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Two Wrongs May Make a Right ... If They Argue Together!Baruch B. Schwarz; Yair Neuman; Sarit Biezuner

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Two Wrongs May Make a Right … IfThey Argue Together!

Baruch B. SchwarzSchool of Education

The Hebrew University, Jerusalem, Israel

Yair NeumanDepartment of Education

Ben Gurion University of Negev, Beer-Sheva, Israel

Sarit BiezunerSchool of Education

The Hebrew University, Jerusalem, Israel

Several studies have investigated the cognitive development of interacting peers.This study focuses on a phenomenon that has not yet been studied: the cognitive gainsof 2 children with low levels of competence who fail to solve a task individually butwho improve when working in peer interaction. We show that this phenomenon(which we call the two-wrongs-make-a-right phenomenon) may occur when (a) the2 wrongs disagree, (b) they have different strategies, and (c) active hypothesis test-ing is made possible. In a preliminary study, 30 Grade 10 low-achieving studentswere tested about the rules they use to compare 2 decimal fractions in a question-naire. The students who were diagnosed as wrongs were invited to solve a task (the6-cards task) with peers. Three kinds of pairs were formed: 7 W1–W2 pairs in whichthe 2 wrongs have different conceptual bugs; 4 W1–W1 pairs in which the 2 wrongshave the same conceptual bugs; 4 R–W pairs in which a wrong interacted with aright student. The 6-cards task was designed to create conflicts between studentswith different conceptual bugs and between wrong and right students. Two days af-ter solving the 6-cards task, the students were asked to answer a similar question-naire individually. The preliminary study revealed the two-wrongs-make-a-right

COGNITION AND INSTRUCTION, 18(4), 461–494Copyright © 2000, Lawrence Erlbaum Associates, Inc.

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phenomenon: Among the 7 W1–W2 pairs, at least 1 wrong became right. In con-trast, in the 4 R–W pairs, only 1 wrong became right, and in the 3 W1–W1 pairs, nochange was detected. In a case study that replicated the phases of the preliminarystudy, disagreement, argumentative operations (such as challenge and concession),hypothesis testing (with a calculator), and the internalization of social interactionsmediated the change of peers from wrongs to rights. We then replicated the initialstudy with 72 low-achieving Grade 10 and 11 students, confirming thetwo-wrongs-make-a-right effect.

Scholars from both Piagetian and sociohistorical perspectives have long discussedthe role of a child’s interactions with age peers in the development of cognition(e.g., Damon & Phelps, 1989; Doise & Hanselmann, 1991; Hartup, 1970;Vygotsky, 1986). This study continues these efforts but focuses on a phenomenonthat has not been systematically investigated—the cognitive gains of two wrongs1

interacting to solve a task they are unable to solve individually.A priori, it seems dubious that when two wrongs interact, at least one of them

should make cognitive gains. At any rate, it seems obvious that in two other kindsof interaction, the wrong should gain more: in modeling studies (in which onechild observes a more competent child) and in active interaction studies in whichthe wrong interacts with a more competent child.

Modeling studies have shown that watching more competent children im-proves subsequent performance on the observed task (e.g., Kuhn, 1972; Murray,1974). Also, interaction studies involving peers with different levels of compe-tence yielded cognitive gains for the less competent students. For example,Botvin and Murray (1975) asked students of different levels of competence(conservers and nonconservers) to solve the task together actively, and othernonconservers were spectators. It was found that nonconservers of both groupsgained from the task and that no significant difference could be detected be-tween the two groups. Therefore, this is the question: Why should two wrongsgain from interaction between themselves? We show in this section that such acognitive gain is theoretically possible. For this purpose, we analyze the reasonsoffered by researchers about why, in modeling and interaction studies betweenchildren with different levels of competence, low-competence students makecognitive gains.

Two factors have been recognized as beneficial in the cognitive development ofinteracting peers and in modeling studies: disagreement and being strategic (i.e.,

462 SCHWARZ, NEUMAN, BIEZUNER

1We use the term wrong to designate an idealized cluster of children using strategies or proceduresthat lead to erroneous outcomes in specific tasks. Therefore, a sentence such as, “This student is awrong,” means that the student belongs to the class of students using such strategies or procedures for aspecific task. In fact, our use of wrongness is never pejorative, as our main claim in this article is that for awrong interacting with a peer, the wrongness of the peer is often preferable over her rightness.


being able to give reasons or arguments2 for a specific solution or offering an oper-ational solution). For example, Miller and Brownell (1975) showed thatconservers influenced nonconservers and not vice versa because they could giveconsistent reasons for their solution when arguing with their peer. In contrast, thenonconservers kept asserting their solution without invoking reasons in favor oftheir assertions. Moreover, these researchers also suggested that simply hearing acontradicting solution plays a major part in the cognitive gains of peer interaction.Therefore, when two interacting solvers disagree, their cognitive gains originatenot only from a pragmatic component—the disagreement—but also from the con-tradicting solution itself. This suggestion was confirmed by a study conducted byDoise, Mugny, and Perret-Clermont (1975), in which students gained from beingpresented with a contradicting solution by an adult, whether the solution was cor-rect or not. In another study, Doise and Mugny (1979) showed that interaction witha less capable child who proposed a contradicting solution led even the more capa-ble child to progress. In the same study, Doise and Mugny showed that when inter-acting students used different strategies, they progressed, whereas when they usedthe same strategies, they did not. A key result obtained by Doise and Mugny wasthat if the ability difference between the two students was too big, low-level stu-dents did not progress. As for modeling studies, Botvin and Murray (1975) showedcognitive gains for children who observed peers expressing and defending differ-ent (counter)arguments. Kuhn (1972) similarly argued that an observer who dis-agrees with the modeler is in a situation of mismatch, and the observer actuallyengages with the modeler in a kind of “tacit interaction.”

In addition to disagreement and being strategic, Doise (1978) showed that hy-pothesis testing (i.e., the manipulation of materials available in the task to check asolution) also leads to cognitive gains. However, the more competent student is of-ten too dominant when doing manipulations; thus, preventing the less competentstudent from making gains.

Glachan and Light (1982) reviewed modeling and active interaction studies tohypothesize that “interaction between inferior strategies can lead to superior strat-egies or, in other words, two wrongs can [italics added] make a right” (p. 258).Glachan and Light grounded their hypothesis on the analysis of the very conditionsthat peer interaction studies have found as affording cognitive gains: disagree-ment, being strategic, and hypothesis testing. Such conditions do not depend onwhether the peers are right or wrong. It follows, then, that the active interactionamong two wrongs can lead to cognitive gains when such conditions are fulfilled.

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2Being strategic and giving reasons for the choice of a solution relate to the cognitive and the conver-sational sides of the same coin. We refer to this condition as simply being strategic although obviouslysome strategic students do not articulate the strategies that lead them to the solution for several reasons(e.g., verbal propensity or tasks that involve manipulations but not verbal explanations).


The researchers suggested that such an interaction can be beneficial because thediffering strategies being pursued by the two children lead to moves inconsistentwith one or both of their strategies.

The child is thus led to (jointly) make moves which he would never otherwise havemade, so that established inefficient strategies are disrupted. As a consequence of thisdisruption one or both of the children may see possibilities for better strategies. Inter-action is thus envisaged as a destabilizing influence. (p. 258)

In sum, Glachan and Light hypothesized that two wrongs can make a right if the stu-dents involved use different strategies and have opportunities to resolve their con-flict.

The destabilizing influence alluded to by Glachan and Light (1982) clearly hasPiagetian roots. It relates to stages of development. However, what about the pro-cess that turns wrongs to rights? The analysis of the studies on adequate conditionsfor cognitive development through peer interaction shows that cognitive gains areassociated with arguments and counterarguments. For example, Miller andBrownell (1975) showed that nonconservers are more likely to yield to conserversbecause conservers produce counterarguments (see also, Doise et al., 1975; Miller,Brownell, & Zukier, 1977). Botvin and Murray (1975) showed that in modelingstudies, the cognitive gains of observing peers originated from listening to solversexpressing and defending different (counter)arguments (see also Kuhn, 1972).And, indeed, subsequent studies (Murray, Ames, & Botvin, 1977) showed that themain cause of cognitive gains was cognitive dissonance resulting from argumenta-tion or observation of conservers’ reasoning.

The role of argumentation in peer interaction and cognitive development re-ported so far in a sporadic way has been examined systematically in several stud-ies. For example, Kuhn, Shaw, and Felton (1997) showed that peer interactionfosters argumentative reasoning. They considered arguments as outcomes ofdyadic interaction and showed that the quality of arguments increased as the levelof interaction increased. Means and Voss (1996) gave a theoretical explanation forthe central role of argumentation in reasoning and learning:

Informal reasoning skills are developed via the learning of language structures, espe-cially those of argumentation, which facilitate the storing and accessing of knowl-edge, and the development of more extensively differentiated situation models, whichlead to better inference generation, problem solving, and learning. (p. 169)

Such a claim is echoed by leading figures from a sociocultural perspective(Leonte’ev, 1981; Vygotsky, 1986), who claim that forms of discourse such as ar-gumentative talk become forms of thinking. In the words of Pontecorvo (1993b),“thinking methodologies are enacted through appropriated discourse practices that



respond to the epistemic needs of a disciplinary topic” (p. 191). The sequence ofclaim followed by opposition followed by counteropposition sparks the need forjustification and finally for explanation. In the classroom, there is a pragmatic needto convince opponents and a need to answer a teacher’s requests for explanations.In this social context, children will express further explanations and will learn, asPontecorvo (1993a) argued:

When there is an opposition of points of view, there is a psychological need for pro-viding reasons and further explanations. If individuals want to preserve a relationship,they must persuade others of the validity of their positions by providing explanationsand reasons (p. 301).

