Classroom interaction as reflection: Learning and ... - Springer Link

ED ELBERS

CLASSROOM INTERACTION AS REFLECTION: LEARNING AND TEACHING MATHEMATICS IN A COMMUNITY OF INQUIRY1

ABSTRACT. This article aims to contribute to the theory of mathematics instruction by highlighting and analysing Leen Streefland’s work as a teacher in a primary school. Core ideas of Streefland’s are discussed as part of a recent wave of educational innovations using the idea of learning communities. I present a case study of a lesson co-taught by Streefland and a primary school teacher (students between 11 and 13 years of age). Streefland addressed the students as ‘researchers’ and gave them realistic problems to work on. Since they provided occasions for increasingly sophisticated solutions, the tasks given to the students stimulated processes of vertical mathematization. The students’ role as researchers required them to construct novel ideas and present these to their classmates. The various solutions proposed by the students were used by the teachers to structure the learning activities in the classroom. The interaction between the common understandings in the classroom and the learning process of individual students shows that the classroom activities amount to a process of collective reflection.

1. I NTRODUCTION

A central issue in the debate on mathematics education is how teachers can help students to participate in the process of knowledge construction (e.g., Lampert, 1990; Ben-Chaim, Fey, Fitzgerald, Benedetto and Miller, 1998; Inagaki, Hatano and Morita, 1998). I would like to contribute to this debate by presenting Leen Streefland’s ideas and practice of mathematics teaching. As a consequence of his untimely death, Streefland’s ideas on mathematics education have failed to receive the attention they deserve. I shall highlight some of his ideas and discuss a case study of a class taught and analysed by Streefland. For a number of years, Leen Streefland and I studied children’s learning and collaboration in mathematics lessons. Streefland was involved in these lessons not only as a researcher but also as a teacher. He believed he needed the experience of teaching schoolchildren to be able to further develop his ideas of realistic mathematics teaching. Leen Streefland started his innovative teaching of mathematics together with the primary school teacher Rob Gertsen as early as 1985. He became so enthusiastic about what he experienced, that he drew my attention to his projects. He asked me to make observations of the unique social and intellectual processes occurring Educational Studies in Mathematics 54: 77–99, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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in the classroom. I participated because of my interest in students’ learning through interaction and collaboration. This research resulted in a number of conference papers and articles (Elbers, Derks and Streefland, 1995; Streefland and Elbers, 1995, 1996; Elbers and Streefland, 1997a, 1997b, 1998, 2000a, 2000b). Our collaboration was episodic: it was concentrated in certain periods when we worked on a topic, mostly because we wanted to present a paper at a conference or hold a workshop for schoolteachers. Just when we had decided to approach the theme of social interaction in mathematics learning more systematically, his illness revealed itself. Leen Streefland conducted pioneering work in creating what we came to call a community of inquiry in the classroom. He encouraged students to carry out research and adopt the attitude of researchers. The task of the teacher was to guide and assist students who had been given considerable responsibility for their own learning. Streefland was convinced that creating a learning community in which students had ample opportunity to produce and discuss ideas would allow their mathematical creativity to blossom. He had a strong belief in children’s talents and thought that their creative abilities had insufficient opportunity to emerge in the conventional classroom. Interaction and collaborative learning would stimulate children to make their own mathematical constructions and to discuss them in what amounted to a social process of reflection. Building an atmosphere of cooperation, instead of a climate favouring competition and individual performance, made it possible for children to feel free to present their ideas and to comment on others’ ideas. In a community of inquiry, students could work together without the fear of failing or disappointing their classmates. Such an atmosphere of mutual trust and cooperation would allow even the least talented children to participate in the process of constructing mathematical meanings. The aim of this contribution is twofold. In the first part, I wish to formulate some core ideas of Leen Streefland’s, as I see them, about interaction and collaborative learning, and to discuss them as a part of a recent wave of educational innovations. In the longer second part, I shall present a case study of a class co-taught by Leen Streefland and Rob Gertsen. In my analysis of this case, I shall particularly be concerned with the relationship between the discussions of the teachers with the whole class and the learning process of individual students. Addressing this relationship will help us to understand the tension, so prominent in inquiry learning, between teacher’s guidance and students’ knowledge construction. Analysing the case study I shall discuss questions such as: How does collective reasoning in the whole classroom influence students’ learning? What is the

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contribution of individual children to the class discussion? What role does the teacher play in all this? 2. L EARNING IN A COMMUNITY OF INQUIRY

