1 Introduction

Of the four stages comprising Pólya’s (1945/1973) model of mathematical problem solving—understanding a problem, devising a plan, carrying out the plan, and looking back—the looking back stage is known to be particularly troublesome in mathematics education research and practice. Pólya (1945/1973) introduced looking back as a stage when a solution to a problem has already been produced but needs to be verified and examined for possible improvements and alternatives. At this stage, argued Pólya, the solvers are expected to address the following questions: (i) “Can you check the result?” (ii) “Can you check the argument?” (iii) “Can you derive the result differently?” (iv) “Can you see it at a glance?” (v) “Can you use the result, or the method, for some other problems?” (p. 27).

Along with arguing for the importance of the looking back stage, Pólya (1945/1973) observed that this stage rarely appears in student problem solving. In his words, “even fairly good students, when they have obtained the solution to the problem and written down neatly the argument, shut their books and look for something else” (p. 27). This observation was reinforced by empirical evidence in several studies (e.g., Goos & Galbraith, 1996; Kantowski, 1977; Mashiach-Eisenberg & Zaslavsky, 2004; Malloy & Jones, 1998; Papadopoulos & Dagdilelis, 2008; Stillman & Galbraith, 1998).

A number of reasons for the rarity of the looking back stage in student problem solving have been pointed out. Kantowski (1977) explained this rarity by strong emotional satisfaction that emerges when a student finds a solution that feels correct. He suggested that this satisfaction makes the looking back stage intellectually unnecessary for a regular student. Stillman and Galbraith (1998) asserted that many students develop trust in their written work, and do not attempt to check their solutions even when they have enough time for this. Nonetheless, past research (Malloy & Jones, 1998; Mashiach-Eizenberg & Zaslavsky, 2004; Papadopoulos & Dagdilelis, 2008; Papadopoulos & Sekeroglou, 2018) has also shown that students can use a variety of verification strategies, as a rule, when properly prompted.

For example, Mashiach-Eizenberg and Zaslavsky (2004) described different occasions where students attempted to verify their solutions while solving combinatorial problems in a task-based interview setting, working individually or in pairs. In their study, the majority of students who had solved the problems incorrectly, were eager to reconsider after the interviewer’s explicit or implicit prompts, the most influential of which was the interviewer’s disclosure of her opinion about the students’ solutions. We learn from this study that the looking-back stage in student problem solving is likely to appear when stimulated by a prompt that is beyond the independent reach of the students.

Being informed by past research that spontaneous engagement in the looking-back stage can hardly be expected from regular students, we recognized the need to understand better the principled design of realistic classroom situations that would be rich in opportunities for the students to engage in the looking-back practices if not in a looking-back stage. By looking-back practices we mean solvers’ engagement with some of the questions (i)–(v) posed by Pólya (see above) in the course of solving problems. We suggested that looking-back practices can be evoked not only after independently solving a problem, but in the course of collaborative coping with specially designed tasks. Specifically, we relied on studies by Swan (2007) and by Rittle-Johnson and Star (2011) on the use of tasks requiring students to compare the validity of two or more worked-out solutions to a problem. Tabach and Koichu (2019) referred to such tasks as Who-Is-Right or WIR tasks. Our wish to examine empirically the potential of WIR tasks for evoking looking-back practices in realistic classroom situations constituted the rationale of our study. Hence, the study aimed to characterize small-group discussions of WIR tasks as a means to support looking back practices.

2 Theoretical framework

The design and conceptual apparatus of the present study were informed by three themes, namely, (1) problem solving as a discursive activity, (2) WIR tasks, and (3) collaborative argumentation and justification.

2.1 Discursively-oriented perspective on problem solving

In this section, we explicate theoretical premises underlying the use of such terms as ‘problem solving’, ‘solution’ and ‘correct/incorrect’ in our study. Generally, we adhered to a spectrum of socio-cultural perspectives on problem solving (Kilpatrick, 1985). In these perspectives a problem is considered as a task given and received in a social situation jointly constructed by the participants involved in it. In particular, the socio-cultural perspectives recognize that problem solvers are continuously involved in interpreting each other’s actions and intentions while conforming to the existing social learning practices, and while contributing to the evolution of these practices (Cobb, 2000). Specifically, we relied on a discursively-oriented conceptualization of mathematical problem solving in instructional situations proposed by Koichu (2019), as follows:

Problem solving in mathematics instruction is a socio-culturally shaped process of not-immediate achieving a chosen or imposed goal, in which the involved individuals enact, through private or public exploratory discourse,Footnote 1 individual or shared resourcesFootnote 2 that they interpret, even for a short while, as appropriate for achieving the goal… [A] solution to a problem is a public narrative, which becomes endorsed, by the problem solvers or the problem proposers, as the one that achieves the goal. (p. 49, footnotes added).

This conceptualization helps us to balance analysis of individual contributions to a group discussion and analysis of decisions made by a group as a whole. In addition, the conceptualization theoretically backs up our decision to equate the solution verification with its endorsement by a group of students, and the solution’s correctness with its endorsement by experts.

