Student teachers' reflections on their learning process through collaborative problem solving in geometry

RAYMOND BJULAND

STUDENT TEACHERS’ REFLECTIONS ON THEIR LEARNINGPROCESS THROUGH COLLABORATIVE PROBLEM SOLVING IN

GEOMETRY

ABSTRACT. This paper reports research that focuses on student teachers’ reflections ontheir learning process in a collaborative problem-solving context. One group of studentswith limited mathematical backgrounds worked on two problems in geometry withoutteacher intervention. We focus on two episodes from the group dialogues. In the firstepisode (section 5) the students basically reflect on two key issues. The first reflectionis related to the concern of making problem-solving tasks too difficult in general while thesecond reflection has to do with the concern of participation in the solution process. Thestudents discuss how they can give hints or introduce particular ideas before presenting asolution in order to stimulate colleague participation, thus promoting the understanding ofthe solution process. The second episode (section 6) illustrates the reflection of studentson their preparation as future teachers of mathematics. They emphasise that the experienceof getting stuck with a problem may help them to better understand the frustration pupilsexperience while working on unfamiliar problems in classroom. Based on the experience ofgetting stuck, the students reflect on how they could motivate themselves as well as pupilsto work on mathematical problems. They suggest that a good strategy is to start workingon an easier problem. If they succeed in coming up with a solution to that problem, theythink that it is then more stimulating to proceed to a difficult one.

KEY WORDS: collaborative small groups, dialogical approach, geometry, problem solv-ing, reflections, student teachers

1. INTRODUCTION

A deep and comprehensive view of problem solving in the school math-ematics curriculum emerged from the work of George Polya. This authorreformulated, extended and illustrated different ideas about mathematicaldiscovery in a way that teachers could see and use (Stanic and Kilpatrick,1989). In the American literature, researchers of mathematical problemsolving have been inspired by the work of Polya (Kilpatrick, 1969, 1985;Lester, 1980, 1994; Silver, 1987, 1994; Schoenfeld, 1985, 1992). Theseresearchers have made great contributions to the development of problemsolving in the school curriculum through their influence on the changingnature of research emphases and methodologies. Linked to the Americanproblem-solving tradition Mason et al. (1982) and Borgersen (1994) have

Educational Studies in Mathematics 55: 199–225, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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expanded Polya’s four-stage model by presenting a framework designed tohelp students to internalise the strategies of Polya and gain access to it forthemselves.

During the 1990s, there has been a trend towards a more situated un-derstanding of cognition in the area of problem solving and a more ethno-graphically inspired approach has been adopted by many researchers (Res-nick, 1991; Lave and Wenger, 1991; Chaiklin and Lave, 1993; Wenger,1998). In recent years researchers have also focused on students’ mathem-atical reasoning in solving problems (Wyndhamn and Säljö, 1997; Lith-ner, 2000; Lithner, 2003) and reasoning processes of students working incollaborative working groups (Artzt and Femia, 1999; Bjuland, 2002).

The aim of this paper is to focus on student teachers’ reflections ontheir learning process through collaborative problem-solving in geometry.We are particularly concerned with two episodes from the dialogue in onegroup of students with limited mathematical backgrounds. The choices ofepisodes are based on the fact that the analyses of the dialogues illustratehow these students reflect on their learning process at the end of the thirdand the fourth small-group meeting respectively. Since such group reflec-tions are important aspects of the collaborative problem-solving activity,the following research questions have been formulated:

• Are students concerned with reflections on their learning process afterhaving worked on geometry problems in small groups?

• Which elements of reflections can be identified in student communic-ation through collaborative problem-solving activity?

We believe that it is important to focus on the concerns of future teachers asthey reflect on mathematical tasks that they have tried to solve themselves.Does the fact that they are preparing for the teaching profession play a rolein this reflection? In other words:

• Do the students reflect on their experience as learners of mathematicsor as teachers of mathematics?

In a previous study (Bjuland, 2002) we have focused on the observation,analysis and interpretation of the discussion among student teachers work-ing collaboratively in small groups in a problem-solving context. Morespecifically, the aim has been to contribute to the understanding of howreasoning processes are expressed in student communication. We havetried to identify how different elements of reasoning are verbalised in thedialogues and have focused on the heuristic strategies used by two groupsof students with different mathematical backgrounds when they worked ontwo problems in geometry. We were particularly concerned with the heur-istic strategy of posing open questions since there was reason to believe

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that these verbalisations worked as triggers for the reasoning process andfor the generation of strategies used in the solution process.

In this paper we will focus on group reflection as an important aspectof the collaborative problem-solving activity. We have chosen to focus onone group of students with limited mathematical background in order toillustrate how they reflected on their learning process.

2. THEORETICAL FRAMEWORK: THE CONCEPT OF REFLECTION

Historically, Dewey is acknowledged as a main originator in the twentiethcentury of the concept of reflection (Hatton and Smith, 1995). Dewey con-sidered it to be a particular form of problem solving, thinking to solve aproblem that involved a careful ordering of ideas linking each with its pre-decessors. According to Hatton and Smith (1995), four key issues emergefrom Dewey’s original work and its subsequent interpretation as far asreflection is concerned. The first issue has to do with whether reflectionis limited to the thought process about action, or whether it is more boundup in action. The second issue relates to the time frames within which re-flection takes place. The third issue is concerned with whether reflection isby its very nature problem-centred or not. Finally, the fourth issue focuseson “how consciously the one reflecting takes account of wider historic,cultural and political values or beliefs in framing and reframing practicalproblems to which solutions are being sought” (Hatton and Smith, 1995,p. 34). This process has been identified as critical reflection (Gore andZeichner in Hatton and Smith, op. cit).

Much attention has been paid to the concept of reflection in the literat-ure in the last 20 years. According to Mason and Davis (1991), the word‘reflect’ means literally ‘to send or to go back’. These authors emphasisethat a person who investigates his own experience in order to specialise agenerality and see if it makes sense to him is concerned with a reflectiveactivity.

