Research in mathematics education has increasingly considered equity issues, and how digital resources can contribute to equitable teaching. Referring to the framework guiding this book, we consider that this research investigates how a Type I variable (digital resources) can be used to counterbalance negative effects of another Type I variable (the socio-economic background) on the relations between processes (Types C and B) and product (Type A, learning outcomes). For example, in a study conducted in a primary school in 'unfavorable' (in socio-economic terms) contexts in Mexico, Sandoval and Trigueros (2021) observed that when primary school teachers create a classroom culture grounded on mutual respect, listening to each other, and combined this with the use of software supporting students' problem-solving activity, all the students can grasp the important mathematical ideas.

Finally, major changes in the mathematics teachers' external context in recent years have been due to the COVID-19 pandemic. From Kindergarten to University, teachers all over the world were forced to teach online of at least some of the time over several months. This dramatic context is also a new theme (or a large set of new themes) for research.

From the first lockdown, researchers in mathematics education launched questionnaires to investigate the consequences of this situation for teachers' practices, including naturally their use of technologies.

Drijvers (2020) and his colleagues, for example, conducted a study entitled "Math@Distance study" in Flanders, Germany and the Netherlands. They asked 1719 secondary school mathematics teachers about their teaching practices during lockdown. The use of digital resources was an important aspect in their study. They observed that the use of video conferencing software drastically increased. More surprisingly, the use of online exercises and online learning environments decreased. During the synchronous video lessons, the teacher presents, the students answer questions; but the collective work of students was scarce. Hodgen et al. (2020) reached similar conclusions, analyzing questionnaires and interviews with 49 heads of mathematics departments in secondary schools in England. Moreover, disadvantaged pupils were less engaged in the teaching due to problems of access, low parental support, and new personal and familial difficulties. Solomon (2021) stressed that equity is one of the most difficult challenges in the COVID-19 context; at the same time this context presented new opportunities for teachers to access student thinking using some of the technologies utilized during distant teaching.

Technological equipment and online teaching practices have changed since these "early" in the pandemic chronology studies. We assume that "Which digital resources can support teachers, and students, in secondary school mathematics for distant or hybrid teaching in a context of pandemic?" will remain an important research question for some years to come. This is because the pandemic unfolds over several years and the research will need several years of setbacks to understand these phenomena.

# **6 Conclusions**

The question leading this chapter was:

How has the evolution of research in mathematics education about digital resources impacted the context of mathematics teachers' professional activity?

Reviewing the relevant literature, we have observed a very large number of changes in the research studies. We have selected and presented particular directions in these that seemed to be the most pertinent. Our focus was not only on digital resources themselves (e.g., e-textbooks, mathematical software, digital platforms, online assessment systems, tools for distant collaboration, videos, and other kinds of digital media), but on various aspects of teachers' external context linked with digital resources: community support, or time economy for example. Moreover, we have shown that the research studies on these topics strongly associate Type I, Type E, Type D and Type C variables.

The changes we observed and insights we gained can be summarized as follows:

• *Evolution in the research about the digital resources*: Concurrent with development and use of new digital resources, new issues have emerged and have been developed. These included the quality of digital resources. Research has produced different kinds of tools for assessing this quality, and revealed the need to re-conceptualize quality, to consider new possibilities for connectivity, and new perspectives on the teachers as designers of their own curriculum (Type D and Type C variables, since the design takes place out-of-class and in-class). It has become evident that new technologies and digital resources necessitate and drive new pedagogical approaches. In other words, questions are not only concerned with how the teacher may be able to suitably integrate resources, but with the digital resources themselves (e.g., digital learning environments) require and force teachers to take a different stance and build their 'teaching' (or coaching) around the new digital environment.

Different causes were combined to produce these changes. Each time that a new digital resource is introduced in school, it is a new element in the external context for teachers and opens the way for research on the potential of this digital resource, on its actual use, on its impact on teaching and on teacher knowledge (Types C, D and E). The general evolution of research on mathematics education has also influenced research about DCRs as part of teachers' context (this can involve any of the variables). New research issues (e.g., assessment; teachers' and students' collective work) encompass studies about digital resources and mathematics teaching. The socio-political turn, and the value of research addressing equity issues is also an important trend in recent research present in the literature we reviewed.

We foresee further evolution in research in all the directions mentioned above that stress the need for more research on:

Moreover, in an external context requiring teachers to become the designers of their own curriculum, more research is needed on educative digital resources for teacher professional development (Type J variable). We contend that digital resources as elements of the external context for mathematics teachers' professional activity are often underestimated, and their affordances, constraints, and potential to drive under-researched. For us, this review was an eye-opener, and we believe that there are many avenues for mathematics education research in this field.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Competency Framework for the Qualification of Facilitators of Mathematics**

**Joyce Peters-Dasdemir , Lars Holzäpfel , Bärbel Barzel , and Timo Leuders**

# **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

J. Peters-Dasdemir (B) · B. Barzel

Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

e-mail<EMAIL_ADDRESS>
B. Barzel e-mail<EMAIL_ADDRESS>
L. Holzäpfel · T. Leuders Institute for Mathematical Education Freiburg, University of Education Freiburg, Kunzenweg 21, 79117 Freiburg Im Breisgau, Germany e-mail<EMAIL_ADDRESS>
T. Leuders e-mail<EMAIL_ADDRESS>
© The Author(s) 2023 A. Manizade et al. (eds.), *The Evolution of Research on Teaching Mathematics*, Mathematics Education in the Digital Era 22, https://doi.org/10.1007/978-3-031-31193-2\_11

311

Data collection and analysis were supported by the expert network within the German Centre for Mathematics Teacher Education (DZLM). We have no conflicts of interest to disclose.

and individual professionalization as a core presage variable in terms of presageprocess–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 metastudy (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.

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

**Fig. 1** Offer-and-use model for research on teachers' professional development. Lipowsky (2014, p. 515), English version: Lipowsky and Rzejak (2015, p. 30)

learning should be considered here. Collaboration should also be supported in inservice 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).

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 "threetetrahedron 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).