Leen Streefland could talk enthusiastically about the mathematical discoveries of his 11 to 12-year-old students. He stimulated them to construct mathematical knowledge, for which he gave them space, rather than asking them to keep to the guidelines set out by a textbook. He wanted his students to engage in mathematical thinking, and this would not be consistent with the performance of mechanical calculations or mindless acceptance of teachers’ instructions. To achieve this aim, Streefland sought to build a participation structure in the classroom (cf. Lampert, 1990) in which students could work and collaborate according to rules different from those of a conventional classroom. By making children responsible for their own learning, he established new norms for classroom life. By telling the students that they could do much better than they thought, if they were to try to do research, he gave them confidence. By transforming the class into a working community, he prompted them to benefit as much as possible from each others’ ideas and productions. He wanted children to learn what mathematics is about and that mathematical understanding is both practical and satisfying as an intellectual enterprise. The motivation to set up the innovative lessons originated from the viewpoint of Realistic Mathematics Education (Freudenthal, 1991; Goffree, 1993; Streefland, 1993; Van den Heuvel-Panhuizen, 2001). The activities designed by the teacher should stimulate students to engage in mathematical thinking and discussion. Streefland organized his lessons around problems which were topical and meaningful for the students, often with a reference to some experience of the students, such as, in one case, the relocation of their school to a different building (Elbers and Streefland, 2000a). The introduction of these problems made it possible for students to formulate their own questions, to answer them, to produce mathematical arguments and to validate their assertions. The classroom activities also included comparing the efficiency of different approaches and reflections on the advantage of one solution over others. Mathematics, taught and learned in this way, loses its character as mere school knowledge, because it helps students to connect everyday knowledge and mathematics and to enrich common sense with mathematical understanding. Three ideas of mathematics education form the basis of these lessons (for a more complete introduction to the principles underlying realistic mathematics education, see Van den Heuvel-Panhuizen, 2001). Firstly, the

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problem given to the students plays a central role: it serves as a source for the class and students to develop mathematical understanding. For working within the domain of the problem the students can use both their understanding of everyday-life situations and the mathematical means already available to them. Secondly, a basic element in realistic mathematics education is to motivate students to ‘mathematize’, that is, to turn everyday issues into mathematical problems and use the mathematics resulting from these activities to address other realistic problems. This task of mathematization also encompasses the construction and manipulation of symbols. Thirdly, part of the students’ task is to develop good arguments to support their solutions. That means that the students should not depend on the teacher to find out whether their ideas are correct but try to show the validity of their approach. Streefland applied these principles to transform the work in the mathematics classroom. He created a learning environment in which the students’ construction of mathematical understanding could occur. Our research (Elbers and Streefland, 2000a, b) showed that students expressed their real interest and were motivated to work on problems. They engaged in mathematical discussions rather than applying algorithms and textbook rules. Encouraging students to work like this helped them to find out how mathematical problems are to be dealt with, and how to formulate a mathematical problem in the first place. They learned to produce arguments, as well as what kind of arguments are valid within a mathematical discourse. They applied the mathematical knowledge that they had acquired in the past when and because they needed it, and this knowledge proved its worth beyond the demands of a textbook. In brief, students learned both mathematical content: knowledge of mathematics, and mathematical practice: knowledge about mathematics (cf. Lampert, 1990). Streefland’s lessons had a basic format (Elbers and Streefland, 2000a, b). They started with a statement about the principle of a community of inquiry: “We are researchers, let us do research!” The students were then given a topic or problem as the subject of their research (often with reference to some example from everyday life, a newspaper clipping, a photograph, etc.). These topics were sufficiently open and general that the students were required to take on the job of asking more specific questions and turning them into mathematical problems. After the introduction of the topic, the students were invited to formulate research questions and develop answers to these questions. Work in small groups of 4 or 5 students, and sometimes individual work, alternated with class discussions in which the results were made available for discussion in the whole class. Often the groups or the individual children had to write down their solutions.


