1 Introduction

The use of digital technologies for mathematics learning and the reflection of its potentials has long been a focus of educational research (Clark-Wilson et al., 2020). According to the consensus view of education scientists, a key factor for the successful implementation of digital technologies in the classroom are the professional digital competencies of teachers (Hegedus et al., 2017). Professional digital competencies are understood in this paper as those that enable a mathematics teacher to evaluate and select digital technologies appropriately and use them in the classroom effectively for mathematical teaching and learning (Dilling et al., 2024). The term digital technologies does not only refer to classic digital mathematics tools such as dynamic geometry software or spreadsheets, but rather to any hardware or software that can support mathematical learning, such as 3D printers and computer-aided design software, virtual reality (VR) glasses and applications, or interactive whiteboards and presentation software.

Competencies in general can be defined according to Weinert (2001) as the cognitive abilities and skills available and learnable by individuals to solve specific problems, as well as the associated motivational, volitional, and social readiness and skills to use in the solution of problems. Models in the context of teachers’ professional competencies therefore usually distinguish between a cognitive and an affective-motivational component (e.g., Blömeke et al., 2010). In this article, a special focus is placed on the two affective-cognitive aspects of self-efficacy expectations and attitudes towards technology use, that are considered in research as important factors for an effective teaching with technology (Thurm & Barzel, 2022).

The development of teachers’ professional competencies has already been investigated in various empirical studies. One major element of successful professional development interventions, highlighted in various studies, reviews, and formulations of design criteria, is subject-related cooperation and collaboration among teachers (e.g., Rzejak et al., 2020). A concept that foregrounds collaborative instructional development is that of the so-called teacher design team (TDT; Handelzalts, 2009; Huizinga et al., 2019):

TDTs are teams of teachers who collaboratively analyse, design, develop, implement and evaluate their shared curriculum (also referred to as the ADDIE model, Gustafson & Branch, 2002; Handelzalts, 2009). TDTs can vary in terms of team size, whether the members are from the same or from different schools, and the courses the members offer (Handelzalts, 2009). (Huizinga et al., 2015, p. 138)

TDTs consisting of two or more in-service teachers have already been successfully implemented in various contexts, including research in the context of digital technology in mathematics education (Agyei & Voogt, 2012; Binkhorst et al., 2017; Kafyulilo et al., 2016). However, the participation of pre-service and in-service teachers in joint TDTs has not been addressed in research so far. This article addresses this desideratum. Through a case study, the characteristics of TDTs consisting of one in-service teacher and one pre-service teacher are examined and related to research on pure in-service or pre-service TDTs. For this purpose, Sect. 2 first provides an overview of research on designing as a fundamental task for teachers as well as collaborative designing within TDTs. Following a detailed presentation of the methodology (Sect. 3), the results of the case study (Sect. 4), and their discussion (Sect. 5), a conclusion is drawn, and implications for future research are suggested (Sect. 6).

2 Literature survey

2.1 Teachers as designers

Many mathematics education traditions provide research and teacher education approaches that focus on the design of learning scenarios and learning units (e.g., “lesson studies,” Isoda, 2010; “resources approach,” Trouche et al., 2019; “design-based research,” Design-Based Research Collective, 2003). In these approaches, the notion of design and the related activity of designing are crucial for creating appropriate learning scenarios and environments, thereby also being fundamental to the profession of mathematics teachers. In this context, Brown (2009) also uses the metaphor of “teaching as design” or “teachers as designers” and describes the tasks and objectives of teachers at various levels in terms of the activity of designing:

Teaching is, in many ways, a design activity. Teachers must perceive and interpret existing resources, evaluate the constraints of the classroom setting, balance tradeoffs, and devise strategies—all in the pursuit of their instructional goals. (Brown, 2009, p. 18)

Thus, the term ‘designing’ is understood rather broadly in this context. It is not only about designing something completely new but also about selecting, adapting, assigning, combining, or redesigning learning environments, even in real time (design-in-use). The so-called design capacity of teachers is crucial to the successful design of learning environments. The term can be traced back to Brown (2009), who understands pedagogical design capacity as follows:

Pedagogical design capacity (PDC) goes beyond the resources that are present in an instructional episode to describe the skill by which the various pieces are put into play. [...] PDC describes the manner and degree to which teachers create deliberate, productive designs that help accomplish their instructional goals. (Brown, 2009, p. 29)

Pepin et al. (2017) describe teacher design capacity consisting of five main components: knowledge of the classroom context; knowledge of the curriculum guidelines; knowledge of the position of the design in the short and the long terms; a set of design principles that can be applied flexibly; the ability to reflect-in-action, which facilitates adaptation to new challenges and contexts. If one compares the components of design capacity described by Brown (2009) and Pepin et al. (2017) with the concept of professional digital competencies, design capacity can be described as an essential part of teachers’ professional digital competencies. Design capacity focuses on creating teaching materials and adapting them for specific teaching situations. Furthermore, the very wide-ranging concept of design also considers the implementation in the classroom.