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):

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

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:

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:

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

**Fig. 4** GRETA competency framework (Strauch & Lencer, 2017). https://ec.europa.eu/epale/en/ blog/greta-competence-model-teachers-continuing-training

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 selfmonitoring, 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,

**Fig. 5** Knowledge in teacher education (Jaworski, 2008, p. 336)

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).

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 domainspecific 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 subjectspecific 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.

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

**Fig. 6** Competency framework of facilitators in mathematics education

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' preexisting 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 openended 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-inaction (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, selfregulation 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.

# **References**

Thema Rechenschwierigkeiten. *Journal Für Mathematik-Didaktik, 40*(2), 227–253. https://doi. org/10.1007/s13138-019-00143-1

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# **Continuing Evolution of Research on Teaching and Learning: Exploring Emerging Methods for Unpacking Research on Teachers, Teaching, and Learning**

**Chandra Hawley Orrill , Zarina Gearty , and Kun Wang**

# **1 Introduction**

In 1987, Medley offered an explanation of the "present state of the art of research in teaching" (p. 105). By doing so, he outlined the categories of variables that could be studied and provided strong guidance for high-quality research on these variables. His guidance suggested that research should seek to find out why teaching quality varies widely and to do that, one must have a conceptualization of what good teaching is, an instrument that is valid for distinguishing good teaching from poor teaching, and a plan for collecting accurate data and for analyzing that data. While aspects of research on the relationship between all of the variables that shape the complex act of teaching students in a formal learning environment have remained unchanged since 1987, much has changed in educational research including the emergence of new methodologies and the increasing presence of technology both as a tool for teaching and learning and as a tool for research, making research on the connections between and among the variables both richer and more flexible than Medley originally suggested.

Z. Gearty

© The Author(s) 2023 A. Manizade et al. (eds.), *The Evolution of Research on Teaching Mathematics*, Mathematics Education in the Digital Era 22, https://doi.org/10.1007/978-3-031-31193-2\_12

C. H. Orrill (B)

Kaput Center for Research and Innovation in STEM Education, University of Massachusetts Dartmouth, 506 Langley St, Fall River, MA 02720, USA e-mail<EMAIL_ADDRESS>
Kaput Center for Research and Innovation in STEM Education, University of Massachusetts Dartmouth, 3 Rumford Road, Norton, MA 02766, USA e-mail<EMAIL_ADDRESS>
K. Wang

Kaput Center for Research and Innovation in STEM Education, University of Massachusetts Dartmouth, 120 Johnson Road, Winchester, MA 01890, USA e-mail<EMAIL_ADDRESS>
In this chapter, our goal is to provide some clear examples of the ways in which the field of mathematics education has been able to pursue the interactions put forth by Medley (1987) thanks to the emergence of new theories, research methods, and technologies. We discuss some of the critical ways in which the research landscape has changed since the original article was published. First, we look at methodologies that have become more widespread since Medley introduced his framework. Second, we explore some examples of research methods that offer new ways of thinking about research questions related to teaching and learning in formal environments. These methods are important because they have opened opportunities to look across the variables in ways that were unavailable to researchers until recently. We finish by considering the opportunities that have been created by technology. These have changed the ways we collect data and the data we collect. We close by briefly discussing how the changes discussed in this chapter specifically relate to presageprocess–product research (Medley, 1987). Our goal in this chapter is not to present a comprehensive review of the literature, rather, we seek to highlight both where the field is now in terms of research methods and tools as well as to provide examples of the ways in which Medley's framework is being pursued in newer research. We have chosen to rely on Fig. 3 in Chap. 1 (Manizade et al., 2023) rather than the original Medley model for the purposes of our discussion, except where noted.

### **2 Changing Methodologies**

Since Medley's framework was originally proposed, the field has seen the emergence of new research methodologies (i.e., scientific frameworks), methods (i.e., specific approaches), and tools (i.e., instruments) that allow innovative lenses with which to make sense of the multi-faceted enterprise that is teaching and learning. In this chapter, we offer brief overviews of just a few of the methods currently available for answering questions related to the presage-process–product model. These tools sometimes allow us ways to look at connections between more than two variable categories (e.g., cultural-historical activity theory) or allow us to conceive of research as a web of interconnected studies all serving to develop a larger theory (e.g., Design-Based Research). Below, we talk about the emergence of Qualitative and Mixed Methods methodologies as well as emerging psychometric models, then we introduce several methods that provide new ways of thinking about the interconnected nature of the variables for teaching and learning. These include teaching experiments, Design-Based Research, cultural-historical activity theory (CHAT), and quantitative ethnography.

# **3 Qualitative and Mixed Methods Research Methodologies**

In 1987, qualitative research was not often used in education. While there were certainly some examples of qualitative research emerging (e.g., Erlwanger, 1973), those studies were not as widely accepted as quantitative studies. However, with the shifts in ontology and increased acceptance of new methods, the field of mathematics education research became more open to—and, indeed, primarily focused on—qualitative research. The reason is simple: quantitative research methodologies are particularly appropriate for a model of teaching and learning that relies on transfer of knowledge from the expert to the novice. As we moved toward theories of learning that were more grounded in constructivism, socio-cultural theories, and critical theories, new questions were being asked. As an array of new learning theories emerged, the definitions of teaching and learning became more diverse and even questions could be pursued. Rather than asking if teachers who took teaching methods courses in mathematics or science are more effective than those who did not, researchers began wondering in what ways particular backgrounds might shape learning experiences (Type E—teacher's competencies, knowledge and skills—and Type F—preexisting teacher characteristics) and their interaction on Type A (student learning outcome) variables (e.g., Manizade et al., 2023; Medley, 1987)) and how teachers conceive of making content learnable for children (Type D variables interacting with Type B and A variables (e.g., Manizade et al., 2023; Medley, 1987)). Further, with the emergence of new theories of learning, the definitions of what constitutes learning, and, thus, how learning is measured, also changed. Consider, for example, Wenger's theory of situated cognition (Wenger, 1998). Within this theory, "learning" is defined as a change in participation, because learning is viewed as becoming a member of a community of practice. Thus, as one learns, one becomes a fully-integrated member of the community of practice. If participation is the goal, a written assessment of content knowledge is no longer an appropriate instrument for measuring learning and new approaches need to be developed. Thus, it is consistent with the rise of the cognitive, socio-cultural, and critical theories that qualitative research would become a critical tool for research.

