1. - edumatedumat.uab.cat/for.pdf · · 2013-04-04the paper we present is a case study which has...

PEDRO COBO and JOSEP M. FORTUNY

SOCIAL INTERACTIONS AND COGNITIVE EFFECTS INCONTEXTS OF AREA-COMPARISON PROBLEM SOLVING

ABSTRACT. The paper we present is a case study which has two objectives: the identi-fication of the interactions between pairs of 16 and 17-year-old students related to problemsolving and the influence of such interactions in their cognitive development. To achievethese aims, we consider problems associated with a definite conceptual structure – prob-lems comparing flat surface areas –, and we focus the analysis of the interactions on thepoint of view of the thematic and interlocutive dimensions of the discourse. Such analysishas allowed us to identify a wide typology of exchanges, and, from this, to outline fourmodels of interaction - alternative, guided, relaunching, and co-operative. In the casesanalysed, these interactions significantly influence the individual development of cognitiveand heuristic abilities in the problem solving process.

KEY WORDS: cognitive benefits, development, discourse, heuristic abilities, models ofinteraction

1. INTRODUCTION

Current trends in educational reform emphasise the role played by so-cial interaction in students’ mathematical apprenticeship. This tendencyagrees with Vygostki’s early ideas. A revealing feature of the Vygotskianapproach is the emphasis on interpersonal processes as a basis for in-trapersonal processes. Following this approach, we adopt the perspectiveexpressed by the Perret-Clermont group (1996) on Social Psychology stat-ing that every individual is a co-author of his or her personal development.His/her potential cognitive elaboration increases when collaborating withothers in tasks designed to develop knowledge. Paradigmatic examples ofcollaborative tasks are those carried out in the process of solving problemsin pairs, which constitutes the focus of this paper.

Contributing with different ideas and confronting distinct points of viewduring the solving process broadens mathematical knowledge and givespower to the heuristic abilities set in motion while working together. Con-sensus in this field is especially sensitive to problem solving. Authorssuch Lester (1994), Balacheff (1990), Forman (1989) and Webb (1989)suggest that research should be directed towards the study of the rela-tions between the interaction characteristics and the cognitive behaviour

Educational Studies in Mathematics42: 115–140, 2000.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

116 PEDRO COBO AND JOSEP M. FORTUNY

of students, since “linguistic constructions and cognitive constructions aredialectally related during the problem-solving process” (Balacheff, 1991,p. 95).

Starting from the assumption that, during the process of problem solv-ing, several interacting forms can be produced between the participatingstudents (Webb, 1989), we outline research that seeks to analyse the natureand quality of the interactions taking place in the process of mathematicalproblem solving among students.

In order to study the nature of interactions, we must delve deeper intothe exchange models produced and into the way in which they combinethroughout the solution process. Therefore, we will take into account threeaspects: the syntactic forms used in interventions – such as assertions,questions, requirements of validation, answers, validations, answers of val-idation – ; the management or directive character in interventions; and therelationship found between some interventions and others, particularly, theways in which transitions between them take place – such as pauses, over-lapping or interruptions –. On the other hand, in order to assess the qualityof interactions, we must relate them to the strategies that the students useduring the solving process of the concrete problems proposed.

The research we present differs from other work related to the studyof interaction in the process of problem solving in the sense that the ana-lysis of the interactions we propose incorporates some characteristic ele-ments of discourse analysis (Gumperz and Hymes, 1972; Roulet, 1987;Kebrat-Orecchioni, 1990–1994; Brown and Yule, 1993; and Calsamigliaand Tusón, 1999).

In this manner, we aim to answer the following questions: What gen-eral models of mathematical interactions take place in problem solvingprocesses between pairs of 16–17 year old students when comparing flatsurface areas? To what extent do such peer interactions influence in anysignificant way the individual development of cognitive and heuristic abil-ities in the problem solving process?

In short, this paper aims at establishing ways in which teachers can dir-ectly observe, in some detail, their students’ interactions, and their relationwith learning.

This will be achieved by generalising from the interpretative results,and embedding them within a theoretical framework of what occurs in thecases considered within this paper.

SOCIAL INTERACTIONS AND COGNITIVE EFFECTS 117

2. ANALYSIS OF THE DISCOURSE AND PROBLEM SOLVING

Following Cesar (1998), we have assumed that peer interactions are asso-ciated with socio-cognitive conflict, and are seen as a way of implementinga co-construction of knowledge. This seems to be a powerful way of con-fronting pupils with each other’s problem-solving strategies, and is onethat obliges them to decentralise from their own position and discuss eachother’s conjectures and arguments.

To understand the role played by peer interaction in the promotion ofmathematical knowledge and strategies, we need to analyse and provideinformation about the mechanisms involved in students’ communication.We will do this by paying close attention to the point of view of bothinterlocutive analysis and the theme of the discourse.

In our context of problem solving, we understand by discourse ana-lysis the use of language that takes place between particular individualswhen talking, with the intention of finding new strategies and generatingknowledge, leading to problem-solving.

This functional focusing not only demands a study of the situationalcontext (the physical frame in which interactions develop, the theme ofdiscussion, the participants’ production channel – oral, written, etc. –, thecode or language used, and the form of the message – argument, conver-sation, etc. – ), but also the analysis of the interlocutors’ individual andsocial characteristics. Among the individual characteristics, we consideron the one hand those making mathematical knowledge meaningful, andthe cognitive profile involved through the inter-subjective work, and on theother hand those contributing to individual modifications of knowledge.

