Collaborative problem solving in a choice-affluent environment

Abstract

For too long, problem solving has been studied as a solitary and isolated activity where individuals make progress based on personal resources acquired from knowledge, previous experiences or moments of illumination; however, in society, and indeed amongst mathematicians, problem solving can be highly collaborative and occurs in spaces where external resources (technology, the internet, social connections etc.) are abundant. In this paper, we present research into what problem solving looks like in classrooms when it is done collaboratively with access to resources that go beyond the knowledge and past experiences of the individual and even the group. Results indicate that in such choice-affluent environments, students will seek out new resources either when their collective or individual resources run low and will do so either by looking or discussing with others. This manuscript also offers a new methodological tool in the form of, what we call, gaze-dialogue transcripts for documenting such resource acquisition.

1 Introduction

When George Pólya published his seminal book, How to Solve It (1949), it was met with huge acclaim by the mathematical community. Mathematicians felt, by and large, that Pólya had captured the essence of what it meant to solve a problem and, in doing so, he had found a way to describe their work. Pólya had written a description of problem solving. But had he really?

A huge part of the problem solving that mathematicians do is collaborative in nature. And not only is it collaborative, but that collaboration is situated within a larger milieu full of resources—other researchers, research articles, technology, media, and so on. Even in Pólya’s time, resources were ubiquitous in the life and work of mathematicians. Yet, How to Solve It is entirely bereft of any mention of the use of resources beyond the solver’s own experiences, abilities, and memories. This focus on problem solving as a solitary activity, bereft of outside resources, shaped much of the next 50 years of research on problem solving, in general, and how students, in particular, solve problems. In this paper we look more closely at problem solving as a collaborative enterprise set within a resource rich environment with the aim to describe what authentic problem solving more accurately looks like.

1.1 Problem solving

Resnick and Glaser (1976) define a problem as being something that you do not have the experience to solve—the situation is new to you. Problem solving then, is what we do when faced with a situation that is unfamiliar to us and for which we do not know how to proceed—problem solving is what we do when we don't know what to do. So, how then do we solve it? If we don't know what to do, what do we do? The answer is, we follow the principles of design.

In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2019). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964; Schön 1987), to produce possible options that will lead towards a solution of the problem (Poincaré 1952). These options are then examined through a process of conscious evaluations (Dewey 1933) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known. And what is known comes from a repertoire of past experience.

This idea of a repertoire of past experience forms the foundation of the four-step process laid out in George Pólya’s book, How to Solve It (1949)—understand the problem, devise a plan, execute the plan, look back—in general and within the heuristics of the devise a plan step in particular.

2. Devise a plan

• Can you find an analogous problem that you can solve or have already solved?

• Can you find a more general problem that you can solve or have already solved?

• Can you find a more specialized version of the problem that you can solve or have already solved?

• Can you find a related problem that you can solve or have already solved?

• Can you change the problem into a problem that you can solve or have already solved?

• Can you find a subproblem that you can solve or have already solved?

• Can you add some new element to the problem to create a problem that you can solve or have already solved?

• Can you decompose the problem and recombine it into a problem that you can solve or have already solved?

• Can you work backwards?

• Can you draw a picture?

The first eight of these heuristics rely a great deal on your familiarity with analogous, related, generalized, specialized, and varied problems and, as such, is an explicit manifestation of relying on a repertoire of past experience. Even Pólya's fourth step, looking back, is a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections "in memory to previously acquired knowledge [..] and further establishes knowledge in long-term memory that may be elaborated in later problem-solving encounters" (Silver 1982, p. 20). That is, looking back is a forward-looking investment into future problem solving encounters. It is a process of building up the repertoire of past experience and prior knowledge for use in the future. Prior knowledge from, and prior experience with, earlier problem solving encounters influences the problem solver's understanding of the problem at hand as well as the choice of heuristics that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem (Mayer 1982; Schoenfeld 1982; Silver 1982).

Pólya’s heuristics, and problem solvers’ abilities to use them, was the basis of Schoenfeld's book, Mathematical Problem Solving (1985). Starting with the premise that knowledge and past experience are the resources that students bring to the problem, Schoenfeld found that for expert problem solvers—those with more resources, like mathematicians—Pólya’s heuristics were an accurate reflection of how they solved problems individually. However, for novice problem solvers—like students—it didn't work. Not only were their resources more limited, but also, their ability to recall, transfer and invoke their resources was not as refined.

In short, Schoenfeld (1985) found that a person’s resources emerge from a repertoire of past experiences and are integral to their ability to solve problems. In essence, what Schoenfeld (1985) found was that in order to be good at problem solving, someone has to have a lot of resources. And to get a lot of resources, they have to have successfully done a lot of problem solving. Where then does that leave novice problem solvers (students)? How can they build up a repertoire of past experiences when they do not have the experience to be successful at problem solving?

Even in cases where there was evidence that an appropriate related problem existed in a students' repertoire, success is not assured. Research on transfer and analogical reasoning (English 1998, 1997; Novick 1995, 1990, 1988; Novick and Holyoak 1991) has shown that heuristics that rely on relating the problem at hand to a problem from their past experience are not overly effective for novice problem solvers and that students had difficulty recognizing the relationship and, if they did, had difficulty mapping the relevant attributes to the problem at hand.

