1 Introduction

Having read all the chapters in the volume “Teachers of mathematics working and learning in collaborative groups” based on ICMI Study 25, I can summarise my reaction in a few words: I am overwhelmed. I am overwhelmed with the broad range of professional development initiatives for mathematics teachers. I am overwhelmed with the described diversity of the kinds and forms of collaborative groups of mathematics teachers. I am overwhelmed with the richness and variety of engagements of mathematics teachers. I am overwhelmed with the variety of rationales and outcomes, as well as of theoretical approaches used to investigate teacher collaboration.

A commentary? As chapters that report the activity of the working groups already summarise and analyse the studies in the submissions, I find it useless to position my commentary as a ‘summary of summaries’ or an ‘analysis of analyses’. As such, I interpret ‘commentary’ as a set of comments, as my personal reflection on what I noticed, what I wondered about and what attracted my attention when reading the chapters.

2 On a Teacher’s Work

Imagine a teacher.

Actually, imagine a mathematics teacher. Has your image changed?

Now imagine a mathematics teacher in an act of teaching.

I suspect that a traditional image puts the teacher in front of a board, with students, if they are part of the image, sitting at their desks and taking notes. In a more ‘progressive’ image, the students are in groups, working on some task, either sitting at their tables or standing by a board, while the teacher is engaged with one of the groups. If the image is dynamic, the teacher is circulating among the groups. Teaching is the main work, or ‘job description’, of a teacher. However, in all these images, while the number of students may vary, there is only one teacher. The teacher teaches alone. And teaching, in its limited interpretation as ‘delivering a lesson’, is an individual activity.

As such, to consider teachers of mathematics working in collaborative groups, we need to reconsider the work of a teacher, or to interpret teaching broadly. In fact, it has been suggested that we consider as teaching all the work that surrounds presenting a face-to-face or online lesson, including planning, communication with students about the taught material and assessing student work (Zazkis & Leikin, 2010). This broad interpretation of teaching practice was suggested when examining the relevance of knowledge acquired in the study of mathematics at the tertiary level for school teaching.

While the suggested breadth would have been applicable for the publications in this volume, the chapters consider teachers’ work even more broadly. In addition to what surrounds the act of teaching, attention is given to teacher learning and professional development, with the explicit or implicit goal of informing and improving teaching. In turn, this goal is geared towards the improvement of student learning outcomes.

3 On Broad Applicability

In reading Chap. 9, by Brodie and Jackson (this volume), based on the plenary presentation, I note that the authors included the WordCloud from the ICME 13 survey (Robutti et al., 2016). So, I considered a WordCloud based on the text of Brodie and Jackson’s chapter. The main featured word, not surprisingly, is ‘resources’, followed by ‘collaboration’, ‘teacher(s)’ and ‘learning’. The word ‘mathematics’ (also ‘mathematical’) is featured, but at a lower level than ‘teachers’ and ‘collaboration’. The authors mention mathematics teaching practices, mathematics concepts, mathematics classrooms, mathematics content, mathematical thinking, mathematical ideas, etc. I wondered, however, what would happen if the word ‘Mathematics’ was replaced with ‘Biology’ or ‘History’ or even ‘special education’. In most places, such a replacement would make sense and the argument would be sustained. Consider, for example, making such a replacement in the following:

Mathematics teachers’ knowledge is a key resource in collaborative learning—what teachers bring to their collaborations will inform the collaboration and its outcomes, and knowledge is obviously a key outcome of collaborative learning.

The facilitator’s expertise in teaching mathematics and in supporting teachers’ learning, as well as their relationship with the teachers, mattered greatly for how they were perceived by the teachers.

I invite you to verify my claim by either omitting ‘Mathematics’ or replacing it with a different subject. Was the intention altered?

In my view, the suggested framework of resources—detailed as representational, knowledge, affective, human and institutional resources—does not explicitly focus on Mathematics. Neither do the two questions that guided the authors:

  • What resources are available to support teacher collaboration? With what effects, both on the collaboration and the resources themselves?

  • What resources are missing for supporting teacher collaboration? How and to what extent can teachers overcome these missing resources?

