1 Introduction

Making mathematics accessible for all is one current challenge for teachers in inclusive settings. Teacher preparation for diversity and equity (and associated research) can be identified as one major field in teacher preparation in the current century (Cochran-Smith et al., 2015; Gervasoni & Peter-Koop, 2020). Including all learners in one classroom might produce new challenges instead of excluding specific groups or persons but at the same time considering the diversity of learners and their individual needs. For the primary school level, different dimensions of diversity have been addressed but students with special needs with specific impairments have been of minor importance. As a consequence, teachers adhere to narrow definitions of mathematical activity and mathematical ability, and deficit views on students’ resources that lead to discrimination and separate learning situations (cf. Chen & Horn, 2022). Although there exists a longer research tradition concerning teaching and learning of students with special needs in specialized schools as well as developing concepts of mainstream schools (cf. Evans, 2004; Nilholm & Göransson, 2017), the professionalization of teachers is challenging: Preparing teachers for both, teaching students with special needs or, more general, for coping with heterogenous learning groups, is of major importance for pre-service and in-service teacher education programs. In particular, more research is needed concerning subject-matter-specific programs for inclusive teacher education, giving all students access to mathematics. Whereas in the past, questions of coping with students with special needs have been dealt with mainly in special education, it becomes more and more the responsibility of the subject disciplines.

While attitudes and beliefs have long been discussed as a significant factor in the realization of inclusion (see for example Avramidis & Norwich, 2002; de Boer, 2012), a subject-specific view has not been taken much here either. Only a few studies which focus on attitudes and beliefs about teaching in inclusive settings concentrate on subject-specific or rather mathematics-specific questions. But subject-specific attitudes and beliefs of future teachers have to be considered especially in mathematics because mathematics might play a specific role compared to other subjects. In different contexts it is assumed that it might be more difficult to teach mathematics in inclusive settings than other subjects, due to the specific structure of mathematics (Seitz et al., 2020), and mathematics being a challenging subject for all students (DeSimone & Parmar, 2006a). However, van Reusen et al. (2001) investigated several factors influencing teachers’ attitudes toward inclusion and analysed if there is an impact of the subject area taught. They did not identify a significant connection between hard sciences (like mathematics), social sciences, or other subject areas and high school teachers’ attitudes toward inclusion. This shows that there is still a need for clarification regarding the relationship between teachers’ beliefs and attitudes and their mathematics teaching, related to the different paradigms and terms towards inclusion and exclusion.

This paper has two goals. The first goal is to discuss different understandings of the term inclusion, and to show an example of how these can be addressed in mathematics teacher education programs avoiding discrimination and marginalization, considering also national conditions in Germany (Sects. 2 and 3). The second goal is to present results of an empirical study in which beliefs and self-efficacy of pre-service teachers about inclusive mathematics teaching have been investigated, considering different points in time within the whole education program (Sects. 4, 5, and 6), followed by further perspectives for teacher education programs and research (Sect. 7).

2 Inclusion—paradigms, terms, and research in mathematics education

Realizing inclusion can be seen as a worldwide paradigm but with heterogeneous concepts based on different historical roots (Köpfer et al., 2021; see also Haug, 2017; Roos, 2019), and literature shows that a consistent understanding of the term inclusion does not exist: Descriptive or prescriptive definitions can be found (Ainscow et al., 2006, p. 14), and different authors discuss the variety of understandings and try to offer a conceptualization. For example, Nilholm and Göransson (2017) analysed European and North American journal articles, referring to four definitions of inclusion. These four definitions range from a placement definition—students with special needs are placed in general education classrooms –, and an individualized definition, either focusing on the needs of special education students, or on the needs all students, to a community definition.

To be able to meet the needs of all students—students with different languages, cultures, abilities, and skills—and to avoid marginalization, inclusion should be used as an overarching notion for support (Bishop et al., 2015). Mainly influenced through the endorsement by many countries of the Salamanca Statement and Framework for Action on Special Needs Education (UN 2006), the term ‘inclusion’ emerged removing more or less the term ‘integration’. Internationally, inclusion is often used to denote what was previously referred to as special education (Nilholm & Göransson, 2017, p. 438). Though, often the notion of inclusion has been connected to special education rather than to a democratic education overall (Allan, 2012). In Germany, the differentiation between a ‘broader’ and a ‘narrower’ term of inclusion emerged, with the former one including all facets of diversity not focusing explicitly on special education (cf. Kiuppis, 2022; Strauß et al., 2023; see also Haug, 2017). The latter one also considers the wide range of diversity but puts a specific focus on special educational needs. The narrow understanding of inclusion focuses on the first two categories by Nilholm and Göransson (2017) and the broader understanding on the latter two categories. Meeting the needs of all students and focusing on all facets of diversity is closely linked to avoid marginalization. Every student is valued and treated as an important person, no matter what kind of support is needed.

