The 3T-Model as introduced in the previous section should not be misunderstood as a “completist” research program. Hardly any research can manage to take into account all different facets and levels simultaneously. Instead, PD researchers have adopted different strategies to consider the multi-level structure and to foreground some faces while keeping others in the background. In the next subsection, we outline three main strategies by which current PD research has realized a multi-level perspective, which we call the lifting, the nesting, and the unpacking strategies.
After this general introduction, the two subsequent subsections provide examples where the 3T-Model is used to analyze two examples of PD design and research, where the three strategies were used implicitly or explicitly by the researchers and locate the foregrounded facets in the three tetrahedrons. In a more overarching approach, the model can be used to understand existing PD research studies and their location, thereby showing the gaps they still leave.
Three strategies for setting PD research agendas dealing with the multi-level structure of PDs
By studying the existing literature on PD research and analyzing it with regard to possible gaps, we identified three general strategies for designing PD and generating relevant research questions, addressing the multi-level structure of PD (see Fig. 4).
A lifting strategy, by which design and research approaches are lifted from the classroom level to the TPD level or to the FPD level. This strategy draws upon the structurally analogous foci of the faces of the tetrahedrons, which allows for using structural analogies of content-related design and research on the three levels. In brief, lifting a design approach means that design principles or design elements developed for the classroom level are implicitly or explicitly transferred (and adapted) to the TPD level (or from the TPD to FPD level). Lifting a research practice means that certain types of research questions and/or methods from the classroom level are implicitly or explicitly transferred (and adapted) to the TPD level (or from the TPD to FPD level) and applied in an analogous way.
A nesting strategy for a research-based PD design, which takes into consideration that, during the design, aspects of the complete classroom tetrahedron (aspects of all vertices and edges) should be nested in the TPD content and aspects of the complete TPD tetrahedron should be similarly nested in the FPD content. This design strategy accounts for the holistic challenges of PD and addresses multifaceted complexities—without claiming any completeness. In brief, the nesting strategy builds the PD design upon the idea of structuring the TPD/FPD content in a self-similar nested structure, taking into account the complexities of the tetrahedrons below.
An unpacking strategy for content-related, design-based PD research which, during research, unpacks the PD content on the TPD level (or the FPD level) with respect to the elements of the respective tetrahedrons of the levels below, for instance, when investigating how facilitators’ knowledge grows during the FPD on the different levels and with respect to different vertices and faces of different tetrahedrons. This research strategy specifically emphasizes the relevance of content-specific research underpinning the chosen design.
The lifting strategy
The lifting strategy capitalizes on the structural analogies between the three levels, that is, the analogous facets of teaching and learning. It is applied when successful design and research approaches are appropriated on the next level for the analogical facets and with structurally analogical questions, instruments, and methods (as, for example, by Jackson et al. 2015). This especially applies to approaches for investigating the phenomenology of teaching and learning of specific content and to theoretical frameworks that can be used as lenses for such research:
Design approaches drawing on video-based material to support teachers’ noticing of relevant classroom incidents can be lifted from the TPD level to the FPD level to help facilitators developing their noticing of teacher learning. Thereby, design elements such as structured or unstructured video clubs can be based on findings on the TPD level (van Es and Sherin 2008).
Research approaches for investigating students’ conceptions and learning pathways for a specific mathematical content (by interview studies or design experiments) can be lifted to the PD level by investigating teachers’ conceptions and learning pathways for a specific PD content area such as language responsiveness in mathematics classrooms (e.g., Smit and van Eerde 2011; Prediger 2019). This resonates with the call for more process focus for teacher learning pathways by Goldsmith et al. (2014).
Research approaches investigating teachers’ classroom practices can be lifted to the PD level to allow for exploring facilitators’ teaching practices. This has been done, for example, for facilitation moves (e.g., Stein et al. 2008; Tekkumru-Kisa and Stein 2017). That is, theoretical frameworks are first moved one level up to guide the research.
Most projects applying the lifting strategy for setting the design or the research agenda can exploit the structural analogies, but at the same time experience its limitations since teachers as learners are different from students as learners, and PD content is usually more complex than classroom content. In order to account for this higher complexity, nesting and unpacking strategies are important.