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Leen Streefland and Rob Gertsen usually worked as a team. They introduced themselves not as teachers, but as senior researchers. This role allowed them to participate in the discussion themselves. Sometimes, in a provocative way, they brought forward or defended different and incompatible hypotheses. More often, one of the them took on the role of manager of the discussion, whereas the other contributed by bringing forward suggestions, more or less on an equal footing with the students. The students knew, of course, that they could expect guidance from their teachers. However, by acting as they did, the teachers gave the students responsibility for their work and made it clear that the validation of their solutions comes from mathematical argumentation and not from the authority of the teacher. Addressing students as researchers and encouraging them to take part in a community of inquiry changes work in the classroom. It asks of students to make their thoughts and solutions explicit. Students have to communicate their ideas and make them ‘visible’ by putting them into words (cf. Mercer, 1995). Verbalizing as a part of problem solving often leads to better understanding (cf. Van Boxtel, Van der Linden and Kanselaar, 2000). Moreover, students have to account for their solutions to their fellow students. The interaction in a community of inquiry amounts to creating a forum for comparing different perspectives. Students know that they are supposed to argue for their ideas and, challenged by others’ comments, to elaborate or improve them (cf. Wells, 1993). The ‘research’ in the classroom leads to the presentation of various ideas and solutions that are discussed and investigated for their plausibility. Participants assimilate insights expressed by others and benefit from the various viewpoints and knowledge brought forward in the discussions both in the small groups and in the classroom as a whole. In previous studies, we showed that children, collaborating as members of a community of inquiry, are motivated to help each other and to learn from each other (Elbers and Streefland, 2000a, b). We observed a lot of identity talk: students began to view themselves in different roles and speak about themselves in different ways. They talked about their new identities and responsibilities as members of a community of inquiry. They criticized each other when somebody fell back into the habits of conventional lessons. They also discussed the role of the teachers and their relationship to them. The students put competition and claims of authorship into perspective. Against these, they emphasized that they should work as a community and that it is the idea that matters, not who came up with it in the first place. In creating awareness of their new identities as researchers, students also invented collaborative and discursive tools for doing their

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work as members of a research community. They started to use new words and expressions which accorded with their joint inquiry. Moreover, their discussions often had a cyclic form in which students proposed ideas, repeated them, explored and evaluated them. Ideas introduced by some students were adopted and expanded by others. These cyclic processes allowed many students to participate in the discussions, and to appropriate what other students had already discovered (Elbers and Streefland, 2000b). In the next section I will present the events of one particular lesson prepared by Leen Streefland and taught by him together with Rob Gertsen. In this lesson, there was an alternation of collective argumentation in the whole class and individual and group work. In presenting this lesson I will address the problem of the relationship between class discussion and individual learning. Because of the alternation of whole class work and individual work, this case presents an excellent opportunity to study the influence of the collective parts of the lesson on individual learning, and vice versa. Learning, according to Rogoff (1994), is the changing participation of students in the classroom discourse. The issue of the next section is then, more precisely, the changes of students’ mathematically significant participation in a community of inquiry. These changes involve the creation and elaboration of new insights, but also the circulation and appropriation of these insights. In studying these changes, I want to contribute to the body of research on the genesis of students’ understanding in the course of one lesson (e.g., Lampert, 1990; Forman, Larreamendy-Joerns, Stein and Brown, 1998; Gravemeijer, Cobb, Bowers and Whitenack, 2000). For that purpose, I will analyse the verbal interaction between teacher and students and the written work produced by students. By comparing individual work with the discussion in the whole classroom we get a view of the progress of mathematical understanding both in the classroom as a whole and in individual children’s minds. 3. A CASE STUDY OF LEARNING IN A COMMUNITY OF INQUIRY

The case study concerns a lesson in a combined seventh and eighth grade classroom at a primary school in the Netherlands with 28 children between 11 and 13 years of age. The lesson was part of a short series of lessons in June, at the end of the school term, when the students had already completed their regular mathematics curriculum for that year. As in most lessons taught by Leen Streefland and Rob Gertsen this lesson consisted of alternate classroom discussions and student work. At the start of the lesson, the teachers told the students that they wanted them to do research.


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Figure 1. The problem in its first shape (P1).

An activity sheet was used with the problem and its variations printed on it, leaving space for the students to write down their own solutions. Although the students had to work on their own solutions, they had ample time to discuss their thoughts with each other. My presentation is based on an analysis of the videorecording of this lesson and a transcript. Moreover, some time before his death, Leen and I watched and discussed this recording. Later on I corrected the transcript. The completed worksheets were mislaid and could not be recovered, but I could borrow information from a short Dutch article Leen Streefland wrote about this lesson (Streefland, 1997)2 . For the sake of clarity, the case is divided into a number of Episodes. The changes from one Episode to the next correspond with the transition from class discussion to individual work, and back to class discussion etc. The lesson started with an introduction of the problem and a class discussion, which I call Episode 1. Odd numbers of the Episodes refer to class discussions, and even numbers to student work. The problem introduced in the classroom is set in a pharmacy and involved calculating the number of tablets prescribed by a physician (see Figure 1). This problem and its variations formed the topic for the students’ work. An overview of the variants of the problem (which I have indicated with P1, P2, P3, P4 and P5) and the solutions (which I have called S1, S2, S3 and S4) is presented in Figure 3. The students, first, had to solve the original version of the problem (P1, starting with 6 tablets), then they had to calculate the number of tablets in a prescription starting with 8 tablets (variant P2: 8 tablets for the first two days, 7 for the next 2 days, etc.) and next the number of tablets in a prescription starting with 10 tablets (variant P3: 10 tablets for the first two days, 9 for the next two days, etc.). The