Independently from the underlying term, special educational needs (SEN) mean different things in different countries:

„In some, it refers only to those students with sensory, cognitive, emotional, communication and multiple disabilities. In others, it also includes those from socially disadvantaged backgrounds. Some countries include, in addition, those from ethnic minority backgrounds; yet others cover gifted children“ (Evans, 2004, p. 32; see also Ainscow et al., 2006).

The different countries also vary substantially in the ways they describe SEN students. In Germany, seven different categories are used to describe SEN students: Disabilities or impairments in learning, mental retardation, language impairments, emotional and social development, physical and motorical development, hearing and communication, and visual impairments (KMK, 2022).

Concerning the specific school systems, Evans (2004) differentiates between (a) special schools, (b) special classes in regular schools, and (c) full inclusion in regular classes. Variants (a) and (c) are common in Germany, whereas variant (b) is not common for students with special needs. In Germany there exists a long tradition of a separated school system. Concerning the concrete mathematics classrooms, special education followed a traditional behavioristic teaching and learning approach (cf. Scherer et al., 2016). So, on the one hand individual special needs were taken seriously in specialized schools. But on the other hand, central research findings in mathematics education and corresponding objectives for teaching and learning mathematics have not been implemented (see for example Scherer et al., 2016; Scherer, 2020). Recently, there is a growing number of inclusive schools. By law, every SEN student has the right to be schooled at a mainstream school in regular classes. In relation to the different categories, the amount of SEN students either being taught at mainstream schools or at special schools differs. While educating SEN students in mainstream schools should redeem the goal of equity, it is important to note that reality often shows a different picture: SEN students are excluded from common learning situations, and traditional behavioristic forms of teaching and learning scenarios can be found (cf. Scherer, 2021).

As a consequence, the framing of inclusive classrooms is of major importance (see also Haug, 2017) and some studies only focus on more general aspects (like illustrating different forms of team-teaching). Apart from general design questions the mathematical content is of major importance. For primary mathematics, a constructivist view enabling students to active acquisition of knowledge, offering individual learning strategies, and considering learners’ individual view on mathematics, can be seen as the underlying paradigm, a paradigm that goes hand in hand with the design of common learning situations for all students and aims at avoiding discrimination and marginalization. This means that active learning, guided discovery, and competence orientation represent central principles, and flexible forms of inner differentiation, open problems and substantial mathematical learning offers should be put into action (cf. Scherer & Krauthausen, 2010; Wittmann, 2001). But with including students with special needs in regular classrooms and possibly different underlying paradigms (see Sect. 3), teaching and learning processes too often are quite different in regular and inclusive classes (cf. Scherer, 2020).

Designing adequate mathematics teacher education programs and settings that avoid discrimination requires the consideration of underlying attitudes and beliefs, experiences, or competencies of (pre-service) teachers, as well as considering country-specific conditions. In the following, we will focus on teacher preparation, illustrating the concrete program at the University of Duisburg-Essen in the context of more general national and international developments, and bring out relevant questions of inclusive mathematics.

3 Teacher education programs and (pre-service) teachers’ beliefs

Coping with heterogeneity in mathematics classrooms can be seen as a main issue for a long time. Mainly influenced through the endorsement of the Salamanca Statement and Framework for Action on Special Needs Education (UN 2006), the term ‘inclusion’ emerged with consequences for the school as well as the teacher education level. In Germany as well as in many other countries, traditionally the two strands—special education and mainstream schooling—represented different fields (cf. Leonhardt, 2018) that have to be brought together in the field of inclusion.

3.1 Teacher education programs preparing for heterogeneity and inclusive mathematics

In Germany, the standards for teacher education (KMK & HRK, 2015) stress the common responsibility of different disciplines: “Educating teachers to embrace inclusive schooling is therefore a cross-sectional task for educational sciences, subject-specific didactics and academic disciplines which must be dealt with cooperatively and mutually agreed upon for all types of teaching” (p. 3). Looking in more detail at the concepts of teaching and learning mathematics, it can be stated that special education followed a more traditional view for a long time, and this proves to be true internationally (Boyd & Bargerhuff, 2009). As a consequence, the concrete situation in the classroom for students with special needs was quite different from the teaching practice in regular schools (Scherer, 2020): Whereas constructivism and investigative learning has been a guiding principle for mathematics education, in special education it was mostly disregarded. Instead, behaviorism remained the central principle. This practice that does not achieve the goals of equity and making substantial mathematics accessible for all students can still be observed in teacher education and school (see Sect. 2).

To avoid marginalization, teacher education programs have to consider and take up those possible differences between mathematics education and special education:

“The challenge to teacher educators is significant. Pre-service mathematics teachers are provided experiences that attempt to broaden their beliefs about mathematics, seeing mathematics as more than just carrying out routine procedures. Mathematics teacher educators help pre-service mathematics teachers see the importance of providing their students with a deeper understanding of mathematics” (Boyd & Bargerhuff, 2009, p. 63; see also Gervasoni & Lindenskov, 2011).

