1 Introduction

In many countries, prospective secondary mathematics (SM) teachers are required to complete extensive coursework in tertiary mathematics (TM), which typically extends well beyond the SM curriculum, and is grounded in ways of doing and thinking about mathematics that are vastly different from the way mathematics is taught and experienced at the secondary level (Even, 2020). While engagement with TM is widely considered to be beneficial for teachers, and an essential component of teacher education, there is a growing interest in revisiting and explicating the potential and realized affordances of studying TM for SM teaching (Biza et al., 2022).

The central role of TM in teacher education is often supported by a long-standing assertion that extending teachers’ mathematical knowledge well beyond the school mathematics curriculum is a good way to enhance the quality of classroom instruction. This assertion has been expressed repeatedly in policy documents (e.g., CBMS, 2012) and is consistent with a long list of studies (e.g., Ball and Bass, 2009; Dreher et al., 2018; Wasserman and McGuffey, 2021). For example, it has been suggested that engaging teachers with specific ideas from the SM curriculum at a level of depth and rigor suitable for TM courses promotes teachers’ awareness of and sensitivity to nuances in the SM curriculum and in their students’ and their own reasoning (CBMS, 2012; Wasserman, 2018). In addition, it has been suggested that engagement with TM promotes knowledge about mathematics as a discipline and thus supports teachers in making judgments about what is mathematically important and in orienting instruction in line with disciplinary values and practices (Ball & Bass, 2009; Hoffmann & Even, 2021).

Until recently there has been very little empirical research on the actual affordances of teachers’ engagement with TM, and this empirical research has found no significant correlation between various measurements of teachers’ knowledge of TM and their students’ success (Wasserman et al., 2018). Over the last two decades, more empirical evidence has accumulated in this area (Wasserman et al., 2018; Zazkis & Leikin, 2010). For the most part, these studies seem to corroborate earlier findings by indicating that prospective teachers taking TM courses find them unproductive and irrelevant for their future profession (Wasserman et al., 2018), and practicing teachers find it difficult to cite specific applications of their learning experiences in TM courses in their teaching (Zazkis & Leikin, 2010). These studies also indicate some positive affordances, such as greater empathy for students’ difficulties or enhanced mathematical confidence, yet it appears that in many cases, TM knowledge may remain a part of teachers’ mathematical knowledge, distinct and separated from their knowledge for teaching (Biza et al., 2022; Wasserman, 2018).

Recognizing that the affordances of engagement with TM for SM teaching are not self-evident, and that realizing these affordances in practice may be far from trivial, researchers have started to reconsider, elaborate, and refine the connections between studying TM and teaching SM. For many years, research in this area has focused on two types of connections (Wasserman, 2018): content connections between specific mathematical ideas in TM and in the SM curriculum, extending the body of work of Klein (1908) on elementary mathematics from an advanced standpoint, and disciplinary practices connections between ways of doing and thinking about mathematics ascribed to mathematicians, and practices that SM students are expected to engage with and internalize (e.g., mathematical habits of mind; Cuoco et al., 1996). However, research suggests that there is no clear path for translating content and disciplinary connections to pedagogical insights that can inform teaching, and thus these connections do not provide, on their own, sufficient motivation for teachers to engage with TM, nor sufficient support for teachers to realize the potential affordances of TM (Wasserman et al., 2018). In recent years, research attention has expanded to other kinds of connections, most notably to classroom teaching connections between specific TM ideas and concrete instructional situations, where mathematics and pedagogy intertwine (Heid & Wilson, 2015; Pinto and Cooper, 2022; Wasserman and McGuffey, 2021; Zazkis, 2020). There is preliminary evidence that when teachers’ engagement with TM is framed and motivated by classroom teaching connections, teachers value this engagement and consider it relevant for their teaching (Pinto & Cooper, 2022; Wasserman & McGuffey, 2021). These findings are encouraging in suggesting viable paths for increasing teachers’ motivation to engage with TM. However, we still know very little about classroom teaching connections.

Existing literature on the affordances of TM for SM teaching provides only limited insight into how SM teachers can be supported in making classroom teaching connections with respect to their own teaching, and how these connections can be utilized to inform teaching. This is due, in part, to teachers utilizing TM in their teaching in ways that are often tacit and highly personal, in the sense that they are inspired by particular TM experiences of individual teachers (Zazkis, 2020). These features make the connections to TM that teachers make in their teaching almost invisible to an outside observer and possibly even to the teachers themselves, and thus difficult to recognize, document and study (Biza et al., 2022; Pinto & Cooper, 2022).

This study is part of a long-term research program – M-Cubed (Mathematicians, Mathematics teachers, Mathematics) – that investigates processes involved in utilizing TM in teaching. In M-Cubed, small groups of mathematicians and experienced SM teachers view videotaped SM lessons and jointly inquire into mathematical and pedagogical issues that they recognize therein. This research setting can be seen as a laboratory for generating and studying implications of ideas and perspectives from TM in consideration of authentic instructional situations. Mathematicians are often oblivious to pedagogical nuances and practical constraints, and hence their implied utilizations of TM may be in tension with the teachers’ goals, values, and professional obligations (Herbst & Chazan, 2020). These tensions can challenge both sides to make explicit and possibly reconsider their positions, thus making public and visible their processes of exploring, elaborating, and refining potential utilizations of TM for informing SM teaching. In prior work (Pinto & Cooper, 2022), we have shown how ideas from infinitesimal calculus were utilized to inform the teaching of topics in the SM curriculum outside calculus. We further observed that making such connections may require subtle and nuanced weaving of knowledge of TM and knowledge of SM teaching.