A harsh critic may point to the lack of specificity in the described research. I would like to adopt a positive tone, focusing on the scope of applicability. A broad applicability of presented claims to teacher community at large, regardless of the subject matter in focus, is where I see the main contribution of the authors. In fact, this breadth exemplifies the leadership of mathematics education community within subject-focused education at large. It could be the case that students’ difficulties with mathematics are those that resulted in a plethora of research in mathematics education, whereas research related to other subjects followed suit.

The initial research initiatives in mathematics education attended to curriculum and student learning, with the development of a more recent focus on teachers—the ‘era of teacher’, according to Sfard (2005)—including teachers’ knowledge, teachers’ pedagogy and … teachers working and learning in collaborative groups. I believe that framework of resources is applicable broadly and could be adopted and modified when attending to teachers of other subjects.

A related claim of broad applicability can be made with respect to the RATE framework—Relevant Actors, Targets and Environments—developed and used by Krainer, Roesken-Winter and Spreitzer (Chap. 8, this volume). The authors described professional development initiatives in seven recent studies chosen from different continents. While the described projects concerned teachers of mathematics, mathematics teacher educators and mathematics classrooms, I suggest that the same dimensions (that is, relevant actors, targets and environments) can be used for comparative analysis of any professional development program, that is, not specific to teachers of mathematics, and possibly applicable to other professionals, not only teachers.

Similar claims can be made about other chapters. In particular, continuing the theme of ‘broad applicability’, I find the discussion of ‘theory’ in Chap. 2 by da Ponte et al. (this volume), to be a valuable resource for any researcher in education. The chapter presents an importance “conceptualising the notion of theory” (p. 7). The authors’ notes on the “heterogeneity in what is called a theory by different researchers and different scholarly traditions” (p. 6) is a valuable summary which highlights the main issues in educational research. While the summary is tailored for research on mathematics teachers’ collaboration, it is applicable to mathematics education and teacher education broadly. The particular section on “Generalities of theories” has been added to the reading list in my doctoral seminar. I trust that it will help novice researchers situate themselves within the web of theoretical constructs and plethora of theoretical models and frameworks.

4 On Mathematics

My appreciation of generality and the applicability of ideas and frameworks beyond the professional development of mathematics teachers also came with a search for specificity. While the word ‘mathematics’ is mentioned multiple times, I explicitly sought specific mathematical concepts or problems. I found the mention of topics, such as ‘logic’, ‘modelling’ or ‘whole number arithmetic’, but the particulars were omitted. This search was based on my selfish approach to the chapters. As I spent long hours on reading the dense material, I wanted to find something ‘useful’ for my work with teachers. I recognised in this selfish approach my frequent critique of teachers: while we (that is, researchers, mathematics educators) attempt to extend teachers’ horizons of knowledge, we are often disappointed that teachers are more appreciative of tricks and whistles that could be ‘used in their classroom on Monday’.

Teachers collaborate on learning mathematics. What mathematics are they learning? Teachers engage in collaborative discussion after watching a video of a mathematics classroom. What was the video about? What was addressed in discussion? Teachers collaborate on reading and discussing research. What was read? What issues were discussed? Teachers are collaboratively working on mathematics problems and sharing their teaching of the problems. What are the problems? What teaching strategies or approaches are shared? I realised that in order to find specificity, one has to examine papers discussed in ICMI Study 25 and published in the conference proceedings (Borko & Potari, 2020), rather than chapters summarising the submissions and discussions.

There were, however, a few specific examples that extended my personal repertoire of tasks and experiences that I intend to bring to my class (more on this in the next section). One such example is found in Chap. 10 by Hollingsworth et al. (this volume), where a teacher described the following:

I was teaching about subtraction of mixed numbers, e.g. 7\( \frac{1}{3} \) − 3\( \frac{2}{5} \). I asked students to explain how they could solve the problem. One student explained, “First subtract three from seven and get four, then subtract two-fifths from one-third.” The student proceeded up to this stage: 4 \( \frac{5-6}{15} \) . And then the student said, “We take one from four, the whole number, and add to five …”. (p. 14)

I suspect that my students (prospective teachers) may attempt to correct the student and insist on the ‘correct way’ rather than to engage with student thinking, but I intend to test and hopefully revise my expectation.

