Introduction

When teachers are working on an educational innovation, they often perceive conflicts in recurrent teaching demands, which can hinder the development of their teaching practices (Jaworski, 2006; Thomas & Yoon, 2014). However, not all of these conflicts are unsolvable dilemmas, as they can be solved by expert teachers (Brodie, 2010). In these cases, empirical research on teachers’ perspectives can substantially contribute to identifying and explaining the teachers’ approaches to new and challenging teaching demands and suggesting possible practices to integrate them into their teaching practice so that professional development programs can provide opportunities for teachers to overcome potential perceived conflicts (Prediger at el., 2015). In this paper, this research agenda is pursued for the case of inclusive mathematics education, which provides a focal point for research on teachers’ practices for dealing with new demands resulting from the complexity of inclusive mathematics education.

Establishing inclusive education was pushed as an educational innovation in many countries following the UN Convention of the Rights of Persons with Disabilities (United Nations, 2006). Inclusive education aims at ensuring all students’ right to participate in quality education addressing their individual needs and abilities (UNESCO, 2009). The implementation of these aims not only requires revised placement policies (e.g., by closing special education schools), but heavily depends on the teachers’ daily practices in mainstream subject matter classrooms for fulfilling two major teaching demands (called “jobs” for short): (a) constituting communities to which all students belong and orchestrating joint experiences and (b) promoting individual mathematical learning by adaptive learning opportunities addressing diverse needs.

According to recent research reviews on inclusive education, few empirical studies have investigated teachers’ practices and orientations for mastering these two jobs (Avramidis & Norwich, 2002; Lambert & Tan, 2020; Van Mieghem et al., 2020). One of the rare video studies on inclusive teaching practices in mathematics classrooms reveals that most of the observed teachers fulfill only job (a) or (b) but much less often both at the same time (Krähenmann et al., 2019). This raises the research question of whether and on what background mathematics teachers perceive a conflict between different demands of inclusive education and which practices are suited to integrate these demands rather than treat them separately or as conflicting.

In this paper, we report on a qualitative study that investigates teachers’ self-reported practices in German inclusive secondary mathematics classrooms that were captured in a professional development (PD) program. Within the theoretical model of content-related teacher expertise (Bromme, 1992; Prediger, 2019), we locate potential conflicts between demands of inclusive education and practices to integrate these demands in an interplay of jobs (a) and (b) with two underlying orientations, which we will introduce as social orientation and mathematical orientation.

For this purpose, the first section presents the theoretical backgrounds on inclusive mathematics education, professional expertise, and teachers’ approaches to new teaching demands. The second section reports on the research context and the methods of data gathering and data analysis. The third section then presents the empirical findings, while the final section discusses achievements, limitations, and consequences for PD programs. We also reflect what this particular case of inclusive education can contribute to other innovative content with new teaching demands.

Theoretical background

Conceptual framework of the study

In this paper, the research on teachers’ approaches to new teaching demands in introducing inclusive practices is conducted within a qualitative approach based on the model of content-related teacher expertise (Prediger, 2019). The model was originally adapted from Bromme (1992) and connects teachers’ practices for dealing with teaching demands (jobs) to the underlying reasoning, using five constructs:

  • Jobs are defined as the typical, often complex situational demands of subject-matter teaching. Whereas Bromme (1992) considers general teaching demands such as planning lessons in general, the content-related model focuses on teaching demands relevant to the innovation content in view (such as inclusive mathematics education).

  • Practices are defined as the recurrent pattern of teachers’ utterances and actions for coping with these jobs (separately or in an integrated way). It is the key idea of the model that teacher practices can be characterized by the underlying categories, pedagogical tools, and orientations upon which the teachers’ actions implicitly or explicitly draw.

  • Pedagogical tools are the concrete, visible tools applied to coping with the job (e.g., differentiated tasks, scaffolding moves, and visualizations).

  • Categories are conceptual (i.e., non-propositional) knowledge elements that filter and focus teachers’ categorial perception, thinking, and evaluation. Bromme’s (1992) construct of categories resonates with Vergnaud’s (1998) construct of concepts-in-action, which explains that they explicitly or implicitly guide the practices when they “cut the world into distinct … aspects and pick up the most adequate selection of information” (p. 219).

  • Orientations refer to content-related and more general beliefs and attitudes that implicitly or explicitly guide teachers’ perceptions and prioritizations of jobs and categories (e.g., beliefs about the content or students’ learning processes; as Schoenfeld 2010, p. 29 explains).

The interplay between these five constructs can be studied for descriptive and prescriptive research purposes and their combination: In a prescriptive mode, the approach of job analysis is used for identifying theoretically what categories, pedagogical tools, and orientations are required for targeted productive teaching practices (Bass & Ball, 2004; Bromme, 1992), which is used for deriving prescriptions for relevant PD content. In a descriptive mode, teachers’ enacted practices are analyzed empirically to identify the underlying categories, pedagogical tools, and orientations that teachers activate. When combining prescriptive and descriptive research purposes, the targeted and enacted practices can be contrasted to reveal more subtle insights into the interplay between practices, categories, and orientations and possible perceived conflicts (Prediger et al., 2015). This also allows design researchers to specify further categories and orientations that teachers should learn in order to leverage their enacted teaching practices into intended teaching practices (McDonald et al., 2013).

Innovation content in view: inclusive mathematics education with its jobs and orientations

Inclusive education became an area of relevant innovation content when many school systems changed their placement policies after signing the UN Convention (2006): In German schools, the rate of students with special educational needs who are placed in regular schools instead of special education schools has increased from 19.8% to 2009 to 42.3% in 2018, with the highest increase in secondary comprehensive schools (KMK, 2020; see also Kollosche et al., 2019). In the same decade, German schools became increasingly aware of equity concerns, as many students from marginalized groups in the regular classrooms were left behind for reasons such as limited access to academic language and low socio-economic and/or immigrant background (OECD, 2006; Stanat 2006). In many countries including Germany, these concerns are subsumed under wide conceptualization of inclusion as bringing together all students (Grosche, 2015; Skovsmose, 2019), regardless of whether the differences are traced back to special educational needs or belonging to marginalized groups. In particular, this study focuses on including low-achieving students and those with mathematical learning disabilities (Scherer et al., 2016), without a narrow focus on specific diagnostic criteria.