2.2 Who-Is-Right tasks and looking-back practices of problem solving

Tabach and Koichu (2019) described Who-Is-Right or WIR tasks as consisting of two parts. The first part is a narrative introducing a problematic situation. The second part consists of different solutions to the problematic situation, which rely on intuitively appealing but contradictory interpretations of the narrative. Each interpretation and corresponding solution consists of a solution-narrative. The solution-narratives are explicitly given and can represent full or partial solutions. The solvers of a WIR task are required to decide which solution-narrative should be endorsed and to support their decision by an argument that might lead to endorsement of the solution also by their peers.

In comparison to regular problem-solving tasks aligned with Pólya’s four-stage model or to regular verification tasks (i.e., solve a problem and verify your solution, hereafter, Am-I-Right or AIR tasks), we see in WIR tasks two advantages for evoking and exploring the looking-back practices. First, AIR tasks and WIR tasks have different referential bases for verification. In an AIR task, the solver’s own solution is usually the only available referent for the solver. If the solver is confident in his or her solution, this confidence can hinder the need to look back, as argued by Kantowski (1977) and Stillman and Galbraith (1998). If a solver is not confident, the referent—some ‘gold-standard solution’—is usually unavailable to her without an intervention by an authority, as in the case described by Mashiach-Eisenberg and Zaslavsky (2004). Therefore, the request to look back at a self-produced solution may feel either artificial or unfeasible to students. In contrast, solution-narratives provided in a WIR task can by themselves serve as reciprocal referents. Second, a WIR task operates with solution-narratives produced by others. Accordingly, the solvers of the WIR task may be more emotionally detached from the solutions than those who produced the solutions first-handedly, and thus they may be more open-minded when verifying them. Hence, we suggest that the request to compare worked-out solutions can be perceived by a WIR task’s solvers as intellectually appealing and feasible.

Furthermore, we hypothesize that WIR tasks can evoke practices germane to the ‘idealized’ looking-back stage, as introduced by Pólya (1945/1973). These practices are not explicitly introduced in WIR tasks, but the tasks allude to them. To this end, we capitalize upon the Rittle-Johnson and Star (2011) analysis, in which they argued that comparison of alternative worked-out solutions to a given problem can involve considerations of correctness, clarity, simplicity, generalizability and applicability.

2.3 Collaborative argumentation

When a WIR task is approached in small groups, the process of endorsing or rejecting a particular solution can be considered as an argumentative activity, that is, an activity in which interlocutors engage in a dialogue characterized by a critical consideration of each other’s arguments. Accordingly, the processes of endorsing or rejecting worked-out solutions can be analyzed by means of conceptual tools developed in prior research on argumentation, justification and socially-mediated metacognition. We consider three studies, which we found particularly relevant to our study.

Goos et al. (2002) analyzed patterns of peer-to-peer social interactions in cases of successful and unsuccessful solving of a word problem in the context of motion. The main data-analysis procedures consisted of characterizing the student conversation moves, particularly concentrating on how the students were engaged in each other’s thinking. The collaborative nature of the interaction was analyzed in terms of categories named self-disclosure (attending to one’s own thinking), feedback requests (an invitation to a partner to attend to one’s own thinking) and other-monitoring (attending to a partner’s thinking). This analysis enabled the scholars to distinguish between discursive patterns characterizing different problem-solving cases. It was found that in successful cases, the students challenged and discarded unhelpful ideas and endorsed useful problem-solving strategies, whereas unsuccessful cases were characterized by the lack of critical engagement with partners’ contributions to the discussion.

Ayalon and Even (2014) explored whole-class discussions of tasks requiring students to determine equivalence of algebraic expressions, for factors involved in shaping opportunities for the students to engage in argumentation. Videotaped discussions were analyzed by means of a scheme of dialogical moves developed by Asterhan and Schwarz (2009). This comprehensive scheme included categories reminiscent of those developed in Goos et al.’s (2002) study, along with additional categories for denoting moves that serve different dialogical functions (e.g., opposing or agreeing with the claim). Analysis in terms of dialogical moves enabled Ayalon and Even (2014) to reveal contributions of different actors to the justification of mathematical claims as well as the structure of the discussions. The scholars also attended to the content of the dialogical moves, and distinguished between student justifications based on mathematical rules, and justifications based on examples.

The dual focus on structure and content of argumentative activity, which can be observed in both Goos et al.’s (2002) and Ayalon and Even’s (2014) studies, was appropriate also for our study. Furthermore, Ayalon and Even’s (2014) study suggested how Asterhan and Schwarz’s (2009) coding scheme, originally developed for analyzing argumentative activity in a non-mathematical context, could be adapted for analyzing discourse in mathematical contexts. The categories developed in all the aforementioned studies served as departure points for developing the data-analysis procedures in our study.

3 Research questions

In the above-explained terms, our study pursues three research questions.

  1. 1.