The current study builds on our prior work and extends it through a fine-grained investigation of the processes entailed in making classroom teaching connections and utilizing TM in teaching in the case of probability, where research has found that teachers often have weak knowledge and understanding of the subject matter and may rely heavily on misleading experiences and intuitions, more so than in other mathematical domains (Stohl, 2005; Batanero et al., 2016). Perhaps as a result, research to date has focused mainly on prospective teachers’ knowledge of subject matter and on ways of developing it, and little is known about the affordances of studying TM for teaching SM probability (Batanero et al., 2016). Compared to other SM topics, activities in SM probability lessons are more often situated in real-life situations, inviting students to explore the relations between mathematical theory, experiments, and prior experiences. These real-life situations may be difficult to mathematize rigorously, which makes it difficult for SM teachers to gauge the validity of student reasoning. The real-life situation we consider in this study is a popular SM probability activity – the River Crossing Game. While at first glance this game may appear relatively simple, it is surprisingly misleading and difficult to mathematize, which makes it particularly difficult to connect its use in SM classes to TM. As such, the investigation of the connections to TM that teachers and mathematicians are able to make in this context can be seen as an extreme case study (Seawright & Garring, 2008) of how TM can inform SM teaching, as we discuss in the next section.

2 The river crossing game

The River Crossing Game (RCG) is referred to in a variety of curricular documents over the world and is apparently quite well known to teachers, from the primary level to higher education (Goering & Canada, 2007; Kazak & Pratt, 2017). In RCG, two players play against each other. The players compete to remove all their tokens from a board with two sets of “pegs” numbered 2–12 (see Fig. 1). After both players have decided on initial configurations of their tokens across their side of the board, a pair of dice is repeatedly rolled, and after each role each player removes a single token from the peg indicated by the sum of the dice (if the peg is not empty), which is said to “cross the river”. The winner is the first to remove all tokens.

Mathematics teachers find RCG appealing since students often turn to intuitions while devising a strategy for spreading tokens and may revisit their intuitions after experimenting with the game (Canada & Goering, 2008; Kazak & Pratt, 2017). For example, studies have shown that students and prospective teachers may spread tokens evenly across all pegs, implicitly assuming a uniform distribution of the sum of two dice, or alternatively place all tokens on 7, which is the likeliest sum. Players familiar with the probability distribution of the sum of two dice (see Fig. 2) may derive from it various heuristics for playing RCG, for example that configurations should be symmetric around 7, or that spreading tokens proportionally to the distribution, or as close to proportional as possible after rounding, is an optimal strategy. Surprisingly, the latter intuition is wrong. For example, in Fig. 1, Configuration A, which follows the distribution of dice outcomes (Fig. 2), has probability 0.31 of winning, lower than the probability of configuration B winning – 0.45 (the probability of a tie is 0.24; all probabilities are approximated) (Goering & Canada, 2007).

Fig. 1
figure 1

River Crossing Game board with two initial configurations of 36 tokens

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Fig. 2
figure 2

Outcome histogram for the sum of two dice

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Investigating the relative merits of various heuristics for playing RCG can be a complicated and misleading endeavor, not only for SM teachers, but also for experienced mathematicians (Goering & Canada, 2007). Given two configurations A and B, A is said to dominate B if, in a game that starts with configurations A and B, the probability that configuration A wins is greater than the probability that configuration B wins. One may assume that domination is a transitive relation and hence that there necessarily exist ‘dominating’ configurations that are not dominated by any other configuration. Moreover, if we denote the expected number of dice rolls needed to clear a configuration A by ED(A)Footnote 1 (the expected duration of A), then a common intuition in relation to RCG is that A dominates B if and only if ED(A)<ED(B), which implies that dominating configurations are exactly those with minimal expected duration. While these intuitions are sensible and provide fairly effective heuristics for playing RCG, both are in fact not true (Goering & Canada, 2007). Thus, RCG needs to be conceptualized as a competition between configurations, meaning that winning RCG is not about producing a dominant configuration but rather about producing a configuration that dominates the other player’s configuration. Existing analyses of RCG involve numerous calculations of the probabilities of the different game results for different pairs of configurations, and such calculations can be an insurmountable challenge even for relatively small numbers of tokens. Most of what is known on RCG is based on approximations obtained using computer simulations and mathematical tools and approaches that are typically not addressed in undergraduate probability courses.

To summarize, RCG is a popular SM activity that invites students to draw intuitively on the distribution of the sum of two dice while playing a game. SM teachers using RCG are likely to encounter students’ intuitions that are not true and may even hold such intuitions themselves, but mathematizing RCG and exploring these intuitions relying only on tools and approaches from undergraduate probability courses can be overwhelming and likely beyond the scope of most SM teachers. Considering that attaining deep understanding of the subject matter is not feasible, it is not clear which TM ideas, if at all, could inform SM teachers’ use of RCG, which raises the question whether and how TM could be relevant and useful for SM teachers in this case. These questions make RCG a compelling extreme case (Seawright & Gerring, 2008) for investigating how SM teachers can make connections to TM and utilize it in their teaching. Specifically, we consider how TM can support SM teachers in addressing different facets of probability (Even & Kvatinsky, 2010) in ways that are productive with respect to the teachers’ goals for using RCG, while also maintaining their professional obligation to the norms and values of the discipline, as represented by mathematicians’ perspectives on the affordances of using RCG in SM education. Accordingly, our investigation is guided by the following questions:

RQ1. What facets of probability do SM teachers seek to address through RCG?

RQ2. What is the mathematicians’ perspective on the use of RCG in secondary probability classes?

RQ3. How can rapprochement between teachers’ and mathematicians’ perspectives inform the use of RCG in SM education?