Yet, good problem solvers do develop. Understanding how this happens, and how to help it along, has been the pursuit of research in problem solving for the last 35 years. Much of this research has focused on heuristics. Schoenfeld (1985) showed quite convincingly that Pólya’s heuristics were not effective for novice problem solvers and the research on transfer and analogical reasoning has confirmed this. As such, other heuristics have emerged—heuristics that go beyond the principles of design.

One such set of heuristics come from David Perkins, who co-directed Project Zero at the Harvard Graduate School of Education along with Martin Gardner from 1971 to 2000. Perkins (2000) put forth a set of four heuristics that deal with situations in which the solver has become stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya (1949)]. Mason, Burton, and Stacey, in their book Thinking Mathematically (1982) likewise put forth a set of heuristics that go beyond design, as do the heuristics of Gestalt psychology (Koestler 1964). Unlike Pólya’s heuristics, however, these heuristics require the solver to go beyond the logical mechanics of design and to rely, in some way or another, on extra-logical processes such as illumination, insight, and creativity in order to go beyond the resources that their repertoire of past experience afford them.

Each of these aforementioned heuristics is predicated on the assumption that problem solving is an individual and isolated activity wherein the only resources that are available to the problem solver are knowledge and past experience. This assumption could have its roots either in the theoretical focus of cognitive psychology or the methodological convenience that is afforded by studying individuals. Or it could have emerged as a result of the fact that Pólya, in his seminal and seemingly definitive work on problem solving, positioned problem solving as an individual and isolated activity. Regardless, not only heuristic research in particular, but also problem solving research in general was dominated by this assumption for much of the first 50 years after the publication of How to Solve It (Pólya 1949).

1.2 Problem solving in collaboration

With the publication of the NCTM Principles and Standards (NCTM 2000), however, the assumption that problem solving, and mathematical work in general, was a solitary activity began to fall away. With its focus on collaboration, this publication marked a shift not only in the teaching and learning of mathematics, but also in the research into the teaching and learning of mathematics. And along with it a shift in the view that problem solving needed to be a solitary activity. Emerging from this research was the realization that collaboration expanded not only the knowledge base available to a group of students, but also the repertoire of past experiences that can be drawn on when faced with a novel problem.

Although problem solving had shifted from being a solitary activity to a collaborative activity, much of this research was still predicated on the assumption that problem solving was still an isolated activity (see for example Ryve 2006) wherein the group was not given access to other groups. That is, resources available to a group of students trying to solve a problem was limited to the union of their collective knowledge and collective experiences within the confines of the group. Although collaborative problem solving within a group is a closer approximation to how problem solving looks outside of the bounds of a classroom, the limitation of additional resources from outside the group artificially constrains the activity.

1.3 Choice-affluent environments

Koichu (2018) sheds this assumption of problem solving as isolated in his research into choice-affluent environments. Building from Lester’s (2013) assertion that research on mathematical problem-solving has been slow and insufficient, remaining largely atheoretical, Koichu (2018) decides to follow Schoenfeld’s (2013) recommendation to move from a framework for studying problem-solving to a model that would describe a theoretical structure for problem-solving. This theoretical structure would connect our understanding of how problem-solving occurs with our understanding of how to improve problem-solving in classrooms.

In doing so, Koichu proposes a model that combines Pólya’s (1949) and Schoenfeld’s (1992) heuristic research with Mason’s (1989, 2008, 2010) shifts of attention. This model of mathematical problem solving asserts that in any problem solving experience, key ideas emerge from a solver’s shifts of attention between the solver’s resources, peers, and external sources of knowledge. In this article, Koichu presents the idea that the solution process is made up of a series of choices that the problem solver negotiates. The model is referred to as the Shifts and Choices Model (SCM) and relies on three premises:

• 1. Even when a problem is solved in collaboration, it has a situated solver: an individual who invents and eventually shares its key solution idea.

• 2. A key solution idea can be invented by a situational solver as a shift of attention in a sequence of his or her shifts of attention when coping with the problem.

• 3. Generally speaking, a solver’s pathway of shifts of attention is stipulated by choices the solver is empowered to make and by enacting the following types of resources:

  • a. Individual resources

  • b. Interaction with peer solvers who do not know the solution and struggle in their own ways with the problem or attempt to solve it together, and

  • c. Interaction with a source of knowledge about the solution or its parts, such as a textbook, an internet resource, a teacher, or a classmate who has already found the solution but is not yet disclosing it. (p. 310–311)

Despite stating that all problems are solved by a situated solver, Koichu introduces the possibility of the situated solver working with others as a result of shifts of attention and choices made in the solving process. In the center of this model, we see the situated solver shifting attention among individual resources, resources from peers and resources from a solution source. These shifts are made in an environment of choices a problem solver is empowered to make: “Among endless conscious and unconscious choices that individuals face when solving problems … the model takes into account the following: a choice of a challenge to be dealt with, a choice of schemata for dealing with a challenge, a choice of mode of interaction, a choice of extent of collaboration, and a choice of an agent to learn from” (p. 309). Koichu presents a model for problem-solving that describes how solvers shift attention from individual heuristics, interaction with peers, and interaction with a solution source when resources are insufficient. Solvers best make these shifts in environments that are affluent in choices.