    What strategies do high-school students use, and how, in small-group discussions of a WIR task in the context of percentages? What are the discursive products of the use of these strategies?

    We answered this question in the spirit of studies by Papadopoulos and Sekeroglou (2018), Malloy and Jones (1998) and Mashiach-Eisenberg and Zaslavsky (2004). Furthermore, the intention of this question is to identify structures of the discussions, by inferring sub-questions attended to by students on the way to addressing the main WIR question, and by considering these sub-questions in light of the looking-back questions described by Pólya (1945/1973).

  2. 2.

    What are the characteristics of dialogical moves in the small-group discussions of students working on WIR task?

    This question was answered by characterizing the extent to which students capitalize on each other’s thinking, in the spirit of Asterhan and Schwarz (2009) and Goos et al. (2002).

  3. 3.

    What mathematical resources are discursively enacted, and how, in the small-group discussions of students working on a WIR task?

Here we attended to the specific content of the arguments, in the spirit of Asterhan and Schwarz’s (2009) and Ayalon and Even’s (2014) studies, and investigated, in particular, what the students relied on while constructing their arguments.

4 Methods

4.1 Participants and settings

A class of 16 11th grade students (16–17 years old) in a school for girls took part in the study. The second-named author of this paper was the teacher of the class. The class studied mathematics in accordance with the highest-level (out of three levels) of the Israeli mathematics curriculum (cf. Movshovitz-Hadar, 2019, for details).

To better understand the context, two comments are in order. First, these students often used different internet resources where solutions to textbook problems are published. Accordingly, they were accustomed to comparing their solutions with the published ones. Second, in Israel, the notion of percentage is first introduced in the 6th grade. From then on, word problems involving percentages are regularly considered in all grades up to high school. From her ongoing work with the class, the teacher was aware of some of her students’ misconceptions with percentage problems and wished to create an opportunity for them to obtained a more nuanced understanding of the topic.

4.2 The WIR task

The WIR task used in our study is presented in Fig. 1.

Fig. 1
figure 1

The WIR task

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The central task condition is ‘the price of 1 kg of apples is 25% higher than the price of 1 kg of pears’. Hila and Sofia interpret this condition in two different ways.

Hila denotes the price of 1 kg of pears as \(x\), and the price of 1 kg of apples as \(1.25x.\) In this approach, the price of 1 kg of pears corresponds to 100% and the price of 1 kg of apples to 125% of the price of the pears, which is a legitimate interpretation of ‘25% higher’. Therefore, the corresponding equation \(2\cdot 1.25x+5x=23\) leads to the correct solution to the problem: the price of 1 kg of pears is \(3\frac{1}{15}\) shekels, and the price of 1 kg of apples is \(3\frac{5}{6}\) shekels.

Sofia denotes the price of 1 kg of apples as \(x\) and the price of 1 kg of pears as \(0.75x\). In this approach, the price of 1 kg of apples corresponds to 100% and the price of 1 kg of pears is interpreted as 75% of the price of the apples. This interpretation does not meet the given condition. Indeed, in this way the price of 1 kg of apples is not 125% of the price of 1 kg of peers but \(x/0.75x=1\frac{1}{3}\) or 133.33%. Accordingly, the corresponding equation \(2x+5\cdot 0.75x=23\) leads to an incorrect solution to the given problem: the price of 1 kg of apples ‘is’ \(4\) shekels, and the price of 1 kg of pears ‘is’ \(3\) shekels.

In fact, Sofia’s equation solves another problem based on an alternative condition ‘the price of 1 kg of pears is 25% lower than the price of 1 kg of apples’. Parker and Leinhardt (1995) showed that the frequent confusion between the original and alternative conditions can be attributed to a variety of factors. These include the ambiguity involved in inferring a reference base for 100% from the problem narrative, and the interference of additive and multiplicative reasoning structures in students’ perception of problems having a ‘differ by percent’ structure (Parker & Leinhardt, 1995, p. 442). Indeed, the two conditions are not equivalent. However, two apparently similar conditions, for example, ‘the price of a book is 25 shekels more than the price of a pen’ and ‘the price of a pen is 25 shekels less than the price of a book’ are equivalent. Accordingly, ‘differ by percent’ problems and ‘differ by quantity’ problems can easily be confused.

The task narrative includes two additional potential traps. First, the word ‘apples’ appears before the word ‘pears’ in the formulation. Here we capitalize on the well-known misconception in solving word problems, namely, denote by \(x\) the first unknown quantity mentioned in the problem formulation (e.g., Clement, 1982), as is done in Sofia’s solution.Footnote 3 Also, Sofia’s solution-narrative leads to two ‘nice’ integers as the answers, while the correct solutions are fractions.

Yet, we assumed that the students in our study might be in a position to correctly solve the task, based on their past experiences with such problems as the following one (hereafter, the market-up/market-down problem).

When first produced, a fancy t-shirt cost 100 shekels. This price was increased by 25% at the beginning of the high season and then decreased by 25% at the end of the season. What was the price of the t-shirt at the end of the season?