2.2 Teacher collaboration in teacher design teams

Collaboration among teachers is considered a key feature of successful professional development (e.g., Peter-Koop et al., 2003). A widely used approach to describing such collaborations are so-called communities of practice (CoP). This approach uses the theory of situated learning, which assumes that experiences and communication are the basis of learning, and that learning does not, or should not, take place in formal contexts but in natural interactions with other people (cf. Lave & Wenger, 1991). According to this approach, learning should take place in CoPs:

A community of practice is a set of relations among persons, activity, and world, over time and in relation with other tangential and overlapping communities of practice. (Lave & Wenger, 1991, p. 98)

Communities of practice can pursue a wide range of goals. One particular type of community of practice is focused on the collaborative design of instruction by teachers, emphasizing the design aspect as previously discussed. This form is often identified by the acronym TDT (see Sect. 1). TDTs are distinct from traditional communities of practice (CoPs) in several key ways. Primarily, TDTs are organized around a collective design effort, while CoPs mainly seek to foster professional development and influence the practices of their members. Moreover, TDTs are deliberately supported and guided by facilitators, unlike CoPs, which tend to evolve organically based on the shared interests of their participants in a specific field. TDTs typically consist of specific members who maintain their roles and specific tasks (rather fixed task distribution) throughout the design process, whereas CoP membership is fluid, with participants able to adopt new roles and tasks based on their evolving interests and needs (rather flexible task distribution). Finally, while a TDT is generally disbanded upon the completion of its design project, a CoP may continue as long as its members feel they can contribute to or gain from the association.

TDTs have two central objectives (Binkhorst et al., 2015): on the one hand, the professional development of the members; on the other hand, the concrete designed concepts and materials. TDTs are typically supported by facilitators, who in most cases are external experts (Huizinga et al., 2015). The facilitators contribute new knowledge and design options to the discussion (Handelzalts, 2009). Similarly, they can design learning environments together with the team and guide the process in an appropriate direction (cf. Becuwe et al., 2016). Becuwe et al. (2016) distinguish three roles of facilitators of a TDT: they provide organizational, esp. logistical, support for the team’s activities; they structure the design process by providing and introducing appropriate tools; they monitor the design process by providing pro-active support, help outline the design process, provide re-active support to ensure readjustment of the design, and intervene when necessary. Huizinga et al. (2015) has emphasized that facilitator support is important at every stage of the design process but especially during the analysis of the teaching situation.

Several empirical studies have examined TDTs as an approach to professional development. Binkhorst et al. (2017) accompanied various teachers in the implementation of six-month TDTs. The participants indicated that they were satisfied with their own professional development resulting from the collaboration and that they had gained new pedagogical and teaching skills as well as professional knowledge through the exchange. Some teachers also reported having changed their teaching approaches as a result of changes in their perspectives and their own further development. Handelzalts (2019) accompanied twelve TDTs over the period of one year and could show that the composition, processes, and results of the TDTs differ greatly. It has been demonstrated that TDTs with a clear objective begin their concrete work more swiftly and successfully achieve their intended results. The study also showed that TDTs that design concrete material jointly and do not just develop general issues and design statements together are also more successful in achieving their goals. The TDT approach has also already been used in relation to the use of digital technologies in the classroom. Kafyulilo et al. (2016) concluded from an empirical study that this approach can enhance the technological pedagogical content knowledge (TPACK) of science teachers, as described by Koehler and Mishra (2009).

Gueudet and Parra (2017) in this context analyzed the collaboration of two mathematics teachers when planning a teaching unit in the field of probability and found that shared notions of learning goals and expectations can promote collaboration and lead to (at least partially) similar materials and modes of using them. In contrast, different notions can lead to different modes of use, even if the design is collaborative and the developed materials are similar. Gueudet et al. (2016) accompanied a team of maths teachers during the long-term design work on a mathematics e-textbook. The results show that a collaborative work where the aim of the activity does not coincide with usual goals of mathematics teachers can lead to the development of shared modes of use. Although different opinions and beliefs of the teachers involved can lead to conflicts, these can be negotiated based on a shared overarching goal. Pepin et al. (2017) accompanied a mathematics teacher in her collaboration with various groups of teachers as part of a multiplier training program. They observed that various modes of use emerged and evolved through collaboration with different groups and in the individual work with the material.

TDTs traditionally are composed exclusively of in-service teachers, and the process is often facilitated by an external person who is not a teacher (e.g., a researcher). However, there is also research on teams in which teachers collaborate with scientists through design processes. For example, a study by Hansen et al. (2016) found that co-constructive collaboration between teachers and scientists in the design of a digital learning environment can result in a specification of the teachers’ technological pedagogical content knowledge. The teachers’ role in the collaboration was particularly to consider how the digital learning environment could be integrated into the classroom or how teachers’ professional development could be enhanced, while the basic design decisions were rather taken by the scientists. First studies have also investigated TDTs consisting of pre-service teachers. Agyei and Voogt (2012) state in an empirical study that pre-service mathematics teachers can work together meaningfully in TDTs and that their technological pedagogical content knowledge can also develop further as a result. However, much more effort and close support from facilitators is particularly important for this. Trgalová and Tabach (2024) made similar experiences in a pre-service mathematics teacher training course.