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variants P4 and P5 left the students a choice as to the number of tablets to start with. To prepare their work the teachers and the students talked about pharmacies. The teachers made sure that the children understood that many pharmaceutical products can be obtained by prescription only. This introductory discussion helped the students to grasp the context of the problem and motivated them to work on it (Streefland, 1997). The activity sheets were then distributed among the students. Episode 1. In a whole class interaction, Gertsen read the first version of the problem aloud and discussed the prescription with the class. Several students volunteered to give the solution to the problem. First, Chris came up with a solution and Gertsen, following the student’s answer, wrote it on the blackboard. Chris solved the problem by multiplying the various numbers of tablets by 2 (2×6; 2×5; etc.) and, next, adding up the results (12 + 10 + etc.; cf. solution S1a in Figure 3). The teachers commented on this solution: Sequence 1. Streefland: Gertsen:

It is right, but I think that it is a laborious way. Does anyone know a more practical way? This is a good way, but it is also a detour. Does anybody have a shorter way, a more convenient way?

In response to the teachers’ comments, Sandra showed that the numbers in the addition can be added up more easily by making combinations of ten and twenty (12 + 8 = 20; 6 + 4 = 10; the addition is then: 20 + 10 + 10 + 2 = 42; cf. solution S1b in Figure 3). Episode 2. After this whole class discussion, the students were asked to write down their solutions to the problem on their activity sheets. During this episode of individual work, the teachers walked around in the classroom and looked at the students’ work. Afterwards, an inspection of answers on the activity sheets showed that all students but two used solution S1a (see for example Alexandra’s solution, Figure 2): they multiplied each number before adding them up. Two children invented a new solution strategy by, first, adding up 6, 5, 4, 3, 2 and 1 tablets and, then, multiplying the sum by 2. One student showed both approaches on his activity sheet. Episode 3. After some time, the teachers called the students’ attention for a further collective discussion of the problem. They asked Yvette, who had applied solution S2a on her worksheet, to show her approach to her classmates. While she explained her solution, Rob Gertsen wrote it on the


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Figure 2. Alexandra’s solution to P1. “The prescription is not right: 2 tablets are missing”.

blackboard. Yvette claimed that her strategy is more practical because it is not necessary to multiply every number by 2. The teachers complimented her for her fine solution, but Streefland added that Yvette’s solution can be improved and that there is a cleverer way to do the calculation. The teachers then introduced the second version of the problem, a prescription beginning with 8 tablets (variant P2). Episode 4. Here, the children were asked to work on the problem starting with 8 tablets and write their answer on their activity sheets. The written work shows that 19 students were influenced by the discussion in Episode 3 and postponed the multiplication (i.e. they used solution S2). An example of this is Alexandra’s sheet (Figure 4). Alexandra made combinations of 10 in order to add up easily; she multiplied the sum by 2. The right side of Alexandra’s work shows that she also calculated the answer by making recourse to her strategy for the previous problem. Many children who applied solution S2 made combinations of 10. Two children invented a new strategy and used the solution of the previous problem as a part of the solution to P2 (see Tommy’s solution in Figure 5). Episode 5. In the following class discussion of P2, the teachers made sure that the new solution, which is in fact solution S3, was presented in front of the whole class. One of the children showed her strategy with the basic form: use the previous answer (the answer to P1) for solving the new problem P2 (starting with 8 tablets): 42 + 2×7 + 2×8. The teachers then asked who had applied a different approach. One of the children

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Figure 3. Overviews of the five variants of the problem and the four solutions discussed during the lesson.

showed that, in order to add up quickly, she had made combinations of tens (8+2=10, 7+3=10, etc.): “I took out the tens”. In the ensuing discussion on this approach, Streefland asked: “What shall we call this approach? Can we invent a name for it?” The children proposed expressions such as “making tens”, “jumping to tens” and “bridges of ten”. Leen Streefland concluded this discussion by suggesting they call this method: “making combinations” and “making clever combinations”.


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Figure 4. Alexandra’s solution to P2.

Figure 5. Tommy’s solution to P2.

