Introduction

Teachers who aim to adopt and develop digital resources for their own teaching should be supported by adequate theoretical and methodological techniques from the educational research community. In the field of education, modern technologies do not only provide services but can also support students’ competency development (König et al., 2020; Miranda et al., 2021). The importance of applying modern technologies at all levels of education is highlighted in both scientific research and national and international educational strategy documents (European Commission, 2020; Lavicza et al., 2022; Weinhandl et al., 2021).

According to Inayat and Hamid (2016), the development of modern technologies supporting learning processes has led to a transformation in education, from which mathematics learning and teaching particularly benefited. In line with Weinhandl et al. (2022), technology use has already been a prominent element in mathematics curricula in different countries. Despite the high potential of using technologies in learning mathematics and that technology use is integrated into various mathematics curricula, many experts (e.g., Clark-Wilson et al., 2020; Thurm & Barzel, 2020) stated that technologies are still underused and the integration of technologies into school-based learning is rather slow. According to Larkin and Milford (2018), there is an increasing diffusion of technologies outside schools and such technologies are already widely integrated into students’ lives. One of the consequences of the proliferation of technologies outside schools is the increasing demand for schools to incorporate technologies into classrooms (Collins & Halverson, 2018; Säljö, 2010). According to Lavicza et al. (2022), new software and hardware or adaptive solutions offer novel opportunities in mathematics classrooms. In line with Sinclair (2020), technologies could change how mathematics is learned and the mathematical content presented in classrooms. Additionally, technologies may change the role of mathematics teachers and make these environments even more complex (Clark-Wilson et al., 2020). In recent years, processes of the design, use, and evaluation of tasks for mathematical learning involving digital technologies have been increasingly investigated from the aspect of teacher training (Clark-Wilson et al., 2020). During an exploratory interview study, Weinhandl et al. (2021) found that mathematics teachers expect advantages but also problems in learning mathematics with technologies or digital devices: one advantage being improved differentiation and individualisation in learning mathematics.

Soboleva et al. (2022) indicate that the design of digital materials in accordance with the characteristics of students increases the benefits of using digital technology in teaching mathematics. In an educational context, mathematical learning and teaching materials supported by different technologies are often designed by teachers, who should consider several factors, the most important of which are as follows: the characteristics of students, tasks, and utilised educational technology (Leung & Baccaglini-Frank, 2017; Trgalová et al., 2018). Sullivan (1999) indicates that while designing tasks, teachers should keep in mind that the design of mathematical tasks can significantly affect the achievement and motivation of students in learning mathematics. Previous research by Venturini & Sinclair (2017) and Cevikbas and Kaiser (2021) concludes that it is of crucial importance to adapt the design of mathematical tasks to the digital and mathematical skills of students. When teachers know well enough the characteristics of their students to whom they create digital mathematics tasks, they enable students to progress in their conceptual understanding (Sánchez-Matamoros et al., 2019); utilising digital mathematics tasks are not in line with skills, knowledge, and expectations of students can lead to difficulties in learning (Johnson et al., 2017). Through a detailed literature review, Cevikbas and Kaiser (2021) determined that teachers should be supported with additional training in designing mathematical tasks with attributes in accordance with the characteristics of students based on scientific and practical knowledge, which also holds true for digital mathematical tasks. The current research literature indicates that developing fruitful digital mathematics learning resources requires extensive knowledge about students’ characteristics, and several studies (e.g., Rocha, 2020; Ratnayake et al., 2016 or Geiger et al., 2018) demonstrated that specifically prospective mathematics teachers struggle with developing fruitful digital mathematics tasks. Using the personas approach (Minichiello et al., 2018; Sundt & Davis, 2017) could offer valuable information about students’ characteristics and needs for prospective teachers. This information should support prospective mathematics teachers in developing digital mathematics resources. Therefore, our study aims to investigate this rather new approach regarding considering students’ characteristics when developing digital mathematics learning resources in basic mathematics teacher training. Previous research indicates that the persona approach can be successfully used both for designing training for teachers and for improving the skills of teachers to create teaching materials in mathematics for students (Madsen et al., 2014; Weinhandl et al., 2022).

Our study focuses on this interface between digital competencies and beliefs of mathematics teachers and the characteristics of students who should broaden and deepen their mathematical competencies with the help of digital resources. For combining mathematics teachers’ competencies and beliefs and the characteristics of students in our study, we used mathematics student personas during the development of resources by prospective mathematics teachers. According to Weinhandl et al. (2022), mathematics student personas are short and simplified descriptions of potential students who might use a digital mathematics learning environment. Personas provide authentic and vivid representations of mathematics students so that the design processes of a digital mathematics learning environment can be more easily aligned with the characteristics of students. In general, and specifically in the context of our study, addressing the teaching and learning of mathematics and related development of digital mathematical learning resources in Austria, it is crucial for mathematics teachers to have extensive digital competencies and to be able to create high-quality digital learning resources adapted to the characteristics of their potential students. This great importance of digital competencies of mathematics teachers and digital learning resources adapted to the needs of mathematics students is because in Austria, since autumn 2021, all students at the beginning of the secondary level (5th grade, 10-year-old students) are equipped with digital devices such as tablets or laptops. For this reason, our study aims to answer the following research question: What aspects of using personas while developing digital mathematics learning resources are relevant for prospective mathematics teachers?

To identify those aspects, we used a case study approach as it is particularly suitable for investigating phenomena that are still fairly unknown (Cohen et al., 2007). We conducted qualitative interviews with prospective mathematics teachers who used personas to develop digital mathematics learning resources and used thinking-out-loud techniques to collect data. We analysed the data in line with grounded theory, qualitative content analysis and qualitative interview study approaches.

Theoretical background and approaches in this research

Since our study aims to shed light on essential aspects of using personas when prospective mathematics teachers are developing digital mathematics learning resources, central components of our research are the learning of mathematics in the digital era, personas as a tool for user experience research, and teacher education for using technologies in mathematics classes, with a special focus on Technological Pedagogical Content Knowledge (TPACK) and Mathematics Digital Task Design Knowledge (MDTDK). In line with Pepin et al. (2017), we define digital mathematics learning resources as learning resources that are made accessible on electronic devices and that often incorporate the dynamic features of digital technologies. As the prospective mathematics teachers work together to develop digital learning resources, communities of practice are also essential as a design element of our study. These different areas are further discussed below.