With Koichu’s recent work on problem solving in choice-affluent environments being mostly a theoretical discussion, we are left with wondering what these environments may look like in practice and how might students navigate the problem solving process within such environments. The choice-affluent environment was presented in 2018 as a possible way to imagine and study collaborative problem solving in action, and there has not yet been any research conducted using this model. In the research presented here we combine Koichu’s (2018) ideas of SCM with Schoenfeld’s (1985) ideas of resources in order to better understand the types of shifts of attention students make in choice-affluent environments. In particular, we want to understand when and how students, working in groups, shift their attention and how these shifts are related to the resources they have available to them at that moment? Clearly, resources are a fundamental unit of interest in this study, so we should be clear on what precisely we mean when using this word. For our purposes, resources can be thought of as ideas. Sometimes, these ideas come from past experiences, abilities or memories (inside resources), but many times these ideas come from outside sources, such as other’s work, conversations, research or even inspiration.

2 Methodology

To answer this question, we needed to study students in a choice-affluent environment. Koichu (2018) referenced a thinking classroom (Liljedahl 2020) as an example of a choice-affluent environment. In the research we present here, we too are looking at problem solving within a thinking classroom context.

2.1 The thinking classroom

A thinking classroom is a teaching framework developed by Peter Liljedahl (2020) in response to the realization that the institutionally normative practices of school are promoting, in both explicit and implicit ways, non-thinking behaviors such as mimicking among students (Liljedahl and Allan 2013). These normative structures that permeate classrooms around the world are so entrenched that they transcend the idea of classroom norms (Cobb et al. 1991; Yackel and Cobb 1996) and can only be described as institutional norms (Liljedahl 2020)—norms that have extended beyond the classroom, even the school building, and have become ensconced in the very institution of school. Much of how classrooms look and much of what happens in them today is guided by these institutional norms—norms which have not changed since the inception of an industrial-age model of public education. Yes, desks look different now, and we have gone from blackboards to greenboards to whiteboards to smartboards, but students are still sitting, and teachers are still standing. Although there have been a lot of innovations in assessment, technology, and pedagogy, much of the foundational structure of school remain the same.

Over the course of 15 years, and through the conducting of thousands of micro-experiments with over 400 practicing teachers, Liljedahl emerged a series of 14 practices that break away from the aforementioned institutional normative ways of teaching that have dominated education for the last 150 years:

• 1. What are the types of tasks used?

• 2. How are collaborative groups formed?

• 3. Where do students work?

• 4. How is the furniture arranged?

• 5. How are questions answered?

• 6. When, where, and how are tasks given?

• 7. What does homework look like?

• 8. How is student autonomy fostered?

• 9. How are hints and extensions used?

• 10. How is a lesson consolidated?

• 11. How do students take notes?

• 12. What is chosen to evaluate?

• 13. How is formative assessment used?

• 14. How is grading done?

Although each of Liljedahl’s 14 practices, on their own and in concert, have been empirically shown to increase student thinking in the classroom (Liljedahl 2020, 2014) the defining qualities of a thinking classroom is that (1) students solve thinking tasks (2) in visibly random groups (3) on vertical non-permanent surfaces.

Thinking tasks If we want our students to think, we need to give them something to think about—something that will not only require thinking but will also encourage thinking. In mathematics, this comes in the form of a problem solving task, and having the right task is important. Liljedahl’s research (2020) revealed that when first starting to build a thinking classroom it is important that these tasks are highly engaging non-curricular problem solving tasks. As the culture of thinking begins to develop, there is a transition to using curriculum tasks. The goal of thinking classrooms is not to get students to think about engaging with non-curricular problem solving tasks day in and day out—that turned out to be rather easy (Liljedahl, 2020). Rather, the goal is to get more students thinking, and thinking for longer periods of time, within the context of curriculum.

Visibly random groups We know from research that student collaboration is an important aspect of classroom practice because when it functions as intended, it has a powerful impact on learning (Edwards and Jones 2003; Hattie 2009; Slavin 1996). How groups have traditionally been formed, however, makes it very difficult to achieve the powerful learning we know is possible. Whether students are grouped strategically (Dweck and Leggett 1988; Hatano 1988; Jansen 2006) or students are allowed to form their own groups (Urdan and Maehr 1995), Liljedahl (2020) found that 80% of students enter these groups with the mindset that, within this group, their job is not to think. However, when frequent and visibly random groupings were formed, within six weeks 100% of students entered their group with the mindset that they were not only going to think, but that they were going to contribute. In addition, frequent and visible random groupings was shown to break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics (Liljedahl 2014).