We hoped that the well-known to the students solution to the above problem (i.e., \(100\cdot 1.25\cdot 0.75=93.75\)) would serve as a resource for solving the current problem, by reminding the students that subsequent adding and subtracting of 25% does not preserve the initial price because the referents for 100% are different. However, the analogy between the WIR task and the market-up/market-down one is limited. Namely, the market-up/market-down problems presume that the first change (i.e., market-up) is applied to some initial quantity, and the second change (i.e., market-down) to the modified or dependant quantity. In our WIR task, the initial quantities in Hila and Sofia’s solutions are independent and acted upon only once.

An additional clue was provided by means of a follow-up task, as follows:

The price of a notebook is 3 shekels less than the price of a pen. The price of three pens is 25% less than the price of 7 notebooks. What is the price of one pen? What is the price of one notebook?

The plan was that during the lesson the groups might return to the WIR task after considering the follow-up task. In the lesson under investigation, the first 15 min were devoted to solving the WIR task in small groups, and the rest of the lesson (which was not part of the current study) was devoted to solving the follow-up task and to the whole-class discussion.

4.3 Data collection and data analysis

The students were working on the WIR task in six small groups, that is, four groups of three and two groups of two; the small-group discussions were audio- and video-taped and fully transcribed (overall, about 12,000 words). The students’ written work and the teacher’s reflective notes served as complementary data sources.

We implemented inductive analysis with partially predefined categories (Dey, 1999; Strauss & Corbin, 1990) as the main analytical approach. The transcripts were first split into episodes consisting of several mutually related consecutive conversational turns. For example, a part of the transcript was called ‘episode’ when the students collaboratively explored a particular argument. The episodes were assigned F-type codes (strategies of exploring the task formulation; see Table 1). In addition, each episode was aggregately summarized (Thomas, 2006) for the purpose of identifying its role in the flow of the discussion. The summaries attended to the central questions discussed and to the main discursive products of the episodes. Examples of such questions are ‘who is right?’, ‘why is Sofia wrong?’, ‘are the solutions indeed different?’ Examples of the discursive products are ‘two students agreed that Hila is right’, ‘the argument about different reference bases for 100% was asserted by one student and ignored by the rest’.

Table 1 Strategies of coping with the WIR task formulation (F-codes)
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The transcripts were then analyzed according to dialogical moves (D-codes). In line with Asterhan and Schwarz’s (2009) approach, we corresponded dialogical moves to conversational turns or parts of conversational turns having different dialogical functionality. Stimulated by Goos et al.’s (2002) study, we developed three types of D-codes: self-disclosing utterances, requests for response (with two subcategories) and other-oriented utterances (with five sub-categories), as shown in Table 2.

Table 2 Dialogical moves (D-codes)
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The third group of codes concerned the enacted mathematical resources (R-codes), see Table 3. In most of the cases, R-codes were applied to self-disclosing dialogical moves, and in some cases also to other-oriented utterances. Note that they are not mutually exclusive: more than one R-code can be applied to one dialogical move.

Table 3 Mathematical resources enacted (R-codes)
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One full transcript was used to develop stable coding schemes. The rest of the transcripts were analyzed by the authors first separately and then together. Disagreements (overall, about 3% out of 1216 analytical decisions) were resolved in discussions among the authors. Most of the initial disagreements concerned the use of the R-AP group of codes (analogical problems). The R-codes were assigned (overall, 387 decisions) according to the following heuristics: while considering a dialogical move in the context of the discussion, we inferred which piece of knowledge could be held by the student as true in order to account for what she actually said. This heuristic conforms to a long-standing tradition of abductive analysis of inferring what is plausible from what is evident (Tavory & Timmermans, 2014). Illustrative examples of the data analysis are presented in the “Appendix”.

5 Findings

5.1 Strategies and discursive products

In this sub-section we address RQ1: What strategies do high-school students use, and how, in small-group discussions of a WIR task in the context of percentages? What are the discursive products of the use of these strategies?

Our first observation is that all the groups used more than one strategy, but employed them in different order. Three groups (G3, G4, and G5, see Table 4) used independently solving the problem (F-ISP) as their first strategy. The exploring the solution-narratives strategy (F-ESN) was used by all the groups, either as the first strategy (G1, G2 and G6) or as the second one (G3, G4 and G5). Accordingly, for G3, G4 and G5 experience of dealing with the WIR task was compatible with that in AIR tasks: first solve and then look back. The strategy of solving the equations (F-SEQ) was used by G4, G5 and G6 at the advanced stages of the discussions. For example, G6 started with F-ESN for describing and evaluating the provided solutions and concluded that they both were right. Only after they suspected that their conclusion might be wrong, they resorted to F-SEQ.