This deeper understanding of mathematics should be applied to all students, and to regular (inclusive) classrooms as well as special education classrooms. With respect to this, the systematic review by Lambert and Tan (2020) showed different approaches and foci for research on mathematical learning that included students with disabilities or not: The former one “lacked sustained qualitative inquiry documenting learning processes of students with disabilities and rarely included the teacher as an explicit focus” (Lambert & Tan, 2020, p. 5; see also Gervasoni & Lindenskov, 2011). There is evidence that the concrete teacher education programs with different orientations also influence pre-service teachers’ beliefs and self-efficacy relevant for inclusive education (cf. Strauß et al., 2023 and Sect. 3.2).

3.2 (Pre-service) Teachers’ beliefs and self-efficacy about inclusive mathematics teaching

Analysing (pre-service) teachers’ beliefs as a part of their competencies for successful mathematics teaching has been a main focus next to teachers’ knowledge in different national and international studies (see for example, Baumert & Kunter, 2013; Blömeke et al., 2014). The huge amount of literature on pre-service and in-service teachers’ beliefs and attitudes towards inclusion often refers to the review of the literature by Avramidis and Norwich (2002), that offers a first overview of the field. Starting with an explanation of the terms ‘integration’ and ‘inclusion’ as well as considering the role of the Salamanca Statement (see above), the review focuses on published studies including students with special educational needs in mainstream schools. The authors especially identified factors (child-related, teacher-related and educational environment-related variables) influencing teachers’ attitudes (Avramidis & Norwich, 2002). For example, they concluded concerning the child-related variables: “teachers seem generally to exhibit a more positive attitude towards the integration of children with physical and sensory impairments than to those with learning difficulties and emotional-behavioural difficulties” (Avramidis & Norwich, 2002, p. 136). Newer studies could replicate these results on the connection between the type of children’s disabilities and teachers’ attitudes (e. g. de Boer, 2012: most negative attitudes for students with learning difficulties and other behavioral problems). Similar results have been identified for pre-service teachers: Their attitudes toward inclusion seem to be positive, but especially their perceived competence is dependent on the types of special educational needs of the students (Avramidis et al., 2000). With respect to teachers’ characteristics, one teacher-related variable influencing teachers’ attitudes is the experience of contact (see also the term ‘Learning opportunities with respect to inclusion’ in Strauß et al., 2023). Some studies prove the ‘contact hypothesis’, suggesting that the experience of contact with children with special educational needs or disabled humans might influence teachers’ attitudes positively (Avramidis & Norwich, 2002). However, as has been concluded by Avramidis and Norwich, “it is important to note here that social contact per se does not lead to favourable attitudes” (p. 138). Newer studies could replicate these results for pre-service teachers. Kunz et al. (2021) showed, for example, that pre-service teachers with contact to humans with disabilities have more positive attitudes toward inclusion, and a higher self-efficacy. Further results concerning, for example, positive effects on teachers’ attitude because of teacher training programs can be read in more detail in the original review (Avramidis & Norwich, 2002). About ten years later, de Boer (2012) concluded similar results in her review of the literature for the variables years of teaching experience (less years more positive), experience with inclusive education (more experience more positive) and teacher training (more training more positive). While the question if pre-service and in-service teachers’ attitudes about inclusion are positive, neutral, or negative, has been answered very differently in several studies, van Steen and Wilson (2020) showed in their meta-review an overall positive attitude towards inclusion. Positive attitudes towards inclusion are linked to a respectful approach to diversity and a valued treatment of all students. Therefore, following a psychological perspective on marginalization (Chen & Horn, 2022, pp. 788), teachers’ beliefs and attitudes should be shifted towards creating mathematics classrooms that allow all students to feel a sense of belonging.