Once qualitative methodologies were established as a norm within the field, it was natural for some researchers to use qualitative and quantitative methods together to better understand a phenomenon. Thus, mixed methods approaches have been used by some researchers to understand the interactions inherent in the learning environment. Grounded in pragmatism (Johnson et al., 2007), mixed methods research is a methodology that combines qualitative and quantitative methods to produce "defensible and usable research findings" (p. 129). For example, a researcher may conduct a survey (quantitative), then conduct interviews with a subset of participants (qualitative) to more deeply understand the findings of the survey (Creswell, 2014). For example, in one recent study (Starrett et al., 2021), researchers used surveys and interviews with teachers and students to understand how teacher's proximity to their school impacted their use of place-based education, thus using mixed methods to connect Type I (external context) variables to Type D (pre- and postactive mathematics teacher activities), Type C (interactive teacher activities), and Type B (student learning activities). Another approach to mixed methods analysis is to use the various analyses to dig into specific aspects of the data. For example, an approach the first author of this chapter has used (e.g., Izsák et al., 2010; Orrill & Cohen, 2016), included mixture Rasch analysis of an assessment of teacher's math knowledge. Our goals was to identify specific mathematics tasks that teachers found difficult or with which different groups of teachers had different experience. From those data, we were able to identify specific items on which to focus in the qualitative analysis of the interviews. Using this approach, we were able to not only see how teachers performed on the assessment, but also to generate assertions about why they performed in these ways, thus providing us with additional information for designing effective instruction. These methods allowed us to more thoroughly understanding Type E variables.

For understanding the ways in which teachers, their experiences, and their actions intersect with student learning, access to a wide array of methodologies is crucial. While quantitative research is still used, studies using qualitative and mixed methods approaches are as accepted in most venues as quantitative research. The critical factor in high-quality research is not the methodologies and methods used, but rather the alignment of the methods and methodologies to the research questions.

# **4 Continuing to Develop Quantitatively: Emerging Psychometric Models**

Methods for conducting quantitative research have also continued to develop since 1987. While quantitative research has remained theoretically grounded in positivism and still adheres to the methodological frameworks that were in place in the 1980s, quantitative research has benefitted tremendously from increased access to computers and the emergence of new models that has been possible because of computers. Now that nearly everyone has access to extremely powerful computing technology, quantitative analysis can be more robust and more accessible than ever. Particularly important for presage-process–product research are the myriad statistical and psychometric models that have emerged in the past few decades. In this section, we briefly introduce four such models that have played a role in our own research on teacher knowledge and student learning: Item Response Theory (IRT), mixture Rasch Models, Diagnostic Classification Models, and Topic Models. We have selected these four models because each provides researchers with different information about learning. Further, we included IRT because of its widespread use. Each of the four offers a way to better connect the variables highlighted by Medley. We do not, however, intend this as an exhaustive list.

Item Response Theory (Baker, 2001; Baker & Kim, 2004) is probably the most influential psychometric model in widespread use as it has largely replaced classic test theory for scoring standardized tests. Rather than simply assigning a score indicating how many items are correct or incorrect, IRT provides various kinds of information. First, IRT provides a score that expresses a participant's performance in terms of the number of standard deviations above or below the mean of participant scores. The second piece of data provided by IRT is information regarding the difficulty of an item, where difficulty is reported as the probability that someone scoring at the identified difficulty level would have a 50% chance of answering the item correctly. IRT provides researchers with information that allows them to consider not just whether "learning" has occurred, but where there may be deficits in aspects of content knowledge as well as a relative weighting of participant's performances. Because of the information that it can return, IRT is currently a critical component in the development of learning trajectories (e.g., Clements et al., 2011; Confrey et al., 2017), which sit at the intersection of student learning (Type A), teacher planning (Type D), and teacher knowledge of students (Types G & H).

Building from IRT, mixture Rasch models (Izsák & Templin, 2016; Rost, 1990) look for latent trends in the patterns of responses among participants to determine whether all the participants should be placed along the same continuum or whether there are groups within the data that performed in ways different from others. This approach has been used to identify patterns of reasoning among teachers. These patterns highlight that performance on an assessment can be tied to patterns in teacher's reasoning about the content (e.g., Izsák et al., 2010; Orrill & Cohen, 2016). The data can also be used to capture a different kind of "learning". Rather than focusing only on whether participants have improved their scores on an assessment, researchers can also determine whether participants have changed latent classes. Such change would indicate a fundamental shift in the ways the participants are reasoning about the mathematics items on the assessment (Izsák et al., 2010). While mixture Rasch has primarily been used for in-depth consideration of Type E variables, we assert that it could be used as a lens for understanding the relationship between Type E and Types C and D (how teachers plan and implement instruction). It could also be readily used to look at connections between Type A and Type C and D variables if an assessment were given to students and correlated to observations of classroom practice.

Another emerging family of models is Diagnostic Classification Models (DCMs; Bradshaw et al., 2014; Izsák & Templin, 2016; Rupp et al., 2010). DCMs require an a priori defining of the specific attributes each item of an assessment measures (e.g., Tatsuoka et al., 2016). From that mapping, analysis is done on participant's performance, and results are reported as the probability that the participant has mastered each individual attribute. For example, in Bradshaw et al. (2014), the authors identified four attributes being measured by the assessment of fractions: referent unit understanding, partitioning and iterating, appropriateness, and multiplicative comparison. The attribute inventories that are returned in place of traditional test scores can provide insights into specific aspects of understanding demonstrated by a given sample, thus providing data that can shape the instructional experiences for participants. As with mixture Rasch models, DCMs provide an opportunity to connect teacher or student understanding of mathematics (Type A or E) to the activity in the classroom (Type B or C).