Table 4 Summary by strategies used and pathways of sub-questions explored
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As for the ‘how’ part of RQ1, of interest is that the above strategies were used in order to address specific questions, which were continuously changing (see Table 4). Most of the time the discussions revolved not around the main WIR question but around more specific questions. As a rule, specific questions (e.g., ‘why is Sofia’s solution wrong?’) re-appeared at different stages of the discussion. In G2, the tendency was to initially consider specific questions and then involve more general questions (e.g., ‘But we’ve done exercises in this way many times, and it was correct… The question is, in which exercises is this way correct?’). The other five groups repeatedly came back to the previously asked specific questions. However, the discussions were not circular: the same specific questions were approached with the help of different resources and strategies. In addition, in four groups (G1, G2, G4, and G6) ‘strategic splits’ were observed: the same questions were considered by different students in parallel but using different strategies.

Table 4 presents the sequences of questions considered in each group. Of note is that the questions are well-aligned to Pólya’s questions for the (idealized) looking-back stage. Indeed, the first two of Pólya’s questions were: (i) “Can you check the result? (ii) “Can you check the argument?” These two questions were at the heart of all the discussions. The third of Pólya’s questions was “Can you derive the result differently?”. Arguably, this question underlined the use of the ISP strategy in all the groups. The fourth question, “Can you see it at a glance?” can be related to the ‘what is the difference’ questions at the advanced stages of the discussions in G1 and G5 as well as to such questions as ‘what is the reference base’ in G4. Finally, the fifth of Pólya’s questions—“Can you use the result, or the method, for some other problems?”—connects to the ‘what is the role of language’ and ‘what class of percentage problems the given problem belongs to’ questions in G2.

As to the group discursive products produced by means of the above strategies, Table 5 shows that most of the groups (correctly) endorsed Hila’s solution, though the final endorsement and rejection decisions achieved different extents of agreement. Four groups (G1, G2, G3, and G6) reached a full agreement that Hila’s solution is right and Sofia’s solution is wrong. Of these, three groups (G1, G3 and G6) agreed on a justification as to why Hila’s solution is right, and why Sofia’s solution is wrong. The agreement in G4 was partial, and G5 did not achieve closure: they only agreed that one of the solutions must be wrong. These findings suggest that the task was quite challenging for all the groups, in spite of the considerably strong mathematical background of the students.

Table 5 Final products of the discussions
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5.2 Dialogical moves

In this sub-section we answer RQ2: What are the characteristics of dialogical moves in the small-group discussions of students working on WIR task? As a reminder, dialogical moves were categorized as self-disclosing utterances, requests for response and other-oriented utterances (see Table 2). As can be seen in Table 6, self-disclosing utterances constituted about 1/3 to 1/2 of the overall number of dialogical moves, which means that, generally speaking, the discussions were not collections of students’ monologues but had a dialogical nature. The general picture presented in the left column of Table 6 suggests that in G1, G3 and G5 the students were more responsive to each other’s ideas than in G2, G4 and G6.

Table 6 Summary by types of dialogical moves and resources
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Of interest is an apparent mismatch between the percentages of the requests for response (D-RR) and other-oriented utterances (D-OO). This finding implies that students often responded to their groupmates when not being explicitly asked to do so, and sometimes did not respond to explicit requests for response.

Following the chains of D-codes, the complex structures of the dialogues is further revealed. In all six groups, there were episodes in which the students readily responded to each other. This observation is reflected in the frequent appearance of the codes of the other-oriented group of codes (D-OO) (e.g., see two consecutive episodes analyzed in the “Appendix”). In contrast, in all six groups there were episodes in which the students asserted their own ideas in parallel. This observation is reflected in the frequent appearance of the self-disclosure code (D-SD). Such episodes often involved a struggle for attention. In Fig. 2, we present part of Episode 6 in G4, in which the struggle for attention became the explicit topic of discussion.

Fig. 2
figure 2

A coded transcript of a part of Episode 6 in G4

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This dialogue exemplifies the reciprocal nature of the struggle: listen now in order to be listened-to later. G4 developed an explicit norm of listening ‘in turns’ whereas in the rest of the groups the norms of listening remained unarticulated. The different patterns of talking and listening observed in the data seem to be connected to the use of mathematical resources, which brings us to the third research question.

5.3 Mathematical resources

In this sub-section we address RQ3: What mathematical resources are discursively enacted, and how, in the small-group discussions of students working on a WIR task? A wealth of mathematical resources was enacted (see Table 6). As a reminder, we refer to mathematical resources as inferences from the students’ utterances which can reasonably explain what the students might temporarily have held as true while constructing their arguments.

Resources that manifested themselves in all groups were related to the use of analogical problems (R-AP) and to someone's specific solutions to the problems (R-TS). The rest of the resources were used occasionally. Attending to analogical problems (AP-MM for G1 and G6 and AP-DP/Q for G3, see Table 3) helped three groups out of six to solve the task fully. Repeated reliance on verbal clues in order to determine a reference base for 100% (R-RB-OF) led G2 to address the general role of language in percentage problems. Referring to the definition of percent (R-DEF) was used as a final argument in G1 and G6. However, the nomenclatures of the enacted resources by themselves do not fully explain success or its lack thereof in solving the task (see Table 5). Indeed, there were groups that used similar sets of resources, but did not all succeed in reaching a (correct) agreement. The differences seem to be related to how the resources were enacted and responded to.

