Abstraction in Context: The Case of Peer Interaction

ABSTRACTION IN CONTEXT II: THE CASE OF PEER INTERACTION Tommy Dreyfus, Holon Academic Institute of Technology, Israel Rina Hershkowitz, Weizmann Institute of Science, Rehovot, Israel Baruch Schwarz, The Hebrew University of Jerusalem, Israel

A final version of this paper appears in: Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2001). Abstraction in context: The case of peer interaction. Cognitive Science Quarterly, 1(3), 307-368.‫‏‬

Abstraction in peer interaction.doc

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ABSTRACTION IN CONTEXT II: THE CASE OF PEER INTERACTION Tommy Dreyfus, Holon Academic Institute of Technology, Israel Rina Hershkowitz, Weizmann Institute of Science, Rehovot, Israel Baruch Schwarz, The Hebrew University of Jerusalem, Israel Abstract We define abstraction as an activity of vertically reorganising previously constructed mathematical knowledge into a new structure. In a previous study, we translated this definition into an operational model whose elements are three nested epistemic actions. We also illustrated the model by means of a case study. In the present article, we validate and refine the model by analysing additional, more complex case studies, we extend the model to a richer context, namely pairs of collaborating peers, and we investigate the distribution of the process of abstraction in the context of peer interaction. This is done by carrying out two parallel analyses of the protocols of the work of the student pairs, an analysis of the epistemic actions of abstraction as well as an analysis of the peer interaction. The parallel analyses led to the identification of types of social interaction that support processes of abstraction. Introduction Abstraction is a central process in learning mathematics; however, it is notoriously difficult to observe. In a previous article (Hershkowitz, Schwarz, & Dreyfus, 2001), we proposed a model for abstraction that is operational in the sense that its components are three observable epistemic actions. We illustrated the use of the model to observe abstraction by means of a teaching interview of a single student having a computerised tool at her disposal. We stressed the crucial importance of context for processes of abstraction and were able to demonstrate it to a limited extent. In particular, the role of the personal history of the student was analysed in some detail. On the other hand, many contextual aspects, including peer interaction, could not be treated because of the specific choices made and the limitations imposed on that case study. It is the aim of the research presented in this article to overcome some of these limitations. Specifically, the present study goes beyond the previous one in that we will • •

Analyse additional case studies by means of the model, and thus validate the model; Refine the model so that it can account for more complex processes of abstraction;

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Show that the model applies in an environment that is rich in social interactions;

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Investigate the distribution of abstraction in the context of peer interaction.


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For this purpose, we will analyse in detail the work of two pairs of Grade 7 on a multiphase algebra problem. A computer with a spreadsheet was available to the students at all times. The problem was designed with the potential for abstraction in mind: The stude nts were led to discover a pattern holding for any initial number, and asked to justify the claim that the pattern is independent of the choice of the initial number. Any justification acceptable from a normative point of view demanded the use of algebra. Within algebra, it required the use of a rule of computation, which was as yet unknown to the students. Moreover, the students had not yet been taught that algebraic manipulations could serve as a way to justify the correctness of a numerical property. The problem thus offered these students opportunities for constructing new mathematical structures at two levels, the level of algebraic computation and the level of algebra as a means for general argument and justification in mathematics. The paper is structured into five sections. In the first section, we present an overview of our model of abstraction. In the second section, we give some background on the students, their school, their classroom and their mathematics curriculum, as well as the setting and task, which were the objects of this research. We also specify our methodology for characterising the interaction between two students as well as between the students and the teacher- interviewer who was present. In the third and fourth sections, we analyse the work of the two student pairs, the interactions among the peers, as well as their epistemic actions in terms of the model. This analysis largely confirmed the adequacy of the model to reflect whether processes of abstraction occur during the different problem solving phases. It also allowed us to elaborate the model so that it can account for processes of abstraction that are nested in more global processes of abstraction. Moreover, we identified patterns in the peer interaction between the students for the different problem-solving phases of the two pairs. For example, the symmetry of the first pair contrasts with the asymmetry of the second pair. This analysis allowed us to identify interaction patterns that are compatible with abstraction processes. The very occurrence of abstraction in collaborating pairs of students raises the issue of the distribution of abstraction among peers. This issue is tackled in the concluding (fifth) section of the paper. What can be said about potential residues of the abs traction activity? To what extent will they be common and the same for the two participants? And to what extent will there be similarities or differences in how the two participants later use those residues as building blocks for further abstractions? These questions are speculated on in light of the data provided in the present study and in light of the theoretical literature on this topic.


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1. The dynamically nested model of abstraction In this section, we briefly review our definition and model of abstraction. The corresponding theoretical background has been discussed in our previous article. A more detailed description of the model can also be found there. Following Piaget, mathematics educators have proposed that abstraction consists in focusing on some distinguished properties and relationships of a set of objects rather than on the objects themselves. Abstraction is thus a process of decontextualisation. According to Davydov (1972/1990), on the other hand, abstraction starts from an initial, undeveloped form and ends with a consistent and elaborate final form. Similarly, Ohlsson and Lehtinen (1997) see the cognitive mechanism of abstraction as the assembly of existing ideas into more complex ones. Noss and Hoyles (1996) go even further. They situate abstraction in relation to the conceptual resources students have at their disposal and see it as attuning practices from previous contexts to new ones. Therefore, according to Noss and Hoyles, students do not detach from concrete referents at all. Leaning on ideas of these and other authors, we define abstraction as an activity of vertically reorganising previously constructed mathematical knowledge into a new structure. We thus take structure to result from the process of abstraction. The use of the term activity in our definition of abstraction is intentional. The term is directly borrowed from Activity Theory (Leont’ev, 1981) and emphasises that actions occur in a social and historical context. It also stresses the inseparability of actions from goals, their meaning being perceivable only within the activity in which overall motives drive individual actions of participants. The reorganisation of knowledge is achieved by means of actions on mental (or material) objects: Mathematical elements are put together, structured and developed into other elements. Such reorganisation is called vertical (Treffers and Goffree, 1985), if new connections are established or some inaccessibility is overcome, thus integrating the knowledge and making it more profound. According to this definition, abstraction is not an objective, universal process but depends strongly on context, on the history of the participants in the activity of abstraction and on artefacts available to the participants. In this sense structure is internal, "personalized". The artefacts are often themselves historical residues of previous activities. They include material objects and tools, such as computerised ones, as well as immaterial ones including language and procedures; in particular, they can be ideas or other outcomes of previous actions (knowledge artefacts). Thus we subsume under the term “artefact” everything that has the potential to mediate students’ learning and their construction of new knowledge structures. This definition of abstraction in context becomes productive through a program of research to experimentally investigate processes of abstraction. As abstraction is an activity consisting of actions, the first step in the realisation of this program is to identify the kinds of Abstraction in peer interaction.doc

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actions involved in the activity of abstraction. The actions we identified in the first study belong to the general class of epistemic actions, actions relating to the acquisition of knowledge (Pontecorvo & Girardet, 1993; Schwarz & Hershkowitz, 1995). For example, appealing to a strategy or inferring a consequence from data are epistemic actions. In many social contexts, such as small group problem solving or teacher-guided inquiry in a whole class forum, participants’ verbalisations may attest to epistemic actions thus making them observable. The three epistemic actions we identified as related to processes of abstraction are Recognising, Building-With and Constructing, or RBC. Constructing is the central step of abstraction. It consists of assembling knowledge artefacts to produce a new structure to which the participants become acquainted. Recognising a familiar mathematical structure occurs when a student realises that the structure is inherent in a given mathematical situation. Recognising may occur in at least two cases: (1) by analogy with another object with the same or a similar structure which is already known the re-cognising subject; (2) by specialisation, i.e., by realising that the object fits a (more general) known (to the subject) class all of whose members have this structure. In terms of actions, the process of recognising involves appeal to an outcome of a previous action and expressing that it is similar (by analogy), or that it fits (by specialisation). Building-With consists of combining existing artefacts in order to satisfy a goal such as solving a problem or justifying a statement. The same task may thus lead to building-with by one student but to constructing by another, depending on the student’s personal history, and more specifically on whether or not the required artefacts are at the student’s disposal. Another important difference between constructing and building-with lies in the relationship of the action to the motive driving the activity: In building-with structures, the goal is attained by using knowledge that was previously acquired or constructed. In constructing, the process itself, namely the construction or restructuring of knowledge is often the goal of the activity; and even if it is not, then it is at least indispensable for attaining the goal. The goals students have (or are given) thus strongly influence whether they build-with or construct. If they solve a standard problem, they are likely to recognise and build-with previously acquired structures. If they solve a non-standard problem, they might be faced with an obstacle that causes them to construct by vertically reorganising their knowledge to overcome the obstacle. The three epistemic actions are the elements of a model, called the dynamically nested RBC model of abstraction. According to this model, constructing incorporates the other two epistemic actions in such a way that building- with actions are nested in constructing actions and recognising actions are nested in building-with actions and in constructing actions. The genesis of an abstraction passes through (a) a need for a new structure; (b) the construction of a new abstract entity; (c) the consolidation of the abstract entity through repeated recognition


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of the new structure and building- with it in further activities with increasing ease. We have argued in the previous article that this model fits the genesis of abstract scientific concepts acquired in activities designed for the special purpose of learning. In such activities the participants create a new structure that gives a different perspective on previous knowledge. The model is then compatible with the dialectical theory of abstraction developed by Davydov (1972/1990). Moreover, the model is operational: It allows one to identify processes of abstraction by observing the epistemic actions and the manner in which they are nested within each other. We stress that the model describes the mechanism of the process of abstraction. As such it contains the main invariant features of abstracting as a thinking process. The mechanism implied by the model is independent of the particular context to which the model is applied. This is what makes the model general. Thus, we claim that it is apt to describe processes of abstraction, whatever the context is, in which they occur. In other words, the model describes general features of the process that are valid in all contexts. This does not mean that the process is context independent, quite the contrary, the model is apt to take context into account and describe each process of abstraction in its particular specific context. Moreover, we note that our definition and model of abstraction are general in the sense that they do not relate specifically to mathematics. Therefore it seems reasonable to assume that it may be appropriate at least to abstraction of science concepts. However, its validity was not yet checked in area outside mathematics. We can now also distinguish processes of abstraction from learning processes, which do not fit our definition of abstraction. For example, learning to mechanically perform a mathematical algorithm is not an abstraction. Children may well learn to subtract four digit numbers without building a structure that allows them to connect their conceptual knowledge of subtraction to the algorithm. Performance is thus generally not sufficient as an indicator of abstraction. Another case in point is children who perform subtraction because a word problem contains a linguistic cue such as ‘less’. Such children are not aware that the abstract arithmetical problem they solve is equivalent to the original one; they operate in the arithmetic realm independently from the referent even if the results lead to aberrations. A further type of learning, which does not qualify as abstraction, is rote learning. The situat ion with respect to problem solving is more complex. Even if challenging, problem solving does not necessarily lead to abstraction. For example, when solving a hard puzzle one may need to capitalise on already learned heuristics in order to monitor and regulate the use of wellknown strategies. But after succeeding to solve such a challenging task, one generally does not feel that one has learned something new. In such cases, the epistemic actions involved are Recognising and Building-With. We thus claim that abstraction occurs during problem


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solving only when the solver elaborates a new method/strategy and thus constructing is involved. In spite of the claims concerning the adequacy of our model for describing processes of abstraction, only one case has been reported in the literature so far: The case of a single student who solved a task in the presence of an interviewer, a suitable computer program being at her disposal (see Hershkowitz, Schwarz & Dreyfus (2001)). Many questions thus naturally arise, concerning the validity of the model in other and richer contexts. Specifically, the following research questions will be treated in this article. 

Is the dynamically nested RBC model of abstraction valid beyond the one case presented in the previous article?

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Which modifications of the model, if any, are suggested by more complex processes of abstraction?

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Is the model adequate for describing processes of abstraction by interacting pairs of students?

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Which patterns of distribution of abstraction occur in processe s of abstraction by collaborating peers?