In the experimental studies following this sociocultural perspective (e.g.,Hershkowitz & Schwarz, 1999; Pontecorvo & Girardet, 1993; Resnick, Salmon,Zeitz, Wathen, & Holowchak, 1993; Schwarz & Hershkowitz, 1995), discourseis analyzed in terms of claims and justifications—not in terms of conclusionsand premises. Also, some studies show how intersubjective processes becomeintrasubjective. For example, Trognon (1993) studied how two students solved alogical problem. He showed how relevant information is provided by one mem-ber of the pair, even if his or her point is incorrect, and how this information isappropriated and transformed by the other member through arguing. It is impor-tant to note, however, that research conducted on argumentation and cognitivedevelopment following the sociocultural approach has consisted primarily ofcase studies.

This study examines Glachan and Light’s (1982) hypothesis that two wrongsin peer interaction can make a right within an experimental design; thus, at-tempting to bridge between the sociocultural approach and experimental meth-ods. In other words, we attempt to investigate empirically whether when twowrongs interact on a task, at least one of them becomes right. We also attempt toexplain how wrongs become rights by studying their peer interactions. In lightof the research on peer interaction, we began study of this phenomenon (whichwe call the two-wrongs-make-a-right phenomenon) with three working assump-tions: (a) The two wrongs disagree on the solution, (b) they employ different so-lution strategies, and (c) they have opportunities to test hypotheses. We do notargue that such working assumptions are necessary conditions for triggering thetwo-wrongs-make-a-right phenomenon—only that existing literature suggeststhat they may increase the likelihood of conceptual change. Therefore, they pro-vide a framework to begin studying the effectiveness of the peer interactions.

We sought a domain and a task that would fulfill the three working assump-tions. The domain we chose—decimal fractions—is known for the strategies usedby students, their consistency, and the conceptual grounds on which these strate-gies rely. Moreover, we designed tasks demanding active hypothesis testing.

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INCORRECT RULES FORCOMPARING DECIMAL FRACTIONS

The consistent use of incorrect rules by students has been identified and usedwidely to analyze errors in performance in mathematics, first in the domain of sub-traction (Brown & Burton, 1978; Brown & VanLehn, 1982). The incorrect ruleswere called procedural bugs (VanLehn, 1989). Similar accounts of proceduralbugs have been given for decimal computation procedures (Hiebert & Wearne,1985) and for elementary algebra (Matz, 1982). Sackur-Grivard and Leonard(1985) found that in fourth- and fifth-grade classes in France, children used threeincorrect rules to compare decimal fractions when the numbers to be compared hadthe same whole-number digit. Resnick et al. (1989) conducted another study inthree countries (United States, France, and Israel) to identify the incorrect rules ofelementary school children and the origins of these rules. The study showed that theincorrect rules found in various classes from the three countries are identical tothose found by Sackur-Grivard and Leonard. The rules are the following:

Rule 1: The number with more decimal digits is the larger—for example, 4.8 <4.68. We designate this as the whole-number rule (WN-rule) because the decimalportion of the number is treated as a whole number.

Rule 2: The number with fewer decimal digits is the larger—for example,4.4502 < 4.45. The decimal part .4502 is less than .45 because the whole is dividedinto more parts that are, therefore, smaller. The students using this rule see the dec-imal part as a division of a whole. We designate this as the fraction rule (F-rule).

Rule 3: When one or more zeros appear immediately after the decimal point inone of the numbers, the child’s judgment is correct. For the remaining numbers, thechild employs the WN-rule. So, when asked to order the three numbers 3.214, 3.09,and 3.8, the child would decide that 3.09 is the smallest, then 3.8, and then 3.214.We designate this as the zero-rule (Z-rule).

Although Resnick et al. (1989) found that the incorrect rules are identical in thethree countries, there were important differences in their distribution. For example,the F-rule was used rarely by the French students (as in the study by Sackur-Grivard& Leonard, 1985), whereas American and Israeli students used this rule often. Incontrast, the WN-rule was used often by French students and less frequently byAmerican and Israeli students. Resnick et al. showed that this difference in rule usehas its origins in instruction. American and Israeli students learned decimal frac-tions just after ordinary fractions, whereas in France the two were separated sub-stantially in time. Resnick et al. suggested that French students’ adoption of theWN-rule originates from the common practice of comparing decimal numbers withthe same number of decimals in French classes (see also Sackur-Grivard & Leon-ard, 1985). They suggested that the invention of the rules derives from “students’



attempts to integrate new material that they are taught with already establishedknowledge” (p. 25). Moreover, they showed that 88% of the students in the threecountries were consistent in the use of one incorrect rule for comparing decimalfractions. Finally, the Resnick et al. study showed that wrong students were able toproduce explanations for their choice. Because of their consistent use by studentsand their conceptual basis, the domain of decimal fractions is a good candidate forstudying the two-wrongs-make-a-right phenomenon. In this study, we call incor-rect rules conceptual bugs, with conceptual referring to the invention of rules aris-ing from integration of new material with existing knowledge and bug referring tostudents’ consistency.

THE PRELIMINARY STUDY

Our aim in the preliminary study was to check whether the two-wrongs-make-a-rightphenomenon is empirically grounded. The independent variable was the type of inter-action. Three types of interaction were considered: pairs consisting of one competent(right) and one noncompetent (wrong) student (R–W pairs), pairs consisting of twononcompetent students with different conceptual bugs (W1–W2 pairs), and pairs con-sisting of two noncompetent students with the same conceptual bug (W1–W1 pairs).The dependent variable was the students’ solution accuracy after the interaction.

Wehypothesized that thedifferentexperimental treatmentswouldyielddifferentcognitive gains. Specifically, we predicted that students in W1–W2 and R–W pairswould be more inclined to repair their procedural bugs than would students in theW1–W1 pairs. This hypothesis was based on the assumption that W1–W1 pairswould not argue because they did not disagree. In contrast, we hypothesized that stu-dents in W1–W2 and R–W pairs would disagree and argue. As the peers justifiedtheir comparison of decimal fractions using different strategic explanations, we pre-dicted they would produce counterarguments that would lead to at least one of thepeers integrating the counterargument into the construction of a new argument and acorrespondingnewcognitiveconstruct.Wealsopredicted that theavailabilityofhy-pothesis testing would facilitate the evaluation of the arguments.

Method

Participants

Thirty 10th-grade Israeli students (18 girls and 12 boys) participated in thestudy. Their ages ranged from 15 years and 11 months to 17 years and 5 months.The students came from the bottom 20% in achievement within the population.We preferred low-achieving high school students to elementary school studentsbecause verbal utterances of older students are easier to analyze.

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Materials

The questionnaire. We used the nine-item questionnaire (see Table 1) de-veloped by Resnick et al. (1989) to classify students according to their conceptualbugs. In each item, two fractions are presented, and the student is asked to indicatethe larger number. For ease of interpretation, the table always shows the correct an-swer as A, although in the actual questionnaire, the position of the correct answerwas randomized. Thus, Column R, representing right student profiles, consists of Aanswers only. The questionnaire differentiates among children using the three con-ceptual bugs consistently. For example, a student who consistently used thewhole-number strategy (WN-student) would choose the B fraction in the first twosets of items but answer the third set correctly.

The six-cards task. The six-cards task was specifically designed for thisstudy to encourage argumentation during the interaction phase. Six cards contain-ing the digits 0, 0, 5, 8, 4, and a decimal point were presented to the student or stu-dents. The goal was to use all cards to construct:

1. The biggest possible number2. The smallest possible number3. The number closest to one4. The number closest to one half

We chose the digits on the cards and the four subtasks to create conflicts result-ing from different conceptual bugs. For example, if we assume that students are


TABLE 1The Questionnaire in the Preliminary Study

Number Pair Consistent Profiles Examples of Students’ Answers

Item A B WN Z F R S1 S2 S7 S10

1 4.8 4.63 B B A A B B A A2 0.5 0.36 B B A A B B A A3 0.25 0.100 B B A A B A A A4 13/100 0.125 B B A A A — B A5 4.7 4.08 B A A A B B A A6 2.621 2.0628790 B A A A A A A A7 4/100 0.038 B A A A B — A A8 4.4502 4.45 A A B A A A B B9 0.457 4/10 A A B A B — B B

Note. The question asked was, “For each pair, circle the number that is bigger (“A” answers arecorrect). WN = whole number; Z = zero; F = fraction; R = right (i.e., competent).


quite systematic in the conceptual bugs they use to compare decimal fractions,WN-students should answer 0.8540 for the third part of the task, whereas F-stu-dents should answer 0.0458. However, the task is more complex than the question-naire: Students need to construct and choose decimal fractions according to theirmagnitude or their closeness to one or one half among a large variety of possiblenumbers. We hypothesized that, although conceptual bugs did not determine an-swers for the four subtasks, students with different conceptual bugs generallywould disagree on the result of each subtask.

Procedure

Classification phase. The 30 students first answered the questionnaire toidentify their conceptual bugs. We matched their answers to those of the consis-tent WN, F, Z, and R-student profiles shown in Table 1. Of the 30 students, 11matched one of the consistent profiles perfectly (e.g., S10 is a consistent F-stu-dent, as shown in Table 1). For each of the other 19 students, we computed thesum of the deviations in the answers from the answers given by WN, Z, F, andR-consistent profiles. For example, the sums of the deviations for S1 from WN-,Z-, F-, and R-consistent profiles are 3, 4, 7, and 6, respectively, and the sums ofthe deviations for S7 are 8, 5, 1, and 3, respectively. Each student was assignedto the conceptual bug with the minimum deviation. Thus, S1 was identified as aWN-student and S7 as an F-student. Three of the 30 students did not evidenceclear conceptual bugs, having minimum distances of 3 or more from all the con-sistent profiles. For example, S2’s distances from consistent WN, Z, F, and Rprofiles are 5, 5, 7, and 5, respectively. These 3 students were interviewed, andtheir affiliation to a conceptual bug was decided according to their most consis-tent explanations. It appears that these students did not understand the meaningof the simple fractions appearing in several of the questionnaire items (4, 7, and9). If we eliminate these items, the picture is clearer. For example, S2 generallyinvoked explanations relating to the whole-number conceptual bug and was thusclassified as a WN-student.

