The Art of Being Specific While Theorising for and from Practice of Mathematics Teachers’ Collaboration

Abstract

Theorising is one of the most endorsed practices in contemporary mathematics education research. At the same time, such questions as “what counts for useful theorising?” and “how can theory inform practice?” are still subjects of debate. Arguably, this is in part because of the theoretical diversity characterising our field (e.g. Sriraman & English, 2010), and in part because of the diversity of approaches to handling the research-practice relationship (e.g. Schoenfeld, 2020).

1 Introduction

Theorising is one of the most endorsed practices in contemporary mathematics education research. At the same time, such questions as “what counts for useful theorising?” and “how can theory inform practice?” are still subjects of debate. Arguably, this is in part because of the theoretical diversity characterising our field (e.g. Sriraman & English, 2010), and in part because of the diversity of approaches to handling the research-practice relationship (e.g. Schoenfeld, 2020).

In her plenary talk at the IMCI Study 25 conference, and in the corresponding chapter of this book (Chap. 6), Susanne Prediger makes an important contribution to the debate, by substantiating the following ideas:

Although PD practices are always content-specific, research papers and particularly theorising processes tend to abstract from these contents; however, we should talk more about content-related theorising. […] Informing the concrete analysis and especially the concrete PD design and facilitation, they [theoretical frameworks] must be elaborated in content-related ways for different PD content. (p. 18)

Prediger uses strong modal words ‘should’ and ‘must’ in the above sentences, while mainly reflecting on her own line of research. However, her central suggestion—more content-specific theorising is needed—is far from being obvious, at least in light of the theoretical landscape reflected in contributions to Theme A of the conference (Chap. 2, this volume). Playing for a moment the role of devil’s advocate, we can imagine a colleague proposing the following line of questions. If the existing, and somewhat overwhelming, diversity of large, intermediate-level and local theories is still not enough in order to provide us with insights for explaining and facilitating mathematics teacher collaboration, perhaps we need less theorising? After all, many decisions that researchers, PD facilitators and teachers make are experience-based.

A possible alternative to ‘classic’ theorising might be what Mason (2002) introduced as a call to produce brief-and-vivid recollections of our observations in ways that would trigger others’ own recollections. These recollections, if properly disseminated, can inform and influence practice (Begg et al., 2003). And if we do need more theorising—for instance, for enhancing our explanatory, design and predictive powers or as a safeguard against policy based on over-generalised anecdotal evidence (Niss, 2007)—perhaps we should move towards the development of unifying meta-theory, as Steen (1999) and Dahl (2010) have reasoned, rather than towards elaboration of the existing theories by developing content-specific elements, as Prediger suggests?

A winning argument is rare in our field, and a counter-argument to our devil’s advocate argument can readily be constructed. To this end, a structural analogy between research on mathematics teacher collaboration and research on mathematical problem solving can be instrumental. (After all, Lampert, 2001, considered teaching as a kind of problem solving.) Let us recall that, under the great influence of Pólya’s (1945/1973) work, mathematics education research protractedly experimented with an idea to teach general problem-solving heuristic strategies to teachers and students. When this idea was found not to be as useful as hoped (Schoenfeld, 1992), the idea to unpack general heuristics into content-specific ones and promote those via appropriate practices was put forward (Schoenfeld, 1985, 1992). By analogy, unpacking teacher collaboration by closely attending to its specific content can usefully complement general characterisations of collaborations in terms offered by big theories.

In addition, the proposal for developing content-specific theoretical elements is well-aligned with Lester’s (2005) suggestion that mathematics education researchers should be equipped with various theoretical tools in the pursuit of solutions to problems at hand. This sort of argument, however, is also disputable. For instance, Lester uses the metaphor of a researcher as a bricoleur, which presumes certain theoretical eclecticism, whereas Prediger strives for theoretical coherence when talking about developing and connecting content-specific theory elements as a worthy goal for current and future research. To me, though the vignette used in Prediger’s chapter for illustrative purposes is truly convincing, it still needs to be elaborated if and how content-specific theorising can be undertaken coherently in different contexts pertinent to different situations of mathematics teacher collaboration.