5 On Content-Related Theorising

Prediger (Chap. 6, this volume) makes a case for content-related theorising. She describes a vignette related to a student difficulty with subtraction, and demonstrates how content-related theory elements may go beyond what is afforded by ‘general’ theories. I appreciate the particular mathematical content, and, when reading the chapter, I wondered whether the notion of ‘content-related’ refers to mathematics-related or particular-mathematical-topic-related. In a way, in his reaction to Prediger’s chapter, Koichu (Chap. 7, this volume) addressed my query. He implemented several theory-elements suggested by Prediger in analysing teachers’ engagement with a particular mathematical problem. Here is the problem:

Epsilon Tower in Zedland is famous among tourists for its wonderful view from the top floor. During the pandemic, it is permitted to use the elevators ONLY on condition that every visitor wears a protective mask AND keeps the distance of at least 1m from the other visitors. Each elevator in Epsilon Tower is designed as a parallelepiped having a square floor with a side of 1.4m.

  1. a.

    How many visitors may simultaneously use one elevator?

  2. b.

    Prove that the number you found in (a) is the maximum number, that is, it is impossible to place any more visitors in the elevator without violating the rules.

This problem satisfied my thirst for the particulars, as I immediately engaged with the problem.

While this problem is embedded in the context of the current pandemic (hopefully just an endemic by the time someone reads this), it reminded me of the ‘4-Trees’ problem, which is to plant four trees such that every pair is exactly one metre apart. Of course, the ‘4-Trees’ are an attempt to create a story line, but the mathematical problem modelled by the 4-Trees is that of finding four points, any two of which are equidistant. The presumed immediate (for most solvers) impossibility of a solution is defeated when the imposed condition of finding ‘Four points on the same plane’ is challenged. The solution is offered by a regular tetrahedron, in which the distance between points is constant for each pair of the four vertices. Bringing the model back to the 4-Trees task, a solution involves planting one of the four trees on a hill.

Before attending to the solution suggested in Koichu’s chapter, a beautiful solution that relies on the pigeonhole principle, I immediately thought of the 4-Trees problem. I imagined the four people in the corners to be kneeling or sitting on the floor, and placed the fifth person in the middle, choosing the tallest person or placing him on a stool, just in case. Note that the elevator in the Epsilon Tower is designed as a parallelepiped having a square floor (so I assume this is a rectangular prism), but that the height of the elevator was not given and restricting that height could influence the solution. Intuitively, having respiratory transmission in mind, I interpreted the distance between visitors as a distance between peoples’ heads. Note that my suggested solution is represented by a square pyramid rather than a tetrahedron, because the requirement is not for equidistant pairs, but for a minimal distance, that is, for each distance to be at least one metre. So, the apex of my pyramid should attend to the distance requirement.

I shared the problem with a colleague, who was not familiar with the 4-Trees problem. She noted, based on the height of the people, and attending to the choice of 1.4m as the size of the base-square side, that there may not be any need for kneeling and stool. That is, if the distance between the mid-point of a square, and the vertex of a square (taken as hypothenuse of a right-angle isosceles triangle with each leg of 0.7) is about is about 0.9899 m, then the middle person has to be ‘just a bit’ taller for the distance between the heads to be over 1m. But this ‘works’ when people or their heads are viewed as points in space. Thinking of people, rather than points, there could be a need to consider the ‘space’ that each person occupies. Turning this thinking to modelling, what if people are represented by circles or spheres? How does this change the problem? In Koichu’s solution based on the pigeonhole principle, what are the assumptions about such representations necessary for the suggested solution to hold? How does changing the given side length of 1.4m effect the solution?

I wonder what insights can be added to my collaborative engagements with a colleague by using the theory-elements suggested by Prediger and used by Koichu. In particular, what content-related theory-elements are applicable in a non-facilitated collaborative work on a problem?

