Using and Developing Content-Related Theory Elements for Explaining and Promoting Teachers’ Professional Growth in Collaborative Groups

Abstract

What kind of theory is required to explain and promote teachers’ professional growth in collaborative groups? Sociocultural theoretical frameworks and models of professional growth have been successfully established as powerful generic frameworks (partially mathematics specific, but not yet content specific). The chapter argues that these generic frameworks should be further refined in content-related ways, so that both types of content, the mathematical classroom content and the mathematics education PD content, are more explicitly part of the research and the designs. Whereas the generic frameworks already exist and can be used, the refinement requires acts of content related theorizing, which means developing and connecting content-specific theory elements. These theory elements refer to several areas that can be located in the tetrahedron model: Theory elements are not only required for explaining and enhancing the PD environment and facilitation practices, but also for explaining and enhancing the content-specific teacher learning that also requires theory elements for the PD content itself to unpack it into relevant aspects of the classroom learning environment, the teaching practices, the student learning, and the mathematical content in view. Different content-related theory elements are presented and exemplified for a vignette showing the professional growth of a collaborative group of middle school teachers aiming at differentiating and fostering at-risk students’ access to mathematics.

1 Introduction: Why Content-Related Theorizing?

Teacher collaborative groups form a widespread and promising environment for teachers’ professional development (PD), and many different variants have already been explored in research and PD practice, as have been shown by an insightful ICME survey (Robutti et al., 2016; Jaworski et al., 2017). However, the ICME survey also revealed that only 85 out of the 316 analysed papers on teacher collaboration explicitly referred to theoretical frameworks on collaboration, with the rest referring to theoretical frameworks for other aspects. This finding shows the need for comprehensive and systematic theoretical foundations for explaining and enhancing various aspects of professional development in collaborative groups (PDCG).

The term theoretical foundation usually refers to local networks of theory elements that serve a certain purpose (here, explaining and enhancing PDCG). Theory elements can be constructs, hypotheses or claims (here, related to mathematics or mathematics education and/or its teaching and learning in classrooms or PD). The local network is usually embedded in a larger (perhaps quite general) theoretical framework.

When starting a research project, researchers usually chose a larger theoretical framework in which to situate the research and design, as well as some already existing theory elements that they can use as background for their research. Further theory elements need to be developed and connected in empirically grounded ways, as an outcome of the research (Mason & Waywood, 1996). The latter process of constructing a theoretical foundation by identifying, refining and connecting theory elements (mostly in an empirically grounded way) is called theorising (Prediger, 2019a). Whereas other papers have focused on how the process of theorising can be conducted (referenced in Prediger, 2019a), this chapter focuses on what should be theorised for providing theoretical foundations for PDCG.

This chapter is an extended version of the plenary paper in the study conference proceedings (Prediger, 2020). The main message of this chapter is that, in order to elaborate a theoretical foundation for explaining and promoting teachers’ professional growth in collaborative groups, various theory elements should be integrated, at both the classroom and the PD levels.

The discussion of a vignette from a PDCG on the PD content ‘differentiating and enhancing access to mathematics for learners with difficulties’ (more briefly referred to as ‘at-risk students’) will exemplify how existing theoretical frameworks (communities of inquiry and models of professional growth) provide a language for explaining the complexities of PDCG, but are mainly generic search spaces. In order to inform the concrete analysis, and the concrete PD design and facilitation in particular, they must be refined in content-related ways, referring to both types of content: the mathematical classroom content and the mathematics education PD content. The need for more content-specific theory elements will become visible as the generic theory elements gain their explanative power when being filled in content-specific ways (Prediger et al., 2019b).

After a brief introduction of PDCG and the construct of practices in Sect. 6.2, an introductory vignette in Sect. 6.3 illustrates why we need more theorising. Section 6.4 disentangles the kinds of theory elements required for a theoretical foundation of PDCG by exploiting the analogy between classroom level and PD level, while, on this meta-theoretical base, Sect. 6.5 specifies the relevant theory elements for the introduced vignette and the PD it stems from, which provides the concrete material for the meta-theoretical reflections in the final Sect. 6.6.

2 Established Generic Frameworks for Explaining Professional Development in Collaborative Groups

2.1 Communities of Practice and Inquiry and the Underlying Sociocultural Theories

According to the ICME survey (Robutti et al., 2016), 80% of the 85 papers analysed that explicitly referred to teacher collaboration in their theory sections addressed the theoretical construct of community, often in the senses of community of practice (Wenger, 1998; Lave & Wenger, 1991) or community of inquiry (Jaworski, 2006). Whereas some articles only referred to community in order to name the PD setting in its practical character, most articles also referred to underlying sociocultural theories of learning that have been generic frameworks and were then specified for mathematics in general, yet not for particular mathematics or PD contents.

Within sociocultural theories, learning is conceptualised as being situated in communities of social practice, where novices are successively drawn into practices, first from a legitimate peripheral position, then to the centre of the experienced practitioners (Wenger, 1998). When these theoretical frameworks are related to teachers’ learning, teaching is thereby conceptualised as a set of social practices, and professional growth is described by increasing engagement and alignment (Putnam & Borko, 2000).

Jaworski (2006) has enriched this theoretical framework by emphasising that PD itself is shaped by a set of social practices in which alignment should not be assimilation but critical alignment. This offers the opportunity for collective professional growth within a community of practice, rather than the stability of only enculturating novices. She is widely cited for the enriched construct of communities of inquiry in which, “participants […] align with aspects of practice while critically questioning roles and purposes as a part of their participation for on-going regeneration of the practice” (p. 190).