Interaction phase. The students worked on the six cards task in two stages.In the first stage, each student was invited to solve, individually, the four subtaskswhile using all cards, with a time limit of 3 min. In the second stage, the studentswere assigned to pairs: four pairs of right–wrong (R–W) students, seven pairs ofwrong students with different conceptual bugs (W1–W2), and four pairs of wrongstudents with identical conceptual bugs (W1–W1). The teacher asked the pairs toreach agreement; in the event that they could not reach agreement, they could use acalculator (to test their hypotheses). Two days after the interaction, each of the stu-dents was given the questionnaire (with different numbers).

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Results

Due to its exploratory nature, the first study involved a relatively small number ofstudents. We consequently use descriptive statistics only to report the results. Table2 summarizes the findings of this study. The two last rows of Table 2 show that theinteraction in the four W1–W1 pairs (three F–F pairs and one Z–Z pair) did not re-sult in any change in conceptual bugs (see columns F–F and Z–Z), except for oneF-student who became a Z-student (see column F–Z). This result supports our hy-pothesis that W1–W1 interactions do little to change students’ conceptual bugs.

The rest of the analysis focuses on changes in the procedural bugs of the 22 stu-dents who interacted in R–W and W1–W2 pairs. Of the 22 students, 13 changed.As shown in the two first rows in Table 2, of the 4 rights, 3 remained right and 1 re-gressed. Moreover, 1 of the 4 wrongs (25%) in the R–W condition became right. Incontrast, the third and fourth rows in Table 2 show that 9 of the 14 wrongs (65%) inthe W1–W2 condition became right. Because the dyad was our basic unit of analy-sis, we checked in each condition the number of dyads in which at least 1 of thewrongs became right. It was found that 100% of the dyads changed after interac-tion in the W1–W2 condition, meaning that at least 1 of the wrongs became right.This result should be contrasted with 25% of the dyads that changed under theR–W condition.

Thus, this study suggests that the two-wrongs-make-a-right phenomenon existsand that, in the case of decimal numbers, the effect is stronger than in wrong–rightinteractions. Another interesting finding is that the 6 Z-students in R–W andW1–W2 pairs became rights. In contrast, only four of the nine F-student pairs be-came rights. These findings show an interesting tendency for students with differ-


TABLE 2Conceptual Bugs of the 15 Pairs of the Quantitative Study Before and After Interaction

F–Z F–R

Pairs WN–R F–F Z–Z Z–F F–Z R–F F–R R–R

4 W–R pairsWN–R — — — — 1 — — —F–R — — — — — — 2 1

7 W1–W2 pairsF–Z 1 — — — — — 3 2WN–F 1 — — — — — — —

4W1–W1 pairsF–F — 2 — — 1 — — —Z–Z — — 1 — — — — —

Note. WN = whole number; R = right (i.e., competent); F = fraction; Z = zero; W = wrong (i.e.,noncompetent).


ent conceptual bugs to gain differentially from interactions with peers. Thesignificance of this tendency could not be checked because of the small number ofstudents in each cell of the analysis.

Obviously, several issues arose from this preliminary study. First, the small sam-ple size leads to questions of generalizability regarding the overall effects of twowrongs interacting and the relative superiority of W1–W2 to R–W interactions. Asecond issue concerns the differential effect of social interaction between studentswith different conceptual bugs. The preliminary study showed that Z-studentsgained most from the interaction and that F-students were more resistant to change.Again, the number of students involved was too small to be statistically analyzed,and a larger study focusing on this issue was needed. However, before undertaking asecond study devoted to these two issues, a more basic question remained unan-swered: Although we had evidence that the two-wrongs-make-a-right phenomenonexists, the processes that govern this phenomenon remained unclear. We decided todeepen our understanding of the two-wrongs-make-a-right phenomenon by con-ducting a case study in which our goal was not to establish external validity but to ex-amine the role of argumentation in mediating the two-wrongs-make-a-rightphenomenon.

A CASE STUDY ON THE INTERACTION BETWEENPAIRS OF DIFFERENT WRONG STUDENTS SOLVING

THE SIX-CARDS TASK

Our goal in the case study was to identify mechanisms explaining the disappear-ance of conceptual bugs in the course of interaction among W1–W2 pairs. We ad-ministered the decimal questionnaire (Resnick et al., 1989) to 30 Grade 10low-ability students. We then selected one Z-student—Ve—and one F-stu-dent—Si—whose deviations from consistent Z- and F-profiles were 1. Because ourgoal was to understand both change from wrong to right and resistance to change,our pairing of a Z-student with an F-student was intentional. (Recall that the prelim-inary study indicated that Z-students were more inclined to change than were F-stu-dents). We asked the pair to solve the six-cards task. The procedure followed thesame sequence as the preliminary study, and, at the end of the task, each student an-swered the questionnaire (with different numbers). As in the preliminary study, theZ-student became right, and the F-student retained the same conceptual bug.

The protocol of Ve and Si solving the six-cards task was transcribed. We pro-vide here excerpts from the protocol for the second subtask (constructing thesmallest number with 8, 5, 4, 0, 0, and a decimal point) and the fourth subtask (con-structing the number closest to one half). We organize the description of the proto-col around mechanisms we identified during the interaction between Ve and Si.An important caveat must be made. We do not claim at this point that these mecha-nisms caused the repair of the conceptual bug Z for Ve, nor its retention for Si, but

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rather that the mechanisms could explain the respective outcomes. We show in alater section that although some of the mechanisms described in the case study arespecific, they belong to general classes.

Mechanism 1: Disagreement With Strategic Reasoning

The first potential change mechanism identified in the interaction resulted from thedesign of the six-cards task. The task offers the opportunity for students to discussand reason strategically in defense of their positions (Doise & Mugny, 1979). Thetask imposes a clear goal: to compose with six given cards a decimal number fulfill-ing a condition. The design of the task leads students with different strategies todraw different conclusions and, hence, to disagree. Students then articulate theirstrategies to convince the peer. For example, at the beginning of the protocol of thesecond subtask, Ve’s strategy is evident immediately after the experimenter pres-ents the task:

Exp20: Now, please, construct the smallest number.Ve21: 04.058 that is 4.58.

Ve’s first try shows that she thinks it is possible to remove a “decimal” zero withoutchanging the value (an alternative to the zero-related bug described bySackur-Grivard & Leonard, 1985). It seems that Ve has a definite strategy as shebegins her interaction with Si. (The two girls agree that the whole number part ofthe smallest number needs to be zero. Si constructs 0.0854 with the six cards, andVe constructs 00.458.)

Ve35: The numbers after the period go smaller. 458 is smaller than 854 andbecause of that my number is smaller.

Si36: If you take a pie and you divide it into 458 parts, every part will be largerthan if you divide by 854. You have 458 parts and I have 854 parts, so Ihave more parts than you do. So mine are smaller than yours are.

In this short excerpt, Ve and Si give different answers and also articulate clear ex-planations of their answers. Si explicitly invokes the main idea on which the F con-ceptual bug relies3 (Si36), and although the number she offers is indeed smallerthan V’s, Si defends it with an erroneous justification. Ve again invokes her buggyrule related to zero (Ve35). When comparing 0.0854 and 00.458, Ve ignores the


3Si expresses a variant of the F conceptual bug, as the fraction bug of Resnick et al. (1989) is based onthe number of digits after the decimal point. More digits means “more pieces” and, thus, a smaller num-ber. Si’s argument is not based on the number of digits. However, as the main idea of the F conceptualbug is expressed, we identify Si with an F-student.


digit zero after the decimal point in 0.0854. As before (Ve21), she removes the zerofrom the decimal part of 0.0854 to compare 0.458 to 0.854. Si and Ve both seemstrategic because they give consistent explanations that function as argumentativemoves. Both explanations appear as counterarguments aimed at challenging the an-swer of the peer, as expressed in “because of that, my number is smaller” and “so,mine is a smaller one than yours.” It appears then that the design of the task elicitsalternative strategies, thereby triggering argumentative activity.

In sum, the first mechanism extends beyond the dyad to include the situation inwhich they interact. The opportunity for disagreement is realized through the task,which leads students with different strategies to propose different solutions.Glachan and Light (1982) already identified disagreement as a condition for con-ceptual change. According to our sociocultural approach, the opportunity (createdby the task) for disagreement between two strategic students is central to the pro-cess of change.

Mechanism 2: Hypothesis Testing as a Way ToEvaluate a Counterargument

The second mechanism, hypothesis testing, also involves the design of the task, butit relies on the initiative of the peers. For example, the peers have a calculator attheir disposal, which they may use for testing hypotheses, and Ve uses it when shecannot refute Si’s counterargument (expressed in Si36):

Ve37: I don’t know. [She takes the calculator. She multiplies 0.458 by 10 andobtains 4.58. She multiplies 0.0854 by 10 and obtains 0.854.]

Ve38: Yours is less but I don’t understand! Why?Si39: I have 854 thousandths and you have 4,580 thousandths. So mine is

smaller because the two are thousandths.