3 Theoretical framework

3.1 Connecting the study of tertiary mathematics and the teaching of secondary mathematics

Wasserman (2018) proposed different points of connection between TM and SM teaching, three of which are relevant for this study. We already discussed content connections and disciplinary practices connections. Classroom teaching connections refer to TM content that has implications for pedagogical decisions, for example making mathematical sense of a student’s question and responding to it or formulating a refutation that explains why a student’s reasoning is invalid (Pinto & Cooper, 2022).

In our study, we conceptualize these connections as ways in which the interweaving of TM knowledge and knowledge for teaching SM can reshape the mathematics that teachers address in their classes. To this end, we draw on a conceptual framework for the mathematics addressed in SM probability lessons (Even & Kvatinsky, 2010). This framework builds on earlier work on SM teachers’ content knowledge for teaching, and is informed by books and textbooks on probability, on research on teaching and learning probability, and on interviews with mathematicians and education researchers on the role and importance of probability theory in mathematics, in other disciplines and in real-world situations, and regarding the place of probability theory in the mathematics school curriculum. This framework was proven useful for comparing the discourses of different teachers on probability and its teaching and learning, and for discerning subtle differences between them. As such, this framework is highly suitable for or study, where we wish to contrast mathematicians’ and teachers’ perspectives on using RCG in SM probability lessons.

The framework comprises five facets of probability, of which four are pertinent for our study:

  1. 1)

    Essential features and strengths of probability theory: Probability deals with uncertain situations whose outcome is not deterministic, and provides mathematical tools for analyzing, explaining and making predictions in these situations.

  2. 2)

    Representations and models commonly used in probability: This facet of probability includes for example tables, diagrams, trees, Venn diagrams, as well more general mathematical representations.

  3. 3)

    Basic repertoire: Powerful examples that illustrate and develop insights and understanding of ideas and concepts in probability (e.g., tossing a fair coin, rolling a pair of dice).

  4. 4)

    The nature of probability theory: This includes ways of reasoning, arguing and establishing ideas and truths, that are unique to probability, due to its non-deterministic nature.

3.2 Conflicting perspectives and rapprochement

As discussed in the introduction, there are many reasons why it is difficult to articulate how TM can be utilized to inform SM teaching. Our long-term strategy for investigating this question relies on bringing together SM teachers and research mathematicians in communities of inquiry (Jaworski, 2003) to discuss and inquire into dilemmas of SM teaching. In this we are bringing together members of two communities that engage professionally in mathematics instruction, yet whose views on mathematics and didactics may be different, and in fact are often in conflict. For example, we have found (Pinto & Cooper, 2022) that in terms of the practical rationality of teaching (Herbst & Chazan, 2020), mathematicians tend to have a strong obligation to mathematical precision and rigor, even when faced with pedagogical dilemmas, while teachers tend to have a strong obligation to the wellbeing of individual students and to the functioning of a class as a social entity. These different obligations can lead teachers and mathematicians to endorse conflicting courses of action. As in our prior work (e.g., Pinto and Cooper, 2022), we view these conflicts as a discursive boundary – “a sociocultural difference leading to discontinuity in action or interaction” (Akkerman & Bakker, 2011, p. 133).

Productive communication across discursive boundaries can rely on boundary objects (Star, 2010) – objects that are robust enough to maintain a common identity across communities who work on them, while flexible enough for these parties to interact with them differently. Previous studies have investigated a variety of such boundary objects, such as methods for performing vertical subtraction (Cooper, 2019), or video-recorded SM lessons (Pinto & Cooper, 2022). Joint work on such boundary objects can generate opportunities for boundary crossing (Akkerman & Bakker, 2011), situations where members of one community have the opportunity to engage with the perspectives of another community. Through boundary crossing, various kinds of dialogical learning can take place. Members on both sides of the boundary have opportunities to learn from and with members of the other community. Such learning may include some of the following elements: Identification, whereby teachers and mathematicians investigate differences in their perspectives and recognized their mutual legitimacy, coordination, whereby they clarify and explicate their conflicting perspectives and attempt to make sense of the perspectives of other, reflection, which may entail members of one community adopting some aspects of the other community’s perspective (called perspective-taking), or conversely, becoming more explicit and aware of why they hold fast to their own perspectives (perspective-making), and transformation, whereby new ideas and ways of acting may emerge through the negotiation of conflicting perspectives, in a process called hybridization. The outcome of this process may or may not resolve the conflict between perspectives, and need not necessarily achieve consensus.

In the present research we are first interested in processes of coordination and reflection around affordances of the RCG for teaching probability. Teachers have some ideas about the mathematics they can address around the RCG, and mathematicians may be critical of them or have other ideas, of which teachers may in turn be critical. These perspectives and differences are the focus of RQ1 and RQ2. Next, we take an interest in processes of transformation, whereby interactions between teachers and mathematicians give rise to new ideas and courses of action. This is the focus of RQ3.

4 Methodology and setting

4.1 Setting and data collection

The design and conduct of the M-Cubed environment draw on the literature on processes of expansive learning through boundary crossing (Akkerman & Bakker, 2011). The videotaped lessons under discussion function as boundary objects (Star, 2010), and are carefully selected by education researchers for the dilemmas that they pose. Many of the discussions are initiated by mathematicians around untaken teaching opportunities that they recognize. As such, these discussions often suggest practices and perspectives grounded in TM. Furthermore, these practices are not limited to what is actually observed in SM classes, but rather draw on mathematicians’ and teachers’ vision of what could take place in principle. Tension between the teachers’ and mathematicians’ professional obligations (Herbst & Chazan, 2020), negotiated through mindful brokering (Cooper, 2019), can challenge participants to cross the boundary between mathematicians’ and SM teachers’ discourse – reconsidering, refining or hybridizing their positions (Pinto & Cooper, 2022).