In both frameworks, practices form the key theoretical construct. Practices in mathematics classrooms can be defined as “ways of acting that have emerged [… that make] it possible to characterise mathematics as a complex human activity and [… that bring] meaning to the fore by eschewing a focus on socially accepted ways of behaving” (Cobb et al., 2001, p. 120). To apply the analogy, teaching practices are ways of acting that have socially emerged to manage typical situational demands at the classroom level, while practices of inquiry refer to those at the PD level (see below).

Putnam and Borko (2000) had already advertised lifting sociocultural, situated theoretical frameworks from the classroom level to the PD level. In order effectively to exploit these frameworks, they underlined the need for further considerations on: (1) where to situate teachers’ learning experiences; (2) the nature of discourse communities for teacher learning; (3) the importance of tools in teachers’ professional learning experiences. In this chapter, I extend their call for further considerations into a call for further theorising, that is not only using additional existing theoretical elements, but also developing new ones in an empirically grounded way.

2.2 Adapted Model of Professional Growth

Another well-established model for explaining and promoting professional development is the interconnected model of professional growth by Clarke and Hollingsworth (2002), which has been widely used, not only for describing and explaining, but also designing PD for promoting professional growth in modes of action and reflection. The model includes four analytic domains: the external domain (with external sources of information, stimulus, or classroom resources); the personal domain (teachers’ knowledge or attitudes),;the domain of practice (in which classroom inquiries can take place); the domain of consequence (with salient outcomes such as students’ learning gains).

The model identifies different mechanisms by which change in one domain is associated with change in another. Rather than claiming simple mechanisms of transmission from the external domain via changing the personal domain to the domain of practice and then to the domain of consequence, they emphasise an “interconnected, non-linear structure” between these domains and identify different “particular ‘change sequences’ and ‘growth networks’, giving recognition to the idiosyncratic and individual nature of teacher professional growth” (p. 947). An example of such an individual change sequence is a teacher experimenting in the domain of practice, monitoring students’ thinking in the domain of outcomes, thereby expanding their knowledge about student thinking in the individual domain, which is then the result of the change sequence rather than its start.

This model resonates well with the ideas of communities of inquiry, as they emphasise teachers’ active roles in connecting their knowledge with teaching practices and evaluating outcomes. More explicitly than Jaworski (2006), Clarke and Hollingsworth (2002) also conceptualise this domain of outcomes as relevant for steering the decisions in the domain of practice. They include the external domain, relevant when the collaborative groups search for external stimuli, for example, from reading or by facilitators, but also when they choose particular curriculum resources or textbooks to adapt for their own work.

As the model was originally articulated mainly for individual teachers and cognitive constructs, two domains in Fig. 6.1 have been slightly adapted:

• With respect to the underlying sociocultural framework, the sociocultural adaptations modified the personal domain (teachers’ knowledge or attitudes) into the collective domain (here conceptualised as including teachers’ shared orientations and categories; see below). In particular, this adaptation can extend the analysis to the connection between individual and collective concerns, which is crucial for the conditions of professional growth in collaboration.

• With respect to the particular focus on communities of inquiry, the domain of practice is adapted into the domain of inquiry (focusing the established practices that teachers reflect on critically and the new practices the teachers experiment with). This adaptation is suitable for PDCG, as it builds upon Putnam and Borko’s (2000) call for authentic situated learning opportunities and includes Jaworski’s (2006) community of inquiry in the domain of inquiry.

The adapted model is still compatible with the original ideas, as it still connects, “the teacher’s professional actions, the inferred consequences of those actions, and the knowledge and beliefs that prompted and responded to those actions” (Clarke & Hollingsworth, 2002, p. 951). In line with Wenger (1998) and Jaworski (2006), teachers’ professional practices have been defined as socially established ways of mastering recurrent situational demands in mathematics classrooms (such as differentiating and fostering low achievers). Whereas Clarke and Hollingsworth connect these practices to the underlying knowledge and beliefs, the DZLM research network (led by the author of this chapter) usually refers to Bromme’s (1992) situated construct of teacher professional expertise and adapts it to a situated sociocultural framework.

Fig. 6.1

Adapted model for professional development in collaborative groups (generic search space)

According to Bromme, practices are visibly characterised by shared pedagogical tools (e.g. tasks and teacher moves), but are explainable only by the underlying orientations (i.e. socially shared beliefs about aspects of mathematics and its teaching and learning) and the activated categories for perceiving, thinking and evaluating. These categories are non-propositional knowledge that filter and focus the categorial perception and the thinking of teachers. Within this model, professional growth is characterised not only by changes in practices, but also in the underlying orientations and shared categories for noticing and thinking.

As will be shown in Sect. 6.4, Bromme’s general framework gains its explanative power for content-related purposes when filled in content-related ways. This is the key idea of the content-related conceptualisation of teacher expertise (Prediger, 2019b). In order to explain teachers’ practices and professional growth for a particular area of PD content, we identify the socially shared, content-related, pedagogical tools, orientations and filtering categories that underlie their utterances and visible behaviour for mastering the self-posed situational demands in the particular area of mathematics education that is the content focus of the PD.

In the following sections, I will try to show why the generic frameworks are highly useful for connecting the different domains (as would also be the general mathematics-specific but not content-specific frameworks such as the Anthropological Theory or Theory of Didactical Situations, which are not dealt with here). But I will also show why the theoretical foundation for a particular PD can substantially profit from a refined framework that allows for more content-specific explorations of particular areas of the mathematical and/or PD content in view.

3 Why We Need More Content-Related Theorising: An Introductory Vignette from a Community of Inquiry

To convince the reader that content-related theorising might be useful, I report a vignette from a PD research episode that took place in the beginning of the design research project Mastering Math (Prediger et al., 2019a), aiming to show where the generic frameworks were not sufficient for explaining the vignette. The teacher group collaborated intensively and grew, but did not succeed in achieving their goals.

