contour lines between a model as a theoretical

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Rachel and Noam didn't yet construct the meaning of events as pairs of numbers (E1), and they don't .... education. In L. Streefland (Ed.), Proceedings of the 9 th.
CONTOUR LINES BETWEEN A MODEL AS A THEORETICAL FRAMEWORK AND THE SAME MODEL AS METHODOLOGICAL TOOL Rina Hershkowitz Weizmann Institute of Science, Rehovot, Israel Research today is quite flexible concerning the methodologies in use. It is very common for researchers to use a combination of research methodologies and for this combination to change from one piece of research to another depending upon their research goals, research interest and the theoretical framework. In addition when a theoretical research model is inferred from empirical data, this hypothesized model serves as a framework to theoretical background and also methodology. In this paper I discuss this issue and related issues via a presentation of two collections of papers; each on a different mental activity concerning learning and knowledge constructing. The second example is then elaborated through classroom research on shared knowledge construction. The definition of shared knowledge in this discussion is quite demanding, so the limitations of methodological tools became very clear. OPENING The flow of theoretical and methodological paradigms which determine the frames for research work in science and mathematics learning has become rich and more and more sophisticated. However, it seems that more than in the past, researchers today do not feel obliged to and/or satisfied with sticking to one methodological paradigm or the other. Research trends in our area are nowadays characterized by flexibility and creativity in combining together research methods and methodological tools, which fit the researcher's theoretical framework and meet the researcher’s goals and needs to explain and answer some "big questions" emerging from his/her explorations. In this paper I would like first to discuss issues concerning the contour lines (boundaries) between the theoretical framework and the methods and methodological tools within the same research. I argue that in more and more research works these boundaries are flexible and even a bit vague in the sense that the same scheme or model may serve as a theoretical framework in one piece of research, as a methodological tool in a second one, and as both of them in a third piece of research. I will discuss these issues via two examples, which illustrate dynamic relationships between theory and methodology in two different research domains of mathematics learning. In each example, the questions will emerge from analyses of the above relationships in a few research papers concerning a particular topic, looking at differences and similarities between theoretical and methodological frameworks. The first example is taken from research on argumentation in mathematics learning, and the second example is taken from research concerning the RBC + C model for abstracting in context. In the section which follows the two examples, I will discuss potential directions in classroom research in which the intention is to trace abstraction processes, and the need to create some new combinations of methodologies and/or new methodologies.

EXAMPLE 1: CLASSROOM CONCERNING ARGUMENTATION

RESEARCH

IN

MATHEMATICS

A meaningful contribution and trigger to research in mathematics education, which investigates mathematics learning in classrooms via two lenses, in parallel: the psychological and the socio-cultural, is the book edited by Cobb & Bauersfeld (1995). It is a cumulative effort of a few researchers in this area of research, who interweave psychological and socio-cultural aspects, and cooperate to explore many aspects of learning mathematics, with the intention to create a theoretical frame for classroom research in mathematics. They investigate their different perspectives in the same context (project), using the same video tapes as sources for their corpus of data. Krummheuer's chapter (1995) in this book, deals in what he calls "the ethnography of argumentation" where his theoretical interest is in the social genesis of argumentation. Hence two crucial goals of his research are: To develop a theoretical framework of argumentation in mathematics, and to incorporate it into a social context. He is interested in "collective argumentation" where the argumentation process is constructed by two or more individuals in the classroom. In this sense Krummheuer considers argumentation as a "social phenomenon; when the cooperating individuals tried to adjust their intentions and interpretations by verbally presenting the rationale of their actions" (Krummheuer, 1995, p. 229). He bases his study of argumentation mainly on Toulmin's argumentation model (scheme) (1969) and shows that this theoretical model is valid and applicable to mathematics classrooms In a sequence of episodes with various students from mathematics classrooms he shows step after step that the elements of Toulmin's model: a claim (a conclusion) and data, warrant and backing, which support, justify and explain the claim, are appropriate theoretical elements for mathematical argumentation and have internal relationships which together present a construct of argumentation and "collective argumentation" in mathematics. The above model elements, in a certain collective argument, might be given by different students in the class and the teacher as well. Does this model serve as a theoretical framework for Krummheuer’s work? As a methodological tool? As both of them? It seems that for Krummheuer, Toulmin’s model is a core for both, theoretical framework and methodological tool. His intention is to create an appropriate theoretical frame for argumentation in mathematics classrooms. But, in order to build it step by step through analyzing the classroom episodes and to show how various elements of the model are taking place in this context, he does not have any choice but to use the model itself first, in order to interpret and explain the argumentation which emerges in the episodes, and doing this gives evidence that this model is appropriate as a theoretical framework for mathematical argumentation. Yackel used Krummheuer work, while emphasizing the aspect of the model as a methodological tool, in investigating the role of teacher in the classroom (Yackel, 2002). Yackel wrote: "…this approach to argumentation is useful as a methodological tool for documenting the collective learning of a class because it provides a way to demonstrate changes that take place over time." (p. 424). She used it as a methodological tool in a research for the purpose of investigating the dynamic role of the teacher in encouraging mathematical argumentation in the classroom. Because her

