of Goodand Brophy (1984) in the USA and Bourke (1984) in Australia, sought to identifyspecific behaviours (such as frequency of student-initiated questions) associatedwith desired outcomes (such as test performance or attitude to the subject). Theinability of such studies to distinguish association from influence, cause, andconsequence limited their explanatory power, while nonetheless drawing attentionto those practices that most characterised certain classroom types and offeringresults that promoted productive speculation abogt causal relationships. Thepersistent tension in the study of classroom practices and outcomes has beenbetween the drive to generalise and the need to understand process.

The complexity of the behaviours being studied challenges any generalisablecategorisation scheme, and conclusions as to causation are undermined by themultiplicity of possible interp.ctive effects. For example, Webb (1989) reviewednineteen published. studies linking peer interactions and achievement in smallgroups learning mathematics and computer science, and highlighted thedifficulties of forming meaningful conclusions on the basis of correlational studies.

Some research on small-group learning in mathematics has been fine-grainedenough to be able to specify the forms of verbal interaction (indicative of cognitiveengagement) which appear to promote learning. For instance, Gooding and Stacey(1991) made a detailed analysis of knowledge development in small groupslearning division. They found that groups that were more effective (showed a netgain on a post-test) were more likely than ineffective groups to show evidencewithin their interactions of particular forms of cognitive and metacognitiveactivity. These included proposing ideas, responding to questions, giving

.explanations with evidence, and refocussing the discussion.Increasingly, the complexity of the practices and outcomes being studied is

leading to interpretive studies in which support for influence and causation issought directly in the data collection process (from informants), rather than beingconcluded by inference on statistical grounds. In such studies, convergence ofinformant accounts and the theoretical coherence of any documented relationshipsare taken as more compelling authorities for causation. This is the approachemployed in the research reported in this paper.

Influences on Cognitive EngagementCognitive engagement is located within a complex and cyclic interplay of

influences. As Ainley (1993) pointed out, characteristics that the individual bringsto the learning context shape and combine with their construction of the task toinfluence learning strategies and learning outcomes. We discuss below how threeinteracting factors--the individual, the learning environment, and the tasksthemselves--impact on cognitive engagement.

138 Helme & Clarke

The individual. Individual students bring to the learning situation numerouscharacteristics that influence their cognitive engagement. These include skills,knowledge, dispositions, aspirations, expectations, perceptions, needs, values, andgoals (Corno & Mandinach, 1983; Meece et al., 1988; Ainley, 1993). Students'reported cognitive and emotional engagement has been found to be influenced bytheir perceptions of their control over academic outcomes (Fullarton, 1996). Asdiscussed earlier, self-regulated learners actively seek out opportunities to learnand systematically use metacognitive, motivational, and behavioural strategies toachieve desired learning outcomes.

The learning environment. Within any classroom, a classroom culture emergescreating conditions that either constrain or promote particular teaching andlearning strategies and particular styles of interaction with the teacher and otherstudents (Edwards & Mercer, 1987). Certain teaching practices have been shown topromote cognitive engagement, while others tend to restrict it (Corno &Mandinach, 1983; Blumenfeld et al., 1992). Current analyses of data collected in theClassroom Learning Project suggest that teaching style and the nature of peerinteractions have a powerful influence on student cognition and metacognition(Lerman, 2001; Holton & Thomas, 2001; Baird, 2001).

Tasks. Classroom learning tasks and activities provide the vehicle for a,student's cognitive engagement. Recent studies into task characteristics such ascomplexity (Williams & Clarke, 1997), challenge (Csikszentmihalyi &Csikszentmihalyi, 1988), familiarity (Helme, 1994), intrinsic interest (Ainley, 2001),and personal meaningfulness (Clarke, 1996) suggest a possible relationshipbetween such characteristics and forms of cognitive engagement. In our analysis,we have endeavoured to locate and explain any evidence suggestive ofinterdependence between these factors and different forms of cognitiveengagement.

Operationalising Cognitive EngagementMany early studies of student engagement appear to have operationalised the

construct in terms of time-on-task (Peterson & Fennema, 1985; Hart, 1989),although the usefulness of this measure has been challenged over a significantperiod of time (Peterson & Swing, 1982). As we have argued above, cognitiveengagement is qualitatively different from time-on-task or student participation.The fine-grained nature of our data collection allows us to investigate more closelythe relationship between time-on-task and cognitive engagement, as we woulddefine the construct.

Very few studies have used direct observation of student classroom behaviourto assess levels of cognitive engagement, and have instead relied on indirectmeasures such as student questionnaires or teacher reports. Meece et al. (1988), forinstance, used a questionnaire which included items which tapped cognitive andmetacognitive strategies and dimensions of self-regulated learning such as "I askedmyself some questions as I went along to make sure the work made sense to me"and "I went back over things I didn't understand"(p. 517).

The unreliability of self-reports of cognitive activity is well documented

Identifying Cognitive Engagement in the Mathematics Classroom 139

(Corno & Mandinach, 1983; Nuthall & Alton-Lee, 1995; Wilson, 1998). But asWilson (1998) pointed out, any study involving children and the examination oftheir thinking processes must involve self-reporting. She has explored ways inwhich self-reporting can be made more reliable, and 'has developed a technique bywhich students' self reports of cognitive and meta-cognitive activity can becombined with other data sources such as observation and video recording.