Personas

Persona development is a widely used tool in user experience research, and personas represent a short and simplified description of a potential user group of specific technologies, services or systems (Minichiello et al., 2018; Sundt & Davis, 2017). This short and simplified description of a potential user group is primarily intended to help developers of technological services or systems put themselves in the shoes of potential users, which can be particularly challenging when the potential users of technological services or systems are adolescents or children (Antle, 2008). Carefully collected and processed data in the process of persona development can serve as homogenisers and equalisers of individuals in the group, and this group is illustrated by a persona (Siegel, 2010; Adlin & Pruitt, 2010). When developing personas, potential users’ needs, desires, fears, and technical experience or access requirements of these fictional users should be given special attention (Lilley et al., 2012; Van Rooij, 2012). Each persona represents an embodiment of the behaviours and incentives of a typical subgroup of potential users and provides a portrait of the potential user group (Maness et al., 2008; Miaskiewicz et al., 2008; Vorvoreanu et al., 2016). Since personas represent archetypes of a specific system or service, personas’ characteristics or demands depend on this context and should be used only in the context for which the respective personas were developed (Antle, 2008; Lewis & Contrino, 2016; Minichiello et al., 2018; Van Rooij, 2012). Personas are already used in tertiary STEM fields, for example, to sketch university teachers’ attitudes towards active learning (Guy, 2017) or redesign an online data visualisation platform for university teachers in STEM subjects (Vorvoreanu et al., 2016). Previous research indicates that persona development can become a useful approach for aligning the design of digital learning materials with characteristics and needs of students (Guy, 2017; Maness et al., 2008; Miaskiewicz et al., 2008; Vorvoreanu et al., 2016).

In the early stage of our study, we developed personas for secondary mathematics students for the first time (Weinhandl et al., 2022). The personas developed in our previous study, i.e. representatives of secondary mathematics student groups, are characterised in terms of their expressions in the categories Goals, Needs, Challenges & Problems, Enjoyment, Fears, and Feelings & Emotions, as well as in terms of their strategies for learning mathematics (see Fig. 1). To develop the personas, we collected in-depth information about various characteristics of mathematics students through a web survey that was sent to 56 in-service and 139 prospective mathematics teachers nationwide from urban as well as rural areas. The ratio of 5:1 between public and private schools corresponds to the overall situation in Austria. Therefore, the sample can be considered representative. Out of this sample, 13 in-service and 61 prospective mathematics teachers, in total 74, submitted a completed web questionnaire. We invited prospective mathematics teachers to complete the web survey because they are considered closer to their student selves but, at the same time, have already gained knowledge in teaching mathematics. The prospective mathematics teachers were asked to describe their former upper secondary school students’ selves and a former classmate who did not enjoy learning mathematics. In this way, we aimed to gain detailed information on both proficient and less proficient mathematics students. The data obtained from these teachers were used to develop the initial versions of the mathematics student personas. These initial versions of the mathematics student personas were then subjected to a validation and improvement process involving 83 mathematics school students, 18 in-service mathematics teachers, four prospective mathematics teachers, and three university staff. In total, five personas were developed and implemented in the study at hand.

Fig. 1
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One of the personas used in our study

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Teachers’ familiarity with their students’ characteristics from the above-mentioned categories is essential when designing mathematical tasks supported by technology, as well as any learning materials in general. Simon et al. (2004) suggest that students solving the same mathematical task may have different goals, and their achievements are greatly influenced by the design of the task itself. Teachers have a professional obligation to adopt “prefabricated” tasks to fit better the local curriculum and the needs and capabilities of their students (Johnson et al., 2017). Esjeholm & Bungum (2013) stated that while creating mathematical tasks supported by technology, it is important to take into consideration problems and challenges that students may face not only in mathematics but also with technology to ensure the understanding and solving of tasks by students. Ainley and Watson (2015) discuss the gap in mathematics teaching between the intentions of teachers as task designers and the experiences of students depending on their individual perceptions. In essence, learning processes are individual for all learners. Personas, as typical representatives of different groups of students, have the potential to address students more individually in mathematical task design and thus foster student learning and understanding of mathematics. In particular, prospective mathematics teachers often have little knowledge and experience of students’ needs or typical problems, using personas by them could be vital when developing digital learning resources.

Teacher training for the use of technologies in mathematics classes

Teachers’ basic education and professional teacher development (PTD) are fundamental to successfully implementing technologies in schools. Prospective mathematics teachers should be prepared to incorporate various computer-based activities into their teaching (Doğan, 2010). Therefore, training and PTD should be designed to enable experiences with technologies, including specific tools and software (Clark-Wilson & Hoyles, 2019b; Doğan, 2012). In the basic education and PTD of teachers, both the use of computers or new technologies and how these technologies could be used and implemented in mathematics classrooms should be addressed (Clark-Wilson & Hoyles, 2019a; Doğan, 2010). The technological tools used in teacher training should be professionally relevant for prospective mathematics teachers, which according to Otterbreit-Leftwich et al. (2012, cited in Gurevich et al., 2017) is often not the case. Out of the aspects mentioned, the focus on tasks and the implementation of one’s knowledge in teaching materials (Weinhandl & Lavicza, 2019) are central activities of the prospective mathematics teachers examined in our study. One of the main tasks of mathematics teachers is to design tasks that match their own pedagogical and epistemological perspectives and the needs and characteristics of their students (Leung & Bolite-Frant, 2015). Joubert (2017) emphasises that the training of mathematics teachers should include planned mathematical learning of students that is adapted to their characteristics and skills and a detailed understanding of digital teaching tools from learning and teaching perspectives. The same author states that such well-designed courses for teachers would help them to develop mathematical tasks according to characteristics of students with maximum use of digital teaching materials. Ratnayake et al. (2020) outline that teachers need continuous support in the form of professional development to design student-centred mathematical tasks reinforced by digital technologies successfully. According to Ratnayake et al. (2020), placing mathematics teachers in the role of task designers has the following advantages over pre-designed tasks: teacher-designed tasks are more sensitive to the way teachers’ own students learn, tasks would be based on their prior knowledge, understanding, and misconceptions. Results of Weinhandl et al. (2022) indicate that the persona concept can assist teachers in developing materials in line with essential characteristics of their students.