Reflection can also be defined as the conscious consideration of per-sonal experiences (Dewey, 1933; Inhelder and Piaget, 1958; Hiebert, 1992;Wistedt, 1994), often in the interests of establishing relationships betweenideas or actions (Hiebert, 1992). In mathematics learning, reflection ischaracterised by distancing oneself from the action of doing mathematics(Sigel, in Wheatley, 1992). According to Wheatley, it is one thing to solvea problem and it is quite another to treat one’s action as an object of reflec-tion. He emphasises that it is not enough for students to complete tasks,but that they must be encouraged to reflect on their activity. For example,being asked to justify a method of solution will promote reflection. This

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may occur in a small-group context when a participant asks: ‘Will thatwork?’ or it may occur in the whole-class discussion when the presenter isasked to clarify an explanation.

In the problem-solving tradition, Polya’s looking-back step (1945/1957)is often related to a reflective activity since it is possible to improve anysolution or the understanding of the solution. By looking back at the solu-tion, students could consolidate their knowledge and develop their abil-ity to solve problems. In our study (Bjuland, 2002) with regard to theproblem-solving tasks and the students’ attempts at coming up with aconvincing argument, we have distinguished between looking back andreflecting. Looking back on the solution process and on the solution itselfis considered to be a heuristic strategy serving to find the solution, whilereflecting refers to the struggle to modify the solution.

Reflection is also considered in a framework particularly used in teachereducation. Schön (1983, 1987) talks about reflection-in-action, implyingconscious thinking and modification of actions virtually instantaneouslyand reflection-on-action, implying conscious thinking upon action after ithas taken place. We have been inspired by Schön’s framework of reflectionrelated to teacher education and have adapted it to our particular contextwhere student teachers are concerned with problem-solving tasks in smallgroups. Looking back on the solution process and reflecting on a solutionjust found are related to the reflection-in-action part of Schön’s framework.

During the work on the problems in the small groups, the students wereexpected to reflect on their group work and their own learning processes atthe end of each meeting. We have considered these reflections as reflection-on-action. It can be argued that reflection-on-action involves looking backupon action some time after it has taken place (Hatton and Smith, 1995).One of Dewey’s key issues with regard to reflection is related to the timeframes within which reflection takes place. Dewey seems to imply thatthe time frames are extended and systematic rather than immediate andshort term (Hatton and Smith, op. cit.). However, in the two episodespresented here we focus on the students’ reflections on the learning processchosen from the third and fourth meeting respectively, implying that thestudents have experienced collaborating in small groups and worked onthese particular problems for three or four meetings.

3. METHOD

Our project is based on a naturalistic research paradigm (Lincoln and Guba,1985; Erlandson et al., 1993; Brenner and Moschkovich, 2000) which as-sumes, as a theoretical premise, that meaning is socially constructed and


negotiated in practice by the participants in the groups. The naturalisticparadigm reflected in our study relies largely on an ethnographic approachtowards research. We believe that it is important to understand more aboutwhat is going on in a particular small group where students are concernedwith reflections on their learning process related to their solution processfor a given problem. The analysis of the dialogues is inspired by an interac-tionist approach (Bauersfeld, 1980; Bauersfeld et al., 1988; Voigt, 1995).This is linked more specifically to the dialogical approach to cognition andcommunication (Marková and Foppa, 1990; Cestari, 1997).

We follow Cestari (1997) in her argument for choosing this approach“because it allows one to analyse the co-construction of formal languageamong participants in a defined situation” (op. cit. p. 41). This means thatthe dialogical approach permits one to identify interactional processes,which, in the analyses of these particular episodes, are the verbalisationsexpressing the students’ reflections.

3.1. Data collection

The data corpus was collected at a teacher-training college in the autumnof 1996 in Norway. Our research project was carried out on a problem-solving course in geometry. This course consisted of two parts: a first partof teaching over a month in September, including group work assisted bythe teacher, and a second part of small-group work (21 groups) withoutteacher intervention over three weeks in October. The corpus was doc-umented over this period in October. It consists of fieldnotes from theobservation of three small groups of student teachers (randomly chosen)when they work on two problems of classical geometry. It also consists ofthe students’ group reports from this collaborative small-group work. Theverbalisations were registered on an audio tape (8 lessons in each group).

3.2. Procedure

In the lectures to the whole class (105 students) during the first part ofteaching, the students were encouraged to create an open learning envir-onment in the small groups. In order to stimulate social scaffolding wefocused on some advice, introduced by Johnson and Johnson (1990), onhow cooperative learning can be used in mathematics: positive interde-pendence, promotive interaction, individual accountability, interpersonaland small-group skills, and group processing. We paid special attentionto the group processing. The students were expected to reflect on theirgroup work and their own learning processes during their work on prob-lems in the small-group lessons. They were also expected to come up witha general evaluation of their experience of working on problem solving

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in collaborative working groups, stimulating for student reflections as apreparation as future teachers of mathematics.

3.3. Problem selection

The problems chosen were the same as the problems given to the studentswho attended a course in geometry at Agder University College in thespring of 1996. Experiences from observations of one student group work-ing on these problems helped us obtain a thorough understanding of theproblems in order to consider whether these problems could be used inthis project or not. The choice of problems has been guided by three maincriteria:

1. The problems had to be relevant to the students’ classroom and small-group experience from the first part in September;

2. The problems had to challenge the students to experiment, to makeconjectures, to reject the conjectures and to prove them if possible.The questions also had to invite alternative proofs and possible waysof generalising the problems or formulating new problems;

3. The questions had to be challenging but within the capacity of thesubjects to solve with existing knowledge.

Based on these criteria three problems were chosen for the collaborativesmall-group work in the second part. During the four group meetings, thestudents had to work on problem 1 and choose problem 2 (see Bjuland,2002) or problem 3 as the second one. Our particular group worked onproblem 1 and problem 3.