As regards social characteristics, we understand that interactions allowus to establish a high degree of semantic and pragmatic coherence fromthe sequence of linked exchanges between interlocutors.

The function of the use of language and the way receivers interpret andunderstand messages from transmitters has been widely studied in math-ematical problem solving (Webb, 1984, 1989 and 1991; Forman, 1989;Lambdin, 1993; Wood, 1996; Yackel, Cobb and Wood, 1991; and Cobband Whitenack, 1996). Communication in problem solving has specialcharacteristics, which make it different from other types of conversation.In this paper, the characteristics mentioned above are related on the onehand to a definite type of geometrical problem solving, the comparisonof flat surfaces areas, and on the other to the fact that the conversationtakes place between two interlocutors. We hope to adduce a new way,identifying and analysing interactions in problem solving between pairs


of classmates based on both thethematicand interlocutivedimensions ofdiscourse analysis.

2.1. Thematic dimension

In the context of problem solving, the thematic dimension of discourseis related to the way in which students contribute to the solving process.In the thematic construction analysis, we differentiate three units, whicharranged from minor to major rank are: intervention, exchange and inter-action. The first of these is a monologue unit, the other two being dialogueunits.

We define intervention as the thematic contribution of an individual tothe development of what is said and about what information will be givenor what position will be taken on the issue in question (Calsamiglia andTusón, 1999). In our context, the thematic contribution each student makesis related to mathematical content used in problem solving. Those them-atic contributions that introduce information liable to become the objectof debate (directive interventions) generally have special relevance. Thistype of intervention, together with those of process management, whoseobjective is to explain what is going to be done, shape the evolution of theprocess.

The exchange is the smallest dialogue unit. We introduce exchanges interms of action-reaction, and we define them in the following way: theintervention of subject L1 produces a reaction in subject L2 if, in theintervention of L2, there is any explicit reference (oral or mimic) to thecontents of L1’s intervention, or to any of its elements.

An exchange can be formed by two interventions. In this case, we callthe first one initiative (action) and the second, reactive (reaction). Thissecond intervention can at the same time initiate another exchange, butwill not do so if the intervention brings no initiative to the theme of thedialogue; this is the case of an intervention which only gives effect to theformer, as long as this validation is not requested in the next intervention.

It is possible that the initiative intervention of a speaker induces noreaction in the interlocutor. In this case, we consider that the exchangeproduced is formed by only one intervention. We call false dialogues ormonologues those generated when the action of one speaker produces noreaction in the interlocutor. This does not mean that the interlocutor isnot able to speak; but in doing so, alternate interventions are produced,in which each speaker makes no reference to the content of their inter-locutor’s intervention. The most common structure of an exchange is theone formed by three interventions (Roulet, 1987) which follow the order:an initiative intervention, a reactive and an evaluative one.


We consider interactions to be sequences of exchanges. In our case, asthe interactive analysis includes the management of the solving process,we propose a delimitation of interactions based on the homogeneity of thenature and purpose of the actions taking place in every moment of thesolution process.

2.2. Interlocutive dimension

The interlocutive dimension of discourse refers to the mechanism of com-munication and to that of the communicative behaviour of each speaker(Calsamiglia and Tusón, 1999).

When we refer to the mechanism of communication we mean the originand way in which speakers take turns talking: if they do so by respondingto their own initiative, that is to say, if there are no explicit or implicitindications in one of the speaker’s intervention that require the interlocutorto speak; or if they do so as a consequence of the requirement of otherindividuals, that is to say, if there are explicit indications (direct questions,imperatives or assertions followed by requirement of an answer) on onespeaker’s side, suggesting a response from the other interlocutor.

We will have to include different aspects in the analysis of the speakers’communicative roles: syntactic, those of management and also ones whichcorrespond to conversational indications generated or activated in problemsolving.

By taking these parameters into account, we will be able to analysethe effects of various interactions between the participants in any dialoguetaking place at any moment of problem solving, and therefore we will alsobe able to bring indications about the development of strategies in problemsolving. We will associate this development of strategies with differenttypes of contribution such as the introduction of new information; the ob-taining of expressions; changes of orientation; a brake block generated bythe problem; the production of new ideas; a relaunching of exploratoryresearch; the improvement of understanding, etc. In this sense we willevaluate, from the interaction identification, to what extent interactions incollaboration acts could meaningfully affect the individual development ofcognitive and heuristic abilities in the problem solving process.

3. CHARACTERIZATION OF INTERACTIVE MODELS

Discourse analysis has opened to us the perspective of being able to inter-pret interaction models such as dynamic structures, characterised as bothprocess and product. On the process side, the interlocutive dimension al-


lows us to characterise these interactive models as a distributed processthat manifests itself through the development of the solving process. Onthe thematic side, however, we could characterise them as a product that isgenerated through shared mathematical knowledge. These two dimensionsare not independent; rather, the thematic, as a product, is function of theinterlocutive, as process. Thus we interpret the interactions as follows:taking these dimensions as reference points, we consider the time as theparameter that marks its historical evolution, and marking the contours ofwhat we refer to as social episode events that take place throughout thecourse of collaborative problem solving.