In this article, we will thus pay a lot of attention to the social dimension of the process of abstraction, a process that has been almost always treated from a solipsistic point of view. The research questions lead us to consider the pair as well as the individual as foci of our study on abstraction. Schwartz (1995) has made a significant contribution to this issue: High and junior high-school students participated in three experiments. on problem-solving activities as individuals or in pairs. The three experiments showed that student pairs constructed more abstract representations of the problems than individual students did. These findings suggest the hypothesis that peer interaction fosters abstraction even though the sense of abstraction adopted by Schwartz is not identical to ours. 2. The Research Design From a methodological point of view, the study of abstraction is subtle. Because of our intention to observe a process while it happens, a qualitative study based on (video) recordings of learning episodes was an obvious choice. In order to ensure a substantial amount of researcher control, we decided on the presence of an active interviewer. But even in such a controlled situation, it is difficult to judge the nature of an epistemic action because this nature is, of course, subjective: Where experts see deep structure in a problem situation, novices often notice only surface structure (see, for example, Chi, Feltovich and Glaser, 1981); while for the experts, seeing the deep structure is a matter of re-cognition, for suitably prepared novices the problem situation might present an opportunity for engaging in a


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process of constructing the deep structure. In order to create a context, in which pairs of students could participate in a process of abstraction, we thus had to: (a) find students who had the habit of dealing with challenging problem situations in pairs; (b) identify some mathematics with a definite potential for abstraction for these students; (c) design an activity, in which this specific abstraction was likely to be used. 2.1 The activity The design of the activity is based on research on algebraic thinking. In cases in which algebraic structures model problem-situations, research has shown that learners use algebra ic expressions only to indicate generality (Arcavi, Friedlander & Hershkowitz, 1990). Other functions of algebra, such as identifying and proving properties, hardly correspond to a psychological need: Learners are generally confident that a property is correct after inductively checking a number of cases. Algebraic manipulations are then not relevant to most students’ informal algebraic reasoning. However, when prompted by questions such as “Why is this property always correct” or “Can you convince your pee rs that the property is correct”, students feel that an inductive justification is not sufficient. This happened even in cases, in which the students did not have algebraic language at their disposal (Arcavi et al., 1990). In this case, students expressed their lack of tools to build an argument. These findings helped us elaborating an activity for observing processes of abstraction in algebraic reasoning. This activity was intended to lead students who had never yet used algebra as a tool for proof into a situation, in which they felt the need to justify a property whose proof requires algebraic manipulation. The translation (from Hebrew) of the corresponding worksheet is reproduced in Figure 1. ------------------------------------------Insert Figure 1 about here ------------------------------------------Questions 3, 4, and 6 relate to two specific properties, which the students may or may not have found earlier in question 2. Throughout this paper, we will call property 6a the diagonal sum property (or DSP) and we will call property 6c the diagonal product property (or DPP). We note that Questions 5 and 6c referred to a third property which does hold for the two given rectangles but not for other rectangles constructed by the same ‘seal’. These questions were included in order to draw the students’ attention to the risk involved in drawing inductive conclusions. Questions 1 through 4 are variants of items already asked by Arcavi et al. (1990). It was shown there that many students at the brink of the learning formal algebra can informally justify the DSP; however, these same students can only give inductive arguments for the DPP, which leaves many of them dissatisfied. Indeed, the DSP can quite easily be justified verbally while the simplest way of justifying the DPP is to use algebraic manipulation and


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compare X(X+8) (the expression for the main diagonal) to (X+6)(X+2) (the expression for the secondary diagonal). 2.2 Participants and procedure The activity was designed for students from whom the use of algebra for proving properties such as the DPP could possibly be expected but who had never actually done it. We “prepared” a class of such students by giving them individual questionnaires at the beginning of the year and closely following the class throughout the year until the day of the experimental interview. At the beginning of the year, a few students used algebra to express generality but most did not. The students’ algebra curriculum consisted of seventeen activities, which were under the researchers’ control (thirteen with spreadsheet, four without computer). Students usually worked in pairs; some of the pairs were working together regularly throughout the year. In these activities students learned to use algebra to express generality. For example, by generalisation from a few numerical examples or from a “story”, they generated an algebraic relation, which they could insert into the spreadsheet. By dragging, they could then obtain sequences of numbers to describe and investigate a phenomenon. On the other hand, students were not asked to justify general properties by using algebraic manipulation. We know from weekly observations and teacher’s reports that the children increasingly used algebra for expressing generality throughout the year. For example, the following fact will be crucial below: The students had experience with the use of the simple distributive laws a (b+c) = ab + ac and (a+b) c = ac + bc but had never yet used the extended distributive law (a+b) (c+d) = ac + ad + bc + bd, which is needed for Question 6. The entire class carried out the ‘seals’ activity toward the end of the school year. This activity is similar in spirit and level of difficulty to other class activities, except for part 6. As pointed out above, the justification of the DPP in part 6 required the use of algebraic manipulation, and this requirement went clearly beyond what the class was used to. Most of the students did the activity alone, using the worksheet. None of the students working alone arrived at an acceptable justification of the DPP. Three pairs of students were chosen for carrying out the activity in an interview situation. These students were chosen upon the teacher’s recommendation because of the their high verbalisation ability, and their habit of working together. The three interviews were videotaped and transcribed. One of the three pairs, Da and Li, two boys with quite high mathematics grades, made no progress beyond the rest of the class. In the concluding section of the paper we will discuss possib le reasons for this lack of progress. The other two pairs did progress far beyond the rest of the class. One of these two pairs consisted of two boys who will be identified as Yo and Ra, or collectively as Yo&Ra; the other pair consisted of two girls who will be identified as Ha and Ne, or collectively as Ha&Ne. The work of Yo&Ra and of Ha&Ne will be analysed in detail below.


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2.3 Protocol analysis In order to study abstraction by interacting pairs of students one needs to study simultaneous cognitive and social processes. For this purpose, we decided to undertake two analyses of the interview protocols, one that analyses the cognition and one that analyses the interaction. Our aim was not to give precedence to either of these analyses, but to carry them out as independently from each other as feasible, and then to compare the resulting pictures. Other researchers (e.g. Pontecorvo & Girardet, 1993; Resnick, Salmon, Zeitz, Wathen & Holowchak, 1993; Sfard & Kieran, in press) also used double coding. The specific ity of our research resides in the fact that the task in which students participated was designed to lead to an abstraction, the construction of specific normative knowledge that is new to the students. For each pair of students, we first produced a coarse segmentation of the protocol into segments (Chi, 1997). Since we aimed to observe abstraction in a setting, where students were engaged in a specific task designed to lead to abstraction, the segmentisation was carried out according to the fulfilment of the task goals, or according a re-evaluation of these goals by the interviewer or the students. Next, we proceeded to two independent analyses within these segments. On the one hand, we identified interaction patterns between the peers according to a system to be explained below. On the other hand, we closely followed the methodology used in the previous article for the identification of the students’ epistemic actions, that is “Recognising”, “Building-With”, and “Constructing”. For pairs of students, this identification poses problems, which were not present in the previous study. Because of the subjectivity of the epistemic actions, what is constructing for one student may be building-with or recognising for the other. We classified such cases as construct ing for the pair. Moreover, peers may collaborate or act independently. We will show cases where a constructing action was undertaken collaboratively and others where one of the peers seemed passive, while the other undertook a constructing action. But eve n when only one student acted overtly, she knew that the peer conferred her the responsibility for undertaking the action thus leaving some commonality to the action. We will stress this point in the two interviews, and return to it from a more theoretical stance in the concluding section of the paper. In the analysis of the interaction, we categorised the conversational moves according to their function into six categories. The categories are based on those used by Resnick et al. (1993) but adapted to our abstraction task. We describe them through the succinct analysis of a central segment in the interview of Ha&Ne. This segment follows their use of the distributive law to develop the expression for the first diagonal of the seal from X(X+8) to XX + 8X. They now turn to the second diagonal (X+6)(X+2): Ha133 Ne134

And this [thinks] ... It's impossible to do the distributive law here. Wait, one can do ...


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Ha135 Ne136 Ha137 Ne138 Ha139 In140 Ha141

This is 6X. This is 6X times X and 6X times 2. Wait, first, no ... Yes. No because this is X plus 6, this is not 6X, it's different. Wait. First one does ... X; then it's XX plus 2X, and here 6X plus 24. Then ... Not 24, why 24? Ah, 12.

This excerpt is described from an interaction point of view in Figure 2. The arrows in Figure 2 point to the nearest referent(s) of any specific utterance. For example, the arrow from Ha135 points to Ha133 because the referent for “this” is clearly the same in the two utterances. The justification for the link between Ne134 and Ha133 is more subtle: In the previous segment, Ha employed the simple distributive law for developing the first diagonal as XX + 8X. When in Ha133 she says “And this …”, Ne knows that she refers to the second diagonal. In Ne’s claim that “It's impossible to do the distributive law here”, the word “here” refers to the same second diagonal. In addition, Ne invokes the distributive law. Her statement therefore also refers to the last utterance in which the term “distributive law” was explicitly articulated. This appears in the diagram as a “long arrow” from Ne134 to Ha121. Similarly, Ne136 has two referents: First, Ha135 as Ne uses the term “6X” first expressed by Ha in 135. Second, Ne134 since Ne elaborates the distributive law that she did not know how to apply in Ne134. Although these identifications are not (and cannot be) free of cognitive details, we stress that these cognitions are of a very local kind and serve as a technical tool rather than being the object of attention. ------------------------------------------Insert Figure 2 about here ------------------------------------------We now turn to the description of the six categories of utterances. Ne134 exemplifies the first category, namely statements that make proposals or plans, ad thus direct a series of actions. Statements in this category will be labelled control statements and designated by the number 1. Statement Ha135 continues and develops statement Ha133: The word “this” refers to the same thing but is articulated in Ha135 as 6X. Ha135 exemplifies the second category, namely utterances that articulate verbally what is done to continue or develop an idea. We call these utterances elaborations and designate them by the number 2. Other examples of this category in the present excerpt are Ne136, which elaborates both Ne134 and Ha135, and Ha141, which elaborates Ha139. The third category contains utterances that comment the results of actions. Frequently, such comments summarise a sequence of elaborations. We call such statements explanations


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and designate them by the number 3. The identification of explanations is made possible by key words like “because” or by their temporality; in our case, they typically appear after a sequence of algebraic manipulations. Ha139a belongs to this category as Ha expla ins “No because this is X plus 6, this is not 6X, it's different” which clearly refers to Ha135 and has the function of explaining. Ha139a also fulfils another function, namely to put in question a previous utterance (Ne138, and by proxy, Ne136). In general this category can originate from a need for clarification, or from a motive to oppose. We call statements in this category queries and designate them by the number 4. Queries are identified by intonation, indicating question or opposition or by expressions referring to negation, puzzlement, etc. In Ha141, Ha agrees with the interviewer’s In140 that the number 24 is wrong. It belongs to the fifth category, namely utterances in which a participant acknowledges agreement or makes a concession. We call this category the category of agreement and designate the statements by the number 5. Such utterances must be differentiated from those that only evidence that the participant attends actions of others. The latter are included in the sixth category, namely interventions that refer to the establishment and the maintenance of the flow of the conversation. It includes two types of utterances: (a) Those that indicate that the interviewer considers how the participants are engaged and whether the conversation can end, including the interviewer’s praise, encouragement or thanks. (b) Short utterances like “Yeah”, “OK”, “wait” that point at the presence of other participants that attend an action. For example Ha137 (“wait, first, no”) is, in addition to being a query, a way to ask Ne to make a pause in her inquiry. We call this category the category of attention and designate them by the number 6. In the two protocols we analysed, the identification of epistemic actions and of conversation categories was validated through independent rating of the protocols by the three researchers. Agreement on both, epistemic actions and conversation categories, between any pair of researchers was high. All disagreements were settled by further analysis and discussion among the three researchers until agreement was reached. 3. The inte rvie w of the first pair (Yo&Ra) As mentioned above, the interviews took place toward the end of an introductory algebra course in Excel learning environment. The two boys, who will be identified as Yo and Ra, or collectively as Yo&Ra, had been working together in exploring problem situations with spreadsheet quite often along the school year. The activity was given to them as a worksheet, and they were supposed to work on it question by question. This learning mode was quite familiar to them. In the following analysis we will relate mostly to the work done in relation to the DPP, the property that the difference between the products of the numbers in the


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diagonals equals 12 (see questions 4a, 4b and 6c in Figure 1). The analysis will be structured into four subsections dealing, respectively, with an overview of the Yo&Ra interview (3.1), with the interaction patterns between the students (3.2), with their epistemic actions of abstraction (3.3), and with the relationships between the epistemic actions and the interaction patterns. 3.1. Overview of the Yo&Ra interview During their work on question 1, Yo&Ra generalise the pattern of the seal. At one stage, they confuse the algebraic notation system with the one of Excel, but they settle quite fast on the Excel notation and complete and verify the generalised seal = A1 = A1 + 2

= A1 + 6 = A1 + 8

Working on question 2, they find many properties, from very simple ones, which are little more than a verbal repetition of the relationships between the cells which they already expressed algebraically, to more complex ones like in the following (Segment 1): Ra91 Yo92

Ra93 Yo94 Ra95 Yo96 Ra97

OK, let’s find something about the products [tries out some numbers and computes something]. Write also that the sum of the diagonals will always be even, because again you add 8 or 4. [Ra passes the worksheet to Yo and continues computing something. Yo begins writing]. Ah, actually the sum of all the square will be even. Did you understand? Wait, I’m verifying something =A1*B2 [enters formula into cell A4], =B1*A2 [enters formula into cell B4]. What are you doing? Wait! Ah, you’re multiplying in order to see if there’s a link? =A4-B4 [enters that formula too]. OK, now let’s change [changes the first number]. Meantime the difference is 12. Here too it’s 12, [changes again] 12. OK, the difference is always 12.