6 On Effects or Products of Collaboration

What are the products or outcomes of teacher collaboration? These are related to the goals of collaboration or of professional development. Most outcomes can be described as teachers’ ‘professional growth’, but such growth can take different forms. The particular outcomes, acknowledged across various chapters in this volume, are related to:

  • enhancement of mathematics teachers’ knowledge of mathematics (for personal enrichment);

  • further development of mathematics teachers’ knowledge for teaching;

  • development of interdisciplinary knowledge needed for STEM education;

  • improvement/change of teaching practice that promotes student learning;

  • extended availability resources, such as particular problems, task sequences, assessment activities or lesson plans;

  • extended familiarity with technological resources, such as particular platforms or websites;

  • extended knowledge of a new curriculum;

  • further knowledge in interpreting student work, student solutions or errors;

  • changing views of the discipline of mathematics;

  • development of teacher professional learning community.

In considering this partial list of possible outcomes, I recall John Mason’s comments on the products of research:

What are the significant products of research in mathematics education? I propose two simple answers: 1. The most significant products are the transformations in the being of the researchers. 2. The second most significant products are stimuli to other researchers and teachers to test out conjectures for themselves in their own context. (Mason, 1998, p. 357)

Here, I paraphrase and extend: what are the main products of teacher collaboration? They are the change in the teacher-collaborator and the stimulus to implement what is learned in their own context. In particular, it is an extended repertoire of mathematical tasks or instructional approaches. It is an extended appreciation of students’ approaches or difficulties. It is an extended awareness of ‘others’—other teachers, other curricular sequences and other professional environments. At least initially, these transformations in ‘being a teacher’ are not directly measurable.

7 The Main Issue

Some background is necessary before I get to the ‘main issue’, so I seek the reader’s indulgence. Simon Fraser University—which is my affiliation for over 30 years—offers a Master’s program in Secondary Mathematics Education (SME). The program, that attracts secondary mathematics teachers, was designed as a collaboration between the Faculty of Education and the Department of Mathematics several years before I joined SFU; the core coursework in the program is from both Education and Mathematics. However, the distinctive feature of the Mathematics courses is that they were designed specifically for teachers. That is, unlike other graduate level courses in mathematics, these courses do not assume fluency with undergraduate content. It is a ‘cohort program’—meaning that the courses are offered in a 2-year sequence and the students take all the courses together.

The first course in the program is titled Foundations of Mathematics and is intended to focus on ‘big ideas’ and ‘great theorems’. For several initial offerings this course was taught by a Professor of Mathematics, the late Dr. Harvey Gerber. However, following Harvey’s retirement, the course has become part of my regular teaching assignment. That is where we get to the ‘main issue’.

Preparing for teaching Foundations of Mathematics for the first time (this was in 2001), I set an appointment with Harvey, expecting to learn his perspective on what ‘foundations’ are essential for secondary school teachers of mathematics and how teachers should be introduced to foundational ideas. I also had my own list prepared, intending to seek feedback from an experienced colleague. Harvey’s response surprised me at the time, and it still resonates with me today. He said: “The choice of a particular mathematics topic is not important. What is important is that students work together. They have to take courses together for two years. Your main goal in the first course is to build community.” I was astonished, not only by the response, but by the fact that this was a response from a mathematician! And it happened before the constructs like ‘professional learning community’ or ‘community of practice’ entered my lexicon.

Meeting and teaching every new cohort of teachers I remind myself of Harvey’s advice. While teacher collaboration is not one of the explicitly stated program goals, it has become an extremely valuable feature and outcome of our graduate SME program. The courses engage students in collaborative problem solving, collaborative task design and collaborative learning of new (for them) mathematics. Using the framework of resources (Brodie & Jackson, Chap. 9, this volume), I suggest that the coursework curriculum attends mostly to knowledge (both mathematical and pedagogical) and representational resources (the latter may include lesson plans, videos of teaching and excerpts from student work). However, as teachers progress in the program, they develop their assembly of human resources, which often compensate for insufficient institutional resources.

As years pass, we learn that many of our graduates continue to collaborate after their graduation. They meet in pairs or small groups to share experiences and ideas, as well as problems and solutions. I wish I could share with Harvey this success.