Some studies have made use of direct observation of student behaviour(Gooding & Stacey 1991, 1993; Leder & Forgasz, 1992). Leder and Forgaszattempted to describe simultaneously students' cognitive and affectiveengagement in mathematical tasks and identified three forms of cognitiveengagement (independent thinking; task/work autonomy; and persistence),providing operational definitions for each. Cognitive behaviours were categorisedas high or low level.

The purpose of the present analysis was to ideptify instances of cognitiveengagement using classroom videotape data as· a primary source. Thus we havefocussed primarily on linguistic indicators of cognitive and metacognitive activity,such as verbalisation of thinking, questions, explanations, and other forms ofcommunication.

As the previous discussion has indicated, the interdependence of cognitive andmotivational aspects of engagement requires that we take account of a range ofbehaviours in our analysis of the video record. We have therefore notedmotivational correlates of cognitive engagement such as active participation,persistence in completing tasks, and resistance to interruptions. Non-verbalcorrelates of cognitive engagement such as gestures, eye contact, and bodyorientation have also been taken into account, as well as indicators of emotionalinvolvement such as expressions of enthusiasm, enjoyment, and satisfaction. Datafrom the present study have strengthened our concerns with studies that makeinferences concerning cognitive activity on the basis of one source of informationonly. A student in a science lesson from the present study who observed herself invideo-stimulated recall commented that her own behaviours could be easilymisinterpreted by the teacher as lack of attention, when in fact she was listening towhat the teacher was saying.

This paper is primarily the report of an investigation of the cognitivedimension of engagement, but we have attempted not to lose sight of the biggerengagement picture which also includes behaviour, motivation, and affect. Bydeveloping a comprehensive and rich understanding of cognitive engagement andits relationship to classroom practices, we hope to be able to deploy our resourcesin a more informed fashion to enhance cognitive engagement in the classroom and,possibly, the quality of student learning.

Method

ParticipantsThe school that participated in the study was a private coeducational school in

an outer suburb of Melbourne. The seven teachers who participated responded to arequest for assistance, the only selection criteria being approval by the school

140 Helme & Clarke

administration. Fifty-four lessons were videotaped over a three year period from. 1994 to 1996: 25 Year 7, 8 and 9 science lessons and 29 Year 7 and 8 mathematicslessons. Four students were selected by each teacher on the basis of theirmembership of a stable social group and on the teacher's recommendation thatthey could be relied upon to talk in a reasonably relaxed fashion in an interviewsetting. At the conclusion of the lesson, two of these students were selected forinterview. Interviewees were selected if they had appeared to be engaged insubstantial interaction with their peers or the teacher, or had displayed visiblesigns of confusion or uncertainty. In total, twenty-four students (14 girls and 10boys) were interviewed throughout the period of the study. One hundred and nineinterviews were conducted, recorded, and transcribed.

ProcedureTwo video cameras were used to record classroom events, one focussed on the

teacher and another one on the group of four students. These images were mixedon-site to produce a composite picture in which the students occupied most of thescreen with the teacher superimposed in a corner. The research~r, seated at the rearof the classroom, was able to view this image and listen simultaneously to both thetarget students and the teacher, whilst recording field notes onto a laptopcomputer. The field notes were time-tagged to the video record, so that events ofsignificance to the researcher or the student could be quickly located during thepost-lesson interview. During the interview, segments of the video record wereplayed back and discussed. Thus students were able to reconstruct theirmotivations, thoughts, and actions, prompted by the video record of the classroomevents.

Interviews with the teacher were conducted at a much later time. The teacherviewed the videotaped lesson and paused the tape whenever she wished tocomment on events in the classroom that showed something important to herabout teaching or learning. The audio record of the teacher's commentaryprovided a further source of data.

Four Year 8 mathematics lessons were fully transcribed, and it is these lessonswhich provided the database for the analysis reported in this paper. The lessonswere chosen for transcription on the basis that they included a significantproportion of student-student communication, enabling us to analyse interactionsmost likely to provide insight into student engagement.

The four transcribed lessons were combined with the field notes taken at thetime and the interview record, to produce one document, or integrated data set, foreach lesson. Each Integrated data set is a single text document incorporatingtranscriptiohs of videotape and interview dialogue together with the researcher'sfield notes, time-tagged to the video record, and supplemented with copies ofstudent written material. For more details of the origins, rationale, andpracticalities of data collection, see Clarke (1998).

Data AnalysisFour Year 8 mathematics lessons, all taught by the same teacher, together with

associated student and teacher interview records, were analysed for evidence ofcognitive engagement. Each integrated data set was examined, and all instances ofstudent on-task activity were identified. Each episode of on-task activity was thenexamined, and an exhaustive list was made of the student behaviours that webelieve could be associated with cognitive engagement. The observed behavioursoccurred in clusters that appeared to be associated with distinct classroomsituations. Consequently, all instances of cognitive engagement were categorisedaccording to the type of social situation that occurred within these lessons.

