Keywords

1 Introduction

As pointed out by Medley (1987), research related to teaching is always focused on student learning outcomes (Type A). The factors relating to teachers which directly influence these outcomes include four online variables. Teachers’ competencies, knowledge, and skills play a critical role in this regard (Type E) and are connected to the variables describing proactive (planning, evaluation in Type D) and interactive (process of teaching in Type C) observable teacher behaviors relating to their performance. The overall basis for teacher competencies are teachers’ pre-existing characteristics which they already have prior to admission to teacher education (Type F).

For optimal performance, it is crucial to strengthen teachers’ competencies (Type E); this is a core goal when conducting mathematics teacher training and reflection on experiences (see Type J). This offline variable is intended to foster teachers’ personal and individual professionalization as a core presage variable in terms of presage-process–product Research (PPPR, see Medley, 1987, p. 108; Manizade et al., 2019). Mathematics teacher training includes pre-service activities at university as well as professional development (PD) programs for teachers already in service.

The core issue in teacher training is linking the cognitive side of knowledge with experience (Type J) gained in either pre-service or in-service training. Schön’s concept of “The Reflective Practitioner” (1983) highlights this link as the initial connection between teachers’ practice and knowledge, and thus also sheds more light on research into professionalization. The aim of any professionalization process is to lead teachers to recognize new ideas and innovative approaches for their teaching and enable them to implement these in the classroom. This process must be initiated and supported during PD programs. It is important that facilitators provide teachers with opportunities to reflect on and enhance all facets of knowledge because—in addition to the central role of knowledge in thinking, acting, and learning—learning is an active, constructive process, with knowledge and learning rooted in contexts and cultures (Brown & Borko, 1992; Putnam & Borko, 2000). It is worth emphasizing here that it is knowledge and beliefs themselves which are the critical targets of change for classroom implementation, since these largely determine what teachers do in the classroom. Accordingly, during PD programs, these components must be considered targets of change (Putnam & Borko, 2000). Borko and colleagues (2014) describe delivering appropriate PD programs as challenging and see facilitators as responsible for appropriately implementing such learning opportunities.

Although facilitators play an important role, a complete description of the competencies required to fulfill the complex tasks they must perform to conduct effective PD is lacking. Even though facilitators are usually experienced teachers, teaching experience in itself does not guarantee that a teacher has the necessary competencies to help other teachers develop their own mathematics teaching (Even, 2005). Our paper follows in the footsteps of Manizade and colleagues’ (2019) framework along with Medley’s (1987) concept of “mathematics teacher training and experience” (Type J). As mentioned in the book’s introduction, over the years there have been many approaches to assessing the quality of teacher education and training and to evaluating (via empirical studies) the influence of corresponding variables on the development of teacher competencies. The importance of PD has also increased in recent decades, and thus corresponding variables such as teacher engagement and participation in PD have also been the focus of research. However, rather than looking at variables such as the design and quality of professional development, the focus here will be on the competencies of facilitators themselves with the aim to describe the range of competencies required by facilitators in mathematics. This chapter’s key aim is to advance teaching research and build on Medley’s framework by focusing on learning outcomes to include the effectiveness of those involved in educating teachers themselves.

Scope of Our Work

To describe the competencies needed by facilitators in mathematics education and understand their responsibilities regarding chain effects on students’ learning outcomes, it is important to build on existing research findings and insights from the field of teacher professionalization (Sect. 2). Epistemologically, learning is an active process of constructing new knowledge (von Glasersfeld, 1998) and it is therefore important to design any learning process as an interplay of content and learners’ perspectives. Facilitators in PD programs must regard teachers as learners. However, unlike students and pupils, teachers are adults and experienced professionals, and facilitators must motivate them and initiate PD in an appropriate way. Therefore, this paper examines the field of general adult education, with a focus on the specific needs of mathematics teachers as adult learners. The first topic under discussion is thus mathematics education. Following this, we start our review with findings and insights regarding adult learning in general (Sect. 2.1) and then specifically from the perspective of mathematics education (Sect. 2.2). An overview of both fields is necessary to examine the various challenges that facilitators must address when leading mathematics teachers to reflect on and develop their practices and teaching routines.

Section 3 provides insights into how the role, tasks, and competencies of facilitators are currently conceptualized. To better understand and categorize our understanding of facilitators, the role is first explicitly defined (Sect. 3.1). Once again, the (overlapping but distinct) fields of PD in adult education and mathematics education are both considered in this part of the paper. In the field of adult education (Sect. 3.2), the review is based on existing competency frameworks already discussed in the literature (Wahlgren, 2016). However, these frameworks require adaptation and further development to be effectively applied within the specific context of mathematics education. Some of the existing competency frameworks within general adult education will also be discussed to provide a historical review of the development of these considerations (Sect. 3.2). This is followed by a focus on the teaching profession in mathematics education (Sect. 3.3). The challenge—as will be shown in the following sections—is to develop a competency framework that fits the professional requirement profile as well as the cultural and structural framework. To emphasize the complexity of this challenge, we will conclude this section by presenting such an evolutionary process based on the literature review performed within the context of the DZLM expert network (Sect. 3.4).

In the outlook, we will summarize the findings and lines of development from the last twenty years and take a look at further challenges in the field of mathematics teacher training and reflection on experiences (Sect. 4). In doing so, technological developments and their influence on teaching and training will be presented as an example.

Methodology of Our Literature Review

Depending on the topic and the existing literature, different types of literature reviews may be appropriate (Higgins & Green, 2008). To provide an overview of the research evidence on the competencies of facilitators in mathematics, we chose to conduct an integrative review to cover the breadth of both fields of interest, general adult education and mathematics education (Whittemore & Knafl, 2005). An integrative review represents a holistic approach offering the possibility of bringing together studies using a wide variety of methods, meaning that qualitative and quantitative methods are considered in addition to theoretical and empirical ones. This enables us to not only depict the current state of research but also to create direct links to possible areas of application. Due to the involvement of several people (the four authors) in the research, a certain level of objectivity can be assumed in the selection and evaluation process.