Social episodes are time intervals in which the students culminate aphase of the process followed by a real resolutor, in the sense of Schoen-feld (1985); or they interpret the statement or they benefit from a greaterunderstanding of the concepts and procedures involved in problem solving,thanks to their confrontation; or else they implement an approach that leadsthem directly to the solution.

In this analysis of the interactive models, we integrate both the them-atic and interlocutive dimensions, defining different types of exchange,therefore we bear in mind the mathematical content of each student’s in-tervention, the syntactic forms and the relationship between different in-terventions.

We indicate our definition (below) of the exchanges in the problemsolving processes analysed; these will help us to characterise the interac-tions that we have identified. We classify exchanges in function of the num-ber of interventions by which they are constituted, differentiating betweenthose made up by only one, two or three interventions, which are the mostfrequent used. Other exchanges shaped by more than three interventionsare rare.

Exchanges of one intervention

It sometimes happens that a speaker (L1) produces an intervention thatdoes not give rise to any reaction on the part of the interlocutor (L2), andthere is a continuation of the discourse on the part of L1; in this case wehave what Kerbrat-Orecchioni (1990) calls a truncated exchange.

We represent the truncated exchange in the following way:


If there was no such continuation we would consider these interventionsas isolated.

The continuation of the monologue on the part of each student cango on contributing with new pieces of information that complement thosealready exposed in their former intervention (we call this type ofprogress-ive discourse); or can be produced as a simple repetition of their formerintervention (repetitivediscourse). In both cases, we say that the studentswork in parallel.

Exchanges of two interventions

Two intervention exchanges are characterised because student L2’s reac-tion brings some new information, gives effect or repeats student L1’sintervention or answers any question asked by L1. Since these reactions arevery different and in order to resume the characteristics of each of them,we will differentiate three types of two intervention exchanges: validation,co-operation and question-answer.

A validation exchange is produced if L2 only:i) gives effect (yes, ofcourse, etc.) to the contents of L1’s intervention affirmatively;ii) values(very well, etc.) the contents of L1’s intervention affirmatively;iii) repeatsthe contents of L1’s intervention using the same words or similar ones;and in the continuation of the discourse on the part of L1, if there is one, itdoesn’t refer to the contents of the former intervention.

We represent them in the following way:

The graphic representation of L2 intervention finishes with a smallstraight arrow to signify that the validation does not contribute with anyinitiative to the dialogue’s theme

If a validation exchange is produced, there is a change of direction in thecontinuation of the mathematical contents of the former interventions. IfL1’s next intervention refers to her first one, the continuation of the themewill be guaranteed and, as a consequence, the linking of such interven-tions, even if it is only on L1’s side. In this case we consider it as a threeintervention exchange (of the type validation-continuation). The purposeof the distinction between these two types of exchanges is to identify themoments in which the students change the direction of the mathematicalcontents of their interventions.

In aco-operativeexchange L2’s intervention in some way modifies thecontents of L1’s intervention, either by:i) introducing some information


equivalent to L1’s intervention,ii) bringing some new information whichwill complete L1’s intervention, oriii) introducing a new element on thedialogue.

We represent it in the following way:

Two students interact in aco-operativeway if most of the exchangesproduced are co-operative ones. In this interaction, there is an equitablecontribution of information on the part of each, even if such informationhas already been introduced in former episodes.

Sometimes parallel reflections result in new contributions that are brieflyanalysed in a co-operative manner. We refer to this type of interaction asthere-launching of solving process.

In the question-answertype of exchange, L2 only answers the ques-tions asked by L1 that do not refer explicitly to the content of the formerintervention. We represent this in the following way:

The fact that student L1 asks a question need not necessarily mean abreak with former interventions, because the introduction of pieces of in-formation that are equivalent – co-operative exchange – can be introducedas a question. We do not classify as ‘question-answer’ exchanges those inwhich student L1 makes a validation requirement – he makes a propositionin a hypothetical way – and expects an agreement as an answer. Theseexchanges are validation types if the answer only gives effect, or are co-operative if the answer is produced in such terms that can belong to thistype.

Exchanges of three interventions

The exchanges made up by three interventions – validation-continuation,explanatory ones – are characterised by the fact that the initial interventionof student L1 has continuation, not only on the part of L2, but also on thepart of L1 himself.

We represent this in two different ways:


A validation-continuationexchange – representation (a) – takes placewhen L2 only gives effect to the contents (or he simply repeats a partof it with agreement) of L1’s intervention and this one goes on with thediscourse referring to her last intervention. We consider, in this type ofexchange, those in which student L1 states a proposition and requires val-idation, that this validation is produced on the part of L2, and so L1 makesanother reference to the statement of her last intervention.

We say that anexplanatoryexchange – representation (b) – is producedwhen L2 only asks for an explanation of (or expresses an agreement with)the contents of L1’s intervention (or with some of its elements), and L1answers this requirement.

Generally, in the exchanges of three interventions, the continuation ofthe discourse on the part of L1 can take place as a simple repetition of thecontents of her first intervention, in which case we would have an exchangethat could be calledrepetitive, or bringing some information which can beequivalent, complementary or introducing some new element; in this casewe would have an exchange that could be calledprogressive.

The combination of exchanges of the types validation-continuation andexplanatory gives rise to a type of interaction we could callguided work.In this, one of the students – L2 – plays the communicative role of makingvalidations or questions on the contents of former interventions, with theintention of inciting her interlocutor – L1 – to continue the dialogue; and onthe other hand, L1 assumes the responsibility of answering L2’s questionsand demands for validation.