Mathematics digital task design knowledge (MDTDK)

The research presented in this paper focuses on prospective teachers’ digital task designs while using personas. According to Tapan (2003) and Leung (2017), for teachers to successfully design mathematical tasks supported by digital technologies, they should have Mathematics Digital Task Design Knowledge (MDTDK), which includes: Knowledge about the Digital Artefact Used, Digital Technological Knowledge, Mathematics Content Knowledge, and Pedagogical Knowledge. MDTDK was created by adapting TPACK (technological, pedagogical, and content knowledge) (Mishra & Koehler, 2006) to a mathematical educational context (Leung, 2017). Gulkilik (2020) implies that MDTDK can be used to design tasks with different cognitive demands using digital technologies in mathematics teaching, aligned with the characteristics of students. A mathematical community of practice can also influence MDTDK (Leung, 2017). The same author states that teachers with high MDTDK can successfully create digital learning and teaching materials. Getting to know the characteristics of students through the persona approach would provide teachers with the information they need to adapt mathematical teaching materials to the characteristics of students and thus would influence their MDTDK.

Teachers who demonstrate a high degree of knowledge and understanding in Technological Pedagogical Content Knowledge most often design purposeful, individualised, and conceptually focused tasks, based on good calling characteristics of their students (Muir et al., 2016; Bobis et al., 2020). It is precisely these knowledge areas, their combinations and the contextual knowledge needed by our study participants while developing digital resources and activities for mathematics learning. The aspects of the personas we researched are thus connected to MDTDK domains while developing digital learning resources for mathematics learning and teaching.

Hoyles and Noss (2009) implied that mathematical learning materials for students should be adapted to their social contexts and their characteristics arising from such contexts. This is in accordance with the concept of learning as a social process and thus within the theory of Communities of Practice (CoP) (Wenger, 1998). According to Wenger et al., (2002, p. 4), CoP can be described as “groups of people who share a concern, a set of problems, or a passion about a topic, and who deepen their knowledge and expertise in this area by interacting on an ongoing basis”. Kynigos et al. (2020) outline the three main characteristics of CoPs: (i) a shared domain that defines the CoP identity, (ii) a community of members who interact and partake in communal activities, and (iii) help each other, share information with each other, and a common practice. Inquiry communities among teachers, educators, and researchers of mathematics education, and thus mathematics teachers and educators who develop instructional digital materials, form CoPs (Voskoglou, 2019). Well-developed CoPs can contribute to the advancement of teachers’ skills, growth as professionals, and the improvement of their teaching practices (Trust & Horrocks, 2017). Accordingly, the existence of a CoP of mathematics teachers collectively participating in the development and improvement of students’ persona profiles would contribute to their ability to perceive the characteristics of different students and thus to better adapt digital learning materials to their students’ needs. In our study, CoP was integrated not as a theoretical frame but as the vital design of the project FLINK in which our study is integrated. Pre- and experienced in-service teachers, as well as teacher educators interacted in the design of digital learning materials, shared information, developed a joint vocabulary and shared practices to follow the principal aim of developing digital mathematical learning resources for a broad range of students.

In summary, our study is embedded in the technological change in mathematics classrooms. To cope with such changing conditions, we need, on the one hand, teacher training or PTD suitable to develop knowledge and competencies in line with the MDTDK framework, and on the other hand, digital materials that students and teachers can integrate into their mathematics lessons. In our study, prospective mathematics teachers develop digital materials for mathematics teaching according to the outlined aims. They work with personas to support them, putting themselves in the shoes of the potential users of materials they are developing. This work with personas and with the aspects that might be relevant are the subject of our investigation. The following section discusses the framework in which our study is embedded, as well as outlines the participants of our study.

Context of our study

To define the context of our study, we explain the project FLINK in which the prospective mathematics teachers are working on developing digital mathematics learning resources.

Our research within the project FLINK

FLINK is a project of the School of Education at the University of Linz. The project aims to accompany and support teachers and students using digital devices in secondary mathematics classes in Austria by developing high-quality digital and interactive teaching and learning resources. One characteristic of these materials is a division into discovery and practice areas. While materials for discovery offer opportunities for interactive exploration of new mathematical content, materials for practice aim at repeating and linking the learned content through many interactive tasks (FLINK Team, 2021; Weinhandl et al., 2021).

In addition to experienced teachers, on average thirteen explicitly selected prospective mathematics teachers have been working on the project FLINK since July 2021. These students were considered for the project based on positive feedback and evaluations from lecturers and university staff. The prospective teachers’ roles in the project are divided into two parts. One group is working on developing resources from didactical and pedagogical perspectives. The other group is in charge of realising the didactical and pedagogical plans of the didactical group digitally. There is a clear predominance of female prospective mathematics teachers in our study. Prospective mathematics teachers were 20 to 27 years old and were mainly at the end of their Bachelor’s degrees or the beginning of their Master’s degree. The four prospective mathematics teachers who were working on developing the materials from a didactic perspective from the beginning of the project were selected to work with the personas and to participate in our study.

Timeline of our study

The creation of digital teaching materials within the project FLINK started in July 2021. At the beginning of October of the same year, the four selected prospective mathematics teachers had their first contact with the developed personas and were asked to consider them while designing their learning materials. They had a month to get to know the personas and engage with them before they were interviewed about their uses for the first time. Almost two months later, the second round of interviews were carried out, whereby the prospective mathematics teachers had already created or assigned concrete learning activities for the personas. The third and final round of interviews, which focused on the prospective mathematics teachers’ more intensive engagement with two personas each, took place after two months working with these study elements. In the third round, the prospective mathematics students had to develop and assign concrete learning activities to specific personas.

During the entire timespan of the study, more than 60 digital mathematics learning resources were designed or revised by the four prospective mathematics teachers, considering the characteristics and needs of personas. As the material development process in the project FLINK is rather complex and lengthy due to multiple reviews and revisions, it cannot be determined precisely how many digital mathematics learning resources were designed or worked on during that time. The prospective mathematics teachers designed the material collections within GeoGebra Books, which included mainly GeoGebra applets and activities on http://LearningApps.org.