research aim was not the investigation of the argumentation process itself, she needed and could use a less refined version of the Toulmin /Krummheuer scheme. (She mostly did not differentiate between the warrant and backing). Can we conclude that this supports an hypothesis that when the "model" serves mostly as a methodological tool for classroom research, where composite interactive processes are taking place, and where the research's aim is not constructing, or elaborating, or confirming the model any more, a simpler dynamic tool is preferable to a very detailed one, and the research tool in use becomes less refined? A complementary finding arises from the paper of Whitenack and Knipping (2002). They use Toulmin's model and Krummheuer's work to learn more deeply how the emerging of collective argumentation is "socially accomplished". In contrast to Yackel research, the (collective) argumentation and argument themselves became again the focus of research and therefore their evolution has to be presented and analyzed in a very refined way. In their microanalysis of classroom work they showed how different students, one after the other, add an additional warrant to support the same conclusion and even the teacher contributes her warrant to the same collective construct. Whether the Toulmin/Krummheuer model is used as a theoretical framework or as a methodological tool, some global methodologies were seen to be used in the research works in addition to the model (e.g., protocol documentation and analyses). The model, at any level of refinement, is used as the lenses through which the data are described and analysed. This analysis suggests that when it is mostly a theoretical framework a process of refinement and confirmation of the theory is taking place. When the research goal is beyond the model itself, the model is used in a more global, unrefined way. EXAMPLE 2: THE RBC+C MODEL FOR ABSTRACTING IN CONTEXT - A THEORETICAL FRAMEWORK AND METHODOLOGY In this example I will elaborate the issue of the flexible nature of the contour lines between a model as a theoretical framework and the same model as methodological framework and/or methodological tool. I will again focus only on a few research papers in which the RBC+C serves either as theoretical framework or as a methodological tool or both. Five years of research and more than 30 research publications, contributed by more than a few people, separate the "birth" of the RBC model as an empirically-based theoretical framework (Hershkowitz, Schwarz and Dreyfus, (HSD), 2001, and Dreyfus, Hershkowitz and Schwarz, (DHS), 2001) from a recent publication, which uses this model as one of two "conceptual frameworks" (Wood et al., 2006), which will be discussed latter. At a first stage, the model was hypothesized out of analyzing piles of data, the three researchers had accumulated. The researchers were led by the need to express theoretically the uniqueness of the findings of these data concerning mathematical thinking and knowledge constructing. In this theorizing process, theoretical contributions of existing theories were considered and examined as well. Abstracting in context was taken as human activity of mathematization, mainly "vertical mathematization" (Treffers and Goffree, 1985). Vertical mathematization represents the process of constructing a new construct in mathematics by learner/s