The relief in the succession of the students’ communicative roles through-out an episode is the characteristic which identifies the interaction calledalternativein the assumption of communicative roles.

These exchanges shaped by two-and-three-intervention are more fre-quent in the dialogue of secondary school students solving problems to-gether. But in certain situations, other types of exchanges, shaped by threeor even more interventions may be produced, such as those interruptingthe intervention of another classmate; those generated by two or even moreprevious interventions; and a disagreement type of exchange (Cobo, 1998).This third category is based on the idea of Roulet concerning ‘interactivecompleteness’ or double action pressure, which indicates the need that


“in order for the negotiation, and consequently the exchange, to draw toa close, it is a requirement that interlocutor reaction and locutor evalu-ation be positive” (Kerbrat-Orecchioni, 1990, p. 237). The exchange endswhen, following a negotiation – if necessary – between both interlocutors,a mutual agreement is achieved.

4. A CASE EXAMPLE

4.1. Problem solving environment

To obtain the objectives proposed, and taking into account the theoreticalframework which we have just developed, we focus on an out-of-classexperimental context in which two Secondary Education students interact(speaking, writing, making gestures. . .) to solve two mathematical prob-lems, for approximately 25 minutes per problem, in the presence of anobserver who does not take part, and of a video and audio camera whichregisters all the solving process. The students’ conversations are recorded,transcribed and analysed, from a qualitative perspective depending on theoral data obtained.

The students resolve a particular type of mathematical problems thatwe have called problems of flat surface area comparison.

Several reasons encouraged us to use problems which compare differ-ent areas: the outstanding place that comparing problems have had in thehistory of mathematics (Hearth, 1981); the lack of specific research oncomparing areas, although there are some on certain associated concepts;the ease of obtaining problems adapted to the level of students of differ-ent ages; and the fact that the resolution of this type of problem can beapproached in several ways (understanding approaches as possible ways,which can combine techniques, or heuristic and techniques, in obtaininga solution, Cobo, 1998). All this facilitates the richness of collaborativesolving processes.

To obtain oral data, we selected the following two problems.The parallelogram problemIf M is any point on the diagonal AC of the parallelogram, What relation isthere between the areas of the triangles in the shaded section in Figure 1?

Figure 1


The triangle problemABC is a triangle (Figure 2) and D is a point of the side AB, dividing itinto two segments which are in the proportion of 2 to 1. If DE and DF aretwo segments parallel to sides AC and BC, respectively, what relationshipis there between the areas of triangles DBE and FEC?

Figure 2

Previous to data collection, we carefully analyse the characteristics ofthese two problems, focusing on the conceptual and procedural contentsinvolved in resolution (Cobo, 1998). To summarise, we can say that bothincorporate graphs within their statements and relate to the areas of geo-metrical constructions (Figures 1 and 2). However, in the problem of thetriangle, the ratio between the elements of the figures is fixed. In the prob-lem of the parallelogram, we establish an implicit relation between thelineal elements of the figures, and ask for the relationship between areas.

4.2. Descriptions of interactions

In order to describe and interpret the interactions in detail, in this paperwe show oral data corresponding to a pair of students: Rosa and Anna. Tofacilitate the communication between students, we chose the pair in sucha way that they have a heterogeneous composition, as each of students inthe pair differs from one another both in her visual-algebraic orientationas in her academic level. Rosa and Anna are two 16-year-old students intheir penultimate year of Secondary Education. Rosa is a particularly giftedstudent in class. She shows this in her interventions, which are not very fre-quent, but are generally very pertinent, and which bring other views of theexercises and problems to those worked on in class. Her type of reasoningcould be qualified as geometric, for her tendency is to express ideas bymeans of graphic representations of the different situations – “I draw”, asshe usually says –, although her knowledge of concepts and specific tech-niques is not very wide. She has certain difficulties in handing algebraicexpressions. Anna is more ‘open’ than Rosa and expresses her opinions inclass in a more spontaneous, impulsive and sincere way. She takes part inall the arguments in the class. She gets better marks in mathematics thanher classmate, although she is not one of the best in the class. She focuseson mathematical problem solving in an algebraic way.

We will now outline the characteristics of the interaction models usedin solving processes for parallelogram and triangle problems.


4.2.1. Characteristics of the alternative modelExploration/analysis social episodeAs a first approach, after reading the instructions and diagram for the par-allelogram, Rosa recognises that there are two triangles sharing the sideAM (Figure 3). Anna takes advantage of this recognition in order to carryout a new representation of the figure in which she identifies through ‘x’the shared side (AM) to the two triangles.

Identifying the shared side is the culmination of co-operative exchange,but the algebraic approach that it points to is abandoned since the dialoguecontinues via three validation-continuation exchanges led by Anna. Onlyone results in a certain kind of progress, namely that in which the twounequal sides DM and AD are identified (Figure 3).

Figure 3.Reproduction made from one of the students’ figure (letters added).

Figure 4.Figure drawn by Rosa.

The search for relationships between the elements in the figure leads Rosato represent the BD diagonal in the ABCD parallelogram (intervention 14).

14. Rosa:And if we put the diagonal here. . . [she traces the diagonal BD, Figure 4].

Having already allowed Rosa’s earlier interventions to influence her, Annatraces the straight line ‘s’, parallel to AB through M (intervention 17).