This is where our research took place. In this study we investigate a 2-hour session that focused on an 8th grade probability lesson, where students – led by their teacher (whom we will call Adam) – played and discussed RCG as well as other probability gamesFootnote 2. The participants were three mathematicians, whom we will call M1-M3, six teachers (T1-T6), and two mathematics education researchers, including the first author of this article, who mediated the meeting. All the mathematicians and teachers have more than a decade of teaching experience. Two of the mathematicians have taught mathematics courses for teachers. All six teachers have experience as students in TM courses, and two of the teachers have a graduate degree in mathematics. In this meeting, the mathematicians recognized the mathematical complexity of RCG, and questioned whether the game was appropriate for SM teaching and what pedagogical purposes it could serve. At first the discussion revolved around the participants’ experiences and perspectives, and then around the use of RCG in the videotaped lesson. The meeting was video recorded, transcribed, and selectively translated from Hebrew by the authors.

4.2 Analysis

Our answer RQ1 is based both on the video-recorded lesson and on the M-Cubed meeting. We first analyzed the facets of probability that were addressed in the video-recorded lesson. For this we drew both on Adam’s utterances and actions (e.g., at the board) – evidence of the mathematics that he was making available in the lesson – and on students’ utterances and work – evidence of mathematics that was present in the lesson, which the teacher may or may not have intended. Next, we turned our attention to the M-Cubed discussion. For RQ1, we restricted our attention to teacher-utterances that were not responses to mathematicians’ criticism – such utterances were considered as potential evidence of boundary crossing, and as such were considered relevant for RQ3. Utterances were classified with respect to Even and Kvatisky’s (2010) framework of the mathematics addressed in probability lessons, to construct a rich description of the mathematical ideas that the RCG affords, from the perspective of teachers.

To answer RQ2 we drew first on existing literature on the RCG authored by mathematicians (Goering & Canada, 2007; Canada & Goering, 2008), once again organizing the mathematical ideas according to Even and Kvatinsky’s framework. Next, we turned our attention to the mathematicians’ utterances in the M-Cubed meeting. This analysis focuses on what mathematicians would bring to a SM lesson in probability, but also (and mainly) on what they would not bring, and why not. For RQ2 we restricted our analysis to what took place prior to viewing the Video-recorded lesson (see supplementary material for an overview of the lesson). In this we focused on mathematicians’ initial perspectives, assuming that processes of boundary crossing – relevant for RQ3 – would commence as soon as they began to relate to Adam’s teaching.

For RQ3 we analyzed teacher’ and mathematicians’ utterances that were phrased as responses to the claims of others. Such utterances could indicate perspective-making when parties reiterated and clarified their initial perspective, or perspective-taking when parties adopted perspectives of others. We were particularly interested in utterances that suggested processes of transformation, in the form of hybridization, where discussants proposed new ideas or courses of action that took into account their own perspectives along with perspectives of others. We view such utterances – authored by teachers or by mathematician, or co-authored in an iterative process – as examples of connections between TM and SM.

By necessity, there was an interpretive element in these analyses. For example, when Adam instructed students to take 30 s to think about how to distribute their tokens, both authors understood this as signaling that the students had some mathematical tools at their disposal to analyze the situation and to make educated decisions in spite of the inherently unpredictable nature of the situation. Hence, this was seen as highlighting an essential feature and strength of probability theory. However, other interpretations are possible. The authors analyzed the data independently, and conflicting interpretations, which were rare, were resolved through discussion.

5 Results (RQ1): SM teachers’ perspective on mathematics that can be addressed through RCG

For the most part, the teaching and learning opportunities that the M-Cubed teachers addressed were not specific to probability. They considered RCG to be fun, engaging, highly motivating, and inviting mathematical intuitions and reasoning. However, the teachers also noted important ideas of SM probability that they believed RCG can help address, some of which were indeed addressed by Adam, as we now discuss.

5.1 Essential features and strengths of probability theory

Before students distributed their tokens on the board, Adam instructed them to think about what arrangement would be beneficial. In this, and in the retrospective discussion on the game, he was positioning probability as a tool for analyzing, predicting, and optimizing outcomes in an uncertain situation. This affordance of RCG was also present, sometimes implicitly, throughout the M-Cubed meeting, where the teachers considered how RCG can suggest ways to draw on ideas from probability for distributing the tokens, and use probability to help devise and analyze winning strategies. For example, while reflecting on her own use of RCG, T1 noted that she expects students to place more tokens on 7 than on 2, and that in her experience there are always students who suggest that configurations should be symmetric around 7 and shaped like a bell. Adam also observed that many of the students’ configurations (Fig. 3) are symmetric and ‘bell shaped’ and discussed these properties in relation to the sum of two dice diagram (Fig. 2), concluding by asking students “so, according to these probabilities [of the sums of two dice], how should you spread the tokens?”.

Fig. 3
figure 3

8th grade students’ RCG configurations in the case of 18 tokens. Each row represents a different configuration

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Adam and the M-Cubed teachers also considered opportunities to address various aspects of uncertainty and lack of determinism. These aspects of the situation were described by T2 as “Even if I know that most chances are that [the sum of the dice] will be 7 and I spread [the tokens] in a nice bell shape, it still does not guarantee that I win”. T1 compared this experience to a more canonic experience of uncertainty – tossing a coin 100 time, where the “expected” outcome of exactly 50–50 hardly ever occurs. In her view, the game is a more engaging way to help deepen students’ understanding that likely events do not necessarily occur. T3 suggested how this message could be driven home: “I’d run a survey, who thinks this strategy is the best? Then we’d split into pairs. One plays the ‘bell strategy’ [i.e., starts with a configuration that is as proportional as possible to the distribution of the dice sums] and the opponent plays something else. We’d see that half of the ones who played that [best] strategy didn’t win”. In practice, this idea of uncertainty and lack of determinism was discussed in the video-recorded lesson, for example when one pair of students tied twice, in spite of differences in their placement of tokens.