Vertical non-permanent surfaces One of the most enduring institutional norms that exists in mathematics classrooms is students sitting at their desks (or tables) and writing in their notebooks. This turned out to be the workspace least conducive to thinking. What emerged as optimal from Liljedahl’s research (2020) was to have the students standing and working on vertical non-permanent surfaces (VNPS) such as whiteboards, blackboards, or windows. It did not matter what the surface was, as long as it was vertical and erasable (non-permanent). The fact that it was non-permanent promoted more risk-taking and the fact that it was vertical prevented students from disengaging. Taken together, having students work, in their random groups, on VNPS had a massive impact on transforming previously passive learning spaces into active thinking spaces where students think, and keep thinking, for upwards of 60 min.

This last practice (VNPS) is the specific practice that Koichu (2018) identified as allowing thinking classrooms to be a choice-affluent environment.

In such a classroom, students are given time to solve mathematical problems in a small group while standing and writing on the vertical surfaces instead of sitting and writing in their notebooks. Accordingly, students all have access not only to the content of the whiteboard of their own small groups but can also see what is written on the other groups’ whiteboards. (p. 320).

Liljedahl (2014) also argues that the frequent randomization of groups creates a porous boundary between groups, across which ideas are more likely to flow. The visible nature of the vertical whiteboard coupled with the close and fluid interactions of the students affords the potential for ideas, hunches, queries and representations to move freely through the room. In a thinking classroom, students are noisily exchanging ideas within groups and between groups, conjectures and diagrams evolve on the whiteboards around the room and shouts of joy can be heard as progress is made. Thinking classrooms are choice-affluent environments.

2.2 Participants

The data for the study presented here comes from a series of video recordings of grade 12 (ages 16–18) students working through curricular problems in a thinking classroom at a high school in western Canada. The course they are enrolled in is Pre-Calculus 12 which is in the academic pathway of mathematics, meaning that most students are planning on continuing their studies at a post-secondary institution (tertiary) upon graduation from high school. Most students in this class are enrolled in Calculus 12 and taking this course concurrently, some students are in their grade 11 year and planning on taking Calculus 12 in their following year, and a small number of students have no plans of taking Calculus 12 in high school. There are also a small group of international students in these classes. In this school district, international students are invited to participate in a Canadian high school experience for a half-year or a full-year. These International students come with a variety of English language competency.

2.3 The activity

The task that they have been asked to work on is the following:

Sketch f(x) = x² − 2 and its inverse. Restrict the domain of f(x) so the inverse is a function.

Students in this cohort have plenty of experience with quadratic functions. Inverse functions were introduced in the previous class, but this lesson is the first time students are faced with contemplating the domain of a function and how the domain relates to the inverse. In the previous lesson, students were finding inverses of linear functions both algebraically and graphically. Students understand the inverse of a function, f(x), as a new function that recovers the input when provided the output. This is achieved algebraically by replacing the input variable, x, with the output variable, y, and vice versa. The graph of an inverse is achieved in a similar approach: the x-coordinate is switched with the y-coordinate, thus switching the role of the input variable with the output variable. In this lesson, students are now, for the first time, being faced with the prospect of the inverse not retaining the characteristics necessary to be considered a function.

In short, this task is brand new to the students. They have some resources available to them in the context of some prior experiences with inverse and domain. But the combination of these two topics is novel and as such meets the definition of a problem solving task (Resnick and Glaser 1976) and a thinking task (Liljedahl 2020).

2.4 Method

Because we were trying to get a sense of how resources were related to shifts of attention within the choice-affluent problem solving environment of thinking classrooms, we chose to capture video data of several groups of students working on vertical whiteboards in close proximity to each other. Because shifts of attention within the thinking classroom can include intra- and inter-group interactions, we set our unit of analysis at the scale of when students are looking, where they are looking, and who they are dialoguing with.

The video clips being analyzed for this article each captures two adjacent groups of students as they work through the aforementioned task to restrict the domain of a quadratic function so that the inverse is still a function (see Fig. 1). Group size in this class is mostly three students; however, in some instances a group is made up of two students.

Fig. 1

Students A, C, D, E and F engaged in a task at the whiteboards

These data were then transcribed into a gaze-dialogue transcript that showed where a student was looking and who they were talking to. This form of transcription was inspired by Liljedahl and Andrà (2014) who added gaze arrows to interactive flowchart transcription (Ryve 2006; Sfard and Kieran 2001) as a way of documenting what and who students are attending to. The gaze-dialogue transcript is recorded in seven columns (see Fig. 2)—one for each participant lettered A through F representing each student in order from left to right and a center column intended to divide the two groups. The center column is also used for the teacher if and when the teacher interacted with either of the groups. For the sake of the analysis, group 1 will refer to students A, B and C, and group 2 will refer to students D, E and F. In each of these columns a small description is written describing what each participant is attending to within the current interval and an asterisk is used to indicate who is currently holding the whiteboard pen (there is only one pen per group). Solid arrows ( →) are used to indicate a directed conversation among participants. A solid arrow with a point on one end ( →) represents a student speaking to another student and a solid arrow with points on both ends ( ↔) shows when both students are speaking to each other. For example, in Fig. 2, one can see that student C is speaking directly to student A while writing an equation with the whiteboard pen. In the other group, students D and E are participating in a conversation and student F also speaks directly to student E.