Identifying Cognitive Engagement irz the Mathematics Classroom 141

Results and Discussion

Behaviours associated with cognitive engagement are listed in Table 1. (Othersituations, such as individuals working alone, or prolonged interactions betweenan individual student and the teacher, were either not present or not captured byour data collection methods.) It is worth noting that the four situations in Table 1can be divided into those situations in which the teacher was an active participantand those in which the teacher was not. The clustering of these behaviours andtheir consistent association with qualitatively different social situations providesan endorsement of the situation-specific nature of cognitive engagement.

Table 1Indicators of cognitive engagement

Classroom situation

Individuals workingin parallel

Collaborative smallgroup activity

Small groupinteractions withteacher

Whole classinteractions withteacher

Behaviour

Verbalising thinkingSelf-monitoringConcentration (resisting distractions or interruptions)Gestures (interpreted as externalising thought processes)Seeking information and feedback

QuestioningCompleting peer utterancesExchanging ideasGiving directions, explanations, or informationJustifying an argumentGestures

Answering teacher's questionsGiving informationExplaining procedures and reasoningQuestions addressed to teacherReflective self-questioning

Asking and answering questionsMaking evaluative commentsContributing ideasCompleting teacher utterances

142 Helme & Clarke

Individuals Working in Parallel

In the text, numbers in brackets refer to utterance numbers in the correspondingtranscripts.

From the interview records we were able to identify a number of additionalpossible indicators of cognitive engagement. These included:

• claims to have made a genuine attempt to learn something;• claims to have learned something in the lesson;• student discussion, communication or recall of details of lesson content

within the interview;• claims to have been engaged during the lesson (e.g., "We really put our

minds to it").

The richest examples (i.e.,' those providing the most comprehensive coverageof forms of cognitive engagement) from each of the four types of classroomsituation are discussed in detail below. The notation used in the transcripts has thefollowing meaning:

TeacherInterviewed studentsStudents not interviewedInterruption (either by self or another person)Researcher's observations or comments[ ]

TB, D, F, .51,52, .

It is common in mathematics classrooms for students to work alongside eachother, interacting in an unstructured yet mutually beneficial manner and seekinghelp and giving assistance as the occasion arises. The task required students tocalculate how many blood cells they had lost in their lifetime up until today. Theneed for each student to produce a unique personal solution (rather than agree ona single solution) promoted a parallel, rather than a collaborative, approach.

T: I want you to tell me how many blood cells to the day you have lost. 1K: [To L] Does that mean how many up to now? [Overlapping talk between 2

T and L for next three lines]T: So you've got to multiply by the years, by the months- 31: And the months, and the days. 4T: By the days. OK? 5L: VVow! 6T: That's when you've finished that. 7K: Seven hundred and thirty million a day, no, per year, times fourteen 8

years. Shit.S1: In your lifetime? 9T: Yes, how many blood cells have you lost in a lifetime. If it's two million a 10

day.S2: I'm not dead yet. VVe've got to do so many months, so many days. Oh oh. 11T: I'll go round and check the rest [T goes around room and checks work]. 121: [Counts on fingers, talks to self] May, June, July, August, September, 13

October. That's six. And how many days have we had in October? [Looksat diary] The nineteenth..

IdentiftJing Cognitive Engagement in the Mathematics Classroom 143

K: [Working aloud] You times that by fourteen, equals one point oh two two 14to the power of ten. Oh yeah, I understand that. One oh two two. One,two, three, four, five, six, seven, eight, nine, ten. Would that be right?[Looks towards L] How many days in a month approximately?

L: [Working aloud] Times nineteen. [looks towards K] 15K: [To L] Would you do approximately thirty days for the month? It would 16

be thirty point five days. No it'd be twenty-nine point nine days orsomething.

L: Hang on. [goes back to own work] Times, times [inaudible]. 17K: Would it be this? Do you reckon, L? Urn, L, can I borrow your calculator? 18

L.L: Yeah [inaudible, working aloud]. 19K: [Talking as she works, L does not appear to be listening] That equals one 20

point oh, oh, oh, no, one point oh two two to the power of ten, make it-L: [To herself] That's wrong. 21K: [To herself]-ten billion, two hundred and twenty million. [Looks toward 22

L] Ten billion, two hundred and twenty million. Is that right? [Noresponse, L is bent over work] Don't worry.

L: , I hate you Mrs B! [Possible emotive response related to frustration of 23challenging task]

Indicators of cognitive engagement. L thought aloud as she worked (13, 15, 17, 19),and her intense concentration was evidenced by her resistance to K's interruptions(17,19). She actively used available resources to help her resolve uncertainty, suchas repeating the teacher's instructions in order to clarify the task requirements (4),and referring to her diary to work out the number of days (13). She made use ofgesture to externalise her thought processes (13), and showed evidence of self­monitoring (21). The emotive dimension of her engagement was apparent in lines 6and 23.