Since reviews usually include all studies that are found to consider the research field, the number of studies included can vary dramatically. In the first search run for the field of general adult education, a very broad constellation was found extending in very different directions. Narrowing the search to the competencies of mathematics teachers and existing research on mathematics facilitators produced relatively focused and clearer results. Due to the wide range of work in both content areas, we decided to focus mainly on articles and studies looking closely at teacher competencies and their development and which attempt to formulate quality standards for facilitators. Furthermore, we also set restrictions in terms of publication date and study design. For example, for general adult education, we looked predominantly at literature from the last twenty years or which (in some cases) introduced changes or innovations to previous frameworks. For the requirements and competencies of mathematics facilitators, such restrictions were unnecessary due to the small field of research. By means of communicative validation, the focus was then condensed to the articles that have been integrated here. The results of the research have been discussed and checked how the authors assess the validity of the results (according to Mayring, 1990). When selecting articles, in addition to general keywords, particular attention was paid to their compatibility with teacher education and, based on the journals consulted in the database search, to high-quality international peer-reviewed journals. More narrowly, the discipline of mathematics was also a key focus, since it soon became obvious that subject-specific content was important.

We searched the literature using various keywords such as professional development, competency framework, competencies of facilitators, and requirements for facilitators in both content areas. We used several databases to include global studies. The databases included ERIC, ELSVIER, and even MathEduc, which was available until December 2019. These journals included the International Journal of Science and Mathematics Education (IJSME), International Journal of STEM Education, Journal of Mathematics Teacher Education (JMTE), Journal of Mathematical Behavior (JMB), Journal for Research in Mathematics Education (JRME), Psychology of Mathematics Education (PME), Mathematics Education Research Journal (MERJ), and ZDM—Mathematics Education. The International Handbook of Mathematics Teacher Education was also consulted, as Volume 4 is specifically addressed to facilitators in mathematics education. With reference to facilitators of adult education more generally, the following journals were reviewed: the Journal of Teacher Educator (JTE), Journal for Research on Adult Education (ZfW), Adult Education Journal, Teaching and Teacher Education, Educational Psychology and Educational Researcher (ER). Finally, in reviewing the articles that were crucial for us, we also consulted others that were listed in these articles if further insights could be generated. It can be noted that all mathematics-specific articles specifically related to PD competencies, rather than PD design and quality, have been listed in this paper. Because there is such variety in the field of general adult education, we primarily selected those that provided an overall view of development cycles or used a Delphi study, as there is much variation to be found at the national level. All 78 referenced articles that were relevant to our work are mentioned and cited in this chapter.

2 The Evolution of Framing Teachers’ Professionalization

2.1 PD in General Education

Medley’s (1987) description of PPPR focused on processes to improve student learning outcomes. The general steps he identifies are nonetheless valuable and provide crucial foundation for further concretization. The modeling he used to describe each teaching and learning process—at the teacher-student level in his case—can be elevated and applied to the facilitator-teacher level too, thus providing a perspective on facilitators’ competencies as well.

Innovations and change processes due to new curricula or new administrative conditions pose many challenges for schools and teachers, who need intensive support and assistance to tackle them. This requires PD programs that reach as many teachers as possible by scaling-up PD programs of a high standard. In Hattie’s meta-study (2009, p. 119 ff.), he calculates the influence of content-related in-service PD programs with an effect size of d = 0.62 and classifies PD programs as an important intervention with significant effects on improving teaching quality. Facilitators of teacher PD programs, in turn, need comprehensive, evidence-based qualifications to be able to successfully design and implement teacher PD programs. The meta-study by Timperley and colleagues (2007) also reported the effects of teacher training on student outcomes. This study discovered that there was an average effect of d = 0.66, but variations were found by school subject and student level. For example, the effect in mathematics was d = 0.50. From the various meta-analyses, it can be concluded that teachers who regularly participate in PD programs sustain and even enhance their professionalism throughout their working lifespan, thus also effecting pupil learning outcomes. For this to occur, Lipowsky and Rzejak (2015) note that in-service teachers need regular PD. This in-service PD must involve a sufficient quantity of high quality learning opportunities planned and implemented by facilitators.

The insights provided by Medley (1987) in his description of state of the art teaching research can also be applied to learning outcomes among teachers following PD programs. In other words, the first question is which competencies teachers need to carry out effective teaching in schools. Based on the answer, the competencies that facilitators should have to deliver effective PD programs should be identified. Simultaneously, research that examines teacher competencies and pre-existing characteristics must also be considered, as this strand of research can shed further light on the knowledge needed by facilitators to adequately enhance the competencies that teachers require.

A detailed illustration which highlights that the design and use of CPD programs are not the only initial and preparatory component of the transfer process in teacher professionalization can be found in Lipowsky’s (2014) “offer-and-use model on PD level” (Fig. 1). It provides a highly differentiated framework to concretize in detail the challenges of the offline variable “mathematics teacher training and experiences” (Manizade et al., 2019) to achieve a successful transfer process and successful CPD at every level (see also Guskey, 2002), including for student learning outcomes.

Fig. 1
A chart presents an offer-and-use model, where successful professional development depends on the facilitator's characteristics, participants' characteristics, and the quality and quantity of learning opportunities. The interplay of these elements in the transfer process determines effectiveness.

Offer-and-use model for research on teachers’ professional development. Lipowsky (2014, p. 515), English version: Lipowsky and Rzejak (2015, p. 30)

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Most research on PD focuses on describing the design of PD programs and assessing their quality. For example, the meta-study by Darling-Hammond and colleagues (2017), which reviewed 35 effective PD, identifies seven characteristics of effective teacher PD, that mainly relate to PD program design and implementation (e.g., content focused, sustained duration, or opportunities for feedback and reflection).

In addition to this focus on content, how the PD is carried out and how spontaneous situations are managed and properly moderated are also important. These areas relate to learning processes in subject lessons and active learning using the theory of adult learning should be considered here. Collaboration should also be supported in in-service contexts, and various models of effective practices should be applied. The format of permanence should be another focus. Finally, regarding feedback and reflection opportunities, coaching and expert support are mentioned. However, like in the illustration of Lipowsky’s model (2014), the competencies of the facilitators themselves are a crucial initial component.