In the above segment, they find the DPP and at the same time verify it inductively by substituting numbers in the left upper cell of the generalised seal which they had built earlier. We note that as a response to Question 6a (justify the DSP), Yo is stuck with the tautological explanation that each diagonal is half of the sum of all 4 cells. Ra, however, justifies the DSP verbally and algebraically using the algebraic expressions X, X+6, X+2, and X+8 (Segment 2): Yo184 Ra185 Yo186

Ah, I know: because the diagonal will always be half of the total sum of the square. No! Yes, and then both will be equal.


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Ra187

Yo188

Ra189 Yo190 Ra191 Yo192 Ra193 Yo194 Ra195 Yo196 Ra197 Yo198

No, because look, here we add 8 to the original number [points at the appropriate cells], here 6 and here 2. 2 plus 6 equals 8. We have the original numbers and in each diagonal we add 8 in both cases. What? Ra, they are asking you why it always will be equal in both diagonals. The answer is because the diagonal will always be half of the total sum of the square. I don’t understand why. I will explain you my way, which is much simpler. We have the original number … The number X Then here it’s X+8. That’s one diagonal. And here it’s X+2. Why X+2? X+4! Here, here, X+2. And here X+6. No, X+4! X+6! [moves the cursor to that cell] Ah, this is X, yes! Did you understand, we have X and in each diagonal we actually have 2X+8. No, I think it’s more logic that “X divided by 2 equals diagonal”. Each diagonal is X divided by 2.

They start question 6c with the declaration that they don’t know how to justify the DPP. While starting from the multiplication of the cells in each diagonal, they have cognitive and technical difficulties. They are distracted by the numbers shown on the spreadsheet for a particular seal (7, 13, 9, and 15) and attempt to compare 13(X+2) and 15X – a mixture of remembered formulae and numbers taken from the specific seal: Yo227

We need to multiply, OK? The first diagonal is x multiplied by 15. Now the second diagonal is x plus 2 multiplied by 13. So why is the difference 12?

The interviewer helps them come back to the formulae in the generalised seal, and shortly they produce the diagonal products algebraically. But again, they are unable to progress, they face an algebraic obstacle comparing the diagonals X(X+8) and (X+2)(X+6), which they can’t overcome. They use a few incomplete algebraic and/or numerical computations to try and figure out why the difference is 12. In In262, the interviewer suggests an end to the interview, but this triggers Yo263 and the main part of the DPP justification that lasts to the end of the interview. It consists of 3 different segments (3, 4 and 5), on which our detailed analyses in the following subsections will be done. Yo263 In264 Yo265 In266 Yo267 In268 Yo269 In270

Oh, I know: You can do the distributive law here. Yes. X times 8 plus X times X ... Yes. Here one can do … Here I’ll write X times 8 … Yes, plus … Plus X times X, yes?


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Yo271 Ra272 Yo273 Ra274 Yo275 In276 Ra277 Yo278 Ra279 In280 Ra281 In282 Ra283 In284 Ra285 In286 Yo287 Ra288 Yo289 Ra290

Now, here one can do X times X … …plus X times … plus X times 6 plus 2 times X plus 2 times 6. Why? You are using too many factors! No, it’s OK. Yes [writes], don’t you agree? No, there are too many factors! It’s correct! We already used the …, I don’t understand what the 8 is for, first of all This? Yes! Ah, yes! We have X times X … Yes! plus X times 6, and then the normal continuation will be X times 2. X times 2, OK! Why X times 2? Because I go according to how you do it, then this is X times X, X times 6, X times 2... No, this is X plus 2. Look, you do, this is the distributive law, and you do X times X plus X times 6; now you pass to the 2; 2 times X and 2 times 6. Ah, logical! I got it.

This third segment starts from Yo’s idea to apply the distributive law to both diagonals, X(X+8) and (X+2)(X+6). During the algebra course they had used the simple distributive law on expressions similar to the first diagonal, but they had never met the extended distributive law. Here they (first Yo in Yo263-Yo273 and then Ra in Ra274-Ra290) construct, step by step this expansion and they are left with the algebraic expressions XX+8X for the first diagonal, and XX+6X+2X+12 for the second. Here they face a new algebraic obstacle, as it may be seen in the following Segment 4: In291 Yo292 Ra293 Yo294 Ra295 Yo296 Ra297 Yo298 Ra299 In300 Yo301 In302

And then, what do you say? Well, so now let’s check whether it makes sense that the difference is 12. No, wait. First of all, the Xs we can’t solve, because we don’t know what X is – it can be any number. 6 times 2 is 12. So let’s solve what we can. So we do X times X which is two X. Here it’s 12; I’ll write this at the top. And then you do 2 times 6 and then it comes out 12. That’s 12, yes. And here it’s X multiplied by 8 because it is X times 8 plus …, impossible to understand this! OK, fine.


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Yo303 In304a

But is it really according to the distributive law? You were really 90% because you don’t know what X times X is.

In the above segment there is no visible progress in the justification. They are led by the need to show that the difference between the above expressions is 12 (see Yo292), and try different sporadic approaches to do that. They try to consider the algebraic expression as an equation to be solved, or to magically receive the desired 12 numerically. The interviewer once again decides to make an end to this interview (In304a). But then he adds a clue (In304b) that starts a new and successful development, as can be seen Segment 5: In304b Ra305 In306 Yo307 In308 Yo309 Ra310 Yo311 Ra312 In313 Ra314 Yo315 Ra316 Yo317 Ra318 Yo319 In320

But this and this – it’s the same thing. So one takes it away; so, in fact, we have only x times 8. And here there’s X times 6 and X times 2. Then, wait, the X times X can it be erased here? Yes. And here; so we have ... X times 2 and X times … we have X … We have 8 Xs times 12 and here 8 Xs. No, this is not times, it’s plus. Ah, plus. We have x plus 6 times 12; here it’s simply X times 8 and here … here we have 8 Xs. and here 6 Xs plus 12. No, in the second one we have 8 Xs plus 12; so, really, the difference is 12 because one can cancel this, and then we have here a difference of 12. X times 6, that’s 6 Xs plus X times 2 that’s 8 Xs plus 12; so this is 8X+12, and here it’s 8X, here it’s 8X without 12; that’s why the difference is 12! You are great! Really nice!

The clue from the interviewer (In304b) releases them from the unfruitful idea of approaching the justification by solving an equation. They start eliminating similar expressions from the diagonals, XX+8X and XX+6X+2X+12, first the XX and then the 8X and the 6X+2X, remaining with 12 for the second diagonal only. Most of the analysis in the following subsections will be focused on the last three segments for the following reasons: (1) These three segments together exhibit the main part of the DPP justification. (2) This justification by Yo&Ra illustrates an abstraction process as it is expressed in the epistemic actions constructing, building-with and recognising and in the way they are nested within each other (subsection 4c). (3) The interaction patterns, especially in the third and fifth segments are typical for much of the interaction in the Yo&Ra collaboration (subsection 4b). (4) Correspondences between the epistemic actions and the interaction patterns are evident in these three segments (subsection 4d).


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3.2 Interaction patterns of Yo&Ra We begin the analysis of the interaction patterns of Yo&Ra with some global comments on the first part of the interview, followed by a detailed analysis of segments 3, 4 and 5. At the beginning of the interview Yo and Ra both have the mathematical curiosity and the drive to complete the mathematical activity. In addition each of them is conscious about his mathematical potential, and likes it to be seen by the other and by the interviewer and even the future video observers. At the same time each of them is quite aware of his friend’s mathematical ability. Both students are assertive and try to convince the other from time to time. This creates the global interaction pattern in the first parts of the interview: Individual work with co-operatively shared results. We may see this, for example, in Segment 1 (Ra91Ra97), where each student raises a different proposal about a possible attribute to be discovered (Ra91 and Yo92), and elaborates what he has in mind individually (Ra93 and Ra97). At the same time, each is quite interested in his peer’s discovery and therefore asks for clarification (Yo94, and Yo96). The detailed analysis of the interaction patterns in segments 3, 4, and 5 will be based on the transcripts presented in subsection 3.1 as well as on the diagram on the left side of Figure 3. (The right side of Figure 3 will be used in the next subsection.). As mentioned in section 2, the arrows in the diagrams point to the nearest referent(s) of any specific utterance only. Since this nearest referent may in turn relate to previous utterances, we may by inference have a dialectic connection from the given utterance to a “few generations” of previous utterances, as will be demonstrated in the following analysis. ------------------------------------------Insert Figure 3 about here ------------------------------------------Segment 3 (Yo263-Ra290). In the first part of this segment (Yo263-Yo273), there is no seen interaction between the peers. Ra is mostly silent. The segment starts with Yo’s proposal (Yo263) for a new strategy to answer the question “why the difference is 12”, that was last raised in Ra252 (category 1 arrow from Yo263 pointing to Ra252). In Yo265, Yo267, Yo269, Yo271 and Yo273, Yo elaborates his proposal (category 2 arrows from each of these utterances, to Yo’s previous utterance). The interviewer supports this elaboration and makes small contributions (category 6 and category 2 arrows, pointing toward Yo’s elaboration flow). Does this sub-segment indicate a situation of ‘no interaction’ between the two students? Looking at the next sub-segment (Ra274-Ra290), we may answer this question negatively. Ra starts (Ra274) with questioning and opposing the main idea that Yo elaborated - the expanded distributive law (category 4 arrow to Yo273). This may be interpreted as evidence that Ra was quite present and involved in Yo’s proposal and elaboration in the previous sub-


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segment. Yo defends his previous elaboration (category 5 arrow from Yo275 to Yo273). Three additional successive oppositions follow, in which Ra again questions the validity of Yo’s conclusion in Y273. This is done either directly (category 4 arrow from Ra277 to Yo273), or indirectly via Yo’s defence, which is also expressed as a re-opposition to Ra (category 4 arrow from Yo278 to Ra277). Within his last opposition (Ra279) Ra starts with the elaboration of the expanded distributive law as well (category 2 arrow from Ra279 to Ra277). This elaboration process continues successively (category 2 arrows from Ra279, Ra283 and Ra285). In a more dialectical way they are all pointing via Ra277 towards Yo’s conclusion of his elaboration in Ra273. During Ra’s elaboration Yo is silent; only the Interviewer supports Ra (category 6 arrows from In280, In282, In284 and In286 toward Ra’s elaboration flow). In Yo287 Yo is active again and queries/corrects both the interviewer and Ra (category 4 arrows). After a short explanatory co-operation (category 3 from Ra288 and Yo289), Ra agrees (category 5 from Ra290) that the process he went through in the last subsegment makes sense. The above segment shows an interaction pattern that is very typical for Yo&Ra. Although the “seen interaction” indicates mostly individual work done by each student, the conclusions are at the end agreed on by both of them. A deep analysis suggests a quite intensive type of interaction between them: At the beginning Yo has a proposal and he elaborates it step by step, while Ra is silent but follows very carefully the proposal and the elaboration. Then Ra starts his elaboration of the same proposal by opposing it. This opposition expresses his struggle to understand the elaboration, which Yo just completed. Only then Ra elaborates it at his own speed. He completes this process by expressing his agreement with the process. During Ra’s elaboration Yo is silent. Each of them has the need to elaborate a given idea in his own way. By remaining silent for a while, each student respects his friend’s need. But both of them reach a point where they refer explicitly to the other’s elaboration and criticise it, until they reach an agreement. Segment 4 (In291-In304a). In this segment the students on the whole do not relate to each other. This can be seen in the diagram where most of the arrows do not reach the peer’s utterance. In addition, there are almost no vertical arrows, meaning there are no indications that the students relate to and continue even their own ideas. This is especially true for Ra. He “shoots” utterances about computations or algebraic manipulations, which seem to belong to an elaboration type, but are sporadic rather than relating to a specific idea that is elaborated. Yo, after relating to the main motive of the justification process (category 1 arrows from Yo292 to Ra218 and In291), tries in Yo294 to propose a strategy, which he unsuccessfully tries to elaborate in Yo296 (category 2 arrow from Yo296 towards Yo294). Until Yo298 his utterances are directed only to himself. There is one short and isolated trial of co-operation


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between the two boys in Yo298 and Ra299. The interviewer ends this segment in In304a by a category 6 utterance directed to both students. Segment 5 (In304b-Ra319). In In304b the interviewer gives a clue to a new strategy for completing this stage of the justification (category 1 arrow from In304b to Yo292). This clue is the trigger for the last segment including its pattern of interaction. During most of this segment, until Yo315, each student elaborates the new idea, almost by himself. Ra starts his elaboration in Ra305 and continues in Ra310, Ra312 and Ra316 as is indicated by the corresponding category 2 arrows. Yo starts his elaboration by questioning Ra and the interviewer in Yo307 (category 4 arrows to Ra305 and In304b). His question seems to express surprise, maybe at the strategy itself and maybe at not having thought of it earlier. Yo then continues his elaboration in Yo309, Yo311 and Yo315 as indicated by the corresponding category 2 arrows. Interestingly, Ra316 serves the elaborations of both students (additional category 2 arrow to Ra315) and constitute the beginning of a short co-operation between the two (Ra316 to Yo319), which completes this segment and the whole interview. In Ra318 Ra concludes and explains why the difference is always 12 (category 3 arrow to Yo292, the last explicit statement of the universality of the DPP, and to Yo317). Finally, Yo reformulates the explanation in detail (category 3 arrows to Yo292 and Ra318). This segment shows a pattern of interaction, which is similar in a sense to the interaction pattern in segment 3. In both segments the peers elaborate an idea mostly individually, and then share the conclusion cooperatively. There is, however a difference: The individual elaborations occurred successively in segment 3 but simultaneously in segment 5. We note that in spite of the differences in the interaction patterns, each of the above three segments starts in a proposal (category 1 arrow) that relates to something said more than a few utterances ago.