In the rest of this commentary, I focus on Prediger’s suggestion that the proposed line of content-specific theorising can provide theoretical foundations for “typical facilitation practices and designs” (p. 10). I examine this suggestion by first attempting to characterise Prediger’s research strategy that affords and includes content-specific theorising. I then engage with some of theoretical elements suggested by Prediger, and check their applicability by attempting to use them to analyse a situation of mathematics teachers’ collaboration that is very different from Prediger’s example, both in its content and in its context. This commentary is concluded with remarks on connecting between content-specificity and content-generality in theorising mathematics teachers’ collaboration for the sake of further accumulation of knowledge and supporting practice. In sum, I address in this commentary the following three questions:

What characterises the research strategy presented by Prediger that affords and includes content-specific theorising?

How can the suggested content-specific theoretical elements be applied to a situation of teacher collaboration that is different in its mathematical, epistemological and PD content?

How can content-specificity and generality of theorising be coherently connected in future research on teacher collaboration and beyond?

2 Research Strategy that Affords and Includes Content-Specific Theorising

2.1 About Content-Specificity and Design Research Practices

In Prediger’s argument, content is a multi-faceted term. As the tetrahedron model proposes (see Fig. 3 in Prediger’s chapter), it includes mathematical, epistemological and PD aspects, and attends to students’ and teachers’ past, current and desirable knowledge and orientations. In the illustrative vignette, mathematical content refers to subtraction algorithms and to underlying mathematical principles (e.g. ‘place value’). Epistemological content consists of teachers’ knowledge about obstacles for performing the algorithm in the case of subtraction with carries.

The PD content is related to the teachers’ apparent lack of knowledge on how to teach subtraction with carries, so that the students would master (and remember) the algorithm in all cases. In addition, PD content relates to the teachers’ inferred belief that short-term success of all students is of primary importance, whereas their long-term learning progress is of secondary importance. Admittedly, all these contents are specific in different meanings: from a specific mathematical topic to a specific teacher belief and knowledge for teaching.

Prediger’s further analysis of ‘where to go from there’ puts forward the purpose for the teachers to recognise the primary importance of the students’ long-term learning progress, which should become at least as important as the teachers’ devotion to the students’ short-term success associated with task completion. This purpose is unpacked into suggestions as to what PD facilitators may work on with the teachers, so that the teachers would work differently with their students in the future. The central suggestion is gradually to help the teachers appreciate the distinction between procedural skills and conceptual understanding, and to equip them with pedagogical tools for promoting conceptual understanding in the context of the subtraction algorithm (i.e. place-value understanding).

This suggestion connects specific contents of the vignette to a big theory of learning via conceptual understanding, among other big theories used in theoretical elements at the levels of teaching and facilitating. Indeed, the above suggestion is essentially informed by research on learning and teaching with understanding (Hiebert & Carpenter, 1992) and on conceptual understanding in the realms of arithmetic operations (Hiebert & Wearne, 1996). In this way, content is not only multi-faceted in Prediger’s argument, but also theoretically-laden.

In addition, Prediger explicitly adheres to practices of theorising that originate in the Design Research perspective (see also Prediger, 2019). In her words, this perspective includes, “iterativity and interactivity between theory-generating and theory-guided experimenting” (Chap. 6, this volume, p. 284). This description is compatible with cross-cutting features of design experimentation. According to Cobb et al. (2003), the purpose of design experimentation is “to develop a class of theories about both the process of learning and the means that are designed to support that learning” (pp. 9–10). The second feature is the use of the interventionist methodology; the third is adhering to prospective and reflective aspects of data analysis; the fourth is iterative design that features cycles of invention and revision; the fifth is accountability of theories developed by means of design experimentation to the activity of design presuming that, “the theory must do real work” (p. 10).

All five cross-cutting features of design experimentation seem to be present in Prediger’s extended example and argument. Furthermore, the very focus on functions of the theoretical elements suggested by Prediger (in her words, the functions are specifying, structuring, explaining, designing and enacting) is well-aligned with the principles of design research. Therefore, it seems to be reasonable that her argument in support of content-specific theorising should be considered first of all in relation to problematics within the reach of design research tradition and methodology.

2.2 About Context-Specificity

The illustration is not only content-specific but also context-dependent, and thus it is indicative of additional interesting elements of the chosen research strategy. Without attempting to re-analyse the vignette, I would like briefly to stop on its two specific features which are not explicitly reflected upon in Prediger’s analysis, but presented as contextual information. First, in contrast to many PDs initiated by university researchers, the teacher collaboration under consideration began as a local initiative having an agenda formulated by teachers (i.e. to afford at-risk students better access to mathematics). The researcher facilitator came to assist and enhance a collaboration, which had already been appreciated as viable by the teacher-participants, rather than being imposed on them ‘from above’. Accordingly, the teachers were from the outset in a position to act as fully fledged stakeholders in the university–school partnership (Krainer, 2014).