2.3 Research questions

Mixed TDTs consisting of pre-service and in-service mathematics teachers have not been considered in educational research so far. The multiple case study presented below addresses this desideratum by exemplifying the roles that pre-service teachers, in-service teachers and facilitators can play in such a collaboration. The following research questions arise from the literature survey and should be investigated by the analysis of the case in this paper:

What characteristics can be identified in a collaboration of pre-service and in-service teachers within teacher design teams in the context of teaching with digital technologies?

  1. (a)

    In terms of task distribution and interactions within the TDT

The term task distribution refers to the types of tasks performed by the different participants in a TDT. This is closely linked to how they interact with each other in the TDTs and thus substantially characterizes their role within the collaboration. There are only a few results on this aspect in the research on TDTs so far. Binkhorst et al. (2015) showed that different roles can develop in pure in-service TDTs, e.g. some teachers take on the leadership of the TDTs. The different roles are associated with specific tasks or a certain task distribution within the TDT. As described above, different tasks are also taken on by the participants in the collaboration between in-service teachers and scientists (Hansen et al., 2016). A hypothesis could be that within mixed pre-service and in-service TDTs, pre-service and in-service teachers might undertake distinct tasks, reflecting their varying experiences and objectives.

  1. (b)

    In terms of the professional development of TDT participants, esp. teachers’ design capacity on integrating digital technologies in mathematics lessons

The professional development of teachers is a central goal of TDTs (Binkhorst et al., 2015). Numerous empirical studies have demonstrated that the design capacity, as discussed by Brown (2009) and Pepin et al. (2017), within both pure in-service and pure pre-service TDTs can be further advanced in the context of incorporating digital technologies into mathematics instruction, as exemplified by research such as Kafyulilo et al. (2016) and Trgalová and Tabach (2024). With regard to the mixed pre-service and in-service TDTs, it can be hypothesised that professional development is also possible in this constellation.

  1. (c)

    In relation to the importance of facilitators accompanying the TDTs

The literature survey shows that facilitators may play an important role in the successful work of TDTs (Becuwe et al., 2016; Handelzalts, 2009; Huizinga et al., 2015). Especially in pure pre-service TDTs, there seems to be a great need for structuring and content-related guidance (Agyei & Voogt, 2012; Trgalová & Tabach, 2024). One hypothesis of this study is that mixed pre-service and in-service TDTs also benefit from strong facilitation.

3 Methodology

3.1 Design of the case study

The questions outlined above are explored within the framework of a case study, in accordance with Yin (2013). This approach was chosen because it allows for a qualitative in-depth analysis of the interrelationships within TDTs. The case under investigation is DigiMath4Edu—a research project at the University of Siegen in Germany in which selected pre-service teachers (so-called digitization assistants) supported in-service teachers from fifteen participating schools in planning and implementing digital technologies in their mathematics lessons (Dilling et al., 2022). The digitization assistants were either advanced undergraduate students or master’s students, who had participated in seminars on the use of digital technologies in mathematics instruction during their academic studies. Additionally, they were prepared for their work through four specific workshops each lasting two days, in which they were trained in the operation of certain digital technologies and practiced the planning of mathematics lessons with focus on technologies. Part of that workshops was the use of hardware such as 3D printers, VR goggles or programmable microcomputers, but also mathematics-specific software such as dynamic geometry software or spreadsheets that each project school was provided with. The pre-service teachers spent 10 h a week working on the project and were financially supported for their work.

Two digitization assistants were assigned to each school for an entire school year. The in-service mathematics teachers at the schools could work with the digitization assistants during this time and involved them in the planning, implementation of and reflection on lessons with a focus on the use of digital technologies. In most cases, the meetings took place face-to-face in the schools, but in some cases online meetings were organized or written communication was used to plan the lessons. The form and intensity of the collaboration as well as the mathematical content worked on was not prescribed externally, but was based on the concrete needs of the in-service teachers and limited by the weekly working hours of the digitization assistants. The goal of the project communicated with the schools was the professional development of the in-service teachers in the use of digital technologies in mathematics lessons. The design teams were supported by project staff from the University of Siegen who served as facilitators that organized training events for the pre-service and in-service teachers, provided the required digital equipment for the schools and helped whenever problems occurred.

The project described provided ideal conditions for examining the collaboration between pre-service and in-service teachers within teacher design teams, aiming to generate answers to the research questions previously outlined. However, it’s important to note that certain specific framework conditions were in place, for example, concerning the selected and specially trained digitization assistants.