Only a few studies which focus on beliefs and attitudes about teaching in inclusive settings concentrate on subject-specific or rather mathematics-specific questions. However, in different contexts it is assumed that it is more difficult to teach mathematics in inclusive settings than other subjects, due to the specific structure of mathematics (Seitz et al., 2020) and mathematics being a challenging subject for all students (DeSimone & Parmar, 2006a). One example for research on subject-specific attitudes towards inclusion is the qualitative study on issues and challenges for middle school mathematics teachers in inclusive classrooms, by DeSimone and Parmar (2006b). The authors interviewed and observed seven teachers, and part of their study focused on teachers’ beliefs regarding inclusion. Exemplary teachers’ statements illustrate especially challenges in including students with learning disabilities in mathematics lessons. For example, the teachers are concerned if mathematics is too difficult for these students and additionally, they do not feel well prepared for using mathematics teaching approaches for effective teaching of students with learning disabilities (DeSimone & Parmar, 2006b). In another survey study, DeSimone and Parmer (2006a) additionally asked teachers about their abilities to adapt their instruction for students with learning disabilities within specific topics in mathematics (17 topics derived from the curriculum, rating options very comfortable, quite comfortable, somewhat comfortable, and not comfortable). For most of the topics (12 out of 17) teachers only felt somewhat comfortable in adaptations for students with learning disabilities. Topics that were most often rated as very comfortable or quite comfortable encompass the equivalence of fractions, decimals, percent, arithmetic operations on decimals and fractions as well as solving one- and two-step arithmetic word problems. When talking about mathematics-specific aspects of inclusion and more traditional views on how to learn mathematics (see Sect. 2), (pre-service) teachers’ beliefs about student achievement, i.e. if mathematical ability is seen as something natural and fixed or not, is another important factor. For example, the teachers’ belief that mathematics requires innate ability predicts a lower intrinsic motivation among low-achieving students (Heyder et al., 2020). Therefore, (pre-service) teachers’ beliefs about student achievement might play an important role for designing mathematic learning opportunities that value all students.

These reported studies concentrate on (pre-service) teachers’ beliefs and attitudes about inclusion often with a narrower understanding of inclusion (see Sect. 2) and try to identify both, other variables influencing beliefs about inclusion and other variables being influenced by beliefs about inclusion. Further studies focus even more on questions about how beliefs and attitudes towards inclusive education can be changed through teacher education programs. For example, Hassanein et al. (2021) and Taylor and Ringlaben (2012) showed an improvement of pre-service teachers’ beliefs about inclusive education during a specific teacher education program. In more general, learning opportunities for inclusive teaching seem to foster self-efficacy for inclusive teaching (Menge et al., 2021). With a special focus on interventional studies, Lautenbach and Heyder (2019) could show within their systematic review that information-based cognitive interventions and interventions with a combination of information and practical field experience lead to more positive attitudes towards inclusion. Interestingly, Greiner et al. (2020) also showed the importance of personal inclusion related experiences for attitudes and the importance of knowledge about inclusion for self-efficacy, but only concerning pre-service teachers at an advanced stage in their education.

3.3 Mathematics teacher education at the University of Duisburg-Essen

The statewide program for teacher education on the primary level prescribes basic studies in the subjects German language and mathematics as well as in another subject of choice. This statewide obligation was implemented by law in 2011 realizing a BA/MA structure (three years for the BA program, two years for the MA program). As a statewide practical element, the MA program contains a six months long practical phase at school accompanied by university seminars. In many teacher education programs, the mathematical courses address on the one hand content knowledge (e.g. arithmetic, geometry, stochastic). On the other hand, the program contains courses in subject specific didactics (e.g. didactics of arithmetic, diagnosis and support, basics of school mathematics). Didactical courses could include real practical elements (e.g. working with children in school or kindergarten), ranging from teaching whole classes to working with an individual child.

Courses at the University of Duisburg-Essen in the BA program for didactics of mathematics for primary in semesters 1 to 4 touch on the topic of inclusion casually, before two practice-oriented courses offered in semesters 5 and 6 focus even more on inclusive mathematics: ‘Learning Mathematics with Substantial Learning Environments’ (SLE) and ‘Diagnosis and Support’. Further courses in the MA program also take inclusive mathematics as a theme to deepen pre-service teachers’ competencies, namely the above mentioned practical phase at school as well as the course ‘Teaching and Learning Mathematics’ (TLM).

The structure and content of this mathematics teacher education program allows to consider both, a broader and narrower understanding of inclusion (see Sect. 2). Some courses focus on students with SEN and others discuss ideas for coping with the heterogeneity of all students in mathematics classrooms. The course SLE, which also considers both understandings of inclusion, has been part of a design-research project that will be presented in the next section. In many courses, pre-service teachers are also encouraged to reflect on their own beliefs and attitudes of inclusive mathematics teaching. Overall, the teacher education program is designed to prepare pre-service teachers for the manifold demands of inclusive teaching, and teaching mathematics for all. Therefore, it is of interest to investigate the pre-service teachers’ beliefs and self-efficacy in the context of inclusive mathematics teaching and how these might change during the teacher education program (see Sects. 4 and 5).

4 The project Mathematics Inclusive within the project ProViel

The project Professionalisierung für Vielfalt (ProViel, Professionalization for Diversity; https://www.uni-due.de/proviel/) was funded by the Federal Ministry of Education of Germany within the frame of a program for teacher education (2016–2023). Diverse sub-projects focused on different aspects of inclusive education, covering the wide array of facets and dimensions in this field (Bishop et al., 2015; Good & Brophy, 2008). Following a design-based research approach, the sub-project Mathematics Inclusive aimed at implementing subject-specific concepts and modules for inclusive mathematics education.