Fig. 2

Excerpt from a gaze-dialogue transcript

Gazes are represented by dashed arrows (⤏) to encode every time a participant is attending to something outside of their group. If the dashed arrow is pointed into the center column, this indicates an individual looking at the board of the other group within the video. If the dashed arrow is pointed to either side of the table, this indicates an individual attending to a whiteboard that is not captured within the video frame as there are eight other groups working at eight other whiteboards around the room. So, for example, in Fig. 2, students D and E are attending to something written on another group’s whiteboard that is not within the viewing window.

To capture shifts in attention the transcript breaks each recording into equal intervals. After some experimentation with multiple transcriptions across a variety of interval lengths, 10 s was chosen for the interval length. We found that when the interval was shorter, there were many intervals with little to no observable dialogue or shifts of attention. When the interval was longer, we found that too much transpired within the interval to effectively document within the gaze-dialogue transcript.

Within these intervals, we coded groups that were working as having adequate resources (Schoenfeld 1985) in the form of knowledge and/or repertoire of past experience. If a group was not working within an interval we coded that group as having depleted their resources. In this coding, we assumed that individuals who were actively participating within a group that was working had access to the collective resources of that group. Likewise, we assumed that individuals that were not actively participating both lacked the individual resources to solve the problem and did not have access to the collective resources of the group.

The transcripts of these video clips were then analyzed through the dual lenses of shifts of attention and resources.

3 Results

In what follows we present the transcripts of the two of these video recordings interwoven with our analysis. We chose only two recordings for reasons of brevity and we chose these two recordings because they present the greatest variety of shifts of attention.

3.1 Recording #1

The first video recording begins at about 1 min into the task, and all students can be seen contributing towards a solution. In both groups, two people appear to be taking the lead (see Fig. 3) as indicated by the solid conversation arrows between students A and C in group 1 and students D and E in group 2. The whiteboard marker also shifts between these two participants at the 1-min mark and then shifts back again, indicating that these two students in each group are more participatory in the collaboration than the third person. This is especially evident in group 1 where student B has completely stepped out of the frame of the camera. There is a moment in group 1 (1:20) where the participant’s attention shifts outside (indicated with a dashed arrow). At this point in both groups there is a flurry of inter-group activities seen as gazes, conversations and gestures in the video evidence. To find out what it is that is driving this activity, we needed to attend more closely to the work at their boards and some of their muffled conversations during these two episodes.

Fig. 3

Gaze-dialogue transcript for 2nd minute interval of recording #1

Both groups 1 and 2 get off to a fine start on the task with a correct sketch of y = x² − 2. Group 2 then begins to sketch the inverse function (correctly) while group 1 is focusing on the equation of the inverse. Group 1 seems to be having some difficulty with the sketch of the inverse and this is when group 1 begins to look outside. There is a pause in group 1, a switch of pens, a switch back, and then a glance by student C to her left. Glancing to her left at the 1:20 mark, student C sees something and then begins to sketch the inverse incorrectly on her board. At this point in the video, it is evident that group 1 is stuck. The pens switched hands from student C to A and back to C indicating an intensity in focus and shifting of attention. Not satisfied, student C finally shifts her attention outside of her group to an external resource, gestures two different possibilities for the sketch, looks to a different group, then commits to a sketch of the inverse on the whiteboard. It is an incorrect sketch.

At the 2:00 min mark in the video (see Fig. 4), group 1 has a correct sketch of the quadratic function and an incorrect sketch of the inverse, and group 2 has a correct sketch of both the function and its inverse. Group 2 appears to be having some disagreement around notation for the domain, as student E continues to write the domain and range of the original function. At 2:15, student B draws student D into her group for a conversation. D can be seen gesturing to her group’s board and then gesturing the shape of the correct inverse. D points and gestures a second time, and then student B enters into the frame and begins to direct students A and C on how to fix their mistake with the inverse sketch. B can be seen pointing to the proper location for a few points on the inverse graph. At this time (2:35), student D appears to be uncertain about group 2′s graph, and she begins to survey other boards around the room. At 2:40 (see Fig. 5), student D points to a board across the room and says, “Oh, they’ve got it.” Seeing this other example that supports group 2’s graphing replenishes student D’s confidence, and group 2 continues to talk about their next steps.

Fig. 4

Gaze-dialogue transcript for 3rd minute interval of recording #1

Fig. 5

Student D saying "Oh, they've got it" (2:30 in Fig. 4)

Throughout this episode, we see moments of inter-group collaboration. We are interested in these moments, as they do not typically occur in non-thinking classroom environments. Usually, problem-solving within groups is a collaboration that stays within groups (intra-group collaboration), and in this short video segment, we see many instances of collaboration across groupings (inter-group collaboration). When we looked more closely at the video evidence in these two episodes, we determined that each instance of inter-group collaboration was preceded by a moment where the group appears to be low on resources.