3.2 Data sampling

The collected data comprise semi-structured interviews conducted in German at the end of the first year of the project with the digitization assistants and the in-service teachers who were especially involved in the project. The pre-service and in-service teachers had different backgrounds and came from an elementary school, two comprehensive schools (Gesamtschule und Sekundarschule), and two secondary schools (Gymnasium). A total of 9 digitization assistants (D1–D9) and 16 in-service mathematics teachers (T1–T16) were surveyed in 23 interviews. The interviews with the in-service and pre-service teachers each had a slightly different focus. The pre-service teachers were asked about their own tasks in the project, the distribution of labour with the in-service teachers, the differences in cooperation between different in-service teachers, the perceived professional development of the in-service teachers and their own professional development in the project. The in-service teachers were also asked questions about their own professional development and the role of the digitization assistants, but there were also questions about the impact of other project aspects such as centrally organized teacher training events or the availability of digital technologies purchased for the project.

3.3 Data evaluation

For the analysis, the videotaped interviews were transcribed according to the rules of Dresing and Pehl (2015). The generated data were categorized inductively using the method of qualitative content analysis according to Mayring (2000). An inductive approach was chosen because no empirical results are available on mixed pre-service and in-service TDTs, which means that the conceptualizations from Sect. 2 cannot be applied directly. Instead, the categories are to be linked to the state of research in the discussion (Sect. 5). The summarizing content analysis was performed in four steps (Fig. 1). First, the data to be analyzed were described in detail. In the present case, the analyzed data included the complete transcripts of 23 guided interviews. The unit of analysis, the smallest analyzed part of the text, was defined for this study as any meaningful statement of the pre-service or in-service teachers. In the second step, the relevant sections of the text were condensed into a form focused solely on content description, paraphrasing the material. The paraphrases are generalized at a defined level of abstraction with respect to the research questions. In the presented multiple case study, the level of abstraction refers to case-specific statements made by a pre-service or in-service teacher concerning collaboration within the TDTs, professional development of participants, or facilitation of the collaboration. These statements are distinct from specific mathematical content or technology. The number of generalized statements was then reduced several times by increasing the level of abstraction and removing statements with the same meaning. In the third step, the statements were compiled in a system of categories that was checked based on the material in the fourth step. The application of these steps is exemplified in Table 1 by the presentation of a transcript excerpt, a paraphrase, the final generalization, and the name of the category.

Fig. 1
figure 1

Simplified illustration of the summarizing qualitative content analysis referring to Mayring (2000)

Full size image
Table 1 Example of applying the method of summarizing qualitative content analysis according to Mayring (2000)
Full size table

The category system is based on three main categories: “C1: Task Distribution,” “C2: Professional Development,” and “C3: Facilitators.” These were deductively derived from the research questions and the theoretical framework on teacher design teams. To further structure the category system, a distinction was made in the first two main categories between categories that refer to pre-service teachers, in-service teachers or to general characteristics of TDTs (see Table 2). Within this framework, the subcategories (see Tables 3, 4, and 5) were inductively created from the interview material using the previously described method. This resulted in a total of 21 categories. In the following section, the different categories are described in more detail by interpreting parts of the data material associated to the categories. The German-language interview excerpts and category names were translated into English for the presentation in this article.

Table 2 Deductively formed system of main categories
Full size table
Table 3 Categories within the main category C1 “Task distribution”
Full size table
Table 4 Categories within the main category C2 “professional development”
Full size table
Table 5 Categories within the main category C3 “facilitators”
Full size table

4 Results of the study

4.1 RQ 1: Task distribution

A total of 11 categories were formed inductively with regard to the distribution of tasks and the interaction between the pre-service teachers (digitization assistants) and the in-service teachers (see Table 3).

4.1.1 Tasks of the pre-service teachers

Five of the categories on task distribution relate to the tasks of the pre-service teachers. For example, one tasks of the pre-service teachers is the technical introduction of digital technologies for the in-service teachers (C1.1). One of the digitization assistants described this part of his task as follows:

D3: We showed the teachers how to use them, what you can do with them. We analyzed the functions and showed them how to use the different functions. For example, Tinkercad classes, how to set it up, how to give the students access, what options the students have.

As an example of the introduction of the functions of digital technologies, the digitization assistant mentions setting up and working with student accounts in the computer-aided design software Tinkercad. Other digitization assistants and in-service teachers also made statements about this task and mentioned for example short workshops that the digitization assistants offered for the in-service teachers during school breaks to give them initial access to the technologies.

Another task is to provide ideas for the use of digital technologies for mathematical learning (C1.2). An in-service teacher from a comprehensive school described this part of the work of the digitization assistants as follows:

T3: The consulting role along the lines of, "What can I do digitally now on this topic?" They were then able to give me examples, advise me on what I could do with the iPad on the topic, perhaps even with the 3D printer.

The in-service teacher explains that the digitalization assistants have made connections between digital technologies and mathematical topics by making suitable suggestions for the implementation of mathematical content with digital technologies in the classroom. Another in-service teacher gives a concrete example of this:

T5: [D8] had found the website Math Learning Center, where you have an online geoboard and that was just great for distance learning, because we could not use a normal geoboard.