The third variant of the problem (P3) was then introduced: the prescription starting with 10 tablets. But here, before the students started their work on the worksheets, the teachers presented a new challenge to the students: Sequence 2. Gertsen:

Streefland:

You can solve the next problem [P3]. (. . .) You just try to solve it: if I start with 10 tablets, how many do I need? The children who have found the answer quickly, then think about this: can I find the answer without doing the sum? I’ll give you a hint: look at, compare the three numbers. When I start with 6. . ., when I start with 8. . ., when I start with 12. . . When you compare, you may come to a conclusion, a discovery. After all, you are researchers, aren’t you? I would like to add that it may be fun to try out different combinations. Try to experiment with combinations. Maybe you’ll discover something surprising. If you discover that, it is a piece of cake.

Episode 6. The children worked on the problem (P3) for some considerable time. There was much interaction between the children while they compared and discussed their solutions. The teachers walked around and occasionally talked with the children about their work. The worksheets show that 16 students used the previous result to find the solution to problem P3 (solution S3). Let’s take Alexandra’s work as an example again. She added 10×2 and 9×2 to 72 (72 is the result of P2). During this episode,

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some students found a novel solution (solution S4) to the problem. The invention of solution S4 was the result of a discussion among a group of four boys. Episode 7. In the class discussion of P3, this new solution (solution S4) was presented by one of the these students in the following way: Sequence 3. Gertsen: Tom:

(. . .) Streefland: Gertsen: Streefland: Gertsen:

Streefland: Gertsen:

Researcher Tom, show your solution on the blackboard. If you start with 42, it is 6×7. If you look at 72, that is 9×8. Beginning with 10, that is 10×11. (He writes these numbers on the blackboard). Be consistent: 6×7, 8×9, 10×11. Put the smaller number first. It is very nice to do it this way. I can tell what the next one is, too, because, look, (pointing at the numbers on the blackboard) here is 6, 8, 10, and the next one should be: 12×13. (. . .) That is very good, but I think that he should show it by writing it out in full. Because it does not appear out of the blue, of course! (addressing Tom) Show it, prove it.

With some help Tom succeeded in showing on the blackboard that 6+5+4+3+2+1 = 3×7 (6+1, 5+2, etc.) and since each number of tablets should be taken for two days, the results should be multiplied by 2: 6×7. Gertsen, commenting on Tom’s contribution, told the class that Tom had made combinations of 7. Next, Gertsen demonstrated this way of calculating for P2: the prescription starting with 8 tablets can be solved by making combinations of 9. With many students participating, the teachers demonstrated the result of the problem starting with 12 and with 14 tablets. Episode 8. After this collective discussion, students started individual work on P4: they had to choose a number to start with themselves. The worksheets show that 6 children applied Tom’s approach, using the multiplicative abbreviation. Alexandra, for instance, began the prescription with 16 tablets; she wrote directly as her solution: 16×17 tablets. Some students chose an odd number to start the calculation with. These students faced an additional difficulty. Making combinations in a prescription beginning with 11 tablets, they found 5 1/2 combinations of 12. Some of the children were able to solve this difficulty.


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Figure 6. Sawina’s solution to P5.

During the same Episode, some students started working on the last problem: they had to write down on their sheet their most practical strategy (P5). The task was explained by one of the teachers: “You have to write down what you understand, so that you can explain your solution to us.” There was insufficient time for all children to answer this last problem. Sawina’s answer to P5 is reproduced in Figure 6.

4. A NALYSIS OF THE CASE

For clarifying what happened during this lesson, I will analyse three aspects: the prescription task, the teachers’ strategies for involving the students in the class activities and the influence of the classroom interaction on students’ learning. The task provides ample opportunities for students’ work. It challenged them to engage in a series of mathematical reflections which amount to a process of mathematizing. Treffers (1986) and Freudenthal (1991) distinguished between two forms of mathematizing: horizontal and vertical. Turning a real life problem into a mathematical form is called horizontal mathematizing; manipulating and reshaping mathematical symbols is vertical mathematizing. In other lessons, Streefland introduced tasks which brought particular difficulties of horizontal mathematizing, such as estimating the height of a tower (Elbers and Streefland, 2000a, b). In the present lesson, he used a task that would not be unduly difficult for the children to solve. The challenge in this lesson was rather for the students to discover that the prescription task could be approached in several ways and that some solutions are more sophisticated and efficient than others. The job was to reflect on the variety of possible solutions and to find clever mathematical expressions for them. We saw that children were not only able to create these various solutions and find increasingly more efficient