17. Anna:So we end up with this drawing[she traces the straight line s, parallel to AB, andindicates the triangles MBE and AMB],don’t we?

Tracing line ‘s’ is the decisive moment in the direction of the process. Thefollowing interventions are erroneous attempts, with their associated recti-


fication, to discover the equalities between the figures obtained by tracingthe parallel ‘s’.

Both students alternate in making proposals: whilst Rosa suggests theequality of DMF and MPD (intervention 20, Figure 5), Anna does the samefor figures AFMB and MBE (intervention 25).

Figure 5.The interaction of Rosa and Anna in the Exploration/analysis social episode ofparallelogram problem solving.

Rosa takes the initial idea of tracing the parallel ‘s’ representing line ‘r’ (in-tervention 28, Figure 5), which is parallel to AD through M. This parallelgives rise to a new attempt (this time successful) to recognise the equalityof other given triangles —BEM and HBM, AHM and AMF (see the endof intervention 28) and FMD & DMG (intervention 30).

Reducing the comparison of the triangle areas ABM & AMD, as pro-posed by Anna, following a co-operative exchange, allows Rosa to conjec-ture the equality of the areas for triangles FMD & HBM.

40. Rosa:I reckon that the area of this triangle[FMD] and this one[HBM] is the same.

We observe that significant progress has been made throughout this epis-ode. This progress may be measured in terms of management and thematiccontribution, as we have done. The management has evolved from the ini-tial, disordered contributions, void of any specific finality, to the orderedsearch for relationships between the triangles that results in the division ofthe parallelogram by the lines ‘r’ & ‘s’. There have been incorrect thematiccontributions (amongst others, 20 & 25), later rectified, and many other


correct contributions, as we indicate in Table I; these include the fact thatstrategies related to straight line tracing essentially mark the geometricapproach implicitly selected by the students.

In general, thematic contributions are produced in the context of a co-operative exchange or after a markedly short pause. Following each oneof these contributions, the student responsible for the suggestion heads asuccession of two or more exchanges of the validation-continuation andexplanatory type, by means of which the dialogue is fuelled with newideas. Whilst this takes place, the other student restricts herself to eval-uating the ideas introduced, or to asking questions related to these newsuggestions.

And so, in the interaction in this episode (partially shown in Figure 5),we could say that Anna and Rosa evenly distribute the roles of questioningand evaluating the contributions made by their partner. This alternation is away of interacting – which we callalternativeinteraction – in which eachinterlocutor has the initiative, in periods of time, in relation to the elementsof mathematical content that she introduces.

In this alternation there is a slight unbalance in the contribution of newideas brought by both student, as we can observe in Table I. In spite ofthis, Anna traces parallel ‘s’, which is fundamental to the development ofthe process as it sets the path to be followed in order to reach a definitivesolution. This tracing in fact imitates Rosa’s behaviour, which initiated thetracing procedure for the lines of the BD diagonal.

4.2.2. Characteristics of guided interactionImplementation social episodeIn order to justify the conjecture of equivalence for triangles FMD & HBM,Rosa develops the idea that will definitively solve the problem (See in-tervention 46, Figure 6). She considers parallelogram GMEC and, basingherself on the areas AFM & AHM, comes to the conclusion that parallelo-grams FMGD and HBEM have the same area (intervention 48), justifyingthe aforementioned equivalence (intervention 50). In intervention 54, andbefore Anna’s request for an explanation, Rosa refers to the parallelogramABCD, and to the equality of the triangles that are obtained by a divisionalong the diagonal AC. Continuing the dialogue, Rosa refers to the con-dition of the parallelograms ABCD, MECG & AHMF in order to justifysuch equalities.

SO

CIA

LIN

TE

RA

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ION

SA

ND

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GN

ITIV

EE

FF

EC

TS

129

TABLE I

Summary of the correct information introduced by Rosa and Anna

ROSA Interventions ANNA Interventions

• She recognises the shared side AM 6 • She symbolically identifies the 7

• She traces the diagonal BD. 14 side AM.

• She traces line r parallel to BC. 28 • She recognises the unequal sides 11

• She identifies the equality of BEM & 28 AD & DM.

HBM, and AHM & AMF. • She traces the line s parallel to 17

• She identifies the equality of FMD & 30 AB.

DMG. • She reduces the comparison of 35

• She conjectures the equality for the 40 the areas ABM & AMD to those

areas of FMD & HBM. of FMD & HBM.


Figure 6.The interaction of Rosa and Anna in the implementation social episode of theparallelogram problem solving.

Rosa’s communicative role consists of ever-more detailed reasoningwith regards to the equivalence of parallelograms FMGD & HBEM, lead-ing to the start of the episode (Figure 6). This reasoning is produced bymeans of a succession of exchanges of the validation-continuation andexplanatory type. It is in this way that Rosa persists in justifying the con-tributions that she introduces, greatly assisted by Anna’s attitude, whichstimulates Rosa’s argumentative process, validating it and introducing newdoubts (articulated through questions: ‘why?’, ‘but how can we be sure?’).

In this type of interaction, which we callguided, the communicativerole of each student is important in itself, although we can’t obviate thefact that the person who takes the lion’s share of the dialogue is Rosa (sheintroduces all the information), because such dialogue will progress or willremain static, depending on whether she provides the discourse with newideas or not. In some way, Rosa is acting as if she were Anna’s tutor.