K's style of engagement took a slightly different form to that of L, in that shetended to rely on another student (in this case L) rather than the teacher forinformation and feedback, asking L for assistance with basic information (14), toclarify the task requirements (2), and to give her feedback about her ideas and herprogress with the task (14, 16, 18,22). Like L, she verbalised her thinking (8, 14, 20,22), showed evidence of self-monitoring (8, 14), and expressed her emotionalinvolvement (8).

L's interview record contained a number of insights into the influence of taskfactors on the quality of her cognitive engagement. Reflecting on the use of real-lifesituations in mathematics, she claimed that they made things clear to her, andmore real.

I: Was there anything new in that lesson for you?L: Not really, it was just urn, the only new thing was using it in sort of real-life

situations.I: Umhm.L: Where, because, usually the numbers don't make any sense 'cause you're

not putting them with anything that you'd think of, but when you're usingstuff like rice and paper and dollars and so on it makes it clear.

I: Yeah, do you feel that you learned anything new from that?L: Urn. Yeah, it just makes it seem more real, if you use it with something you

know about. OK?

144 Helme & Clarke

L's interview record suggested that task novelty, context, and personalmeaning acted to heighten her participation and cognitive engagement. Howeverthe link between cognitive engagement and successful learning has yet to be madein a convincing' and empirically grounded fashion. Although the data available tous cannot confirm that cognitive engagement facilitated learning in this particularsituation, both the video record and the teacher's interview clearly indicate that Lwas a highly successful student. As evidence of this, the teacher considered theproblem above to be quite challenging, "It's quite a tricky task", and her highopinion of L's competence was evidenced by the fact that she addressed Lpersonally when she initially set the task. Pending further analysis, we suggest thatL's consistently high cognitive engagement contributed significantly to her succe~s

and to the quality of her learning.

Collaborative Small Group ActivityIn this task, F and M were required to select from a set of graphs the one they

thought best fitted the v-t graph of a ball thrown into the air (see Figure 1). Becauseof the requirement to reach agreement on their solution, the situation necessitatedsome form of collaboration.

M: [To F] Ready, what do we do with the ball thrown into the air? 1F: This [points to material on table]. 2M: Which one? 3F: I reckon [selects an alternative by tapping her page with pencil]. 4M: Uh uh, I don't reckon. It would go up-it wouldn't go up fast, and it 5

would come down real fast, so.F: But it doesn't come down real fast. 6M: It does [nods head]. 7F: [Shakes head] 8M: It comes down faster than it goes up. 9F: No. What happens, is it goes really fast [with pencil traces what appears 10

to be trajectory of ball thrown into air] and then it slows down once itgets to the top, and comes -it comes up and then slows down [repeatsfirst gesture], and stops when it turns around. And then it comes upagain [pencil moving up] it goes down quite fast and then it slows downwhen it gets to the bottom [sketching second half of parabola] because ithas to-

M: [Looking at book] Yep, this one. Because it doesn't, but it doesn't, that's 11like it goes up and then [speaking slowly] kind of !TI0ves real [speakingslowly, gesture with hand], you can see how it. But that one just kind ofgoes like that.

F: But it doesn't go Go-op [moves pencil upwards from desk in straight 12diagonal line] slow down suddenly, it sort of gradually slows. [Anotherstudent briefly interrupts about another matter] I reckon-

M: D. 13F: D. 'Cause it doesn't really just go uomp [moves ruler up from desk in 14

steep curve with a jerk] and then slow down straightaway [moves rulerslowly in arc] and then-

M: All right, do D. [Makes side comment about pen] 15

Identifying Cognitive Engagement in the Mathematics Classroom 145

o

A. E.

V V

t0

t0

B. F.V

V

t0

c. G.V

o "---------. t •

V

o

D. H.

V

o

V

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Figure 1. Velocity-time graphs presented to students.

Indicators of cognitive engagement. F and M employed various resources to helpthem with the task. Both students used gestures to create a slow-motion version ofthe path of the ball and to help them work out how the velocity changed during itsflight 00/ 11, 12, 14). Gesture here seems also to have enabled them to create ashared representation (Clarke, 1996) which they could both visualise and modifyuntil they reached agreement. (For a comprehensive discussion of the role ofgesture in mathematical problem solving using this episode, see Reeve &Reynolds/ 2001).M used her voice to enhance this process: As she described thechanges in the ball's velocity, her speech slowed down and speeded upaccordingly (1). The episode consisted largely of a negotiative event (Clarke &Helme, 1997), in which the students initially disagreed and used argument andexplanation to eventually reach consensus (4-15). There is evidence of monitoringof task requirements in line 16, in which F reminded M that they had to draw thegraph.

146 Helme & Clarke

As stated earlier, our analysis here is not intended to demonstrate a connectionbetween cognitive engagement and learning outcomes, but to identify indicators ofcognitive engagement prevalent in a particular classroom situation with aparticular teacher, and to infer their relationship to features of that situation. Wewould argue, thought that the challenging nature of the task pushed these studentsto make full and creative use of their cognitive resources to help them makeprogress on what was for them a very difficult problem.