Category 1 - Control

Ra

In

Yo

Total

5

(36%)

4

(28%)

5

(36%)

14 (100%)

2 - Elaboration

17

(41%)

4

(10%)

20

(49%)

41 (100%)

3 - Explanation

2

(40%)

0

(0%)

3

(60%)

5 (100%)

10

(44%)

3

(13%)

10

(44%)

23 (100%)

5 - Agreement

4

(50%)

1

(13%)

3

(38%)

8 (100%)

6 - Attention

3

(14%)

18

(82%)

1

(4%)

22 (100%)

41

(36%)

30

(27%)

42

(37%)

113 (100%)

4 - Que ry

Total

Table 1. Distribution of utterances into categories for Yo&Ra


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The pattern of interaction and the symmetry between the students receive additional validity by counting the categories of the different utterances during the whole justification segment Yo217 –Yo319 (see Table 1). The two students show a very similar division of the utterances among the different categories: few proposals (category 1), a large number of elaborations (category 2), few explanations (category 3), quite many queries/oppositions (category 4), very few agreements (category 5) and attentions (category 6). The symmetry between the peers becomes even more apparent when one looks at the distribution of the utterances within one category (in percentages). For categories 1, 2 and 4, this distribution is almost the same for the two students. As can be seen from Table 1 as well as from diagram in Figure 3, the interviewer’s utterances are mostly of category 6 type, supporting the students and controlling the situation. 3.3 The epistemic actions of abstraction of Yo&Ra In this sub-section we will go through the interview again, highlighting what we see as a process of abstraction by the students. This will be analysed in the terms of the nested RBC model of abstraction. In our analysis we will relate to Yo&Ra’s justification of the DPP first in a global way, and then to segments 3, 4 and 5 in more detail. From the very beginning of the justification process, three main points are very clear: (a) The students know that their goal is to justify (to prove, to explain) why the difference is always 12. We may conclude this from numerous statements, which were made along the relevant parts of the justification, including: Ra218 Yo219

It is 12. How to explain that, I don't know. We have to explain. One moment [both students watch at the computer screen] one diagonal is X plus 8. Right?

Yo227

We need to multiply. OK? The first diagonal is x multiplied by 15. Now the second diagonal is X plus 2 multiplied by 13. So why is the difference 12?

Ra252

No! We ask why the difference is 12.

Ra258

I try ... why it is 12, the difference.

Yo292

Well, so now let’s check whether it makes sense that the difference is 12.

Ra318

No, in the second one we have 8 Xs plus 12; so, really, the difference is 12 because one can cancel this, and then we have here a difference of 12. X times 6, that’s 6 Xs plus X times 2 that’s 8 Xs plus 12; so this is 8X+12, and here it’s 8X, here it’s 8X without 12; that’s why the difference is 12!

Yo319

These statements unify the whole part of the interview in which Yo&Ra struggle to justify and construct an algebraic proof to the DPP (Yo217 -Yo319). While Ra218, Yo219, Yo227, Ra252, Ra258 and Yo292 show that during the work the students are explicitly aware of what they are looking for, Ra318 and Yo319 show that after


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completing the justification, they are aware of the fact that the DPP was proved. The regularity with which these statements recur shows that the whole process is controlled by the students’ need to justify the DPP. (b) During the beginning stages of the process, the students are aware and admit that they don’t know how to justify (see, for example Ra218). (c) The students do not consider an inductive argument as a proper justification, and they are very naturally looking for an algebraic argument. For example, Ra discovers and verifies the DPP by substituting numbers in the generalised seal (Ra93 – Ra97 in segment 1 above), but when asked to justify its universality, he states that he does not know how to explain it (Ra218). As was pointed out in section 2, the students were expressing generalisations in algebraic symbols and considering these algebraic expressions as standing for all numbers since the beginning of the year. On the other hand, they did not have any experience in formulating a justification algebraically. But their tendency to link generalisations to algebraic notation, combined with their need to justify these generalisations naturally led them to link justification to algebraic notation. This link is evidenced from the very beginning of their efforts to justify (Yo219) and along the whole paragraph (Yo217 - Yo319) of their justification. In the previous article we showed that constructing is a combination of the three epistemic actions where recognising actions are nested in the two others, and building- with actions are nested in constructing actions. In this paper, we show that the nested model is even more intricate. The constructing action may be quite long and contain shorter segments, which themselves are constructing actions. For example, in the case of Yo&Ra the main constructing action (which we will name C 1 ) takes the entire segment Yo217-Yo319 and includes the expansion of the distributive law (which we will name C 2 ). Building-with and recognising actions are nested in C 1 as well as in C2 . In other words, constructing actions, like building-with and recognising actions, may be nested in a more global constructing action. This shows that the nested RBC model may describe a dialectic way of mathematical thinking, in which there are ‘pockets’ of breakthrough progress (mostly the C 2 segments) and segments of local small progress, and segments with no progress at all. Figure 3 (right hand side) describes the model for segments 3, 4 and 5. Level 1 represents the whole process of C 1 , in which the epistemic actions of level 2 are nested. These level 2 actions are constructing (C 2 ), building- with and recognising actions. There are also level 3 actions, which are the building-with and recognising actions that are nested in C 2 constructing actions. More specifically, segment 3 is the C 2 construction of the extended distributive law, Segment 4 which is a “no progress” segment, and Segment 5 is again a C 2


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construction, the completion of the proof of the DPP. We now present a more detailed, protocol-based description of the RBC model for these three segments. The extended distributive law (Segment 3). This construction starts with Yo’s breakthrough suggestion to use the distributive law. The students both recognise the structure of the expression X(X+8) as appropriate to apply the (simple) distributive law. Yo then builds-with the elements of (X+6)(X+2) and obtains the correct expansion (Yo273); in other words, he constructs the expanded distributive law. Although he makes no explicit reference to the simple law, we surmise that the immediately preceding use of the simple law has guided him in building the more complex expression. Now Ra takes over. After being momentarily confused by the many addends, he goes through the same constructing process Yo went through, step by step building up the expanded law with the elements of the expression (X+6)(X+2). This C 2 construction does not occur in the void but as a crucial part of the justification of the DPP. It is therefore nested in the C 1 construction of the algebraic justification of the DPP. The ‘no progress’ segment (Segment 4). This starts with evidence that the students are aware of the C 1 level – given the new expressions XX+8X and XX+6X+2X+12, they check “whether it makes sense that the difference is 12” (Yo292). But then they raise various isolated ideas, which are mostly recognising actions, like seeing the above expressions as an equation to be solved (Yo294 and Yo296) or as a basis for numerical computations that will show the difference of 12 (Ra295 and Ra299). These recognising actions are not joined together to a more global structure, even to a building-with one. The segment ends with the interviewer’s “You were really 90% because you don’t know what X time X is” (In304a), in which he indicates an end to this interview. Completing the proof of the DPP (Segment 5). This C2 construction segment starts, like the previous one, from a breakthrough idea. In this case, it is triggered by the interviewer’s clue in In304b: “but, this and this - it’s the same thing.” Yo&Ra adopt this idea and eliminate, step by step, the similar expressions from the two expressions for the diagonal products. These steps require recognising (the similar expressions) and building-with (the elimination) actions, led by an explicit awareness of the need to complete the construction of the justification. When they complete it and are left with only the number they are aware not only of having reached the aim but also of the important stages they have passed. They can explain the proof explicitly repeating the whole process of construction (Ra318 and Yo319). This C2 construction completes the algebraic justification, and thus it also completes the C1 construction in which it is nested. This C 1 construction, in spite of its dialectic nature, and especially when it is expressed by means of the C 2 actions nested in it, is carried out within the norms of mathematical proof through its algebraic and Excel notations. Hence, we claim that, in addition to the construction of the algebraic justification of the DPP itself, the C1 Abstraction in peer interaction.doc

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constitutes a more global and deeper construction, namely the realisation that justification may be expressed algebraically and that some algebraic computations (including the C 2 constructions) have to be done while accomplishing it. In this sense C 1 and C2 in the Yo&Ra case have different nature: From the beginning of their struggle to construct the DPP justification, the students are at the level of the C 1 construction. On this C 1 level their progress is controlled and monitored by their awareness and their need to accomplish the DPP justification. During this process, they face algebraic obstacles, which are quite unfamiliar to them. Overcoming these obstacles necessitates the construction of new mathematical structures, which are the C 2 level constructions. These C 2 constructions are controlled only indirectly by the motive of the C 1 Construction. The students enter these C 2 level “adventures” without any knowledge about the needed mathematical structures, and they have to discover as well as to construct them. The C 2 Constructions thus make the C 1 level into a deep holistic construction, which goes beyond the specific construction of the DPP justification, and in which the constructions of unfamiliar algebraic structures are nested. In this sense C 1 is an activity of vertically reorganising previously constructed mathematical knowledge into a new mathematical structure, which fits our definition of abstraction. A comparison between the justification of the DPP and the justification of the DSP (see segment 2 in section 3.1, Yo184 – Yo198), might make the characteristics of the above C 1 construction quite clear. Ra justifies the DSP verbally and algebraically in a very accurate way. He combines algebraic elements he already recognises, in order to build-with them the algebraic justification of the DSP. He does not have to construct new mathematical structures. Therefore we consider his algebraic justification of the DSP only as building- with rather than as constructing. 3.4 Relationships between epistemic actions and interaction patterns for Yo&Ra The Yo&Ra interview is a case of collaboration between the two students. This collaboration finds its expression in the students’ cognitive RBC actions on one hand, and in their pattern of interaction on the other hand. The RBC flow and the flow of the interaction patterns are developing in parallel. Both of them are d ifferent “indications” of the single collaborative process revealed in the interview. Our understanding of this process is dependent on our understanding of both, the RBC flow and the interaction patterns, as well as the relationships between them. In this subsection we will try to throw some light on these relationships. Globally, Yo&Ra share the activity, because they share the motives of searching for the mathematical properties of the “seals” and of justifying these properties; they also share the justification processes themselves, as well as their conclusions. We focused above on the DPP justification flow. There the global perspective of sharing is expressed in the cognitive Abstraction in peer interaction.doc

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flow on one hand, and in the interaction patterns on the other. In the preceding subsection, we analysed globally the nested RBC actions produced by the students within the cognitive flow. We claimed that the students constructed a global structure of meaning for algebraic justification (the C 1 action). From the interaction perspective, the long-range category 1 and category 3 arrows can be considered as the “glue” that ties this justification process together. What is the role of these long-range arrows? 

They are distributed equally between the two students; each student contributed three category 1 statement and later one category 3 statement from which the long-range arrows emanate. This indicates that the students equally share the control of the justification process and the ability to explain it.



These statements either concern the question why the DPP holds (the first six statements, arrows of category 1), or they explain the DPP (the last two statements, arrows of category 3). The corresponding arrows are connected to the beginning and/or end of the three analysed segments, or their sub-segments. Examples: – The category 1 arrow from 263 (beginning of segment 3) is pointing to 252. – The category 3 arrow from 289 (the end of segment 3) is pointing to 273, and therefore indirectly to 263 and 252. – The category 1 arrow from 292 (the beginning of segment 4) is pointing to 218. – The category 3 arrow from 319 (the end of segment 5) is pointing to 292 and therefore pointing indirectly towards 218.

These comments and examples reveal some ways in which the interaction patterns and the RBC cognitive actions are tied together. Long-range interactions occur between statements that are milestones in the RBC flow. It seems that in a sense the interaction pattern has nested characteristics similar to the RBC flow, where various patterns of interaction are nested in the overall global collaboration. And the cutting edges of the interaction patterns are those that at the same time define the different segments of the RBC flow. In other words, the cognitive segmentation we started out from fits the interaction as well. Now we are in a position to analyse locally the match between the two flows in segments 3, 4 and 5. The following examples demonstrate that this match is evident locally as well as globally: -

Both, segments 3 and 5 are C2 type segments from the RBC point of view. Their patterns of interaction are quite similar: In both segments the peers elaborate an idea mostly individually (either one after the other, where the peer remains silent as in segment 3, or the two in parallel as in segment 5) and then share co-operatively their conclusion.