In this excerpt, Ve, who cannot refute Si’s counterargument (“I don’t know”[Ve37]), evaluates both arguments with the calculator. She uses a strategy thatavoids the interpretation of decimal numbers (Ve37): She multiplies the two num-bers by 10 and obtains 4.580 in which the whole-number part is 4, and 0.854 inwhich the whole-number part is zero. Ve then concedes that Si is right (because sheknows how to compare decimal numbers with different whole-number parts), butshe does not understand why (Ve38). Si gives a justification (“I have 854 thou-sandths and you have 4580 thousandths.4 So mine is smaller because the two are

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4Although Si mistakenly compares thousandths when she should compare ten thousandths, this detailis not relevant because she preserves the proportion 10 between the two. The same mistake occurs inSi41, but again, it is not relevant to the argumentative activity that takes place.


thousandths” [Si39]). It appears, then, that hypothesis testing is capitalized on in anargumentative context as a means to evaluate arguments. In this case, the hypothe-sis testing was preceded by a counterargument (Si36), and it yielded a justification(Si 39) and a concession (“yours is less”) followed by a query (“but I don’t under-stand! Why? [Ve 38]).

Mechanism 3: Inferring New Knowledge ThroughChallenging and Conceding

As the exchange continues, Ve’s hypothesis testing leads to a third potential mech-anism of change—challenge and concession:

Ve40: So … [she changes her number to 0.4580]. Now mine is smaller.Si41: You did not change anything. Zero does not change anything because

it’s possible to take it off, 0.4580 is 4580 divided by 1,000 that is 458 di-vided by 100 and this returns to 0.458.

Ve42: And zero in the middle. Say, 0.0458?Si43: Here it’s impossible to take it off. It’s 458 divided by 1,000.

Ve44: And you, what do you have?Si45: 854 divided by 1,000. So your number 0.0458 is smaller.

Ve46: That is, that zero at the beginning and at the end does not count and inthe middle does count.

According to our initial screening, Ve originally showed evidence of the Z concep-tual bug described by Sackur-Grivard and Leonard (1985). The Z conceptual bug isrelated to the WN conceptual bug, so that for Ve the length of a number is an indica-tor of the magnitude of the decimal part of the number. The length criterion raisessome questions for Ve about the values of 0.458 and 0.4580. Her opposition (“So …[she changes her number to 0.4580]. Now mine is smaller” [Ve40]) to Si’s explana-tion (Si39) functions as a way to get clarity on a piece of knowledge about whichshe is not confident. We interpret Ve’s move in Ve40 as a challenge–an inferencemechanism in which Ve both challenges Si’s explanation and attempts to infer newknowledge. This interpretation is strengthened by Ve’s query “And zero in the mid-dle. Say, 0.0458?” (Ve42) following Si’s explanation “Zero does not change any-thing” (Si41). Si’s explanation and final justification (Si43 and Si45) are followedby Ve’s concluding reason that supports the argument “0.0458 is the smallest num-ber”: “That is, that zero at the beginning and at the end does not count and in themiddle does count” (Ve46). Ve’s statement reflects both a concession to Si and theconstruction of new knowledge. Before learning this new rule, Ve violated it—a vi-olation that is perfectly consistent with her previous mistakes. We see then that herconstruction of the correct rule is driven by argumentative operations (challenge,concession).



Ve’s newly constructed knowledge is evident as the two girls move to the taskof finding the number closest to 1. Si proposes 0.8540 and Ve 00.854. They agreethat the numbers they each proposed are the same, and Ve restates her new rule:“This is the same number, because zero at the beginning or at the end does notchange” (Ve51).

Mechanism 4: Internalization of Social Interactions

So far, we have observed synchronous links between interactional and reasoningprocesses (e.g., challenge leading to inference and agreement leading to rule appli-cation). Here, we observe how peers capitalize on previous social interactions inlater activities. For this purpose, we present the discussion of the last subtask, form-ing the closest number to one half:

Ve58: 00.548.Si59: 0.4580. If for 1, we had to reach 1,000, for one half we need to reach

500. And 458, the zero doesn’t count because it is at the end, is closer to500 than 548.

Ve60: From my number, to reach 500, one needs to go down 48, and fromyours, one needs to go up [Ve takes the calculator and computes 500 –458 to obtain 42]. So yours is closer to one half.

Si61: That’s correct.Ve62: One moment! [She changes her number to 0.5048.] This is the same?

Here also, we have to take off 48, but you said that zero in the middlechanges!

Si63: It’s not the same. Now we have to take off 48/1000 and before it was48/100, so this number is smaller.

Ve64: One second! Zero in the middle always makes it less? And if we’ll put ithere [changes the number to 0.5408].

Si65: Now, we have to take off 408/1000, and before it was 48/1000, so0.5048 is closer to one half.

Ve66: That is to say that zero in the middle, the more it’s to the left, the smallerthe number is, that is, 0.0548 is smaller than 0.5048?

Si67: I don’t understand what you mean, but 0.0548 is less than 0.5048 forsure.

Ve68: Doesn’t matter. I got it!Si69: So the closest number to one half is 0.5048.

Ve70: [Moves the cards, mumbles.] 0.5048 is the closest to 0.5000.

This excerpt involves iterations of earlier aspects of the girls’ interaction: A dis-agreement (Ve58–Si59) leads to hypothesis testing (Si59–Ve60) and Ve’s con-struction of new knowledge (Ve66). The excerpt, however, that occurs at the end of

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the interaction between Ve and Si also illustrates the internalization of a social in-teraction. From Ve’s comments, we can see a general mechanism acquiring fleshand bones. When Ve changes her number to 0.5048, she attributes the rule “zero atthe beginning and at the end does not count and in the middle does count” to Si (Vedeclares “but you said that zero in the middle changes” [Ve62]), although Ve her-self formulated it (Ve46). Ve first formulated the rule as a result of a number of jointargumentative moves: challenge (Ve40), counterchallenge (Si41), query (Ve42),implicit challenge (Ve44), concession (Si45), and claim (Ve46). Consequently, Vemixes up what she inferred in Ve46 with what led her to this inference: Si’scounterchallenge (Si41), Si’s justification (Si43), and concession (Si45).

Mechanism 3: Generalized Version: Inferring NewKnowledge Through Argumentative Operations

The girls’ interaction during the fourth subtask (finding the number closest to onehalf) shows again how an argumentative operation mediates the inference of newknowledge. This time, however, the mechanism seems much broader than the spe-cific challenge–inference mechanism described previously. Ve’s query “This is thesame?” (Ve62) is superfluous for somebody who already has acquired the rule “thezero in the middle does count.” However, Ve asks for clarification from Si, and Sioffers a justification (Si63). (As previously, although Si mistakenly compares hun-dreds and thousands when she should compare thousands and ten thousands, thisdetail is not relevant because she preserves the proportion 10 between the two.) Vefurther investigates the influence of moving the zero digit by querying Si (Ve64),and Si provides a new justification (Si65). Ve elaborates Si’s justification by articu-lating a new rule (“Zero in the middle, the more it’s to the left, the smaller the num-ber is” [Ve66])—a rule that Si does not understand (Si67). Just as Ve had previ-ously inferred new knowledge through challenging and conceding, she was alsoable to construct new knowledge following other argumentative operations, suchas justification. In sum, various classes of argumentative operations led to infer-ence of new knowledge during the interaction between Ve and Si.

Case Study Conclusions

Clearly, Ve repaired her conceptual bug, as evidenced by utterances in the protocol(e.g., Ve66) and confirmed by the answers she gave in the final questionnaire. Wehave identified several mechanisms that accompanied this repair. First, the designof the task led to disagreement between Ve and Si in the second and fourth subtasks(no disagreement occurred in the third subtask, however). During interactions sur-rounding these two subtasks, Ve revised her understanding of zero and the magni-



tude of decimals. Second, Ve and Si engaged in a process of argumentation, as eachof them invoked principled (although sometimes buggy) explanations for their de-cisions about the construction of numbers through argumentative operations (chal-lenge, counterchallenge, justification, and concession). In addition, when Ve andSi saw that they could not reach agreement, they evaluated their arguments by turn-ing to a hypothesis-testing device, a calculator on which they undertook manipula-tions. Si’s opposition to Ve made the position of the zero relevant to the decisionabout the magnitude of the decimal numbers. Ve’s hypothesis testing clarified howthe position of the zero influences the value of the decimal fraction. Si offered Ve apiece of information (the place of the zero) that took hold in her cognitive environ-ment, and Ve decided that the place of the zero merited examination. The projectionof an intersubjective process (i.e., the argumentation between Ve and S)5 led Ve tointernalize her interaction with Si and construct a new rule regarding zero in deci-mal numbers. The answers Ve gave in the questionnaire after the six-cards taskshowed that her gains achieved during peer interaction were maintained.

Ve’s repairofconceptualbugswas thusmediatedbyherdisagreement, argumen-tation, hypothesis testing, and internalization of her social interactions with Si. Thequestion is, then, if Ve and Si eventually reached agreement on the four subtasks andfulfilled them successfully, why did Si not repair her bug as well? (According to Si’sanswers in the postquestionnaire, she remained resistant to change.) At first glance,it seems that Ve had more difficulties than Si. In many of their interactions, Ve askeda question, and Si gave an explanation that was not challenged by Ve (e.g.,Ve42–Si43, Ve44–Si45, Ve62–Si63, Ve64–Si65, and Ve66–Si67).