Small-Group Interactions With TeacherK and L worked on a problem in which they had to calculate how many sheets

of graph paper they would need to obtain one million millimetre squares. Thefollowing interaction occurred when the teacher approached the group, .havingpreviously accepted a different answer from another group, and was trying todecide which of the two was correct. The change in social dynamics-a shift awayfrom enquiry and problem solving to explanation and justification of theirsolution-was accompanied by a very different pattern of interaction, with theoverwhelming majority of student utterances (twelve out of fourteen) beingresponses to teacher questions.

T: Uh, question 1. Explain your working. 1K: Question 1. We did length times width. 2T: Of? 3K: Of the sheet of paper. 4T: And what was your length? 5K: Length was two hundred and fif-two hundred and fifty what? 6T: Yeah, two hundred and fifty whats? 7L: [quietly] millimetres. 8K: Urn. Yeah, millimetres. 9T: She hasn't had the pure joy of me with units yet, has she? Hm Hm. 10

Times?K: Yeah, yeah. Minus the outlay, border thing [makes page-shaped gesture 11

in air], that didn't have graphs on it, the graph on it, times a hundred andeighty 'millimetres-

T: Where'd you get a hundred and eighty from? 12K: Width. Equals forty- 13T: Why did you multiply them together? Why not add? 14K: To get the area. I know that much. 15T: [to L] You've been tutoring her? 16K: Equals forty-five thousand, therefore you'd need-oh how'd I get that? 17T: Forty-five thousand? 18K: Forty-five thousand. That's what we got. 19T: Forty-five thousand? Can you press that- can you press that- can you 20

do that again? Two hundred and fifty times a hundred and eighty? Oh,hang on, hang on, I think you're right. Hang on. I think they're wrong.

K: Yep,they're wrong, we're right. 21L: [Holds up calculator] 22T: What is it? Oh dear. And what's the answer? 23L: Forty-five thousand millimetres on one piece- 24~ Y~. ~

Identifying Cognitive Engagement in the Mathematics Classroom 147

L:

T:L:

-and then you divide, you divide a million by that, and it gives youtwenty-two point two, so therefore you need twenty-two and a fifthsheets.Excuse me. [goes to group at front of room]Aha! We're right.

26

2728

Indicators of cognitive engagement. As stated above, the overwhelming majorityof student utterances in this episode (twelve out of fourteen) were made inresponse to questions and requests initiated by the teacher. These included"responses to requests for information (2, 4, 6, 8, 13, 19), and giving an explanationwhich was either procedural (2) or some combination of procedure and reasoning(11, 17, 24, 26). Thus the social dynamics of this episode were very different tothose in the instances of peer interaction discussed previously, where questionsand responses were more equitably balanced. But some aspects of self-regulationwere present, such as reflective self-questioning (6, 17). K used gesture to enhanceher explanation of how she worked out the area of the page (1). Line 17 capturedthree" exemplars of cognitive engagement: explanation of procedure, verbalisingreasoning, and reflective self-questioning. The emotive dimension of engagementis evident in line 28, in which L clearly expressed pleasure and satisfaction in beingright. .

This episode illustrates some important differences between peer interactionsand student-teacher interactions in this classroom. With the teacher askingvirtually all the questions, there was little opportunity for students to initiate ideasor spontaneously express and resolve uncertainty. Nevertheless, the teacher's useof some open-ended questions in this situation 0, 12, 14) may have provideduseful cognitive and metacognitive scaffolding. (For a comprehensive discussion ofscaffolding in relation to this particular episode, see Holton & Thomas, 2001).

Whole-Class Interactions With TeacherM and D participated in a teacher-led class discussion about the v-t graph of a

bouncing ball that is initially thrown into the air. (This is the same problem as inthe above discussion): The teacher drew a set of axes and a diagonal line from apoint on the y-axis representing the initial velocity to a point on the x-axis. Thiswas a very challenging task (for the teacher as well as the students), and bothteacher and students appeared to be confused about whether they were discussingspeed, velocity, displacement, or the actual motion of the ball itself.

T:

M:T:M:5s:

There are two different ways it can go. It can go here to zero so I've, I'vestarted it off. Right? It's slowing, slowing, slowing, slowing [gesturesupward movement which stops] and then at z-something is zero. WellI've got the zero and then it's going to"change direction, and get fasterand faster and fasterand faster. Now basically because it changesdirection, it-you could argue that it [referring to graph] will go down[gestures below x-axis]. And then, and then, somehow or other­[Whispers] It can't go down.-It has to hit the ground. When it hits the ground what happens then?Can't.It bounces.

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148 Helme & Clarke

T: OK, it bounces so it's going to start going - 6D: Up again. 7T: -Back up again. So, is the graph like that [hand gesture up from point on 8

x-axis].SI: It is. 9T: Or is the graph- 10D: Yeah, it's because the speed [points]. .11M: Yes. 12S2: It's the velocity, not"the way it bounces. 13T: -like that? [Hand gesture going up from point below x-axis] 14D: It's the speed. 15

Indicators of cognitive engagement. Because of the manner in which the teacherwas controlling the flow of the discourse, there was little opportunity for studentsto ask questions and express and develop their ideas. Within these constraints,however, we were able to identify several indicators of cognitive engagement, andobtain some insight into the strategies that students were using to make sense ofthe teacher's argument. These included responding aloud to teacher's questions(l2),.completion of teacher utterances (7), and spontaneous reflective comments (2,4, 13, 15). .