-

In segment 4 (the ‘no progress’ segment from the RBC point of view), the students are doing only recognising actions which do not join together, not even to a Building-With


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structure. From the interaction point of view, these actions are expressed in sporadic elaboration type utterances, which do not join together to any individual or shared elaboration, meaning that there is no co-operation between the two boys in this segment. -

In In304b the interviewer’s clue is a trigger to a new C2 construction as well as to a new interaction pattern. On the whole we conclude that the symmetry between the two students is expressed

cognitively as well as by means of balanced interaction patterns. Each individual constructs the notion of algebra as a tool to justify, as well as the lower level C 2 constructions, and their contributions to the construction process as expressed in the interaction patterns are globally and locally symmetric. 4. The inte rvie w of the second pair (Ha&Ne) We now turn to the analysis of the work on the DPP of the second pair, two girls called Ha and Ne or, as a pair, Ha&Ne. Like in the case of the first pair, we ask how the flow of epistemic actions is shared by the two students, as well as how each student contributed to this flow. The section will be structured into four subsections parallel to those of Section 3 and dealing, respectively, with an overview of the Ha&Ne interview, the interaction patterns of the pair, their epistemic actions of abstraction and the relationships between the epistemic actions and the interaction patterns. 4.1. Overview of the Ha&Ne interview A brief description of the interview stages leading up to the proof of the DPP will serve as appropriate background. In response to Q uestion 2 (“Find as many common properties as possible for this type of seal”, see Figure 1), Ha&Ne produced ten properties, most of them concerning constant differences (such as B1-A1=6) or comparisons of sums or differences (such as the DSP). They also included a divisibility property (the sum of all four elements is divisible by 4) but not the DPP. Interestingly, in response to Question 3 ("Compare the sums of the diagonals in the seal; do you think the property you found is true for all seals of this type?"), Ha&Ne stated the DSP without showing any urge to explain why it holds for all seals. After they stated the DPP in response to Question 4, the issue of explanation came up, and they proceeded to prove the DSP rather than the DPP: HaNe62 [They read Question 4a and compute mentally.] Times 5, how much is that? 45, 3 times 11, 33, 45 minus 33. Ha63 They have a difference of 12. Ne64 Perhaps a multiple of 4? Ha65 Ah! That the difference is a multiple of 4?


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Ne66 Ha67 In68 Ha69

Ne70 Ha71

In72

How much is 13 times 9, it’s 117, and 7 times 15 is 105. 117 minus 105 is also 12. The difference between the products is always 12. Do we have to explain? Do you feel that you have to explain or is it clear that it is so? Wait, here it’s X plus X+8, so it is like 2X+8; and here, this is X+6 and this X+2, which is 6 plus 2, which is 8. That is, in fact, the same thing, because it is 2X+8 in both. 2X+8 plus 2X+8? No, wait, the 3 is the X, so there is an X here. This is X+8, right? So together it is 2X+8. Here there is X+2 [pointing to the lower left cell], and here there is X+6. So together this is also 2X+8. [Pointing to the DPP and addressing I:] To explain this? Here it is not necessary.

We note that like for Yo&Ra, this proof is fully algebraic. Moreover, Ha not only produces an argument in Ha69 but also reacts to Ne70 with an explanatory reformulation of the argument. We further note that Ha&Ne ask the interviewer about each property, whether they should explain it (Ha66, Ha70). This may be interpreted as showing that from the beginning, they never questioned the universality of the properties. They come back to the DPP when reading Question 6c, where they are explicitly asked to justify their claim that the property is universally true. The ensuing work constitutes the main focus of our analysis. From a cognitive point of view, it decomposes, quite naturally, into five segments, in each of which Ha&Ne focus on a different issue. The first segment starts with a discussion whether or not they had justified the DPP earlier: Ha111 Ha112 Ha113

We didn't want to think about this before. This is really somehow like that. Because this is the first diagonal, and that is the second, right? No. Yes. So, like, see: X plus 8, X, like, then it is 8X plus XX.

Here they embark on the computation (8+X)X = 8X + XX for the first (main) diagonal. Along the way, they explicitly mention the use of the distributive law: Ha121

So, like, one does the distributive law.

The second segment starts when they turn their attention to the other diagonal: Ha133 And this [thinks] ... Ne134 It's impossible to do the distributive law here. Wait, one can do ... Ha135 This is 6X. Ne136 This is 6X times X and 6X times 2. Ha137 Wait, first, no ... Ne138 Yes. Ha139 No because this is X plus 6, this is not 6X, it's different. Wait. First one does ... X; then it's XX plus 2X, and here 6X plus 24. Then ... In140 Not 24, why 24? Ha141 Ah, 12. Abstraction in peer interaction.doc

26

In this second segment, they thus deal with the applicability and application of distributive law to a more complex case, the extended distributive. The issue of the extended distributive law arose because of the need to compare the two diagonals. In the third segment, they are momentarily uncertain how to continue. They wonder whether they need to know the value of X to proceed. Ha proposes to use the computer to check a large number of cases. The interviewer reminds them that they were able to deal with the first diagonal without knowing X, nor using the computer: In149

You started well. That is, you wrote, you said that the first diagonal is XX plus 8X, right?

This leads to the fourth segment: Ha152 Ne153 Ha154 In155 Ne156 Ha157 In158 Ha159 Ne160 Ha161 Ne162 Ha163

Ah, it's XX plus 8X, but I don't know, like, how this will also be XX plus 8X. Like, it has to be. Is XX a square root? I have the first part. This is XX, so this is OK. Yes. Is XX a square root? … plus 8X. Here I have 6X ... Yes. Ah, and 2X, can I do this? Because 6X ... Is XX a square root? You can write this. Ah, yes, XX is X to the power 2, because it is X times X. Wait. XX is X to the power 2 plus 8X, wait ... Write this. Wait, it's X to the power 2 plus 6X, plus 8X, but there is also, like, plus 12. Ah, so, like, plus 12 because this is bigger by 12. Understand?

In this segment, the girls apply the extended distributive law to the comparison of the two diagonals. Moreover, a question of the interpretation of the algebraic expression XX appears to intervene. The segment culminates in Ha163 where the difference of 12 between the two diagonals becomes significant. Ha163 Ne164 Ha165 Ne166 Ha167 Ne168 Ha169 Ne170 Ha171 Ne172 Ha173 In174

… Understand? Like, yes, it's the same thing but this is bigger by 12. This is XX. Yes. This is 6X plus 2X. This is the same thing. This is 8X. Yes. And this is plus 12, because of that, it's bigger by 12. [Nods.] OK, we can explain each one and leave the formula. Yes.


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Ha175

[Writes down, with Ne's help, what they found.] X to the power 2 plus 8X, and here, here it is, wait, these two are X to the power 2, this is 2 times X, plus 6 times X, plus 6 times 2. So together it is 8X and 2 times 6 equals 12.

From Ha173, we have entered a concluding stage, in which the students, together with the interviewer, put some closure on their work. We have chosen the second, fourth and fifth segments of this interview for detailed analysis of the epistemic actions and the interaction patterns for the following reasons: The interaction patterns during the fourth and fifth segment are typical for much of the interaction in the Ha&Ne collaboration. The interaction pattern during the second segment, though exceptional for Ha&Ne occurred at a crucial moment. These patterns will be exhibited in Subsection 4.2. In the second segment, a clearly identifiable constructing action occurs and in the fourth segment a clearly identifiable building-with action occurs. The nature of these epistemic actions will be demonstrated in Subsection 4.3. Finally, some correspondences between the epistemic actions and the interaction patterns will be established in Subsection 4.4. 4.2 Interaction patterns of Ha&Ne In this subsection, like in the parallel subsection about Yo&Ra, we describe the interaction patterns between the two girls and with the interviewer on the basis of the arrow diagrams. We remind the reader that these arrow d iagrams show, for each statement, to which previous statements it refers and what its discursive role is. Just as for the first pair of students, an overall impression of the patterns of interaction can be gleaned from a count of the interaction categories which covers the entire relevant part of the interview (108-175). This count is presented in Table 2.

Category 1 - Control

Ha

In

Ne

Total

6

(67%)

2

(22%)

1

(11%)

9 (100%)

2 - Elaboration

13

(76%)

0

(0%)

4

(24%)

17 (100%)

3 - Explanation

14 (100%)

0

(0%)

0

(0%)

14 (100%)

4 - Que ry

7

(41%)

2

(12%)

8

(47%)

17 (100%)

5 - Agreement

1

(9%)

0

(0%)

10

(91%)

11 (100%)

6 - Attention

3

(20%)

7

(47%)

5

(33%)

15 (100%)

44

(53%)

11

(13%)

28

(34%)

83 (100%)

Total

Table 2. Distribution of utterances into categories for Ha&Ne From Table 2 we can see an asymmetry between the two girls, Ha contributing quite a bit more than Ne to the activity. This asymmetry becomes even more pronounced, when one


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focuses on the different categories: Ha gives many explanations (category 3) whereas Ne gives none at all. Ha is also more active than Ne in producing elaborations (category 2) and in monitoring the flow of the argument (category 1). Ne mainly questions and confirms. The asymmetry between the roles of the two girls merits attention especially in comparison to the almost complete symmetry between Yo and Ra (Section 3). The interviewer is even less active than in the Yo&Ra interview, but the composition of his contributions is similar. One might remark that he confirmed a little less, and questioned a little more than in the Yo&Ra interview. ------------------------------------------Insert Figure 4 about here ------------------------------------------We will now examine the interaction patterns in the different segments separately and in detail on the basis of the diagrams presented in Figure 4 on the left side. Since the second segment is less characteristic for the Ha&Ne interaction than the others, we start with the analysis of the fourth, and then the fifth segment. The conspicuous features of the arrow diagram for segment 4 (Ha152-Ha163) are the following:  Many arrows point to statement Ha152; this shows that Ha152 is a statement on which much of the ensuing action focuses. We observe that moreover, Ha152 is not only an elaboration of previous statements (category 2 to Ha139, In149) but also has a role in controlling the flow of the discussion (category 1 to Ha111).  The roles of the two students are very different. With a single exception, all references are to Ha’s statements; this is true for Ha’s own references, for Ne’s references, and even for those of the Interviewer.  Ha’s role is to develop the argument; most of her statements are elaborations (category 2) of previous statements, either the immediately preceding one, or Ha152. Only at the very end (Ha163) does she emerge from the sequence of elaborations to relate to the global flow of the argument (category 1).  During the elaboration, Ha shows some uncertainty, twice questioning herself (loops of category 4). The Interviewer encourages her (category 6).  It is difficult to assess Ne’s role from the diagram; clearly, it is a minor role. She is repeatedly questioning Ha, but without relating to the content, we cannot assess the significance of the questions. A closer inspection shows that the three questions in Ne153, Ne156 and Ne160 are, in fact, identical, and relate to a technical detail rather than to a central point of the argument. Their intention might simply have been a plea for Ha to slow down.  Only toward the end of the segment (Ha161) does Ha accede to this request and give an explanation (category 3). To this, Ne reacts with by contributing to the control of the action.