Although the rate of inclusive placement increased in Germany, institutional conditions did not necessarily improve accordingly. In most cases, students with and without special educational needs are schooled alongside “regular” students by teachers with little preparation for inclusive mathematics education, and only very limited time resources exist for support by special education teachers. Possibly due to these conditions, regular teachers often perceive inclusion as a valuable yet impossible task. This makes inclusive mathematics education a promising area of research for investigating teachers’ approaches to dealing with new teaching demands, especially as much of research itself is centered around possible incongruencies and contradictions in the concept of inclusion.

Although inclusion has a long tradition in special educational needs research, it does not have a single agreed-upon definition (Graham–Matheson, 2012). One reason for this could be that researchers tend to conceptualize inclusion through the use of very different concepts on different levels, which impedes establishing connections between research (Grosche, 2015). Some research is related to the orientations of inclusion. Roos (2019) traces the missing definition of inclusion back to two general strands of discourse that pursue different goals: First, a discourse of ideology about the normative foundations of inclusion, mostly related to concerns of equity. Second, a discourse of ways of teaching about the ways in which inclusion can be realized in classrooms. Such distinctions are here subsumed under two different orientations:

  • Following a social orientation, inclusion is connected to the general “ethical commitment [of inclusion as project in society] to find ways to live and learn together, to deal productively with otherness” (Kollosche et al., 2019, p. 4).

  • Following a mathematical orientation, inclusion is concerned with creating suitable learning opportunities for every student and with creating mathematical learning communities.

Another approach for conceptualizing the demands of inclusion is to focus on the jobs required for inclusive teaching. Similar to the distinction between two orientations, jobs can illustrate a different kind of possible conflict. On the one hand, UNESCO (2009) described inclusive education as an “ongoing process aimed at offering quality education for all, while respecting diversity and … differing needs and abilities” (p. 18). This definition calls for teachers to teach in such a way that individual students are respected and the learning of every individual is ensured. On the other hand, researchers are also interested in how teachers orchestrate their classrooms and how they create collective learning (Höveler, 2019; Krähenmann et al., 2019). To conceptualize this distinction, two jobs have been identified:

  • Promoting Individuals Adaptively, which means focusing on individual needs and promoting individual mathematics learning through adaptive learning opportunities and.

  • Establishing Joint Experiences, which means forming communities to which all students belong and orchestrating joint experiences.

The interplay of the jobs and orientations can be used to specify the demands of inclusive education. From this perspective, the UNESCO definition of inclusion is mostly concerned with Promoting Individuals Adaptively under a mathematical orientation. Taking into account the orientations and jobs can also illustrate the potential conflicts that teachers face by highlighting the conflicts articulated in research. For example, Grosche (2015) identified two different goals of inclusion. The first goal is the recognition of students with special educational needs, enabling them to participate in society and to be appreciated socially. The second goal is remediation, meaning promoting the learning of students with special educational needs as much as possible so that they reach the basic qualifications for participating in society. Grosche (2015) considered inclusion as bearing an antinomy, because these goals can sometimes be contradictory: remedial teaching can lower the recognition of students with special educational needs, and recognition does not automatically lead to remediation. Here, Grosche (2015) outlined a conflict between Promoting Individuals Adaptively under a mathematical orientation (as he was concerned with the mathematical learning of the individual) and Establishing Joint Experiences under a social orientation (as he was concerned with the social place of special needs students in classrooms).

In contrast, other researchers have been concerned with Establishing Joint Experiences under a mathematical orientation by tracing how students cooperate in the classroom and how mathematical ideas can emerge from classroom activity (Höveler, 2019; Krähenmann et al., 2019).

However, the meta-review by Van Mieghem et al., (2020) revealed that most empirical studies on inclusive education have mainly concentrated on students’ social participation, whereas academic participation in specific subject matter learning processes has mainly been investigated with regard to enrollment rates and rarely in more detail within classrooms. Thus, Establishing Joint Experiences as a job has mostly been interpreted under a social orientation, although it should also be interpreted under a mathematical orientation.

In contrast, Promoting Individuals Adaptively has mostly been interpreted under a mathematical orientation. To maximize the individual learning growth of all students, adaptive teaching that addresses all students’ diverse needs and abilities is crucial (Corno, 2008; Hardy et al., 2019). In the past, attempts were made to achieve adaptive teaching using ability grouping or supplemental instruction outside class. As these practices have grown to be considered a contradiction to the job of Establishing Joint Experiences under a social orientation, /they have increasingly been substituted by practices of differentiated instruction (Tomlinson et al., 2003) or adaptive teaching (Corno, 2008) within class. These adaptive practices can use curricular adaptations (modifying what is learned), instructional adaptations (modifying how it is learned, e.g., by additional scaffolds or simplification), or alternative adaptations (modifying the what and the how to a completely different learning setting; Janney & Snell 2006). Due to the cumulative nature of mathematics understanding, secondary mathematics education cannot rely only on instructional adaptations (as foregrounded in inclusive instructional approaches such as Universal Design for Learning; Rose & Meyer 2006), but also requires curricular adaptations to each student’s mathematics learning needs (Krähenmann et al., 2019), for instance, to the understanding of basic concepts.

In summary, research on inclusive education has tended to equate Promoting Individuals Adaptively with a mathematical orientation and Establishing Joint Experiences with a social orientation. By integrating the jobs and orientations, inclusive education is here considered to be a space of categories spanned by orientations and jobs (Fig. 1). Not only do teachers have to integrate social and mathematical orientations, they also have to do this for both jobs. The job of Promoting Individuals Adaptively needs to be implemented differently for each orientation. Following the social orientation, students have the individual need to feel recognized, to conceive of themselves as belonging to the class as a legitimate part of the whole. Following the mathematical orientation, students have individual learning goals. Establishing Joint Experiences can also be interpreted socially as enabling social integration without taking into account the learning of the mathematical content. Following the mathematical orientations, joint experiences can relate to the form of cooperative learning without explicitly creating social integration.

Fig. 1
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Inclusive mathematics education is a space spanned by orientations and jobs

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The two jobs need to be enacted following two orientations, which creates four combinations of concretized demands of inclusive education. None of the combinations can be considered superior to the others, and they likely cannot be addressed together in every moment of teaching. Instead, teachers have to navigate this space of demands for inclusive mathematics education. Within mathematics education research, instructional approaches have been developed for addressing different demands at the same time, for example, using the same rich open-ended tasks for all students with “natural differentiation”, in other words, tasks allowing students to work on different levels according to their abilities (Lawrence-Brown, 2004; Scherer, 2013), and the approach of “mutual learning” by which heterogeneous small-group communication is initiated on different levels (Höveler, 2019). However, like for many other innovations (Century & Cassata, 2016), the implementation of instructional approaches depends on the teachers’ practices and orientations.