4.1 Exemplary courses ‘Learning Mathematics with Substantial Learning Environments’ and ‘Teaching and Learning Mathematics’

In the following two courses will be sketched: ‘Learning Mathematics with Substantial Learning Environments’ (SLE) (third year, BA program for primary mathematics), and ‘Teaching and Learning Mathematics’ (TLM) (second year, MA program for primary mathematics). The developmental work within the project ProViel mainly concentrated on the BA course, whereas the MA course integrated the topic of inclusion as one selected focus theme. The design process for the BA course started in 2016, and the first course ran during the winter semester 2016/17, followed by a yearly repetition.

The course SLE contains a weekly two-hour lecture combined with a weekly two-hour seminar. The lecture covers theoretical backgrounds following a constructivist view on teaching and learning mathematics, and illustrating that this approach is suitable for inclusive mathematics classrooms making mathematics accessible for all students (see Sect. 2). Topics are characteristics of learning offers and the concept of natural differentiation in contrast to more traditional concepts of differentiation, examples for concrete learning arrangements as well as the analyses of concrete interview or classroom situations for various mathematical contents (cf. Scherer, 2020). The didactical concept, and by this realizing a natural differentiation is in line with a constructivist understanding of teaching and learning (cf. Wittmann, 2001), and has been proved to be suitable for heterogeneous learning groups in primary mathematics (cf. Scherer & Krauthausen, 2010). The course should enable pre-service teachers to design common learning situations within a common mathematical topic. Moreover, the course also aims at addressing and changing beliefs concerning inclusive mathematics (cf. Beswick, 2007/2008).

The corresponding seminars offer different focal points like differentiation, difficulties in language or inclusive mathematics (see also Scherer, 2021), addressing a broader or narrower understanding of inclusion. Within small groups the pre-service teachers have to design and carry out clinical interviews with primary school students—with or without special needs. The pre-service teachers should offer one and the same substantial learning environment to different students, and have to analyse concrete learning processes, and students’ existing competencies as well as existing difficulties. The practical phases and seminar reflections should take into account that subject matter knowledge as well as pedagogical content knowledge play an important role for using tasks and problems in mathematics education and supporting students’ learning (Sullivan et al., 2013, pp. 15). Reflecting on equity and diversity in mathematics classrooms and avoiding discrimination, is of major importance. In particular, this course provides a ‘content related learning opportunity for inclusive professionalization’ (Strauß et al., 2023).

As illustrated in Sect. 3.3, further courses follow, addressing inclusive mathematics. Especially, ‘Diagnosis and Support’ (last year, BA program), and ‘Practical studies in school’ (first year, MA program) provide further practical field experiences for coping with heterogeneity and inclusive mathematics in schools that might have a central influence on pre-service teachers’ beliefs and self-efficacy which will be in the focus of our study. After spending six months at school, the MA course ‘Teaching and learning Mathematics’ (TLM) follows, focusing on different topics and representing central research fields in mathematics education, like models for teaching and learning, didactical materials and representation, or inclusive mathematics. For example, the latter topic stresses concepts like universal design for learning (cf. CAST, 2011), avoiding a deficit view on students’ learning. The course aims at reflecting on these selected themes, analyzing own documents (e. g. videos/transcripts of interviews or classroom situations) gained in practical courses. So, the second point in time of data collection at the beginning of this course (see Sect. 5) represents more or less the end of the teacher education program, embracing substantial further learning opportunities with respect to inclusion compared to the initial data collection.

4.2 Previous research results and current research questions

The project results showed that the BA course conception SLE could address intended objectives (Scherer, 2021), as the differences of the self-assessment before/after the course showed high significances with different aspects, in particular: knowledge about the characteristics and the use of suitable learning arrangements, clinical interviews, and the analysis of students’ learning processes. These self-reported developments of the pre-service teachers are in line with the findings for in-service teachers (cf. Scherer et al., 2019). In addition, pre-service teachers’ affective attitudes (emotions and feelings) concerning teaching mathematics in inclusive settings have been analysed (Bertram & Scherer, 2022). Results showed in average more optimistic and comfortable than pessimistic and uncomfortable affective attitudes of the pre-service teachers, which hardly changed during the course SLE. At the same time, some pre-service teachers showed bigger and some smaller or no differences in their affective attitudes concerning teaching mathematics or other subjects in inclusive settings being able to consider the specific needs of all students. Furthermore, pre-service teachers in TLM, i.e. at the end of the MA program, feel less confident and more often rather helpless about the idea of teaching mathematics in an inclusive classroom than in SLE, i.e. at the end of the BA program (Bertram & Scherer, 2024). At the same time, they feel less optimistic and more often neutrally pessimistic/optimistic.