The digitization assistants thus also carried out searches and looked for suitable technologies for implementing mathematical topics—in this case, a digital alternative to geoboards for distance learning. Other in-service and pre-service teachers also mentioned the role of idea generation in the interviews.

Based on the generated ideas and the agreements with the in-service teachers, the digitalization assistants also designed parts of the digital lessons (C1.3). An in-service teacher from a secondary school reported the following:

T14: They created lots of instructional videos for the students, and for us too, of course. They wrote instructions that the students can now use to work step-by-step with GeoGebra. That was very professional.

As examples of teaching materials created by the pre-service teachers independently, the in-service teacher mentions instructions and instructional videos for the students on the use of digital technologies such as GeoGebra. Other pre-service and in-service teachers give further examples, some of which also involve the joint creation of teaching materials.

The digitization assistants were also involved in the implementation of the digital lessons. They provided technical support for the students and the in-service teachers during the lesson (C1.4), as one digitalization assistant also described in the interview:

D5: And if they felt overwhelmed or didn’t feel confident implementing it, they’ve always asked me if I can help: Especially when new things are used, like 3D printing or 3D pens, my help was needed because the teachers themselves didn’t know how to handle it and so they seemed very unconfident with the students.

In the excerpt, it is described that the in-service teachers receive technical support from the digitalization assistants when implementing new technologies in the classroom. In other interviews, in-service and pre-service teachers stated that the digitization assistants sometimes also introduced the technology directly to the students during the lesson.

In their collaboration with the in-service teachers, the digitization assistants ultimately assumed the role of professional exchange partners (C1.5). A teacher from a secondary school explains this as follows:

T16: You could observe the students in class and see where there are still difficulties or suddenly you saw that there are also completely different exciting possibilities and the digitization assistants were pre-service teachers relatively at the end of their university studies, so you could also have good conversations about didactics, methodology about modeling in math, about performance measurement, about all sorts of things and that was a fruitful exchange.

This section shows that the digitization assistants were seen as competent discussion partners—even on topics that did not directly concern digital technologies. It is also clear from the various interviews that the joint follow-up of digital lessons is an important task of the digitization assistants.

4.1.2 Tasks of the in-service teachers

The tasks of the in-service teachers within the TDTs were described in less detail in the interviews, probably since these are quite close to the normal tasks of a mathematics teacher. Four categories refer to the tasks of in-service teachers. From all interviews, it emerged that the in-service teachers are responsible for preparing the lessons (C1.6) and primarily conducting them on their own (C1.7). This can be seen, for example, in the following short transcript excerpt in which a secondary school teacher describes the limits of the responsibilities of the digitization assistants:

T6: [They are] definitely not lesson planners for our lessons and are supportive and not directive in the execution. That’s how it should be, I think.

The support in the field of digital technologies enables the in-service teachers to focus on the overall structure and the methodological and mathematical design of the lessons (C1.8), as described by various in-service teachers in the interviews, e.g.:

T16: I could then plan the lessons around it with the mathematical focus and think about a good lesson structure and good methods.

A final task of the in-service teachers in their collaboration with the pre-service teachers is learning how to use digital technologies in mathematics lessons (C1.9). A digitization assistant describes a prototypical situation in this regard:

D5: I showed [T16] from the beginning how to create applets in GeoGebra. He didn’t know that either. And on his own initiative, he created lots of really great and complex applets and then always sent them to me and asked: Do you have any suggestions for improvement? Does it work like that? Did I do it right with the commands?

4.1.3 General characteristics of the task distribution

In addition to describing the areas of responsibility of the pre-service and in-service teachers, the interviews also included statements on general characteristics of the cooperation. Some of the digitization assistants stated that their support of the in-service teachers gradually decreased (C1.10) because they had learned to work more independently with the digital technologies:

D6:Many teachers have also taken us into the classroom several times for the same things and yet you noticed that each time it became a little more independent, that the teachers themselves recognized the mistakes and pitfalls when something went wrong and I think that the support has simply been such a process.

Furthermore, the form and intensity of the support varied depending on the in-service teacher (C1.11):

D4:I would say it varies a bit depending on the teacher. Some teachers just have a small technical question or perhaps want to get an idea and then implement it relatively independently. Others, I had the feeling, can’t really imagine it or perhaps don’t want to work out an idea that might not suit them so well. In other words, you had to come up with a kind of overall concept to convince them.

4.2 RQ 2: Professional development

A total of six categories relate to the professional development of the pre-service and in-service teachers as participants in the TDTs (see Table 4).

4.2.1 Professional development of the pre-service teachers

With regard to the pre-service teachers, the professional development includes in particular the positive change in self-efficacy expectations on technology usage (C2.1). In the following section, one digitization assistant briefly describes how he has become more confident in using the technologies and has learned to use them in the classroom:

D1: That you are much more confident with the digital technologies yourself and have been able to learn a lot about how to use them in class and then on what to pay attention to when using them.