Existing research on teachers’ inclusive teaching practices

As several research surveys on inclusive education have revealed, teachers’ inclusive practices and orientations have mainly been investigated with respect to external aspects: Most studies on inclusive teaching practices have documented organizational practices for co-teaching and peer support, but little is known about classroom practices for realizing inclusive subject-matter education (Van Mieghem et al., 2020). This lack of empirical research on teachers’ inclusive practices has also been confirmed for mathematics classrooms (in the research overviews by Lambert & Tan 2020; Roos, 2019). Similarly, teachers’ orientations about inclusion have mainly been investigated with respect to external factors, focusing on teachers’ acceptance of inclusion, and rarely with respect to the main quality criteria they pursue (Avramidis & Norwich, 2002; de Boer et al., 2011; Van Mieghem et al., 2020).

Most studies on inclusive practices in mathematics classrooms have focused on the job of Promoting Individuals Adaptively and have analyzed the teachers’ adaptation practices. They have consistently shown that more teachers make instructional adaptions (with more scaffolds or simplified texts) than curricular adaptations (i.e., setting differentiated learning goals; Kurth & Keegan 2014; Strogilos et al., 2020). Such a strategy becomes problematic for learners who have not yet gained access to the necessary conceptual foundations for the learning content of the “regular” class, as it can lead to a lack of learning opportunities for these conceptual foundations. Thus, it seems that Establishing Joint Experiences through instructional adaptations stands in conflict to Promoting Individuals Adaptively when the individuals have very different learning requirements.

Cases have been examined in which teachers address different demands of inclusive mathematics education. In a study on teachers’ practices for including students with Down syndrome, Faragher & Clarke (2020) illustrate different themes in teachers’ practices. In the beginning, the participating teachers were mostly concerned with Promoting Individuals Adaptively under a social orientation by focusing on fostering independence in learning. In the course of the study, they shifted their considerations to the mathematical orientation and to Establishing Joint Experiences by trying to engage the students in learning on “the same basis.” However, the study does not provide insights into whether the teachers switched from one demand to another or whether they learned to integrate different demands of inclusive mathematics education.

In their video study on Swiss inclusive primary mathematics classrooms, Krähenmann et al., (2019) captured 34 teachers’ practices for the jobs of both Promoting Individuals Adaptively (operationalized by curricular adaptation practices) and Establishing Joint Experiences (operationalized by establishing joint learning settings). The analysis reveals a high negative correlation of −0.69 between these two practices, which means that teachers work on either the first or the second job while rarely working on both at the same time by means of integrative practices that fulfill both jobs (e.g., Mutual Learning; Höveler 2019).

This observed incompatibility resonates with a perceived conflict that many teachers articulate in acceptance studies, in which they have admitted not feeling able to promote the learning of students with special needs while at the same time Establishing Joint Experiences. These teachers often ascribe this challenge to a conflict between the ambitious aims and external factors such as resources and working time. (Avramidis & Norwich, 2002).

However, recent research has indicated that the source of the perceived conflict might also lie within the teachers’ reasoning. Prediger & Buró (2021b) found in an interview study that teachers reported fulfilling the job of Establishing Joint Experiences mainly by practices of whole-class teaching without differentiation, whereas Promoting Individuals Adaptively was mainly realized by individualized learning with differentiated worksheets. Within the conceptual framework of content-related teacher expertise (see first subsection), they theorize that the perceived conflict between the jobs might be traced back to a perceived conflict between the social and mathematical orientations: Similar to research studies that have focused either on individual learning growth or social participation as their categories for evaluating the outcome of inclusive practices (van Mieghem et al., 2020), teachers also tended to choose between the two without finding ways to combine them.

Research questions

The jobs that need to be mastered and the orientations under which they can be pursued create a challenging space of demands for inclusive mathematics education that teachers need to navigate. Although truly inclusive mathematics education would need to integrate these demands, acceptance studies and observation studies have shown that teachers address these demands separately. This points to a perceived conflict between subjectively contradictory demands of inclusive education, which is a phenomenon well known in research on mathematics teachers: While working on an educational innovation, teachers can perceive conflicts in occurring teaching demands (jobs) that can hinder the development of their teaching practices (Jaworski, 2006; Thomas & Yoon, 2014). Some of these conflicts are real, unsolvable teaching dilemmas (Brodie, 2010; Lampert, 1985). Others are solvable for expert teachers who do not perceive the demands as incompatible (Lampert, 1985). Well-designed research-based professional development can support the teachers to overcome these kinds of conflicts.

This study aims to support the teacher PD effort in inclusive mathematics education by identifying practices through which teachers can integrate the demands of inclusive mathematics education that might otherwise be perceived as conflicting. Once identified, such practices might guide PD facilitators in specifying the content of PD courses and in understanding participants’ needs (Prediger et al., 2015). The study follows the general research agenda of educational reconstruction (Kattmann et al., 1998) that was lifted to the level of teachers: Qualitative empirical research was conducted to explain teachers’ perceived conflicts and to identify ways of integrating demands (in this study, of inclusive education). For this, we chose teachers’ discussions as a data source for pursuing the following research questions:

(RQ1) By which kinds of practices and underlying orientations and categories do teachers navigate the space of demands for inclusive mathematics education?

(RQ2) Which kinds of practices are used for integrating different demands of inclusive mathematics education?

Methods of the qualitative study

Methods of data gathering

General methodological approach

In order to identify practices for navigating the space of demands for inclusive mathematics education and the underlying categories, we chose a qualitative methodological framework due to the explorative nature of the research question. Whereas observation data (e.g., from videotaped classrooms; see Prediger & Buró, 2021a) can help to characterize teachers’ practices, their reasoning is more accessible when they talk about the practices (as shown also in the preceding interview study in Prediger & Buró 2021b). For the purposes here, we chose teachers’ discussions in a PD context as the dialogic nature made the reasoning accessible.