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The conspicuous features of the arrow diagram for segment 5 (Ha164-Ha175) are the following:  This segment is different from the other segments discussed in this article in that there are no category 1 arrows at its beginning. This expresses that the segment is a direct continuation of the preceding one, without any intervening reorientation. On the other hand, the segment is similar to the preceding one in that the initial statements (in this case Ha163 and Ne164 constitute the focus of attention for almost the entire segment (down to Ha171).  Like in the preceding segment, the roles of the two students in this segment are different from each other, but they are also different from their roles in the preceding segment.  Ha’s role clearly is to explain (as opposed to elaborate in the preceding segment) statements Ha163 and Ne164 to Ne and, to a lesser extent, to herself.  As they go along, Ne regularly confirms or supports Ha’s explanations.  The category 1 references from Ha173 to Ha111 and from Ha175 to Ha67 shows that a (concluding) reflection is taking place. Although the arrow by itself (without content) does not convey this, in fact the students ask: Have we finished, are you satisfied? Does this argument satisfy the socio-mathematical norms of proof? These two segments are typical for the manner in which Ha and Ne collaborate. There are several other instances in the Ha&Ne interview, where a segment like the fourth, in which Ha develops an idea, is followed by a segment like the fifth, in which the two students discuss and attain a more profound grasp of the idea. It is also typical that even when there is significant interaction between the two students, Ha still leads, explaining her ideas to Ne, while Ne participates with questions and comments. The second segment (Ha133-Ha141) shows a very different kind of interaction pattern, and one that is less typical for Ha&Ne. Although it is rather short, this segment consists of four clearly distinct parts (see Figure 4):  At the beginning (Ha133, Ne134), there are several control statements (category 1) referring to earlier statements (Ha111, Ha121). Apparently, the students situate their present focus of attention within the larger problem they are working on.  Next, they produce a first elaboration (category 2) of this topic.  This leads to a dispute (category 4), which is ended by Ha with a (presumably convincing) explanation in Ha139a.  This explanation leads to a revised elaboration and a small correction in the last part. The most prominent feature of this segment, as compared to the ones analysed before, is the considerable symmetry between the contributions of the two students. In all except the fourth part, the students contribute equally, and with statements of the same category. Also, at least until Ha139a, there are almost as many references to Ne’s statements as there are to Ha’s. Abstraction in peer interaction.doc

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The interaction patterns of the Ha&Ne interview show, in addition, a feature that transcends the three segments and links between them. These are the "long" arrows that connect statements Ha173 to Ha67 and Ha111, statement Ha171 to Ha163, Ha163 to Ha67 and Ha152, Ha152 to Ha111 and Ha133 to Ha111. Most of these arrows belong to category 1, and the remainder to category 3. The category 3 arrows appear mainly toward the end of the interview. In other words, the long arrows direct the flow of the activity and close it at the end. It is not surprising then, that many of the statements, which are linked by these arrows, begin or end segments; in other words, these statements are the ones which define and/or begin a new stage in the activity or, alternatively, end and summarise such a stage. 4.3 The epistemic actions of abstraction of Ha&Ne For this subsection, we turn our attention away from the interaction and on processes of emergence of knowledge structures, which we deem to be new and give a new perspective for Ha&Ne. We show how the nested epistemic actions of constructing, building- with and recognising are indicative for such processes. As we did in section 3.3 for Yo&Ra, we do this for three separate constructions, for the extended distributive law, for the proof of the DPP, and for the general perspective that algebra can serve as a tool for the justification of general properties. The extended distributive law. The second segment of the Ha&Ne interview (see Subsection 4.1) follows the application of the simple distributive law to the first diagonal of the seal. In Ha133 and Ne134, the students focus on the second diagonal, recognising that it has a more complex structure than the first, and wondering whether it is possible to apply the distributive law here as well. While this is not something they know from earlier experience, they are motivated by the need to simplify this expressio n in order to progress toward their aim of justifying the DPP. The recognition of the expression 6X (in Ha135), allows them (in Ne136) to use distributive law: 6X(X+2)=6XX+6X2. Although it is not clear from the transcript why the expression 6X arose, recognising it as a single unit gave the girls the necessary point of view to realise the applicability of the law, though not to the correct expression for the second diagonal. The correction of the mistake forces them to separate the 6 and the X (in Ha139); in spite of this added complexity, they are now able to obtain the correct simplification by building separately with the X and separately with the 6 and adding the results together. To the best of our knowledge, Ha&Ne had not seen the extended distributive law applied before this interview. Alternating Recognising and Building-With existing knowledge has thus allowed them to construct a new knowledge structure, the extended distributive law. This process is represented in rows Ha133 to Ha139 of the right side of Figure 4 as follows: At the lowest level (level 3), recognising and building-with actions alternate; there is no strict association of each statement to one of these actions since some statements (such as Abstraction in peer interaction.doc

31

Ne138) cannot be categorised, whereas others (such as Ha139) include both, recognising and building-with. Therefore, at level 3, the diagram only indicates the alternation between these two epistemic actions. However, at the next higher level (level 2) these actions, taken together over the entire exchange from Ne134 to Ha139, constitute the construction of the extended distributive law. This is indicated in the diagram by C 2 : a constructing action at level 2. Concerning the relationship of levels 2 to level 3, we note that no single action at level 3 has been identified as a constructing action; rather, constructing is a composite action in which (alternating) recognising and building- with actions are nested. With respect to the relationship between level 2 and level 1, we only note at this stage that what happens at level 2 cannot be fully understood without taking into account what happens at level 1; indeed, we have given a description of the students’ actions without asking what drives these actions. The picture will be completed at the end of this subsection. The proof of the DPP. By the start of the fourth segment (Ha152-Ha163), most elements needed for the proof of the DPP had been attended to but the students may not have been presently aware of all of them. Some of these elements are technical such as the simplified expressions for the first and second diagonals (last mentioned in In149 and in Ha139/Ha141, respectively). Others are important for the flow of the argument such as the statement of the DPP (in Ha67) and the plan to inspect the two diagonals (mentioned in Ha111). In Ha152, the two diagonals again become the focus of attention. The plan is now more specific and detailed than was possible in Ha111. There is an explicit reference to a comparison of the two diagonals and to the fact that the two corresponding expressions have to match. In the sequel, Ha systematically uses the elements which have been prepared to build-with them the completion of the proof of the DPP. She alternates recognising (e.g., Ha154 and Ha161) and building-with (e.g., Ha157, Ha159) actions at level 3, all of which are familiar to her. She combines them artfully but without the need to restructure her existing knowledge. This passage culminates in Ha163 where the number 12, which was briefly mentioned in Ha141 as a detail of computation, becomes significant for the students as being the difference between the two diagonals. Since this corresponds precisely to the claim, which had been made in Ha67, it completes the proof by combining all the elements that were available but not necessarily recognised as significant in Ha152. According to our interpretation that this proof has been achieved by combining previously known elements without the need to restructure knowledge, we classified this segment as building-with at level 2. The culmination of the Construction in Ha163 is followed by the fifth segment, which is of quite a different nature: The students review what Ha has developed during the previous segment. They arrive, in Ha175, at a clear and convincing formulation of the proof. As a pair they are not constructing something new, nor even building-with the acquired elements but rather recognising, in the literal sense of re-cognising step by step the previously developed


32

argument. As an individual, Ne might at this stage be constructing the proof of the DPP but we do not have enough evidence to make this claim. This is the reason why we did not include a RBC diagram for this segment. Algebra as a tool for the justification of general properties. Just like for Yo&Ra, and for the same reasons, the (algebraic) proof given by the students for the DSP is a building- with rather than a constructing action. Similarly, most of what has been stated with respect to Yo&Ra about the nature of C 1 and the role of the completion of the proof of the DPP in the students’ realisation that algebra can serve as an efficient tool for justification applies here as well. We now accumulate evidence to show that Ha&Ne have constructed this encompassing knowledge structure during the interview. The first time Ha&Ne connect a justification to algebraic computations is in Ha67-Ne69, where they react to the interviewer’s question about the need to explain by producing an algebraic computation. The fact that the question referred to the DPP and the computation to the DSP does not alter the fact that they link justification and algebraic computation. This recurs in Ne103-Ha105 (not cited) as response to question 6a, and again in Ha111-Ha113 as response to question 6c. Although the students do not exp licitly refer to justification in these instances, there is no need for them to do so since the questionnaire explicitly asks them to justify their claims and try to convince others. To this requirement for justification, they react by starting algebraic computations. It is their need or wish to justify the DPP that drives their computations, whether these computations amount “only” to building- with as in segment 4 (the comparison of the two expressions for the two diagonals) or whether they result in constructing as in segment 2 (the extended distributive law). Interestingly, between these two segments, which are similar and somewhat technical in character, the students raise, and rather quickly discard again, the possibility that the computer could be used to convince the interviewer that the DPP is true. We interpret this as a sign that some uncertainty prevails as to what counts as a mathematically convincing argument; this reinforces our claim, based on the teacher’s report, that algebraic proofs are a novelty for the students. The students complete the algebraic structure in Ha163, where they obtain two expressions that are identical except for the term 12 that appears only in one of them. The link between algebra and justification now works in reverse direction: The students realise that the term 12 is significant for the justification of the claim. Once again, the (result of) the algebraic computation is linked immediately and explicitly to the claim that had to be justified. Finally, in 171-173, the link between explanation and algebraic formula is mentioned explicitly. We thus infer that the didactic strategy was successful for this pair of students and


33

that their construction of the specific DPP proof led, in all likelihood, to the more general realisation that algebra is a useful tool for proving certain types of general mathematical claims. This realisation constitutes a new perspective that has arisen in the course of the interview and was driven by the students’ desire to achieve the specific proo f of the DPP. The links between justification and algebraic computation, which were acted out during the earlier stages, became conscious and explicit during the later stages. We also notice that because of the encompassing nature of the construction, this is a process going on during the entire interview rather than in specific identifiable segments. Nevertheless, we were able to identify a number of specific statements in which the students’ awareness of the power of algebra for justification became more and more apparent. We claim that the students’ actions throughout the entire episode comprising all five segments contribute to the establishment of the link they make between algebra and justification. In fact, almost all their actions throughout the entire episode contribute to the justification and thus to their realisation that such a justification can be achieved by algebraic means. It is this realisation which led to the explicit statements at the end. Therefore, the epistemic action C 1 of constructing the power of algebra as a tool for justification is composed of all the other epistemic actions. These nested actions include not only recognising and building-with actions but also the lower level constructing action C2 for the extended distributive law. All these actions taken together constitute C 1 (Figure 4). 4.4 Relationships between epistemic actions and interaction patterns for Ha&Ne In the two previous subsections, two separate and largely independent descriptions of the same interview have been given. The first description in Subsection 5b had a social focus on interaction patterns and was kept content independent as much as possible. The second description, in subsection 5c, had a cognitive focus on epistemic actions and was therefore closely related to the mathematical content of the students' activity. In this final subsection, we consider the previous two descriptions in parallel and analyse what relationships between the two descriptions emerge from this parallel presentation. It is for this purpose of parallel analysis that the RBC diagrams and the interaction diagrams have been presented in parallel in Figure 4. With respect to C 2 , it has been shown in Section 3 that C 2 is compatible with at least two different formats (or patterns) of interaction. In segment 2 of the Ha&Ne interview, we discover a third interaction pattern which is compatible with C 2 : Here the construction arises out of contributions, in short succession, from both girls: First Ne insisted that the application of distribution to the present complex situation was possible and started to apply it. The fact that Ne's insistence and application of distribution were based on a misinterpretation is not relevant - it nevertheless did move the construction forward. Ha is then correcting the mistake and constructing the correct extended distributive law. There is no point in asking what


34

would have happened without Ne's intervention. It is, however, worthwhile to contrast this construction from those by Yo&Ra where we have observed a certain need for isolation at the very moments of C 2 construction. This need, if it exists at all for Ha, is very limited, at least for this particular construction. She clearly achieved the construction with minimal isolation. We have no indication to what extent Ne also constructed the extended distributive law. We know much less about Ne because of her limited active participation: She contributed less than 40% of the statements and close to 75% of her contributions were queries, attentions or otherwise minor (categories 4, 5 or 6). As a consequence, our interpretation of her constructions remains somewhat speculative but we present them nevertheless, and we present them in this subsection because they are intricately related to the interaction between the girls. For this purpose, we focus on segment 5 of the interview, which follows Ha's question to Ne whether she understood, at the end of Ha163. This segment has a very particular pattern of interaction where all arrows point to the top two statements, each statement thus contributing to explain what was claimed in Ha163 and Ne164. We note that in Ne164 Ne appears to confirm having understood Ha's Ha163 but Ha nevertheless decides to re-explain all of it to Ne, and presumably to some extent also to herself. The interaction between the girls during this segment is intensive, each being fully attentive to the other. Although we have no definite evidence of Ne's construction, we believe that construction is at least possible during such an "explanatory" interaction: One of the participants explains to the other what she has constructed earlier and thus reconstructs and consolidates it while the other has an opportunity to make the construction. Finally, we focus on C 1 and by implication on the entire global picture. According to the analysis of the epistemic actions, C 1 is not located in a specific part of the interview. On the contrary, the entire analysed part of the interview as well as some earlier statements revolve around C1 . The specific references to C 1 have been listed above. Many of these references display the students' awareness of C 1 . The analysis of the interaction between the girls (subsection 4.2), on the other hand, reveals that these same (groups of) statements are connected by a considerable number of long-range category 1 and some category 3 arrows. In other words, these are not random references to the proof of the DPP or to the means to achieve that proof, but they aggregate into a structured growth of the proof of whose progress the girls are aware. For example, in Ha163 Ha did three things: She summarised the somewhat technical development that had been elaborated between Ha152 and Ha163, she stressed that the resulting number 12 is significant as the difference between the two diagonals - the claim that had to be proved, and she asked Ne whether she also understood the significance of the result. Thus Ha163 refers back to the start of that technical development (Ha152), as well as to the articulation of the claim that the difference is 12 (Ha67), and Ha163 is itself being referred to from the following sequence of statements.