A Vignette from a PDCG

The very first phase of our PD design research project (Mastering Math) involved researcher facilitators from our Dortmund research team (i.e. PDCG facilitators who are researchers at the same time) visiting active schools in which teachers had started collaborations for developing their teaching practices with the goal of providing better access to mathematics for so-called at-risk students (e.g. students with special needs or students at risk of failing due to limited support at home and/or in earlier school years).

The vignette took place in a school where mathematics and special education teachers worked collectively on differentiating, in order to foster all students’ achievement in their grade 5 classroom. The outcome of the first meeting was that the teacher group joined a university–school partnership for 18 months. At the moment of this first meeting, the teachers had already spent 9 months on finding ways to differentiate their teaching material, in order to adapt to students’ diverse mathematical abilities. After 9 months of intensive collective work, they were proud to have substantially changed their teaching practices in order to adapt to all students’ abilities, mainly in task-based, individualised settings.

When the researcher facilitator first met them, Paul, the mathematics teacher, reported about Suleika, one of their students with learning difficulties, and showed two of her products on multi-digit subtraction (shown in Fig. 6.2):

Paul: Suleika can calculate the subtraction well, only the carries pose problems for her. But we can handle this successfully by differentiated tasks: I only give her subtractions without carries.

Although the community agreed on the success of their changes, it took 3 months for one of the members, Maria, a special education teacher, to convince her colleagues to participate in a PDCG program with the university, as she saw an additional problem to be tackled:

Maria: I tried to teach them subtraction with carries several times, but they always forget it.

Fig. 6.2

Snapshot from a community—monitoring Suleika’s learning

Analysis Within the Generic Framework

Without doubt, Maria, Paul and their colleagues formed a community of inquiry in Jaworski’s (2006) sense. They collaborated intensively, questioned and developed new practices of differentiation (domain of inquiry in Fig. 6.1). However, Paul’s comment was a characteristic expression of this community’s adaptive differentiating practices (that the group articulated repeatedly in different utterances): students receive individualised tasks that are optimised on a level that they can master. Within the frame of these teachers’ collective evaluation (domain of consequence in Fig. 6.1), the collectively established differentiation practice in their domain of inquiry proved to be successful, as Suleika was able to complete her tasks.

However, in spite of the intensive engagement of these teachers’ community of inquiry, they could not develop more productive practices for enhancing Suleika’s learning. Although her second product reveals serious struggle with place value understanding (see Fig. 6.2), this was not treated by the teachers. Maria’s additional concerns that students “always forget” had also not yet entered the teachers’ collective domain and was not part of their shared space of discourse.

What the Analysis Leaves Out

This brief analysis reveals the importance of shared orientations of what the teachers considered relevant in their community of practice. However, the language of analysis provided by the generic theoretical framework from Sect. 6.2 is not yet well-enough elaborated to identify the critical points in the teachers’ PDCG in more detail. In this case, it is not the theorising of the mathematics content in view that is lacking, but rather the PD content.

A rough analysis outside the given theoretical framework reveals that, for the teachers, differentiation meant adapting to students’ abilities rather than really strengthening their learning, so their shared category for evaluation was reduced to task completion rather than learning progress. The reduction to this evaluation category also reduced the need for critical alignment with their own teaching practices. At the same time, the teacher community did not distinguish between procedural and conceptual knowledge, as they did not problematise whether teaching Suleika the algorithm with carries might miss the conceptual base.

These aspects (procedural and conceptual knowledge and the difference between learning progress and task completion) occur to be crucial for reaching the PD goal that the collaborative group gave itself, namely providing access to mathematics for Suleika. For the researcher facilitator who started a university–school collaboration with the group, these aspects had to be a substantial part of the reasoning about the PD content of this PDCG.

This indicates that a theoretical foundation which can really inform facilitators’ work in supporting the development of this community of inquiry will need to include these aspects of the PD content. The theoretical foundation should also provide a term for explaining why Maria was not able to introduce her ‘students forgetting’ category into the shared discourse. Again, this has both a generic part and a content-specific part that is tied to the particular PD content in view, which in this case is differentiating and fostering mathematics learning of at-risk students. But what exactly is meant by theoretical foundation? For which functions is it needed?

Section 6.4 will introduce the meta-perspective on generic and content-specific theory elements and its functions for the first steps towards a content-related sociocultural theory of professional growth. Section 6.5 then discusses the concrete elements that can explain the vignette and guide facilitators’ actions in supporting a community’s professional growth.

4 What Kind of Theory Elements Are Required for a Content-Related Theoretical Foundation for PDCG?

This section starts with deepening some structural, meta-theoretical clarifications mentioned in the introduction: what is a theory, and why do I only speak about theory elements? What kind of theory elements are required for an empirically grounded theoretical framework for explaining and promoting teachers’ professional growth?

The role of theories for educational research studies is two-fold. On the one hand, theories influence (but do not determine) the design decisions and the methods and perceptions in the empirical investigations of the teaching–learning processes that have been initiated (theories as a framework for research and design). On the other, empirical investigation aims at generating and eventually testing or refining theories (theoretical contributions as outcomes of research). The interplay between theories as frameworks and outcomes of research applies to all kinds of research in mathematics education (Mason & Waywood, 1996; Prediger, 2015). In design research, in particular, it is fuelled by the iterativity and interactivity between theory-generating and theory-guided experimenting.