Both M and D discussed this episode at length in their interviews, lendingfurther support to our contention that both students were significantly cognitivelyengaged with the problem in the lesson. It could also be argued that theirengagement with the problem ha~ extended beyond the immediate classroomsituation, in that they continued to express uncertainty and to elaborate and justifytheir ideas in the interview itself.

Excerpt from D's interview:

I: Do you remember, remember the stuff about the graph, about Mrs Bdrew the thing about the bouncing ball, and she said did it go like this ordid it go below the axis like this?

D: Oh, yeah.I: Umhm.D: I, I think it's the top one.I: OK.D: Because you can't go under the graph because that means that it would

just be floating or...1: How do you know it would be floating? Can you tell me that quickly?D: It-:-hang on, I've got to think about this one. Urn, the ball goes, so she's

throwing the ball up in this case, right?I: Yeah.D: With this graph it would have to be thrown, no dropping the ball, yeah,

dropping the ball, so it goes up, hits the urn ground so, and then it goesback down again.

I: Yeah.D: And then it's up in the air and it stops, then it goes back up again.I: Yeah.D: So it wouldn't go straight through.1: OK.D: Because that would mean it would be through the ground virtually.

Identifying Cognitive Engagement in the Mathematics Classroom

Excerpt from M's interview:

149

I: OK, why can't it go below the graph?M: Because it's speed and you can't get below zero speed.I: Right, but actually.M: That's what I thought.I: Yeah, yeah, so what do you think now?M: Urn. Well, it's the bit I didn't understand, I didn't understand how it could go

below.I: Urn hm. OK, so you still don't know, you still don't know on that one?M: Yeah..I: OK. [Plays tape: teacher says "slowing, slowing, slowing"] Does that make any

sense to you?. M: No.

I: OK. [Plays tape] So which one are you saying no about?M: The bottom one where it goes below the line.I: Yeah, that's what I thought, that's kind of what I thought.M: 'Cause she's'sort of saying that balls, the way the ball bounces.I: Yeah.M: Goes up and down.I: Yeah.M: It's not the way it goes, it's the speed going up and then it slows down, it's

going back down.

When the students were given the opportunity in the interview to reflect onthe lesson and elaborate their ideas, they revealed far more information about theiruncertainties and difficulties than was evident in the lesson excerpt reproducedabove. This finding demonstrates the need to base conclusions on more than onesource of data. The episode convincingly illustrates the constraints that whole-classinstruction can place on the form and expression of students' cognitiveengagement. Nevertheless, it did provide a powerful stimulus for subsequenthigh-level cognitive engagement.

The Case ofH: /I We Really Put OUf Minds to It"

H made this statement in her interview after a lesson in which studentsworked in groups of four to name some rates that affect their life. This activity wasused as an introduction to a series of lessons on rates. It was not one of the fourlessons used as the basis for this paper, but came to our attention because of H'scomments in her interview. This was the only occasion in over 100 interviewsrelating to 55 lessons in which any student made a spontaneous positive claimabout their cognitive engagement. Our immediate interest was in the nature of theclassroom episode that facilitated H's cognitive engagement, which was evidencedby H's ability to form a conceptual link between a mathematical entity such as arate and its practical application in a real world context.

We believe that the distinctive features of the episode for the student were itsnovelty and the emergent connection with her personal experience. Below is ashort excerpt from a single videotaped group discussion..

150 Helme & Clarke.

Conclusions

Cognitive engagement is valued by the education community (as evidenced bythe studies cited earlier in this paper) and widely held to facilitate learning.

H participated actively in the group discussion, and her contribution to thediscussion of child maintenance rates was clearly about something quite personaland important in her life. She also appeared to value the opportunity to work in agroup:

H: I think it was good the way we went into groups and kind of thought about it,'cause it was easier to do than just to give Mrs B feedback, just from sitting inour seats by ourselves.

I: Yeah. So which bit about the groups part was good?H: Well, we, I thought it was good that we didn't get into friendship

groups... 'cause if we get into friendship groups it's just like you talk all thetime and you don't actually do your work, but. So we were with people, BandI were with D and M, and we kind of really put in together, and that workedrE;ally well.

I: So can you tell me what made it, what was good about it, what worked foryou?

H: Uh, I think, urn, it was just that we all really put in together, we really put ourminds to it, thought about it.

This last statement of H's reflects the dynamic interplay between cognition andmotivation discussed earlier, and supports Csikszentmihalyi and Rathunde's(1992) contention that cognitive engagement can be intrinsically satisfying andmotivating. Our conjecture is that it was the novelty of the task ("because wehadn't done anything like that before in maths") and its connectedness with H'spersonal experience that facilitated H's cognitive engagement. The followinginterview excerpt appears to support this interpretation.

I: What were you thinking about then?H: Well, water rates, 'cause I remember my brother did an assignment on that,

urn, child maintenance rates because urn, my father has to pay them for mybrother and I, urn.