within the mathematics itself and by mathematical means; like reorganization of previous mathematical constructs, interweaving them, into one process of mathematical thinking, with the purpose of constructing (vertically mathematizing) a new mathematical construct. Abstracting in context has emerged and was described at the first stage, by means of illustrative examples in different contexts, which differ by their mathematical content, social setting and research setting (problem solving by individuals, and dyads' actions in problem solving). For the detailed description and theoretical analysis of this model, see Dreyfus (this workshop). At that stage, the RBC researchers found themselves in a situation, where theory is constructed from analyzing data and the analyzed data themselves serve as evidence for the theory validation. They were quite aware of this situation and explained: "This definition (of abstraction) is a result of the dialectical bottom-up approach…a product of our oscillations between theoretical perspective on abstraction and experimental observations of students' actions, actions we judged to be evidence of abstraction.” (HSD, 2001, p. 202).

It is clear that for analyzing the above actions, the researchers had to use some basic methodologies which fit protocol analyses of an individual and the more complicated analysis of a dyad’s cognitive and interactive work. Within the protocol analyses, the three epistemic actions and the dynamically nested relationships between them were hypothesized as the main building blocks of the model. They were also used as the lenses and compass to evaluate the model from the protocol analyses of the data. In a sense the above situation is similar to the situation of Krummheuer in his theoretical/empirical work. Such a situation holds for the first steps of validating the model as a theoretical framework. Since then, the RBC model has been validated (both as a theoretical framework and a methodological tool) and its usefulness for describing and analyzing processes of abstraction of various mathematical contents, in various social settings and learning environments, has been established by a considerable number of research studies. The settings considered by the researchers include classrooms, group interviews, tutoring situations with single students, and even introspective self reports of single learners. The age range of the learners extends from elementary school to adult experts and the longitudinal dimension varied. And indeed research made it clear that the RBC+C model can be extended to processes of abstraction and consolidation on a medium term time-scale, where Consolidation is a process by which the construct becomes progressively more self-evident, the student's awareness of the construct increases and the use of the construct becomes more flexible (Dreyfus et al., 2006; Hershkowitz et al. in press ). It is interesting to follow the role of the RBC+C model in different pieces of research and how it is changing with the changes of the research goal and the researcher/s interest: In Kidron and Dreyfus (2004) and in Dreyfus and Kidron (2006), the researchers used the RBC model as a methodological tool for the analysis of constructing knowledge with two “new” features: Very advanced mathematics and a solitary learner. But then came the surprise, namely that the model turned out to be only partially sufficient and the analysis of the data turned into theory building,

namely the extension of the model to interacting parallel constructions, the refinement of the epistemic actions, and the connection between these two. The paper of Tabach et al. (2006) presents an example of knowledge constructing within the context of peer learning in a working classroom. It showed how the design of the tasks and the computerized tools available to the students afford the constructing of conceptual knowledge (the phenomenon of exponential growth and variation, as it is expressed in its numerical and graphical representations). The researchers trace the constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment, and show that knowledge is constructed cumulatively, each activity allowing for consolidating previous constructs. This pattern indicates the nature of the processes involved in creating a new abstract entity: knowledge constructing and consolidating are dialectical processes, developing over time, when new constructs stem from old ones already consolidated, and old constructs are further consolidated through the new constructing. As the aim of this research was beyond the model in itself, to trace the global Constructing and Consolidating of specific conceptual knowledge along successive dyadic interactions, with a few months time interval in between, the main function of the RBC+C model in this research was to serve as a methodological tool. The researchers could not allow themselves to have the “space” (and may be not the interest) to polish and refine the model theoretically. In research published recently (Wood, Williams & McNeal, 2006), where the goal was to examine "the relationship between the patterns of interaction that exist in the classroom and children’s expressed mathematical thinking" (p. 228) in classes from different cultures. The model serves as the “conceptual framework employed to examine the quality of students’ expressed thinking.” (p. 225). I think that the term conceptual framework, when applied to a certain model, expresses the flexibility, in which this model may be applied as a framework for both theory and methodology. And indeed in Wood, Williams, and McNeal’s research, it seems that on one hand the authors believe theoretically that the model with its three epistemic actions: Recognizing, Building-with and Constructing may expresses quite accurately the level of mathematical thinking that children have. On the other hand they use the model for analysing the protocols of the class members and identifying the levels of thinking expressed by class members in whole class interactions. They accumulate these data for the purpose of quantitative analyses of the levels of thinking available to class members in discussions in the different classrooms. This research shows some maturation of the model as a theoretical framework and as methodology. The authors need two conceptual frameworks in their study and the model is one of them. The model is not anymore the focus of the study. It allows the researchers to determine levels of thinking that are available for inspection in the classroom. The last example in this section is Dooley's classroom research (2006) concerning what I would like to call collective abstraction processes, which emerge in one lesson, mostly during the last phase of the lesson where a whole class interaction takes place. The researcher’s aim is to show how the class community as one entity reaches "sophisticated constructions". She explains: “… One pupil's 'recognizing' led to 'building with' by another and to 'constructing' of new ideas and strategies by others.” Again, this is a situation where the researcher uses the RBC model as a methodological tool to explore the existence and nature of collective abstraction and its existence, and by doing this she confirms the RBC model as a conceptual