Two of these learning resources are presented below (translated into English). The learning resources were developed for Grade 1 secondary school students and are thus aimed at students aged 10 to 11 years. The first learning resource, shown in Fig. 2a, is specifically designed for a persona who needs, among other things, step-by-step learning instruction and who also has motivational problems. By creating a real-world connection (Fig. 2a left), the learner’s motivation should be increased, and a step-by-step explanation of the problem (by clicking on “Next” in Fig. 2a right), students with mathematical learning difficulties should be supported by presenting the problem-solving process in small steps. The students can then perform a similar calculation—again step by step. Figure 2b shows two learning resources for the same content that differ in difficulty—Fig. 2b on the left shows a standard problem in line with the curriculum, and Fig. 2b on the right shows a task primarily intended for interested and gifted mathematics students.

Fig. 2
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a Learning resource “How tall will I become one day?” about text equations (simple equations). The learning resource is only shown in part here, the full version is available at: https://www.geogebra.org/m/dqcdrcmx. b Learning resources “Decimal number square—Level 1” and “Decimal number square—Level 2”. The full versions are available at: https://www.geogebra.org/m/e3rvx5km, https://www.geogebra.org/m/nu675kkw

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Methodological framework

In terms of methodological approaches, we combined case study, grounded theory, and qualitative interview study approaches in combination with Mathematics Digital Task Design Knowledge (MDTDK) and CoP in our research. As research tools and data sources, we used qualitative interviews, thinking-out-loud techniques, and learning resources of prospective mathematics teachers and analysed them according to the methodological approaches above.

A case study approach to explore the use of personas

We used the techniques and principles of case study research. Case studies investigate a clearly defined limited system consisting of real people in real situations when specified interventions occur in this system (Cohen et al., 2007). The defined bounded system of our study consists of prospective mathematics teachers who develop digital mathematics learning resources as part of the FLINK project. Our intervention included prospective mathematics teachers’ use of personas when developing digital mathematics learning resources. As our intervention aimed at discovering something currently unknown in the limited system of our study, according to Yin (1984), our case study can be characterised as an exploratory case study.

Methodological approaches

Within our case study, we employed grounded theory approaches (GTA), qualitative interview study approaches (QIS), and qualitative content analysis (QCA) to collect and analyse data, and to develop our research findings. Regarding grounded theory, our research can be categorised as grounded theory in line with Strauss and Corbin (Khan, 2014) and constructivist GTA (Charmaz, 2006). Following Khan (2014), it is typical for GTA in line with Strauss and Corbin (1997), that prior knowledge of the researcher and the current body of knowledge are included in the theory building. According to Charmaz (2006), a constructivist interpretation of GTA means that the theory developed in the course of the research is grounded in the collected data and developed from the data by the researchers; i.e. dependent on researchers’ perspectives. In our research, both the development of the clearly defined limited system, the intervention taking place within it and the data analyses integrated the prior knowledge of researchers and the current findings from the literature. On the one hand, the authors’ prior knowledge and scientific findings were used to design the prospective teacher training and resource development settings. On the other hand, the authors’ scientific experiences and recent research literature played a significant role in collecting and analysing the data and developing findings. These features of our research illustrate that our study followed a GTA, according to Strauss and Corbin (Khan, 2014) and constructivist GTA (Charmaz, 2006).

Data collection and analyses

During the data collection, we followed the principle of theoretical sampling, which is one of the special features of GTA. According to theoretical sampling, new data are always collected in line with the latest state of research. In this process, the types of data or the tools for collecting data should always depend on what is needed for the next step of developing the results (Charmaz, 2006; Glaser & Strauss, 1999; Strauss & Corbin, 1997). To identify aspects of personas relevant for prospective mathematics teachers while developing digital mathematics learning resources, we conducted qualitative interviews, analysed developed digital learning resources, and used the principles of thinking-out-loud.

For data analysis, we used techniques from GTA and QIS. The detailed process of data collection and processing in individual steps is shown in Fig. 3.

Fig. 3
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Data collection and processing in our research

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By using the summary technique by Mayring (2015), it should be possible to reduce the extent of the data and leave the essential content of the data unchanged. Following the principles of QCA, according to Kuckartz (2019), categories, associated subcategories and a category system were formed in this process. The following section explains the research procedure in the individual phases of our study and the associated development of our category system.

First phase of the analysis and the development of results

In the first phase, all five personas created for the purpose of the study were presented to the prospective mathematics teachers, who were then given the task of using the personas when developing resources. They were also asked to take notes concentrating on the elements of personas they found particularly helpful. The main purpose of this phase was to familiarise prospective mathematics teachers with the uses of personas. At the end of the first phase, which lasted for a month, qualitative interviews were conducted with all prospective mathematics teachers. Prototypical questions in the first interview were “How did you use the personas in developing and revising materials?” or “How did the personas change your approach to material development and revision?”. All questions from the first interviews and all further interview guidelines can be found in supplementary material 1. Interviews were then transcribed and analysed according to the techniques of the above section. This made it possible to develop the first individual codes of the first four authors (see Table 1) and codes of a higher level of abstraction (see Table 2). All individual codes, codes of a higher level of abstraction, and categories and associated subcategories (from the second phase onwards) can be found in the codebook (supplementary material 2).

Table 1 First individual codes
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Table 2 Higher level of abstraction codes
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Second phase of the analysis and the development of results

In the second phase, prospective mathematics teachers were given the task of mapping developed digital mathematics learning resources to one or more personas. For this mapping process, they were provided with a table. In this table, the prospective mathematics teachers had to document the digital mathematics learning resources that were assigned to particular personas and to the elements of the learning resource, as well as to the categories of the personas that were essential for this assignment.