I Right.H: All, all the rates that we pay, bank rates, horne loans, we just bought a new

horne and stuff like that.

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10

Bank rates. [D writes, T approaches then moves off] Child maintenancerates. Affects me, 'cause my dad has to pay it.What?Child maintenance. [To D] Child maintenance rates. [D writes] Stoplaughing M, it's true.School rates.[Nods] School.[To D] Up there. [D writes]Write school fees. [D writes]Pay TV rates.Does anyone here have Pay TV?Doesn't affect me.

B:H:

B:H:B:H:D:H:B:

H:

Identifying Cognitive Engagement in the Mathematics Classroom 151

However, most studies that have investigated this link tend to rely on impreciseand indirect measures of cognitive engagement such as time-on-task, or studentself-reports. Our purpose in the present study was to develop some empiricalindicators of cognitive engagement so that its role in learning can be investigatedmore precisely.

.On the basis of the data we have examined, we argue that cognitiveengagement, as we have defined it, is observable (by inference) in classroomsituations, and can be recognised by specific linguistic and behavioural indicators.We examined cognitive engagement in four distinct classroom situations-paralleland collaborative classroom activity, whole class instruction, and student-teacherinteraction in a small group-and observed that different patterns of cognitiveengagement appeared to characterise each type of activity. We found that, in thisclassroom, student-student interactions appeared to offer more scope formanifesting high,-level cognitive engagement than teacher-student interactions,both in whole-class instruction and in interactions with small groups. The socialrules governing these different activities appeared to playa pivotal role in the formand expression of cognitive engagement.

We have also found evidence to support the view that task characteristicsinfluence cognitive engagement, as do individual factors, such as construedpersonal meaning. Emotional and motivational aspects of engagement are alsoevident in our data, which locate cognitive engagement within a complex interplayof particular aspects of the classroom situation, of the task, and of the individual.

However, the link between cognitive engagement and learning has yet to bedemonstrated empirically. We are hopeful that current research that broadens thisenquiry to include students and teachers in a variety of classroom settings(including involving observation of students as they learn over a sequence ofconsecutive lessons) may provide us with data that may usefully address thisissue.

Acknowledgment

The work reported in this paper was carried out with the support of a grantfrom the Australian Research Council. We thank the teachers and students whoparticipated.

ReferencesAinley, M. (1993). Styles of engagement with learning: Multidimensional assessment of their

relationship with strategy use and school achievement. Journal of Educational Psychology,85, 395-405. .

Ainley, M. (2001). Interest in learning and classroom interaction. In D.J. Clarke (Ed.),Perspectives on practice and meaning in mathematics and science classrooms (pp. 105-130).Dordrecht, The Netherlands: Kluwer.

Baird, J. R. (2001). From ignorance to understanding: Thinking, feeling and acting. In D.J.Clarke (Ed.), Perspectives on practice and meaning in mathematics and science classrooms (pp.255-290). Dordrecht, The Netherlands: Kluwer.

Blumenfeld, P. c., Puro, P., & Mergendoller, J. R. (1992). Translating motivation intothoughtfulness. In H. H. Marshall (Ed.), Redefining student learning: Roots of educational

152 Helme & Clarke

change. Norwood, NJ: Ablex.Bourke, S. F. (1984). The teaching and learning of mathematics. Canberra: Australian Council for

Educational Research.Clarke, D. J. (1996). Refraction and reflection: Modelling the classroom negotiation of

meaning. RefLecT 2(1),46 - 5l.Clarke, D. J. (1998). Studying the classroom negotiation of meaning: Complementary

accounts methodology. In A. Teppo (Ed.), Qualitative research methods in mathematicseducation (Journal for Research in Mathematics Education Monograph No.9). Reston,VA: National Council of Teachers of Mathematics.

Clarke, D. J., & Helme, S. (1997). The resolution of uncertainty in mathematics classrooms. InF. Biddulph & K. Carr (Eds.), People in mathematics education (Proceedings of the 20thannual conference of the Mathematics Education Research Group of Australasia, pp.116-123). Hamilton, NZ: MERGA.

Como, L., & Mandinach, E. (1983). The role of cognitive engagement in classroom learningand motivation. Educational Psychologist, 18, 88-108.

Csikszentmihalyi, M., & Csikszentmihalyi, 1. S. (Eds.). (1988). Optimal experience:Psychological studies of flow in consciousness. New York, NY: Cambridge University Press.

Csikszentmihalyi, M., & Rathunde, K. (1992). The measurement of flow in everyday life:Toward a theory of emergent motivation. In J. E. Jacobs (Ed.), Developmental perspectiveson motivation. Nebraska Symposium on Motivation, 40, 57-97.'

Edwards, D., & Mercer, N. (1987). Common knowledge: The development of understanding in theclassroom. London, UK: Methuen.

Fullarton, S. (1996, November). Transition to secondary school: A help or hindrance to engagementI in mathematics? Paper presented at the joint conference of the Australian Association for

Research in Education and the Educational Research Association of Singapore,Singapore.

Good, T., & Brophy, J. (1984). Looking in classrooms (3rd ed.). New York, NY: Harper andRow.