Another essential feature of probability theory brought up by the teachers is that probability can be complicated and counterintuitive. T3 suggested that the game could convey this message – which is often left implicit – demonstrating that intuitions that first come to students’ mind (e.g., placing all tokens on 7 or spreading tokens evenly on all pegs) are not necessarily valid.

5.2 Probabilistic representations and models

Several representations and models were addressed and connected in the recorded lesson, for example a table of possible outcomes and a distribution diagram. With respect to the table of outcomes, Adam noted that the different outcomes are arranged in diagonal lines of different lengths and used this representation to model and to calculate the probabilities of different outcomes (see Fig. 4). Adam used the game as an opportunity to discuss this representation and its relevance for choosing a strategy. He highlighted two features – 7, which occurs on the diagonal, is the most likely outcome, and all other outcomes are symmetrical around this diagonal. To highlight these features, Adam asked students to show with a gesture the probabilities of the different sums and identified this gesture as ‘the shape of a bell’, which he drew on the board (Fig. 4). The table and the bell-shaped diagram were used in Adam’s and the students’ reasoning for placing more tokens on 7 than on any other number, and for maintaining symmetry around 7. In the M-cubed discussion, teachers referred to similar representations and models when reflecting on their own use of RCG in class.

Fig. 4
figure 4

Different representations of probability in the Video-LM lesson

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5.3 Basic repertoire of examples that illustrate and develop insights and understanding

It would seem that Adam, and many of the M-cubed teachers, saw the RCG as an example upon which to develop insights and understandings. The students in the recorded lesson had learned about the distribution of the sum of two dice in the lessons the preceded the RCG activity, which suggests that Adam may have considered RCG an opportunity to draw insights from this example and further develop students’ understanding. Teachers in M-Cubed addressed similar opportunities. For example, T1 saw RCG as an opportunity to move from the familiar case of rolling a single die to the more complex case of two dice, and T4 anticipated that some students could actually generate the distribution for the sum of two dice on their own.

5.4 The nature of probability theory

Both Adam and the M-Cubed teachers considered RCG as a good context for engaging students with mathematical reasoning, in general, and in the special case of probability. In the recorded lesson, the scene was set by Adam, who gave students time to think about how they would like to distribute the tokens before they start doing it, signaling the expectation to reason about their strategy. Then, after the students played twice, Adam asked students to share whether they changed their strategy after one game, and if so explain why, perhaps in order to bring into the discussion some empirically based reasoning. Adam elicited explanations for various strategies, and for the most part avoided judgment.

Most of the probabilistic reasoning in the videotaped lesson occurred after seven strategies, proposed by students, had been written on the whiteboard, when Adam asked students to find similarities and to explain the sense in them. This course of action was also suggested by some of the M-Cubed teachers before watching the lesson. One similarity students discovered was a kind of ‘monotonicity’, in the sense that pegs with lower probability did not have more tokens than pegs with higher probability (e.g., the number of tokens next to 5 should not be greater than the number of tokens next to 6 or 7). Another similarity was ‘symmetry’, in the sense that the same number of tokens (with possible rounding errors) is placed next to numbers that are equidistant from 7 (e.g., 6 and 8). Adam phrased a reasoning task as “how can I convince myself” that these strategies are sensible. After drawing the table of outcomes on the board (Fig. 4), Adam directed students toward recognizing the monotonicity and symmetry properties in the table, and elicited this as a warrant for RCG strategies, stating that the decision how many tokens to place on each number should rely on the probability of that number as the outcome of rolling the dice. Eventually, he guided students toward the conclusion that if there had been 36 tokens, they should have been placed according to the frequencies of outcomes in the table (Fig. 2), and that the strategy for 18 tokens should be to divide the case of 36 by 2 (rounding where the number of tokens is odd). Adam’s justification was “I think it’s clear to everyone why”, and some of the teachers in M-Cubed agreed with this conclusion and that it is clear and requires no further justification given the distribution of the sum of two dice.

The validity of empirical reasoning was also discussed. The warrant “that’s what happened to us” was dismissed by Adam and by students based on the fact that one trial is inconclusive. However, some observations, such as “2 never came up”, were not challenged, perhaps because it is consistent with the low probability of this outcome. Furthermore, some empirical evidence (e.g., “I’ve been waiting for 2 forever”) is in fact consistent with a sophisticated analysis of the game (i.e., while minimizing the expected duration may call for a token on 2, it may nevertheless lower the chances of winning because of the high probability of a long sequence of rolls without 2). Notably, the empirical conclusion of a student (based on one game) that “9 and 10 are most likely” was discredited by Adam based on the frequency table, signaling a clear hierarchy – empirical reasoning can be trumped by deductive reasoning.

6 Results (RQ2): mathematicians’ perspective on the use of RCG in SM

In the M-Cubed meeting, after RCG was introduced to the participants and before viewing the recorded lesson, M1 immediately recognized that the mathematical complexity of the game was daunting:

I don’t understand, what is the optimal strategy? This is a super complex game if you want an optimal strategy. I think that without a computer I don’t stand a chance calculating an optimal strategy […] It’s possible that there is a strategy with a slightly better expectation and slightly worse variance, and another strategy that is the other way around. It’s a very sensitive game.Footnote 3

Here, M1’s reference to expectation and variance suggests that he has started mathematizing the game and may be conceptualizing configurations, or strategies, in terms of random variables, possibly the number of turns needed for a configuration to be emptied. Furthermore, M1 quickly recognizes the important and subtle role that variance plays in RCG, which suggests that there is more to RCG than minimizing the expected number of turns needed to clear the board. Having realized the complexity of RCG, the mathematicians’ immediate reaction was to dismiss it as unsuitable for SM education. They rejected the idea that it can be used to demonstrate the strengths and applicability of probability. For example, M2 argued that mathematical intuitions cannot be endorsed by the teachers if the teachers cannot justify to themselves and explain to the students why these intuitions are true. He concluded that since he does not feel he has ‘good intuitions’ about the RCG, he would not take it to class. The mathematicians also rejected the idea that RCG can be used to convey an experimental approach to probability, arguing that the difference between strategies can be very subtle and would not be discernable in a few rounds, while pointing out that playing the game to conclusion may take a long time.