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solutions, but we also witnessed the sheer pleasure of the children and their eagerness to make and show their mathematical inventions. Because of the students’ contributions to the class’s work, the teachers faced a problem originating from a double role. On the one hand, they were in charge and responsible for the students’ activities. They decided what topics would be worked on and they had their own ideas of what knowledge students should acquire during the lessons. On the other hand, the teachers wanted the students to find out for themselves: to invent solutions to problems and to prove their validity. They did not want to frustrate children’s creativity by using their authority for supporting certain answers instead of others. To solve the problem originating from this double role and to channel the discussion in certain directions, the teachers used three strategies. Firstly, they stimulated variation in solutions. Students trying to discover a different solution from one already found were rewarded with compliments and enthusiasm by the teachers. Even after having worked out a correct solution, there was no reason for students to stop, since there was always another and more efficient solution to be found. Obviously the teachers’ strategy was to tell students not to be satisfied with one solution but to realize that the problem could be approached in various ways. By doing so, they were laying the foundation for the development of students’ answers in the direction of a solution with combinations (solution S4). An example is in Episode 1, when Streefland comments on one student’s solution: “It is right but I think that it is a laborious way.” Secondly, the teachers made suggestions, although they were global and general suggestions. Episode 5 shows, for instance, that the teachers suggested that a more convenient solution can be found by comparing the results of the various problems P1, P2 and P3. Another example is the discussion about the name to give to the procedure of taking certain numbers together in order to make easy additions. The children, who at this stage had combined numbers in order to make tens, proposed: taking out tens etc., but Streefland taught them to use the term: making combinations. In this way he paved the way for making combinations that add up to numbers other than ten. Streefland made use of the collective understanding at a particular moment in the classroom for bringing up suggestions which the students could use to transform their understanding and invent new solutions. Thirdly, the teachers selected the students who were asked to give a presentation in front of the whole class. During the parts of the lesson in which students worked on their activity sheets, the teachers walked around and sometimes asked individual students to show and explain their work.


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TABLE I Numbers of students who invented new solutions in comparison to their previous answers. Data borrowed from Streefland (1997) Episode 4: Episode 6: Episode 8:

19 17 17

During these episodes, the teachers observed what solutions the students were trying to work out and they were able to single out students with novel solutions rather than familiar ones to present their work to the whole class. This happened in all episodes following student work (Episodes 3, 5 and 7). An instance of this is in Episode 7, where Rob Gertsen gives the floor to ‘researcher’ Tom. Using these three strategies the teachers could direct the discussion, and at the same time value the children’s own constructions. The teachers’ educational approach was primarily aimed at directing the discussion by exploiting the best solutions the students had invented. In all Episodes with class discussions the teachers made sure that a novel strategy for solving the problem was presented (see the discussion of Episodes 3, 5 an 7 above). Streefland and Gertsen were confident that the presentation and discussion of the class’s best solutions would impress and influence all or most children’s ‘research’ and understanding. One way of showing the success of the lesson is to look at the number of students changing their answers from one episode to the next. The teachers urged the students to use practical and smart solutions, but did the students actually find increasingly smart approaches to the various versions of the problem? The activity sheets demonstrate that the majority of students indeed did so. Table I shows that the majority of students, from one episode of individual work to the next, abandoned their previous strategy and applied a new approach that was more practical in terms of the four solution strategies of Figure 3. They acted in line with the teachers’ encouragement to continue finding other, more efficient and clever solutions. Another way of demonstrating that the teacher’s approach led the children to mathematizing activities is in looking at the influence of the solutions discussed in the whole class on the individual work of children immediately following the previous class discussion. An analysis of the activity sheets shows that many students followed the line set out in the class

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TABLE II Numbers of students adjusting their solutions (as shown on their activity sheets) to the previous class discussion. Data borrowed from Streefland (1997) Episode 2: Episode 4: Episode 6: Episode 8:

26 19 16 6

discussion: they answered a problem by adopting the solution discussed just before in the whole class (Table II).

5. G ENERAL DISCUSSION

The choice and arrangement of learning activities made the classroom into a forum for exploring the merits and validity of various solutions to the problem. This process of exploration was guided by the conventions of a community of learners in which ideas matter, not who wins an argument or raises an idea for the first time. Classroom interaction was valuable and fruitful because it required the participants to seriously consider many ideas. Challenged by their teachers or their schoolmates, students had to argue and account for their ideas. They asked questions, made comments and brought up arguments in favour of or against particular points of view. In short, interaction in this classroom was a collective form of reflection. This process of reflection is not constrained to discussing the content and the plausibility of solutions to problems, but also goes further. The collective reflection occurring in inquiry classrooms bears upon two other aspects. Firstly, it challenges students to think about the tools of argumentation they were using. They learn not only to reason by talk but also to talk about their reasoning. One observation often made in studies of communal learning is the abundance of phrases that show metacognitive awareness, such as “I think, that . . .”, “my hypothesis is . . .”, “I want to comment on Claire’s solution” (Lampert, 1990; Wells, 1999; Elbers and Streefland, 2000a). Secondly, working as members of a community of inquiry leads students to an awareness that it is not sufficient to have a good answer or the right solution. Students learn that they also have to be convincing (cf. Forman, Larreamendy-Joerns, Stein and Brown, 1998). It follows from