Problem 1A. Choose a point P in the plane. Construct an equilateral triangle such

that P is an interior point and such that the distance from P to thesides of the triangle is 3, 5 and 7 cm respectively.

B. Choose an arbitrary equilateral triangle �ABC. Let P be an interiorpoint. Let da, db, dc be the distances from P to the sides of the triangle(da is the distance from P to the side opposite of A, etc.)

a) Choose different positions for P and measure da, db, dc each time.Make a table and look for patterns. Try to formulate a conjecture.

b) Try to prove the conjecture in a).c) Try to generalise the problem above.

Problem 3Given a right-angled triangle �ABC (� B = 90◦) and a semicircle �, withcentre O and diameter AQ, where Q is a point on AB. The points P (P �=


A) and R on � are given so that P is on AC and OR is perpendicular toAB.

a) Find � APR and � QPC.

b) Prove that � BQC = � BPC.

c) Prove that if B,P,R are collinear (are points on a line), then BC andBQ are of equal lengths.

d) Formulate the converse of the theorem in c). Investigate whether thisformulation is a theorem.

Problem 1 is an open form of Viviani’s theorem, originally introduced bythe Italian mathematician Vincenzo Viviani (1622–1703). This theoremstates that the sum of the lengths of the perpendiculars from an interiorpoint P to the sides of an equilateral triangle equals the altitude h. Byintroducing this historical problem in an open form, we aim that the studentteachers are able to find the value of the distance sum based on drawings,measurements, and constructions of conjectures, and finally give a prooffor it.

With similar intentions, an open form of Viviani’s theorem and its gen-eralisations has also been used in Hungarian classrooms for investigationsat different school levels (Kántor, in press).

Maybe problem 3 does not offer the same nice setting for explorationas that of problem 1. However, based on observation of student teach-ers working with these problems in small groups, we believe that theseproblems are so difficult that the students are dependent on one another inorder to come up with a solution. A study carried out by Borgersen (1994)shows that problems in geometry are ideal for use in small groups in orderto let students experience the entire problem-solving process. Based onfindings from the students’ group reports, Borgersen (op. cit.) claims thatit is possible to achieve interesting results at all levels of knowledge eventhough students have limited mathematical background and experience.

In this respect, these problems are useful from the point of view ofour research objectives. Taking part in the process of working on one ofthese problems may stimulate reflections on their experience as learnersof mathematics. It is also possible that their preparation for the teachingprofession plays a role in this reflection.

Since our group of students chose to work on problem 3, a brief recon-struction of their solution process for this problem is presented below.

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4. RECONSTRUCTION OF THE SOLUTION PROCESS FOR PROBLEM 3aAND 3b

As a background for the analysis of two episodes concerning student re-flections on their learning process, a brief reconstruction of the solutionprocess for problem 3a and 3b is introduced based on the data corpus.

4.1. Drawing a figure of the problem

The students have a 25-minute discussion on problem 3 during the secondmeeting. They read the problem and discuss what is meant by the mathem-atical symbols ‘�’ and ‘�=’. They decide to draw a figure of the problem.The students agree on placing point B arbitrarily on a line through point A.They go on to draw a right-angled triangle arbitrarily chosen. They observefrom the text that AQ is a diameter of the semicircle �, and they agree onplacing point Q arbitrarily on the line segment AB. Since point O is thecentre of �, it follows that O is the midpoint of AQ. The students haveso far been concerned with the placements of points A,O,Q,B and C

respectively. They draw the semi-circle � and observe that point P liesboth on � and on the line segment AC. The students help each other toplace the different points on the figure. They need some time to place pointR, but they realise that R lies on �, so that RO is a perpendicular on linesegment AB.

4.2. Two possible solutions on angle � APR emerge from the figures

After having come up with a figure, the students agree on measuring angle� APR with a protractor in order to get an idea of the size of the angle.The measurements from four of the students show that angle � APR is 45degrees. However, one student’s measurement suggests that angle � APR

is 135 degrees. The students compare this figure with the other figures, but


they do not find any mistake. They observe that point P is placed to the leftof point R.

One of the students focuses on � APR and � AOR. The argument isintroduced that the angle � AOR at the centre is double the angle � APR

at the circumference since they both subtend the same arc AR of the circle(Thales’ theorem). The group members do not focus more on the fact thatthere are two solutions to the problem, depending on how the points P andR are placed in relation to each other. They just change the figure whichcomes up with the 135-degree angle of � APR. The students agree thatthey have come up with a solution in which angle � APR is 45 degreesdue to Thales’ theorem.

4.3. The solution process of finding angle � QPC

Some individual and introductory attempts have been made at the end ofthe second meeting in order to find angle � QPC. They observe that theangle is 90 degrees but without giving an argument for this. The studentsstart the third meeting by recapitulating the reason why angle � APR is45 degrees before they go on to find an argument which shows that angle� QPC is 90 degrees. They are concerned with the semi-circle and observethat the points A,P and Q all lie on �. One of the students also focuseson the points A,O and Q. When one of the other students focuses onThales’ theorem, she observes that the angle at the centre is 180 degrees.She comes up with the argument that angle � APQ is 90 degrees sinceit is an angle at the circumference. The students agree that the argumentfollows directly from Thales’ theorem. From a mathematical point of view,it is then easy to conclude that angle � QPC is 90 degrees. However, thestudents choose a cumbersome way to come up with this conclusion (formore details, see Bjuland, 2002).

4.4. The solution process for problem 3b

The development in the solution process is shown in the three consecutivefigures below.