The nesting strategy
As the PD content on the TPD level usually encompasses aspects of the complete classroom tetrahedron (and the FPD content consequently aspects of all vertices of both tetrahedrons), the nesting strategy builds the PD design upon the idea of capturing this complexity in a self-similar nested structure (e.g., Zaslavsky and Leikin 2004; Carroll and Mumme 2007; Perks and Prestage 2008; Ball 2012; Luft and Hewson 2014; Wood and Turner 2015) and of exploiting the research findings in the tetrahedrons below for shaping the TPD and FPD content. This provides the opportunity, for example, to rely on research results for the classroom level in order to specify the TPD content:
When designing a TPD for a specific classroom innovation (e.g., using digital tools for exploring algebra), the PD also has to relate to students’ typical learning pathways in exploratory situations and to misconceptions for algebra.
Analogically, the FPD preparing for leading such a PD course should take into consideration all relevant aspects on the classroom level, but also activities by which the teachers can reflect on these issues and empirical insights into typical teacher learning pathways, such as typical challenges that teachers face when implementing these exploratory situations in their algebra classrooms.
These examples show that the nesting strategy for research-based design imposes research questions: When teachers’ learning pathways toward digital tools in algebra classrooms are considered to be a relevant part of the nested FPD content, then they should be investigated. First, for example, when designing a TPD for teaching in heterogeneous mathematics classrooms, one can exploit the research on the classroom. However, when specifying the content of the PD, one realizes that although there is research on, for instance, the effectiveness of student grouping strategies (Slavin 1987; Deunk et al. 2015), there are only few empirical findings on content-specific whole-class differentiation (e.g., Klieme et al. 2009, on the Pythagoras theorem). Thus, such content-specific research is required in order to inform facilitators on the nesting strategy, which draws on research-based design aspects for the PD level.
On the second level, specific empirical insights are required into how teachers can best be introduced to these content-specific whole-class differentiations in their mathematical topic. Again, the nesting strategy for research-based design imposes new research questions, also on the TPD level.
So far, the nesting strategy has been introduced in the literature in a rather generic way without specifying what nesting means given that a specific content area is subject of the PD. From our viewpoint, the nesting strategy indicates research gaps on the levels below, which should be closed in order to inform the content specification in the research-based design of PD courses.
The unpacking strategy
Before generally describing the research strategy of unpacking, the example of designing PD for a specific classroom innovation from above is continued: A content-related research approach investigating the effects of a designed FPD course not only asks generic questions on facilitators’ general acceptance of the classroom innovation but unpacks different aspects from the FPD content in order to capture what the facilitators drew from the FPD and might implement in their PD practices. This will refer not only to mathematical knowledge about exploring mathematical theorems but also to knowledge about typical classroom tasks (knowledge about CR) and facilitators’ knowledge about teachers’ typical learning pathways when implementing the explorations (TC-T and T-CR).
This shows the complementarity of lifting and unpacking strategies: Whereas the lifting strategy exploits the structural analogies for design and research but treats every learning content in an unpacked way, the unpacking research strategy takes into account the nestedness of the PD content and goes down to aspects of the tetrahedrons below. For example, the necessary facilitation moves in TPD might not be the same in classrooms as the PD content is much more complex than the classroom content and takes into account the mathematical classroom content as well as didactical aspects on learning the mathematical content. Hence, the strategy of unpacking the nested PD content is fundamental to allowing the PD research to be really content specific rather than only generic. By unpacking the PD content along the tetrahedrons, a language is provided for capturing the complex interplay of different aspects. This is specifically important for design-based PD research but is also important for content-related PD research that does not start from a design perspective.
Although the three strategies can be distinguished structurally, they usually appear in intertwined ways in many research projects. In the following two sections, we provide examples and discuss how the 3T-Model enables disentangling the elements of PD design and research and the respective strategies in these projects.
