Who-Is-Right tasks as a means for supporting collective looking-back practices

Abstract

The looking-back stage is rarely observed in students’ problem solving in spite of its recognized importance. The importance of this stage is attributed to practices of engagement with queries on verification of the obtained solution(s), comparative consideration of alternative solutions, and formulation of implications for future problem solving. We refer to such practices as looking-back practices. In the present study we explored the hypothesis that the looking-back practices can be evoked in small-group classroom discussions of controversial worked-out solutions to word problems. Such tasks are known as Who-Is-Right tasks. The data consisted of audio- and videotapes of six small groups of high-school students working on a Who-Is-Right task in the context of percentage. The data analysis, informed by a discursively-oriented perspective on problem solving, attended to strategies, dialogical moves and mathematical resources enacted by the students towards attempted agreement as to which of the solutions should be endorsed and why. The findings imply that Who-Is-Right tasks have undeniable potential for supporting collective looking-back practices. In addition, the study contributes to the literature on enactment of mathematical resources in problem-solving discourse and on patterns of students’ dialogic participation in small-group problem solving.

Appendix: Illustration of the data analysis

We illustrate the coding processes based on two episodes (out of 11) from the work of Group 2, consisting of three students, A., B. and M.

In Episode 1, the students read the problem formulation and the solution narratives and attended to the central task question ‘who is right?’. A. quickly suggested that Hila’s solution was right because this was what she would have done herself (‘That’s it. I’d write, $x$ is a kilogram of pears. As Hila wrote.’). B. and M. agreed with A. However, A. also suggested that Sofia’s solution was wrong, and this assertion launched a discussion. A. argued that Sofia’s approach was wrong by pointing out that it was based on ‘decreasing’ the initial price. This argument appeared to be unconvincing to M. A. attempted to formalize her argument by using algebraic tools. She asserted that the price of 1 kg of apples plus the price of 1 kg of pears in Hila’s approach and in Sofia’s approach were different ($2.25x$ for Hila and $1.75x$ for Sofia). B. opposed A. without entering deeply into what A. had said, and suggested her own line of exploration in Episode 2 (Fig. 4).

Fig. 4

A coded transcript of Episode 2 in G2

As for strategies, a change in strategy, from exploring the solution narratives (F-ESN) to independently solving the problem (F-ISP), was inferred for A. and B., whereas M. was likely to be still captured by exploring the solution-narratives (F-ESN) strategy (# 27). B.’s question at the end of the episode (# 37) suggests that her numerical example was aimed at addressing the question ‘why is Sofia’s solution wrong?’ This question differs from Episode 1′s central question ‘who is right and why?’.

In terms of dialogical moves, B. introduced her own ideas starting in #24. Note that she alternated between disclosing her line of thought (D-SD) and requesting her peers to respond by using rhetorical questions (D-RR-RQ). Both A. and M. followed B.’s argument, as reflected in the frequent appearance of the ‘other-oriented’ (D-OO) group of codes. Note the difference between a simple agreement move (D-OO-SA) by B. in #28, and a reasoned agreement move (D-OO-RA) by A. in #30.

Regarding mathematical resources enacted, an R-AF code (self-produced arithmetic/algebraic facts) was assigned to most of the conversational turns in Episode 2. An additional resource, an analogy to differ-by-percent problems (R-AP-DP, #31) was employed only by B.

In Episode 3 (Fig. 5), the group continued to explore the ‘why is Sofia’s solution wrong?’ question. What had been a peripheral strategy in Episode 2 became a central strategy in Episode 3. M., who had taken the lead, continued exploring the solution narratives (F-ESN). She suggested that Hila’s referent for 25% is correct and Sofia’s referent is wrong, but B. opposed her. In response, M. modified her argument and, in addition to attending to the role of the word ‘of’ as a clue (R-RB-OF), resorted to an analogy with differ-by-percent problems (R-AP-DP).

Fig. 5

A coded transcript of part of Episode 3 in G2

The students were attentive to each other’s ideas in Episodes 2–3 (operationally, the D-OO codes were relatively frequent).

The sequence of two episodes is illustrative of the following phenomenon: an opposition to an incomplete or not-understood argument results in an attempted elaboration of the argument by repetition and enactment of additional resources. In this way, a resource that had been peripheral for a while could be put forward again. Simultaneously, a resource, which was central in one episode (i.e., R-AF in Episode 2) could be ‘forgotten’. Realization that none of the employed resources led to an unequivocal resolution of the question under exploration may lead to enactment of additional resources, but up to some saturation point. At that point the ‘forgotten’ resources can come into play again, sometimes in a slightly modified way.