Relevant subconfigurations from the initial and complex figure havebeen found in order to make sense of the problem. The students havefocused particularly on quadrilateral QBCP on a separate figure. Theidea of the cyclic quadrilateral has been discussed, by searching for thecharacteristics in a textbook. After some discussion, the students haveconcluded that QBCP is a cyclic quadrilateral. The subconfiguration hasbeen modified by a circle which roughly circumscribes QBCP . One of thestudents draws special attention to this circle. The student, who has comeup with the extended figure, suggests that this is not constructive for the

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.


solution process. However, since the other group member is concernedwith her circle, she focuses more closely on it and suggests how theycould construct the centre of this particular circle in order to obtain a moreaccurate figure. The construction of the circle triggers the breakthroughin the solution process. It has been observed that angle � BQC and angle� BPC are both angles at the circumference, subtending the same arc. Thetheorem of Thales has been used in order to justify that these angles areequal.

5. STUDENT REFLECTIONS ON THE LEARNING PROCESS (FIRST

EPISODE)

The aim of the following episode is to illustrate how the students reflecton their learning process at the end of the third meeting. The analysisof the students’ dialogue focuses on verbalisations that show evidence ofthis reflection process. During the first three meetings, the students haveworked on problem 1.

The analyses of the dialogues from our group (Bjuland, 2002) give adetailed description of how the students approached and made sense ofproblem 1B. One major difficulty for these students was to consider howto measure the distances da , db and dc from a point P to one of the sidesof the triangle. Some of the students thought that they had to measure thelengths of the perpendiculars from P to their intersections with the sidesof the triangles, but one of the group members had an alternative way ofdoing it. She thought that they should measure the distance along the linethrough P parallel to the base of the triangle. This idea challenged theother students to focus on an alternative way of doing the measurements.The elaboration of the two different perspectives led to agreement. Thestudents realised that they had to do the measurements in the same way inorder to find a pattern. The students made several efforts to come up witha conjecture da + db + dc = constant. They did not find a proof for theconjecture.

The students are now in the process of working on problem 3. Theyhave come up with a reasonable solution for problem 3b, and they havestarted to work on problem 3c. After having read the formulation of theproblem and drawn a figure as a starting point, the students decide to go onworking on problem 3c at the fourth meeting. They spend the last minutesof the third meeting on group reflections.

Before the episode presented below, the students have started this re-flection process, and they have been concerned with the problems givenin the group work. Mia and Gry claim that the problems are too difficult,

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while Unn suggests that they make the problems too difficult. She saysthat they work too quickly and introduce too many ideas in the solutionprocess.

The students’ reflections on the learning process are organised in fivethematical segments, which have emerged from the analysis of the stu-dents’ conversation.

5.1. Making problems too difficult

1538. Unn: Yes. . . but I think that we generally make all the problems toodifficult. . .

1539. Roy: What can we do about it then?. . .

1540. Unn: No. . . I don’t know. . . (laughs quietly). . .

1541. Liv: I think that it just has to be like this. . . well. . . since we are notvery good at this. . .

Unn’s initiative (1538) is a continuation of the conversation in which thestudents reflect on the problems given in the group work. There is a signof frustration which could indicate limited experience of working on suchproblems. Unn is concerned with the fact that they make the problemstoo difficult. She has introduced this consideration in an earlier discussionlinked to the fact that they introduce too many ideas in the solution processwithout spending any great time focusing on each idea. The word ‘gen-erally’ suggests that Unn’s statement is not only linked to the particularexperience from the ongoing solution process for problem 3. Unn seemsto suggest that they have a general difficulty in their way of approachingand making sense of a problem.

Roy’s open question (1539) follows up the prior verbalisation, encour-aging a focused discussion on this statement. The question is an indicatorof the supportive and constructive atmosphere established in the group.The students not only focus on the difficulty in the solution process, but arealso concerned with finding ways to improve their sense-making processon a problem given. The uncertainty and the quiet laughter (1540) indicatethat this process is not easy. Liv follows up (1541) by suggesting thatthey have to work on these problems in this way, reminding the studentsof their limited mathematical backgrounds. This verbalisation is probablylinked to Unn’s earlier statement on making the problems too difficult.Liv seems to emphasise the fact that they need to bring a lot of ideas intothe discussion since they do not have a very clear way of approaching theproblem. The students seem to be conscious of their limited knowledgewith mathematical problem solving.


5.2. Perceiving differences in their participation in the solution process

1542. Mia: However. . . I feel it’s unpleasant when you (Liv and Roy) sit andthink out everything. . . and we know you’ll say. . . yes what arewe going to do now, then?. . . what are we going to write now,then?. . .don’t you agree?. . .

1543. Liv: Yes I get so incredibly keen because eeh. . .

1544. Mia: Yes but you understand it and you. . .

1545. Liv: Well. . . we do spend a lot of time on it. . .

1546. Gry: We would never have come up with a solution I think. . . if ithadn’t been for the fact that someone maybe thought a bit bythemselves and thought aloud. . . and they found out a bit. . .individually. . . if we had all sat here thinking about everything. . .

then we wouldn’t have . . . I don’t think we would have finished asingle problem yet. . .

1547. Mia: But everybody has to do their bit. . . well it. . .

Mia’s verbalisation (1542) shows a shift in the students’ reflections onthe learning process: from their considerations of their limited experi-ence of working on problems to their perceptions of differences in theparticipation on the solution process. Mia’s frustration is related to herself-perception of being too little involved in the solution process. All herquestions indicate that she is just told what to do, implying that she hasdifficulties in participating in the mathematical discussion. Mia remindsthe other group members of the importance of involving all the participantsin the discussion in order to achieve the expected synergetic quality for thecommunication. Being involved in the discussion is also important for thestudents’ learning processes.

Working on these problems also provokes excitement (1543). This af-fective expression is probably related to the solution process for problem3b in which Liv came up with the final step in the solution process. Theprogress and understanding of the problems (1544) seem to be relatedto their willingness to spend considerable time on the solution process(1545). The student’s excuse of spending a long time on the problems(1545) could be an indicator of willingness and perseverance to work in-tensely on the problems. These affective involvements could be crucial inorder to succeed in solving a mathematical problem.