Prospective mathematics teachers had two months time for this process. At the end of the second phase, qualitative interviews were conducted again, and thinking-out-loud techniques were used in the joint analysis of the digital mathematics learning resources assigned to the personas. Typical guiding questions of the second interview cycle were “What information about the personas did you pay particular attention to when assigning the digital mathematics learning resources?” or “At which point in the process of material development do you see the most profitable application possibilities of the personas?” (see supplementary material 1). During the thinking-out-loud, prospective mathematics teachers were asked to show the selected digital mathematics learning resources and to explain which elements of the digital mathematics learning resources were crucial so that they could be assigned to respective personas. The thinking-out-loud process was conducted using an online conferencing tool so that prospective mathematics teachers could access the digital learning resources and show and explain them. The thinking-out-loud was audio-visually recorded, and screenshots of the digital mathematics learning resource presentations were included in the transcripts where appropriate. For the data analysis, GTA and QIS techniques were used and, for the first time, the principles of QCA were employed, especially for the development of subcategories and summaries. While analysing the data collected in phase two, using the techniques of GTA and QIS, the first four authors used the common codebook from the end of phase one as a starting point. The first four authors developed different codebooks by individually coding the phase two data. Using QCA techniques, i.e. developing subcategories and summarising, the different codebooks were merged into one codebook and the first subcategories were developed into preliminary categories. The preliminary categories and associated subcategories were documented in the codebook of our research (supplementary material 2) and are presented as examples in Table 3.

Table 3 Preliminary categories and associated subcategories from Phase 2
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Third phase of the analysis and the development of results

In the third phase of our research, each prospective mathematics teacher was assigned two personas and asked to create a digital mathematics learning resource or a collection of digital mathematics worksheets (i.e. GeoGebra book) for each of these personas. Again, the prospective mathematics teachers were asked to take notes on which elements of the learning resources were adapted to specific categories of the personas. Prospective mathematics teachers had two months to complete this process. Similar to the data collection in the second phase, qualitative interviews were conducted with prospective mathematics teachers in the third phase and thinking-out-loud techniques were used to collect data. In the third phase of data collection, at least two authors were always present during the interviews to be able to document as diverse information as possible; the third phase also proceeded in the same way as the second phase concerning the other technical aspects of data collection. Typical guiding questions of the third data collection cycle were: “Which aspects of the activities (GeoGebra book) have been specifically developed or adapted for the corresponding personas?” or “What information about the personas do you look at first when developing the materials? Why?”. The full set of guiding questions can again be found in supplementary material 1. While analysing the data, the techniques of GTA and QIS, as well as the principles of QCA, were utilised. Through employing GTA and QIS techniques, the codebook of our research was expanded. Analysing the data in the third phase differs from the second phase as there was a stronger focus on QCA techniques. Through the technique of summarising, it was possible to reduce the gathered information to a manageable size. Central categories and associated subcategories were formed parallel to reducing the abundance of information by using summarising techniques. Exemplary categories and associated subcategories are presented in Table 4; all categories and associated subcategories can be found in the codebook (supplementary material 2). Through this process, the first system of categories could be developed in our study.

Table 4 Exemplary categories and associated subcategories from Phase 3
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Fourth phase of the analysis and the development of results

In the fourth phase of our study, the existing data were further analysed, and an attempt was made to raise the already developed categories and associated subcategories to a higher level of generalisability. A higher level of generalisability was to be achieved in the techniques of selective coding (GTA), reviewing the themes after thematic analysis (QIS) and summarising (QCA). In the first step, the already developed categories were condensed by summarising them according to QCA. The first four authors grouped those categories with similar definitions and similar passages in the transcripts and then gave a new keyword, a definition of the categories and a prototypical passage in the transcripts. Then, the interdependencies and influences of the newly developed categories were elaborated by selective coding according to GTA. This elaboration of interdependencies and influences was carried out both between and within the categories (i.e. between the subcategories). This allowed us to develop the core categories of our study, which are described in detail in the Results section.

Validation processes in our study

To validate the themes of our study, i.e. the core categories, and thus to keep the quality of results developed in our study high, we followed the recommendations of Jonsen and Jehn (2009). They suggested that qualitative and explorative studies use mixed methods, different sources and different coders. This use of mixed methods, different sources and coders should help to balance bias and increase the quality of the results. In our study, the recommendations of Jonsen and Jehn (2009) were implemented in that a mix of grounded theory approaches, qualitative interview studies and qualitative content analysis was conducted. Similarly, different sources were used to develop our results, namely (a) interview data, (b) data from thinking-out-loud processes, and (c) data from developed digital mathematics learning resources. During the interview and thinking-out-loud data collection, different authors carried out the data collection process or the qualitative interviews, which should also contribute to a lower bias within the data and to a higher data quality. Data collected in the study were analysed individually by four authors, then compared and summarised similar results to improve reliability. To ensure intercoder reliability and codes’ ordination into categories and subcategories in our research, we used Cohen’s Kappa (Cohen, 1960), and Miles and Huberman’s (1994) formula. In this procedure, reliability was first calculated between the members of the research team who performed the coding process and then among the members of the research from one side and external experts on the second side. Two external experts in qualitative research were asked to arrange the codes into subcategories and categories. The Cohen’s Kappa in our study has a value of 0.8912 between the members of the research team and 0.8104 between the members of the research team and external experts. The correspondence between the members of the research team in the ordination of codes to the subcategories and categories was 89.8%, and between them and external experts 81.1%. These data indicate that the results obtained in this research can be considered reliable.

Through this constant and rigorous quality control, the biases in our study should be low, and results presented in the upcoming section should be valid.

Results

This study explores those aspects of personas that might be relevant to prospective mathematics teachers in developing digital mathematics learning resources. Based on the collected and analysed data, the following aspects could be identified: (A) personas as representatives of real people, (B) personas as planning & feedback tools for material development, (C) professionalisation of prospective mathematics teachers (by using personas), (D) differentiation/individualisation for personas through digital learning resources, and (E) motivational elements of digital mathematics learning resources. These results of our study are to be explained in detail in the following section and supported with quotes from the interviews. We translated the quotes into English as interviews were conducted in German. Furthermore, we used pseudonyms and added a number in round brackets to the quotations to indicate from which interview round the quotation originated in order to illustrate how much experience the interviewees already had with working with personas at the time of the quotation.

Personas as representatives of real people

First of all, it emerges from the data that the prospective mathematics teachers perceived personas as representatives of real people. Recognising the multidimensionality of prototypical learners plays a central role here. In particular, it is a matter of perceiving a young person in several dimensions through the work with personas and recognising them first as an adolescent, then as a student and subsequently quite specifically as a mathematics student. According to their feedback, seeing a young person as an adolescent is about considering general information about them, such as gender and age. The extracurricular interests and hobbies, such as sports, music or art, also seem worthy of attention. If the young person is viewed from a school perspective, motivation, interest, competitiveness, performance and ability play an essential role according to the responses of prospective mathematics teachers. When observing students from a mathematical or, more precisely, school-mathematical perspective, the main focus seems to be on the areas of motivation, interest, competitiveness, performance, and ability. However, these areas are specifically related to the subject of mathematics. The prospective mathematics teachers know that there are also intersections between these three dimensions of young people, characterising multidimensionality.