Whereas the role of theories as frameworks for research have often been discussed (e.g. Cobb et al., 2001; Mason & Waywood, 1996), the role of theoretical foundations for designs includes describing, explaining and predicting what can happen (Prediger, 2019a). The process of theorising is worth further methodological and strategical reflections. As already defined, theorising is the process of developing new theory elements in an empirically grounded way, including activities such as identifying an interesting phenomenon and developing constructs for describing and explaining it, refining constructs in order to increase their explanatory power, connecting two descriptive elements to explanatory elements, transforming an explanatory theory element into a normative element or into a conjecture for a predictive theory element, and connecting elements to new explanatory and predictive theory elements (Prediger, 2019a).

From the methodological and technical side, the process has been intensively reflected upon (e.g. Glaser & Strauss, 1967). However, limited explications exist so far on what kind of theory elements are to be generated with respect to PDCG. This section therefore makes a suggestion about the kinds of theory elements required, based on distinctions in their logical structure, their size and, in particular, their function (Prediger, 2019a).

Niss (2007) characterises a theory by its logical structure as an “organized network of concepts (including ideas, notions, distinctions, terms, etc.) and claims about […] objects, processes, situations, and phenomena” (p. 1308). The claims can be basic hypotheses, statements logically derived from the fundamental claims or empirically grounded propositions about connections and mechanisms. The logical structure of theory elements can therefore entail constructs, basic assumptions and empirically grounded connections.

Theories vary in size. Some encompass a well-elaborated theoretical framework with a complex network of constructs and propositions (such as sociocultural theory), while others are reduced to single constructs or claims (such as the communities-of-practice construct). Rather than networking complete theoretical frameworks (Bikner-Ahsbahs & Prediger, 2014), this chapter focuses on the local integration of several constructs and claims. This networking strategy is more suitable for fields that are not yet mature enough for big theories (as Jaworski, 2006, stated for the field of PDCG).

In order to decide which theory elements have to be integrated for a theoretical foundation for PDCG, distinctions based on their functions in the design and research process are useful (Prediger, 2019a). On the classroom level, there is a long tradition in developing theory elements that can support the design of classroom learning environments and learning trajectories (e.g. Gravemeijer & Cobb, 2006). Theory elements are necessary for at least four main functions (Mason & Waywood, 1996; Prediger, 2019a):

• C1-content: theory elements for specifying and structuring the mathematical learning content (e.g. constructs describing relevant parts of the learning content and their relationships);

• C2-learning: content-related theory elements for explaining mechanisms of mathematics learning (e.g. a hypothetical content trajectory, hypothesising relevant steps in the learning progress with respect to the specified aspects of the learning content);

• C3-teaching: content-related and generic theory elements for explaining the nature and background of mathematics teaching;

• C4-classroom environment: content-related and generic theory elements for designing and enacting learning environments (e.g. derived design principles and their justification by C2 and C3).

As the lower part of Fig. 6.3 visualises, these functions refer to different parts of the didactical tetrahedron on the classroom level and have increasing complexity: C1-content refers to the mathematical content alone; C2-learning to the edge between students and content; C3-teaching refers to the faces (e.g. among teacher, students and content); C4-classroom environment to the whole tetrahedron. (Of course, theory elements on other single vertices or edges have also been established insightfully in the mathematics education research community, but to keep the theoretical foundation manageable here, we focus on the main four elements in increasing complexity, where C4 also includes C3 and so on.)

Fig. 6.3

Lifting theory elements from the classroom to the PD tetrahedron. (Prediger et al., 2019b)

Experience with theory elements and theorising on the classroom level can be lifted to the teacher PD level, as the teacher PD complexities can be grasped with a structurally analogous tetrahedron that relates the teachers (now in the role of learners) to the PD content, the PD resources and the facilitators (Prediger et al., 2019b). Similar to Putnam and Borko (2000), who lifted the generic socio-cultural framework from the classroom level to the PD level, we can lift the assumed need for content-related theory elements to the PD level:

• PD1-content: theory elements for specifying and structuring the PD content;

• PD2-growth: content-related theory elements for explaining mechanisms of teachers’ professional growth;

• PD3-facilitating: content-related and generic theory elements for explaining the nature and background of facilitating PDs, if a facilitator exists;

• PD4-PD environment: content-related and generic theory elements for designing and enacting PD environments.

The structural analogy between the classroom tetrahedron and the PD tetrahedron has two major limitations:

• 1. the ways that teachers engage with the PD content is not simply called ‘learning’, but is referred to as ‘professional growth’ to indicate the teachers’ much stronger agency and, in PDCGs, the facilitators’ job is not teaching, but only facilitating in a narrow sense, indicating again the teachers’ higher agency;

• 2. even more importantly, the nature of the PD content is much more complex than the classroom mathematical content. As the grey lines between the classroom tetrahedron and the PD tetrahedron indicate, the complete classroom tetrahedron is nested in the PD content (Prediger et al., 2019a). This means that all theory elements at the classroom level can potentially be a relevant part of the PD content. For content-related research on teachers’ professional growth, the nested nature of the PD content makes it necessary to unpack PD1-content into C1-content, C2-learning, C3-teaching and C4-environment.

In the next section, I exemplify how these different content-related theory elements can support the understanding of the initial vignette and how they could inform the facilitators’ reasoning and actions.