Research context

The data corpus of the current study stems from the research context of the large PD research project Matilda on inclusive mathematics education in German Grade 7 classrooms. We chose mathematics teachers and special education teachers from secondary comprehensive schools (Grades 5 to 10) as these schools have had a recent increase in integrating students with special needs (KMK, 2020) and a particular need to work on inclusive practices for ensuring more equity for all students. The PD program introduced 37 volunteer teachers to instructional approaches for combining Establishing Joint Experiences and Promoting Individuals Adaptively under social and mathematical orientations. Neither the instructional approach nor the PD project as a whole are the object of the current study; however, interested readers can find more details about them in other papers (Büscher, 2021; Prediger & Buró, 2021a; Prediger & Buró 2021b).

Data corpus

Data collected in the Matilda project consisted of videos from all small-group discussions and plenary discussions across all PD sessions. Most discussions revolved around the participants’ reactions towards new professionalization content, thus providing only limited insights into the teachers’ own practices. However, in two episodes, discussions were initiated by the research team with the intent to elicit reports on teachers’ practices in dealing with the demands of inclusive mathematics education. The first episode consists of small-group discussions from the very first moments of the first PD session. Here, the participating teachers were asked to discuss the challenges and successes of their inclusive teaching. The second episode stems from a plenary discussion in the last PD session. Here, teachers reflected on their challenges and successes with the new instructional approaches. Both episodes provide windows into the jobs and orientations that teachers hold when dealing with the demands of inclusive mathematics education. The two episodes form the focus data corpus of 120 min of video data, which were fully transcribed.

Participants

The 37 mathematics teachers and special education teachers involved worked in socially underprivileged urban areas and had varying teaching experience in general (between 4 and 20 years) and in inclusive classrooms (between 2 months and 3 years). All teachers participated voluntarily as they felt the need for PD for inclusive mathematics education, and were highly motivated for improving their teaching. Due to varying attendance in the course, only 20 teachers were present in at least one of the two episodes. All 20 teachers’ utterances were analyzed with the method described below. The teachers gave their informed consent that the video data of their discussions were gathered and analyzed.

Methods of qualitative data analysis

The qualitative analysis procedure of the transcripts was based on the model of content-related teacher expertise (from the first subsection) and identified both the teachers’ talk about the practices for integrating the demands of inclusive education and the underlying categories. It combined the deductive and inductive steps of qualitative data analysis (Mayring, 2015):

Step 1. From the transcript data, utterances were identified that concerned teachers’ stories of success or challenge with inclusive mathematics education. This was done by focusing on the language employed and looking for explicit descriptions of teaching practices and more implicit descriptions encoded in emotional responses to teaching situations.

Step 2. Each of the teachers’ utterances was assigned to the job and orientation that were relevant to that utterance. Promoting Individuals Adaptively was assigned when teachers referred to teaching approaches or evaluation criteria focusing on individuals (or small groups that were considered homogenous) and their individual needs, while Establishing Joint Experiences was assigned when teachers referred to the class as a whole and emphasized the collective. A mathematical orientation was identified when teachers referred to the mathematics learning of students and the mathematical content, a social orientation was identified when teachers referred to social or pedagogical factors such as the well-being of students in the group or to the social role or place of students.

Step 3. From the utterances, the encoded categories were identified inductively. By systematically comparing and contrasting the different categories across the utterances, the categories were further refined. The categories were then located in the coding scheme in Fig. 2. This allowed the location of teachers’ explicitly or implicitly articulated conflicts, as a single utterance often referred to multiple jobs or orientations, with the categories being assigned uniquely to one of the cells in Fig. 2.

Step 4. Each configuration of jobs, orientations, and categories thus represented a practice for navigating the space of demands for inclusive mathematics education. All practices were then again systematically compared and contrasted to find similarities and differences in the co-occurring jobs, orientations, and categories. This resulted in identifying four kinds of typical practices that teachers employed for dealing with the demands of inclusive mathematics education.

Fig. 2
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Deductive analytic scheme for jobs and orientations with example entries for inductively identified categories

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Empirical results: teachers’ practices of integrating challenging demands

By comparing and classifying the analyzed utterances, we identified four kinds of typical practices for integrating the demands with or without perceived conflicts (research question RQ1) that will be presented in four subsections: practices of explicating conflicts (Sect. 3.1), practices of reducing complexity (Sect. 3.2), practices of alternating between jobs (Sect. 3.3), and practices of flexibly using visual models (Sect. 3.4). These kinds of practices illustrate different approaches for integrating the various demands of inclusive mathematics education (RQ2). It is typical for the whole data corpus that the fourth kind of practice was only articulated in the second episode.

The practices are illustrated through examples of six teachers: Patricia and Ursula are special education teachers who have limited formal training in mathematics education and work in a multi-professional dyad with a regular mathematics teacher. Their co-teachers did not attend the PD program. Dora, Hanne, Zoey, and Tina are regular mathematics teachers with little formal background in special education. They also work in dyads with a special education teacher, and Tina’s co-teacher also attended the PD program. Some teachers are used to speaking about “weak students,” an informal term by which they refer to students who have not yet found access to the regular classroom learning content or who require additional support. To contextualize this wording, it is important to note that this is a common way to refer to low-achieving students in Germany and does not imply a belief in unproductive ideas like “weak students are unable to learn mathematics and thus cannot be helped.” Instead, the teachers acknowledged that these “weak students” face different learning conditions and assumed responsibility for their mathematics learning. However, they also articulated feeling underprepared for enacting this responsibility, a major reason for voluntarily participating in the PD program.

Practices of explicating conflicts between demands of inclusive education

When reporting their experiences with inclusive mathematics education in the first PD session, the teachers identify numerous demands, prompting them to posit that inclusive mathematics education is hard or even impossible. Rather than a general rejection of inclusion, they locate them in the conflict between specific demands:

Patricia::

[Our weak students] only remain in the classroom if they can manage things to a certain extent. Like if they—compared to the regular students—perform adequately. That is, those that almost lose their official status as student with special educational needs. The others, they don’t make it. And that is a real shame, because inclusion, which is an issue we work on, does not happen this way.

Dora::

Isn’t it obvious that they need different materials, but if it is too different, then it automatically excludes? I mean, work sheets have to be structured clearly, right, and with a picture on them, like for all other children. Who, when they are done, can color in the picture. But if it looks wildly different from what the other children do, then they feel excluded, right?

Here, both teachers report conflicts between the two jobs of Promoting Individuals Adaptively and Establishing Joint Experiences (see Fig. 3 for the coding scheme).