Analyses of the interactions revealed that most of the discussion between Veand Si was around how to interpret the digit “0” in the beginning, the end, and the“middle” of a decimal number. Ve, whose conceptual bug involved zero, engagedin tasks in which the explanations she encountered were in conflict with her hy-potheses. She then inferred several new rules: “Zero at the beginning and at the enddoes not count and in the middle does count” (Ve46), “Zero at the beginning or atthe end does not change” (Ve51), “Zero in the middle, the more it’s to the left, thesmaller the number is” (Ve66). In contrast, Si’s conceptual bug, which she articu-lated quite clearly (Si36), involved mapping the decimal digits inaccurately ontothe denominator (rather than the numerator) of the related fraction. However, thistopic never became the focus of the interaction. Si used her F rule to explain to Ve

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5This result can be compared with a study by Trognon (1993), in which he examined two students in-teracting to solve a selection task (Wason & Johnson-Laird, 1972). At the end of a session of 5 min, oneof the two students succeeded in solving the selection task. Two interactional mechanisms were in-volved in arriving at the right solution: the mechanism from which the refusal to choose the 4 card re-sulted, and the mechanism from which taking the 7 card into consideration was induced. Therefore, itappears that one interlocutor offered the other a piece of information that took hold in her cognitive envi-ronment: The first interlocutor discovering the relevance of the 7 card merits examination because it is aprojection of an intersubjective process, the debate between the two interlocutors.


why 0.458 is different from 0.0458 (Ve37): She invoked thousandths, hundredths(e.g., Si39), and fractions (“0.4580 is 4,580 divided by 1,000 that is 458 divided by100 and this returns to 0.458” [Si41]). Thus, Si encountered no conflict with herconceptual bug. She continued to use fractions (now correctly) to show why Vewas wrong. Ve constructed a new rule on the basis of Si’s knowledge (Si67), buteven though Si engaged in interactive argumentation with Ve, Si’s F conceptualbug was not at stake. Their joint conversation was not about testing Si’s fractionrule. It is, thus, not surprising that Si did not repair her conceptual bug.

This case study has revealed several mechanisms we claim to be general. This isnot to say, of course, that the interaction between students with specific conceptualbugs leads to inevitable effects. For example, after Si clearly expressed her con-ceptual bug in Si36, Ve might have objected, but she did not. Thus, the nature ofthe interaction is critical in determining whether a given student’s bug will becomethe focus of the argumentative exchanges. The preliminary study, indeed, showedthat interactions between Z- and F-students do not necessarily lead to consistenteffects. For example, in two of the six Z–F pairs in the preliminary study, both Z-and F-students became right (see Table 2). These findings merit further researchand are considered in the concluding section.

An Unexpected Finding

An interesting finding arises from Ve’s explanations when comparing decimalnumbers in which zero digits appear. Her explanations do not match the conceptualbug proposed by Sackur-Grivard and Leonard (1985). Rather, two different expla-nations could be identified in Ve’s interactions with Si. In Ve21, when asserting“04.058 that is 4.58,” she clearly expresses a buggy rule; that is, removing zero fromthe decimal part of a decimal fraction does not change its value. This rule is also im-plied in Ve35. In Ve46, she expresses a new rule: “Zero at the beginning and at theenddoesnotcountand in themiddledoescount.”This rule is specialized inVe64andVe66to“zero in themiddle, themore it’s to the left, thesmaller thenumber is.”Thesetwo kinds of explanations are not consistent with a student holding the Z or the WNconceptual bug. To check that the two kinds of rules given by Ve were not idiosyn-cratic, we interviewed the 5 (of 30) students diagnosed as Z-students in the question-naire. They were asked how they compared 4.7 and 4.08, as well as 4.4502 and 4.45.Their explanations showed two distinct conceptual bugs, Z’ and Z”:

• Conceptual bug Z’: If a decimal fraction contains a zero in its decimal part, it issmaller than any number that has the same digits to the left of the zero and does nothave a zero in the same place; otherwise, the number with more decimal places isthe larger (WN conceptual bug). For example, according to rule Z’, 4.7 > 4.08 and4.4502 < 4.45 (because the two first digits in the two decimal parts are identical andthe third digit in 4.4502 is zero).



• Conceptual bug Z”: If a decimal fraction contains a zero in its decimal part, re-moving the zero does not change the value of the number. Otherwise, the numberwith more decimal places is the larger (WN conceptual bug). For example, 4.7 <4.08 (because 4.08 = 4.8) and 4.4502 > 4.45 (because 4.4502 = 4.452 and 4.452 hasmore decimal places than 4.45).

It is important to note that in the explanations collected from the 5 students“with problems with zeros” (these students are no longer called Z-students), theformulation of the conceptual bug Z” was uttered explicitly (as in Ve21 and Ve35).In contrast, the conceptual bug Z’ was not formulated explicitly. Rather, the expla-nations given were similar to “zero in the middle, the more it’s to the left, thesmaller the number is” (Ve66) or of the kind “there is a zero here.”6 Such explana-tions may lead to the conceptual bug Z’ if they are used as general rules. And in-deed, we could distinguish two conceptual bugs on the basis of the answers givenin the questionnaire.

Rules Z’ and Z” are different but are both variations of rule WN andovergeneralize a correct rule. Rule Z’ is correct in many cases (e.g., 4.7 > 4.08,4.509 < 4.58). It results in incorrect comparisons, however, when the decimal frac-tion containing the zero—if truncated at the zero—is equal to the decimal fractionbeing compared (e.g., 4.4502 < 4.45 and 4.4 > 4.402). Rule Z” overgeneralizes therule that adding zeros at the end of a decimal number does not change the value.

It is surprising that in the two previous studies on incorrect rules in comparingdecimal fractions (Resnick et al., 1989; Sackur-Grivard & Leonard, 1985), re-searchers did not identify these bugs. The Z’ conceptual bug is a generalization ofthe Z conceptual bug identified previously. However, the Z” conceptual bug istruly new. There are two possible reasons why Z” was identified here. First, ourpopulation consisted of low-achieving high school students rather than youngerstudents. Moreover, a consistent Z” student gives answers identical to a WN stu-dent for eight of the nine items in the questionnaire developed by Resnick et al.The two bugs can be distinguished only in the comparison of 0.25 and 0.100: AWN student would assert that 0.100 is bigger, whereas a Z” student would regard0.25 as bigger. Thus, it seems that the questionnaire constructed by Resnick et al.needs to be modified to be sensitive to the four possible conceptual bugs—WN, Z’,Z”, and F—especially for older students.

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6From the interaction between Si and Ve, it appears that Ve expressed explanations in which it is pos-sible to identify the Z’ conceptual bug (in Ve64 and Ve66) as well as the Z” conceptual bug (in Ve21 andVe35). The question is then to decide about Ve’s conceptual bug. It seems clear that at the beginning, sheis a Z” student. However, by the end of the interaction, she learns that the rule is incorrect, as shown byher correct answers in the questionnaire asked 2 days after the interaction. In Ve64 and Ve66, which oc-cur at the end of this learning process, Ve became right. Thus, these explanations do not reflect a Z’ pro-cedural bug because nothing in these explanations and in the questionnaire given 2 days after peer inter-action evidences that Ve is not a right student.


THE QUANTITATIVE STUDY

The quantitative study was designed to investigate two research questions thatarose naturally from the preliminary study and the case study. The first question ad-dressed whether the Z conceptual bug can be replaced by the two conceptual bugs,Z’ and Z”, detected in the case study. The second question concerned the effect ofpeer interaction condition (W1–W2 vs. R–W) on conceptual bugs, with a relativelylarge number of pairs. In particular, we examined whether W1–W2 interactions aremore likely to lead at least one of the students to become right (i.e., thetwo-wrongs-make-a-right effect).

Theorists in pragma-dialectics such as van Eemeren and his colleagues (vanEemeren et al., 1996) give two good reasons why R–W pairs may be less engagedin argumentation than W1–W2 pairs. First, the right student is often more compe-tent and more confident on the correctness of his or her standpoint. As Zarefsky(1995) suggested, argumentation takes place when one has to justify decisions un-der conditions of uncertainty. Confidence in the correctness of a standpoint obvi-ates negotiation because the opponent is seen as incompetent rather than as a partyin argumentation. Second, Zarefsky considered argumentation a practice of justi-fying a standpoint in contrast to “proving” something. The more competent stu-dent may be more familiar with the norms of the knowledge content discourse (inthis case, mathematical analytical) and, hence, may be more inclined toward con-sidering it as self-evident.

Method

Sample

The quantitative study included 72 low-achieving Grade 10 and Grade 11 Is-raeli students: 45 girls and 27 boys. Their age ranged from 15 years and 5 monthsto 17 and 5 months. They belonged to the 20% lowest achieving population of highschool students.

Materials

To differentiate between Z’ and Z” conceptual bugs, we added new items to thequestionnaire developed by Resnick et al. (1989). The extended version of thequestionnaire appears in Table 3. Items 8, 9, and 10 in the extended questionnairewere added to differentiate Z” from WN students. For example, a consistent WNstudent would decide that 3.4803 > 3.4095, whereas a Z” student would decide that3.4803 < 3.4095 because 3.4803 = 3.483 and 3.4095 = 3.495 and 3.483 < 3.495.



Item 13 was inserted to better differentiate between F and R students. Thesix-cards task was used again in the dyad interactions.

Procedure

The procedure was similar to the preliminary study. The 72 students completedthe extended questionnaire. Of the 72 students, 64 were categorized according toone of the conceptual bugs (see the following classification subsection). These 64students were invited to solve the six-cards task individually. At the interactivestage of the six-cards task, they were arranged into 10 R–W pairs and 22 W1–W2pairs. Pairs were asked to reach agreement on each of the four subtasks. When stu-dents felt they could not progress any further, they were allowed to use a calcula-tor. At the end of the six-cards task, the students again were asked to answerindividually the extended questionnaire (with new numerical values).