5 Generic and Content-Related Theory Elements for Explaining and Enhancing PDCG

Coming back to the vignette from Sect. 6.3, the researcher facilitator has to obtain a profound understanding of the collaborative group’s practices and challenges before changing their role from observer/analyser to facilitator of the PDCG and before supporting the group’s professional growth. These dual practical goals—the facilitator’s understanding and then intervening—have a counterpart on the theorising side. The repeated (and much more systematic) analysis of these kinds of vignettes can enhance the researchers’ theoretical understanding by processes of empirically grounded theorising. Systematically connecting the theory elements can generate a theoretical underpinning for typical facilitation practices and designs for PDCG. That is how our research group aims to find a theoretical foundation for enhancing teachers’ professional growth. For this theorising purpose, it proved to be highly relevant to unpack the theory elements on both the classroom and PD levels, not only by means of generic theory elements (which apply to all classroom and PD content), but also by means of content-related theory elements. In this case, the classroom content was understanding multi-digit subtractions and the PD content was fostering at-risk students’ access to mathematics.

Although it is not possible to demonstrate the theorising process with all its details here, this section’s intent is to show the power of working with articulated theory elements unpacked down to the level of the mathematical content (C1-content). I will successively introduce theory elements C1–C4, and then PD1–PD4, and use them for the analysis. The analysis starts at the classroom level to build the ground for analysing the group’s teaching practices and the group’s processes of professional growth later at the PD level. It ends with a look at how the facilitator reacted and how this experience informed the PD design for future collaborative groups.

5.1 Theory Elements on the Classroom Level for Explaining the Vignette

Introducing Established Theory Elements for C1-Content

On the classroom level, the theory elements for C1-content relevant for specifying the classroom content in view are printed in Fig. 6.4. The table entails not only the classical distinction of conceptual understanding and procedural skills (Hiebert & Carpenter, 1992), but also an additional distinction that emerged as necessary when working with at-risk students: the actual learning content and its foundations from previous years (Prediger et al., 2019a).The examples in Fig. 6.4 relate to the classroom content for multi-digit subtraction and its conceptual underpinnings (discussed in Hiebert & Wearne, 1996). Multi-digit subtraction with carries was the actual procedural skill to be learned, with the procedure being based on the conceptual understanding of regrouping units while subtracting. The regrouping of units is based upon the place-value understanding of the meaning of the digits and carries in the subtraction.

Fig. 6.4

Theory elements of C1-content for specifying the classroom content, with content trajectory for structuring it (C2-learning)

Analysing the Vignette with Respect to C1-Content

In the earlier vignette, Suleika mastered the basic skills of subtraction facts up to 10 and used them for multi-digit subtraction without carry. Subtraction with carry, however, is based on the conceptual understanding of decomposing numbers into digits. Suleika could not build on her mastery of multi-digit subtraction without carry due to limited fundamental place-value understanding (which becomes visible in her decomposition of 443 into 400 − 400 − 300 rather than 400 + 40 + 3; see Fig. 6.2).

Introducing Established Theory Elements for C2-Learning and C4-Environment

The generic mechanism of learning (theoretically described in C2-learning) that helps to explain Suleika’s challenge in remembering the procedure of subtraction is that sustainable learning always requires connecting to previous knowledge (Hiebert & Carpenter, 1992, p. 67). These connections can best be accomplished with consolidated understanding of the foundations of earlier years. Hence, students who have no understanding of specific basic concepts cannot continue learning along the content trajectory. In other words, the actual conceptual understanding or the procedural skills building upon these basic concepts are not accessible.

This proposition about the generic structure of content trajectories has been empirically proven for the case of arithmetic in long-term assessment studies (e.g. by Moser Opitz, 2007) and resonates with the disentangled content aspects for multi-digit subtraction (Hiebert & Wearne, 1996). In Fig. 6.4, the resulting proposition for a content trajectory is indicated by the arrow. Understanding of basic concepts often underpins basic skills, and these are necessary for building further conceptual understanding, which then underpins the new procedures.

Based on these empirical findings on typical content trajectories in arithmetic and the high relevance of the basic concepts, particularly for students at risk of missing access to mathematics, our design research group has designed learning environments in Mastering Math that enable teachers formatively to assess and foster students’ understanding of basic concepts (Prediger, Fischer et al., 2019a).

The theory elements of C4-environment underlying these learning environments include three design principles (for their empirical and theoretical justification, see Moser Opitz et al., 2017):

• (DP1) focusing on conceptual understanding;

• (DP2) monitoring students’ learning progress using diagnostic tasks;

• (DP3) promoting discourse.

This learning environment was shown to be effective for giving students safe access to the understanding of basic concepts, with significantly higher learning gains than the control group (Prediger et al., 2019a). However, the learning gains varied substantially and relied heavily on the teachers, so further PD research is required for optimising support for all collaborative groups.

Analysing the Vignette with Respect to C2-Learning and C4-Environment

The teachers in the vignette did not know about the learning environments designed and evaluated by the university at the time of the first meeting, and were not aware of the relevance of place-value understanding for multi-digit subtraction with carries. In the vignette, they reported that they had chosen other curriculum resources that provided learning opportunities for Suleika that were not aligned to the trajectory (C2-learning) in Fig. 6.4. The fact that the teachers tried to teach her later stages of the content trajectory, without taking into account the earlier stages, might explain why she always forgot the content. In this way, the theory element of the content trajectory entails the generic mechanisms of learning (procedural skills are acquired sustainably when they are connected to the underlying understanding of basic concepts) and their content-specific substantiation (multi-digit subtraction with carry should be connected to decomposing numbers and the meanings of the digits in place value understanding).

However, the teachers’ practices for differentiating relied neither on the categories from C1-content (Fig. 6.4), nor on the propositions about the learning trajectory (C2-learning) and design principles in C4-environment. Instead, they were based on the practices that the community of inquiry collectively developed, independently from the design research team. Rather than blaming the teachers for not using this unknown approach, theory elements of C3-teaching enabled the researcher facilitator to explain the teachers’ practices and their logical consistency with the approach of differentiated tasks and to describe their forms of inquiry, acknowledging the enormous efforts and achievements in their community.