Second, the vignette represents an episode that occurred at the beginning of an 18-month university–school partnership. Such a long collaboration period provided the researcher facilitators with an opportunity to manifest a great deal of patience and sensitivity while striving, along with the teachers, for sustainable change in teaching practices. It is notable in the vignette that the researcher refrained from immediate actions, but preferred to study the situation carefully. In Prediger’s words, “the researcher facilitator has to obtain a profound understanding of the collaborative group’s practices and challenges before changing the role from observer/analyser to facilitator of the PDCG and before supporting the group’s professional growth” (Chap. 6, this volume, p. 287).

This intention conforms with Swan’s (2011) notion that working with teachers should rely on recognising their current values, beliefs and practices. (Notably, this idea is well-aligned with the above-mentioned principles of design research.) Also, this researcher facilitator intention is indicative of not falling into the trap of a deficient discourse on teachers, as frequently happens when the researchers strive for educational change while perceiving the teachers as ‘lacking something’ (Adler & Sfard, 2016). It is worthy of attention that Prediger first decided to attend to what the teachers were proud of (adaptation to all students’ abilities) and to what was disturbing for them (i.e. ‘the students forget …’), rather than deciding to reveal quickly to the teachers what was disturbing for her (i.e. the teachers’ overly strong commitment to students’ short-term success associated with task completion). These decisions, which perhaps were natural for Prediger as a researcher facilitator having a design-research orientation, are far from obvious for many of us.

2.3 Summary of Prediger’s Research Strategy

In this sub-section, the above comments are summarised by means of a list of characteristic features of a research strategy that affords and includes content-specific theorising of teacher collaboration, as it is reflected, at least for me, in Prediger’s example and argument (but see Prediger, 2019, for the first-handily produced list of theorising steps in the context of design research).

• Adhere to theorising practices of design research as an overarching perspective.

• Recognise teachers as fully-fledged stakeholders in school–university partnership, respect their current achievements, beliefs, aspirations and agenda for professional growth.

• Observe existing practices patiently and interpret them without falling into the trap of deficiency-discourse on teachers. Produce accounts and pause on events that are disturbing for the teachers or for yourself.

• Do not act upon what disturbs you too quickly. Begin from what disturbs the teachers and introduce additional goals gradually, in order to nurture a school–university partnership towards sustainable educational change in the long run.

• First analyse the accounts simply while trying to make sense of what is going on and why. Then analyse them more systematically, by means of hierarchically organised theoretical elements attending to different but interrelated types of content.

• While analysing cumulative data in depth, use structural analogies, in order to connect phenomena at the levels of students, teachers and facilitators.

• Be functional while theorising at different levels: specify, structure, explain, design activities, support enactment.

• Be explicit about and continuously reflect upon your own theoretical premises, preferences, beliefs and results of theorising.

• Look for the general in the particular, but do not over-theorise.

3 Same Theoretical Elements: Different Situation

In this section, I engage with some of the theoretical elements suggested by Prediger in the context of an episode from the project Raising the Bar in Mathematics Classrooms (RBMC; Cooper & Koichu, 2021). This project is very different from Prediger’s Mastering Math project in its key characteristics. The project’s goal, which was predefined by a research university and a philanthropic foundation, is to motivate and support middle-school teachers of so-called ‘excellence’ classes in Israel to incorporate problem solving systematically in classrooms while focusing on developing students’ autonomy and high-level mathematical competences (OECD, 2018). The teachers are recruited to the project’s communities and to collaborate towards creating and enacting opportunities, in order to challenge their students with increasingly demanding problem-solving activities at an increasing frequency. Thirty teachers were recruited in 2021; the project seeks to reach two hundred teachers over 4 years.

RBMC develops around the core community comprising six experienced mathematic teachers, three teacher facilitators and two researchers. One of the roles of the core community is to design problems and ways of discussing them for use in general communities and for subsequent enactment in the teacher participants’ classrooms. The episode in question occurred in the core community 3 months after RBMC had been launched.

3.1 An Episode

In preparation for one of the core-community meetings, the teachers were asked to solve several candidate problems, including the Elevator Problem in Fig. 7.1.