Patricia articulates a typical organizational conflict: Inclusion is equated with Establishing Joint Experiences, which she interprets in a purely social orientation, according to which the fulfillment of this job is evaluated by the category of permanent presence in classrooms (“remain in the classroom”). However, this is in conflict (“that is a real shame”) with fulfilling students’ individual learning needs that are considered addressable for some students mainly in supplemental instruction outside classrooms (“The others, they don’t make it”).

In Dora’s school, the supplemental instruction outside classrooms is reduced, but she articulates a similar conflict between both jobs as an unsolvable dilemma: For Dora, inclusion involves the jobs of both Promoting Individuals Adaptively and Establishing Joint Experiences. The mathematical orientation leads her to calling for addressing individual needs with differentiated worksheets with instructional and/or curricular adaptations (“obvious that they need different materials”). But the social orientation calls for making all students to feel that they are being treated similarly. Her evaluation category for the social orientation focuses on the individual sense of belonging (“then they feel excluded”), not on the social participation itself. Promoting Individuals Adaptively in the social orientation is hence evaluated by the sense of belonging, and this is—in her eyes—an unsolvable conflict with the mathematical orientation, as it requires being treated similarly.

Fig. 3
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Visual summary of analysis for Patricia’s and Dora’s utterances on perceived conflicts (dashed arrows indicate the conflicts)

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Similar to Patricia and Dora, many teachers in our sample of 20 teachers articulate conflicts between the different demands of inclusive mathematics education. Across the conflicts found on the whole sample, a pattern emerges: It is the mathematical orientation for Promoting Individuals Adaptively that lies at the root of almost all conflicts perceived by the teachers (Fig. 4). The need for individualization was perceived as contradictory to social considerations like individuals’ sense of belonging to the class and the idea of providing common instruction for all students, as well as to mathematical considerations like following a curriculum and the collective “progress” of class.

Fig. 4
figure 4

Widening the summary to all conflicts explicated by 20 teachers: Most conflicts were between Promoting Individuals Adaptively under a mathematical orientation and other demands (dashed arrows indicate the conflicts)

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Whereas Patricia and Dora consider the perceived conflicts unsolvable dilemmas in the cited situation, they find other practices to deal with the perceived conflicts in further situations. The same applies for the other teachers in the data set. Kinds of practices are not types of teachers.

Practices of reducing complexity

An alternative practice for navigating the space of demands for inclusive mathematics education is to reduce the complexity of this space. In this practice, only a single orientation is activated for a single job, as illustrated by the following teacher.

Ursula::

In the last 2 years, I made the experience that it can work for many things. Especially if you can make it comprehensible through other means. Things that you can present differently, or with a completely different approach. So that they can connect to it. … For example, with measuring. We went outside, and we measured cars and stairs, and eventually there was a conception of what is a meter, what is 2 m, what is 1 centimeter.

Here, Ursula reduces the demands of inclusive mathematics education. She evaluates her inclusive teaching via the category of individualization. She is oriented towards the mathematics, as her story is about the learning of measuring. Her story is a success, because she satisfied the individual needs of her special education students through this individualized approach. She does not refer to any kind of joint learning with all students, nor does she include socially oriented categories. Thus, this is an example of reducing the complexity involved by focusing on a single job and a single category.

A similar practice of dealing with perceived conflicts is employed by Hanne, recounting a story that happened while her school was under review by the school inspection board. Such reviews are conducted regularly, and success in reviews is considered highly desirable.

Hanne::

Well, there was a school inspection review at our school, and we again were at the same point in the materials, and we did the introduction on addition and subtraction of decimal numbers. The included kids were in my class, and we were working beautifully … and then she [the inspector] came into the classroom … and said “please tell me what you did,” which I did. “Oh, this is an inclusive class, but I will recognize the included kids, you don’t have to tell me” … and after 3 min she turns towards me and says “Mrs. Ufer, where are the included kids?” [laughs] Yes! And that’s my high point and if things are getting bad, I think of this lesson: Remember, Hanne Ufer, they were not discovered, they all worked the same way.

This story gives Hanne hope that inclusive mathematics education is possible. Her evaluation is interpreted as belonging to the social orientation (Fig. 5), as Hanne focuses on the fact that the students all “worked the same way.” It remains unclear whether the students also worked on the same content or they had different materials. The point Hanne stresses is that from the outside perspective, it was not possible to observe differences between the students — the “included kids” were invisible. Other demands are not relevant to her in this utterance, and she enacts a practice of reducing the number of categories employed for evaluating her teaching. This reduced complexity allows her to find success instead of frustration in her inclusive teaching, an important step in overcoming potential conflicts between different demands.

Fig. 5
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Visual summary of analysis for Ursula and Hanne’s utterance exemplifying practices of “reducing the complexity”

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Fig. 6
figure 6

Summary of categories for all 20 teachers’ practices of “reducing the complexity” (in each practice, the teacher focused only one of these categories)

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These episodes serve only as two examples of practices commonly employed by the observed teachers. The practice of reducing can also draw on other categories, focusing on other jobs and orientations (Fig. 6). It is interesting to note that the mathematical learning of the individual is not the dominating focus here; often, teachers contend themselves with creating opportunities in which special needs students simply participate in activities together with other students. This shows a social orientation, as the mathematical point of the activities for the special needs students tends to be ignored.

Although these practices could be interpreted as a lowering of ambitions, this would not do justice to the teachers. The teachers are well aware that inclusion is a larger concept than shown here. But by reducing the complexity, the teachers are able to actually find success stories rather than having to surrender before the daunting demands of inclusive education. Therefore, such practices can serve as a first step on which PD programs can build.

Practices of alternating jobs

A way of combining multiple orientations or jobs are practices of alternating between different jobs, in order to fulfill them separately. One example is given in the following.

Zoey::

That’s a ritual in my class, that we work on 10 calculation tasks and then every one of the students receives their work sheet. We introduce them together, and then we see that we work in some way on the same content. She [her special education colleague] prepares materials for students with special education needs, depending on where in the textbook they are … and then we see that we each work in our own domain … and if I have the time, I go to them [the students with special educational needs].

In this success story, Zoey reports on a kind of practice employed by many teachers. By alternating practices in her classroom, she alternates between different categories for evaluating her teaching. Her lessons begin with a ritual of equalizing her heterogenous class by solving the same procedural tasks together. She does not explicate what kind of tasks these are, and if they possibly allow for Promoting Individuals Adaptively. Therefore, this category only relates to the job of Establishing Joint Experiences, without explicitly referring to students’ individual needs. After this ritual, ability grouping is enacted by handing out different tasks for different groups of students, so that these individuals can be Promoted Adaptively, yet without initiating joint experiences. Both jobs are explicated in a mathematical orientation (Fig. 7).