Introducing New Theory Elements for C3-Teaching

Teachers’ professional practices are socially established ways of mastering recurrent situational demands in mathematics classrooms; in our case, the situational demands are differentiating and fostering at-risk students’ access to mathematics. Qualitative analysis of the teaching practices for specific situational demands is done with respect to the shared pedagogical tools (e.g. curriculum resources, tasks and teacher moves), the underlying orientations and the activated categories for perceiving, thinking and evaluating (Prediger & Buró, 2021, 2024). Empirically identified categories are usually marked by ||...|| and orientations by <...>. In these studies, we found that teachers’ differentiation practices are not always guided by the idea of enhancing students’ ||learning progress||, but sometimes simply by the category of ||task completion||, a category that Gravemeijer et al. (2016) have also identified and called ‘task propensity’. Rather than focusing on how they can leverage their students’ understanding to the next zone of proximal development along the content trajectory, some teachers optimise their differentiating practices in a way that all students can succeed to complete the task, even if no ||learning progress|| is initiated.

Analysing the Vignette with Respect to C3-Teaching

In our vignette, the practices to be analysed using the theory elements of C3-teaching are Paul’s and Maria’s (and their colleagues’) differentiating practices dedicated for at-risk students such as Suleika. The teacher community was driven by the shared inclusive orientation <a good inclusive classroom is adaptive to students’ abilities>, and they realised it using the pedagogical tools of differentiated tasks and activity settings of individualised learning. In line with design principle DP2 (monitoring students’ learning progress using diagnostic tasks), they used diagnostic information, but only about Suleika’s procedural skills, not about the underlying conceptual understanding. The design principle reflects a <diagnostic orientation> to initiate adaptively students’ ||learning progress|| that entails going back in the content trajectory first to ensure she gets access to earlier steps in the content trajectory and then can make progress along the trajectory.

This was not relevant for the teacher group, because they chose their practices guided by the shared category of ||task completion||. Applying this category, the teachers evaluated the short-term success by assessing whether a student was able to complete the task with the given support and simplifications. In this case, the teachers were successful in creating classrooms in which all students worked and completed tasks. And, indeed, simplifying Suleika’s mathematical demands to digit-bound subtractions without carries proved to be efficient for fulfilling their evaluation category ||task completion||. However, this category is guided by a <short-term orientation>, whereas a <long-term orientation> would refer to ||learning progress||, in other words, amplifying Suleika’s skill and understanding. The case of Paul in the vignette resonates with empirical results found for many teachers (Prediger & Buró, 2021, 2024).

5.2 Theory Elements on the PD Level for Explaining the Vignette and Taking Actions of Facilitation

Deriving Theory Elements for PD1-Content

The content-related analysis in Sect. 6.5.1 reveals not only that the teachers held certain orientations and categories that guided their practices of differentiating, but also which orientations and categories were relevant in the case of Suleika and the mathematical content of multi-digit subtraction. What was exemplified in this vignette was also found more systematically in many other vignettes, and revealed a specification of further content-related categories and orientations that could bring the teachers’ collective inquiry forward. In a process of empirically grounded theorising, researchers who observed several teacher groups finally structured the relevant categories and orientations as shown in Fig. 6.5, which includes, as a nested core, the categories for specifying and structuring the classroom content.

Fig. 6.5

Specifying the PD content—Teachers’ categories and orientations (PD1-content)

For facilitators, Fig. 6.5 later turned out to be a useful tool for communication with teachers (although it did not exist during the vignette itself). The theory elements C1-content and C2-learning, namely the categories from Fig. 6.4 and their structuring in a content trajectory, can help teachers make decisions about learning goals and assess students’ learning pathways along the content trajectory. This also requires a general <conceptual orientation>, in order not only to focus skills in a <procedural orientation>. However, the choice between <short-term orientation> and <long-term orientation> determines whether teachers focus on ||learning progress|| or on ||task completion|| as their major category of thinking, perceiving and evaluating their practices. Only when the category ||learning progress|| is involved can the content trajectory become relevant.

Introducing Theory Elements for PD2-Growth

To explain the mechanisms of teachers’ professional growth, the adapted model of PDCG from Fig. 6.1 is used and systematically substantiated with the unpacked PD content (in Fig. 6.6).

Fig. 6.6

Content-related substantiation of adapted model for PDCG (PD2-growth and PD4-environment)

Analysing the Vignette with Respect to PD2 and PD1

Until the facilitator came in and provided external sources, the vignette can be analysed in the three lower domains without the external domain: the collective domain, the domain of inquiry and the domain of consequences. Based on the status quo in the collective domain (as presented in the analysis by C3-teaching), the community of inquiry was working hard on new differentiation practices (their domain of inquiry). However, their current focus on the evaluation category ||task completion|| had substantial impact on the way they perceived their success in the domain of consequence.

With the theory elements of the specified PD content in mind (see Fig. 6.5), the researcher facilitator noticed that the teachers were not concerned about Suleika’s place-value understanding, although her written product in Fig. 6.2 provided strong evidence of its peculiarity. Thus, the researcher facilitator noticed that the teachers did not activate the category ||understanding of basic concepts|| for assessing Suleika’s work. The researcher facilitator assessed that the community of inquiry was driven neither by the <conceptual orientation> nor by a <long-term orientation>, which would have led them to focus on her understanding of basic concepts, rather than trying to teach her multi-digit subtraction without any place-value understanding. A first rough approximation of this analysis allowed the researcher facilitator to explain why the teachers’ enormous efforts had not yet led to satisfying long-term results (see Fig. 6.6).