In the literature, the understanding of inclusion is discussed, as well as the question of the design of associated teacher education programs. The given example of a specific teacher education program in Germany in Sect. 3—which elements might be transferable to other universities and other countries—as well as previous project results offer first new insights into professionalisation for inclusive mathematics. Due to the specific role of beliefs about and self-efficacy for inclusive teaching, the lack of research on mathematics-specific aspects, and the design of the teacher education program (see Sect. 3), the study, which is going to be presented next, concentrates on these research questions:

  1. 1)

    Do pre-service teachers’ beliefs about student achievement differ in the last year of the BA program and the last year of the MA program?

  2. 2)

    Do pre-service teachers’ cognitive beliefs about inclusive mathematics teaching differ in the last year of the BA program and in the last year of the MA program?

  3. 3)

    Does pre-service teachers’ self-efficacy for inclusive mathematics teaching differ in the last year of the BA program and in the last year of the MA program?

Here, the end of the BA program refers to pre-service teachers attending the course ‘Learning Mathematics with Substantial Learning Environments’ (SLE), and the end of the MA program refers to pre-service teachers attending the course ‘Teaching and Learning Mathematics’ (TLM).

5 Methods—sample, instruments, and data analysis

The sample consists of 92 (86 female, 6 male; age: M = 22.61, SD = 3.141) primary pre-service teachers. Data collected in the winter semesters 2016/17, 2017/18, 2018/19, 2019/20 in the course ‘Learning Mathematics with Substantial Learning Environments’ (SLE) have been matched with data collected in the winter semesters 2018/19, 2021/22 and 2022/23 (with two missing years due to online teaching within the covid pandemic without data collection) in the course ‘Teaching and Learning Mathematics’ (TLM). Pre-service teachers’ beliefs and self-efficacy were measured with a paper-pencil questionnaire before attending the respective courses (see Sect. 4.1).

In the questionnaire three scales were used to assess pre-service teachers’ beliefs about and self-efficacy for inclusive mathematics teaching. The first scale comprises items measuring pre-service teachers’ beliefs about student achievement. The concrete items were taken from TEDS-M (Laschke & Blömeke, 2014) and are based on Stipek et al. (2001) and Schmidt et al. (2011). The second scale was used to assess pre-service teachers’ cognitive beliefs about inclusive mathematics teaching, whereas the third scale was used to assess pre-service teachers’ self-efficacy for inclusive mathematics teaching. Items for both scales were taken from Meyer (2011), who used different existing national and international questionnaires about this topic to design the scales.

The concrete items for the first scale ‘Beliefs about student achievement’ are formulated more or less generally (Item: Some people are good at mathematics and some aren’t.), or represent a broader understanding of inclusion, for example related to gender (Item: In general, boys tend to be naturally better at mathematics than girls.), or ethnicity (Item: Some ethnic groups are better at mathematics than others.). This scale does not focus explicitly on the narrow understanding of inclusion like specifying SEN students or selected special needs (like mental retardation or down syndrome).

In contrast, some of the items for the scales for cognitive beliefs and self-efficacy with respect to inclusive mathematics address specific groups and focus on a narrow understanding of inclusion. For example, students with mental retardation or severe physical impairments are reflected on (Item with respect to cognitive beliefs: If students with mental retardation attend the class, the quality of teaching for the other students suffers. Item with respect to self-efficacy: No matter how hard I try, I will not succeed in making students value the achievements and contributions of students with severe physical disabilities.).

In summary, the three scales make it possible to assess pre-service teachers’ beliefs and self-efficacy in relation to inclusive mathematics teaching in terms of the different underlying paradigms and subject-specific aspects. Table 1 gives information about the number of items and summarizes the reliabilities, which range from acceptable (α > 0.60) to good (α > 0.80).

Table 1 Instruments with number of items, reliabilities (Cronbach’s α), sample size, and example items
Full size table

For data analysis the paired t-test was used to analyse if there are differences between the two dependent samples (SLE and TLM). The criteria for using the paired t-test, especially a normal distribution of the differences, were met.

6 Results

The first research question focused on differences in pre-service teachers’ beliefs about student achievement at two different points in time within the teacher education program, on the one hand at the end of the BA program in the course ‘Learning Mathematics with Substantial Learning Environments’ (SLE) and, on the other hand, at the end of the MA program in the course ‘Teaching and Learning Mathematics’ (TLM). Between these two courses an intense six-months practical phase at school takes place. The results, using the t-test, are shown in Table 2. At first sight, the pre-service teachers in our sample at both points in time disagree to the beliefs about student achievement stating that mathematical ability is naturally given and unchangeable. However, comparing SLE and TLM showed significant changes in the pre-service teachers’ beliefs: The beliefs about mathematical ability being naturally given and unchangeable (i. e. connected to gender, age, ethnic groups) are significantly more strongly disagreed with in TLM (M = 2.141), at a later point of time, than in SLE (M = 2.595), at an earlier point in time in the teacher education program (t(87) = -6.508, p  < .001). The effect size shows a medium effect. It seems as if pre-service teachers’ existing beliefs in SLE are strengthened during the program and the practical phase at school until participating in TLM.