Classification of Students

We identified students’ conceptual bugs according to the following procedure:(a) We first calculated the distances from the five consistent profiles (WN, Z’, Z”,F, and R); (b) we then chose the minimal distance; (c) if this minimal distance was2 or less, the student was categorized as belonging to the nearest conceptual bug;

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TABLE 3The Questionnaire in the Quantitative Study

Number Pair Consistent Profiles Examples of Answers

Item A B WN Z’ Z” F R S1 S6 S13

1 4.8 4.63 B B B A A B B A2 0.5 0.36 B B B A A B B A3 0.25 0.100 B B A A A B A A4 13/100 0.125 B B B A A A B B5 4.7 4.08 B A B A A B B B6 2.621 2.0687986 B A B A A B B A7 4/100 0.038 B A B A A A B B8 3.4803 3.4095 A A B A A A B A9 7.6707 7.6085 A A B A A A B B10 2.3402 2.3075 A A B A A A B B11 4.4502 4.45 A B A B A A A B12 0.457 4/10 A B A B A B B B13 5.8607 5.86 A B A B A A A A

Note. The question asked was, “for each pair, circle the number that is bigger (“A” answers arecorrect). WN = whole number; Z = zero; F = fraction; R = right (i.e., competent).


(d) if the minimal distance was more than 3, the student was categorized as non-strategic; (e) if the minimal distance was 3, the student was interviewed to estab-lish a conceptual bug. Table 3 shows that the distances between consistent profilesrepresenting different conceptual bugs are more than 5, except for the distances be-tween Z’ and F, between F and R, and between WN and Z” profiles, which are 4, 3,and 4 respectively. For this reason, the value 3 was chosen as a critical minimaldistance to decide whether a student was strategic or not. For example, student S1in Table 2 deviated from WN, Z’, Z”, F, and R profiles by 3, 5, 6, 7, and 6 respec-tively; he was identified as a WN-student because his deviation, 3, was minimalfor WN and because an interview with him confirmed this conceptual bug. StudentS6 was classified as a Z” student without interview because her distance from theZ” profile was 1. In contrast with the preliminary study, we excluded students whowere not sufficiently consistent in the use of one specific conceptual bug (e.g., S13in Table 3). Similar to the consistency reported by Resnick et al. (1989), 89% ofthe students could be assigned to a conceptual bug using this conservative classifi-cation.

To validate the classification procedure, we compared the results of our classi-fication with the judgment of experts. The experts were three graduate students inmathematics education who learned how to identify bugs as part of a graduatecourse. We gave the experts the audiotapes of all pairs discussing the six-cardstask and asked them to identify the initial conceptual bugs of each member of thepairs. Interjudge reliability was measured by comparing the experts’ judgmentsacross the pairs. The judges reached complete agreement concerning all the proto-cols. A complete overlap also was found between the conceptual bugs identifiedby our classification procedure and those identified by the experts

Results

Of the 64 students left in the study, 15 were WN-students, 4 were Z’-students, 15were Z”-students, 20 were F-students, and 10 were R-students. We are aware thatthe questionnaire possibly failed to uncover new unknown strategies for some stu-dents identified as nonstrategic. However, if such new strategies exist, they areprobably rare.

Our success in identifying consistent strategies confirms that the Z conceptualbug should be replaced by two conceptual bugs: Z’ and Z”. This result is furthersupported by the deviations from consistent profiles on the postintervention ad-ministration of the questionnaire. Six students differed from the Z’-rule by 2 orless and from any other rule by 3 or more, and 15 students differed from the Z”-ruleby 2 or less and from any other rule by 3 or more. This finding does not contradictthe results of the studies undertaken by Sackur-Grivard and Leonard (1985) and byResnick et al. (1989). As mentioned previously, Z is a variation of Z’, and stu-



dents’ explanations for Z’ and Z were similar. The difference between the twocould be detected only through special comparison items, such as 4.4502 and 4.45.

For the second research question, we used the dyad as our basic unit of analysis.Table 4 shows how the 32 pairs were constituted and how individuals changed fol-lowing interaction. At least 1 of the students became right in 17 of the 22 dyads inthe W–W condition (in nine dyads, both students became rights). A chi-squareone-sample test indicated this result was statistically significant, χ2(1, N = 22) =6.54, p = .05. In contrast, in the W–R condition, at least one of the students becameright in only 5 of 10 dyads, yielding a nonsignificant chi-square. In summary, thesefindings show that argumentation between two wrongs leads to a significantchange among the wrongs. However, discussion between W–R pairs does not leadto a significant change among wrongs.

An interesting question concerns the rate of change from wrongs to rightsamong each of the groups. Under the null hypothesis, the probability according towhich a wrong becomes right after the six-cards task is 1 in 5 (e.g., a WN-studentcan remain WN, or become a Z’-, Z”-, F-, or R-student). Twelve of the 15 WN-stu-dents became right (80%), p < .001. Of the 4 Z’-students, 3 became right (75%), p= .03. Of the 15 Z”-students, 8 became right (53%), p = .03, and 8 of the 20 F-stu-dents became right (40%), p = .05; that is, in each group, the rate of change wassignificant beyond what could have been expected under the null hypothesis. Thisresearch design does not allow us to compare the rate of change across the sub-groups. However, this question is theoretically important and should be examinedin future studies.

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TABLE 4Conceptual Bugs of the 32 Pairs of the Quantitative Study Before and After Interaction

Z”–R

Pairs R–R WN–R Z”–R R–Z” R–F Z”–F WN–WN WN–F F–F

10 W–R pairsWN–R 1 — — — — — — — 1Z’–R 1 — — — — — — — —Z”–R 1 — 1 — — — 1 — —F–R 2 2 — — — — — — —

22 W–W pairsWN–Z’ 1 — — — — — — — —WN–Z” 2 — — 1 — — — 1 1WN–F 4 — — — 3 — — — —Z’–F 1 — — — — — — — 1Z”–F 1 — — — 4 1 — — 1

Note. R = right (i.e., competent); WN = whole number; Z = zero; F = fraction; W = wrong (i.e.,noncompetent).


GENERALITY OF MECHANISMS IDENTIFIED INTHE CASE STUDY

The quantitative study largely confirmed the two-wrongs-make-a-right effect. Italso showed that the gains of wrongs are significant in W1–W2 pairs but that thesegains are not significant in W–R pairs. In the case study, we claimed that the mecha-nisms we identified explained change (for Ve) as well as resistance to change (forSi) for a particular pair of W1–W2 students. An obvious issue concerns the general-ity of the conclusions drawn from the case study. This issue is complex and in-volves many related questions, including:

1. Are the mechanisms identified in the protocol of Si and Ve characteristic ofW1–W2 pairs in which one of the peers becomes a right?

2. Are additional mechanisms involved in solving the six-cards task amongW1–W2 pairs in which both wrongs become rights?

3. Are similar mechanisms involved in solving the six-cards task among R–Wpairs in which the wrong becomes right after their interaction?

4. Are alternative mechanisms involved in solving the six-cards task amongpairs (W1–W2 or W–R) who do not gain from their interaction?

Full consideration of such questions demands the analysis of a large number of pro-tocols. In this study, we limit ourselves to a succinct analysis of short excerpts fromthree pairs chosen to represent various pairings and outcomes. We preview furtherresearch, to be published separately.

To examine the problem of generality of mechanisms leading to change, we se-lected three pairs of students who solved the six-cards task in the quantitativestudy. One pair, Mo and Ha, was similar to the pair Ve and Si: Mo was a Z”-studentbefore interaction and Ha was an F-student. After interaction, Mo became a rightand Ha remained an F-student. In the second pair, the two students (Da and Re)also had Z” and F conceptual bugs before interaction, but both changed to rightsafter interaction. As for the third pair, the two students (Ad and Ra) were Z”- andR-students who remained the same after interaction.

We identified four mechanisms in the interactions between Ve and Si: (a) in-duced disagreement, (b) hypothesis testing as a way to react to a counterargument,(c) challenge and concession as two ways to infer new knowledge, and (d) inter-nalization of social interactions. We noted that the third mechanism is a specialcase of inference of new knowledge through argumentation and argued that the in-ternalization of social interactions is more general. We suggested that disagree-ment, hypothesis testing, argumentation, and internalization mediated Ve’scognitive change. We also suggested that, although Si actively participated in thesix-cards task and was part of each interaction, she did not experience cognitivechange because the interactions were irrelevant to her own conceptual bug. We be-



gin the discussion with the pair Mo and Ha, whose initial and final states are identi-cal to those of Ve and Si.

The Pair Mo and Ha: Revisiting Mechanisms of Change

The first salient phenomenon in the interactions between Ha and Mo is their initialdisagreement and their reliance on different conceptual strategies, brought to thesurface by the design of the six-cards task. The following is excerpted from theirdiscussion of the second subtask:

Exp16: Construct the smallest possible number.Ha17: 0.8540.Mo18: 0.0854.Ha19: For decimal fractions, what seems the bigger is the smaller.Mo20: We constructed the same number. The zero does not change anything.

There is nothing like zero 854 parts, one counts 854.Ha21: Thousandths. It turns to thousands. See there’s nothing like 0.0, but if

there is 0.0 and something after, this is more than zero.Mo22: It’s more than zero, but you don’t count the zero.Ha23: It’s true [pause]. Why? If you think.Mo24: Mine is smaller because there are no thousands and no wholes, and

there are only hundredths.Ha25: Why? The parts of the thousands are smaller. One moment. Mine is

854/1,000 and yours is 854/10,000, yours is smaller because we di-vided in more parts.

Mo26: But when we have 0.5 and 0.50, it’s the same.Ha27: Zero at the beginning or at the end does not make change because they

can be removed. 0.5 is 5/10. So if we enlarge the fraction, it’s 50/100,which is 0.50, so 0.5 is 0.50, which is equal to 0.500.

Mo28: So zero at the beginning or at the end does not count but in the middle itcounts.