At the same time, outside the shared collective domain, Maria (the special education teacher) put a second evaluation category on the table by saying, “I tried to teach them subtraction with carries several times, but they always forget it”. This category of ||forgetting|| in Maria’s utterance is a remarkable one. While it refers to the category identified as crucial in PD1-content, ||learning progress||, it does not relate students’ ||learning progress|| to teachers’ practices, but explains the failure of ||learning progress|| solely within the students. In this way, it had not yet initiated a focused reflection in the domain of inquiry for several months in the collaborative group, but a call for external support. Indeed, Paul and other colleagues articulated that they were not interested in ||forgetting||, as this could not guide their teaching and assumed too much responsibility by the students (Jackson et al., 2017). Here, the incompatibility of teachers’ orientations explains why the community of inquiry could not adopt Maria’s concerns.

This analysis in line with Fig. 6.6 gave the researcher facilitator an idea about how to enter the discourse with the collaborative group, in order to turn Maria’s concern into a collective and productive concern.

Consequences for PD3-Facilitation and PD4-Environment in the Continued Vignette

In order to draw consequences for PD3-facilitation and PD4-environment, the content-related substantiation of the adapted model for PDCG provided a helpful framework. The facilitator researcher’s practice for enhancing professional growth started with listening to the teachers and analysing their collective efforts in all three domains, by means of the theory elements PD1-content (Fig. 6.5) and PD2-growth (Fig. 6.6).

Following the principle of building upon teachers’ collective starting points, she realised that the <conceptual orientation> was not the ideal starting point to discuss in this community, as it was too far from their actual collective concern. Instead, she chose <short-term versus long-term orientation>, in order to relate Paul’s and Maria’s points and build upon Paul’s intention to consider aspects that they could influence in order to offer a new orientation. The following utterances are synthesised from a longer conversation:

Facilitator: Paul says you can handle Suleika’s difficulties successfully by giving her only subtractions without carries. However, Maria does not seem to be satisfied with the learning outcome. What is the problem with Suleika always forgetting the procedure, Maria?

…

Facilitator: You also seem to be interested in the long-term learning. Can we go back some steps and check what Suleika can master on her learning pathway towards multi-digit subtraction? I see how she decomposed the numbers (443 = 400 – 400 – 300); do you think this could have any impact on her ability to remember the procedure of managing carries?

After 30 min of discussion, Paul, Maria and their colleagues collectively decided that they needed to go back in the content trajectory, in order to stabilise Suleika’s learning pathway in a more sustainable way. It took them much longer to realise that they needed to provide learning opportunities for the understanding of basic concepts, and that this might also be the more productive practice of differentiation. Thus, the first external offer provided by the facilitator could strengthen Maria’s implicit <long-term orientation>, which also opened the teachers to other evaluation categories for their teaching success. However, to enact teaching practices towards the new evaluation category ||learning progress|| in ||understanding of basic concepts||, they required further external offers, namely pedagogical tools for formatively assessing students’ ||understanding of basic concepts|| and teaching material for enhancing them.

The externally offered curriculum materials for formative assessment and remediating sessions provided them not only with the required pedagogical tools, which they could now integrate into their practices, but also with access to the detailed pedagogical content knowledge on understanding of basic concepts for other mathematical topics, such as place-value understanding on the number line and meanings of multiplication and division (see Prediger et al., 2019a).

Once the teachers had incorporated these categories and orientations into their collective domain, their inquiries resulted in bigger changes of their practices and a closer approximation to the newly set goals: enhancing all students’ ||understanding of basic concepts|| in order to assure adaptation practices to ||individual learning progress||. Based on these experiences, it took another year before a <conceptual orientation> really started to guide their work instead of a purely <procedural orientation>.

Interestingly, it entered their collective domain via the domain of inquiry, when experimenting with the curriculum materials for all students and experiencing “lovely aha moments, when students say ‘now, I really got it!’” In this way, the teachers’ pathways of long-term collective professional growth reflected an interesting interplay among the four domains, with a growth pathway that was not at all linear.

Tentative Content-Related Theorising on PD2-Growth and PD4-Environment

As these considerations illustrate, the adapted model for PDCG can serve as a theoretical framework for explaining teachers’ professional growth (PD2-growth) and for offering external sources to the PDCG environment to promote the professional growth (PD4-environment). Looking back on the vignette and its illustrative analysis, the term generic search space can be further unfolded. The general model only provides the framework for necessary content-related theorising. The analysis of this vignette and many further cases with the same PD content (e.g. Prediger & Buró, 2021, 2024) resulted in the first tentative theorising about teacher communities’ content-related growth pathways towards striving for differentiating practices (PD2-growth) and the roles of external resources such as classroom material in supporting the process (PD4-environments).

When communities of inquiry work on innovative practices (in this case, on their differentiating practices), their evaluation categories in the domain of consequence might be the most crucial to develop as it has important impact of all practices. As an interview study on self-reported practices revealed (Prediger & Buró, 2021), teachers’ leading evaluation categories influence their monitoring practices because they have an impact on what teachers intend to notice in students’ work. For fostering students’ learning in differentiated ways, the leading evaluation category influences whether teachers adopt compensation practices (while aiming at circumventing weaknesses by supporting students to work around limited abilities in the leading category ||task completion||) or enhancement practices (starting with setting differentiated learning goals along the content trajectory and guided by the category of adaptive ||learning progress|| along the content trajectory). The difference between compensation and enhancement practices is also visible in a video observation study on classroom practices (Prediger & Buró, 2024).