One final emergent psychometric model is Topic Modeling (e.g., Blei, 2012). Topic Models allow a statistical analysis of qualitative data to show a change in patterns of language usage. Topic models rely on looking for particular words in natural text or natural speech to see their patterns of co-occurrence. From those patterns, groupings are created that separate participants. For example, in Kim et al. (2017), the researchers found three main themes in their analysis of student's responses on a science assessment. The first theme featured answers that included appropriate technical terms for middle grades science students (e.g., change, variable, dependent). The second theme was discipline-specific terms (e.g., energy, population, kinetic). The third theme focused on everyday language (e.g., put, stronger, big, think). Across four assessments, the participating middle school students shifted from using the everyday language topic to the other two topics (Linking Type B to Type C variables). These results suggested that students were learning about the discipline. Topic Modeling is particularly important for measuring learning through a socio-cultural lens.

New psychometric models do not fundamentally change the design and limitations of quantitative research. Thus, they have some limitations outlined by Medley in terms of looking at relationships between the variables. However, the new models allow measurement of different kinds of learning (e.g., change in participation or in natural language use, rather than acquisition of knowledge), and they open opportunities for mixed methods approaches such as those described in the discussion of mixed methods above. So, even within the realm of quantitative research, there are more tools available to support asking questions in new ways and looking at relationships through different theoretical lenses than was possible in 1987.

### **5 Emerging Research Approaches**

In this section, we introduce four of the approaches to research that have changed the ways in which we can answer questions about teaching and learning in formal (and informal) contexts. As with the discussion above, we do not assert that these are the only approaches appropriate for research in the presage-process–product framework. Rather, these are tools that have been used, or are emerging in use, in mathematics education and the learning sciences to answer research questions related to the variables in Medley's framework and the connections between them. Certainly, there are many other approaches that could also be used for this purpose. Below, we discuss: Teacher Experiments, Design-Based Research, Cultural-Historical Activity Theory (CHAT), and Quantitative Ethnography. Teaching experiments are featured because of their prominence in mathematics education research over the past three decades, while the other approaches were selected because they offer robust and diverse pathways for making sense of the complexity of learning environments through their analytical lenses or through iterative implementation. For each, we describe what it is, some benefits and limitations for the approach, and some examples of studies done with the approach.

### **6 Teaching Experiments**

## *6.1 What is It?*

Teaching experiments stem from Piaget's clinical interviews (Steffe & Thompson, 2000) and have roots in Russian education research (e.g., Davydov, 1975). Teaching experiments are fundamentally constructivist and have been used with a wide variety of constructivist perspectives, ranging from radical constructivism to social constructivism (Cobb, 2000). In a teaching experiment, the researcher serves in the role of a teacher and conducts a series of teaching episodes, usually working with a small group of students or one individual (Cobb & Steffe, 1983). The key goal is to develop a "living model of student's mathematics" (Steffe & Thompson, 2000, p. 284), testing and revising instructional activities designed to support student learning (McClain, 2002).

Teaching experiments go beyond the scope of a clinical interview by aiming to help the researcher understand the change and progress of a student's mind rather than just the current state of the mind. The teacher-researcher constructs a conjecture about student's mathematical knowledge, then tests the conjecture with teaching episodes designed to move the student's understanding forward, reformulating the conjecture after each episode (Cobb & Steffe, 1983; Steffe & Thompson, 2000). Initial hypotheses can be abandoned based on student's responses as it is vital that the teacher-researcher allows the student's contributions to guide the trajectory of the teaching episode.

Typically, a teaching experiment consists of the teacher-researcher, the student(s), and an observer. The role of the observer is to witness and document student's reaction and behavior. The teacher-researcher, constantly interacting with the students and instantaneously reacting to the students, may not be able to capture all relevant observations (Cobb, 2000). The teacher-researcher engages the students in instructional tasks or activities to observe and promote mathematical learning and reasoning by posing tasks and asking follow-up questions (Steffe & Thompson, 2000). The data that is collected is qualitative and meant to record the models of student's mathematical understanding. The teacher-researcher uses this data to revise conjectures as teaching episodes progress and to revise the activities in the episodes. Ultimately, a model of student thinking about the specific content or topic is generated by the researcher, with a variety of qualitative data to support the model (Cobb & Steffe, 1983). From the perspective of the framework of research on teaching mathematics adapted from Medley (Manizade et al., 2023), teacher experiments allow conjectures to be made about how student mathematics learning outcomes (Type A) are shaped by the interactive mathematics teacher activities (Type C) that are pre-active activities (Type D) from considering student's engagement in previous mathematics learning activities (Type B). In a sense, while the outcome of the teaching experiment is a theory about student's learning, the experiment itself is an iteration of the relationships of Type A, B, C, and D variables conducted by a person with competency, knowledge, and skill (Type E) in the mathematics, in student learning, and in designing instructional interventions.

### *6.2 Benefits and Limitations*

One clear benefit of teaching experiments is the insight they provide about how to support a student or small group of students to move forward in their understanding of specific concepts. A unique feature of teaching experiments is that the researcher is directly involved with the teaching. Therefore, the researcher should have teaching experience and the ability to interact and engage with students (Steffe & Thompson, 2000). The goal is to elicit and support thinking during these interactions; teacherresearchers should be cognizant of how their actions and language are perceived by students (Tallman & Weber, 2015).

Teaching experiments are powerful tools for understanding learning, however, they are also very challenging. One clear challenge of this method, particularly for inexperienced researchers, is that data collection and data analysis are simultaneous during the series of episodes (Tallman & Weber, 2015). Additionally, to demonstrate the evolution of the conjectures and model building, the researcher-teacher needs to maintain on-going documentation of the reasoning for decisions and the interpretation of student's thinking (McClain, 2002). Self-reflexivity becomes a key assumption, where the researcher-teacher acknowledges that he/she is an active participant of the student's constructions (Steffe & Thompson, 2000; Tallman & Weber, 2015). A common data analysis method after the series of teaching experiments is retrospective analysis, changing and revising the hypothesized model (Cobb & Steffe, 1983).