Another aspect mentioned by the digitization assistants is the change in their professional personality (C2.2). Through their practical experience at school and by accompanying different in-service teachers, they were able to further develop their own personality in relation to the teaching profession:

D5: You know how to stand in front of students, how to explain it, also explain it to teachers. […] You can’t just learn something like that in theory. You just have to find your own style.

Finally, some digitization assistants stated that, in addition to using digital technologies, they had gained a lot of subject-independent knowledge about schools, lessons and the tasks of teachers (C2.3):

D4: Then you have learned a lot about how teachers organize everyday school life, what their tasks are and how things generally work at school.

4.2.2 Professional development of the in-service teachers

The in-service teachers also state that they have learned a lot in their collaboration with the digitization assistants. Initially, this also concerns the positive change in their self-efficacy expectations regarding the use of digital technology (C2.4):

T8: The fact that you are a bit less afraid to try something new and also solve some things via 3D printing that you previously did with pen, paper and scissors and can now apply them over several years.

Many other in-service teachers also describe how they have gained knowledge about the use of digital technologies and have thus been encouraged to use them. In general, many in-service teachers also state that they integrate digital technologies into their lessons more frequently as a result of the collaboration in the TDTs (C2.5):

T9: What I can say across the board is that we have integrated digital technologies into lessons much more often than before.

For many teachers, digital technologies have become a standard and have lost their status as something special. Finally, some in-service teachers also stated that their attitude towards the use of digital technologies in mathematics lessons had changed positively (C2.6), as can be read in the following transcript excerpt from a secondary school teacher regarding the use of 3D printing in calculus:

T16: I hadn’t considered the use of 3D printing to be particularly useful before this project year and that has changed. [...] I had already attended a training course and thought it was more of a gimmick and now I have gained insights into the teaching sequence on solids of revolution in the advanced course with the digitization assistant, which I will now also carry out with my advanced course in two weeks and that also offers a connection to the local industry, to things that they really do with the production of mechanical components and the precise adaptation […]. This is modelling that is really close to reality and not as artificial as many other tasks are, and that is also convincing for the students.

The teacher has had positive experiences with 3D printing through collaboration with the digitization assistants and recognizes significant potential in its application to mathematical modeling. These experiences have shifted his initially negative perception of the technology.

4.3 RQ 3: Facilitators

The main category concerning the methods by which facilitators support the TDTs includes four sub-categories (refer to Table 5).

The in-service teachers mentioned in the interviews networking with schools and the university (C3.1) as an important aspect. In the excerpt below from an interview, a secondary school teacher explains how collaboration with the university has offered fresh insights into the use of digital technologies:

T7: The didactic exchange once again with the university, I always really appreciate that. The new impetus for mathematics teaching with regard to digitization.

Another aspect mentioned in the interviews, which the university project team used to support the in-service teachers as participants in the TDTs, are teacher training events (C3.2). According to the teachers, these form the more general technical and didactic basis for the subsequent more concrete collaboration with the pre-service teachers in relation to the current lessons:

T15: Basically, we had such a shift through the training courses. I would say that the digitization assistants always reacted on a daily basis and then responded to the specific needs of the teachers they were working with. And the training courses that we did here, for example, were a bit more general [...]. So, I think the mix was fine.

Another task of the university project team as facilitators is the selection and training of the digitization assistants to accompany the in-service teachers (C3.3):

T16: You [interviewer] really found a way to make a selection, they were really excellent people. They were incredibly professional, it was really great.

In addition, the supportive tasks of the project team also included the selection and provision of digital hardware and software (C3.4), which could be used in the collaboration between the pre-service and in-service teachers:

T1: Keep it up, that you take over the purchases for the schools, because I think a lot of new things can be created through this. I think that we would have decided in favor of what we know and it’s good that we didn’t have control.

5 Discussion of the results

The previous sections present an approach to professional (further) development of mathematics teachers in the context of the use of digital technologies. In the case study, teachers design, implement and reflect on lessons together in TDTs in the context of digital technologies. This approach in general has already been successfully described for this purpose in several empirical studies (e.g., Agyei & Voogt, 2012; Kafyulilo et al., 2016). The special feature of the case presented here is the collaboration of pre-service and in-service teachers in mixed TDTs, which has not been the focus of mathematics education research so far. The following research questions were posed:

What characteristics can be identified in a collaboration of pre-service and in-service teachers within teacher design teams in the context of teaching with digital technologies?

  1. (a)

    In terms of task distribution and interactions within the TDT.

  2. (b)

    In terms of the professional development of TDT participants, esp. teachers’ design capacity on integrating digital technologies in mathematics lessons.

  3. (c)

    In relation to the importance of facilitators accompanying the TDTs.

The collaboration occurred under the unique circumstances of the DigiMath4Edu research project at the University of Siegen in Germany. Here, pre-service and in-service teachers collaborated, mostly in pairs, to plan and implement mathematics lessons. Results pertinent to the research question were derived from the qualitative content analysis of 23 guided interviews, which will be discussed subsequently.