An example for existing research in lifting, nesting, and unpacking strategies between the classroom and TPD tetrahedrons
An example for PD research that intertwines the different strategies is Swan’s (2007) often-cited paper on a task-based PD. As Fig. 5 illustrates, its focus can be located in the 3T-Model (here only the two lower tetrahedrons).
The starting point for the PD, extensively described in Swan (2007), was the research-based design of rich algebra tasks (including video clips showing the tasks in use), which were developed in order to foster low-achieving students’ algebraic understanding. This classroom resource (CR) was iteratively optimized to engage students in cognitively demanding mathematical activities and rich mathematical discussions. The material was then used to initiate change in teaching practices of the 44 teachers, who had participated in 4-day PD program, spread over 6 months. After an initial phase of reflecting the participants’ contexts for working, the task types were introduced and discussed, and in a later session, the teachers’ experiences in adapting and using the tasks were shared.
The paper as a whole delivers the core message that the careful design of classroom resources (CR) can be an important tool for PD courses to represent the real or intended practice on the classroom level. Tasks are typical classroom resources, and here they are also considered a major element of TPD, as they can engage teachers in the important activity of experimenting with these tasks in their own classrooms and thus become integrated in TPD resources (TR). Hence, the main design focus for the PD was designing rich activities around the tasks and videos. The strategy visible in this PD amounts to the author having lifted his own major design focus (engaging learners in rich activities) from the classroom to the TPD level—without the author stating so explicitly in their reports. The fact that he also included videos from classroom situations in which students worked on the tasks shows how he nested the classroom tetrahedron in the PD content: Not only the tasks as resource, but also practices of dealing with the tasks were considered by the nesting strategy for design.
The PD research reported in the paper corresponds to a second lifting strategy, this time for research. Rather than considering what students learned by the activities (lower face CR-CC-S), which is a typical research perspective in design research, the research accompanying the PD investigates what teachers learned during the PD (lifted lower face TR-TC-T). However, due to the nesting of the PD content (TC), the research cannot be organized in strict parallelism because it has to take into account the much more complex character of the TPD content (TC). In this paper, the unpacking amounts to conceptualizing teacher learning as change in teacher practices and teacher beliefs, which is captured by questionnaires and interviews. A deeper way of unpacking could have included the learning of students, e.g., by analyzing products from the classrooms of the PD participants.
The interesting question is how the nested TPD content is unpacked in order to capture this change. A careful reading of the paper shows its limitation in this respect: Beliefs are only considered in a generic way, addressing mathematics (CC), teaching (whole tetrahedron), and learning (lower face CC-CR-S) with respect to transmissive, connectionist, and discovery stances. The practices are captured more content specifically with respect to the tasks (back face CR-T-S), but without explicitly taking into account aspects of algebra learning (in the lower edge CC-S). In this way, the unpacking remains restricted to generic aspects and does not include the focus on the mathematical learning content (CC). For the purpose of informing facilitators, however, the teachers’ change with respect to algebra learning would have also been of great interest.
The preceding analysis showed how locating the focus of the research within the 3T-Model enables us to structurally understand the research approaches, the intertwinement of applied research strategies, and how to identify potentially important gaps in the research.
An example for existing research in lifting, nesting, and unpacking strategies between the classroom, TPD, and FPD tetrahedrons
An example for FPD design and FPD research that (implicitly or explicitly) combined the different strategies is Borko et al.’s (2014) multi-year design research project on FPD to promote the Problem-Solving Cycle in TPD. Figure 6 shows how the focus of the project can be located in the 3T-Model and how all three levels are integrated.
The TPD activities in the Problem-Solving Cycle approach include multiple cycles of interconnected PD workshops that center around mathematically rich tasks. In each cycle, teachers first solve the given problem on their own and develop a lesson plan for teaching it in their classrooms that considers their students’ specific learning needs. The teachers then implement the problem, immerse their students in solving it, and video-record the classroom situation. The TPD facilitator carefully selects video clips that show key examples of both teacher instruction and student thinking. The video clips take a prominent role within the TPD as they are used as starting point for discussing how to elicit and build upon student thinking and what instructional aspects push students toward the learning goals.