### *6.3 Examples*

Many researchers used teaching experiments as exploratory tools, usually as part of larger projects. Simon and colleagues (2018) conducted a teaching experiment with a single student as part of the Measurement Approach to Rational Number (MARN) Project. Their goal was to understand how instruction could promote student's construction of the concept of multiplication with whole number and fractions, and to develop a hypothetical learning trajectory based on their analysis. The participant was a fifth-grade student, Kylie, that the research group had been working with for two years, conducting various clinical interviews and teaching experiments. One of the teaching experiments involved the use of a computer application called Java Bars as the instructional tool. Simon served as the teacher-researcher and posed several multiplication tasks designed to explore Kylie's changing conceptions about the meaning of the multiplier. The research group initially used the concept of generalizing assimilation, as defined by Piaget (1952) as the theoretical base for their hypothetical learning trajectory. They hypothesized that the progression of their instructional tasks would stimulate changes in the assimilatory structure of the student. However, during the initial teaching episodes, the researchers were not seeing any evidence of conceptual change and modified the teaching tasks. Using retrospective analysis of the data collected, Simon and colleagues revised their hypothetical learning trajectory conjecture to rather be stimulated by reflective abstraction, also a construct defined by Piaget (1952). When reporting their findings, the authors included a detailed description of the progression and rationale of the learning trajectory. The research group went on to conduct more teaching experiments with the revised instructional tasks and hypothetical learning trajectory (Simon et al., 2018). This research is focused on the interaction of student mathematics learning activities and student mathematical outcomes (Type B and Type A)—that is, how does an instructional intervention affect learning. Because it was a teaching experiment, though, it extended to Type C, interactive mathematics teacher activities, because one of the outcomes of this work was a hypothetical learning trajectory which could be used to guide other teachers for supporting student learning. Finally, consistent with the Framework of research on teaching mathematics as shown in Fig. 3 of Manizade and colleagues (Manizade et al., 2023), Type G research (individual student characteristics, abilities, and personal qualities) is also tacitly happening as the teacher-researcher is consistently assessing the student's abilities and understandings to make the instructional decisions that lead to particular learning activities.

Teaching experiments can also be done with larger groups of students. A study conducted with 299 undergraduate calculus students by Wagner and colleagues (2017) consisted of eight teaching episodes which were designed to study the change in student's ability to generate examples for the purpose of understanding novel concepts. The researchers formulated a hypothesized learning sequence and developed an instructional sequence of tasks and questions. The learning progression was broken down into intended student's awareness and behavior on specific skills and views. Over the course of eight teaching episodes, the teacher-researcher introduced the tasks, which progressed from more rigid to more open-ended to allow students to show their views and ability to generate examples. Data analysis was done using emergent codes from the participant's words and phrases from the reflections and written tasks. The evidence showed a progression of positive changes in the student's views of generating examples. The researchers revised their proposed learning sequence for the third iteration of the teaching experiment, where new students were chosen who had not yet learned calculus material, to test their revisions.

As with the Simon et al. (2018) example, this research approach follows a Type C-B-A flow, moving from considering interactive mathematics teacher activities to student mathematics learning activities to student mathematics learning outcomes. The researcher plays the role of a teacher, so data can be gathered on Type C, interactive mathematics teacher activities. Second, the research group designs a set of activities based on data from Type G (individual students characteristics, abilities, and personal qualities) and Type A (student learning outcomes) to simulate student learning experiences that could occur in the classroom, allowing the study of Type B (student mathematics learning activities) variables. Finally, the researchers analyze data and make assertions on how the activities impact student learning. Both groups used iterative design to develop a learning theory, ultimately laying out a trajectory of activities for teacher to implement and student to experience (Type C and Type B). Simon et al. (2018) implicitly also study the individual student Kylie, providing details about her abilities and personal qualities (Type G). In summary, teaching experiments are an extension of clinical interviews and align to constructivist learning theories. The researcher becomes a teacher in this methodology where he/she formulates, tests, and revises a hypothesis of a model for a change in student thinking as a response to some instructional sequence (Cobb, 2000; Steffe & Thompson, 2000).

# **7 Design-Based Research**

### *7.1 What is It?*

Design-Based Research (DBR) approaches originated from a desire to pursue research questions that cannot be answered in a laboratory setting (e.g., Brown, 1992; Collins, 1992). Underlying the development of DBR was a desire to develop an approach that overcame the issues in attempting to apply results from education laboratory studies into actual classrooms (Cobb et al., 2003; McKenney & Reeves, 2013). Over time, Brown's and Collin's notions of "design experiments" matured into an approach known by many names, that we refer to as Design-Based Research (DBR; Design-Based Research Collective (DBRC), 2003). DBR focuses on the development and refinement of theory along with the development and refinement of innovations that embody that theory (e.g., Barab & Squire, 2004; Cobb et al., 2003; DBRC, 2003; McKenney & Reeves, 2013). DBR is an approach to research that relies on a trajectory of inter-connected studies conducted, often over several years, rather than a single study (Cobb et al., 2003). It is inherently grounded in partnerships between researchers and practitioners.

A unique feature of DBR is the dual goal of generating a theory and developing and refining a particular intervention that embodies that theory (McKenney & Reeves, 2013; Sandoval, 2014). Researchers focus on both problematizing the context and on using theories to generate usable knowledge (DBRC, 2003). The development of such theories is a key component of DBR (Cobb et al., 2003). Through the iterative processes, conjectures are made, tested in the natural setting, revised based on the outcome, and tested again. The theory becomes emergent through this process and is refined at the end of the project (Barab & Squire, 2004). Because of this focus on developing theory and innovation together, DBR projects tend to include serious consideration of student learning outcomes (Type A), student mathematics learning activities (Type B), interactive mathematics teacher activities (Type C) and pre or post-active mathematics teacher activities (Type D). Many also include consideration of teacher competencies, knowledge, and skills (Type E); external context variables (Type I); and teacher development and experiences (Type J).