  1. (a)

    The first research question highlights the distribution of tasks and the interactions within the TDTs. These have been reconstructed in detail on the basis of the interviews. In our special case, the pre-service teachers (who had no prior teaching experience) had supportive tasks in the planning and implementation of the lesson—providing input in the planning phase (e.g., online geoboard for distance learning of Geometry), independently designing parts of the lesson regarding digital technologies (e.g., instructional videos for students to work with GeoGebra), or providing support in the event of technical difficulties in the lesson (e.g., troubleshooting for 3D printers in the classroom). In contrast, the cooperating in-service teacher remained responsible for the lesson the entire time—deciding which ideas to implement and to use for planning and conducting the lesson. The pre-service teacher’s support allowed the in-service teachers to focus on essential elements of teaching, such as the mathematical content, an appropriate structure, or the methodology. Therefore, the tasks outlined in the interviews represent typical activities within TDTs, aligning with the comprehensive understanding of teacher design as presented by Brown (2009) at the outset.

    The role and task distribution within the TDTs examined was clear and has been distinctly described in the interviews by both pre-service and in-service teachers. However, according to the pre-service teachers, the intensity of support varied depending on the in-service teacher from answering a few technical questions to preparing detailed concepts and materials. The orientation of support towards the needs of the in-service teachers can largely be attributed to the systematic approach of the DigiMath4Edu project. The main objective of the project, which was also communicated to the participants, was the professional development of the in-service teachers. Accordingly, the in-service teachers in the cooperation also had the role of a learner who wanted to expand their skills in the use of digital technologies. In order to achieve this goal, the support provided by the pre-service teachers was gradually reduced so that the teachers finally had the competencies to use the digital technologies independently in mathematics classes. As shown in the studies by Gueudet and Parra (2017) and Gueudet et al. (2016), a common overarching goal—in this case, the communicated goals of the DigiMath4Edu project—is beneficial for the collaboration of teachers. Furthermore, as in Pepin et al. (2017), the collaboration and outcomes differ depending on which in-service teachers the pre-service teachers work with. As shown in the studies by Binkhorst et al. (2015) and Hansen et al. (2016), there can be considerably different roles and associated tasks within TDTs, which in this case are allocated relatively clearly to the pre-service or in-service teachers. Nevertheless, as described above, it is not appropriate to describe this as a hierarchical structure—both the pre-service teachers and the in-service teachers have areas in which they are experts and take the lead.

  2. (b)

    The second research question deals with the professional development of the TDT participants and especially their design capacity (Brown, 2009; Pepin et al., 2017). The pre-service and in-service teachers reflected on their own learning processes within the Project DigiMath4Edu in the interviews. Both the pre-service and in-service teachers declared a positive development in self-efficacy expectations with regard to the use of technology. Thus, both seem to have had positive experiences with the use of digital technologies during the collaboration and also expect that this will continue to be successful after the collaboration. The aspects addressed by the pre-service and in-service teachers relate to important elements of design capacity according to Pepin et al. (2017), in particular design principles in relation to the digital technologies tested by the TDTs and reflection-in-action when using the technologies in the classroom. The positive experiences can also explain why the in-service teachers assume that they use and will use digital technologies more frequently in mathematics lessons than before participating in the TDT. They feel confident in applying their knowledge about digital technologies in other situations, which could be an indication of an adequate design capacity. The change in attitudes towards certain digital technologies described by some teachers can also be attributed to the positive experience in accomplishing their instructional goals with their designs (e.g., fostering mathematical modelling competences through 3D design and printing) (Brown, 2009). The pre-service teachers also mentioned the enhanced development of their professional personality and the knowledge they had gained about schools, lessons and the tasks of teachers. It is important to emphasize that the statements of the pre-service and in-service teachers not only refer to the preparation of the lessons, but also focus on the successful implementation in accordance with the concepts of design capacity (“the skill by which the various pieces are put into play”, Brown, 2009, p. 29; reflection-in-action, Pepin et al., 2017). The described aspects of the professional development of pre-service and in-service teachers in DigiMath4Edu largely coincide with the findings presented in the literature review in Sect. 2. Binkhorst et al. (2017) report the development of pedagogical and teaching skills as well as the professional knowledge. This corresponds to the development of the pre-service teachers and, with specific reference to digital technologies, the development of the in-service teachers in this case study. The studies by Kafyulilo et al. (2016) and Agyei and Voogt (2012) found that in-service and pre-service teachers can build up extended technological pedagogical content knowledge in TDTs. The self-reported developments of the teachers from this case study can also be categorized in this category. Overall, in line with the current state of research, the participants of the TDTs work together successfully especially if there is a clear common goal, within a structured framework for the development of concrete teaching material.