4.2.3. Interactive model of relaunching of the solving processExploration/analysis social episodeAt the beginning of the triangle problem solving process, Rosa and Annatry to understand the statement of the problem and they interpret the pro-portion – 2 to 1 – of segments AD and DB (Figure 7 and Figure 8).

Figure 7.Reproduction made from one of the students’ figure (letters added).

Figure 8.Final figure drawn by Anna and redrawn by Rosa.

The exploration/analysis episode begins with the identification of theequality of triangles FDE and FEC (Figure 7). In what remains, we havebeen able to observe students’ performance that repeats itself three con-secutive times. This performance has four parts: The identification of theproblem objective (to reach the point of relating the areas of the trianglesFEC and DBE); parallel work with gesture references to the elements ofthe figures or with long pauses (more than 30 seconds); the introductionof new information is always related to the layout of lines parallel to thesides of the triangles; and the exploratory search of relations between thetriangles obtained with the division of the original figure and between thelineal elements of such figure.

We interpret that when the students do not know how to continue, theymake the objective of the problem explicit, with the purpose of keeping inmind what they want to achieve (interventions 26 and 27).

26. Anna:Relation between. . . [she points at FEC and DBE, Figure 7].27. Rosa:Between this one here[FEC] and the other one there[DBE] [pause (35)]


The reference to the objective marks the spearhead of a blockade situationthat is solved with the individual reflection of each, without any gesture(e.g., the pause in intervention 27). After this individual reflection, thestudents make contributions that are all of the same nature: the layout ofparallels to the sides of the triangle (intervention 28).

28. Anna: [She draws a parallel line from F to AB, Figure 7].If we draw . . . [she pointsat the parallel line in Figure 8].This [FME] equals that[FLE], does not it?, they will beequal. Is this one, at the top[FLC], not equal to that?[DBE], is it like this or not? I wouldsay it is[she points at the figure from the statement and pretends to draw a parallel line toAB from F] isn’t it?

This contribution opens a new view for the search of relations betweenelements of the new triangles obtained. In this case, this search for rela-tions is centred on comparing the triangles (FME and FLE, FLC and DBE,intervention 28) and the segments produced (interventions 31 to 34).

31. Rosa:How would this part here[DB] be with this one?[DE].32. Anna:I would say that this part from here to there[AN] is equal to that part from hereto there[DE], of course, because they are parallel lines, a parallel. . . [she points at DEand AC].33. Rosa:Yes, yes, this one here[DE] would be the same as that one there[FL].34. Anna:I would say that this part[FL] would be the same as that[DB], wouldn’t it?Because, if we start from here to there. . . [AN].

This succession of interventions starts with a question-answer exchange. Init, Rosa asks for the relation between segments DB and DE (intervention31), and Anna answers this question (intervention 32), although, in thiscase, Anna’s intervention is not a direct answer to Rosa’s question, prob-ably because there is no definite relation between the segments DB andDE. The contents of Rosa’s question do not refer to that of Anna’s formerintervention – ‘Yes, it should be’ (intervention 30) –, in which she onlycorroborates the equality of the triangles FLC and DBE that she introducedin intervention 28.

The communicative role assumed by the students in this part of thesearch for relationships, after intervention 28, begins with a validation-continuation exchange, that she initiated herself, and it is followed by aco-operative dialogue (exchanges 32 to 35), which begins with Rosa’squestion (intervention 31). This co-operative dialogue brings forward somepieces of information which appear for the first time in the solving process,although some of them are not correct (intervention 33). This co-operativedialogue degenerates into a succession of two exchanges (interventions 35to 39) in which Rosa takes the initiative and Anna only gives effect to herclassmate’s affirmations followed by demands of validation.

35. Rosa:If we draw this parallel line[parallel to AB from F, Figure 7],this part that wehave here, this part here[FME] is the same as this one here[FLE], isn’t it?


36. Anna:Yes.37. Rosa:And this one here. . . [DBE], this one here[MDE] and this one here[FLC] areequal, aren’t they?38. Anna:Yes.39. Rosa:We need to find the relation between this one[FEC] and that[DBE].

The two exchanges led by Rosa finish (intervention 39) with the identi-fication of the objective, which starts a new repetition of the performancedescribed.

As we have already observed, in this episode the students make threecontributions to the strategic layout of the parallel lines. This is a newheuristic ability.

15. Rosa: [Draws a parallel from E to AB and points at triangles MDE and DBE, Figure 7].28. Anna: [Draws a parallel line from F to AB] (. . .).41. Rosa: [Draws the parallel to AC from M again].And this one here, like this[She drawssegment LM and points at triangle MEL]equals this one[MDE], does not it?

These three contributions have their origins in parallel work or individualreflection. The relations between the elements of the figures obtained asa consequence of such a contribution help, in a definite way, to obtain asolution to the problem.

The interaction we have identified in this episode – which we callre-launchingof the solving process – is one of the forms which the studentsuse to face the search for new relationships between the elements of the fig-ures. Rosa and Anna reproduce ways of performance based essentially onobjective explanations and individual reflection, after which they usuallyintroduce new pieces of information that are analysed jointly to generateother new pieces of information.