The enacted resources had different influence on the endorsement/rejection decisions. Some were ignored. Sometimes, an argument that did not immediately sound right to the interlocutors was put aside and another argument was put forward instead. Some of the resources were ‘peripheral’ in one episode, though became central in the next episodes (e.g., R-AP-DP in Episode 2 in G2, “Appendix”). Simultaneously, a resource that was central in one episode could be ‘forgotten’ in the next one (e.g., R-AF in Episode 2 in G2, “Appendix”). Sometimes the use of a particular resource was short-lived in the sense that it helped the students to make some progress and then ceased being useful. For example, relying on the fact that the solutions of the equations in Hila and Sofia’s solutions were different (R-EQ-DA) enabled the students of G2, G5 and G6 to conclude that at least one of the solutions must be wrong. However, it was not enough for determining who was wrong and why. On this basis, student N. from G6 put into play a new resource, analogy with market-up/market-down problems (R-AP-MM), which eventually led to their solving of the task. Also, some resources could subsume each other in the discussion. For example, a self-produced solution to the problem in G2 (R-TS-SP) was a resource that underlined the decision to endorse Hila’s solution (‘Let’s say that we do the task. How would we do it? A kilo of the pears equals \(x\). Correct?’). Then reliance on Hila’s solution being true (R-TS-HS) became a resource on the way of evaluating Sofia’s solution.

In many cases, a student who brought to the discussion a particular resource tended to keep on using it, with little attention to the resources brought by the other students, though, as illustrated, they sometimes struggled for each other’s attention (see Fig. 2 and “Appendix”).

In sum, the kaleidoscope of different resources and a limited readiness to work with others’ resources for more than several moments in a row might explain the inconclusiveness of the discussions in G2 and the lack of agreement in G4 (see Table 5). Success of G1, G3 and G6 in solving the task seems to be related to moderate attention to a considerable number of each other’s ideas. The story was different in G5, which ended the discussion without endorsing or rejecting any solution. In this group, when a student brought to the discussion a particular resource (see Table 6), her peer joined her in a non-critical way (i.e., a typical response consisted of simple agreement, D-OO-SA). Furthermore, G5 was the only group in which the number of the other-oriented dialogical moves (D-OO) was approximately twice the number of the self-disclosing utterances (D-SD). Accordingly, the inconclusiveness of the discussion in G5 can be attributed to over-collaboration based on the relatively shallow repertoire of enacted resources.

6 Discussion

The goal of the study was to characterize processes involved in small-group discussions of Who-Is-Right (WIR) tasks as a means to support, in realistic classroom situations, practices known as germane to the looking-back stage of problem solving. To address this goal, we designed a WIR task in the context of percentage and enacted it in a regular mathematics lesson, in which high-school students worked on the task in small groups.

The study was driven by three research questions. In response to RQ1, concerning strategies of coping with the WIR task, we identified three strategies used in the groups in different orders. Those groups who began from independently solving the task and then moved, sometimes repeatedly, to exploring the solution-narratives and examining the algebraic parts of the provided solutions, arguably experienced the task as if it were a regular ‘solve and then look back’ task (referred to as ‘Am I Right’ or AIR task). In all cases of coping with the WIR task in our study, the use of the above three strategies was driven by attempting to address specific sub-questions. Many of these questions appeared to be well-aligned with five questions reserved by Pólya for the (idealized) looking-back stage. Therefore, our central claim was that all the participants in our study were engaged in looking-back practices, if not always in a looking-back stage. In response to RQ2, concerning dialogical moves, we showed that some parts of the discussions were of a collaborative nature, whereas in some other parts the students tended to ignore each others’ arguments. In response to RQ3, concerning mathematical resources enacted, we identified a wealth of mathematical resources, but also showed that success with the WIR task cannot be associated with mere use or non-use of certain resources, but rather with ways of enacting some of the resources (most notably, the use of analogical problems) in a dialogue.

We begin the discussion by elaborating on the above-formulated central finding of our study. It is based on the distinction between a looking-back stage in AIR tasks, and looking-back practices in WIR tasks. An initial comparative discussion of these two types of tasks was provided in Sect. 2.2. We are now in position to say more. First, as our data show, independent solving of a given problem takes place in both cases, though in AIR tasks it is necessarily the first stage and in WIR task it can be either the first or the intermediate stage. Second, as past research shows (see Sect. 1), the appearance or non-appearance of looking-back in AIR tasks frequently depends on prompts provided by an authority. In WIR tasks, even if there is no designated looking-back stage, the solvers engage in verifying solutions and arguments, comparing alternative solutions, and sometimes in considering how the considered means of solution corresponds to that used in solving other problems. These practices are in remarkable accordance with Pólya’s (1945/1973) vision for the looking-back stage. Importantly, they were not solicited, at least not explicitly, in the context of the WIR task. Third, even when the looking back stage is present in students’ coping with regular AIR tasks, it frequently ends with mere endorsement or rejection of the constructed solution (Malloy & Jones, 1998; Mashiach-Eisenberg & Zaslavsky, 2004). In our WIR task, endorsement of a particular solution came relatively early (e.g., from G1: ‘Hila is right because this is how everyone does it’), but then the main challenge began. Beyond endorsement of a particular solution, a WIR task requires students to engage with a series of subtle ‘why-questions’ stemming from the need to compare the provided and the self-produced solutions. Of note is that AIR tasks, which as a rule operate with one self-produced solution, do not include this opportunity.