framework for collective abstraction as well. . The above process is in analogy to the collective argumentation of Krummheuer, (1995), which was investigated also by Whitenack and Knipping (2002). In both cases the process is distributed among quite a few students in the same classroom. The above collective abstracting raises many questions, like: What can we learn from the above kind of research about abstracting in classrooms? What can we say on the individual students in the classroom and/or on the classroom community not only as one entity but as a community which consists of all the individuals who belong to this community? Do we have a methodology/methodological tool by which we will be able to perform the kind of research which gives some answers to such questions? These questions and others led us to start with a RBC+C based classroom research, whose long range goal is to learn about and understand the development of shared knowledge in a community where the shared knowledge consider ALL individuals of this community. I will discuss some of these dilemmas through the example of our resent research on knowledge shared by a triad (Hershkowitz et al., in press), which uses the conceptual framework of RBC+C model. SHARED KNOWLEDGE DILEMMA The relationships between the constructing of knowledge by individuals and the shared knowledge of the group that is constituted by these individuals is a fascinating issue, both from the cognitive and socio-cultural points of views, especially when the group is an integral part of a working classroom. Typical groups in classrooms include dyads, small groups, and the entire class. Researchers who plan to observe and analyze in detail micro-processes of constructing knowledge in a given context such as a classroom, along a time segment that may range from minutes to weeks, and do not want to ignore the knowledge that the individuals constructed, face great difficulties: The observation and documentation processes are complicated, data are messy and massive and there are no systematic clear-cut methodologies for analyzing them (Schoenfeld, 1992). Many researchers have been aware of the above difficulties. In their efforts to analyze learning of a classroom community in mathematics, Cobb and colleagues focused on the evolution of mathematical practices in classrooms (Cobb et al., 2001). For this purpose, they combined "a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices" (p. 113). They discussed the notion of taken as shared activities of the students in the same classroom: “We speak of normative activities being taken as shared rather than shared, to leave room for the diversity in individual students' ways of participating in these activities. The assertion that a particular activity is taken as shared makes no deterministic claims about the reasoning of the participating students, least of all that their reasoning is identical.” (p. 119)

In our research we decided to investigate the knowledge constructing of groups in the classroom, where knowledge constructing of the group is based on the knowledge constructing of individuals who form the group and on interactions among them. This implies that we had to observe and investigate in detail cognitive and interactive processes of knowledge constructing by groups where the interactive flow of constructing knowledge is going from one individual to another one in the group until they reach shared knowledge. We define Shared knowledge as a common basis of