Fig. 7.1

The Elevator Problem. (Adapted from Andžāns & Johannesson, 2005)

Three teachers (Maria, Peter and Rachel) and a researcher (Baruch) discussed this problem and possible scenarios of its use with students during 40 min in a Zoom breakout room. The discussion began from considering the teachers’ own solutions to the problem. They confidently answered question (a)—the maximum number of visitors is four—and vividly discussed question (b) (see Fig. 7.1). Maria shared with the group that she approached the problem “wearing students’ shoes” and suggested a ‘solution’ presented in Fig. 7.2a. Peter observed that this ‘solution’ is constrained by an assumption that four visitors are placed in the corners of the square whereas there are endless placement options. He then presented his solution based on the pigeonhole principle (also tagged as the Dirichlet Principle, e.g. Andžāns & Johannesson, 2005), a synopsis of which is presented in Fig. 7.2b.

Fig. 7.2

Approaches to solving question (a) Synopsis of Maria’s solution (b) Synopsis of Peter’s solution

Rachel approached the problem by placing the visitors in the corners and drawing areas forbidden for the other visitors (i.e. circles of radius 1). Both Maria and Rachel endorsed and appreciated Peter’s solution. Then the following dialogue took place.

Peter:

Actually, this is one of those questions that when you know the right method, it takes 2 min to solve. I have encountered 10–15 problems like this [the Elevator Problem] in the past, so I just saw that the pigeonhole principle would work [...] Therefore, it was quite a technical question for me, like an exercise.

Maria:

And if you were not familiar with the pigeonhole principle, how would you approach the problem, say, as a ninth-grade student?

Peter:

Apparently, I’d do what you did [referring to the ‘solution’ in Fig. 7.2a] […] And then I’d begin moving the first four [visitors] from the corners somehow.

Maria:

So, this would not be a technical question.

Peter:

Not at all! I think that it would be an impossible question […] This is why I don’t know what to do with it [in a classroom …] Question (a) is OK, it would be some trial-and-error, but question (b) would be a real obstacle.

Maria:

Baruch, do you have another solution to (b), without the pigeonhole principle?

Baruch:

Actually, I don’t. We posed this problem having the pigeonhole principle in mind. However, I think that the pigeonhole principle is just a nice name, but the logic of the principle is quite intuitive.

Peter:

[nods in agreement]

Baruch:

Peter, I understand that it is difficult for you to depart from your knowledge of the pigeonhole principle, but how did you solve the first problem of this type? Had somebody just told you this principle? Or did you succeed to somehow discover the idea behind it and later somebody just told you the right name?

Peter:

I don’t remember […] It was many years ago. […]

Baruch:

Let’s assume that the intended solution [to the Elevator Problem] is by the pigeonhole principle, and that the students don’t know it. So, with all our pedagogical wits, how can we help students without telling them that there is some mathematical principle that solves the problem? …

Maria:

I don’t know […] Maybe, we can decompose the problem? To show them a square divided into small squares as a hint? I am not sure where they would take it.

Baruch:

I think that a serious obstacle with this problem for students is: how can one prove that something is impossible?

Peter:

[nods enthusiastically in agreement] Yes! Yes! That’s it! If I’d have an opportunity to teach this […] Now I really wish to use this problem in my classroom! First, I’d hear their solutions, and I would somehow convince them that they are wrong […] Then I’d focus on the most problematic aspect of this problem, which indeed is how to prove that something is impossible. They know how to prove [that something is possible], but they don’t know how to prove that something is impossible. Are there such things in the curriculum?

Maria:

Well, there are proofs that begin from ‘assume that’ and arrive at a contradiction.

Peter:

Proofs by contradiction!

The discussion continued, and eventually the teachers developed a lesson plan consisting of three stages that would circumvent the need to reveal the pigeonhole principle. The first stage consisted of a guideline for how to help the students to appreciate the difficulty of the problem. Here, Rachel’s idea about “forbidden circles” was recalled and implemented by means of a GeoGebra applet for exploring the problem. The next stage—in Peter’s words, “a meta-mathematical stage”—was devoted to proofs by contradiction in general. For the last stage, the teachers composed specific hints that were intended gradually to lead the students to construct a contradiction for the Elevator Problem by partitioning the square (see Fig. 7.2b). After the meeting, all three teachers successfully implemented the problem with their students, though the actual lessons deviated essentially from the plan. However, this is another story.