2.2 PD in Mathematics Education

Focusing on the context of mathematics, Sztajn (2011, p. 221) pointed out that the attention paid to research in the field of mathematics teacher education increased significantly in the 1990s. When this is connected to Medly’s framework (1987) and its adaptation in the context of research on mathematics teaching and mathematics teacher education by Manizade and colleagues (2019), it becomes apparent how this field has evolved to reflect current research.

In their meta-study, Timperley and colleagues (2007) looked at the effects of teacher training on student outcomes in a differentiated way, depending on school subject and student level. In mathematics, they speak of an effect size of d = 0.50 for student learning outcomes, although this differentiated view relates only to 11 core studies. For mathematics specifically, stronger effects were found in studies that focused on building teachers’ content knowledge and pedagogical knowledge than studies that looked only at content knowledge. As reflected in several research papers on PD programs and most notably in the review by Sztajn and colleagues (2017), there is a growing body of empirical research that reveals the structure, content, and impact of effective CPD in mathematics education. Predominantly, these studies provide insights into the characteristics of PD programs that provide appropriate learning opportunities for teachers.

Other studies indicate that PD opportunities for mathematics teachers are recognized as a critical factor in increasing student achievement. Figure 2, which depicts the chain of effects from the competency level of facilitators (teacher leaders (TL)) to student learning outcomes, illustrates that the design and use of PD programs is not the only initial and preparatory component in the transfer process within teacher professionalization. Borko and colleagues (2014, p. 149 ff.) see the quality of mathematics instruction as the central factor influencing student learning. In this regard, the emergence of teacher competencies for quality instruction is seen as starting from high-quality PD programs. Accordingly, facilitators must be able to consider and implement all aspects of PD programs so that their influence on the quality of mathematics instruction is sufficient. This also requires establishing qualification standards for this group of adult learners (PD for TLs).

Fig. 2
A flow diagram presents a problem-solving theory. Improved student learning can be achieved through P D for T Ls, P D for teachers, and improved quality of teaching. Professional development for T Ls and teachers involves an increase in knowledge and skills.

Implementing the problem-solving cycle: theory of action (Borko et al., 2014, p. 152)

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As can be seen in Fig. 2, frameworks usually describe facilitators from the classroom level (see Carroll & Mumme, 2007; Perks & Prestage, 2008; Jaworski, 2008; Hauk et al., 2017; Prediger et al., 2019). Thereby, knowledge on the lower level is always a component of the level above. Here, Carroll and Mumme (2007) see mathematical knowledge at the classroom level: a teacher’s mathematical knowledge for the mathematics teacher educator within a larger context. That includes, for example, knowledge of teachers’ professional learning. Perks and Prestage (2008) add an additional aspect they call “professional traditions” describing a structure similar to Carroll and Mumme’s in which each level is nested within the next. This new area includes knowledge of school curricula or practices as well as research at classroom level. At teacher PD level, this is expanded to include knowledge of systems, institutions, and teachers’ own research efforts.

However, the framework focuses more on the types of knowledge that must be brought into the classroom and less on the interactions between people and content. Thus, despite very similar nesting in both frameworks, the focus lies on different areas. In the work of Hauk and colleagues (2017), the relationships between knowledge and thinking types associated with the development of mathematical knowledge for teaching are presented. To illustrate this, they have used a concrete case of a specific type of elementary-middle school. Like Carrol and Mumme, the “three-tetrahedron model (3TM) of professionalization research” (Prediger et al., 2019) revisit the interaction between individuals—actors—, but rather than focusing on a specific type of school, it is intended as an overarching framework (Fig. 3, Prediger et al., 2019, p. 410). Following this, the 3TM instead describes rather the mesh of the different levels and is not to be understood as a process model as in Medley (1987).

Fig. 3
An illustration of three tetrahedrons. The model describes the flow of knowledge from facilitators to students. Three levels are facilitator P D level, teacher P D level, and classroom P D level. Facilitator's P D resources change into F C, T R, T C, C B, and C C.

Three-tetrahedron model (3TM) for content-related PD research (Prediger et al., 2019, p. 410)

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In Germany, these research topics are the focus of the work of the German Center for Teacher Education in Mathematics (DZLM), a nationwide institution for the development and research of in-service PD programs for mathematics teachers (Prediger et al., 2019), concentrating on the qualification of facilitators. In the 3TM, individual levels are described and related to one another (Prediger et al., 2019). The elements of the three-tetrahedron model are the most important reference points for facilitator activities (see in more detail Prediger et al., 2019):

  • The lowest level is the classroom tetrahedron, in which the pedagogical triangle has been extended by the corner “classroom resources.” This classroom tetrahedron, as a whole, is PD content.

  • The teacher PD level regards teachers learning this content—so here too, there is a tetrahedron with the relevant actors (facilitators as teachers; teachers as learners) and a corner for resources, especially resources seeking to further education.

  • Facilitators themselves are “learners” at the top level. Here, facilitators are involved in continuous qualification programs, which vary greatly in quality and quantity in different systems depending on local framework conditions.

Regarding the development of a competency framework for facilitators, it is important to realize that facilitators are related to all elements of the 3TM, since they are both learners on the qualification level and teachers on the PD level, while the classroom level, as the PD content, is always on the facilitators’ minds. In addition, depending on the framing of the educational system in question, facilitators may act as teachers as well. In most systems, facilitators are also teachers and are active in the development of their own schools. They can act as colleagues among peers, accompany a quorum as a regular guest, or provide impetus as external experts.

For the qualification of facilitators, all relevant aspects of the individual must be considered (e.g., Bromme, 1992; König & Blömeke, 2009). This involves a cognitive perspective on the facilitators’ knowledge and orientations and a situated perspective on their work as a facilitator (Prediger, 2019). These perspectives are complementary and may be located within a continuum from disposition to performance (Blömeke et al., 2015; Depaepe et al., 2013). In this paper, the focus is on the cognitive perspective. We will discuss competencies, which represent the latent characteristics required to perform effectively as facilitators (Weinert, 2001, p. 27). These cover facilitators’ knowledge, beliefs, and attitudes, which can be learned and improved through institutional learning opportunities (Klieme et al., 2008; Weinert, 2001).