Fig. 7
figure 7

Visual summary of analysis for Zoey and Ursula’s utterance exemplifying practices of “alternating jobs”

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Ursula::

Sometimes, it depends on the availability of rooms, I pull out some of the kids. Not only those with status as special needs, but simply those where you see that the learning content is a bit rough. Just so that you can explain it a little bit better. Of those four special needs kids there are two that are relatively strong in math. So that they can manage relatively well in regular class, and sometimes they receive easier tasks. The other two… well, it really depends on the content. Partly they really need dedicated materials.

Ursula, who previously explicated practices of reducing the complexity (see preceding subsection), also provides an example of an alternating practice. She sometimes creates pull-out groups of students to provide special, individualized instruction (to “explain a little bit better”). She does so under considerations concerning the mathematical learning (the “rough” learning content), thus promoting individual learning adaptively under a mathematical orientation. This is alternated with the establishment of a “regular class” in which the students otherwise participate. The point of regular class is that students “manage”. This is in this context interpreted as following a social orientation, as Ursula does not make explicit reference to any aims that students should work cooperatively on the content. Sometimes, some special needs students receive “easier tasks”. Again, this seems to be a way for her to allow those students to “manage” regular class, instead of consisting of focused individualized instruction. Thus, she here Establishes Joint Experiences under a social orientation by providing tasks that let the students feel that they belong to the regular class. For the two other special needs students, the case is different: They “really need dedicated materials” and thus cannot take part in the joint experience of regular class. Instead, they are Promoted Adaptively under a mathematical orientation. This excerpt shows how two very similar tools—providing easier tasks and providing dedicated materials—can implement different jobs under different orientations.

Like reducing the complexity, alternating the jobs is a kind of practice that reduces the simultaneous demands faced by teachers. Here, inclusive mathematics education is decomposed into different jobs that are fulfilled subsequently.

For other observed teachers, the explicated practices of alternating always refer to an alternation between the two jobs within the mathematical orientation as well (Fig. 8). No case occurred in which a teacher would explicitly alternate between, for example, promoting individuals mathematically and socially. Similar practices occurred in the interview study by Prediger & Buró (2021b) when teachers reported enacting whole-class teaching as long as all students could follow and then switching to individualized differentiations.

Fig. 8
figure 8

Two other identified versions of the practice of “alternating jobs”

Full size image

Although the normative demands of inclusive education posit a more holistic kind of instruction, again, practices such as those collected under the label of alternating jobs represent interesting steps toward increasing the integration of orientations and jobs.

Practices of integrating jobs and orientations by flexible use of visual models

In the last session of the PD course, the teachers were asked to recount the successes and challenges in implementing the new course materials. Here, another practice for navigating the space of demands for inclusive mathematics education surfaced that simultaneously integrates demands of different combinations of social and mathematical orientations with the two jobs. Ursula, who in the first session has articulated practices of both reducing the amount of evaluation criteria and alternating, now articulated a more integrated approach.

Ursula::

When we approached the tasks, the first thing we did was to have the weaker students look at which values are given and which could be marked on the percent bar. And the stronger students took over the calculating process, the solution path. And that also triggered something within the weaker students. So that they felt safer, they participated more, they were able to participate more. And they experienced that as a very positive thing.

In the last session of the PD, Ursula is able to integrate both orientations with both jobs (Fig. 9). For the social orientation, she emphasizes that every individual student needs to feel safe, while there should be participation in discussions. For the mathematical orientation, she addresses different individual learning goals: Low-achieving “weaker” students should have the necessary conceptual understanding of the situation involving percentages and be able to extract the necessary information in the correct way, while “stronger” students should also be able to carry out procedures. But these student groups also interact with each other by learning from each other, as the “weaker” students can not only observe the “stronger” ones, but also provide information and interaction. The teachers’ practice here is not a reduction nor an alternation, but instead an integrated approach. She uses a visual model—the percent bar—for integrating the different goals. It is the percent bar that enables the “weaker” students to fill in values that can be taken up by the “stronger” students. Thus, it is the percent bar that provides the common ground for participation in the classroom. Through this flexible use of a visual model, orientations and jobs that previously seemed antinomic can be integrated.

The same kind of practice is also explicated by Tina, another teacher:

Tina::

We looked for moments of joint learning and found them in those moments where the students presented their results at the end of the lesson. The stronger students eventually realized that from the whole procedure [working with the percent bar], which they understood by the second time, that they could take a shortcut. They only looked like “aha, here is the task, there is the value. I do not need to partition but simply mark my solution, done,” while the weaker students partitioned the bar. And they even could react to the others or discover errors. Those were the moments where two working methods have led to results. And sometimes the stronger students got careless because they were bored from this procedure, whereas as you said, the weaker students could profit from it all. Regarding their confidence as well as motivation, so I believe they really liked it.

Fig. 9
figure 9

Visual summary of analysis for Ursula’s and Tina’s utterance exemplifying practices of integrating jobs and orientations by flexible use of visual models

Full size image

Similar to the previous excerpt, Tina here integrates the demands of different orientations and jobs (Fig. 9). Through the practice of integrating orientations by flexible use of the percent bar, individual goals could be addressed, which were included in situations of learning from each other. This time, however, it is not only the “weak” students that learn from the “strong” ones, but also the other way around: Through this practice, “weaker” students can spot errors in the “stronger” students’ reasoning as well. This all leads to increased confidence and motivation in the social orientation. It is also worth noting that the moments of joint learning, the common presentations, take place not in ritualized introductions at the beginning of the lessons, but rather at the central part in the end, where different results are shared in the classroom.

Similar to Ursula and Tina, other teachers also developed their practices of mastering both jobs together. We observe that in the course of the PD and through their own teaching experiences with the new materials, the group of observed teachers were increasingly able to integrate social and content-related orientations with the two jobs and thereby could overcome the perceived conflicts between the demands of inclusive education. Although the data is not at all sufficient for declaring these tentative observations to be a methodologically sound proof of the effectiveness of the PD (see Discussion section), it is interesting to see these first indications of development.