knowledge, which allows the students in the group to continue together the constructing of further knowledge in the same topic. We consider understanding the relations between the constructing of knowledge by individuals and the constructing of the group's shared knowledge as crucial in research concerning learning processes in the classroom, and our aim is to understand its mechanisms. Four elements are very important for us: 1. The micro-perspective: To provide detailed evidence for the group’s shared basis of knowledge, for the manner in which it emerges from the individuals’ knowledge constructing processes, and for the way in which it constitutes a shared basis that allows the students to continue constructing further knowledge together. 2. The continuity/longitudinal perspective: We tie the data and their analysis together along a time span of several lessons in order to observe and analyze students constructing new mathematical knowledge in one activity, and observe and analyze in detail if and how they use this knowledge in further activities. 3. The theoretical perspective: Students are able to go through an abstracting process - to make a vertical mathematization and construct a new piece of knowledge, and by interaction to trigger other student/s in the group to have the above construct. 4. The methodological perspective: For the analyses in 1, 2 and 3 above, we use the RBC+C model of abstraction. We show how this model enables a deep analysis of a group's constructing shared knowledge. Like Cobb and his colleagues (2001), we were aware of the diversity of knowledge constructing among individuals, as well as of the methodological problems mentioned above. Therefore we chose to start from student triads, working in a classroom. In addition to the triad, we also documented (with another camera) the global happenings in the whole class. We will bring part from one example (story) to illustrate the level of the analyses of constructing knowledge processes, in a group of three interacting students, whereby personal diversity, the unique nature of each individual, is observed and analysed. We will emphasize the flow of knowledge from one student to the others, until they have a common basis of knowledge, which allows them to continue together. The fact that students in the group keep on constructing knowledge collaboratively and actualizing it in further activities, allows us to identify this shared knowledge. By demonstrating this analysis we will also demonstrate how the RBC+C model is used as a conceptional framework in both aspects, the theoretical and the methodological (elements 3 & 4 above). Yael, Rachel and Noam in constructing E1 In this partial story the discourse among the three 8th grade girls shows the constructing process of a shared knowledge concerning the concept of a simple event in 2 Dimensional Sample Space (2D SS). Such a simple event consists of a simple event in each dimension (principle E1).

The three girls just started with the 2D SS topic and they discuss the first question (Question 1a): 1a: Yossi and Ruti throw 2 white dice. They decide that Ruti wins if the numbers of points on the 2 dice are equal, and Yossi wins if the numbers are different. Do you think that the game is fair? Explain

The knowledge evolves by Yael, who somehow constructed principle E1 before this activity or at its very beginning. She soon tries to count pairs of numbers to find the number of all simple events and of all relevant simple events in order to calculate chances. She reaches 27 pairs for both Ruti and Yossi together. Noam 21: What are you doing? Aa? Yael 22:

I did what are the chances… one, one; two, two; three, three; four, four; five, five; six, six; and one, one; one, two; one, three; one, four and two, one; two, two…

Rachel 23: I don't understand what you are doing? Yael 24:

It is because I have to know what is our whole, as if it is, what are all the possible outcomes that might be, and all these outcomes are either (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) or (1,2) (1,3) (1,4) (1,5) (1,6) and then 2 (meaning that now she has to count the combinations of 2 with the other numbers).

Noam 25:

Yael, I don't understand anything of what you are doing.

Rachel 26:

Yes, nor am I.

It seems that Yael is already busy in counting all the relevant pairs for calculating Ruti's and Yossi's chances to win, and wants to know what are all possible simple events. Rachel and Noam didn't yet construct the meaning of events as pairs of numbers (E1), and they don’t understand what, how, and why Yael is counting. Yael explains again: Yael 27:

Listen, there are some possibilities that 1 will appear: (1,2) (1,3) (1,4) (1,5) (1,6) and we finished with 1. Nnow 2: (2,3) (2,4) (2,5) (2,6).

Rachel 28:

O.K., O.K. we understood that, but why are you adding? I don't understand.

Although it is not evident from Rachel’s utterances, that she has already constructed E1, we may assume that she already shares with Yael that one has to count pairs (E1), because now she only asks about Yael’s counting system (28), and not about the nature of simple events in 2D SS. Along all the discourse, Noam tries to understand the situation, and she is still in the process of struggling to construct E1. She is doing it by confronting her friends with her misunderstanding (58-62): Noam 58:

Look you don't… you did as if one side of the die is 3 and the second side is 4 and you did 3 plus 4 and it is as if…

Yael 59:

I didn't do 3 plus 4. I will tell you exactly what I did…

Noam 60:

No, one second, second. That’s what I understood of what you did.