Due to the enriched referential basis embedded in the WIR task formulation, handling the ‘why’ questions naturally required considering the question ‘why is the other solution wrong?’, which is not the same as considering the question ‘why is the chosen solution right?’ Answering the former question requires the students to use various looking-back practices (e.g., connecting problems by considering analogous problems, formulating implications for future problem solving, and comparing alternative mathematical points of view for explaining their endorsement and rejection decisions to the peers), whereas the answer to the latter question can be socially-based and not necessarily mathematically based (e.g., G1: ‘this is how everyone does it’). The differences between AIR and WIR tasks are schematically summarized in Fig. 3.

Fig. 3
figure 3

Comparison of AIR tasks and WIR tasks

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To recapitulate, WIR tasks and AIR tasks are essentially different with respect to the opportunities they provide for the use of looking-back practices. In particular, we argue that in addition to affordances inherent in comparing worked-out solutions (Rittle-Johnson & Star, 2011; Swan, 2007), the WIR task contains affordances provided by a regular AIR task. Our hypothesis that WIR tasks can be a valuable tool for engaging students in looking-back practices was partially based on the relative richness (in comparison with AIR tasks) of the space of referential bases for verification. This hypothesis is supported by our findings, which, in turn, calls for future research: it would be interesting to find out if students who systematically engage in WIR tasks tend to employ more verification strategies in regular AIR tasks than students who are exposed only to AIR tasks.

We now turn to discussing our findings in light of past research. The identified strategies of coping with the WIR task—exploring the solution-narratives (ESN), independently solving the problem (ISP) and solving the given equations (SEQ)—are in line with verification strategies identified in past studies. For example—referring to Malloy and Jones’ (1998) and Mashiach-Eizenberg and Zaslavsky (2004) lists of strategies—ESN corresponds to rereading the problem, checking the plan and adding justifications to the solution, ISP to redoing the problem, and SEQ to checking calculations and comparing answers. A novel (yet not surprising) finding is that all small groups in our study used several strategies in the context of the WIR task. As mentioned, the looking-back stage in the context of AIR tasks is rarely observed, as well as the phenomenon of voluntarily using more than one verification strategy (Mashiach-Eisenberg & Zaslavsky, 2004). We also note that our study documented the student strategies in a realistic classroom situation, and not in an interview setting, as had been done in the prior studies (e.g., Malloy & Jones, 1998; Mashiach-Eisenberg & Zaslavsky, 2004; Papadopoulos & Dagdilelis, 2008).

In addition, our findings imply that comparing worked-out solutions in the WIR format raised the level of difficulty of the task for the students. Indeed, the task in the context of percentages, which could have been offered to middle-school students, evoked hot discussions among high-school students studying mathematics at the highest level of the Israeli curriculum. How so? This may seem particularly surprising in light of the fact that almost all students who adhered to the ISP strategy reiterated Hila’s (correct) solution. Furthermore, based on close familiarity of the second author with the students as their mathematics teacher, we suggest that most of the students could easily have answered the question included in the task-narrative (see Fig. 1) if it were given in a standard format. We deem that the student difficulty with the WIR task can be explained by pointing out that explaining why Sophia’s solution was wrong was a higher-challenge request than producing the correct solution. This is because producing the correct solution could have been solely based on procedures memorized while solving similar problems in the past, and understanding why Sophia’s solution was wrong required a nuanced understanding of the problem situation, as the task analysis provided in Sect. 4.2 shows.

Returning to Pólya’s argument concerning the importance of looking back, our next claim is about the potential of the WIR task for creating connections between problems with an eye to future problem solving. Evidence for this claim consists of the use of analogical problems in all groups and the tendency (observed only in G2) to consider questions of a more general nature than the original task question towards the end of the discussion. G2 discussed which class of percentage problems the given problem belongs to, and compared the difficulties they encountered in the given task with difficulties in tasks from the past. To our knowledge, documentation of a gradual shift of focus in a problem-solving discussion towards unsolicited generalization (see G2 in Table 4) is a novelty in empirical research on problem solving.