The preparation of teacher leaders to facilitate the Problem-Solving Cycle included two and a half years of courses and support in changing the role from teacher to facilitator. The specific design of the facilitator preparation was developed and investigated in a multi-year design research project. In the FDP, the novice facilitators were also immersed in the Problem-Solving Cycles described above, so the designers adopted a lifting strategy for the design. In addition, the FPD comprised a summer leadership academy to initially introduce the key characteristics. Also, when conducting the TPD on their own, the facilitators received ongoing support from the research team.
Preparing the facilitator to lead such a TPD with the Problem-Solving Cycle with integrity was based on the following key characteristics: using a rich mathematics problem and involving teachers in the solution process; facilitating productive discussion on the mathematics content, students’ thinking, and instructional practices; focusing the discussions on diverse representations and solution strategies; and carefully selecting video clips from teachers’ classroom practice.
As overarching design element, it is clear that the Problem-Solving Cycle as a teaching approach has been lifted from the classroom to the teacher and facilitator PD level to immerse teachers and facilitators in discussing tasks they first approached themselves. Carefully designed tasks serve as resources to engage students (classroom resources CR), teachers (TPD resources TR), and facilitators (FPD recourses FR) in rich mathematical discussions (and address them in their double roles as educators and learners). Both approaches of implementing the Problem-Solving Cycle on the PD level and the classroom level are grounded on prior research that provides empirical evidence for positive effects on student learning.
In addition, the project designers implicitly drew on the nesting strategy for specifying the TPD content as they to consider all faces of the classroom tetrahedron in the PD: The face of content-specific learning support (right face on classroom level) is integrated through videotaping teachers’ classroom instruction, and the content-specific learning pathways are displayed by the students’ artifacts the teachers collected and also what could be observed in the selected video cases (lower face of the classroom level). The structuring of the content (TC) and the realization of resources (TR) was particularly embedded in the TPD design as teachers developed a lesson plan together during one of the sessions. The nesting strategy can also be observed for the design on the FPD level, as the tetrahedron on the TPD level was additionally taken into account to specify the FPD content. Using mini-cases to simulate implementation of the Problem-Solving Cycle, the facilitators’ discussion was directed to the interplay of teacher learning and instructional practices.
The dominant research strategy in this study was unpacking from the FPD to the TPD level. Besides investigating whether the main program goals were reached in the Problem-Solving Cycle the facilitators provided, the research concentrated on unfolding what characteristics of the approach on the TPD level were enacted particularly well or inadequately:
The facilitators performed well on sustaining a professional learning community (content-independent pedagogy) and were able to select videos to enact rich discussions on important aspects relevant to classroom practice (TR). Regarding the latter, a facilitator’s guide providing key characteristics of “rich” video clips and instructional support meetings where the selection of video cases could be probed together with colleagues proved to substantially inform facilitation practice.
However, the research shows limitations with respect to TPD content enacted by the facilitators: The facilitators had difficulties in “support[ing] deep analysis in discussions to foster both SCK [specialized content knowledge] and aspects of PCK [pedagogical content knowledge]” including knowledge of content and students … and knowledge of content and instruction” (Borko et al. 2014, p. 164), indicating that they struggled with providing content-specific learning support (right face on the PD level) for the participants of their PD with respect to the mathematical content itself (CC). The researchers concretized that although the FPD included many opportunities to facilitate such “high-press exchanges,” the participating facilitators struggled with “engaging teachers in discussions about the relationships, affordances, and constraints of representations and solution strategies” (ibid., p. 164). Grasping these limitations was only possible by the (implicit) research strategy of unpacking the PD content (TC) into its component of the classroom tetrahedron (of course, phrased in other terms by Borko et al. 2014).
In light of the 3T-Model, one could infer that more insights into teachers’ content-specific learning pathways could have informed the facilitator preparation in this respect. Again, unpacking seems to be a valuable research strategy for identifying these possibilities and to draw consequences for the FPD content which transcend the simple lifting strategy for design and take into account the nested structure of PD contents. In this way, lifting, nesting, and unpacking can be combined.