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their investigating activities that they have to find ways of supporting their claims in order to see their solutions accepted by others. In Elbers and Streefland (2000b) we gave an example of how children gradually persuaded others into accepting their point of view. They succeeded because they cleverly made use of knowledge shared in the classroom, rather than formulating their arguments without appealing to common knowledge. These two aspects, the presence of metacognitive expressions and the need to be convincing, contrast the inquiry classroom with the modes of verbal interaction observed in more conventional lessons, such as those described by Mehan (1979) and Edwards and Mercer (1987). In the inquiry class, students, from a mathematical point of view, are introduced into the mathematical discourse (Lampert, 1990). They learn what is important in mathematics, what it means to mathematize, and they develop an initial understanding of how mathematical hypotheses are built and what arguments are appropriate in this context. Students are initiated into the tools and conventions of mathematical reasoning in a process of guided reinvention. Moreover, they have to come to terms with the fact that the character of knowledge changes under circumstances of a community of inquiry, since the aim of the lessons was not the individual reproduction of knowledge, but the joint construction of knowledge by listening to others, presenting convincing arguments and looking at a problem from various points of view. Working as ‘researchers’, students begin to understand that knowledge is not something fixed or ready-made, but that it always needs further elaboration and is dependent on the perspective taken. Inagaki et al. (1998) view learning through whole class discussions as taking place at two levels that are organized hierarchically. Their analysis assumes that “through interactions, some discussed and negotiated meanings and understandings are first constructed collectively, and then participants incorporate this ‘information’ individually for generating, elaborating, and revising their comprehension.” (p. 524). At first sight, this theory would seem to apply here, since the case study showed that the majority of students adjusted their solutions on the worksheets to the previous collective argumentation in the whole classroom. However, on closer inspection, Inagaki et al.’s account is not sufficiently precise, because it is based on too schematic a theory of internalization (cf. Elbers, 1996). Inagaki et al. assume a clear division between collective and individual argumentation, but it is not so easy to tell where the collective work ends and individual learning begins. Collective argumentation and individual work, in the classroom of the present case study, took place within a discursive structure formed by social and sociomathematical norms (Yackel and Cobb, 1996). The students and teachers used rules such as: find out for

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yourself, present it understandably, invent a solution that is as economical as possible, make combinations. These rules structured the students’ work, both in the whole class discussions and in their work on their activity sheets. Therefore, students’ learning is best understood by referring to this discursive structure. In their individual work, students apply the same kind of arguments that they have to use in the class discussions. The individual work is to be considered as an anticipation of a class discussion, or, following a class discussion, as a reconstruction of it. Given this discursive structure, there is no priority for individual or collective work: they are two sides of the same coin. Students’ creation of novel solutions can illustrate this. These solutions were invented by students alone or with one or two classmates in a small group, before they were discussed in the whole classroom. Students developed these answers as a preparation and anticipation of the discussion in the whole classroom. The same applies to students making use of the results of class discussions for writing an answer on their sheets. Students did not just internalize or incorporate the results of the class discussion, they had to reconstruct them (cf. Elbers, Maier, Hoekstra and Hoogsteder, 1992). Even the use of the outcomes of class discussions for writing an answer on their sheets was not a reproduction, but demanded creativity. We get a glimpse of the intellectual effort underlying these reconstructions, by looking at Sawina’s answer to P5, when students were asked to write down their most practical solution. She chose almost the simplest variant of the problem to show a general point (Figure 6). The variant of the problem starting with 4 tablets had not been discussed with the whole class. Streefland’s experiment, however original, is not an isolated enterprise. It is part of a wave of innovations in mathematics teaching and in education in general. These innovations are based on the understanding that children’s learning is not so much an individual, but a communal activity (cf. Cobb, Gravemeijer, Yackel, McClain and Whitenack, 1997). Addressing students as researchers who form a community of inquiry, in this and other innovative projects, creates “new cultures of schooling” (Scardamalia and Bereiter, 1997), in which understanding, rather than the reproduction of knowledge, arises as the main focus. Educational projects creating learning communities in the classroom, such as those reported on by Rogoff (1994), Brown and Campione (1994), Scardamalia and Bereiter (1997) and Brown and Renshaw (2000), have met with success and shown that students are able to take upon themselves co-responsibility for the events in the classroom. They make a considerable number of spontaneous contributions that are not responses to questions by the teacher, but are initiatives in an ongoing dialogue. The teacher provides support and guidance without