Our next point is about enacting mathematical resources in the group problem-solving. We recall that mathematical resources were conceptualized in our study as discursively-enacted pieces of knowledge held by a student as temporarily true, and which served as anchors for constructing arguments. It was not the resources themselves, which were more or less the same in the most of the groups, but rather the students’ patterns of responding to the enacted resources that seemed to account for success or failure with the task. Indeed, resources that could lead to a quick solution (e.g., analogy to differ-by-percent problems) were sometimes enacted but ignored, then repeated, being subsumed under the other resources, modified, forgotten or otherwise put forward.

Asterhan and Schwarz (2009) described a similar phenomenon when writing about shifts of the epistemic status of ideas in small-group discussions. Our study contributes to this line of research. In particular, we found that, as a rule, each student was an enactor of a small set of her ‘favorite’ resources that she repeatedly returned to throughout the discussion. Our data enable us to suggest that one’s ‘favorite’ resource had a chance to influence further discussion (namely, to gain response from others) if and when it appeared at particular moments, namely, when it resonated with the resources enacted by the others. Abdu and Schwarz (2020) showed that cooperation in small-group problem solving can be as important and useful as collaboration. The above suggestion about how and why the particular resources can influence the discussion sheds light on how and why collaboration and cooperation in problem solving are interleaved.

We now discuss a related finding: request for response was the least frequent dialogical move in all the groups, and the other-oriented dialogical moves were relatively frequent, especially in G1, G3 and G5. To interpret this finding, let us recall that in written dialogues—for example, in classic Socratic or Lakatosian dialogues (Zazkis & Koichu, 2018) or in dialogues written for learning purposes (Koichu & Zazkis, 2018)—the number of requests for response is roughly equal to the number of responses. An idealized dialogue develops in a linear manner, from one idea to another. The lived dialogues reported in our study are much more complex, and include instances of simultaneous use of several strategies or ideas. A vivid dialogue among teenagers seems to be shaped by their free choice regarding what to respond to, when to respond, and how, in accordance with their own interests and lines of reasoning. To this end, our findings challenge one of the conclusions of Goos et al.’s (2002) study. In that study, successful problem-solving cases were characterized as those in which the students discarded unhelpful ideas and actively endorsed useful problem-solving strategies, whereas unsuccessful cases were characterized by the lack of critical engagement with the partners’ contributions to the discussion. Our findings reveal a more complex picture: it seems that, as a rule, the students were unaware of which idea should have been endorsed or rejected. Accordingly, success or failure in completing the task seems to be related to the extent to which one’s ideas could be combined or contrasted with others’ ideas, in the quid-pro-quo struggle for attention. In addition, our data corpus includes a case (G5) where the lack of success with the task may be attributed to over-collaboration.

The phenomena discussed in the preceding paragraphs were revealed through the dual coding of dialogical moves (D-codes) and of mathematical resources (R-codes). The former coding scheme is of generic nature, and the latter coding scheme is task-specific. We believe that the codes we developed, along with the way of using them in tandem, can be useful in future studies.

This being said, it is important to discuss limitations of our study. Obviously, the use of one WIR task in one classroom does not enable us to generalize our claims to other tasks and classrooms. Another limitation is related to the fact that our analysis, as complex as it was, did not attend to individual differences among the students, and only narrowly attended to social roles of the interlocutors. With more comprehensive analytical tools we might be able to refine some of the explanations of the observed phenomena. Next, the complexity of the analysis presumes that, in spite of our attempts to assure proper reliability of coding, some mistakes could occur. As a measure of dealing with this inevitable threat, we formulated our findings using qualitative rather than quantitative language whenever possible. Thus, we hope that our conclusions are immune to occasional mistakes in coding.

We conclude by pointing out implications of our study. First, the findings seem encouraging for practice, which gives us an opportunity to reinforce a previous call for the greater use of specially designed WIR tasks in school setting (e.g., Rittle-Johnson & Star, 2011; Swan, 2007; Tabach & Koichu, 2019). To this call we would like to add that WIR tasks have deeply entered the school reality even when not intentionally designed. For example, students encounter spontaneous WIR tasks in various internet resources, in which a wealth of unverified worked-out solutions are discussed (Koichu et al., 2018). Next, we propose that the study provides some ideas for how to increase the feasibility of regular AIR tasks for students. For example, an AIR task can be followed by a WIR task or be included in a sequence of mathematically related AIR tasks for the sake of enriching the space of possible reference bases for verification.

As to theoretical implications, our study creates grounds for re-thinking the looking back as the desirable (but hardy achievable) last stage of problem solving (Pólya, 1945/1973), into a set of collaborative looking-back practices that can be evoked at different stages of problem solving. In line with the discursively-oriented perception of problem solving (Koichu, 2019), we offer the following (tentative) conceptualization: the looking back practices in problem solving are socially-shaped processes of endorsing or rejecting a particular set of solution-narratives to a given problem, in which the solvers discursively enact resources available to them prior to or during solving the problem, and also resources developed in collaboration, cooperation or exposure to solutions, which do not belong to the initial set. We hope that this conceptualization will support discussion of and research attention to the (still underexplored) phenomenon of collective looking back in mathematical problem solving.