    It is clear from the distribution of tasks and their own descriptions of professional developments that the aspect of using digital technologies in lessons determines the cooperation between the pre-service and in-service teachers. This is also reflected in the framing of the project DigiMath4Edu. However, in terms of professional development, a comparison between pre-service and in-service teachers reveals differences. While all three categories developed for in-service teachers relate directly to technology, two of the three categories for pre-service teachers are of a more general nature. One reason for this difference could be seen in the prerequisites of the individuals prior to the collaboration. The pre-service teachers were specifically trained in the use of technology and were to some extent experts in this field. However, they were not yet fully educated teachers and had little practical experience so far. In contrast, some of the in-service teachers had less experience in using digital technologies to learn mathematics, but they were able to draw on extensive practical experience of teaching mathematics in general.

  3. (c)

    The third research question examines the importance of having facilitators to accompany the TDTs. The support within the project DigiMath4Edu included networking with schools and the university, the provision of digital equipment, the selection and preparation of the pre-service teachers and the support of the in-service teachers by introductory workshops by the project team of the University of Siegen. Therefore, a crucial aspect, so it seems, involves preparing both groups of teachers—in-service and pre-service—while considering the specific needs of each. The special preparation was perceived as beneficial by all participants.

With regard to the research by Becuwe et al. (2016) on facilitators of TDTs, the support of the project team can be located in our case in particular at the organizational and structuring level. There is no extensive monitoring of the design process, but substantial content-related input was provided during the preparation of the participants. This means that the TDTs examined involved close preparation and support, which was rated positively by the participants of the interviews. Huizinga et al. (2015) also emphasize that the facilitation is beneficial at every stage of the design process. Although the focus in this case study was on the facilitation of the preparation phase, support was also provided in other phases whenever necessary, in particular through the continuous exchange between the university project team and the pre-service teachers. It can be stated that in addition to pure pre-service TDTs (Agyei & Voogt, 2012; Trgalová & Tabach, 2024), mixed pre-service and in-service TDTs also benefit from close support.

6 Conclusion and outlook

6.1 Summary

This article examined teachers’ professional development in the context of digital technologies through the collaborative design, implementation, and reflection of lessons in so-called TDTs. The focus was on a case study on the participation of pre-service teachers and in-service teachers in joint TDTs in the project DigiMath4Edu. The statements of the mathematics teachers interviewed indicate that the tasks in the collaboration were clearly distributed and that based on this the in-service and pre-service teachers each fulfilled different roles. In terms of professional development, both groups of teachers made positive self-assessments. However, the areas of their professional development certainly differed, which could be attributed to the different prerequisites. Finally, the mathematics teachers emphasized the positive effects of close support and extensive preparation, so that it can be concluded that appropriate support and preparation had a positive effect on the work of the TDTs.

The impact of the digital technology context and its connection to mathematics teaching on the TDTs’ work warrants explicit reflection at this juncture. We believe the influence of digital technologies is distinctly visible in the categories derived. The uniqueness of this context as a foundation for collaboration might lie in the in-service teachers’ need for further professional development and support in this area, which the specially trained pre-service teachers can offer. In contrast, the link to mathematics teaching is only sporadically evident in the transcript excerpts presented. This could be partly explained by the study’s design, which primarily focuses on the overarching aspects of cooperation between the TDTs rather than on specific content-related decisions in concrete lesson planning and execution. Nonetheless, the incorporation of digital technologies in mathematics lessons possesses certain distinctive features that likely vary from those in other subjects (e.g., the use of math-specific software). Hence, any generalization beyond mathematics teaching should be approached with caution.

6.2 Limitations

The findings from the DigiMath4Edu project can provide valuable insights for further systematic research on pre-service and in-service TDTs, particularly in the realm of integrating digital technologies into mathematics education. It’s important to highlight that the methodological approach used does not allow for making valid generalizations beyond the case analyzed. One reason is that the functioning of the TDTs and key factors in the study were not systematically varied using control groups, nor was there a systematic assessment of competencies. Instead, the results are based on reports and self-assessments of the participating teachers in interviews and are therefore affected by social and psychological phenomena. Additional limitations stem from the project structure of DigiMath4Edu, which already outlines a framework for the teachers’ collaboration, thus not allowing for a completely autonomous design process. The fact that the interviews were conducted in German and have been translated for the purposes of this article can also lead to a bias in the presentation. In short, the result of the case study is a detailed and systematic description of TDTs that are classified as successful by the participants.

6.3 Outlook

The explorative results show that it seems promising to engage in more systematic research on mixed pre-service and in-service TDTs. With regard to the task distribution, it should be investigated whether the clear allocation of roles and tasks to pre-service and in-service teachers is also evident in other settings. In the case study discussed in this article, professional development was reconstructed on the basis of self-assessments, only. Further studies should use valid test instruments to investigate the potential of mixed TDTs for professional development in general. In addition, it should be examined whether the learning outcomes of the pre-service and in-service teachers are also located in different areas beyond this case study. Further investigation into the role of facilitation seem warranted, possibly by systematically varying the different forms of support, such as organization, process structuring, and content input.