The mathematicians also criticized Adam’s use of the word ‘bell’ and the ‘bell’ diagram he drew on the board (Fig. 4), noting that the diagram should be in fact triangular. They also criticized the connections Adam made between the symmetry and monotonicity of the bar chart (Fig. 3) and these properties in an optimal RCG strategy, arguing that it is not clear what sort of reasoning could substantiate these connections. M1 went so far as to claim that the table and histogram that Adam drew on the board are not really connected to playing RCG, and at best could provide heuristics for devising strategies. He questioned whether deriving such heuristics is related to learning probability, or even if it could be considered doing mathematics. In addition, M1 suggested that the insufficient reasoning in the recorded lesson may lead students to develop a misconception regarding probability games: the optimal strategy is always given by a distribution histogram. More generally, this may reinforce a known tendency of students to emphasize the role of central tendency over the role of variability when making predictions (cf. Canada and Goering, 2008). Finally, M1 also voiced a concern that students might independently run simulations of RCG and discover that distributing tokens in accordance with the distribution of dice outcomes is in fact not optimal, leading them to doubt probability as a valid theory for analyzing and predicting outcomes.

7 Results (RQ3): rapprochement of perspectives on the river crossing game

Following the initial recognition of the complexity and subtlety of the game, the mathematicians’ stance was generally that they would not bring to class a game they cannot understand, because it is likely to place them in situations where they may accidentally condone invalid statements and potentially entrench misconceptions. Moreover, the mathematicians criticized the analysis of RCG in Adam’s lesson, framing it as “not mathematics” and “not really related to probability”. Teachers, on the other hand, were generally reluctant to fully dismiss the didactic potential of the game. They stressed that in their experience, which in some cases spanned more than a decade, the game is not only fun, but it also engages students with important probability ideas. However, the teachers found it difficult at first to respond to the mathematicians’ query about what mathematics is addressed in the discussion of RCG, beyond suggesting that it can foster useful intuitions and connect abstract mathematical ideas with and concrete real-life situations. Conversely, the mathematicians could not specify practically harmful pedagogical implications of using RCG as a school activity that the teachers found convincing. For example, the teachers flatly rejected the possibility, suggested by M1, that students may run simulations at home, find out that claims about RCG were not true, and end up doubting probability, saying that nothing like this has ever happened in their many years of experience. Consequently, the mathematicians conceded that students were unlikely to suffer consequences from invalid statements being voiced. The tension between the mathematicians and teachers’ perspectives pushed both sides to explore the didactic potential of the game in a process of boundary-crossing, reflecting on their different – sometimes conflicting – perspectives, making them explicit, or adopting new perspectives. This exploration led eventually to four concrete suggestions for using RCG in SM education that drew on the diverse perspectives and expertise in the group.

One suggestion, addressed on several occasions both by mathematicians and teachers, was to clearly and explicitly distinguish between two kinds of mathematical discourse – one that is precise and endorsable, and one that is more colloquial, which in one case elicited the response in the title of this article – “but this is not mathematics!”. Teachers suggested that the more colloquial discussion can encourage student intuitions without endorsing false claims. For example, T3 argued that the recorded lesson contains a “mathematical part” that has “graphs and tables and calculations”, and another part that is “non-mathematical”, which he considered “much closer to popular-science”, and that the students cannot recognize on their own when the rules change. He therefore suggested that before discussing intuitive connections between the distribution of the sum of two dice, the teacher convey to the students that “we’re now taking a break for a moment from mathematics”, thus signaling a shift to a different kind of reasoning. Similarly, M1 noted that “the histogram is mathematically precise, just not clearly directed to the game”, and proposed to explicitly refrain from talking about connections between the histogram and RCG in terms of conclusions, and instead use terms such as intuitions, heuristics, and educated guesses.

A second suggestion was to extend the repertoire of examples and engage students with games that invite counter intuitions, so as to preempt misconceptions. For example, concerned that students may deduce that distribution histograms always provide winning strategies, M1 proposed the following problem, which he has been using in tertiary probability courses:

A fair dice with numbers 1,2,3,4,5,5 (5 appears twice) is thrown six times. If you correctly guess the sequence of outcome, you win a prize. What should you guess?

M1 explained that typically students’ intuition is influenced by the distribution histogram and they may guess 1,2,3,4,5,5, which can easily be shown to be 16 times less likely than 5,5,5,5,5,5.

A third suggestion was to carefully explore, before brining RCG to class, how some heuristics for playing RCG could be highlighted and discussed while hiding complexities, thus avoiding endorsement (even if only implicit) of false statements. The following exchange exemplifies an exploration of how to explain in a SM classroom the sense in symmetric configurations:

T3: You can see [in the videotaped lesson] that almost all the students proposed symmetric configurations […] we can discuss this […] ask why almost everyone did this.

M1: Why is symmetry good? How do you explain this?

T3: Because the histogram is symmetric. The probability for 2 and 12 is the same, so if you place a certain number of tokens on 2, then it would be strange not to put the same amount on 12.

M1: But by the same reasoning, I can freely move tokens from 6 to 8 because they have the same probability. But this is a very bad idea. How do you explain that?