3.2 Analysis by Means of Prediger’s Content-Specific Theoretical Elements

In the spirit of Prediger’s ideas, the goal of the forthcoming analysis is to explain the above-episode in terms that would inform the PD design towards enactment of challenging problems in a classroom. A simple account might be that the change in teachers’ attitude occurred due to the intervention of the researcher aimed at encouraging the teachers to move beyond the telling/not-telling dilemma. Let us now see what the (inevitably concise within the space constraints of this commentary) use of Prediger’s theoretical elements may bring.

3.2.1 C1 (Classroom Content)

The episode does not contain actual student data (cf. Suleika, in Prediger’s example). However, the teachers plausibly suggested that their students would approach the problem by focusing on its particular case (visitors in the corners), while being unaware that there are additional cases to consider. This suggestion can be theoretically backed, for example, by Buchbinder and Zaslavsky’s (2019) framework for characterising logical structure of mathematical statements and, in particular, by their research on the role of examples in students’ proving. In addition, the content at C1 level includes the pigeonhole principle and the associated question of how the problem can be solved without explicitly referring to the principle. Of note is that the teachers did not initially attend to mathematical ideas behind the principle.

3.2.2 C2 (Learning/Problem Solving)

The mechanisms underlying the students’ hypothetical response to the Elevator Problem may be unfolded in terms of an educational perspective on intuition (Fischbein, 1987), by conception of a problem situation image evoking in students’ tentative solution starts (Selden et al., 2000), or by a discursively-oriented conceptualisation of problem solving that puts forward the interplay of students’ existing and emerging discursive resources (Koichu, 2019). Of note is that elaborated theorising in terms of these perspectives would lead us to consideration of the relationships between mathematical knowledge and competences needed to solve the problem and the actual students’ knowledge and competences as developed in their past problem-solving experiences. These relationships might have been presented in a way structurally similar to the mention of learning trajectory (Prediger, Chap. 6, this volume, p. 290).

3.2.3 C3 (Teaching)

Given that all the teachers in the core community of RBMC had strong positive orientation towards problem solving, how can their initial difficulty with didactical handing of the Elevator Problem be explained? Two possibilities come to mind: the teachers’ close attention to their own experiences with the problem (‘technical’ versus ‘impossible’) and the perceived gap between the knowledge needed to solve the problem and their students’ actual knowledge. In alignment with Prediger’s analysis of C3, the teachers’ initial perception of the problem was not based on a deep analysis of its mathematical content. We can also borrow from Prediger’s analysis a suggestion that a successful scenario of using the problem in a classroom for all three teachers was initially associated with completion of the task. An alternative that emerged later was to use the Elevator Problem as an opportunity to discuss important mathematical ideas (see “meta-mathematical stage” offered by Peter). Theoretically speaking, these suggestions bring us to a classic distinction between teaching for problem solving versus teaching via problem solving (Schroeder & Lester, 1989) as a big theory (paralleling teaching for conceptual understanding in Prediger’s case).

3.2.4 C4 (Environment)

Though it is not explicitly evident in the presented excerpt, classroom environments promoted in the RBMC project are designed bearing in mind the following principles: (1) the problem-solving environments should be emotionally safe for students, and in particular detached from formal evaluation; (2) students can fruitfully engage with atypically challenging mathematics when feeling praised for effort and for sharing incomplete solution ideas; (3) student discourse and dialogue are not only means but also a goal of problem-solving activity (see Goldin, 2009; Koichu, 2017; Schwarz & Baker, 2016; for theoretical justification of principles 1–3, respectively).

3.2.5 PD1 (Content) and PD2 (Growth)

As mentioned, the PD content of RBMC is oriented towards motivating and supporting teachers to incorporate challenging problems in their regular teaching. As follows from the preceding analysis, this process requires a gradual change from valuing teaching for problem solving to valuing teaching via problem solving (cf. a desirable shift from task-completion orientation to learning-progression orientation in Prediger’s analysis). Furthermore, in RBMC, we adopt a perspective on educational change worded by Swan (2011) as follows: “We do not seek to change teachers’ beliefs so that they behave differently, but rather offer opportunities to behave differently so that their experiences may give them cause to reflect on and modify their beliefs” (p. 57).

In the above episode, this perspective is manifested by effort invested in noticing deep mathematics involved in the problem (i.e. existential versus universal statements), explicit formulation of learning opportunities for the students (i.e. deciphering the pigeonhole principle as a case of proof by contradiction without necessarily telling the principle) and in creating an imaginary scenario of a successful lesson, which begins from didactical handling of the students’ anticipated responses (i.e. a GeoGebra applet for ‘forbidden areas’).