In the first minute of the transcript (see Fig. 3), group one begins with lots of engagement and activity. Student C is not only writing, but she is also involved in direct conversation with student A. At 0:40, there is a noticeable decline in activity, no more writing and longer pauses indicating that the group has exhausted their resources with the group. Student C is noticeably stuck on drawing the inverse. At 1:20, we observe her gesture two possibilities for the inverse function—one correct and one incorrect. Group 1 is at an impasse; they have reached the end of their resources. At the 1:25 mark, both students A and C look outside of their group and observe another group’s work. In this moment, their resources are increased, and they are able to continue working on the task.

From 2:00 to 3:10 of the recording (see Fig. 4), there were two examples of inter-group collaboration as a result of diminished resources. Group 1 has just completed an incorrect sketch of the inverse function. Student C is continuing with some writing on the whiteboard, but student B is not satisfied with their sketching. B calls student D (from group 2) over to her group for a conversation about the sketch of the inverse. We can see student D gesturing the correct form for the inverse and pointing to the board for group 2. Students A and C had just recovered some resources from the earlier episode, but their graph was in error. Student B is not satisfied with the sketch but is apparently unable to convince the other two group members—resources are low. B calls over student D from group 2 for an outside opinion—resources are increasing. This interaction results in group 1 erasing their previous sketch and drawing the correct inverse function. Immediately after this interaction between students B and D, student D is now uncertain with her group’s sketch of the inverse. This conversation with group 1 has reduced her confidence in their work, and this confidence cannot be replaced within her group—resources are low. She scans the room until she finds another group’s work that aligns with her own, seeking more resources. She finds it, and then her group is able to continue working through the task.

3.2 Recording #2

The two groups featured in the second recording have unique compositions. Group 1 has only two students and contains an international student who has limited English language skills resulting in low participation in the task and almost no collaboration within her group. Group 2 contains students who are all experiencing a high degree of difficulty with the course material resulting in many false starts and a higher reliance on external resources. These low resourced groups provided rich opportunities for observing how they managed their progress.

Within the first minute of the recording (see Fig. 6), both groups make false starts and use information gleaned from board work around the room to correct their courses. The teacher makes a comment towards student A concerning the incorrect start to his graph. This comment causes student A to search around the room for ideas before correcting his sketch. In group 2, they make a similar error, but this is not noticed until near the end of the first minute as student D glances at group 1’s correct sketch at the 0:40 mark (see Fig. 7).

Fig. 6

Gaze-dialogue transcript for 1st minute interval of recording #2

Fig. 7

At 0:40, students A and D glace at each other's work

In the second minute of the video recording (see Fig. 8), both groups have incorrect work in some form on their boards. Group 1 has a correct graph of the function, but the inverse is opening in the wrong direction. Group 2 has an incorrect sketch of the function and spends most of this minute spinning their wheels. Student E (with the pen) makes a side-ways glance to group 1’s work, pauses, and then steps back, passing the pen to student D. While D stalls for a time not really contributing anything new, student E is seen looking at various other boards in the room. Then, in a surprisingly confident move, E moves back in and erases the incorrect sketch. He then grabs the pen and draws the correct sketch of the function.

Fig. 8

Gaze-dialogue transcript for 2nd minute interval of recording #2

Inter-group collaboration is not often seen in problem-solving environments. In this small sample of video, we observed frequent examples of groups working with other groups and groups observing work from other groups. In thinking classrooms, students have choices when their resources dwindle. Instead of becoming stuck and not making progress until a teacher introduces more resources, students in this classroom were able to increase their group resources by drawing on resources around the room. These resources can include the teacher, but more often, they come from other groups. When a group’s resources are low, they have lots of choices available: students working next to them, and multiple other examples of student work on whiteboards around the room.

3.3 Summary and discussion

As can be seen from the three transcripts above, the shifts of attention from within a group to across groups happens at multiple points in the transcript. Closer examination of these transcripts, as well as others within our data, shows that these shifts always occur when resources are running low and the group (or individual) seeks to gather more resources. Interestingly, we found that the gathering of resources takes two different forms—what we have come to call passive resource gathering and active resource gathering. Passive resource gathering is done through gazes or overhearing of other’s conversations. Active resource gathering, on the other hand, starts with a gaze but also involves more active discussion with another group or individual. Regardless of whether the gathering of resources is passive or active, the data showed that the transaction of resources can occur between groups, between individuals, or between a group and an individual. Taken together the types of resource gathering can be coupled with who the transaction of resources is between to form a 2 × 3 table (see Table 1). This table represents all possible types of transactions captured in the data.

Table 1 Resource acquisition transaction types

The reason that passive individual–individual resource transaction is omitted is that such interactions were not, and cannot be captured, in the data. Such a transaction would either consist of an individual gazing at an individual's work or an individual overhearing another individual. In the first case, we define any of the resources displayed on a white board, irrespective of who created it, as resources belonging to the group. Likewise, in the second case, we define any utterances made by an individual to their group to be a group discussion. Taken together, a passive individual–individual transaction is impossible.

In what follows we discuss each type of transaction represented in Table 1. For purposes of brevity not all transactions are exemplified with excerpts of transcripts.

3.3.1 Individual-group passive

Individual–group passive transaction of resources occurred in the data in situations where an individual within a group was either not following what is happening in their group, or when an individual was seeking outside validation for a solution that was emerging within their group. In these situations, that individual would look from her group to other groups to gather resources, either for their own use or for the benefit of their group.