Interest in PD in mathematics has increased, with consideration given to location and structure as well as interrelationships within the impact chain, but as Sztajn and colleagues (2017) pointed out, in terms of what is known about PD, the knowledge gap still includes what facilitators should be required to know and be able to do and what is associated with their preparation of PD. This field of interest has become more significant in more recent working groups. Since 2011, there have been several working groups at PME addressing this field, and attention has also been paid to mathematics facilitators at CERME and ICME. There have also been three major international publications, namely JMTE (2018, Vol. 21(5)), The International Handbook of Mathematics Teacher Education (2020, Vol. 4), and the book “The Learning and Development of Mathematics Teacher Educators” by Goos and Beswick (2021) which specifically address the knowledge, skills and development of facilitators in mathematics education.

Although the topic has been identified as important, there is still little research on this phase of teacher education. This is at odds with the attention given to the design of high-quality learning processes for teachers and facilitators, as previously mentioned. It is interesting to see which research strands have stood out in this area of mathematics teacher training over the past 20 years. While comprehensive and accurate knowledge of facilitator competencies was neglected in the past, it is now recognized as being an area of great interest which is worthy of specific consideration. We therefore aim to address the competencies of facilitators as a new presage variable.

3 Facilitators’ Competencies—A New Presage Variable

As initiating PD with adult learners presents specific challenges for facilitators to overcome, we first focus on our understanding of facilitators as well as their role and then take a closer look at the field of general adult education. Following this, we look in detail at findings in the field of facilitators in mathematics education and highlight developments in this area of research.

3.1 The Role of Facilitators

In most school systems, there are people entrusted with the task of planning, organizing, and carrying out CPD programs for in-service teachers. In many (but not all) school systems, these people have worked or still work as teachers and are often, in a sense, “self-made” (Zaslavsky, 2008, p. 93). They usually devote themselves to these activities in addition to their work as teachers and often have few systematic qualifications for this activity. The many different designations in use—such as mathematics trainers, moderators, multipliers, teacher educators, didacticians, specialists, coaches, or facilitators (see also Bernhardsson & Lattke, 2011, p. 19)—show the heterogeneity of the work they carry out. Henceforth, we will call them facilitators, as we feel this term best expresses their role in guiding teachers to undertake change processes more easily.

A “facilitator” is a person who opens new possibilities and accompanies others on development processes; in contrast, “teachers’ leader” or “teachers’ educator” designates a hierarchical relationship (e.g., in teacher PD programs with structurally conditioned relationships of dependence). Lunenberg and colleagues (2014) identify six different roles: teacher of teachers, researcher, coach, curriculum developer, gatekeeper, and broker. A look at these various roles once again highlights the manifold requirements for facilitators in terms of both knowledge and competencies. The diverse roles a facilitator might perform require expertise, skills, and special abilities such as accompanying, demonstrating, counseling, mentoring, evaluating, empowering, cooperating, and so on (Shagrir, 2013). Smith (2005) and Zaslavsky (2008) concretized these requirements by listing facilitators’ typical characteristics (e.g., Shagrir, 2013; Smith, 2005; Zaslasvsky, 2008). Smith (2005) identifies specific qualities and behavior for facilitators, who he states should:

  • be self-aware to reflect on their actions and discuss them,

  • have in-depth professional knowledge based on theory (on testing in practice),

  • be involved in research (to be involved in creating new knowledge) and in the writing processes of the curriculum,

  • be good teachers and have experience in different age groups (school levels),

  • have a comprehensive understanding of the education system and

  • have reached a high level of professional maturity and autonomy.

These requirements, as Shagrir (2013) points out, obligate facilitators not only to establish clear work procedures at each stage, but also to sustain the relationship between the field of practice and teacher education institutions. Interestingly, these characteristics are all interdisciplinary and their application to mathematics and mathematics teachers is only implicit. Therefore, it is unsurprising that similar issues are also discussed in adult education research, with the added discourse of the respective professional field.

3.2 Facilitators in General Adult Education

Why do we need such competency frameworks in adult education at all? The necessity arose over time. Just as with other professional training strands, as adult education became an increasingly important field of action, professionalization had to occur, since the demands on teachers were constantly growing. Adult education represents a significant challenge and cannot be undertaken lightly, as pointed out by MacKaye back in 1931 when he described it as an “act of war” for which one must prepare tactically (as cited in Rossman & Bunning, 1978, p. 140). Delivering it is a challenging task which therefore requires in-depth qualifications. In the first half of the twentieth century, numerous programs were developed in which studies were conducted to identify the central requirements and the core activities of adult trainers. As Rossman and Bunning (1978) noted, one of the first texts on training adult educators was published in 1948 by Hallenbeck. Since then, more and more attention has been paid to this topic, and over time, awareness has arisen that this activity must be taken up as a profession.

The literature shows that over several decades, various frameworks have been developed for adult education to teach skills, knowledge, and competencies (see Wahlgren, 2016). During this period, the question of adult educators’ competencies has been studied from different perspectives and in different contexts. Three main positions can be distinguished according to Wahlgren (2016): Delphi studies, national curricula for adult educators, and studies on competencies for vocational educators. Wahlgren (2016) gives an overview of these developments and draws attention to the fact that these findings were mostly gained through Delphi studies by experts in this field. It is noteworthy that the differentiation between skills and knowledge is no longer made in more recent studies, and that these two concepts are no longer even the focus of research. Instead, attention is focused on the concept of competency: “In the more recent study, a distinction between knowledge and skills is no longer made, but the concept of competencies is still used.” (Wahlgren, 2016, p. 346.) In addition, Wahlgren emphasizes that communication skills and the related ability to identify students’ needs and experiences have consistently been found to be essential, even though different studies identified different emphases and rankings (Wahlgren, 2016, p. 346). One of the most recent large-scale Delphi studies on adult learning relating to facilitators’ core activities was published by Bernhardsson and Lattke (2011), and makes it possible to compare different competency frameworks both over time and across countries, as Wahlgren (2016) has done.

There are numerous efforts underway to formulate uniform frameworks that can be used not only across occupational groups but also across countries. However, this is particularly difficult for teachers, as not only does each country have specific school qualification frameworks, but teachers as a professional group differ greatly from other professional groups such as lawyers or police officers. In this context, the prominent project QF2TEACH is worth mentioning. In QF2TEACH, the core competencies of teachers for continuing education are developed in relation to the European context (see Bernhardsson & Lattke, 2012). Such projects usually focus on comparing different activity profiles, but also on the degree of concretization, which varies across frameworks. Our aim is thus to identify the overarching structures that are considered relevant.