Gry (1546) points to the fact that in order to make progress in the solu-tion process, the students must allow time for individual considerations.Mia (1547) follows up and makes it clear that all the group members mustbe involved in the solution process.

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5.3. Suggesting improvements in the learning process

1548. Roy: Well. . . but is the difficulty simply that we. . . eeh. . . before every-body in the group has understood the point. . . we go on with thenext thing?. . .

1549. Liv: I could maybe have hung on for a while . . . with the solution(for problem 3b). . . if I had made it out. . . or. . . or perhaps. . . oryou or anybody. . . that. . . we maybe should just give some hintshow. . .

1550. Unn: which (idea) we’ll use. . .

1551. Liv: what we can do in order to find it out. . . before just introducingthe solution at once. . .

1552. Roy: Yes. . . but on the other hand. . . probably we’re all glad now wefound a solution. . .

1553. Mia: Yes of course. . . that’s quite obvious. . .

1554. Roy: instead of spending extra time on it. . .

1555: Mia: Yeah, yeah, yeah. . . that’s obvious. . .

1556. Roy: But. . . eeh. . .

The difficulty and also the frustration brought into the discussion earlierby Mia (1542) are handled in a supportive and constructive way. Roy’sopen question (1548) is linked to Mia’s concern about not being involvedin the solution process. He focuses the discussion by inviting the otherstudents to mention explicitly what the difficulty in the students’ learningprocess is. In his question, Roy also suggests what the difficulty is andthe verbalisation triggers suggestions of improvements for the students’learning processes.

Liv follows up (1549) by looking back on the solution process for prob-lem 3b, reflecting on her own way of presenting the solution for the otherstudents. Liv criticises herself for coming up too quickly with the solution.Her reflection can be understood from the special case for problem 3bto a more general way of improving the learning process when they aredealing with problem solving. Giving hints (1549), pointing to the ideathey want to use (1550) instead of presenting the solution at once (1551)are the students’ concrete suggestions for improving their solution processfor a particular problem.

After having introduced some general improvements for the learningprocess, Roy and Mia agree on the fact that they are glad they found asolution for problem 3b (1552)–(1555). Roy’s initiative (1552) is probablylinked to Liv’s considerations for her presentation of the solution, sup-porting her for the fact that she came up with the last step in the solutionprocess. The difficulty in the learning process, the differences in student


involvement, has been elaborated on by concrete improvements and Mia’sfrustration, brought into the discussion, has been followed by satisfactionwith coming up with a solution.

Roy’s verbalisation (1556) suggests a shift in the dialogue which isfollowed up by Unn’s initiative in the next segment (1557).

5.4. Perceiving how to solve the problems

1557. Unn: But there is something about these problem-solving tasks. . .when you don’t know which methods you’re going to make useof. . .isn’t there?. . .

1558. Liv: But it’s. . .

1559. Roy: That’s what we do by trial and error all the time. . .

1560. Unn: Yes. . .

1561. Liv: That’s why we have to use a lot of. . . (methods). . .

Unn’s verbalisation (1557) triggers a shift in the reflection process whenshe focuses on the nature of the problems. This reflection is linked tothe dialogue of the first segment in which the students focused on theobstacle of making the problem-solving tasks too difficult. Unn’s initiativein this segment probably focuses on the particular problems given for thegroup work, but her verbalisation also suggests a consideration of howto solve problem-solving tasks in general. The students’ perceptions of amathematical problem are linked to the fact that they do not have a readilyaccessible mathematical algorithm for the solution process. The studentsseem to be aware of the complexity of the problem-solving tasks. They areconfronted with the fact that they need to find methods or generate ideasin order to make progress in the solution process. In this sense-makingprocess they realise that they have to solve the problems by trial and error(1559), (1560).

In the introduction to this episode, we pointed to Unn’s suggestion thatthe students introduced too many ideas in the solution process. Liv (1561)makes it clear that they need to introduce a lot of ideas or methods in thesolution process due to the nature of the problems.

5.5. Elaborating on improvements in the learning process

1562. Unn: Yes but. . . suppose if one (of us) maybe comes up with asolution. . . okay we’ll use that method. . . then it’s maybe easierfor the others to understand why it’s like this. . . so that theyget to try out the method before. . . before the whole solution ispresented. . .

1563. Mia: Yes maybe. . .

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1564. Roy: Yes. . . if. . . if one of us discovers the solution. . . then we’d ratherintroduce the method. . .

The students put this idea in concrete form by focusing on thesolution for problem 3b.

1579. Roy: But the fact that we could have. . . eeh. . . we could all have con-structed that circle. . . and then seen what it was like. . . that weagreed on the fact that now we’re constructing a circle whichcircumscribes the cyclic quadrilateral. . .

1580. Unn: Mmm. . .

1581. Roy: That’s what we could have done. . . and then we could. . .

1582. Unn: And what do we see then?. . . or something like that. . .

The students elaborate on the discussion from the third segment in whichthey suggested improvements for their learning processes. Unn (1562) fo-cuses on the relation between being involved in the solution process andtheir mathematical understanding. The students are not satisfied just withcoming up with a solution, but they are concerned with the fact that allthe group members must understand ‘why’ the solution is like this. Miaagrees to some extent (1563), while Roy (1564) repeats Unn’s initiative,confirming that they introduce the idea or method for the other studentsbefore presenting the whole solution.

The students put this general way of dealing with problem-solving tasksfor later discussions in concrete form by focusing on the solution for prob-lem 3b (1565)–(1578). Roy focuses on the specific suggestion of construct-ing a circle which circumscribes the cyclic quadrangle QBCP (1579). Thedevelopment in the solution process is shown in three consecutive figures(see 4.4). After having finished this construction, the students could thenindividually get the opportunity to observe that angle � BQC and angle� BPC are both angles at the circumference, subtending the same arc anduse Thales to conclude that these angles are equal.