T3: I would say: Imagine now I have only 6 and 8, and say I have 6 tokens, what do I do? And, with a bit of hand waving, it is very reasonable to conclude [that there should be an equal number of tokens on 6 and 8].

Here, M1 identified a weakness in T3’s line of reasoning – while it sounds reasonable, taking the interchangeability of 2 and 12 to an extreme shows that they are not completely interchangeable even though they have the same probability. Thus, one cannot simply infer that 2 and 12 should receive the same number of tokens. While it may be unlikely that SM students would propose this counter argument or even question T3’s reasoning, M1’s response did push T3 to revisit and refine his argument, restricting it to a simpler case, albeit without providing a justification.

A fourth suggestion was that at some point, students may use computer simulations to empirically test and refine intuitions and statistical inferences. When proposing this approach, M2 noted that he can reason for the mathematical validity of drawing conclusions from simulations that run the game thousands of time, but he is not sure about the ‘pedagogical legitimacy’ of such an approach. T4 responded by claiming that students, at least at first, need something concrete they can feel, and that “a button someone presses would not have the same effect”. Moreover, one teacher also suggested that in the context of RCG, helping students gain confidence in their mathematical intuitions may be more important than refining their intuitions, particularly if they lack tools for making sense of these refinements. However, she noted there could be various pedagogical advantages for seeing results of computer simulations after students have developed and discussed their intuitions.

These four suggestions emerged in interactions between mathematicians and teachers, drawing on the rigor of TM and on pedagogical realities of SM. They present classroom teaching connections between TM and SM not as mathematical knowledge prescribing pedagogy, but rather as a process in which the weaving of ideas grounded in TM and in the pedagogy of SM can suggest courses of action that are pedagogically productive without sacrificing mathematical validity.

8 Discussion

The point of departure for our research was that the affordances of tertiary mathematics (TM) for the preparation and professional development of secondary mathematics (SM) teachers are not well understood. As discussed in the Sect. 1, prior research has shown that identifying connections between TM and SM teaching and utilizing these connections in practice are not straightforward, as they entail subtle and nuanced weaving of knowledge of TM and knowledge of SM teaching. We have proposed boundary encounters between mathematicians and SM teachers as a methodology for generating and studying such weaving. In the current research we have investigated how TM can be utilized to inform SM teaching in an extreme case, in the sense that the knowledge and approaches needed to analyze and understand it mathematically extend well beyond the undergraduate level, which raises the questions whether TM is relevant in this case, and how TM can inform teachers about using the River Crossing Game (RCG) in class.

The M-Cubed discussion about RCG demonstrates how TM may inform SM instruction. The mathematicians quickly recognized that mathematizing RCG is not easy, and that they are unsure about basic properties of RCG, for example whether there exist optimal strategies, and if so, what they are. Accordingly, the immediate reaction of the mathematicians was to dismiss RCG, oriented by a strong sentiment that without proper mathematical understanding, using it in class would likely lead to endorsement of statements that the students and the teachers cannot substantiate, and that might even be incorrect. The mathematicians’ reaction represents the most obvious affordance of TM for SM teaching in the context of RCG, namely that TM knowledge may highlight for teachers their implicit assumptions and gaps in their own reasoning. This affordance was evident, for example, when initially some of the teachers considered the optimality of ‘bell shaped’ configurations to be an immediate consequence of the distribution of the sum of two dice and felt no need for any further justification. However, through the exchanges with the mathematicians who were hesitant to make claims about RCG, the teachers became more self-conscious and careful in their reasoning. Thus, our analysis shows how even in extreme cases such as RCG, TM may still support SM teachers in orienting instruction in line with disciplinary values and practices (Ball & Bass, 2009; CBMS, 2012).

However, our analysis also suggests why teachers’ TM may remain a part of their mathematical knowledge, distinct and separated from their knowledge for teaching. The mathematicians’ sentiment did not seem to resonate as strongly with teachers, who were reluctant to discredit a game that they considered, based on many years of experience, to be pedagogically productive. While they were surprised that many of their intuitions were imprecise, they still considered students’ engagement with these intuitions worthwhile. Notably, the mathematicians’ argument that their ‘analysis’ of RCG from a probabilistic perspective is not faithful to the mathematics did not in itself discredit the game in the teachers’ eyes, who asked specifically how this could be harmful for their students, suggesting that teachers and the mathematicians may interpret differently the professional obligations of a teacher to the discipline (Herbst & Chazan, 2020). As such, the mathematicians’ perspective did not appear to be useful for the teachers, since it provided very limited insight about how RCG can be used in class. This demonstrates why teachers may find knowledge of TM to be irrelevant, or at least unapplicable, with respect to their actual needs (if it discredits an activity they consider productive), and why the potential affordances of TM for SM teaching that are discussed in the literature are not realized in practice (TM does not reconcile conflict between mathematical and pedagogical considerations).

Our analysis shows that rapprochement of the teachers and mathematicians’ perspectives began when the focus shifted from whether or not to use the game to how to use it responsibly and productively, and for what purposes. The ground for this rapprochement was laid through identification of the teachers and mathematicians’ different perspectives. On the teachers’ side, this entailed exploring and making explicit specific and concrete opportunities for learning that RCG affords, beyond more generic affordances such as being fun, engaging, and highly motivating. As our analysis in Sect. 5 shows, the opportunities proposed by the teachers spanned across all five features of probability discussed by Even and Kvatinsky (2010). The mathematicians’ identification of their perspectives focused on deriving practical pedagogical implications from their disciplinary values, towards explaining not only what is mathematically important but also specifying why, as discussed in Sect. 6. Identification of concrete mathematical ideas and pedagogical implications enabled the mathematicians and teachers to try coordinating their perspectives, and finally to engage in hybridization – searching for ways to leverage opportunities the game provides, while lowering the risk of cultivating unintended and unproductive mathematical ideas and practices.