35

Moreover, Ha163 is revealing for the relationship between the two students: It is clearly important to Ha that as a pair they come to the conclusion of the task. This conc lusion is arrived at in Ha175 where the entire argument is summarised, thus explaining Ha173 and, indirectly, Ha163 and Ha67. Another way to describe how the cognitive action develops overall within the interaction pattern is therefore to look at the long-range arrows as showing the interaction/construction C 1 at the global level, within which the more local interaction/constructions C 2 are nested segment by segment. This analysis has been carried out in more detail in section 3. 5. Discussion: The distribution of abstraction among peers In the research presented in this article, we had several related goals. One goal was to validate and refine the nested RBC model of abstraction that we had proposed and illustrated in the previous article. The two research questions that related specifically to this goal were: 

Is the dynamically nested RBC model of abstraction valid beyond the one case presented in the previous article?



Which modifications of the model, if any, are suggested by more complex processes of abstraction? At the most basic level, we have shown that the model is a useful tool to analyse

processes of abstraction in two additional case studies. This by itself provides a powerful validation for the model, especially since the cases were more complex than the one analysed in the previous study, both cognitively and socially. The cognitive complexity gave rise to a potent refinement of the model. In fact, we discovered that during processes of abstraction a constructing action might be nested within another constructing action. Specifically, for the student pairs whose interviews we analysed, constructing the expanded distributive law was nested in the higher level construction that algebra is a tool for justification of certain general assertions. More generally, the fact that a constructing action can be nested in a higher level constructing action gives the model a recursive nature. A number, in principle unlimited, of levels of construction nested within each other may be envisaged. The model thus acquires additional depth and power. The other research questions focused on processes of abstraction by interacting pairs of students. Before drawing conclusions concerning these research questions, a theoretical digression is in order. In sections 3 and 4, we considered the question, how the common RBC flow is shared by the two students in the pair. This question raises at least three problematic issues. The first one concerns the pair as a coherent entity, a dyad. In other words, can we relate the epistemic actions to a social being? The second issue, intimately related to the first one, concerns the part of the individual in the activity of the dyad. The third question concerns the learning of the individual, or any inference we can draw from the present


36

activity of the dyad on further activities in which the individuals will participate. Each of these questions is general and complex. Although research on interaction is abundant, it is surprising that the questions we raise on interaction in conjunction with learning have been discussed in few articles only (Baker, 2000; Kieran & Dreyfus, 1998; Schwarz, Neuman, & Biezuner, in press; Sfard, 2000). 5.1 The coherence of the dyad The location of the question of the coherence of the dyad as an entity at t his stage of the article may appear somewhat surprising. How could we deal with the validation of the RBC model for collaborating pairs if this model cannot be linked to a coherent entity, if a recognition, or a construction can only be attributed to an individual rather than to the dyad within which this individual acts? Giving an acceptable general answer to this question is a complex endeavour. In the case of the two dyads Yo&Ra and Ha&Ne, our approach was intuitive. The peers were responsive to most of the individual actions: They agreed, objected, or questioned. The coherence of the two dyads was then a working hypothesis. We briefly review in this paragraph how the problem may more generally be coped with, and we confirm that the two dyads were coherent. Two ways have been adopted to cope with coherence. The first one consists in analysing products that are the outcomes of groups of students collaboratively solving problems. This is the case for the study conducted by Schwartz (1995) in which representations co-constructed by dyads were analysed. In the same vein, Schwarz, Neuman, Gil, and Ilya (in press) studied maps and essays collaboratively constructed by triads. In both studies, collective outcomes were analysed by using methods for analysing individual outcomes: The coherence of collective outcomes was established by considering these outcomes as if they were constructed by one student and by studying their “quality” (criteria for defining such quality were elaborated). For example, in the study by Schwarz and colleagues, collective arguments were analysed by looking at several characteristics such as the acceptability and the relevance of reasons invoked to support the standpoint claimed, or the quality of the reasons (abstract, personal, vague, etc.). The coherence of the group was then measured a posteriori, through a final product. The rationale for this assertion was that the very existence of a product of a group originates in an agreement by the members of the group that this product is acceptable. According to this approach, if the collective product is of high quality when considered as elaborated by an individual, the group is coherent. Schwartz showed that interacting dyads use more abstract representations than individuals. By showing that the product of the group is of higher quality than that of the individuals, Schwartz concluded that the group was coherent. Schwarz and colleagues obtained parallel results. They found that when triads


37

discuss a moral issue together, the essay they write collaboratively is generally more abstract than the essay of the best student in the triad 1 . Such an approach could have been applied in the case of Yo&Ra and Ha&Ne. The answers they gave at the end of the interview were more abstract than those at the beginning (e. g., in question 4 when the dyad is asked to explain why DPP is always true). They were also of higher quality than the arguments given by the students who answered the questionnaire in an individual paper-and-pencil format. This approach is problematic though. If the final outcome is of high quality, does this mean that the peers functioned as a coherent entity? And conversely, if the outcome is of low quality, and shows internal inconsistency, does it necessarily mean that the dyad did not function coherently? This question can be answered by testing students individually, after the interaction (e.g., Kieran, 1999; Schwarz, Neuman & Biezuner, in press). However, this methodology does not really answer the question of whether during their interaction, the students formed a coherent entity. The other method to check coherence of the group is to select a few utterances or segments to show that the group has adopted some construct or norm. This is the methodology chosen by Yackel and Cobb (1996) to identify the establishment of sociomathematical norms in a class. In the protocols selected, only some students interact, while the others are silent or passive. According to this method, in the case of Yo&Ra and Ha&Ne, coherence can be shown as follows. At the end of the interview of Yo&Ra, Ra (in Ra318) as well as Yo (in Yo319) relate to a previous utterance (Yo292) concerning the validity of the DPP and produce good final explanations (category 3) of this validity. Similarly, the Ha&Ne interview ends when the two peers agree to accept the final explanation. However, even this approach is far from being satisfactory, as the RBC model is based on epistemic actions and the question of the coherence of the dyad concerns a succession of actions. We chose to tackle the issue of the coherence of the dyad by referring to activity theory (Leont’ev, 1981): The term activity is reserved to designate the very framework in which individual actions are meaningful. Deciding whether students in dyadic interaction participate in the same activity, means that in the course of their actions they share the same motive. As shown in sections 3 and 4, for the two dyads we described, each of the peers was conscious In the two studies, the meaning of the term “abstract” was quite different, and at any rate was different from the activity-based, dynamic meaning of abstraction we adopt in this paper. However, as a first approximation, the term “abstract” in the two cited studies corresponds to what we could call “abstracted”. The question is who abstracted the abstracted product. Children may use a formula, a principle (moral or scientific), or an external representation at their disposal as cultural tools that are abstracts of human activity. However, this use does not always point to abstraction (in our sense) of formulae or principles. The two studies are relevant though. They stress the progressive participation of children working in small groups, at least to use abstracted outcomes. 1


38

the overall goal imposed by the task. Evidence for this can be found in the control category conversational moves throughout the interview: Utterances such as Yo292, and Ra318 show that the dyad Yo&Ra is aware of its overall motive, especially after Ra gave an inductive verification of the DPP during the work on question 2 (Ra97). Concerning the second dyad, the need for a justification is felt by Ha already in Ha67 when she declares “the difference between the products is always 12. Do we have to explain?” Her query follows numerical computations on several seals, a fact that proves that she has a personal need for a more principled explanation. Ne is often passive, and it is not easy to establish that she shares the same motive as Ha. However, two facts show that she is concurring with Ha during their work on question 6. First, her attention and agreement conversational moves are numerous (15 out of 31, as shown in Table 2). Second, when Ha gives (in Ha163) an acceptable explanation for the DPP, Ne reacts by confirming with a partial explanation, which leads Ha to re-articulate hers in even more detail. 5.2 The part of the individual in the abstraction of the dyad We have established that the two pairs are coherent dyads, and that as dyads they made abstractions. In order to decide on the part of the individual in the proc esses of abstraction, more needs to be known about the nature of the interaction among the peers and its relation to construction of knowledge. In a pioneering study, Baker (2000) has characterised interactions by identifying three dimensions: alignment, symmetry and agreement. The degree of alignment expresses “the extent to which the peers are in ‘phase’ during problem-solving moves”. Symmetry, expresses “the extent to which the responsibilities of the partners … are the same or different”. And the degree of (dis)agreement refers to “the extent to which partners’ publicly manifested propositional attitudes differ”. According to Baker, coconstruction can occur only when interaction is symmetric and aligned, both in situations of agreement and of disagreement. A crucial point in Baker’s methodology is that the dimensions are established at the level of episodes, and not at the global level encompassing the whole interaction. The attempt to apply Baker’s model to our two dyads leads to a partial characterisat ion of the role of the participants in the interaction and construction. For example, in segment 3 of the Yo&Ra interview, the interactions in the dyad are symmetric, in the sense that both peers feel the need to construct the extended distributive law, and both also achieve this construction (C 2 ). They are not aligned in this segment because they construct the extended distributive law one after the other rather than simultaneously. This is reflected in the conversational moves in the segment: The symmetry is indicated by the fact that both students use the same categories. The non-alignment is expressed by a chain of arrows that link between successive elaboration utterances of each student in the sub-segment reflecting his construction (see Fig. 3). In the fifth segment, the situation is slightly different. The


39

interaction is symmetric again, and the categories of utterances of the students are similar. The students do not seem to be aligned, in spite of the fact that they are simultaneously going through a similar process of construction (the elimination of expressions from both diagonals’ products). This time the reason for non-alignment is that they do not collaborate during the construction; this is expressed in two parallel chains of elaboration utterances ending in an explanation for each student. In section 3, we concluded that in the DPP justification the dyad Yo&Ra constructed three abstractions (C 1 and twice C 2 ). In the first part of this section, we showed that Yo&Ra could be considered as a cohe rent cognitive entity. Our conclusion is that for both cases of C 2 (in segments 3 and 5) there is symmetry but not alignment, because the students have the need to be left alone in order to construct a structure at the C 2 level. As for C 1 , the students are aligned concerning the general interaction and abstraction during the whole interview. They take the same responsibility (symmetry) concerning reaching their specific goal, justifying the DPP and thus the construction of a global structure of meaning for algebraic justification. This is expressed in the long category 1 and category 3 arrows which, as we mentioned before, tie the justification process together and are distributed equally between the two students. The alignment on the C 1 level is expressed in category 3 arrows from one student to the other at the end of each segment (see Fig. 3). As for the Ha&Ne dyad, the characterisation of the segments in which abstraction occurred is different. In the second segment, a C2 level abstraction occurs while there is alignment and symmetry. The peers alternatively agree and disagree, as it appears in the categories of the conversational moves in Figure 4. On the other hand, for the C1 level abstraction, which also includes the fourth and fifth segments, the peers are aligned, but for most of the time there is a clear asymmetry among them. For example, in the fifth segment, they are in complete agreement but the interaction between them is asymmetric, as the query category is predominant for Ne, and the explanatio n category dominates for Ha. In summary, the dimensions defined by Baker are useful to characterise interactions within segments. However, they are too local to be taken into consideration when describing co-construction or abstraction, as the dimensions o f alignment and symmetry vary along the activity. It may evolve along the activity, leading to different degrees of alignment and symmetry. Also, the nested nature of construction makes it impossible to use alignment to decide whether co-construction is accomplished along an activity, since students may be aligned concerning one level of construction, but non-aligned concerning another level of construction. Thus one needs to identify patterns of interaction with which abstraction may develop. Conceivably, such patterns are manifold and numerous. Our empirical study led the identification of some such patterns. We describe them below.