For example, in the first 30 s of recording #2, there are many instances of individual–group passive collaboration (see Fig. 9) which may be a result of the low initial resources in both groups. Student A is virtually on his own within the group due to the language barrier between the two. After being prompted by the teacher, student A searches the room for ideas, and then corrects his sketch of the function. In fact, student A is observed looking at another group’s work in almost every 10-s interval during this first minute.

Fig. 9

Individual–group passive resource acquisition in 1st minute of recording #2

In the same recording, Group 2 is observed participating in individual–group passive collaboration as well (see Fig. 10). They have started the activity with an incorrect sketch of the function. After some significant pausing, student E seems to shift his attention to group 1’s board work and then rapidly steps away from the board passing the pen to student D. At this point, student E can be seen looking at multiple boards around the room. Through these passive collaborations, E then confidently re-engages with his group erasing the incorrect sketch and replacing it with a correct sketch.

Fig. 10

Individual–group passive resource acquisition in 2nd minute of recording #2

3.3.2 Group–group passive

Group–group passive transaction of resources occurred in the data when whole groups looked at board work belonging to other groups in the room. These instances almost exclusively follow periods of inactivity where group resources are depleted and each member in a group is seen glancing in unison at one or more examples of student work around the room. This type of resource acquisition is different from the individual–group described above, as it represents a moment where the entire group is stuck, and the entire group is shifting attention in unison to other spaces in the room.

In the recording #1, after a brief instance of individual–-individual inter-group collaboration (discussed later) that increased the resources for group 1, the conversation introduced some doubt into group 2 concerning their sketch of the inverse function (see Fig. 11). Student D looks at boards to her left and then towards the camera where she sees something that affirms group 2′s work. She says “Oh, they’ve got it,” and the rest of her group (students E and F) look with her at a group’s work near where the camera is situated. Group–group passive transactions can serve many purposes when problem solving in a public space. In this situation, it is used as affirmation for a group that is not so confident in their solution. Group–group passive transactions are also observed when groups are looking for ideas, diagrams or clarifications when their resources within their group become depleted.

Fig. 11

Group–group passive resource acquisition in 3rd minute of recording #1

Earlier in the same recording, there is another instance of group–group passive collaboration for resource acquisition (see Fig. 12). Group 1 pauses after sketching the original function (1:00–1:20), apparently uncertain about the sketch of the inverse function. Students A and C both look at other groups to their left and then towards the camera. In terms of correct solutions, this resource search was not very successful, as the resulting sketch turned out to be incorrect.

Fig. 12

Group–group passive resource acquisition in 2nd minute of recording #2

Individual–group passive and group–group passive resource acquisition are the most commonly observed in the choice-affluent environment afforded by thinking classrooms. When resources become low within a group and they need to look outside of their membership in order to find new resources (be resourceful), the passive type of resource acquisition requires the least amount of social energy. Groups or individuals are merely glancing at other’s work—a relatively low level of social disruption. This type of resource acquisition requires very little effort on the part of low-resourced; and as such, it is observed very frequently in thinking classrooms.

3.3.3 Individual–individual active

Individual–individual active transactions of resources occurred in the data when an individual, low on resources, leaves her group and engages in dialogue with another individual from another group in order to increase her resources and share back to her own group. These instances may follow periods of inactivity within a group but also result after an individual is not directly involved with her own group’s work. An individual may be excluded from intra-group collaboration for a number of reasons: the individual is falling behind in the collaboration, the individual may not agree with the group’s thinking, or the individual does not feel that they have anything to contribute at the moment. For many students who fall out of intra-group collaboration, the result is lower overall engagement leading to disinterest and distraction. Some students, however, move out of their group to actively find ideas by collaborating with other individuals in the class.

In this episode from the first recording (see Fig. 13), we observe student D in a conversation within her group. She appears to be dissatisfied with a notation that student E is using to denote domain and range (2:00). At the same time, group 1 has finished with an incorrect sketch of the inverse function, and student C is writing the equation of the inverse on the board. Student B is trying to convince her group that their inverse sketch is incorrect, but she is not making much progress. In order to increase her resources and support her argument, she enters into a direct conversation with student D. D makes a gesture indicating the correct shape of the inverse graph and points to her groups sketch a couple of times. Group 1 was low in resources and unable to make progress on their own, and student B sought help from student D. In this active exchange, D is seen supporting B’s argument increasing group 1’s resources and helping group 1 fix their incorrect sketch.

Fig. 13

Individual–individual active resource acquisition in 3rd minute of recording #1

Student B did not agree with her group’s thinking and was not making any headway with her partners. Seeing that the neighboring group was making better progress, student B entered into a direct conversation with student D from group 2. This conversation increased student B’s resources and in turn increased the resources of group 1 leading to a correction in their graph. Because of the rich selection of choices available to her, B’s shifts of attention included attending to the work of her neighboring group followed by an active conversation with an individual within the group. This interaction was a key step in group 1’s problem solving process.