The last sequences of verbalisations (1579)–(1582) show that the stu-dents agree on Roy’s specific suggestion. The students’ reflections fromgeneral considerations of introducing ideas or methods before presenting asolution to put this in concrete form with the particular case with problem3b, indicate a willingness to really make an improvement in their learningprocesses. By moving from the general to the particular, their idea of deal-ing with the solution process is exemplified for a concrete problem. Roy’sspecific suggestion of introducing the construction of the circle is expan-ded by Unn since she focuses on the strategy of posing open questions(1582). The question stimulates each of the group members to discover thelast step in the solution process themselves.


6. STUDENT REFLECTIONS ON THEIR EXPERIENCE AS FUTURE

TEACHERS OF MATHEMATICS (SECOND EPISODE)

In the previous episode we observed that the students reflected on theirlearning process after having worked on two geometrical problems. Thestudents reflected on their experience as learners of mathematics. The aimof the following episode is to illustrate how the students reflect on theirproblem-solving experience as future teachers of mathematics. The epis-ode is selected from the end of the fourth and last meeting.

Throughout the whole meeting the students have been concerned withworking on problem 3c, but they do not manage to come up with a solution.Two of the students claim that they have come up with a proof for thetheorem based on measurements. However, two of the other students claimthat it is not good enough to measure the line segments BC and BQ on fivedifferent figures and conclude that BC and BQ are of equal lengths. Theyneed to find a proper mathematical argument. The experience of gettingstuck with the problem has led to frustration among the students. A detailedreconstruction of the students’ attempt at solving problem 3c can be foundin Bjuland (1997).

The students’ reflections are organised in three thematical segments,which have emerged from the analysis of the students’ conversation.

6.1. Experiencing getting stuck and understanding pupils’ frustration

1361. Roy: Mmm. . . (14 sec. . . low voices). . . then we have maybe learntquite a lot about. . . how the pupils often may feel maybe. . . likefrustration and so on. . .

1362. Liv: Mmm. . .

1363. Mia: Yes. . .

1364. Roy: There is certainly a good deal to learn from that. . .

1365. Liv: Mmm. . .

1366. Mia: Yes. . .

1367. Roy: We know something about our situation. . . (experience fromsolution process). . .

1368. Mia: Yes that’s for certain. . . frustration and. . .

Roy (1361) is concerned with what they have learnt from the experience ofgetting stuck with problem 3c. Working on this geometrical problem hasprovoked frustration among the students. Instead of remaining frustrated,the students reflect on the experience as future teachers of mathematics byconsidering pupils’ frustration in classrooms. Roy’s initiative of focusingon the perspective of pupils is followed with agreement (1362), (1363),

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stimulating Roy to repeat the fact that they can learn much about pupils’affective involvement based on their own experience from the solutionprocess (1364), (1367). Both Liv (1365) and Mia (1366), repeat their shortresponses, and Mia strengthens her agreement with this reflection (1368).

6.2. Motivating pupils in order to deal with frustration

1369. Roy: What could we have done when. . . when we were thatfrustrated. . . in order to pass this situation when we look back?. . .what could we have done differently. . . for example. . . Mia?. . .

1370. Mia: No, then I’d probably have had to. . . for example if I had beenbetter prepared in advance. . . personally, for my part. . .

1371. Roy: Usually a pupil in school hasn’t (done) that. . . if he is goingto learn something new. . . he gets his homework. . . but he’s sofrustrated that he doesn’t seem to be able to face working on hishomework. . .

1372. Liv: It’s like you say. . . that you don’t see the point with it. . . there areprobably quite a few pupils who have the same feeling. . . how. . .

what can we say in order to. . .

1373. Unn: motivate. . .

1374. Liv: in order to motivate?. . .

Roy’s question (1369) seems to be triggered by Mia’s confirmation, indic-ating frustration in the solution process (1368). The question shows theimportant shift in the reflection process from the state of being frustratedwhile getting stuck with the problem to an invitation to the other studentsto participate in discussing how to learn from this situation. The questionis repeated, elaborated on, and particularly directed to Mia. Mia criticisesherself for not being sufficiently prepared for the small-group work (1370),stimulating Roy to focus on the situation in school. He suggests that pupilsin general are badly prepared for learning something new and easily giveup working on their homework when getting frustrated (1371).

Earlier in this discussion, Mia has told the other group members thatshe cannot see the point of working collaboratively in small groups withthese geometrical problems (see Bjuland, 1997, p. 195). Liv (1372) bringsMia’s opinion into the conversation and links it with pupils’ frustration inclassrooms. The dialogue shows how the students’ reflections on their ex-perience as learners of mathematics trigger reflections on their preparationas future teachers of mathematics by bringing their perspectives on pupilsinto the conversation.

The open question (1372) stimulates the students to focus on how tomotivate pupils in classrooms in order to deal with frustration (1373),


(1374). Liv’s question could also be an invitation to focus on their ownmotivation as learners of mathematics.

6.3. Working on another problem when getting stuck

1375. Roy: I think it’s important maybe when we get stuck (with aproblem). . . then it’s very important that we can start workingon another problem. . .

1376. Liv: Mmm. . .

1377. Roy: that we manage. . .

1378. Unn: Mmm. . .

1379. Roy: in order to increase our self-confidence. . .

1380. Mia: Yes if you manage that (problem). . . then you have made someprogress. . .

In the previous segment (6.2), we identified that the students’ reflectionsas learners of mathematics and their reflections as future teachers wereboth present in the dialogue. Based on their own experience from thesolution process, they used their frustration at getting stuck to reflect onthe situation of pupils in schools. Instead of elaborating on the perspectiveof pupils’ frustration in classrooms, Roy’s response (1375) to Liv’s ques-tion (1372) shows that Roy seems to be more concerned with their ownexperience as learners of mathematics. His suggestion about working onanother problem when getting stuck (1375) is an important strategy whichcould help them to improve their problem solving. It is also possible thatRoy thinks of this suggestion as a strategy to be used in his future work asa teacher of mathematics.