Yael 61:

I will explain…

But Noam wants to explain it herself:

Noam 62:

One minute! No! You have to do 3 and 4; it is one possibility, and 4 and 5 is a second possibility, so it is two (possibilities).

Rachel 64:

That is what she did; (3,4) is one possibility and (5,4) is one possibility.

We can conclude now that Noam (62) has constructed E1. Rachel (64) who explains that it is exactly what Yael did, by repeating Noam's explanation, uses E1 which she had constructed before to build-with it her explanation, and by this she provides evidence that she consolidated E1. At this point of time it seems that principle E1 is a shared common basis of knowledge for the group. Comments: 1. The repeated questions of Noam and Rachel forced Yael to repeat and count pairs. While organizing the counting, Yael shows that she consolidates the E1 principle (21-27). 2. Similarly, Rachel's explanation to Noam forced her to express things explicitly and by doing so she consolidated her knowledge, and not just agreed with Yael (62-64). 3. Noam puts the blame for her mistakes on Yael (Noam 58), and then corrects herself (Noam 62). She not only realizes that she has to relate to pairs but also can explain her mistakes (in 58-62). Thus she has constructed E1 as well. Even though the above is only a part of a story (see Hershkowitz et al., in press, for a more complete report), it can be observed that the shared knowledge of the group is characterized by its diversity – often each partner expressed her own way of constructing a piece of knowledge. This variability evolves from different needs, at different points in time, and shows the uniqueness of constructing of each individual. Yet all three students may benefit from this multifaceted shared knowledge in their common work, and may go on constructing new constructs and/or consolidating constructs in follow-up and assessment situations, where the consolidated construct has an individual flavor as well. The above example illuminates some of the important elements we listed above: 1. The level of refinement of the analysis, which we would like to have in order to trace the interactive flow of knowledge constructing from one student to her friend in the triad. The shared knowledge of the triad concerning a given element is interactionally constructed when each student constructed it in his/her unique way, and not only when the triad as one entity reached a collective abstraction distributed among the all three. 2. The example is only a partial story, and is limited in expressing the longitudinal element which will be raised again at the end of this talk. 3&4. The above example shows that once again we are in a situation where the RBC +C model is theoretically expanded to fit the exploration of a new social context in the classroom, the shared knowledge, and like in the previous situation it is done with the RBC+C model as a methodological tool, meaning the lenses through which we look on the protocols and decide what level of refinement to use and above all the lenses which guide us to analyze and interpret the data.

CONCLUDING REMARKS The two research models by which I tried to exemplify the flexible contour lines between the model as a theory and the model as methodological tool discussed above have some common features: Both of them deal with a model which is aimed to serve as a framework for describing, analyzing and interpreting a human mental activity. Both of them are appropriate for exploring individual mental activity as well as for exploring collective mental activity which is distributed in the classroom among different individuals (e.g . Whitenack & Knipping, 2002, concerning argumentation and Dooley, 2006, concerning the RBC+C model) The elements of Toulmin's model for argumentation, as well as the three epistemic actions of the RBC+C model, have a very general nature (general in a sense that it can be used in many and varied contexts). The relationships between the elements of Toulmin's model and the nested relationships among the epistemic actions of the RBC+C model, are global as well. The elements of both models are observable and can be identified. Therefore they lend themselves easily to be adapted and to contribute to research in many different contexts of argumentation or abstracting knowledge. In the RBC+C model based research for example, the generality of the three actions, and the dynamic nested nature of the relationships among them, make it possible to use the model in various situations, where each situation in a certain context is expressed as a certain combination of all the above "ingredients", which differ by the ways in which the epistemic actions are nested in each other and by the ways in which reorganization and consolidation is expressed. This creates the possibility for finegrained descriptions and analyses of various abstracting processes in various contexts. I would like to finish this talk with a question: How can we expand RBC+C based microanalysis, done on the protocols of the of the small group work in the classroom in one or two activities (Hershkowitz et al. in press) to include these students’ protocols from the entire learning unit, which contains a sequence of activities and a three different post-tests to get some useful information and understanding concerning the constructing of abstract shared knowledge of the group and without loosing the traces of the individual constructing of knowledge? REFERENCES Cobb, P., & Bauersfeld, H. (1995). The Emergence of Mathematical Meaning Interaction in Classroom Cultures. Lawrence Erlbaum Associates. Cobb, P., Stephan, M. McClain K., & Gravemeijer, K. (2001). Participation in classroom mathematical practices. The Journal of the Learning Sciences 10 (1&2), 113-163. Dooley, T. (2006) Consruction of knowledge by primary pupils: the role of wholeclass interaction. Accepted to CERME 5. Dreyfus, T., Hershkowitz R., & Schwarz B. B. (2001). Abstraction in context II: The case of peer interaction. Cognitive Science Quarterly 1, 307-368.