Ha29: Yes.Mo30: So the smaller number is my number.

As with Ve and Si, this excerpt shows that the design of the task led to disagreement(Ha17 and Mo18) followed by two arguments (Ha19 and Mo20) reflecting the F andtheZ”conceptualbugs. In fact,Mo20alreadycontains thebeginningofanargumen-tative process in the utterance “there is nothing like zero 854 parts, one counts 854,”which isachallenge toHa’s initial argument.The interactiondevelopsasa richargu-mentative exchange: Ha21 is a counterchallenge and Mo22 challenges it in turn.Mo24 may represent a change in Mo’s thinking. Although Mo is a Z”-student, so ze-ros should not count, she attends to the zeros in her number and Ha’s to argue that her

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number (Mo’s) is smaller: “Mine is smaller because there are no thousands and nowholes, and there are only hundredths.” Mo appears here to be treating the portion ofthe number to the right of the deciaml as a whole number (in which case there is in-deed a zero in the “thousands” place). In Ha25, the utterance “mine is 854/1,000 andyours is 854/10,000, yours is smaller because we divided in more parts” is a refuta-tion of the argument stated in Mo22 that zeros do not count. The most central utter-ance is in Mo26: “But when we have 0.5 and 0.50, it’s the same.” This statementchallenges the refutation Ha25, but it seems that Mo is already convinced by Ha’srefutation. After she listens to Ha’s answer in Ha27, Mo immediately infers the newrule “so zero at the beginning or at the end does not count but in the middle it counts”(Mo28), which is a concession to Ha. It appears then that the challenge “but when wehave 0.5 and 0.50, it’s the same” (Mo26) serves as a way to infer new knowledge ar-ticulated inMo28.Thisexcerpt shows then thatMo’sarticulationofa repaired rule isembedded in twogeneralmechanismswepreviously identified in the interactionbe-tween Si and Ve: disagreement and argumentation. The particular argumentativeoperation inwhich this repair is embedded isachallenge (of thewrongwhobecomesa right) followed by a concession.

The question is again why the Z”-student repairs her conceptual bug, whereasthe F-student is resistant to change. As in the case study, the F-student provides theZ”-student with the justification for abandoning the conceptual bug (Ha27). Whythen does the F-student not repair his bug? We contend that the six-cards task pro-vides more opportunities to discuss issues about zeros. For example, in Mo20,Ha21, Mo22, Mo24, Mo26, Ha27, and Mo28, the term zero or no is central. InHa19, Ha expresses his F conceptual bug, and in Ha25 he reiterates it in a correctversion: “Mine is 854/1,000 and yours is 854/10,000. Yours is smaller because wedivided in more parts.” The fact that Mo accepts his explanation does nothing tochallenge Ha’s conceptual bug.

In the third subtask, Ha and Mo disagree, and the reason invoked by Ha reflectshis conceptual bug, as shown in the following excerpt:

Exp31: Compose the closest number to 1.Ha32: 0.4580.Mo33: 0.8540.Ha34: Because it is bigger. The smaller it is, the closer to 1 it is.Mo35: I think the contrary, the bigger the numbers are, the closer they are to 1.

[Mo takes the calculator and computes 0.4580 ÷ 100 and obtains0.0048, and 0.8540 ÷ 100 and obtains 0.00854.] It doesn’t help me.

As shown in this excerpt, Ra is an F-student (as it appears in Ra34). Resolving thissubtask is done by using the calculator and doing manipulations: dividing by 10,100, and 1,000, and this leads to an impasse. We do not present the end of the proto-col of the third subtask, which is very similar to the beginning: The two peers sub-



tract the two alternative solutions from 1 to decide which of them is closer to 1. Thisstrategy leads again to an impasse because neither can decide who is right. This in-teraction is again initiated by a disagreement, and manipulations on the calculatordo not succeed in settling the disagreement. In the last subtask, we again see themechanisms in which change and resistance to change are embedded:

Exp43: What is the closest number to half?Mo44: 0.0548.Ha45: 0.4850.Mo46: The issue is whether it passes one half? Mine passed one half.Ha47: One half is 0.5, and you have 0.05. You’re not at all close to one half.Mo48: Why not? 548, and one half is .500.Ha49: You didn’t reach even 0.1. You’re still at 0.0. You can’t overlook the

zero.Mo50: Why? Ah! We said that zero in the middle changes. It makes it smaller.Mo51: [Changes to 0.5480.] Now mine overcame one half.

This protocol again illustrates several mechanisms we have previously identified.First, the subtask elicits disagreement (Mo44 and Ha45) and reveals different con-ceptual bugs. Ha47 challenges the Mo44 argument. Mo48 counterchallenges.Ha49 begins with a counterchallenge, and “you can’t overlook the zero” is a refuta-tion. The utterance in Mo50 is particularly interesting. Here, Mo infers new knowl-edge and articulates a revised rule regarding zero in the middle. This inference isreached through concession. The origin of the inference can be traced to Ha27, inwhich Ha states “Zero at the beginning or at the end does not make change becausethey can be removed,” and to Mo’s reaction (Mo28): “So zero at the beginning or atthe end does not count but in the middle it counts.” It seems reasonable to argue thatthe utterance in Mo28 (expressed by Mo as a concession to a refutation) is internal-ized when she states (Mo50) “Ah! We said that zero in the middle changes, it makesit smaller.” This interpretation is strengthened by the fact that before Mo50, the ruleexpressed in Mo28 is not invoked and is explicitly violated in Mo48.

It appears, then, that the inference of new knowledge is triggered here by dis-agreement, developed through argumentative moves, and inferred via a concession.This concession can be traced back to the internalization of a social interaction(which ends with a refutation in Ha27 and a concession in Mo28). Again, the F con-ceptual bug is not discussed and not even referenced. It is, therefore, not surprisingthat Ha remains an F-student. The analysis of the interactions between Ha and Moleadsus to identifydisagreement,argumentation,andthe internalizationofsocial in-teractions as general mechanisms in which the change of conceptual bugs is embed-ded. Different argumentative moves (concession and challenge) can mediate theinference of new knowledge. Hypothesis testing does not lead here to any change.However, we claim that it creates opportunities for change.

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To deepen the explanation of the differential effect of the six-cards task on stu-dents with different conceptual bugs, we turn to a brief discussion of two addi-tional pairs. Da and Re constitute a pair of Z”–F students who both turn to rightafter interaction; Ad and Ra constitute a pair of Z”–R who remain the same.

The Pair Da and Re: How Both WrongsCan Become Rights

Da and Re are girls, with Da holding the Z” rule and Re holding the F rule. Theirconceptual bugs are reflected, as usual, in the initial statements of their discussionof the subtasks. We present here an excerpt of the third subtask (finding the numberclosest to 1). In this subtask, Re has chosen 0.8540 as the number closest to 1:

Da23: If we move your last zero to the beginning, the number will be closer.Re24: I think that you are right. Let’s check [Re computes 1 – 0.8540 and 1 –

00.854 on the calculator]. It’s the same, it does not make a change.Da25: So, it’s possible to take off the two zeros. Let’s see [Da computes 1 –

.854].Re26: At the end they make change!Da27: They don’t. See, we took them off from the beginning and we got the

same result. At the end or at the beginning before the dot, they do notmake any change.

Da28: [Computes 0.0854 – 0.00854 and obtains a positive number.] Why it’snot zero?

Re29: Because the zeros in the middle do make change. For example, .85004is bigger than .80054 because eighty-five thousands and four is morethan eighty thousands and fifty-four.

Da30: And if we’ll do .85040?Re31: It’s bigger because it’s eighty-five thousands and forty.Da32: So let’s change to .85400.Re33: This what we did at the beginning. I don’t know.Da34: Let’s check again on the computer. [Da enters 85400 – 85040 in the cal-

culator and obtains a positive number.]Re35: But it’s possible to erase the zeros at the end. [She enters .854 – .8504 in

the calculator and obtains the same result.]Re36: So the closest number to 1 is .85400, or it’s possible to write it .854.Da37: That is [pause] one moment, I want to make some order [pause]. When

the zeros are at the end of the number or at the beginning before the dot,they do not make any change, but if there are others or before themnumbers to the left of the dot, they make a change.

Re38: And something else, the more there are zeros one beside the other, thesmaller the number is. For example, 0.0085 is smaller than 0.085.



We focus here on the new phenomenon, the disappearance of the conceptual bugF, which accompanies the disappearance the conceptual bug Z”. The protocolshows that this task presents a conflict for Re. On one hand, as an F-student,numbers with more digits are smaller (Re states this rule explicitly in previousutterances). On the other hand, the manipulations that Re undertakes on the cal-culator show her that taking off zeros at the end of the number does not changethe number. The conflict is visible in Re24 (in which Re thinks that transferringa zero from the end of a number to the beginning makes the number closer to 1),in Re26, and in Re33 (The “don’t know” expresses Re’s confusion after her ma-nipulations on the calculator show that deleting zeros from the end does notchange the value of a number). The conflict is resolved through the use of thecalculator for evaluating an argument (in Re24 and Re35) and for constructingan argument (in Da28) and through argumentative operations (challenge inRe26, refutations in Da27 and Re29, challenge to refutation in Da30). A centralfinding is that the repair of the two conceptual bugs is collaboratively inferred inRe 36, Da37, and Re38: Re offers a statement that clearly violates the F rule, Dacontinues by inferring a rule that repairs her Z” conceptual bug, and Re elabo-rates Da’s utterance with another rule about zeros that is correct. This rule servesto repair her F conceptual bug.