Flora (1): [...] that it is interesting - not only related to mathematics - what the main interests of the students would be, for example. Sometimes the interests lie in something completely different.


Furthermore, responses from the prospective mathematics teachers showed that they recognise and question stereotypes through their work with personas. It seems particularly relevant here that the prospective mathematics teachers become aware of different stereotypes through their work with personas and are willing and able to communicate these concepts.

Olivia (2): In the practical training [in school during university] I think I did this frequently, that I thought of three types: the person who is very motivated, something in the middle and a person who needs small-step assistance. I think the biggest learning effect was that there is a lot in between.

In contrast to recognising stereotypes, a renewed formation of stereotypes was also observed. According to the responses of prospective mathematics teachers, personas can lead to stereotyping insofar as it leads to a categorisation into certain types of students.

Antonia (2): [...] I now see or am reminded more often that there are quite a lot of different types of students who have different needs.


Moreover, it was found in the data that working with personas could lead to personification. In our data, this personification led to a coding or technical language within the community. Personas seemed to become codes or abbreviations for certain characteristics, thus carriers of implicit knowledge. On the one hand, according to the data, this led to the holistic and personal interpretation that there is a “real” person behind each persona. The portrait on the personas sheets also played a vital role in reminding prospective mathematics teachers that it represents a “real” person. On the other hand, the data showed that personification goes hand in hand with limiting personas to certain characteristics. This means that personas are potentially no longer seen by prospective mathematics teachers as the sum of all the items described but were reduced to certain characteristics.

Antonia (3): This [learning material] is for Johannes [Persona], because it contains mathematics for particularly interested students.


According to the feedback, recognising the multiple dimensions and their interconnections of prototypical students, as well as the personification of personas, thus led to a perception of the personas as representatives of real persons. While working with personas reveals and questions stereotypes, on the one hand, personification particularly could lead to a reduction of the personas—and as representatives of real persons, of students—to specifically salient characteristics.

Personas as a planning & feedback tool for material development

The data showed that personas can be used as a planning and feedback tool for developing learning materials. On the one hand, according to responses of the prospective mathematics teachers, personas can be used for planning a learning module for specific curriculum content. They describe the use of personas in planning as particularly helpful when considered throughout the entire process.

Flora (3): I think it’s best to do this [use of personas in material development] at the very beginning. So, if you [the material creator] make a book [collection of materials] from scratch and think at the beginning: What would I like to have included? That you [the material creator] think about it right away: there are different types of students and I make sure that there is something for everyone. I think that’s the most reasonable variant.


Corresponding to the prospective mathematics teachers’ feedback, especially at the beginning of the planning process, a look at the personas was considered useful. Within such a learning module, different activities could be planned by the prospective mathematics teachers with the help of the personas insofar as all or at least many needs of the different personas are covered as far as possible.

On the other hand, the data showed that personas can also be used as feedback for the developed materials or during their revisions. Personas can be used to check whether the goals of the planning have been achieved and whether the planned materials meet as many needs of the different personas as possible. Furthermore, during the interviews, it became clear that it can be determined whether or not aspects may be missing, and corresponding activities can be added subsequently.

Nora (2): [...] we had a book [collection of materials] that we now have to adapt, and there were many tasks in it that were relatively difficult, and we made sure to divide them into levels so that we can adapt it [the level of difficulty] for the different personas.


For the prospective mathematics teachers in our study, it was crucial that within material development personas can thus be helpful both in the planning of a new learning module and in the sense of a feedback tool to give feedback on the one hand, about the achievement of the planned goals, and on the other hand, about the suitability for the target group.

Professionalisation of prospective mathematics teachers (by using personas)

The data also highlighted that using personas in creating materials could lead to the professionalisation of prospective mathematics teachers. The following three aspects seem to be particularly important considering their professionalisation when working with personas:

Firstly, the data suggested that using personas could create professional authority and distance among prospective mathematics teachers. Professional authority here means that, on the one hand, the competence to be a “bad cop” as a teacher and thus to be able to ignore some areas such as the interests of students, one’s interests as a student or the need to be liked by the students if the situation requires it.

Antonia (1): But I do think that it is important that there are different levels [in a collection of materials], that you [the material developer] maybe also really create difficult ones [tasks], where you [pupils] really have to sit down and think: “How can I solve this now?”.


On the other hand, the knowledge of legal and administrative frameworks and the knowledge of duties as a teacher also play an important role in the development of professional authority.

The data demonstrated that building a professional distance may refer to recognising heterogeneity within the student body through perceiving the different personas, the high number of personas and also the diversity within the personas and creating a distance both to one’s own former student-self and to other students.

Olivia (2): [...] that the person [persona] wants to get away with the minimum of effort, that is something I don’t know from personal experience and what I also find difficult to understand. I think the personas have made it a bit clearer to me that I have to have a broader view than my own.

Antonia (1): I think all [personas] are important because they represent very different types and I think it’s good to be reminded that maybe some [students] are not so dedicated [to mathematics] or that mathematics is not as important to them [as it is to me]. I really like mathematics and I think it’s good that I am reminded that there are others who don’t see mathematics as important.


Secondly, according to the feedback from the interviews, the personal development needs of prospective mathematics teachers can be revealed and identified. The development needs can be of a mathematical-didactical as well as a solely mathematical nature. Concerning mathematical-didactical development needs, gaps in the knowledge about different approaches to curriculum content or typical student errors can be revealed through the responses of the prospective mathematics teachers. According to the feedback, mathematical development needs can be identified when creating the material, for example, if difficulties arise in the simple explanation of content.

Olivia (2): Well, for us, for me personally, it was difficult, for example, the subtraction algorithm or the addition algorithm, that I prepare such things in a way that they [students] know why it [the algorithm] works. But at the same time, it [the material] is easy to understand. I had the feeling that we didn’t really address this at university, or only briefly. Yes, that was somehow completely new.