Four concrete suggestions for using RCG in class (elaborated in Sect. 7) started to emerge through exchanges between the teachers and the mathematicians. Three are cases of hybridization in the sense that they reconcile teachers’ perspective – using RCG to invite, test and refine intuitions – with the mathematicians’ reluctance to endorse statements that cannot be substantiated. The first was merely to distinguish between ‘disciplinary’ discussions, where the expectation is that claims can be mathematized and substantiated, and ‘colloquial’ discussions. Whereas both T3 and M1 proposed such distinction, neither stated explicit rules for this ‘colloquial’ discourse. Yet, based on findings of Sect. 5, we gather that mathematical narratives should be plausible, though we may be unable to formalize and prove them, and in fact we may be uncertain of their correctness. This suggestion can be seen as a basic kind of hybridization, as it does not seem to draw in substantial ways on mathematicians and teachers’ diverse expertise.

More substantial hybridization is evident in the second and third suggestions, where two new pedagogical moves were co-developed. The second suggestion was to complement RCG with additional chance games chosen specifically to challenge and refine the (flawed) intuitions RCG invites, thus “defusing” the misconceptions that RCG might encourage. The third suggestion is to refine the reasoning around RCG, for example by drawing attention to special cases (less pegs, less tokens) where the reasoning is more straightforward, or at least less inaccurate. Hybridization in these two suggestions is more substantial in the sense they were formulated and refined through interactions between mathematicians and teachers, while drawing on both sides’ expertise. This indicates that implementing these two suggestions may require interweaving TM and knowledge for teaching SM.

The fourth suggestion is subtle. It relies on experimental evidence, and the basic unreliability of empirical data was, from the teachers’ perspective, one of the mathematical ideas that discussing RCG was considered to afford. Yet empirical evidence produced from a very large number of repetitions can be useful, even in disciplinary research, to test and refine hypotheses, if not to substantiate them. Notably, not all mathematicians considered the use of simulations in relation to RCG to be appropriate, pointing to the lack of probability tools that can be used to estimate how (un)likely it is that empirical results will diverge significantly from what is theoretically anticipated. The teachers were also hesitant about moving from the concreteness of playing RCG to computer simulations, and about undermining students’ confidence in their mathematical intuition, which may be “contradicted” by simulation. The combination of enthusiasm and hesitation from both mathematicians and teachers indicates that this suggestion needs to be further refined, drawing on both TM knowledge and knowledge for teaching SM to explain how inferences about RCG can be tested through computer simulations in a way that maintain mathematical integrity while maintaining alignment with the SM teachers’ goals and obligations.

Intuition is a central theme in probability education, and research has shown that SM students and SM teachers rely heavily on misconceptions and misleading experiences when reasoning about probability (Stohl, 2005; Batanero et al., 2016). Realizing this may lead SM teachers and mathematicians to hold that intuitions should be avoided in SM probability lessons, which can be discouraging for learners, and seems to go against the generally accepted maxim that sense-making is an essential aspect of doing and learning mathematics. One may claim, as some of the M-Cubed teachers implied, that intuitions need not be mathematically valid in order to be didactically productive, but this is controversial. Our analysis has illustrated four approaches for addressing intuitions in probability lessons more responsibly that TM knowledge can support: recognize when intuitions cannot be mathematized and justified, and explicitly frame their discussion as ‘colloquial’; recognize when students’ reasoning might be relying on misconceptions or misleading experiences, and propose counterexamples; direct discussion towards intuitions that can be explained and avoid endorsing intuitions that cannot; use computer simulations to test and refine intuitions. We stress that these approaches do not dictate a single course of action – with respect to RCG or teaching SM probability in general. In fact, we do not even claim that they offer a course of action that all teachers and mathematicians would condone. Rather, they propose ways in which individual teachers can draw on TM mathematics to attend to achieve pedagogical goals without compromising mathematical validity. Thus, our use of the term rapprochement draws inspiration from literature on boundary crossing, where members of different communities (i.e., mathematicians and teachers) can collaborate – to gain insight on teaching SM probability – without achieving consensus.

The above suggestions were derived specifically for the use of RCG in SM education, yet they offer insight with regard to our overarching question of how TM can be utilized to inform SM teaching, of probability and in general. In particular, our analysis suggests that TM knowledge can be utilized to inform teaching even in situations where content connections (Wasserman, 2018) are not attainable, namely, where teachers cannot apply their TM knowledge to sufficiently understand the subject matter. In fact, even the mathematicians, by their own admission, did not fully understand the nuances of RCG. Thus, our study shows that content connections may not be necessary for realizing the potential affordances of TM for SM teaching.

We now come to consider implications of this research for teacher education and professional development. Our study adds another layer to the discussion about the TM mathematics that SM teachers should learn and how it might be learned, providing further empirical evidence about the particular needs of SM teachers compared to future STEM professionals (Biza et al., 2022). Prior research has shown that providing concrete and elaborated examples of how TM may inform SM teaching is crucial for motivating teachers to learn TM, and for supporting teachers in making their own personal connections between TM and SM that might later inform their teaching (Wasserman, & McGuffey, 2021). This study suggests how teachers may benefit from a TM perspective on the SM content that they teach, yet this perspective needs to be sublimated into productive didactic activity. The case of RCG illustrates how this sublimation may take place through discussions with mathematicians. Of course, this is not a viable model at scale, however, discussions that take place in the M-Cubed setting can provide inspiration for creating opportunities for similar discussions to take place in more conventional teacher preparation and PD settings. Eventually, teachers who hold such discussions with others may come to hold similar discussion with themselves.