4.2.4. Model of co-operative interactionThe local assessment social episodeBoth students maintain a dialogue formed basically by co-operative ex-changes (we present part of this dialogue in Figure 9). In this, both studentsoften refer to the results obtained previously with expressions such as:“We’ve just drawn. . .”, “that way, we got. . .”, “we said that. . .”, etc. Thisco-operative way of interacting connects, in a definite moment (interven-tion 61), with the introduction of new information.

In this co-operative dialogue (Figure 9) the students request each other,in an explicit way, to speak -by means of demands of validation-on twodifferent occasions (interventions 54 and 55), and another three occasions(interventions 51, 52 and 57) in an implicit way – leaving the sentencesunfinished –.


Figure 9.The interaction of Rosa and Anna in the local assessment social episode of thetriangle problem solving.

The succession of co-operative exchanges guarantees the equality ofthe contribution made by the two students to the progress of solving pro-cess, because each intervention brings some contribution either new orequivalent to the former one (see Figure 9).

Here we have a type of interaction in which the students resort to thereproductions of what they have reached up to now, assuming equally theresponsibility in the continuation of the dialogue. This way of acting servesas a springboard to generating new ideas. This is an interaction basicallyformed by co-operative exchanges that we callco-operativeinteractionmodel.

5. INFLUENCE OF INTERACTIVE MODELS

DISCUSSION AND CONCLUSIONS

We analyse and discuss this section from an interactionist approach fol-lowing the emerging view of the issue expressed by Yackel and Cobb(1996), and Hershkowitz and Schwarz (1999), in which it is assumed thatsocial interaction facilitates learning. When pairs of students collaboratein problem sharing, a series of learning chances are generated. In our case,these chances or opportunities which we express in terms of cognitive be-nefits, are related to the likelihood of improvement in the argumentative


processes; to heuristic abilities, largely those of drawing straight lines;to ways of approaching problems; and to ways of generating new ideaswithin the problem solving process. It is in this way that each participantappropriates aknowing-actiongenearated by each pair. The interactionamong students affects the cognition of each, as if it were a genuine socialsystem. This inter-actionist perspective is in accordance with the ideas ofKieren (2000) in considering that the domain of mathematicalknowinginclassroom situations grows throughout various kinds of intrapersonal andalso interpersonal interactions.

In accordance with the objectives that we have set out in the introduc-tion to this paper, in the following paragraphs we analyse the influence ofthe interactive models described in section 4.2 on the cognitive develop-ment of the students. We relate the thematic contributions and interlocutiveprocesses in which such contributions are generated to the cognitive benefitof the students (see Table II).

Sometimes, requests and challenges made by one of the students ob-liged her classmate to offer explicit details of her thoughts. It occurs, forinstance in theguidedinteraction of the implementation social episode inthe parallelogram problem solving (see Section 4.2.2). Here Anna playsthe communicative role of making validations or questions regarding thecontents of former interventions, with the intention of inciting her inter-locutor -Rosa- to continue the dialogue; on the other hand, Rosa assumesthe responsibility for answering Anna’s questions and demands for valida-tion. In this episode, Rosa answers Anna’s ‘demands’, bringing ideas to thedialogue and providing greater details to the type of reasoning that is beingcarried out. Rosa provides thematic contributions in an interactive con-text that is characterised by a succession of three-intervention exchanges,which she herself directs.

Our interpretation is that both students benefit from such reactions,since Rosa’s explanations oblige her to reflect on the issue of thoughtand strategy, structuring her own thought more effectively, whilst Anna’scognitive benefit is associated with the possibility of learning both fromthe conceptual and procedural arguments that are involved (the breakingdown of the parallelogram into others, the division of the parallelograminto triangles through the diagonal, the equality of the triangles, etc.).

The relief in the succession of the students’ communicative rolesthroughout an episode is the characteristic feature that identifies the so-calledalternativeinteraction in the assumption of communicative roles. Inthe case of the exploration/analysis episode for the triangle problem solv-ing (see Section 4.2.1), Rosa and Anna alternate their communicative rolesdepending on the mathematical contents they introduce, and in the same


episode, briefly and successively, reproduce forms of interaction similar tothose of the guided work.

In this interaction, the thematic contributions are broad indeed, andare principally related to heuristic abilities such as the strategic drawingof lines parallel to the sides of the parallelogram, and to the consequentidentification of the resulting figures. Furthermore, the students reduce theproblem to the search for equivalence between triangles that are differentto those outlined in the instructions, establishing a conjecture. These them-atic contributions are produced after short pauses, or within a co-operativeexchange, marking the borderline between successions of two-interventionor three-intervention exchanges, guided alternatively by each of the twostudents. In this succession of exchanges, the students continue to contrib-ute. Sometimes the contributions prove to be erroneous, the consequenceof the exploratory process in which they find themselves. Such thematiccontributions are then rectified by the students themselves, in the same in-terlocutive context in which they have occurred (principally in explanatoryinterchanges, see Figure 5).

Alternation in communicative roles, and the progress that results asa consequence of the successive contribution made by each student (seeSection 4.2.1), contributes to the benefit of both participants, above all tothe heuristic abilities related to drawing parallel lines. In Rosa’s case, thisbenefit deepens her geometrical view of problem solving. In Anna’s case,it enriches her way of approaching problems, inclining her more frequentlythereafter to geometrical approaches.