40

5.3 The learning of the individual: consolidation of abstractions The present study focuses on abstraction during interactio n. Like those of the previous study, the findings of this study cannot give an empirical answer concerning cognitive gains of the individual, the learning of the individual after abstraction in dyadic interaction. We have no information about whether the students realised that they used a rule which they never saw before, nor whether they would be able to use that rule again, to reproduce it as a formula, or to state it in other general terms. This issue is an important concern for further. Our remarks on this issue are thus no more than speculations. We speculate that under favourable circumstances the students would be able to either reconstruct the rule, or reapply it by recognising a similar structure in a different context. We further speculate that they would be able to state the rule, either at this stage or after reconstructing it and/or reapplying it and/or after a suitable summary discussion in class. In other words, we speculate that after one or several repeated encounters with the same structure in different contexts, the structure is internalised and becomes a recognisable entity for the students, and they will use it with increasing facility and flexibility for building-with in different contexts. We use the term consolidation for this progressive familiarisation and the increasing ability to recognise and build-with. The entire sequence from the first construction to the ongoing consolidation forms the process of abstraction. Further research will be needed to establish the entire sequence empirically. As expressed by Perret-Clermont (1993), in addition to the gains of the individual, a natural question about abstraction or any learning in interaction is to check the functioning of the group in further activities. Another exciting program of research is then also to check consolidation of abstraction of the dyad in further activities, and the part of individuals in these activities. 5.4 Concluding remarks: Abstraction and patterns of peer interaction In conclusion, we argue that we have identified several types of social interaction that give opportunities to processes of abstraction. Our arguments stem from the protocols we analysed but are nurtured by previous research and theoretical findings. To begin, we claim that abstraction can result under varied and numerous patterns of interaction. We seek tendencies, conditions that favour abstraction. We cannot prove our claims, as they are based on two interviews only, but we exemplify them in the two protocols and root them in theoretical findings or considerations already raised by other researchers. All patterns of interaction about which we claim that they give opportunities for abstraction have in common the coherence of the dyad. Coherence was analytically established for the two dyads we analysed. We claim here that coherence is a characteristic of interaction that strongly favours abstraction; similarly, lack of coherence inhibits abstraction. Further


41

evidence (in addition to the evidence provided by the Yo&Ra and Ha&Ne interviews) for this claim can be found in interview of the third pair, Da and Li, that was briefly mentioned at the end of section 2.2. This pair began their work on question 6c in a quite similar way as Yo&Ra, but when they reached an impasse (like Yo&Ra), and the intervie wer declared that they did a good job, the students were satisfied and did not continue. This is different from Yo&Ra for whom the interviewer’s intervention triggered the most significant segment concerning abstraction. The persistence of Yo&Ra to continue evidences an inner drive to clarify the situation together. The two students need each other to consider a new (algebraic) perspective and to concretise it through a new rule they construct. They accept their mutual dependence. In the case of Ha&Ne, the two girls need each other but for totally different reasons: Ha needs Ne because Ne’s queries afford her an opportunity to reflect upon her own construction, and Ne needs Ha because of the leadership she provides in the process of abstraction. In this case also, the two girls accept their mutual dependence. In contrast, Li and Da, act collaboratively only for relatively easy tasks. For more challenging tasks, they are both afraid to uncover their weaknesses. They are then not eager to collaborate on challenges. Their interactions evidence tension that impairs co-ordination of perspectives or coconstruction of a new rule. The important role that coherence plays in collaborative abstraction, does not imply that, at a local/specific level, students should always collaborate. We saw in the present study that co-ordination of actions is sometimes too demanding, and even student who need each other may at times prefer to work separately. During such episodes, the two students preserve alignment on a general level, but may break it at a lower level. For example, in the case of Yo&Ra, while they tacitly agree that the justification for DPP must be algebraic (C 1 , the general perspective), the segment that leads to the construction of the extended distributive law (C2 , the specific construction), show that they are quite non-aligned. Also, in the case of Ha&Ne, Ha at least at one stage momentarily ignores Ne’s queries (Ne153-Ne160). Abstraction in dyadic interaction seems then to gain from an interaction pattern of alignment concerning the overall perspective, while for the specific construction that embodies this perspective, each of the peers may opt for an independent course of action. Let us now turn to the specific patterns of interaction that led to abstraction. The case of Ha&Ne exhibits two patterns. The first one fits Vygotsky’s ideas of development of higher mental functions: Development proceeds from asymmetric interactions between an adult and a child, or between a more competent and a less competent person, as reflected in the central idea of zone of proximal development (Vygotsky, 1978; we replaced the term “capable” used in the translation of Vygotsky’s writings by the term “competent” that points to superiority concerning a specific task). In the Ha&Ne dyad, Ha is the more competent. Ne extends her ZPD by asking queries, until Ha’s explanations satisfy her. At the end of the interview, the


42

queries are more focussed, a fact that seems to indicate that Ne gradually gains more control on the new constructs. Ha’s progress is also compatible with Vygotsky’s idea, but in a less orthodox way: Ha explains to Ne a construct she does not completely master herself. In the course of the interaction with Ne, Ha achieves the construction, a feat that she possibly could not have achieved alone. Although Ha is more competent, she extends her own ZPD while collaborating with a less competent student. Ne’s queries force Ha to self-explain her actions, a type of action that has been recognised as improving learning through problem-solving (Chi, DeLeeuw, Chiu & LaVancher, 1994; Neuman & Schwarz, 1998, 2000). Ha’s selfexplanations are instructional explanations for Ne; thus Ne pushes Ha to express the explanations in a away that is meaningful not only to Ha herself, and thus to co mmunicate her ideas in a new perspective. As noted by Neuman & Schwarz (1998), such (self)explanations may lead the student to uncover the deep structure of the task. In summary, combining guidance with (self)-explanation is particularly fruitful concerning abstraction. The protocol of Ha&Ne provides a second pattern of interaction leading to abstraction in a short segment that plays a decisive role in the students’ C 2 construction. This is segment 2, in which the interaction is symmetric. In this segment, Ha and Ne engage in a short but fruitful argumentative activity. Ha integrates Ne's elaboration of an idea Ha had previously stated. As noted by Trognon (1993) and Pontecorvo & Girardet (1993), the social and the cognitive intermingle in the course of the actions of the dyad. Such symmetric argumentative interactions have been recognised as leading in general to construction of knowledge (e.g., Kuhn, Shaw & Felton, 1997; Hershkowitz & Schwarz, 1999). We claim that such interactions create opportunities to consider multiple perspectives on the same idea. Theorists such as Hundeide (1985) made similar claims. The case of Yo&Ra provides two more patterns of interactions that led to abstraction, both symmetric. The third pattern occurs in segment 3. Yo and Ra in turn, one after the other, accomplish a specific level 2 construction. This pattern will be called asynchronic collaboration. Yo and Ra are not aligned concerning this C 2 , although they do pay attention to the actions of each other. Following research by Kuhn (1972), we claim that in this case, students may be constantly in tacit interaction. Such tacit interaction seems plausible, as the students share the goal to elaborate the same construct and moreover are aligned concerning C1 . The fourth pattern of interaction appears in the fifth segment. It is similar to the asynchronic pattern, except that the two act synchronically, in parallel. It is probable that Yo and Ra pay attention to the mutual results they reach. Paying attention to the success or the failure of peers has been recognised as significant in solving Piagetian conservation tasks in peer interaction (Botvin & Murray, 1975). We suggest that such interactions provide opportunities for abstraction.


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We are now in a position to answer the third and fourth research questions, both of which concern abstraction by interacting peers, namely: 

Is the model adequate for describing processes of abstraction by interacting pairs of students?



Which patterns of distribution of abstraction occur in processes of abstraction by collaborating peers? Concerning the third research question, we showed the centrality of coherence. We

established coherence of the two dyads that participated in the research. These dyads accomplished a justification of the DPP, a property we designed to lead to abstraction. Although the protocols showed discrepancies between the peers of each dyad concerning their epistemic actions as well as dimensions of their interaction, when examined segment by segment, coherence was found globally along the sequence of the segments that constituted the entire activity. This coherence found its expression in the sharing of the global motives and perspectives and induced the dyads to vertically reorganize their knowledge into new structures. Coherence thus establishes the RBC model as an efficient model to describe processes of abstraction by interacting pairs of students. Concerning the fourth research question, we identified four patterns of interaction that led to abstraction, and we are aware that many other patterns could be identified with other dyads working on other tasks. For example, in our previous study, we showed how a single student abstracted when working with a teacher- interviewer, who occasionally guided her, thus establishing abstraction in a guidance pattern of interaction. The patterns we found in the present research are guidance/self-explanation, symmetric argumentation, asynchronic collaboration and collaboration in parallel. In all these interaction patterns the peers shared a common perspective toward the new knowledge structure, and were driven by a common motive. We stress again that this is true in particular for the asynchronic and parallel collaboration, which on superficial inspection could appear to lack interaction. The solut ion of this apparent paradox is found in the fact that the common motive and the common perspective toward the new knowledge structure led to a very deep interaction at the level of the activity as a whole and thus enabled the process of abstraction to unfold. Acknowledge ment We would like to thank Ms. Shirley Atzmon for help with the transcription and analysis of the interviews. References Arcavi, A., Friedlander, A., & Hershkowitz, R. (1990). L’algèbre avant la lettre. Petit x, 24, 61-71.


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Baker, M. (2000). Forms of cooperation in dyadic problem-solving. Interaction & Cognition Research Reports, N0 IC-1-2000, GRIC Laboratory, CNRS & University of Lyon 2, France. Botvin, G. & Murray, F. (1975). The efficacity of peer modelling acquisition of conservation. Child Development, 46, 796-799. Chi, M. T. H. (1997). Analysing the content of verbal data to represent knowledge: A practical guide. The Journal of the Learning Sciences, 6, 271-315. Chi, M. T. H., DeLeeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self-explanation improves understanding. Cognitive Science, 18, 439-477. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152. Davydov, V. V. (1972/1990). Types of Generalisation in Instruction: Logical and Psychological Problems in the Structuring of School Curricula. Volume 2 of J. Kilpatrick (Ed.), Soviet Studies in Mathematics. Reston, VA: NCTM. Hershkowitz, R. & Schwarz, B. B. (1999). Reflective processes in a technology-based mathematics classroom. Cognition and Instruction 17 (1), 65 – 91. Hershkowitz, R., Schwarz, B. B. & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education 32 (2), 195-222. Hundeide, K. (1985). The tacit background of children's judgments. In J. W. Wertsch (Ed.), Culture Communication and Cognition, Vygotskian perpsectives (pp. 306-322). Cambridge, UK: Cambridge University Press. Kieran, C. (1999). Mathematical learning in a collaborative problem-solving setting: entering another’s universe of thought, patterns of interaction and the role of questioning. Symposium paper presented at the annual meeting of the American Educational Research Association, Montreal, April 1999. Kieran, C. & Dreyfus, T. (1998). Collaborative versus individual problem solving: entering another’s universe of thought. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Annual Meeting for the Psychology of Mathematics Education, Vol. III (pp. 112119). Stellenbosch, South Africa. Kuhn, D. (1972). Mechanisms of change in the development of cognitive structures. Child Development, 43, 833-844. Kuhn, D., Shaw, V., & Felton, M. (1997). Effects of dyadic interaction on argumentative reasoning. Cognition and Instruction, 15, 287-315.


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Leont’ev, A. N. (1981). The problem of activity in psychology. In J. V. Wertsch (Ed.), The Concept of Activity in Soviet Psychology (pp. 37-71). Armonk, NY, USA: Sharpe. Neuman, Y. & Schwarz, B. B. (1998). Is self-explanation while solving problems helpful? The case of analogical problem solving. The British Journal of Educational Psychology, 68, 15-25. Neuman, Y. & Schwarz, B. B. (2000). Substituting one mystery for another: The role of selfexplanations in solving algebra word-problems. Learning and Instruction, 10, 203-220. Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings: Learning Cultures and Computers. Dordrecht, The Netherlands: Kluwer Academic Publishers. Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of abstract ideas. International Journal of Educational Psychology, 27, 37-48. Perret-Clermont, A-N. (1993). What is it that develops? Cognition and Instruction, 11(3 & 4), 197-205. Pontecorvo, C., & Girardet, H. (1993). Arguing and reasoning in understanding historical topics. Cognition and Instruction, 11(3&4), 365-395. Resnick, L. B., Salmon, M., Zeitz C. M., Wathen, S. H. & Holowchak, M. (1993). Reasoning in Conversation. Cognition and Instruction, 11(3&4), 347-364. Schwartz, D. L. (1995). The emergence of abstract representations in dyad problem solving. The Journal of the Learning Sciences, 4, 321-354. Schwarz, B. B., & Hershkowitz, R. (1995). Argumentation and reasoning in a technologybased class. In J. D. Moore and J. F. Lehman (Eds.), Proceedings of the Seventeenth Annual Meeting of the Cognitive Science Society (pp. 731-735). Mahwah, NJ: Lawrence Erlbaum Associates. Schwarz, B. B., Neuman, Y, & Biezuner, S. (in press). Two wrongs may make a right…If they argue together! Cognition and Instruction. Schwarz, B. B., Neuman, Y., Gil, J. & Ilya, M. (in press). Construction of Collective and Individual Knowledge in Argumentative Activity: An Experimental Study. Journal of the Learning Sciences. Sfard, A. (2000). Steering (dis)course between metaphor and rigor: using focal analysis to investigate an emergence of mathematical objects. Journal for Research in Mathematics Education 31 (3), 296-327. Sfard, A. & Kieran, C. (in press). Cognition as communication: Rethinking learning-bytalking through multi- faceted analysis of students’ mathematical interactions. Mind Culture and Activity. Abstraction in peer interaction.doc

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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. Trognon, A. (1993). How does the process of interaction work when two interlocutors try to resolve a logical problem? Cognition and Instruction, 11 (3&4), 325-345. Vygotsky L. S. (1978). Mind in Society - The Development of Higher Psychological Processes (ed. by M. Cole, V. John-Steiner, S. Scribner & E. Souberman). Cambridge, MA, USA: Harvard University Press. Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education 27, 458-477.


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Figure Captions Figure 1.

The seals activity worksheet

Figure 2.

The interaction categories

Figure 3a.

The Yo&Ra interview – Part 1

Figure 3b.

The Yo&Ra interview – Part 2

Figure 4.

The Ha&Ne interview

Abstraction in Context: The Case of Peer Interaction

Abstraction in Context: The Case of Peer Interaction

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