There are certainly substantial differences between the various occupational fields that may explain the far greater differences in the related formulation of competency frameworks. Facilitators must be experts in their specific domain (in this case mathematics education) and must know about the individual characteristics that teachers need to successfully carry out their profession. In the frame of DZLM, we aimed for a competency framework for facilitators of mathematics education. As a starting point we choose the GRETA competency framework of the German Institute for Adult Education (DIE, Leibniz Centre for Lifelong Learning) in the field of general adult education, for two reasons. First, it fits well with the existing requirements (the abbreviation GRETA in German stands for “basics for a standardized process for recognizing teachers’ and trainers’ competencies in adult and continuing education”; www.die-bonn.de/greta; see Fig. 4), which are as follows:

Fig. 4
A circular illustration. The center is labeled professional teachers' Competencies. The outermost circle is labeled professional self-monitoring, knowledge, and skills, content and field-specific knowledge, and respect of professional values and beliefs.

GRETA competency framework (Strauch & Lencer, 2017). https://ec.europa.eu/epale/en/blog/greta-competence-model-teachers-continuing-training

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  • The cultural context (cf. Wahlgren, 2016) emerges from it.

  • The specific requirements of this professional field are considered.

  • The level of detail corresponds to that which appears to be suitable for the later use of the framework.

  • Docking with the required subject area of mathematics is possible.

The GRETA framework offers an interdisciplinary structural competency framework covering all the basic competencies required to be able to teach well in adult and continuing education and it highlights the importance of including content-specific competencies. This framework uses Baumert and Kunter’s framework (2013) as orientation and was developed through a Delphi-process with educational practitioners and stakeholders. The comprehensive framework of Baumert and Kunter (2013) refers specifically to mathematics teachers. It relies on Shulman’s (1986) structural knowledge dimensions: content knowledge (CK), pedagogical content knowledge (PCK), and general pedagogical knowledge (PK). Baumert and Kunter (2013) also included existing organizational knowledge, coaching knowledge to communicate professionally with parents (see Bromme & Rambow, 2001), and beliefs and values as separate categories, with fluid transitions.

The GRETA framework was developed in a Delphi-process with educational practitioners, researchers, and stakeholders, meaning manifold perspectives were involved. Facilitators for every subject area must consider aspects of general adult education because program participants are adult professionals. In addition, they must be experts in the specific domain (in this case, mathematics education) and must know about the individual characteristics that teachers need for the successful accomplishment of their profession. In this regard, the DIE has developed an interdisciplinary structural competency framework covering all the basic competencies required to be able to teach well in adult and continuing education.

The GRETA framework is designed to identify all relevant competency aspects (outer ring), domains (inner ring), and facets (middle ring) via an assessment procedure (Lencer & Strauch, 2016). The framework comprises an even more holistic understanding by providing four aspects of competency (see Fig. 4): professional knowledge and skills, content and field-specific knowledge, professional self-monitoring, and professional values and beliefs. The framework is applicable to all fields of adult education so the areas pertaining to subject-specific competencies have been left blank and must be filled in for the discipline under consideration (mathematics education in our case).

3.3 Facilitators in Mathematics Education

As noted in Sect. 2.2, several working groups have been at PME since 2011, and facilitators in mathematics education have also been a focus for CERME and ICME since 2021. In addition, since 2018 there have been three major international publications specifically dedicated to this field.

In contrast to the previous section which dealt with adult education in general, this section now focuses on the target group of facilitators and on facilitators in mathematics education specifically. A general adult education framework is insufficient as a competency framework for the qualification of facilitators in mathematics education, as content-specific concretizations must be made. For example, in the meta-study by Timperley and colleagues, they note that, when it comes to the school sector,

Experts need more than knowledge of the content of changes in teaching practice that might make a difference to students; they also need to know how to make the content meaningful to teachers and manageable within the context of teaching practice. We are calling these skills provider pedagogical content knowledge. (Timperley et al., 2007, p. xxix)

This makes it clear that, above all, domain-specific knowledge is also highly relevant to the content in which one is acting.

Here, we take a more concrete approach and focus on teachers and facilitators in mathematics. By adopting this specific focus, more concrete facets of interest can be identified, which can then be elaborated as individual categories—always in comparison to an underlying general framework of adult education. Consequently, a concrete examination of the subject-specific challenges involved in the discipline of mathematics must take place. Teacher beliefs sometimes play an important role in mathematics education because they guide actions determining how the subject is taught (Kunter et al., 2013). For instance, mathematics as a discipline may be conceptualized in a more receptive way, with a focus on algorithms and automation. Alternatively, mathematical thinking and problem-solving may be the focus (e.g., Rott, 2020). The content of qualification programs for facilitators would differ according to these perspectives.

At the same time, it must be kept in mind that such a framework does not apply equally to every facilitator. The many designations used for people who carry out this role not only testify to the heterogeneity of their tasks but also make it clear that very different roles and activities are linked to the diverse requirements and competencies. Facilitators also play decisive roles in PD in terms of the extent to which teachers are motivated and supported in their learning (Linder, 2011). To the areas mentioned by Smith (2005, see Sect. 3.1), Zaslavsky (2008) adds further requirements for facilitators in mathematics education such as adaptability and conscious selection of methods and media. It should be noted that facilitators are often expected to be very good (or even the best) teachers (as Smith notes in his list above). However, the role of a facilitator can be compared to that of a soccer coach, in that someone who may not be (or have been) the best player or teacher may be able to successfully train others. Carroll and Mumme (2007) also suggested that facilitators should have detailed subject content knowledge, information about the participating teachers as well as the students of those teachers, knowledge of how to teach students and adults, and knowledge of how to use materials for training to create a productive learning environment.

Various studies have set different priorities for the content knowledge that facilitators should have. However, all studies emphasize that facilitators’ knowledge must exceed the teachers’ knowledge to enable the former to encourage the latter to grow and acquire new knowledge, i.e., fulfilling a similar role that teachers must play for their students (Borko et al., 2014). This goes hand in hand with the 3TM already described (Fig. 3, Prediger et al., 2019).