DBR projects usually span several years. The reason for this is twofold. First, DBR focuses on iterative in design (Cobb et al., 2003; DBRC, 2003), which involves multiple implementations, data collection and analysis cycles. Second, to understand an innovation and the theory it embodies in a way that moves toward generalizable knowledge, different grain sizes must be considered. Studies may focus on a single tool with a few students, then that tool used in a classroom, then that tool used in the context of the delivery of a piece of curriculum, etc. Because the studies focus on a series of related questions of different grain sizes, they often benefit from mixed methods approaches across the lifespan of the research (DBRC, 2003). Rather than having confined control variables, multiple dependent variables such as classroom environment and learning outcomes are examined to generate a deep understanding of the issues and the effect of the intervention (Cobb et al., 2003).

In DBR, researchers partner with various stakeholders to achieve the goals of refining theory and refining the intervention. Interventions can be educational products, policies, or programs (McKenney & Reeves, 2013). Examples of stakeholders are teachers, school leaders, coaches, and subject matter experts. These participants become an integral part of the development and implementation of the design, sharing their expertise to collaboratively work through the project (DBRC, 2003). Much of the work is conducted in a natural authentic setting, such as schools and classrooms; the context is problematized and studied as a vital part of understanding the learning and teaching that occurs (Barab & Squire, 2004).

The overall structure of DBR is flexible and iterative, but also systematic. It is a sequence of approaches rather than just one approach (Barab & Squire, 2004). Several models of approaches have been offered by researchers (e.g., Eljersbo et al., 2008; McKenney & Reeves, 2012; Reeves, 2011). Most of these models include the initial phase of exploring and analyzing a problem, followed by the construction of a design and then reflection and evaluation. Since the entire process is iterative and usually non-linear, most researchers using DBR work back and forth through those phases. Theories are developed and tested throughout the process until enough evidence and data is gathered for a mature theory and usable knowledge. Usable knowledge can be declarative knowledge, such as describing a certain phenomenon or prescriptive knowledge, such as ways to facilitate learning with a certain intervention (McKenney & Reeves, 2019). In the initial phase, researchers study a setting and develop testable conjectures about how to address an educational problem or how to influence a change in students learning (Cobb et al., 2003). Data analysis becomes an ongoing process as both researchers and practitioners aim to deepen their understanding of phenomenon that occur in the natural setting (Barab & Squire, 2004). This collective partnership and iterative design process can be seen as unique features of DBR whose purpose is to close the gap between educational research and classroom practice and to further theoretical knowledge that can influence change in settings facing similar problems (DBRC, 2003; McKenney & Reeves, 2019). The DBR approach has been applied in various sectors of education such as learning sciences, curriculum development, instructional design and teacher professional development (See special issues of journals such as *Educational Researcher (*2003, 31(1)), *Journal of the Learning Science* (2004, 13(1)) and *Educational Psychologist* (2004, 39(4)).

## *7.2 Benefits and Limitations*

One of the key benefits of DBR is its ability to inform the development of a usable intervention while also yielding a generalizable theory. This two-faceted benefit ensures that both the immediate outcome of the project (the intervention) has educational merit while also ensuring that there is something beyond a single application of the theory that can support teaching and learning. This ensures that the theory can continue to inform practice beyond the lifespan of the intervention.

Due to its multi-faceted design, DBR can be a challenging approach even for experienced researchers. The role of the researcher is less defined and more fluid; she can be the designer and the implementor, which can introduce threats to validity and objectivity (Barab & Squire, 2004). Furthermore, the researcher needs to anticipate and communicate means of support for the various groups of people involved in the project, who often can have different opinions and perspectives on educational issues (Cobb et al., 2003). Time and personal commitment are devoted to maintaining close and respectful relationship with partnerships (Cobb et al., 2003). Another source of difficulty arises from the various sources of data and the extended period of collection time (DBRC, 2003). Various techniques for data collection and analysis are often required along with an appropriate balance between rich data and a surplus of data to ensure validity (McKenney & Reeves, 2013) and often retrospective analysis is needed for theory development (Cobb et al., 2003). Despite these limitations and challenges, researchers have found DBR to be useful for a wide range of studies. A few such studies are highlighted in the next section.

### *7.3 Examples*

Barab and colleagues (e.g., Barab et al., 2010) combined DBR and socially responsible design to create an intervention that would help students develop their identity both as individuals and members of their community along being educated to be knowledgeable citizen of the world. The project spanned over five years and included several iterative components, ultimately designing a video game that became known as Quest Atlantis (QA), with teachers, students, community members and web designers as part of the research partnership. Key to the DBR methodology, the research group developed a theory about transformational play: that video games can serve as effective mediums for deep and sustained learning by providing engagement not possible in the classroom (Barab et al., 2010). This theory has also been used by other researchers (e.g., de Sousa et al., 2018) as a framework to their work. One project that grew from the theory developed by Barab and colleagues is the extension of the *Adventures of Jasper Woodbury* mathematics curriculum work that has been undertaken by Gresalfi and her colleagues (e.g., Gresalfi & Barnes, 2012). Like the development of Quest Atlantis and the development of the original *Adventure of Jasper Woodbury* series (e.g., CTGV 1992; 1994), Gresalfi has undertaken the new work related to the *Adventures of Jasper Woodbury* by engaging in a DBR approach that looks at the relationship of student mathematics learning activities (Type B) and student mathematics learning outcomes (Type A), but also incorporates variables of Types C (interactive mathematics teacher activities) and G (individual student characteristics, abilities, and personal qualities). Her group designed the Boone's Meador mission as an activity that provides insight into Type B variables (student mathematics learning activities). In Boone's Meadow, students are tasked with making calculations and decisions regarding how to reach a destination in order to save an endangered eagle. The game measures student learning outcomes based on student's responses and decisions made throughout the activity, shedding insight into Type A variables (student mathematics learning outcomes). The game includes feedback that is meant to reflect the actions of a teacher, thus including Type C variables (interactive mathematics teacher activities), as well as how that feedback affects the activity of the game (Type B—student learning activities) and student learning (Type A—student learning outcomes). Gresalfi and Barnes (2012) did two iterations of DBR to design and explore the effect of consequential feedback, which is feedback that is embedded in context and requires students to evaluate their mathematical reasoning based on the outcome of their decisions. The two rounds of implementations spanned across two years; data sources included videotapes of discussions and student work. Several rounds of data analysis were done using both a priori and emergent codes. The team