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controlling all interactions during the lessons. The instructional discourse in a community of inquiry is, as Rogoff (1994) called it, “conversational” rather than based on the usual format of questions put by the teacher and answers given by the students. Working in a community of learners was designed in various ways in the experiments referred to in the previous paragraph. There is, however, a common finding in all projects that build on students’ inventions and collaboration: children need guidance in order to work according to the ground rules of this form of learning (Mercer, 1995; Joiner, Littleton, Faulkner and Miell, 2000). One example of such guidance is provided by Brown and Renshaw (2000), who taught students to work on mathematical problems according to a ‘key word format’ of collective argumentation. The students were encouraged to organize their work in phases corresponding with the terms in the key word format: “represent”, “compare”, “explain”, “justify”, “agree” and "validate”. First, students had to try to work out (represent) the task alone, then compare their answer with other students in small groups, explain and justify their solutions to each other, and, if possible, reach agreement. The validation phase refers to the discussion in the whole class, where groups presented their ideas for acceptance by their peers and the teacher. Lampert (1990), too, described how she helped students to use appropriate expressions. For instance, she taught children what terms to use for presenting an argument or for questioning somebody’s hypothesis (see the analyses by Blunk, 1998, Rittenhouse, 1998, and Weingrad, 1998.) Lampert taught students to associate terms like “know”, “think”, “explain” with the mathematical activities they engaged in. By stimulating the students to use these words, she also demonstrated them their new roles and responsibilities. (For other examples of scaffolding activities in the context of collaborative learning, see: Bennett and Dunne, 1991; Rogoff, 1994; Brown and Campione, 1994; Mercer, 1995; Rojas-Drummond, Hernández, Vélez and Villigrán, 1998). Streefland had another way of helping students to become researchers. Rather than teaching a word format or particular expressions, Streefland introduced the metaphor of a research community. This metaphor proved very useful for the teachers to structure the children’s activities and it created a solution to the ‘didactic tension’ (Jaworski, 1994) between teachers’ objectives and students’ own initiatives and constructions. By referring to a research community, the teachers could help the students articulate their roles. Each lesson started with the instruction: “From now on we are researchers”. In fact, emphasizing the students’ identities as researchers was repeated several times during the lesson (see, for instance, “Researcher

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Tom, show your solution on the blackboard” in Sequence 3.) Referring to this metaphor, the teachers could explain that researchers try to solve problems, that they have to justify and compare their solutions and discuss them. The metaphor also led to discussions among the students about what it means to be researchers and to arguments about the roles and responsibilities of the students and the teachers (see Elbers and Streefland, 2000a). Emphasizing the idea of a community of inquiry was Streefland’s way of bringing the rules regulating interaction in the classroom close to the standards for argument in the scientific, and in particular mathematical, research communities (Van Amerom, 2001).

6. C ONCLUSION

By addressing the students as researchers, and giving them realistic problems to work on, Streefland’s aim was to engage all students in the process of collective thinking. He was convinced that students benefit from participating in social interaction and collective knowledge building activities, and that even the least gifted students learn when they feel free to express their ideas and are invited to discuss them with their classmates. For Streefland the classroom was “a milieu that is child centred but in which the teacher functions as a guide, creating structure with the help of the students themselves” (Halliday, quoted by Wells, 1999, p. 26). The last part of this quotation was vital in Leen Streefland’s thinking about education. In a passage written by him in a Dutch article of 1995, he commented on educational theorists who want to change learning by placing the responsibility in the learners’ hands. For him, this was insufficient. Giving responsibility to students is certainly a step forward. But the important thing is to achieve a more radical change of perspective: students should take responsibility upon themselves. They should feel responsible for their own and their classmates’ learning (Streefland and Elbers, 1995). The case study showed how students in an atmosphere of collaboration and interaction contributed to their learning, and how the teachers exploited the various productions and constructions by the children to structure the learning process.

N OTES 1. Comments by Koeno Gravemeijer and Marja Van den Heuvel-Panhuizen on an earlier version are gratefully acknowledged. 2. My description of the case was based on the video recording, the transcript and the article by Streefland (1997). I also wanted to provide information on the activity sheets,


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but they have been lost. When in my description of the case I give information on the students’ written work, I borrowed this from Streefland’s (1997) article, which contains data on the students’ written work. The pictures of students’ work have also been taken from this article.

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