Although, according to these three levels, facilitators should have extended knowledge compared to teachers, it should be emphasized that there are also knowledge elements that are relevant for teachers but not for facilitators (Beswick & Chapman, 2015). These include, for example, detailed knowledge of school curricula or background knowledge about individual students. For facilitators, only general knowledge of educational standards and curricula is important, as well as relevant empirical findings from (current) research. This is also expressed in Jaworski’s framework of knowledge in teacher education (2008, Fig. 5).

Fig. 5
A Venn diagram titled Systemic and cultural settings and boundaries within which learning and teaching are located. The circle on the left is labeled A, educators' knowledge. The intersected part is labeled B, knowledge shared by educators and teachers. The circle on the right is labeled C, teachers' knowledge.

Knowledge in teacher education (Jaworski, 2008, p. 336)

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The extended knowledge of facilitators refers not only to new knowledge of mathematical content and the relevant pedagogical aspects aimed at PD-level, but also to their pedagogical knowledge of adult education. This includes, for example, knowledge of teachers’ existing practices (Even, 2005; Even et al., 2003) and current views on PD programs in mathematics education (Borko et al., 2014).

In addition to current views on PD programs, Borko and colleagues (2011) also highlighted knowledge about mentoring (i.e., the accompanying support in the implementation of training content) for PD in mathematics. Facilitators should be able to stimulate productive mathematical work in teachers and lead discussions about student reasoning and instructional practices while encouraging reflection, as well as build professional learning communities. In this context, mentoring is seen as a special form of individual support and, unlike coaching, the individual’s interests are seen as the absolute priority. Particularly in the second key aspect of mentoring discussed by Borko and colleagues (2011), leading discussions about student reasoning and instructional practices as well as effective use of video-based PD programs can contribute as forms of facilitation (e.g., Ebers, 2020; van Es & Sherin, 2010; van Es et al., 2014; Zhang et al., 2011). In this regard, communication about video cases is an important component in training teachers’ awareness and ability to analyze. Content should be purposefully related to the teaching and learning of mathematics, ideally contributing to more reflective classroom practice.

Certainly, several of the interdisciplinary competencies mentioned earlier can be directly applied to the roles of facilitators for mathematics teachers. However, it is also apparent from the formulation of individual areas of competencies that a domain-specific focus is needed. As Tzur (2001), in describing in his own development as a mathematics facilitator, states, “… a development from a lower to a higher level is not a simple extension, that is, doing more and better of the same thing. On the contrary, development entails a conceptual leap that results from making one’s and others’ activities and ways of thinking at a lower level the explicit focus of reflection” (Tzur, 2001, p. 275).

As the following example shows, it is not enough to swap out content, rather the entire spectrum of competencies mentioned is needed because they are all systemically interrelated. In one teacher training program on mathematical modeling, the focus is the student task “There is a 3 km traffic jam on the motorway. How many vehicles are caught in this traffic jam?” (see Peter-Koop, 2005, p. 6). It quickly becomes clear that there is no standard procedure or clear solution to this problem. Some of the participating teachers loudly reject the task as irrelevant to mathematics teaching since the task is not in line with their idea of mathematical thinking. The teacher training situation is now challenging for the facilitator in several respects. First, the subject matter “modeling” needs to be taught—meaning that a subject-specific PD program is essential (e.g., Bardy et al., 2021; Dreher et al., 2018). At the same time, it is also important to question participants’ skeptical attitude towards the task and to address their basic beliefs about teaching and learning mathematics. At this point, competencies are required of the facilitator that go beyond pure content or pedagogical content knowledge.

The example shows the need for facilitators to have a wide range of competencies: subject-specific competencies at the classroom level and subject-specific competencies regarding mathematics teachers’ particular needs, problems, and obstacles as well as social competencies as adult educators. All these must be deployed as needed in a fluid interplay of PD activities. As Koster and colleagues (2005) concluded, references to PD activities should be made in addition to a competency framework. In this article, a competency framework is understood as part of a more comprehensive concept of the professional profile, which is additionally linked to activities. These activities define the purpose of the competencies (see Koster et al., 2005). In other words, a competency framework is a working repertoire of expertise that provides orientation and enables someone to perform professionally. As the previous PD activity example showed, facilitators need different competencies for different PD activities on specific PD topics.

In the field of mathematics education, plenty of studies exist which describe the differentiation of facilitators’ knowledge towards teachers in mathematical education (e.g., Ball et al., 2008; Beswick & Chapman, 2015; Beswick & Goos, 2018; Borko et al., 2014; Lesseig et al., 2016; Smith, 2005). However, a systematic description of a framework is still missing. This is essential to adequately support and, potentially, qualify facilitators. As early as 1999, Even stressed the importance of holding frequent planning meetings with facilitators learning a new mathematics PD program to develop their knowledge and leadership skills and to create a professional reference group. She described such meetings as the cornerstone for the “development of a common vision and feeling of shared ownership” (Even, 1999, p. 20).

Reviewing the relevant literature on the generation of competency frameworks in adult education enabled us to identify the relevant categories that underpin these frameworks. Furthermore, we were able to deduce the important stakeholders for this process. With these findings, we laid the foundations for implementing a Delphi study, which involved researchers in mathematics education and key stakeholders (senior administrators). Through cycles of design, evaluation, and redesign, the framework was evaluated for holism, integrity, and practicality. It was important to strike the right balance between general adult education and mathematics education. We briefly report on this process in the following section.

3.4 Facilitators’ Competencies Framework

Here, we give a brief insight into the study undertaken and, above all, present the result of the Delphi study. Based on our results, we will once again take up the findings from the literature review. A detailed version of the Delphi study can be found in the PME paper by Peters-Dasdemir and colleagues (2021). As with other reported studies, it was important for us to identify a competency framework designed to fit the present setting, local requirements, and stakeholder acceptance to qualify facilitators in Germany appropriately. For this purpose, we also included the literature review shown above and embedded it in the Delphi study.