One further observation is that the articulated integrative practices seems to build not only on adopting social and mathematical orientation at the same time, but also upon an intense decomposing of the learning content of percentages, because only this decomposition enables teachers to set individual learning goals that still provide the grounds for Establishing Joint Learning Experiences. Central to this seems to be the innovative ways of teaching percentages through using the percent bar, which became a common tool for students to show different types of reasoning and referring back to conceptual foundations of the same content. Teachers’ ability to decompose learning goals cannot be taken for granted: A small case study from the same larger project (Büscher, 2019) revealed that in the beginning of the PD, teachers struggled to specify learning goals for their students that support Establishing Joint Learning. When percentages were set as the shared learning content in the early PD sessions, the teachers specified learning goals for their lowest-achieving students that often were not actually related to the content of percentages at all. Only at the end of the PD were they able to identify those conceptual foundations of percentages (such as multiplicative reasoning by counting in steps along the percent bar; Büscher 2021; Prediger & Buró, 2021b) that are also relevant for all students.

Discussion and conclusion

Discussion

Teachers’ professional development is in part shaped by challenges and sometimes even conflicts that are perceived between new teaching demands in educational innovations (Jaworski, 2006; Thomas & Yoon, 2014). This is particularly important for inclusive mathematics education where new demands can be characterized by the interplay of the two teacher jobs of Promoting Individuals Adaptively and Establishing Joint Experiences and a social orientation and mathematical orientation. In order to support teachers’ PD for inclusive education, it is important to identify ways to integrate both teaching demands to overcome possible conflicts (Prediger et al., 2015; Thomas & Yoon, 2014), which is the core idea of educational reconstruction lifted to teachers (Kattmann et al., 1998).

Drawing on the model of content-related teacher expertise (Prediger, 2019, in adaptation from Bromme 1992), the current study investigated 20 teachers’ discussions in a PD course to understand which practices were reported for dealing with the demands of inclusive mathematics education (RQ1). Four kinds of practices were identified in the process of qualitative analysis: (1) practices of explicating conflicts between demands, (2) practices of reducing complexity, (3) practices of alternating jobs, and (4) practices of integrating jobs and orientations by flexible use of visual models.

These practices can be located on a spectrum for various degrees of integrating the demands of inclusive education (RQ2). The first practice of explicating conflicts between jobs or orientations is reported by only some teachers. When such a conflict is explicated, it is mainly when the first job is exclusively understood in mathematical orientation and the second in social orientation. Two additional kinds of practices include reducing the complexity of demands to a single job and orientation and alternating between jobs. Although these practices do not directly integrate different demands of inclusive education, their recurrent use can still lead to classrooms in which various demands are addressed. Finally, some teachers are able to fully integrate all demands by orchestrating common learning, for example, with the visual model of the percentage bar. This seems to be a very challenging kind of practice that requires the ability to decompose learning goals.

The kinds of practices identified can also provide directions for understanding the existing results of research from a new perspective. Previous research has documented practices such as providing additional scaffolds or simplifying tasks for certain students rather than curricular adaptations (Kurth & Keegan, 2014; Strogilos et al., 2020). In light of the results of this study, such kinds of practices can be interpreted as ways to reduce the complexity of inclusive teaching to increase all students’ access to joint experiences while neglecting the individual learning progress. Krähenmann et al.’s (2019) video observations of practices showed that the jobs of Promoting Individuals Adaptively and Establishing Joint Experiences were strongly negatively correlated. The teachers observed by Krähenmann et al., (2019) also seemed to engage in practices of reducing the complexity or alternating between jobs, possibly because conflicts between various demands were perceived. This result seems to confirm the tentative observation that practices of integrating the demands are highly challenging for teachers.

Suitable instructional approaches for this integration have been already established in the research community (e.g., open differentiation tasks by Lawrence-Brown 2004, or Scherer 2013; or the approach of “mutual learning” by Höveler 2019). However, their successful enactment in classrooms seems to largely depend on teachers’ integration of demands that can be perceived as exclusive. Although we have no evidence about how the self-reported practices in this study match the teachers’ enacted practices in the classroom, their reports nevertheless provide highly important insights into how the possible conflicts can be overcome: It seems that with an adequate visual model and unpacked learning content, individuals can be promoted at the same time as joint experiences can be established under both social and mathematical orientations.

Teachers’ professional development is in part shaped by challenges and sometimes even conflicts that are perceived between new teaching demands in educational innovations (Jaworski, 2006; Thomas & Yoon, 2014). This is particularly important for inclusive mathematics education where new demands can be characterized by the interplay of the two teacher jobs of Promoting Individuals Adaptively and Establishing Joint Experiences and a social orientation and mathematical orientation. In order to support teachers’ PD for inclusive education, it is important to identify ways to integrate both teaching demands to overcome possible conflicts (Prediger et al., 2015; Thomas & Yoon, 2014), which is the core idea of educational reconstruction lifted to teachers (Kattmann et al., 1998).

Drawing on the model of content-related teacher expertise (Prediger, 2019, in adaptation from Bromme 1992), the current study investigated 20 teachers’ discussions in a PD course to understand which practices were reported for dealing with the demands of inclusive mathematics education (RQ1). Four kinds of practices were identified in the process of qualitative analysis: (1) practices of explicating conflicts between demands, (2) practices of reducing complexity, (3) practices of alternating jobs, and (4) practices of integrating jobs and orientations by flexible use of visual models.

These practices can be located on a spectrum for various degrees of integrating the demands of inclusive education (RQ2). The first practice of explicating conflicts between jobs or orientations is reported by only some teachers. When such a conflict is explicated, it is mainly when the first job is exclusively understood in mathematical orientation and the second in social orientation. Two additional kinds of practices include reducing the complexity of demands to a single job and orientation and alternating between jobs. Although these practices do not directly integrate different demands of inclusive education, their recurrent use can still lead to classrooms in which various demands are addressed. Finally, some teachers are able to fully integrate all demands by orchestrating common learning, for example, with the visual model of the percentage bar. This seems to be a very challenging kind of practice that requires the ability to decompose learning goals.