In order to deal with frustration and to stimulate their motivation in theproblem-solving process, the students suggest starting to work on an easierproblem. By succeeding in solving the new problem (1377), they believethat this positive experience will increase their self-confidence (1379),(1380).

7. DISCUSSION

We have, in the first episode, focused on one group of students and their re-flections on their learning processes after having collaborated with problem-solving tasks in a small-group context for three meetings. Through thedetailed analysis of the students’ conversation, we have identified someof the students’ reflections organised in five thematical segments: Makingproblems too difficult, perceiving differences in their participation in the

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solution process, suggesting improvements in the learning process, per-ceiving how to solve the problems, elaborating on improvements in thelearning process.

Our analysis has been based on the perspectives of the students beingstudied. From a methodological point of view, this is one principal char-acteristic of much observational research (Bryman, in Silverman, 1993).However, we have to take into consideration, as Silverman points out, that“any attempt to base observation on an understanding of how people ‘see’things can speedily degenerate into a commonsensical or psychologisticperspective” (op. cit, p. 31).

7.1. Focusing on difficulties in the learning process

If we focus more carefully on the identified group reflections which haveemerged from the analysis of the students’ conversation in the first epis-ode, we observe that the students basically reflect on two key issues. Thefirst reflection is related to the concern about making problem-solvingtasks too difficult in general (segment 1). The second reflection has todo with the concern of participation in the solution process (segment 2).These reflections-on-action (Schön, 1983, 1987), more specifically on theproblem-solving activity, indicate the conscious thinking upon action afterit has taken place. The general concern identified in the first reflectionindicates an extended and systematic reflection since it is not only linkedto the particular activity on the third meeting. Even though the studentshave been introduced to problem-solving in small groups throughout theteaching part of September and they have been working on the group workfor three meetings in October, the students realise that they lack knowledgeand experience of working on such problems. They need to introduce a lotof ideas into the sense-making process since they do not have a very clearway of finding a direction for the solution process.

The students do not reflect on the fact that two possible solutions onangle � APR emerge from the figures in problem 3a, depending on howthe point P is placed related to R. They just change the figure which comesup with the 135-degree angle of � APR. An important reflection at theend of this group meeting would have been to focus more closely on thisparticular figure. Maybe it is new to the students to come up with morethan one solution to a problem. One the other hand, the two figures withdifferent angles of � APR have perhaps made them aware of two possiblesolutions, but they are satisfied with one solution. It is also possible thatthe students want to have the same starting point for the other parts of theproblem.


The second reflection, related to the perception of differences in par-ticipation on the solution process, is closely connected to frustration. Itis possible to argue for the fact that this affective involvement has beentriggered from experiences throughout the three meetings. However, basedon knowledge of the students’ solution processes with the two problems(Bjuland, 2002), we believe that this reflection is relatively immediateand related to the activity at the third meeting in which the students wereconcerned with solving problem 3b. This reflection is constructive for thelearning processes in the group since the students are reminded of theimportance for all group members to participate in the mathematical dis-cussion in order to achieve the expected synergetic quality in the commu-nication.

7.2. Elaborating on reflections to improve their cooperation on thesolution process

The two reflections, identified in segment 1 and segment 2, focus on theirdifficulties in the learning process. However, these reflections are discussedand reflected upon at a meta-level throughout the other segments of thedialogue. The students are concerned with finding general ways to improvetheir cooperation and in this way, the understanding of the problems. Onthis general meta-level, the reflections are also related to their learningprocesses.

The first reflection is expanded in segment 4 in which the students focuson the nature of problem-solving tasks in general. Their perceptions ofwhat they mean by a mathematical problem coincide with definitions inwhich it is defined as a task for which the student does not have a readilyaccessible procedure/method/algorithm sufficient to answer the question(Blum and Niss, 1991; Schoenfeld, 1993). We follow Borgersen (1994)when he emphasises that teachers must make an effort in order to find andtest potential problems which are suitable for cooperation in small groups.

The students’ second reflection is related to this synergetic quality ofthe discussion. This reflection is expanded in segment 3 and segment 5. Inorder to stimulate student participation, they reflect on how they can givehints or introduce particular ideas before presenting a solution. The stu-dents move from general considerations of improvements for their cooper-ation on the problems and put their ideas into concrete form by focusingon the solution for problem 3b. This shift from the general to the particularshows that the students are focused on improvements for their learningprocess. This becomes even more clear since they use the particular ideaof constructing a circle which circumscribes the quadrilateral for problem3b as a starting point to reflect on how general strategies like posing open

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questions can stimulate participation and their learning processes for futurework on problems and as future teachers.

In a previous study (Bjuland, 2002), we have seen that questions in thestudents’ discussion can contribute to the process of sense-making whenthey are confronted with a problem. This is also supported by Cestari(1997), who claims that questioning could be a vehicle in a small-groupcontext which enhances students’ building of argument. In the analysisof the dialogue concerning the student reflections on the learning processpresented above, open questions are used in a supportive atmosphere tofocus the reflections. The questions stimulate the shift in the reflectionsfrom introducing the difficulties in the solution process to an elaboratedattempt at improving their learning processes.

7.3. Reflecting on their experience as future teachers of mathematics

Based on the analysis of the first episode, the student teachers’ reflectionsillustrate that they are concerned with their own experience as learnersof mathematics. The fact that these students are preparing to enter theteaching profession does not seem to play a central role in their reflection.Why are such reflections not present in the first episode?