The process consisted of three consecutive rounds in which 61 experts with different professional backgrounds participated. In these discourses, the most important stakeholders were invited to evaluate the framework regarding its practical applicability. We involved 33 researchers, 28 stakeholders, and several teachers with experience in CPD. We completed three cycles of further development. All researchers involved were experts in the field of CPD in mathematics education for primary and secondary levels and were asked to use this expertise to point out key competencies for facilitators. The selection of the people involved in the Delphi study was carried out along the 3TM so that all levels involved in the facilitator activities were included. This was in line with the basic idea of a Delphi study which should include all experts with different backgrounds. The results of the Delphi study showed that experts from different fields were able to develop a common understanding of the competencies necessary for the qualification of facilitators who are responsible for the CPD of mathematics teachers. It leads to the DZLM framework covering four areas, which are concretized from the perspective of mathematics education (see Fig. 6): (1) Professional Values and Beliefs, (2) Professional Self-Monitoring, (3) Competencies at the Professional Development Level, and (4) Competencies at the Classroom Level.

Fig. 6
An illustration of concentric circles.

Competency framework of facilitators in mathematics education

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Like the GRETA framework, we have chosen to structure these aspects and the related competencies in a circle format to symbolize the dynamic fluidity and interconnection between all competency domains in the inner ring. Besides the Competencies on the Classroom Level with PK-C, CK-C, and PCK-C, there is an extra level for Competencies on the Professional Development Level. All surveys focused on the question of the differences and similarities that exist between the competencies of teachers at the classroom level and those of facilitators at the PD level. This is similar to the work of Borko and colleagues (2014), who based their framework on the work of Ball and colleagues (2008) on “Mathematical Knowledge for Teaching (MKT)” analogously for facilitators. There was an intense debate about whether these two aspects should be separated as equal parts or whether competencies at classroom level are an integral part of competencies at PD level. This led to the specification in the two areas on competencies at classroom and PD levels. The final agreement was that content knowledge at PD level (CK-PD) would cover all aspects of teachers’ knowledge. This is in line with Beswick and Chapman’s (2015) and Jaworski’s (2008) similar views but in our case was expanded to include knowledge domains. Looking at the established specifications CK, PK and PCK at the PD level, PK-PD and PCK-PD for facilitators need to be further specified (Wilhelm et al., 2019). Both consider the specific focus on teachers as learners, either from a general adult education perspective (PK-PD) or in a subject-related way (PCK-PD) (Prediger, 2019). PCK-PD encompasses all “skills to engage teachers in focused activities and conversations about these mathematical concepts and relationships and to help them gain a better understanding of how students are likely to approach related tasks” (Jacobs et al., 2017, p. 3). This also includes, for example, the possible learning hurdles when teaching mathematics (Rösken-Winter et al., 2015). Furthermore, the PD and qualification programs developed and implemented within the framework of the DZLM were also examined as examples. A subject-specific view was appropriate for working out specific requirements and then classifying them into a larger framework. As with Borko and colleagues (2014), a PD excerpt for problem-solving in mathematics was chosen to gain insight into the facilitators’ concrete tasks. As a result, both perspectives (“tasks” and “activities and competencies”) are set in relation to each other.

To cooperate efficiently with higher authorities such as ministries or learning communities, facilitators require competencies similar to the coaching knowledge required of teachers communicating with laypersons. As a result, the clear structure of the framework with four key competency areas (Competencies on the Classroom Level, Competencies on the PD Level, Professional Values and Beliefs, and Professional Self-Monitoring) needed to be changed to become five by adding Professional Social Competencies. Another reason was that “communication and cooperation” must be considered at all levels and is a competency relevant to all actors (teachers, facilitators, stakeholders).

An intermediate result of the evaluation of an online questionnaire with a response rate of 34% was that the domains of Communication and Cooperation as well as Coaching and Counseling are often only perceived as a level between facilitators and teachers. This is due to the representation of Competencies on the PD Level. Therefore, a small change was made here so that both aspects were placed in the new area of Professional Social Competencies. Thus, as in the 3TM, the relationship of these competency aspects to facilitators, to teachers, and among facilitators themselves should be better emphasized. Essentially, no significant discrepancies occurred here. A comparison with the literature on general adult education reveals that the competencies of facilitators of mathematics teachers can also be divided into the four essential areas of social and communicative competencies, personal competencies or self-competencies, values and beliefs, and field competencies. In terms of different levels, however, content knowledge and (content) pedagogical knowledge are subdivided into PD and classroom levels.

The aspects of Professional Values and Beliefs and Professional Self-Monitoring were essentially retained but required partial restructuring due to specific characteristics of facilitators of mathematics. Respect of Professional Values and Beliefs can be taken as an example: it may be useful to consider beliefs about teaching and learning mathematics (Grigutsch et al., 1998) in all PD courses. However, if a facilitator is mainly concerned with the subject of language sensitization, then their beliefs on language sensitization would be important or, in the case of the subject of digitization, beliefs on use of digital media would matter. In the same course, changes in the focus on self-efficacy beliefs or in the knowledge of frequent teacher problems occur when delivering different PD content, and it is therefore not possible to present competency facets in great detail. The areas of Professional Values and Beliefs and Professional Self-Monitoring have a strong interplay. Above all, the dual role and one's own understanding of one’s role as a facilitator has an increased influence. Thus, Role Identity was included alongside Professional Beliefs and Professional Ethics. The concept of motivational orientation was replaced by Self-efficacy Beliefs, as this aspect of PD is more relevant to mathematics (Bandura, 1999; Thurm & Barzel, 2020). At times, the question arose in the discussions as to whether one’s own experience should be included as a competency domain. For the stakeholders, it was important to include Professional Experience to explicitly promote appreciation of teaching practice. In addition to formal learning pathways, it is often the informal paths (practice experiences) that strengthen facilitators’ competencies (Zaslavsky & Leikin, 2004).

4 Outlook

Teacher PD plays an important role in the continuous development of mathematics teaching and learning. Medley (1987) articulates this in the context of teachers’ pre-existing characteristics (Type F) and their required competencies (Type E). In this chapter, we focused on facilitators as core actors and on professionalizing mathematics teachers (Type J). One can also note that while Medley ends with instructors as teachers, it is clear to see that the chain of effects extends beyond this point. To strengthen teacher competencies, we need to start one level higher (e.g., Lipowsky, 2014; Prediger et al., 2019). As Medley states, teacher competencies need to be strengthened, and by extension, research on the competency development of facilitators needs to be undertaken in the field of teacher education. This is necessary to achieve real improvement in the quality of PD programs and their implementation and thus for Type J to positively impact on Type E. The framework developed within this study is designed to be used for this purpose.