Table 2 Beliefs about student achievement, cognitive beliefs about inclusive mathematics and self-efficacy for inclusive mathematics in SLE and TLM
Full size table

The second research question focused on differences in pre-service teachers’ cognitive beliefs about inclusive mathematics teaching at different points in time during the education program (courses SLE and TLM). The results of the t-test are shown in Table 2. Pre-service teachers do have positive cognitive beliefs about inclusive mathematics teaching, but these do not differ significantly between SLE and TLM (t(75) = 1.812, p = .074). These positive beliefs indicate what pre-service teachers think about inclusive mathematics education and, as the following examples show, the fact that these positive beliefs have not changed could be seen as a good result. For example, on average, pre-service teachers agree that teaching disabled and non-disabled students can meet the needs of all students through appropriate methods. They also agree that a heterogeneous classroom promotes acceptance of differences, or that SEN students are not socially excluded in mainstream classrooms.

The third research question focused on differences in pre-service teachers’ self-efficacy concerning inclusive mathematics teaching. In total, the pre-service teachers in our sample agree to the items stating a higher self-efficacy for inclusive mathematics teaching. The SLE and TLM comparison, again, showed significant changes in the pre-service teachers’ self-efficacy: The self-efficacy for inclusive mathematics teaching is significantly higher in TLM (M = 3.339), at a later point in time, than in SLE (M = 3.135), at an earlier point in time in the teacher education program (t(83) = 3.798, p  <.001). The effect size shows a weak effect. Again, it seems as if pre-service teachers’ existing self-efficacy in SLE increased during the program and the practical phase at school until participating in TLM.

The research in the field of changing beliefs in a teacher education program on the one hand allows to assume positive effects concerning inclusive teaching (see for example Hassanein et al., 2021; Taylor & Ringlaben, 2012; Lautenbach & Heyder, 2019), but on the other hand states more generally, that “teacher preparation programs appear to play little role in shaping beliefs” (Burroughs & Schmidt, 2014, p. 279). Compared to other subjects or general surveys, also the school level seems to be an influencing factor: Lojewski et al. (2022) found that pre-service teachers on the primary level showed more positive attitudes towards inclusive education than pre-service teachers on the secondary level. This finding could be shown before and after the six-months practical studies at school at the end of MA program. Moreover, pre-service teachers on the primary level showed slightly more concerns after the intense practical element which might be caused by more insight in the further development and complexity in inclusive school systems. This ambivalent perspective on whether and how beliefs and attitudes change is underlined by our results. Beliefs about student achievement—connected to various aspects of students’ heterogeneity—changed during the teacher education program, while cognitive beliefs about inclusive mathematics teaching did not change significantly. This might be connected to the underlying paradigms and understandings of the term inclusion being represented by the different scales, but it might also be connected to the mathematics-specific perspective. It remains an open question why cognitive beliefs did not change. Perhaps, in our case, cognitive beliefs are more stable over time than beliefs about student achievement, or further participation during the teacher education program and the intense practical phase at school have a stronger influence on pre-service teachers’ self-efficacy than on cognitive beliefs. Therefore, future research could also focus on the links between these three constructs (see Sect. 7).

Considering previous project results, our findings are of major importance: In former surveys it turned out that after two years of attending the BA program with practical elements at school, only about 50% of the pre-service teachers have had school-related experiences whereas the others had no experiences or had experiences out of school or in other fields (see Scherer, 2019). For school related experiences, some of the pre-service teachers’ statements lead to the interpretation that their observed teaching and learning settings more or less represented an exclusive setting with separate learning situations and the risk of discrimination rather than inclusive education. The courses SLE and TLM explicitly offered alternative experiences to avoid exclusion and marginalization like realizing substantial mathematics for all, or common learning situations in inclusive settings, and these learning opportunities might have influenced pre-service teachers’ beliefs and self-efficacy with respect to inclusion until the end of the MA program in a positive way.

7 Discussion

Professionalisation for inclusive mathematics and addressing equity is a central topic in the development of and research on teacher education programs. Because of different underlying terms and paradigms in the field of inclusion (see for example Roos, 2019), teacher education programs can be designed between focusing on specific students or taking a look at all students, or considering both. Regardless of the focus, teacher education has to avoid discrimination and address the needs of all learners. This paper reported on the teacher education program at the University of Duisburg-Essen, more or less representative for the national education program and with elements of practical studies possibly comparable to international programs (see for example Schmidt et al., 2011; Strauß et al., 2023), as one example of how to address different understandings of inclusion. Research results of the project ProViel were presented, especially on beliefs about student achievement and self-efficacy for inclusive mathematics teaching. In this last section, we discuss the results, show limitations, and concentrate on implications.