The kinds of practices identified can also provide directions for understanding the existing results of research from a new perspective. Previous research has documented practices such as providing additional scaffolds or simplifying tasks for certain students rather than curricular adaptations (Kurth & Keegan, 2014; Strogilos et al., 2020). In light of the results of this study, such kinds of practices can be interpreted as ways to reduce the complexity of inclusive teaching to increase all students’ access to joint experiences while neglecting the individual learning progress. Krähenmann et al.’s (2019) video observations of practices showed that the jobs of Promoting Individuals Adaptively and Establishing Joint Experiences were strongly negatively correlated. The teachers observed by Krähenmann et al., (2019) also seemed to engage in practices of reducing the complexity or alternating between jobs, possibly because conflicts between various demands were perceived. This result seems to confirm the tentative observation that practices of integrating the demands are highly challenging for teachers.

Suitable instructional approaches for this integration have been already established in the research community (e.g., open differentiation tasks by Lawrence-Brown 2004, or Scherer 2013; or the approach of “mutual learning” by Höveler 2019). However, their successful enactment in classrooms seems to largely depend on teachers’ integration of demands that can be perceived as exclusive. Although we have no evidence about how the self-reported practices in this study match the teachers’ enacted practices in the classroom, their reports nevertheless provide highly important insights into how the possible conflicts can be overcome: It seems that with an adequate visual model and unpacked learning content, individuals can be promoted at the same time as joint experiences can be established under both social and mathematical orientations.

Additionally, the video data from the discussions only capture shifts in teachers’ collective discourses but not their individual personal growth. PD courses are highly complex social events, and it is not easy to distinguish individual orientations that truly are held by teachers from those that arise mostly through negotiations in group discussions or are considered socially desired within the particular PD program. Still, the fact remains that sophisticated integrations of demands only took place after multiple PD sessions and after the implementation of the new curriculum materials. And even if only a few teachers managed such integrations, the mere existence of an authentic report of their success can enable further growth of the PD group. Future studies should find methodologies to capture the personal professional growth more systematically.

As has often been problematized, practices articulated in PD do not necessarily correspond to the practices really enacted in classrooms (Mullens & Gayler, 1999). Within our larger PD project, the video analysis of practices enacted in classrooms revealed few occasions of combining both jobs at the same time (Prediger & Buró, 2021a), but the limitations of the video data did not allow us to infer back to the orientations. Future studies should combine the research in the PD sessions with observations of the classroom video data more thoroughly.

Limitations of the study

Due to methodological limitations of the study, the first indications of the teachers’ pathways towards integrating the demands of inclusive education must not be overinterpreted as a methodologically sound proof of the effectiveness of the PD course. Teachers’ participation in plenary discussions was of heterogenous intensity, and the teachers who reported integrating practices in the last PD session were not always the same as the teachers who reported perceived conflicts in the first session. It might also be the case that teachers articulated very productive practices for integrating the demands of inclusive education outside the analyzed data. This study does not claim to adequately represent the practices of all 20 teachers in the PD program. Instead, it provides an exploration of possible practices to inform the design of future PD programs.

Additionally, the video data from the discussions only capture shifts in teachers’ collective discourses but not their individual personal growth. PD courses are highly complex social events, and it is not easy to distinguish individual orientations that truly are held by teachers from those that arise mostly through negotiations in group discussions or are considered socially desired within the particular PD program. Still, the fact remains that sophisticated integrations of demands only took place after multiple PD sessions and after the implementation of the new curriculum materials. And even if only a few teachers managed such integrations, the mere existence of an authentic report of their success can enable further growth of the PD group. Future studies should find methodologies to capture the personal professional growth more systematically.

As has often been problematized, practices articulated in PD do not necessarily correspond to the practices really enacted in classrooms (Mullens & Gayler, 1999). Within our larger PD project, the video analysis of practices enacted in classrooms revealed few occasions of combining both jobs at the same time (Prediger & Buró, 2021a), but the limitations of the video data did not allow us to infer back to the orientations. Future studies should combine the research in the PD sessions with observations of the classroom video data more thoroughly.

Implications for PD design

This study aims to provide a contribution to the effort for teachers’ PD for inclusive education. When conducting PD for inclusive mathematics education, PD facilitators are quickly confronted with a peculiar situation. Many teachers hold inclusion to be a worthwhile goal, but they often perceive more conflicts between the demands than possibilities for actualizing inclusion in their classrooms (Prediger & Buró, 2021b). In our experience, these conflicts often reveal the prominent points of discussion at the beginnings of PD courses on inclusion. Yet although there are many real hindrances to inclusion (e.g., Avramidis & Norwich 2002), some teachers manage to succeed in actualizing the complex demands of inclusion. This study shows that conflicts may be overcome with certain practices. The findings suggest that the more sophisticated the practices employed by the teachers for integrating the demands, the less conflicts between them surfaced in their discussions during PD. By paying close attention to the jobs and orientations articulated by teachers, PD facilitators can draw on the variety of practices to show similarities and differences, to provide points of discussion, and to show how conflicts can be solved through new practices in which the mathematical and social orientations can be harmonized. By demonstrating that inclusive mathematics education is not a monolithic and unsolvable problem, but rather a space of jobs and orientations to be navigated within, PD facilitators might steer discussions towards more productive practices for dealing with the complexity of inclusion. Integrating the demands of inclusive education through flexible use of visual models is one such practice that is very demanding and was reported only by a small number of participants. It can thus be considered a goal of PD, but is not the only practice for dealing with the demands of inclusive mathematics education. PD facilitators might introduce simpler practices like reducing the complexity and alternating jobs as legitimate approaches towards inclusive mathematics education if their limitations are taken into account. The rough outline of PD progressing from explicating conflicts to reducing complexity, alternating jobs, and finally integrating jobs and orientations through flexible use of visual models might also provide a guideline for PD facilitators to locate participants on their personal pathways and to suggest next steps for their professional growth.

Moving beyond the PD content of inclusive mathematics education

Whereas the current project treated the specific PD content of inclusive mathematics education, the same research approach might also apply to other areas of PD content: For understanding what hinders teachers in realizing innovations (Century & Cassata, 2016) and developing their professional practices, a focus on possible conflicts arising from different practices for dealing with various teaching demands might be promising. The model of content-related expertise (Prediger, 2019) can inform qualitative investigators in disentangling the complex interplay of jobs, orientations, categories, and practices both to specify the PD and to conceptualize differences in teachers’ approaches and development needs. Future studies might transfer this research approach to other areas of PD content and explore how the identification of the underlying challenges can inform the design of the PD programs. In our experience, this approach can support PD design researchers’ consideration of teachers’ perspectives not as deficient but as plausible perspectives that can be taken seriously and at the same time be enriched.