3.2.6 PD3 (Facilitation) and PD4 (Environment)

In Prediger’s example, the teachers were those who formulated the content and agenda for collaboration, and the researcher facilitator acted as a silent observer in preparation for the future facilitation. In RBMC, the content and agenda of teacher collaboration were offered by the project team and thus can be seen as (arguably) fragile. In addition, the researcher (Baruch) intervened in the teacher discussion in the episode in question. Despite these differences, teacher collaboration is vivid in both examples. To this end, we can observe that Baruch first listened carefully to the teachers’ discussion and entered the conversation only when asked by one of the teachers (Maria). Furthermore, his first intervention was shaped as a question to another teacher (Peter) aimed to deepen his reflection on his past experience.

The second intervention (about the main difficulty of the problem) was prepared by Maria’s previous assertions about “the students’ shoes”. Similar facilitating behaviour can be seen in many additional episodes of the project. Therefore, I believe that I am in position to argue that PD3 and PD4 in RBMC can be analysed by means of the same framework that Prediger offers for analysing PD3 and PD4 in her Mastering Math project. Namely, the focus on the interplay of external domain, collective domain, inquiry domain and the domain of consequences (Figure 7.1, p. 307) seems to be relevant also for RBMC.

To conclude this section with a personal note, I would like to acknowledge that writing the above two pages deepened my understanding of Prediger’s ideas, as well as my understanding of the presented RBMC episode. I have now clearer ideas about how further to run RBMC and to explore additional episodes while preparing myself for future theorising the content-specific phenomena identified. Therefore, thank you, Susanne!

4 Remarks on Connecting Content-Specific and General Modes of Theorising

Prediger argues for content-specific theorising as a way to enrich and complement insights that may stem from the use of big theories. Indeed, several big theories are referred to in the theoretical elements of unfolding student learning, teaching and professional growth. So, in what way can content-specific and general modes of theorising be coherently connected? To contemplate this question, let me revert to a thought experiment.

One of the big theories used in the analysis of Prediger’s vignette is a theory of learning via conceptual understanding. Let me now imagine that some other research group might have considered alternative theoretical approaches for dealing with the same content. For example, some would prefer to focus on the development of appropriate discursive routines where the ritual performance precedes explorative participation (Nachlieli & Tabach, 2019), as informed by the commognitive theory (Sfard, 2008). Some would consider helping teachers design sequences of exercises highlighting similarities and differences between different cases of the use of the same algorithm (Watson & Mason, 2006), as informed by the variation theory of learning (Marton & Booth, 1997). And some might choose to work on gradual change of classroom norms and patterns of students’ engagement (Goldin et al., 2011), as informed by socio-cultural theories relying on Vygotsky’s legacy. However, in all these hypothetical cases, which would apparently lead to different explanatory and PD-facilitation decisions, the researchers might benefit from the multi-level and multi-faceted, functionally oriented scheme of analysis in terms of C1–PD4. (At least, this worked for me in the context of an episode from another project.)

Therefore, I suggest that a functionally oriented scheme of analysis, which is informed by design research as an overarching perspective, is one feature that makes Prediger’s ideas transferable across different mathematical, epistemological and PD contents, though, as mentioned, specific PD-facilitation decisions may depend on the big theory being used. Another feature is the explicit bottom-up approach that begins with in-depth engagement with mathematical content, and gradually unfolds by theorising questions on learning, teaching and PD-facilitating.

Aligned with a design research paradigm, connection between the content-specific and the general is achieved by considering particular content as a case of something more general (e.g. a specific teaching decision for Suleika is seen as an instantiation of a specific teacher belief, which, in turn, is seen as a basis for long-term facilitation in a specific direction). The scopes and time-lines of the observed phenomena are different, but theorising remains content-specific. In turn, general characteristics of teacher collaboration (e.g. a repeatedly made observation that the collaborating teachers formed a community of inquiry) are mentioned as conditions that can enhance or hinder the work towards specific changes in practice.

To summarise, for me it is Prediger’s decision to focus on functions of theoretical elements that connects the specific and the general. As to coherence, of special note is a systematic implementation of the proposed functionally oriented scheme of analysis for contemplating and acting upon phenomena at the levels of student learning, teaching and PD-facilitating. This is without declaring the use of one big theory as an umbrella, which relates nicely to Prediger’s past work on networking theories (e.g. Prediger et al. 2008). Elegantly done!