Because of the non-experimental design of our study, there could be different explanations for the changes in pre-service teachers’ beliefs. Hunter et al. (2020) found changes in in-service teachers’ beliefs, who participated in a project for equity, accompanied by redevelopments of teachers’ pedagogical practices as part of the project participation. Compared to our group of pre-service teachers, one could also see several practical phases in school (subject specific field experiences) during the whole teacher education program as suitable content related learning opportunities for inclusive professionalization (Strauß et al., 2023) that might support changes in beliefs and especially in pre-service teachers’ self-efficacy. Future research could focus on the question whether practical elements of teacher education programs and related learning opportunities foster a positive development of pre-service teachers’ beliefs and self-efficacy, despite the fact that they may have had less school-related experience of inclusive teaching before, particularly in mathematics. Those findings can also be proved for other universities with similar education programs (especially connected to practical studies at school, see also Lojewski et al., 2022). Our program addresses explicitly the topic of inclusion, in theoretical as well as practical courses. The underlying paradigm pleads for a competence-oriented view aiming at equity and avoiding discrimination and marginalization, and our findings confirm a positive attitude towards this paradigm. Moreover, a positive development during the program and after important practical studies in school have been found. The concrete results are a further starting point for accompanying and reflecting on practical studies.

One limitation of our study refers to the used questionnaire, because we did not assess, what exactly pre-service teachers associate with inclusive mathematics teaching, and if for example avoiding marginalization is in the focus. Based on additional interview data we have hints to assume how pre-service teachers define inclusive mathematics teaching: It can be suggested that they notice the different terms and paradigms, ranging from a more ‘narrower’ to a more ‘broader’ understanding of inclusion. Such comparisons of teacher education programs should also be addressed in international research and take into account the different traditional contexts concerning inclusion. Another limitation is the rather small number of pre-service teachers in our sample (compared to all pre-service teachers enrolled in the entire teacher education program), which is a result of the matching process over a longer period of time, and not least due to the missing data collection during covid pandemic. Thirdly, the focus on two specific courses (SLE and TLM, see Sect. 4.1) might be seen as a limitation. Although only two courses provide the basis for our data, the complete teacher education program with two central points in time connected to practical elements are of major importance. By this, our results might be transferable to other programs, also to an international context.

Future research can analyse the given data with respect to further interesting questions, like “How are the pre-service teachers’ pre-experiences and their competence development connected to pre-service teachers’ beliefs?”. Because subject-matter questions of inclusive teaching are relevant not only in mathematics but also in other subjects, future research can compare teacher education programs. Moreover, it would be interesting to compare pre-service teachers’ beliefs in mathematics with other subjects. Although primary teachers are more generalists teaching several different subjects (Burroughs & Schmidt, 2014, p. 280), mathematics might play a specific role when aiming at giving subject-specific access to all students (see DeSimone & Parmar, 2006a; Seitz et al., 2020; see also Sect. 1). Of course, corresponding results will have to consider the different perspectives or positions on inclusion the specific subjects pursue in their teacher education program. With a focus on the presented teacher education program, further projects could concentrate on comparing programs for different school levels (see also Lojewski et al., 2022) as well as pre-service teachers’ beliefs depending on whether they are enrolled in the regular mathematics teacher education program or the special teacher education program. Formerly, the University of Duisburg-Essen focused on teacher education for the regular school system including inclusive education, and started with a new special education program in 2022. First hints about different changes in attitudes toward inclusion whether focusing on general or special education pre-service teacher, presented by McHatton and Parker (2013) are an interesting starting point for further research (see also Strauß et al., 2023).

In general, the above presented project results confirm that a program addressing the different categories of competence development (as content knowledge, pedagogical content knowledge, pedagogical knowledge; cf. Baumert & Kunter, 2013) are relevant for preparing teachers for coping with diversity. In particular, addressing students’ individual needs and aiming at giving all students access to mathematics, should be important program objectives. Just as much, making pre-service teachers’ beliefs to the subject of discussion, in our view is of major importance, although some research caution against expecting too much of this phase (cf. Burroughs & Schmidt, 2014, p. 282). Furthermore, existing research does not only take into account knowledge, beliefs, self-efficacy for inclusive teaching per se, but also its interconnections which could be investigated for inclusive mathematics as well, like: Do pre-service teachers with a high pedagogical knowledge in mathematics have a higher self-efficacy for inclusive mathematics teaching? Do positive beliefs about student achievement or about the teaching and learning of mathematics influence the self-efficacy of pre-service teachers for inclusive mathematics teaching?

What remains an open question is, if and in how far the gained knowledge and competencies as well as attitudes and beliefs will be implemented optimally and consistently in everyday classroom practice (Burroughs & Schmidt, 2014, p. 282; Boyd & Bargerhuff, 2009), and if in-service teachers will be successful in avoiding discrimination and marginalization when confronted with the manifold challenges in school. With respect to this, life-long learning and the phase of in-service teacher training remains an important field.