Dreyfus, T., Hadas, N., Hershkowitz, R., & Schwarz B.B.: Mechanisms for consolidating knowledge constructs. In J Novotna, H. Moraova, M. Kratka, & N. Stelinkova (Eds.). Proceedings of the 30th International Conference for the Psychology of Mathematics Education, Vol. 2 (pp. 465-472). Prague, Czech Republic: Charles University in Prague. Faculty of Education. Dreyfus, T. & Kidron, I. (2006). Interacting parallel constructions: A solitary learner and the bifurcation diagram. Recherches en didactique des mathématiques 25(3). Dreyfus, T., & Tsamir, P. (2004). Ben's consolidation of knowledge structures about infinite sets. Journal of Mathematical Behavior 23, 271-300. Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education 32, 195-222. Hershkowitz, R., Hadas, N., Dreyfus, T., & Schwarz B.B. (In press). Processes of abstraction, from the diversity of individuals’ constructing of knowledge to a group’s “shared knowledge”. Mathematical Education Research Journal. Kidron, I., & Dreyfus, T. (2004). Constructing knowledge about the bifurcation diagram: Epistemic actions and parallel constructions. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Conference of the International Group for Psychology of Mathematics Education (Vol. 3, pp. 153-160). Bergen, Norway: Bergen University College. Krummheuer, G. (1995) The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The Emergence of Mathematical Meaning Interaction in Classroom Cultures, pp. 229-269. Lawrence Erlbaum Associates. Schoenfeld, A. H. (1992). On paradigms and methods: What do you do when the ones you know don’t do what you want them to? Issues in the analysis of data in the form of videotapes. The Journal of the Learning Science, 2(2), 179-214. Tabach, M., Hershkowitz, R., & Schwarz, B. B. (2006). Constructing and consolidating of algebraic knowledge within dyadic processes: A case study. Educational Studies in Mathematics 63(3), Toulmin, S. (1969). The uses of argument. Cambridge, England: Cambridge University Press. Treffers, A., & Goffree, F. (1985). Rational analysis of realistic mathematics education. In L. Streefland (Ed.), Proceedings of the 9th International Conference for the Psychology of Mathematics Education, Vol. II (pp. 97-123). Utrecht, The Netherlands: OW&OC. Tsamir, P., & Dreyfus, T. (2002). Comparing infinite sets – a process of abstraction the case of Ben. Journal of Mathematical Behavior 21, 1-23. Whitenack, J. W., & Knipping, N. (2002). Argumentation, instructional design theory and students' mathematical learning: A case for coordinating interpretive lenses. Journal of Mathematical Behavior, 21, 441-457. Wood, T., Williams, G., & McNeal, B. (2006). Children’s mathematical thinking in different classroom cultures. Journal for Research in Mathematics Education 37 (3), 222-255. Yackel, E. (2002). What we can learn from analysing the teacher’s role in collective argumentation. Journal of Mathematical Behavior 21, 423-440.

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