feedback was embedded in context and given prior to the end of the game. An extension of DBR that arose in the early 2010s is Design-Based Implementation research (DBIR), in which implementation becomes the vital focus of theory development and analysis (Penuel et al., 2011). DBIR often includes the combination of learning sciences research and policy research. One such example of DBIR is the work of Cobb and colleagues (e.g., Cobb et al., 2013), who partnered with four urban schools to improve the quality of mathematics instruction with an 8-year project titled Middle School Mathematics and the Instructional Setting of Teaching (MIST). The focus of improving mathematical instruction was broken down into increasing the learning of conceptual understanding, justifying solutions, and explicit connection between multiple representations. The researchers believed that a reorganization of teacher's instructional practice was necessary for these improvements to occur.

saw an increase in the use of mathematical justification and critical engagement when

The research partnership consisted of school and district leaders, math coaches, teachers, and researchers. The iterative design process consisted of yearly cycles of data collection, analysis and feedback: they documented district's improvement strategies, collected and analyzed data to assess the implementation of the strategies, and recommended revisions of strategies for the following year. Additionally, a secondary level of focus was on gathering data to test and revise conjectures about supports and accountability measures that the research group had generated from literature. Examples of data collection methods include audio-recorded interviews, district organizational schedules, evaluation forms, online surveys, classroom observations, and student achievement data. Cobb's team (Cobb et al., 2003) analyzed their recommendations for each district, looking patterns and similarities to find potential generality. After the third year of data collection, retrospective analysis was conducted to provide evidence for conjectures about major components of their emerging theory of action for instructional improvement in mathematics.

As part of the theory development, the researchers designed, tested and modified conjectures about instructional improvement, more specifically on methods of both supporting and holding teachers accountable for reorganization of practices. They developed an interpretative framework that captured four general supports that the districts used in the improvement strategies: new positions, learning events, organizational routines and tools. The research on the ways that mathematical instruction improved map to Medley's Type E variables, focusing on teacher competency, knowledge, and skills. The researchers also attended to the ways in which a change in teacher's instructional practice, including variable D, pre and post-active mathematics teacher activities, and variable C, interactive mathematics teacher activities, was tied to mathematics teacher's competence, knowledge, and skills (Type E). Further, the MIST research team was able to provide recommendations for the district on how to support teachers. The four recommendation areas focused on variables of Type I (external context variables) and J (mathematics teacher development and experience). Therefore, MIST was able to study the relationship of variables of Type E (teacher's competence, knowledge and skills), D (pre and post-active mathematics teacher activities), and C (interactive mathematics teacher activities) by surveying and observing teachers. Recommendations are also focused on variables Type I (external context variables) and J (mathematics teacher training and experience) as the research team partnered with the school leaders to influence those categories outside of teacher control. MIST has continued to work with schools as partnerships in implementing strategies to improve math instruction and teacher practices (see Cobb et al., 2018 for more detail).

# **8 Cultural Historical Activity Theory**

### *8.1 What is It?*

Cultural-Historical Activity Theory (CHAT) is a theoretical framework for conducting sociocultural research. CHAT supports the analysis of human interaction while considering how an individual or group of individuals and their interactions with the environment affect their activities (Kuutti, 1996; Cole & Engeström, 1993; Engeström, 1993). The basic idea of CHAT is that humans should not be separated from their participation in various activities; therefore, rather than focusing on the individual as the unit of analysis, CHAT instead focuses on the activity in which people participate (Cole & Engeström, 1993; Engeström, 2004). Thus, the unit of analysis includes both the setting and individuals. The activity system refers to a collective concept—it is object-oriented, designed to think about the phenomenon in terms of the inner relations of activity and collaborative relationship between people (Roth, 2012). For the purposes of this chapter, we are focusing solely on the psychological aspects of CHAT and not on economic or materialist aspects. We assert that is most productive for this chapter's focus.

CHAT was initiated and developed by Russian theorists who saw behaviorism and analytical psychology lacking in its ability to describe cultural realities. The pedigree of CHAT can be traced back to dialectical materialism, and then to Lev Vygotsky who founded the first-generation of activity theory in the 1920s, centering it around his core idea: cultural mediation that is graphically expressed as a triangle with subject, object, and mediating artifact/tool comprising the vertices (Cole, 1998). The basic elements could be described as:

Beyond the prevailing behaviorist theories about the stimulus–response association at that time, Vygotsky's mediation triangle, as a semiotic process between subject, mediating artifact, and the object of an activity, was a revolutionary way individual make meaning of the world (Cole, 1996; Cole & Engeström, 1993; Yamagata-Lynch, 2007).

The first-generation theory was critiqued, because the unit of analysis still focused on individuals. To overcome it, Alexei Leont'ev (1981), Vygotsky's colleague and disciple, along with his colleagues, created a second generation of CHAT, which took into account inter-relationships between the individual and the community, history, context, and interaction of the situation and activity. According to Leont'ev (1974): "activity is…a system possessing structure, inner transformations, conversations, and development" (p. 10). Thus, the consequences of events and activities that occur during the activity can qualitatively change the participants, the participant's participation purpose and motivation, the social environment of the activity, and the activity itself (Rogoff, 2008; Rozin, 2004; Yamagata-Lynch, 2010).

According to Engeström (2004), Leont'ev never graphically expanded Vygotsky's original model to illustrate a collective activity system. In addition, Leont'ev and his colleagues did not adequately address the methodological challenges for capturing, analyzing, and presenting activity-based data. To address these shortcomings,

Engeström created the third generation of CHAT, offering a foundation for understanding and designing learning as a transformation of human activities and organizations. Engeström and his colleagues developed CHAT as an analytical framework by introducing a descriptive model of activity, which can be used in analyses of complex qualitative data. Compared to other sociocultural learning theories, Engeström's theory of expansive learning puts the primacy "on communities as learners, on transformation and creation of culture, on horizontal movement and hybridization, and on the formation of theoretical concepts" (Engeström, 2010, p.2). Cole and Engeström (1993) further detailed the representation of modeling human activity as a system form in the diagram of expanded mediational triangle, shown as Fig. 1. This is the triangle that typifies CHAT research.