The data showed that by working with personas, personal gaps with varying degrees of urgency, depending on their relevance to the current activity, can be uncovered by the prospective mathematics teachers and, if necessary, closed.

Thirdly, working with personas can also unveil the need for developing these personas. In this regard, it was noted during the interviews that additional information on personas, such as learning preferences, information on preferred social forms, and contextual information on the teaching and learning situation (e.g., tablet or laptop), would assist prospective mathematics teachers.

Flora (3): What I think would be useful would be maybe teaching methods or learning settings or something like that that a persona likes. How they prefer group work, or whether they like frontal teaching or free work phases.


In summary, working with personas could lead to the professionalisation of prospective mathematics teachers, on the one hand, through the emergence of professional authority and distance to the student and the former student-self, and on the other hand, through the recognition of the need for the development of one’s own mathematical-didactic and mathematical expertise as well as of the personas themselves.

Differentiation/individualisation for personas through digital learning resources

The terms differentiation and individualisation were continuously used with similar meanings. For this reason, we now use the terms synonymously. Individualising for the five personas has a differentiating effect in that each persona stands for many students and groups them into a cluster.

Through the data collected, it can be shown that differentiation/individualisation for personas can be achieved through digital learning resources in two dimensions: differentiation in-depth and differentiation in breadth.

According to the responses, differentiation in depth means that it occurs within a material or a collection of materials. Based on the collected data, it can be determined that differentiation within a collection of materials usually occurs through the diversity of information. Basic information is presented in the materials, which is the same for all students. However, additional small-step explanations are often added adaptively and automatically if a student needs further explanations for working on the tasks. In some cases, students can also decide whether they want to follow the small-step path with additional explanations.

Olivia (2): [...] for Diana [persona] the FLINK material is often suitable because my interpretation was: she wants to be able to repeat [what she learnt] and review it again slowly. For me, here the differentiation aspect was always very strong, and I see it strongly in the use of technology, in that I [as a student] can do the applets [digital learning activity] again and again and at my own pace and get help, hints and solutions.


Another possibility for differentiation within a collection of materials is differentiation by difficulty. This can either be automated by computers, or students can choose tasks with different difficulty levels, which are marked transparently.

Flora (1): There are basic tasks and then more difficult, Level 2 tasks - for example, for the students who already know the basics well and want to know more in addition.

If, on the other hand, differentiation occurs across the board, this describes a differentiation between different materials or collections of materials. The data also showed that it makes sense to offer different learning packages that train similar competencies so that students can ultimately reach the same goal in different ways, i.e. the same competencies. During the prospective mathematics teachers’ interviews, it became clear that it is important for them to recognise the plurality of preferred task formats and social forms and to offer as many different formats as possible; e.g., visual vs auditory, active vs inactive, open vs closed, discovery-based vs guided.

Nora (1): For most of the personas, it is mentioned how much they like mathematics or how much effort they want to put into the mathematics work. This was important because we then developed some materials for interested students who want to invest more time and effort and some for students who want to get to the goal [learning goal] faster and invest less effort.


Overall, differentiation in material development can therefore take place in two dimensions: it is possible to differentiate in depth by differentiating within a material package through adaptive adjustment of the information presentation and selection of different levels of difficulty. Differentiation in breadth is also possible by offering different learning packages that train the same competencies.

Motivational elements of digital mathematics learning resources

As described earlier, recognising the different student preferences for task formats and social forms are closely related to the motivational elements of digital mathematics learning resources. The data revealed that the different learning approaches can have a significant impact on students’ motivation. Prospective mathematics teachers understand motivational learning approaches in particular approaches such as gamification, collaborative learning, and discovery-based learning.

The data collected also suggested that the prospective mathematics teachers interviewed recognised the plurality of preferred learning actions of the different personas and students. For example, while some students prefer to perform calculations according to clearly prescribed schemes, others have a need to discover something new. According to the responses of the prospective mathematics teachers, perceiving and responding to these preferences can be understood as a motivational element.

Flora (2): What I have noticed recently is that I have realised that I have to respond to different interests, also in terms of design. That not all students have the same interests as me or our team, for example, but that you [the material creator] also have to incorporate different things. Maybe something that’s a bit cooler or something like that.


The data highlighted that reward can be another key element in motivating students in a digital environment. In terms of reward, it is important for the prospective mathematics teachers in our study first to recognise that different personas may have different needs regarding reward. Subsequently, it can be considered that there are different reward systems and that these can be implemented in the learning materials.

Antonia (2): For Aurelia [Persona], I think it is also mentioned that she wants a reward or something like that, or just positive feedback. I think it’s easy [to implement] when you [the material creator/the teacher] do little challenges or you [the student] compete against someone else in mental arithmetic, for example, and Aurelia [Persona] can then also show what she can do and then get positive feedback.


Another motivational element emerged from the data was the fun factor. It was considered essential by the prospective mathematics teachers that students should enjoy learning and have fun with mathematics. Corresponding to the prospective mathematics teachers’ feedback, this can again be made possible, for example, through the use of playful elements or challenges.

Flora (2): What stood out for me was the motivation factor, that he [Manuel, persona] has a bit of a hard time and that maybe things [learning materials] can help with motivation that are a bit more playful or fun.


Across personas, it was assumed by the prospective mathematics teachers that visual, technical as well as digital implementation, where appropriate, can have a positive impact on students’ motivation by making learning materials visually appealing and interactive.

Flora (3): For example, we have a maths snake. Since Manuel [persona] often has a bit of a motivation problem, something like this [motivating learning material] can perhaps have a positive effect.


In general, a distinction between the two types of motivating elements could be uncovered. On the one hand, motivation can be thought of from the bottom up, for example, through the targeted introduction of interactive elements, challenges or tasks tailored to the preferred learning actions. On the other hand, motivational elements can also be integrated across the material collections, independent of topic and personas. These include, in particular, a visually appealing design or visual feedback.

Discussion

The research presented aims to shed light on developing digital teaching materials for mathematics using personas by prospective mathematics teachers. It was found that (A) personas as representatives of real people, (B) personas as planning & feedback tools for material development, (C) professionalisation of prospective mathematics teachers (by using personas), (D) differentiation/individualisation for personas through digital learning resources, and (E) motivational elements of digital mathematics learning resources are central for prospective mathematics teachers in developing digital learning resources when using personas.