The question is why such a change happened for this pair but not for Ve and Si,nor for Mo and Ha. We already acknowledged that relations between social inter-actions and cognitive development are not completely deterministic. However, wepropose here a reasonable explanation for the gains made by Da and Re. We ar-gued previously that the design of the six-cards task allows F-students to avoidcognitive conflict. However, Da and Re violated “the rule of the game,” that is, touse all six cards. For all subtasks, except finding the biggest number, the correctanswer begins with a zero followed by a decimal point. In most cases, the studentsreached this conclusion. Therefore, students generally manipulated numbers inwhich the decimal part has a constant length of four digits. According to the F con-ceptual bug, when comparing numbers with the same integer part and with differ-ent lengths of decimal parts, the number with the longer decimal part is the smaller.This rule cannot be expressed or discussed easily when all numbers manipulatedhave the same length. In the interaction between Re and Da, however, the peersused the calculator to compare numbers in which the decimal parts had differentlengths (Re24, Da25, and Da28). Thus, they violated the rules of the six-cards task,and, as a result, Re encountered conflict.

In conclusion, the interactions between Da and Re showed again that changewas mediated by disagreement (triggered by the task) and argumentative opera-tions. The role of the calculator was richer here. It helped not only to evaluate argu-ments but also to construct them. Re’s conceptual change originated from amodification of the task that led her to encounter conflict and to resolve it in collab-oration with her peer.

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The Pair Ad and Ra: Why a Wrong OftenDoes Not Learn From a Right

We turn now to the last pair, a pair that shows no postintervention change. We pres-ent here an excerpt characteristic of the interactions in this W–R pair. The excerptrelates to the last subtask (the number closest to one half):

Ad27: 0.0458.Ra28: 0.5048. One half and a little more. Yours doesn’t get at all close to one

half.Ad29: One half is point five.Ra30: How much is missing to one half?Ad31: A lot! [Changes to 0.4058.] It’s closer.Ra32: It’s closer than what was before. But mine is closer to one half. You

miss 42.Ad33: Mine is closer. I miss 42.Ra34: No. If here you have 42 then instead of zero there will be 1, 0.4100.Ad35: [Changes to 0.4850.] Now I miss 150.Ra36: [Changes to 0.5084.] I have to lessen by 84. Eight tens and four units.

0.0084 and you have to add 0.0150. Mine is closer.Ra37: [Takes the calculator and computes 0.4850 + 0.0150 and obtains 0.5.

He computes 0.5084 – 0.0150.] I took one half and I added to it thesmallest number. If I had taken 0.04, I should have taken off a lot more.

This protocol shows interactions of a very different kind from the other pairs we en-countered. These interactions are asymmetric. Ad’s exchanges are very short. Ra’sinterventions are much longer, and he also monopolizes the calculator. Additionally,the interaction, which is initiated by a disagreement, does not go on as an argumenta-tive process. Rather, Ra demonstrates in a patronizing way why Ad is wrong. For ex-ample, “How much is missing to one half?” (Ra30) is a leading question to which Raknows the answer, as evidenced in Ra34 and Ra36. In fact, in Ra32, Ra coaches Adwith the utterance: “It’s closer than what was before.” Also, when Ra changes hisnumber in Ra36 to 0.5084, this not to improve his result but rather to show Ad that sheis wrong by adding an additional example. Similarly, in Ra37, Ra does not use thecalculator to evaluate an argument but to demonstrate to Ad that she is wrong. Thelast utterance, “If I had taken 0.04, I should have taken off a lot more” (Ra37), showsthat Ra continually had in mind Ad’s proposed solution 0.0458. The Z” conceptualbug is not expressed by Ad or discussed by Ra. Rather, Ra dominates the interactionand demonstrates that he is right. In conclusion, the interactions in the W–R pair, Adand Ra, were asymmetric, and no conceptual change was evident. Disagreement ini-tiated the interaction, but it was followed by a series of coaching questions, and thecalculator was used for demonstration rather than for evaluation of arguments.



As mentioned before, the three additional cases we analyzed in this section donot constitute all possible interactions. However, we argue that the cases reveal es-sential mechanisms that can explain other possibilities. These mechanisms are re-viewed in the concluding section.

CONCLUSIONS

The quantitative study clearly showed the two-wrongs-make-a-right effect: Whentwo wrongs interacted, at least one of them turned to a right.

In the case study and in the three additional protocols, we identified generalmechanisms that explained conceptual change or resistance to change. The firstgeneral mechanism was disagreement. The task elicited disagreement in the sensethat it led peers with differing conceptual bugs to disagree. In most of the subtasks,these conceptual bugs led peers to state initial arguments that included differentconclusions and different reasons supporting their conclusions.

The second mechanism we identified in all but the W–R pair (Ad and Ra) is ar-gumentation leading to the construction of new knowledge. In each of the threeW1–W2 pairs, several argumentative moves preceded the inference of a new rule.In the case of Ve and Si, Ve challenged an argument previously uttered by Si. Inthat way, Ve achieved two goals: (a) to seemingly defend her own argument and(b) to construct new knowledge. In another instance, the argumentative operationwas concession following a counterchallenge. In the case of Mo and Ha, inferenceof new knowledge occurred through concession following a challenge. In thesetwo cases, and in the case of Da and Re, the inference of new knowledge followeda chain of argumentative operations, including challenge, counterchallenge, refu-tation, concession, and justification. In contrast, the resistance to change in theW–R pair (Ad and Ra) seemed to result from the dominance of the right. This dom-inance suppressed a genuine argumentative process, although the dyad sometimesengaged in processes of claim followed by opposition followed bycounteropposition followed by justification. However, the objects of discussionwere not authentic arguments but demonstrations that the right was indeed correct.

We observed that what Glachan and Light (1982) called active hypothesis test-ing was often intermingled with the construction of new knowledge. Examiningthe role of the calculator, we identified active hypothesis testing integrated in argu-mentation in two ways. The first consisted of argument evaluation, as in the case ofVe and Si. In the case of Da and Re, the calculator also was used collaboratively toconstruct a new argument. Although hypothesis testing as argument evaluation orconstruction was central for Ve and Si as well as for Da and Re, it did not play arole with Mo and Ha.

The internalization of social interactions is a general mechanism we identifiedin the learning pairs; it captures how the history of interactions is integrated whenpeers collaborate. Ve reconstructed previous oppositions and challenges in her

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reasoning, and Mo’s concession to a refutation in the second subtask became anautonomous statement in the fourth subtask.

Finally, although the two-wrongs-make-a-right phenomenon was embedded inthe mechanisms we identified, these mechanisms do not explain why the gainswere often asymmetric. We argue that change occurs for peers when the interac-tions are relevant to their own conceptual bugs. We showed that in two cases ofZ”–F pairs, interactions were relevant to Z”-students but not to F-students. Weshowed that the design of the task was largely responsible for such differences inrelevance. When a Z”–F pair violated the constraints of the task (to use all sixcards), the discussion became relevant to the F-student and both students turned torights. In all examples, relevance of the interactions led students to repair theirconceptual bugs, via specialization of the conditions of the conceptual bug. Thefundamental idea of repair explains why it is easier for wrongs who are strategiststo turn to rights; no repair can be initiated ex nihilo.

New Directions

The case study and the excerpts from additional protocols open interesting vistas.Of course, the first concerns the study of more kinds of dyads, for example,nonlearning W1–W2 pairs, W–R pairs in which the wrong becomes a right, W–Rpairs in which the right regresses, or dyads in which one of the wrongs is a WN- orZ’-student. Still, it is quite reasonable to hypothesize that the mechanisms we iden-tified in the four pairs have some generality. For example, it seems that dominanceinhibits learning. The role of disagreement and of authentic argumentative opera-tions in knowledge construction showed that these mechanisms are general and thatthey play a central role in knowledge construction. Of course, it is not necessary toengage in argumentation to infer new knowledge. Modeling studies (e.g., Kuhn,1972) have demonstrated that less competent students may experience cognitivegains while observing “experts” because they engage in tacit interaction. However,we suggest that the case of Ad and Ra is more typical. Ra did not engage Ad in au-thentic argumentative activity, and Ad did not learn from her interaction.

We claim that hypothesis testing also can facilitate constructing or evaluatingarguments. Devices that support hypothesis testing, such as calculators, are notnecessary for inferring new knowledge (as in the case Mo and Ha), but they oftenhelp. The genesis of internalization of social interactions is the most difficultmechanism to investigate. It splits into two different questions: What is internal-ized, and when? To answer this, we have undertaken a full analysis of additionalprotocols of peers interacting on the six-cards task. The study has both a qualita-tive and a quantitative character, and it is needed to investigate the role of argu-mentation in the elimination of conceptual bugs in particular and the role ofargumentation in the construction of knowledge in general.



Another important future direction concerns characterizing tasks that facilitatethe two-wrongs-make-a-right phenomenon. As shown descriptively in Table 4, thesix-cards task, as currently designed, was beneficial for some conceptual bugs (e.g.,Z”) but not for F. We argued that the six-cards task does not afford F-students oppor-tunities for cognitivedissonance.An interestingdirection for research, then, is tode-fine theaffordancesa taskprovides,as related tostudents’cognitiverepresentations.

Finally, the two-wrongs-make-a-right effect has potential educational implica-tions: Small groups of students with different levels of competence are basic socialorganizations of the classroom. The study of the two-wrongs-make-a-right effectmay allow educators to facilitate cognitive gains among peers interacting aroundinstructional tasks. Thus, it is crucial to extend the scope of this study to variousdomains. We chose the domain of decimal fractions because the strategies usedwere well-known and because the elimination of conceptual bugs duringshort-term peer interactions seemed a reasonable undertaking for a first study.However, studying the two-wrongs-make-a-right effect in other domains shoulddeepen our understanding of this effect and should help evaluate its possible appli-cations for educational practice.

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