We have seen that the students have been so concerned with solving theproblems by encouraging one another to participate in the mathematicaldiscussion, stimulating all the students to focus on one particular idea, astrategy or a mathematical concept in order to understand the differentsteps in the solution process. It is likely that it is difficult for them to focuson their future role as teachers simultaneously. We also have to take intoconsideration that the students were just implicitly told to focus on theirrole as teachers when they were expected to come up with a general eval-uation of their experience of working on problem solving in collaborativeworking groups.

However, the analysis of the second episode at the end of the fourthmeeting shows that the students bring this kind of reflection into the dis-cussion. Working on these particular problems has provoked frustrationamong the students. They emphasise that the experience of getting stuckwith a problem may help them to better understand the frustration pupilsexperience while working on unfamiliar problems in classroom. By sug-gesting working on an easier problem in order to promote motivation andself-confidence, the students are aware of an important problem-solvingstrategy, especially if that new problem is related to a difficult one. If theysucceed in achieving a solution to the easier problem, it is then more stim-ulating to go on to work on the difficult one. Polya (1945/1957) claimsthat in order to solve a mathematical problem it is fundamental to relate


the problem to formerly acquired mathematical knowledge. This is alsosupported by Dreyfus and Eisenberg (1996), claiming that an extremelyimportant key in developing mathematical reasoning ability is to train one-self to look for analogies. The students do not seem to elaborate on thisreflection since there is no indication of accommodating the easier prob-lem into existing knowledge along with the difficult one. The students’reflection on the experience of getting stuck is more linked to motivation.

According to Mason and Davis (1991), it is important to recognisebeing stuck and to acknowledge it. These authors claim that this is a pos-itive state since it offers a possibility to learn something about yourselfand maybe about pupils in school. We have seen that the students in ourparticular group respond positively and productively on getting stuck. Re-flections on their own learning processes trigger reflections on their pre-paration for the teaching profession.

The analysis of the second episode has shown that the students reflecton their experience both as learners of mathematics and as future teachersof mathematics. Why are both these perspectives present in the secondepisode and not in the first episode? Maybe it is easier for them to focuson their future role as teachers at the end of the last meeting when they areto finish the collaborative small-group work.

7.4. Reflecting on experience in small groups: can we take it for granted?

The analysis of these particular episodes has shown that the students withlimited mathematical backgrounds really have been concerned with reflec-tions on their action (Schön, 1983, 1987). In the teaching part in Septem-ber, in order to stimulate social scaffolding, we focused on some advice,introduced by Johnson and Johnson (1990), on how cooperative learningcan be used in mathematics. Since we paid special attention to group pro-cessing, it is probable that the students’ reflections on the cooperation, ontheir learning processes, and on their preparation as future teachers havebeen stimulated by the activity from the first part of the project. It could betempting to ask: What about the reflections in the other groups? Based onthe background knowledge about the subjects in one of the other groups,categorised as a group of students with average mathematical background(Bjuland, 2002), the reflections in this group are very brief and not muchelaborated. The students start to reflect on their problem-solving activity,but they quickly run into the process of solving the problems. They are sofocused on the problems, which implies that they do not spend much timeon group reflections.

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8. FINAL REMARKS

In order to answer the first research question, in section 1, we have, throughthe analysis of the student conversation, illustrated that the group of stu-dents with limited mathematical backgrounds are really concerned with re-flections on their learning process after having worked on geometry prob-lems in small groups. Even though students are stimulated for social scaf-folding and expected to reflect on their group work, we learn from thegroup of students with average mathematical background that it cannot betaken for granted that students will focus on group reflections at the end ofa meeting. Problem-solving tasks trigger affective involvement like will-ingness to spend considerable time on the solution process. It could there-fore be difficult to change focus from working on a problem to reflectingon the problem-solving activity.

Another research question was to identify elements of reflections in stu-dent communication through collaborative problem-solving activity. Ba-sically, the students reflect on two key issues. The first reflection is relatedto the concern of making problem-solving tasks too difficult in generalwhile the second reflection has to do with the concern of participationin the solution process. These reflections are discussed in order to stim-ulate colleague participation promoting the understanding of the solutionprocess.

The third question focused on whether the students reflect on their ex-perience as learners of mathematics or as teachers of mathematics. Wehave seen that the students mainly reflect on their own learning process.However, at the end of the last meeting, the students bring the perspect-ive of pupils’ frustration in classroom into the discussion, indicating apreparation for the teaching profession.

The analysis of the dialogue of the group of students with limited math-ematical background gives evidence to support the argument that groupreflections at the end of a meeting are an important part of the problem-solving activity. These reflections have emerged in the conversation withoutteacher intervention.

One possible direction for future research would be to focus more closelyon observation, analysis and interpretation of the conversation of pupils atlower levels in the school system (for example pupils aged 12–16) work-ing collaboratively in small groups in a problem-solving context. Morespecifically: Which elements of reflections can be identified in pupils’communication at lower levels in the school system during collaborativeproblem solving in small groups?


As a pedagogical implication, teachers should stimulate their studentsto reflect on their cooperation in small groups. The presentation of episodesfrom student dialogues in collaborative working groups can be useful incourses at teacher-training colleges or in in-service training of teachers inorder to provide opportunities for students and teachers to observe howgroup reflection develops in student groups.

ACKNOWLEDGEMENTS

I would in particular like to thank the leadership at NLA College of TeacherEducation which provided the grant for doing this research. Thanks also goto the student teachers, the teachers and the administration at the teachertraining college where the observations have been carried out. I would alsolike to thank my colleagues Professor Maria Luiza Cestari and Associ-ate Professor Hans Erik Borgersen at Agder University College for theirhelpful comments on this paper.

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Agder University College,Faculty of Mathematics and SciencesDepartment of MathematicsServiceboks 4224602 KristiansandNorwayTelephone +47 51 43 20 95E-mail: [email protected]

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Student teachers' reflections on their learning process through collaborative problem solving in geometry