As the leading evaluation categories are revealed to be so crucial across various case studies from PD research projects, they are therefore an important focus for theorising. In the project, we hypothesise that potential growth trajectories of teacher communities’ learning of the PD content ‘fostering at-risk students’ access to mathematics’ might be characterised as a successive extension of evaluation categories in four stages (see Fig. 6.7). Teacher communities often start with evaluating only ||work intensity|| (no matter on what), but quickly turn to ||task completion|| to incorporate a first mathematics education perspective. ||Task completion|| stays an important category for the evaluation in <short-term orientation>, but should be complemented in <long-term orientation> by ||learning progress||. Often, teacher communities first focus on the evaluation category ||learning progress|| only in <procedural orientation>, which can later also be extended to including both ||procedural and conceptual learning progress|| when the <conceptual orientation> can be established.

Fig. 6.7

Content-specific theory element for PD2-growth—hypothesis on trajectory of growth by successively extending the repertoire of evaluation categories for fostering practices

Hence, initiating shifts in the evaluation categories might be the most crucial external input required to allow teacher communities to continue their independent inquiries. However, orientations and evaluation categories cannot simply be ‘taught’, so facilitators need to find entry points for their successive extension. Offering curriculum materials for monitoring (formative assessment) and enhancement sessions may not only provide a pedagogical tool for developing teaching practices, but might also be a key external source for indirectly influencing the collective domain when the teacher group starts to adapt and appropriate the curriculum materials for their purposes. In our situation, the curriculum materials seem to have strengthened the shared pedagogical content focus on further understanding of basic concepts in different mathematical content (see Fig. 6.6). Although both hypotheses (the hypothesised trajectory of growth and the hypothesis on the effects of the curriculum material) emerged from the qualitative analysis of various case studies, they will require further systematic investigation before they can count as stable empirically grounded theory elements.

Zooming out from the specific PD content ‘fostering at-risk students’ access to mathematics’ to a more generic perspective, three main lessons learned might be derived for PDCG in general, with respect to Putnam’s and Borko’s (2000) three areas of consideration: (1) where to situate teachers’ learning experiences; (2) the nature of discourse communities for teacher learning; (3) the importance of tools.

• 1. It is worth situating teachers’ learning experiences in communities of inquiry with emphasis on the content-specific domain of inquiry.

• 2. The nature of the discourse in PDCG is heavily influenced by shared orientations and categories, specifically by evaluation categories. Extending these categories in the domain of consequence seems to be an important step in the trajectory of the community’s growth, to be taken into account by the facilitator.

• 3. Classroom materials are relevant tools that can support the teachers’ monitoring and enhancement practices and, at the same time, serve as tools to extend the communities’ shared orientations and categories specific for the mathematical content in view.

6 Meta-theoretical Reflections on the Necessary Topic-Specific Theory Elements

The intent of this chapter is to contribute to developing theoretical foundations for explaining and promoting teachers’ professional growth in collaborative groups. Building upon the general sociocultural framework on teachers’ practices (Wenger, 1998) and the construct of communities of inquiry (Jaworski, 2006), as well as an adapted model of professional growth (adapted from Clarke & Hollingsworth, 2002), it used the exemplification in one vignette in the community of inquiry to show the following.

• Theory elements of PD content (PD1), teacher growth (PD2), facilitating (PD3) and PD environment (PD4) are all useful and necessary for explaining and promoting professional growth in collaborative teacher groups. Of course, other vertices and edges of the tetrahedron also serve theorising, but these four may be considered a minimal set.

• The PD content comprises the classroom mathematics content (in our case, multi-digit subtraction, for which a substantial research background exists), but also the specific teaching practices that the communities of inquiry have chosen to work on—in our case, differentiating and fostering at-risk students’ access to mathematics. Even if an area of PD content is usually not sharply defined in communities of inquiry but successively emerges, unpacking it with respect to the underlying orientations and categories that the teacher community implicitly or explicitly refers to is crucial.

• 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.

• The structures of the big theoretical frameworks (communities of inquiry and models of professional growth) are helpful in understanding the complexities and intertwinement of different domains. However, they mainly provide a generic search space. Informing the concrete analysis and especially the concrete PD design and facilitation, they must be elaborated in content-related ways for different areas of PD content (Prediger et al., 2019b). Of course, the large body of mathematics education research on the mathematical content and on mathematics teaching and learning has revealed many candidates for theory elements, most importantly for C1–C3, but also for C4.

• The generic theory elements from the PD level gain their explanative power when being filled in content-related ways, and this also requires the nesting of corresponding theory elements (C1–C4) from the classroom level into the PD level. The more this nested structure is unpacked, the more we learn in content-related ways about the PD content.

Finally, let me briefly respond to two questions. Is the meta-theoretical reflection here (a) specific to mathematics and (b) specific to PDCG, or could it have been suggested with respect to any form of PD for any subject matter?

• (a) Whereas other colleagues have emphasised the need to be specific to mathematics, I emphasise that we need even further substantiation in content-specific ways. At the same time, the model itself is not mathematics specific and might be applied also for other subjects, with their particular content.

• (b) Whereas the above meta-theoretical arguments are not specific to the form of PD in view here (PDCG), the main result of the current case study might be characteristic for the collaborative setting and relevant for supporting collaborative groups. The shared evaluation categories seem to be the crucial point, more than the shared knowledge or orientations as a whole. As long as individual evaluation categories have not really entered the collective domain (as Maria’s ||forgetting||), they cannot exert their influence, and this can also hinder the professional growth of the individual within the collaborative group. This interplay of individual and collective learning in particular will require substantial further empirically grounded theorising.

However, both claims of potential specificity or generalisability will require further PD research, in order to be explored in depth by comparing across content, across subjects and across particular PD settings.