3.3.4 Individual–group active

Individual–group active was another form of gathering resources observed in the video data. This form of resource acquisition showed in two ways. Sometimes, when resources were low in a group, the whole group would enter into a discussion with an outside individual. More commonly, however, during a pause in activity, indicating low resources, an individual from the group would go out, wander the room, and enter into another group to discuss the problem. This was also observed as a teacher directed interaction. Instead of the teacher spending a considerable amount of time within a group instructing or providing hints on a problem, the teacher was observed directing a student to go and visit another group to get some ideas. This not only freed the teacher up to visit other groups, but it also created an opportunity for the students to develop independence and autonomy in their own learning. In this choice-affluent environment, the students and the teacher can all leverage the rich resource space to improve progress in their problem solving.

3.3.5 Group–group active

Group–group active was the least common but the most exciting to observe in the video data. On occasion, when a group was showing signs of struggle indicating lower resources, the whole group would move in with another group and work together to solve the problem. Pruner (2016) referred to this type of collaboration as forming a “super-group”. This type of resource acquisition is not limited to two groups and it does not rely on only one group being low on resources. Sometimes, three groups were observed working together for extended periods of time. Another distinguishing characteristic of this type of resource acquisition was that these groups often remained together for the duration of the task. It appeared that once the virtual walls that divided groups were breached and the groups combined, like water breaching a dam, they were not likely to go back to their original groupings until a new task began.

4 Conclusions

Much of the research on problem solving has positioned problem solving as a solitary and isolated activity. Problem solving outside of education or among mathematicians is seldom a solitary activity. Our goal in this paper was to describe what problem solving looks like when it is neither solitary nor isolated—when it is collaborative and situated within a choice-affluent environment. We constructed data from video recordings made in thinking classrooms to help us describe what problem solving looks like in these choice-affluent environments. In such environments we saw that problem solvers shift their attention among a variety of resources—other learners, other boards, technology, media, etc.—while traversing their path toward a solution. Providing environments for students that contain an abundance of choices will not only improve problem solving performance but also help students in developing their personal repertoire of resources. Characteristics of thinking classrooms, such as randomized grouping and vertical non-permanent surfaces, make these classrooms good candidates for observing problem solving in choice-affluent spaces.

Through video analysis, we noticed that students would shift their attention outside of groups when resources became low. When group resources are depleted, group activity, such as conversing and writing on the boards, decreased significantly. After some time with this pause in action, individuals and groups began to look outside of their group to increase their resources. This inter-group resource acquisition was either active or passive, and it was carried out by an individual or by the whole group.

As a result of this study, we are able to provide a glimpse of what classroom problem solving looks like in choice-affluent environments. We have seen that when students are engaged in problem solving within spaces that approach real world problem solving environments (choice-affluent spaces), students will take advantage of external resources when necessary. This is more akin to authentic problem solving in the real world and in the world of research mathematicians. Pólya described problem solving at an individual level based on prior knowledge and prior experience. Schoenfeld (1985) expanded on this by suggesting that good problem solvers needed to have a lot of resources and in order to gain resources, they needed to have lots of success in problem solving. Mason (1989, 2008, 2010) introduced the collaborative into the problem solving process by suggesting that the solver’s shifts of attention among personal resources (heuristics), peers, and sources of knowledge is a good framework for problem solving. Koichu (2018) created a model that encompasses all of these ideas—the Shifts and Choices Model. In this model, the situated solver shifts attention among internal and external resources and is able to make choices between schemata, challenge, sources of knowledge, etc. We offer here an opportunity to envision what this model might look like in a real classroom. In a thinking classroom, students demonstrate problem solving that comes close to real world problem solving and demonstrates what Koichu describes as a choice-affluent environment.

This study also provides a method for analyzing classroom video of problem solving activity. The gaze-dialogue transcript honors what an observer would see in the classroom. Short of actually watching the video or observing the classroom live, the gaze-dialogue transcript tells the story of the problem solving episodes by not only describing the written work of the students but also their actions, attentions, and directions of their gazes. This type of transcript codifies everything that is observed in a concise and analytical form that allows for future analysis and discussion of the actions that transpired.

Moving forward, future studies can expand on this transcription method. It would be interesting to capture the problem solving of an entire class by recording each group’s work in video and producing a gaze-dialogue transcript for a full class. Instead of noticing that a group or individual is attending to a different group’s work, it would be interesting to be able to document whose work is being attended too, and to what degree are resources being distributed. Such a transcription would, perhaps, allow new heuristics to emerge—heuristics that include problem solving in choice-affluent spaces where resources are shared when needed and problem solving transcends the individual into the public space.

We know that problem solving is dependent on heuristics, experience, and personal resources and that in real life, problem solvers rely not just on their own resources but also on external resources: peers, technology, social networks, etc. As such, we argue that authentic problem solving is best learned and practiced in choice-affluent environments where the situated solver makes shifts of attention within spaces that afford an abundance of choice for engagement. We presented here an opportunity to envision what this might look like in practice. In thinking classrooms, students have been observed to move outside of their group when their group’s resources are diminishing. Thinking classrooms provide one potential model that demonstrates choice-affluent environments for students to exercise and build their problem solving resource repertoire.