Gooding, A., & Stacey, K. (1991, June). Aspects of knowledge development throughinteraction in small groups. Procedings of the fifth conference on the Theory of MathematicsEducation, Paderno, Italy.

Hart, L. E. (1989). Classroom processes, sex of student, and confidence in learningmathematics. Journal for Research in Mathematics Education, 20 (3),242-260.

Helme, S. (1994). Mathematics embedded in context: The role of context in task perceptions,performance and the solution methods of adult women students. Unpublished Master ofEducation Thesis, Australian Catholic University.

. Holton, D., & Thomas, G. (2001). Mathematical interactions and their influence on learning.In D.J. Clarke (Ed.), Perspectives on practice and meaning in mathematics and scienceclassrooms (pp. 75-104). Dordrecht, The Netherlands: Kluwer.

Leder, G. c., & Forgasz, H. J. (1992). Inside the mathematics classroom. Paper presented at thejoint conference of the Australian Association for Research in Education and the NewZealand Association for Research in Education, 1992.

Lerman, S. (2001). Accounting for accounts of learning mathematics: Reading the ZPD invideos and transcripts. In D.J. Clarke (Ed.), perspectives on practice and meaning inmathematics and science classrooms (pp. 53-74). Dordrecht, The Netherlands: Kluwer.

Meece, J. L., Blumenfeld, P. C. & Hoyle, R. (1988). Students' goal orientations and cognitiveengagement in classroom activities. Journal of Educational Psychology, 80, 514-523.

Nolen, S. B. (1995). Effects of a visible author in statistical texts. Journal of EducationalPsychology, 87, 47-65.

Nuthall, G., & Alton-Lee, A. (1995). Assessing classroom learning: How students use theirknowledge and experience to answer classroom achievement test questions in scienceand social studies. American Educational Resem;ch Journal, 32, 185-223.

Identifying Cognitive Engagement in the Mathematics Classroom 153

Peterson, P. L., & Fennema, E. (1985). Effective teaching, student engagement in classroomactivities, and sex-related differences in learning mathematics. American Educationalresearch Journal, 22,309-335. .

Peterson, P. L., & Swing, S. R (1982). Beyond time on task: Students' reports of their thoughtprocesses during classroom instruction. The Elementary School Journal, 82, 481-491.

Pintiich, P.R (1989). The dynamic interplay of student motivation and cognition in thecollege classroom. In M. C. Maehr & C. Ames (Eds.), Advances in motivation andachievement: Vol. 6: Motivation enhancing environments. Greenwich, CT: JAI Press.

Pintrich, P. R, & De Groot, E. V. (1990). Motivational and self-regulated . learningcomponents of classroom academic performance. Journal of Educational Psychology, 82,33-40.

Reeve, R, & Reynolds, F. (2001). The role of gesture in collaborative problem-solving: Thecase of mathematics. In D. J. Clarke (Ed.), Perspectives on practice and meaning inmathematics and science classrooms (pp. 231-254). Dordrecht, The Netherlands: Kluwer.

Skinner, E. A., & Belmont, M. J. (1993). Motivation in the classroom: Reciprocal effects ofteacher behaviour and student engagement across the school year. Journal of EducationalPsychology, 85, 571-581.

Skinner, E. A., Wellborn, J. G., & Connell, J. P. (1990). What it takes to do well in school andwhether I've got it: A process model of perceived control and children's engagement·and achievement in school. Journal of Educational Psychology, 82, 22-32.

Stipek, D. J. (1996). Motivation and instruction. In D. C. Berliner & R C. Calfee (Eds.),Handbook of educational psychology. New York, NY: Simon & Schuster Macmillan.

Swing, S. R, Stoiber, K. C. & Peterson, P. L. (1988). Thinking skills versus learning time:Effects of alternative classroom-based interventions on students' mathematics problemsolving. Cognition and Instruction, 5, 123-191.

Webb, N. M. (1989). Peer interaction and learning in small groups. International Journal' ofEducationalResearch, 13, 21 - 39.

Williams, G., & Clarke, D.,J. (1997). Mathematical task complexity and task selection. In D.Clarke, P. Clarkson, D. Gronn, M. Horne, L. Lowe, M. Mackinlay, & A. McDonough(Eds.), Mathematics: Imagine the possibilities (Proceedings of the 34th annual conferenceof the Mathematical Association of Victoria, pp. 406-415). Melbourne:MAV.

Wilson, J. (1998). Metacognition within mathematics: A new and practical multi-methodapproach. In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching mathematics in new times(Proceedings of the 21st annual conference of the Mathematics Education ResearchGroup of Australasia, pp. 693-700). Gold Coast, QLD: MERGA.

Zimmerman, B. J. (1990). Self-regulated learning and academic achievement: An overview.Educational Psychologist, 25, 3-17.

Authors

Sue Helme, Department of Science and Mathematics Education, University of Melbourne,Parkville VIC 3052. E-mail: <sueh@netspace.net.au>

David Clarke, Department of Science and Mathematics Education, University of Melbourne,Parkville VIC 3052. E-mail: <d.clarke@edfac.unimelb.edu.au>.