The triangle provides an organizer to support researchers in mapping complex human interactions that take place in collective settings. The uppermost sub-triangle is identical to Vygotsky's basic structure of mediated action. In addition to the basic components of Subject, Tools and Object presented in the basic first-generation triangle, the expanded mediational model also includes the following three elements:

The Rules, Community, and Division of Labor in the bottom portion of the triangle model add the sociohistorical collective nature of mediation that was not addressed by Vygotsky (Engeström, 1999a, 1999b). The outcome is the results or consequences that the subject finds once the activity is completed (Engeström, 1993, 1999a). Engeström (1999a) explained that the relationship between components of an activity system is two-way as people not only use instruments, but also renew them, they not only use rules, but also reformulate them.

The interactions among the components of the triangle model highlight tensions that are inherent in human activities; researchers find tensions in activity systems when elements from one or more components pull participants away from achieving the purpose of the activity thus cause changes in activities, so tensions may either promote or hinder human activities. (Cole & Engeström, 1993; Engeström, 1993, 2004; Yamagata-Lynch, 2003). We argue that from the perspective of Medley's (1987) variables, CHAT is appropriate for looking at relationships between any subset of the variables, depending on the data to be collected. Because CHAT was developed to consider complex systems, it is particularly suited to the task Medley was conceptualizing in the development of the presage-product-process perspective (Medley, 1987).

### *8.2 Benefits and Limitations*

The primary benefit of CHAT is its inherent ability to make sense of a complex system in a way that accounts for the actors and mediators at work in that system (Yamagata-Lynch, 2010). As exemplified in the examples below, CHAT provides a means for making sense of external context variables and the effects they have on instructional activities for teachers and students. This is important if the field wants to extend beyond Mendeley's (1987) original assertion that only two adjacent levels of variables can be considered at one time.

The limitations of CHAT are important considerations. First, it is not appropriate for considering human thinking, as it relies on observable activity (Yamagata-Lynch, 2010). This has implications for the kinds of growth that can be considered, how knowledge is characterized, and other elements of consideration that can only be observed by proxy. Further, the triangle model, while supporting sense-making about human activity systems, also oversimplifies those systems (Yamagata-Lynch, 2003, 2010). That is, complex human interactions are summarized to the point that they are "…not as rich and complex as real experiences" (Yamagata-Lynch, 2010, p. 33). Finally, CHAT is complicated to learn. This is because it requires the researcher to be proficient in qualitative methods, to understand and honor the complexity of collecting trustworthy data, and the ability to bring all of that together within a very specific theory (Yamagata-Lynch, 2010).

### *8.3 Examples*

One of the challenges of STEM education is to integrate activities, content, and tools in a meaningful in-class activity. Using CHAT models as a basis for analyzing learning, teaching, and in-class interactions between different subjects calls for the transformation of authentic scientific/mathematics practices into classroom activity systems. Here we provide two examples that have relied on Engeström's theorizing of CHAT. We also invite the reader to look at the work of Schmittau, who used cultural-historical theory as put forward by Vygotsky to make sense of student's mathematical learning (e.g., Schmittau, 2004, 2005, 2011).

CHAT has become an important lens in mathematics education research because it has "power to deal with complexity in educational systems" (Jaworski & Potari 2009, p. 222). To build on early research with use of activity theory in mathematics teaching–learning, as well as with a focus on the classroom tasks and their related macro-social setting, Jaworski and Potari (2009) considered teaching as activity in their study in the 10th grade classroom of a UK secondary school where students in this grade group are considered "lower achievers". They used CHAT to consider the role of the social framework within which classroom teaching is situated. They had two primary goals. The first was to understand the relationship between teacher-student interactions and the ways in which cognition is evident in classroom dialogue. The second was to analyze the relationships between classroom interaction and cognition within the broader cultural context in which learning occurs. They employed triangles from EMT to characterize the "subject" to be any teacher or pupil learning in this setting, each with their goal or object for their activity.

Specifically, Jaworski and Potari (2009) analyzed teacher-student interactions through classroom dialogue, which they viewed as a micro-analysis. In an episode offered by the authors, the teacher, Sam, had planned a didactical inquiry including in-class activities and relevant resources. In the implementation of this plan, Sam met with some "tensions" (p. 229). For example, students who had not done their homework completely derailed Sam's lesson plan. Jaworski and Potari suggested explanations behind the homework issue. They talked about the task that teacher assigned to students: from a teacher's perspective, the activity seemed "logico-mathematical" and reasonable in "didactical communities"; however, for student peer and family communities, it is "strange and unreasonable".

The representation of the application of CHAT to allow analysis from both teacher's and student's perspective. For example, through their analysis, Jaworski and Potari (2009) determined: teacher's object could be "understanding of basic statistical terms and associated concepts," while pupil's object would be "classroom survival"; teacher's rules could be "teacher/student authority structures," while pupil's rules are "homework expectations within the school"; and the outcomes for teachers are "Non achievement of object due to pupils not taking the required responsibility, tension in the classroom", for pupils are "survival by ignoring terms of homework, contravening rules and contributing to classroom tension" (p. 231). By illustrating the descriptive power of CHAT for making sense of the observable activities in mathematics classrooms, the researchers framed teacher's mathematics teaching as inconsistent with their socio-cultural histories. Further, they found that the teaching did not match nondominant student's learning. This highlighted, for the researchers, the lack of opportunities for mathematics teachers to challenge privilege-oriented activities. Without these opportunities, many well-intentioned mathematics teachers may unconsciously continue to perceive, explain, and respond to the classroom activities and specific learners through the dominant discourse system, which triggers the equality that they originally desired to abolish. (Jaworski & Potari, 2009).