Asking questions that do not refer in an explicit way to the contentsof the former intervention gives a new orientation to the dialogue if thenew information introduced becomes the object of discussion. We ana-lyse this case in the exploration/analysis episode of the triangle problemsolving (see Section 4.2.3). Here, Rosa and Anna reproduce forms of in-teracting, based, essentially, on the identifications of the objective followedby individual reflection, after which they usually introduce new pieces ofinformation – generally in the form of questions –, jointly analysed andused in the search of relations between the elements of the figures. We callthis interaction arelaunchingof the solving process. In this way, Rosa andAnna unblock the difficulties encountered in the process.

The thematic contributions in this interaction tend to be related to draw-ing lines parallel to the sides of the triangle, and to the exploratory searchfor equality amongst the resulting segments and triangles. In the first case,they arise after individual reflection; in the second, they come about inco-operative dialogues or at the direction of one of the students.


The relevance of drawing lines parallel to the sides of the triangle inthe solving process, and the similarity of this strategy to that developed inthe parallelogram problem solving, means that the learning opportunitiespresented in this episode are of the same type as those we remarked uponin the alternative interaction.

The co-operative review of the results previously obtained by the stu-dents generates new ideas, which contribute in a positive way to the de-velopment of the process. This is the case of the students’ performance inthe local assessment social episode of the triangle problem solving (seeSection 4.2.4). Here, the co-operative exchanges are formed by interven-tions whose contents have already appeared in the solving process (and thestudents merely repeat them). Thus, in this social episode, the dialogue isproduced in a fluid and rapid manner, since the students share the meaningof the contributions made. Additionally, they assume similar communic-ative roles, given that their interventions are assertions, followed in mostcases by requests (explicit and implicit) for dialogue continuation.

We believe that all the interactive models that we have identified canform part of the students’ learning process. Co-operative interaction maybe easier to assimilate, since new thematic contributions are only producedat the end of the interaction and not (as is the case for the other threeinteractions that we have detailed) whilst it is in mid-process. Without un-dermining the importance of this last type of co-operative interaction, webelieve that it is very useful to foster co-operative dialogues in which everystudent makes contributions that have not previously appeared in the con-text of solving. These considerations have implications on the role of theteacher, which is not only to propose problems, but is, rather one that oc-cupies a central point in the network of interactions, promoting the devel-opment of mathematical knowledge in class, administering knowledge-in-action produced in the solving process, as the necessary final reflection onnew contributions that allow each student to appropriate new knowledge,generated socially.

Table II summarises the general models of interactions discussed aboveand their effects on students’ learning opportunities:

These contributions to the identification of interactions and the eval-uation of productivity in problem solving are the consequence of threedecisions we took on the design of the research presented in this paper:the grouping of students into pairs, to the non-intervention of the observerin the solving process and to the decision of taking observations out ofthe classroom. The variation of those limits should open perspectives tonew research. To sum up, questions such as the adaptation and amplifica-tion of the type of exchanges proposed to more than two students, and to


TABLE II

Resume of interaction types and of their influence

INTERACTION THEMATIC EXCHANGES COGNITIVECONTRIBUTIONS BENEFITS

Guided Decomposition of a Thematic contribu- Cognitive reorgan-parallelogram into tions are generated isation. Opportunityothers. Division of in exchanges of to improve the argu-a parallelogram into three interventions mentative processtwo equal triangles. and conceptsArgumentative and proceduresprocess involved

Alternative Recognition of In isolated Generating ofequality and co-operative a geometricalinequality in lineal exchanges, after approachelements and brief pauses, or in Heuristic abilitiestriangles. Strategic three-intervention (drawing straightdrawing of straight exchanges lines, conjectures,lines. Reduction problem reduction)of the problem.Conjecture

Relaunch Identification of In individual reflec- Generating of atriangle equal- tions followed by geometrical ap-ity. Segment brief dialogues (co- proachcomparison. operative or guided) Opportunity toDrawing parallels. improve the searchExploratory search for new equalities.for triangle equality Heuristic abilities

(drawing parallellines)

Co-operative Review of previous In the co-operative Learning oppor-results exchanges tunities to generate

new ideas

those with the teacher, and the identification of new forms of interactionbetween pairs of students in problem solving, among other issues, are opento debate.

It is too early to establish the didactic implications extensive to thecontext of a classroom from the cases analysed. We should also be awarethat it is not enough to sit students together in order to solve problems:to establish successful collaboration between them, it is also necessaryto teach themhow to collaborate, in order that they get a clear idea ofwhat we expect of them (Mercer, 1995), thereby increasing the quality andefficiency of the teaching-learning process in mathematics.


Teachers need to learn how to keenly observe the ways in which inter-subjectivity is created within the unit of a pair, and to comprehend exactlywhat kind of benefits can undoubtedly be gained in relation to mathem-atical ability. Student pair-work collaboration in problem solving createsa dynamic that would appear to destabilise or disturb their usual mode ofworking. This forces them to focus both closely onto and away from theirconcepts, to make conjectures or provide well-argued reasons. In this way,it gives rise to cognitive and social progress.

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PEDRO COBO

IES Pius Font i Quer08240 Manresa (Barcelona)SpainE-mail: [email protected]

JOSEPM. FORTUNY

Universitat Autònoma de BarcelonaFacultat d’Educació08193 Bellaterra (Barcelona)SpainE-mail: [email protected]

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