There are many competency frameworks for adult educators or facilitators generally, but they are not always usable in specific cases, either because they lack focus or because they are too non-specific. In such cases what is needed is adaptation; for our purposes, a framework that is tailored to the teaching profession and which focuses on the didactic perspective of mathematics. But what is new about this competency framework compared to existing frameworks as reported in the overview by Rossman and Bunning (1978), Wahlgren (2016) or the framework by Bernhardsson and Lattke (2011)? The challenge was to use these rather general frameworks as a starting point to develop a specific competency framework for facilitators in mathematics education, and even more specifically to the context of the DZLM, and to develop such a framework in cooperation with key stakeholders in school administration to ensure a systemic strategy.

First, the professional field of teaching must be considered as a specific feature here. The general competency frameworks that have emerged from adult education are too broad in their orientation for this and do not emphasize aspects relevant to the teaching profession or inadequately differentiate individual competency facets (see GRETA framework; Wahlgren, 2016). However, this differentiation is necessary to address the relevant domains.

A further specification that had to be considered here was the discipline of mathematics because even within the teaching profession, there are significant differences across subjects. In addition to the differences between scientific disciplines such as mathematics and physics, professional cultures in the field (teachers, engineers, etc.) also vary considerably. Therefore, specific areas have emerged, which also account for attitudes towards the subject.

Lattke and Zhu (2010) drew attention to another reason for developing a specific framework for mathematics facilitators: cultural context is key. Cross-cultural studies show that cultural norms significantly influence views of what constitutes “good” mathematics teaching (Dreher et al., 2021). Of course, a different focus can be applied in terms of cultural context (whether this is regional, national, or global). However, in a field where close cooperation with local authorities and schools is required, local challenges must be considered. Close dialogue with stakeholders in the local system should therefore largely determine which components are included in such a framework. Local, cultural anchoring is always present when such a framework is developed with stakeholders on the ground—and the resulting competency framework would in all probability look different if it were to be developed elsewhere.

The continuous changes taking place in the school system require teachers to act dynamically and to respond in a differentiated way to changing needs. Facilitators, through the relationship between the field of practice and the teacher education system, can build an important bridge here by considering social changes and responding to them within the educational system and, accordingly, by responding to these innovations in PD programs. To do so they need to be competent and perform their role properly. As we have seen, facilitators in mathematics education have professional status and must therefore have a wide range of competencies.

The developed competency framework provides a research-based systematic overview for research and development of what should be kept in mind, especially the affective and self-regulatory competencies, in addition to the central competencies PK-PD, PCK-PD, and CK-PD. For communication at educational administration level, the competency framework is useful to sensitize facilitators to the fact that performing their role effectively requires not only being well versed in CK, but a multitude of other facets as well. Alternatively, the competency framework can be helpful for quality assurance at the level of educational administration.

The need for the whole range of competencies in the frame of PD programs became obvious when (for example) we looked at integrating technology in mathematics classrooms (Barzel & Biehler, 2020; Thurm et al., accepted), possibly due to the fact that PD aims relating to technology are manifold. Teachers must familiarize themselves with the latest technology and consequently rethink their tasks, teaching routines, and practices. All these aspects also touch on underlying beliefs about mathematics and teaching mathematics with technology (Clark-Wilson & Hoyles, 2019; Thurm & Barzel, 2020).

Covering all these areas requires facilitators to have not only mathematical competencies at classroom level but also competencies at the PD level. For example, regarding attitudes and beliefs, awareness of the latest research on teachers’ attitudes and beliefs about teaching with technology is relevant to ensure facilitators are adequately prepared to address teachers’ diverse knowledge and beliefs on the matter in a PD program. Especially for teachers with a more traditional, instructor-centered teaching approach, teaching with technology often means a greater challenge and loss of control than for those teachers who are more used to managing more open-ended approaches (Simonsen & Dick, 1997; Zbiek & Hollebrands, 2008, p. 291). Therefore, research results highlighting the importance of fostering self-efficacy to be able to teach mathematics with technology are not at all surprising (Thurm, 2020). Specific activities such as classroom trials are suitable as the basis for reflection-in-action (Schön, 1983), a promising method at PD level for the implementation of new teaching routines and innovations (cf. Arsal, 2014; Hattie, 2009; Lipowsky & Rzejak, 2015; Thurm, 2020). This has been identified as an especially important element in PD concerning technology in mathematics (Thurm, 2020). Besides general design principles for effective PD (Barzel & Selter, 2015), all these aspects demand highly developed moderation skills (PK-PD) to deal with disruptive situations in PD, self-regulation to organize the different fields of requirements, and strong professional ethics to ensure they always consider teachers in their thinking. The fluidity of all these facets in the competency framework is essential for facilitators to achieve their aims when enabling teachers to integrate technology into their everyday classrooms.

The success of PD from the facilitator to the student outcomes could not be identified in various research studies and have not been the subject of investigation. What can be assumed, however, is that the levels of success of PD as shown in Fig. 1 can be extended upwards and can lead to a change in student outcomes if facilitators train teachers adequately.

The presented competency framework offers a starting point for making the competency level of facilitators measurable. In a further step, instruments need to be developed to measure competencies. The competency framework could be a good way to map and check which competencies are addressed in a qualification program. For example, there are already approaches to designing PD for mathematics education with technology to foster teacher and facilitator noticing (see Thurm et al., accepted), which could be researched in more detail regarding implementation, where the interplay of competencies could also be considered.

It has been shown that this field of research, “mathematics teacher training and experience” (Type J), is still relatively young area of mathematics didactics, and it can be assumed that research in this area will continue to sharpen. Already in the various mathematics conferences, as described above, there are more and more workshop groups discussing this topic. In different countries with differing structures, the need for quality standards for facilitators is still present and qualification programs must be specifically developed and implemented to ensure the nationwide success of PD programs. Our findings show that facilitators are a central factor in such success and that their own training must therefore not be neglected, as they play a crucial role in the PD process for mathematics teachers.