Our study shows that a holistic view of students is of central importance. This involves perceiving young people in three dimensions: as adolescents, as students, and as mathematics students. Essential to developing such a holistic view of students for prospective mathematics teachers is information about the personas that goes beyond the information important for mathematics teaching, i.e. background information and hobbies of the prototypical learners. This is an extension of Lewis and Contrino’s (2016) account that personas should only be used in the context for which they were developed. Context is also relevant when using personas in our study, but it shows that non-context-specific elements such as background and hobbies can also be relevant. This is in line with Mishra’s (2019) extended TPACK model highlighting the importance of contextual knowledge, which goes beyond the subject and the classroom. Our findings about the benefits of the personas approach can be of great use to designers of professional teacher development for mathematics teachers who face obstacles when adapting mathematical digital materials to the characteristics of students (Cevikbas & Kaiser, 2021). This is especially important considering that according to Leung and Baccaglini-Frank (2017) and Traglová et al. (2018), the alignment of mathematical digital teaching tools with the characteristics of students can affect their achievement in learning mathematics. Whereas teacher education and training should be designed to offer direct experiences with technologies, including specific tools and software (Clark-Wilson & Hoyles, 2019b; Doğan, 2012), our results show that when developing digital mathematics learning resources, experiences with personas in PTD can also be significant for the use of technology in teaching (e.g., considering students’ extra-mathematical interests).

Our study showed that using personas while developing digital learning resources can be connected to the MDTDK framework (Leung, 2017) in mathematics education: (A) personas as representatives of real people are relevant for pedagogical knowledge, (B) personas as planning & feedback tools for material development could be relevant for mathematical knowledge for teaching as well as for knowledge about the digital artefact used, (C) professionalisation of prospective mathematics teachers (by using personas) is related to the development of their MDTDK, (D) differentiation/individualisation for personas through digital learning resources is related to mathematical knowledge for teaching(differentiation) and digital technological knowledge (realisation of differentiation/individualisation with the help of technologies) and further fosters knowledge about the use of digital artefacts, and (E) motivational elements of digital mathematics learning resources could be related to mathematical knowledge for teaching.

Results of our study indicated that by working with personas, personification may occur (?). This results in technical language or coding within the community and suggests that using personas could contribute to the creation of a community of practice as summarised by Voskoglou (2019). Since the creation of a technical language takes place through interaction over a longer period of time, personification can be attributed to the characteristic of the community as well as to the characteristic of forming a common practice. According to Kynigos et al. (2020), the members of CoP may help each other, and share information with each other, and develop a common practice. These data may be of particular importance for policy makers and curriculum developers in the field of mathematics education, considering that Leung (2017) believes that CoP can influence MDTDK. In the context of our study, this indicates that the formation of a CoP of mathematics teachers for resource development can potentially improve the detailed description of personas and contribute to a better adaptation of mathematical teaching materials to the characteristics of students. These findings suggest that future research should explore this assumption and provide future information.

It was also found that prospective mathematics teachers can better empathise with their potential students while working with personas. This suggests that the persona approach can be used successfully with MDTDK indicating that teachers should know the Mathematical Digital Artefact Used and how to adapt them to the characteristics of students (Leung, 2017). According to our results, personas can serve as representatives of real people and provide data on extracurricular interests and hobbies, such as sports or music, which can be important for teachers when adapting teaching materials to the characteristics of students, which is in line with Pedersen et al. (2021) and Leung (2017) main feature MDTDK.

Results of numerous research (e.g., Gurevich et al., 2017; Weinhandl & Lavicza, 2019) recommended that teacher training for using technology in mathematics education should work on and with concrete materials, relate them to practise and thereby assess one’s competencies concerning developing and using digital mathematics learning resources and the quality of the resources. To conduct assessments of digital activities, prospective mathematics teachers need a tool or standard to support them in this process. In our study, it became clear that personas could be such an assessment and feedback tool. Results of our study indicate that personas could therefore be a tool that allows prospective mathematics teachers to assess their digital activities and thus also their digital teaching competencies in line with MDTDK. The assessment of one’s own competencies goes hand in hand with another result of our study, namely that the personal development needs of prospective mathematics teachers can be identified when developing digital mathematics learning resources from a personas perspective. These data obtained in our research can be of particular importance if it is having in mind that according to Ratnayake et al. (2020) and Clark-Wilson et al. (2020), depending on their primarily digital competencies, mathematics teachers would make decisions on adopting digital teaching materials or using their students’ characteristics or use them in existing forms.

For the prospective mathematics teachers in our study, the aspect of differentiation and individualisation could be important when creating digital learning resources with the assistance of personas. Our data showed that the prospective mathematics teachers strongly attach the differentiation aspect to the use of technologies. This is in line with the results of Weinhandl et al. (2021), in which mathematics teachers expect improved individualisation and differentiation with technologies or digital devices.

The described aspects of improved differentiation and individualisation, which prospective mathematics teachers associate with using technologies, can be assumed that the advantages of utilising technologies are recognisable by using personas when developing digital resources. This is confirmed by Weinhandl and Lavicza (2019) and Gurevich et al. (2017), who suggested that the benefits of using technology are primarily recognised by working with specific teaching materials and by working in practice. Working with personas can possibly also be seen in part as engaging practice, as practice includes students and their needs.

Conclusions and implications

The development of personas could contribute to the field of mathematics teacher education from several different perspectives. Well-developed personas could lead to substantial improvements in teacher education. Through this, prospective teachers could develop skills to target specific groups with more suited learning resources and pedagogical approaches to students’ needs. All of these should lead to the improvement of differentiated mathematical digital learning resources. In addition, from the methodological and theoretical perspectives, we believe that our study utilised a widely applied methodological approach from UX research and adopted it for educational research purposes. Thus, the adopted UX research methodology assisted us in developing our mathematics student personas that we utilised as methods to develop digital learning resources and assist prospective teachers. Our results showed promising outcomes, and based on this, we aim to improve further and theorise our approaches with personas in mathematics education. To do this, we will collect additional quantitative and qualitative data directly from secondary school mathematics students to evaluate our theoretical assumptions.