1 Introduction

The widespread availability of digital educational technologies and resources will increasingly influence mathematics teaching and learning in the coming years. How will this growing influence affect teachers and students of mathematics? Which digital and analogue resources will be used in the classroom? How will the use of digital technology affect the theory of mathematics education and the implementation of ideas in the classroom? And finally, what are the consequences for ongoing and final classroom assessments? These questions have been discussed extensively over the last decades in diverse mathematics education journals (see especially ZDM issues 55(1), 2023; 52(7), 2020; 52(5), 2020; 49(5), 2017; 44(6), 2012; 42(7), 2010; 41(4), 2009; 34(3), 2002) and at many conferences, especially the biennial Congress of the European Society for Research in Mathematics Education (CERME), which began in 1999. In 2018, a group of participants who were deeply involved in the CERME Thematic Working Groups on Technology took the initiative to launch the ERME Topic Conference for Mathematics Education in the Digital Age (MEDA). This conference sought to expand discussion of theoretical and practical knowledge and experience in the context of digital technologies and to look ahead at how mathematics education ought to be shaped in the coming years. The articles in this issue are significantly influenced by the three MEDA conferences and in particular are based on papers given at the 2022 MEDA3 conference held in Nitra, Slovakia (Weigand et al., 2022).

We begin by explaining the terminology we use in this article, followed by our methodological considerations. The literature uses various terms and expressions to refer to digital devices and digital content for mathematics teaching and learning, the most common of which are digital tools, digital technology, digital resources, and digital media. Because these terms are often used interchangeably, here we discuss each of them briefly to support our decision regarding the terms we use in this paper. Monaghan et al. (2016) broadly define a tool as “something you use to do something” (p. 5), thus distinguishing a tool from an artefact, defined as an object (material or symbolic) made by humans for a specific purpose. According to these authors, “an artefact becomes a tool when it is used by an agent, usually a person, to do something” (ibid., p. 6). Hence, according to the instrumental approach a tool is called an instrument (Rabardel, 2002) in that it cannot be conceived without considering a user and a purpose. The term digital technology usually refers both to tangible devices such as computers or tablets and to intangible means such as software. In the previous conceptualization, digital technology was considered an artefact or a tool when someone used it to achieve a given goal. In the last decade, researchers have begun referring to resources available for teachers and students rather than to technology: “Resources might be software, computers, interactive white-boards, online resources, but also traditional geometry tools and textbooks” (Trgalová et al., 2018, p. 146). The term resource was conceptualized by Adler (2000), who suggested thinking of it “as the verb re-source, to source again or differently” (p. 207). This perspective emphasizes the user-resource relationship in which the user’s activity is nourished by the resource and, in turn, the resource can be modified and adapted by the user. Yet another perspective views mathematical knowledge as mediated by artefacts (Vygotsky, 1978). To stress the mediating role of a digital artefact, some researchers use the term digital media. Borba et al. (2023) suggest considering media in a broad sense; for them “computers, videos, paper-and-pencil, regular classrooms, homes, and libraries are examples of media” (p. 2). According to the theoretical construct of “humans-with-media”, digital technologies are thought “to be coparticipants in the production of mathematical knowledge” (ibid., p. 3).

In this paper, we use the term digital technology to refer to hardware or software. Furthermore, in referring to other objects used in mathematics education, such as videos, e-textbooks, and interactive whiteboards, we have chosen to adopt the term resources (whether digital or non-digital) that is widely used in mathematics education research.

After defining our terminology, we can now explain our methodological approach to the topics of teaching, learning, and assessment in the digital age. Like any survey, this survey paper must set the boundaries of what to retain and what to omit. In setting these boundaries, we have adopted two guiding principles. On the one hand, we have attempted to avoid repeating information from recent publications. One example is the chapter on “Technology and Resources in Mathematics Education” (Trgalová et al., 2018) in the book Developing Research in Mathematics Education (Dreyfus et al., 2018), released to celebrate the 20th anniversary of the European Society for Research in Mathematics Education (ERME) and its tenth congress (CERME). The chapter reviews the last 20 years of technology use in research and practice, describing the development of technology and resources and the design and implementation of digital mathematical tasks and theories and approaches concerning technology and resources. Another example is the book Mathematics Education in the Digital Age (Clark-Wilson et al., 2021), which emerged from the intense communication and collaboration during and after the MEDA1 conference. Finally, the recently published handbook on Digital Resources in Mathematics Education (Pepin et al., 2023) “presents the state-of-the art scholarship on theoretical frames, mathematical content, learning environments, pedagogic practices, teacher professional learning, and policy issues related to the development and use of digital resources in mathematics education” (p. 2). On the other hand, we have sought to be sensitive to the contributions appearing in this special issue. These two considerations reflect our view of the purpose of this paper: to position the contributions of the special issue in the broader context of the field. We categorised these contributions into three topics inspired by the recent MEDA3 conference: teaching, learning, and assessment with technology. These contributions propose new theoretical developments for tackling emergent issues and employ a variety of research methods, leading to further methodological elaborations: case studies and other qualitative methods; quantitative studies; and mixed methods. In the survey we have also taken into consideration publications from the last five years in journal papers and at leading conferences such as CERME, ICTMT, and MEDA.

Despite highlighting these three main aspects—teaching, learning, and assessment with technology—we are aware of other fields of interest related to technology in mathematics education. While we do not explicitly refer to these, we do touch upon them when relevant to this issue, e.g., technology and curriculum design, technology in instruction processes, technology and embodiment, and technology and beliefs of teachers and students. Moreover, many published studies have focused on young students working with digital technology, especially with tablets and special apps (e.g., Cavalletti et al., 2023; Kortenkamp et al., 2020) using Scratch or TouchCount (Bakos, 2023). In this issue, we concentrate on the use of digital resources in secondary education as all the papers in this special issue address this school level.

We begin our discussion with an overview of recent developments in teaching with technology, with specific emphasis on teachers’ digital competencies in using technology effectively in the classroom and on the role of teachers as designers of digital resources. The section on learning with technology begins with an overview of the use of multiple resources while learning mathematics and then discusses new developments such as augmented reality and video technology. The section on assessment with technology emphasizes the meaning of formative and summative assessment and discusses automatic feedback and adaptive assessment, conceptual understanding, and task design in the development of assessment profiles. In discussing each of these topics, we anticipate the possible future developments emerging from the present state of research to provide further perspective on the use of digital resources in the digital age.

2 Teaching with technology

In a recent survey paper on teaching mathematics with digital technology, Clark-Wilson et al. (2020) noted that in the last two decades, the focus of research studies has shifted from how technology can foster students’ learning to “how teachers can make practical use of different types of digital technology to provide students with activities that will enhance their mathematical learning” (p. 1223). This shift reflects researchers’ acknowledgement of the fact that “of all the factors influencing student activity it is the teacher who most influences learning” (ibid.).

Topics that address teaching with digital technology and resources include the investigation of obstacles and levers to digital technology integration (e.g., Thomas & Palmer, 2014), teachers’ knowledge, beliefs, and competences and the way these influence their digital practices (e.g., Remillard et al., 2024), digital task design to support teachers’ use of technology (Mariotti et al., 2023), design and evaluation of teacher education, and professional development programs (e.g., Thurm et al., 2024).

In this section we provide an overview of three ‘hot’ research topics in the area of teaching mathematics with digital technology that are also the focus of papers in this special issue: teachers’ digital competencies, teachers as designers, and innovative teaching practices leveraged by digital technology.

2.1 Teachers’ digital competencies

The belief that teachers are one of the most important factors influencing student learning when it comes to integrating digital technology in mathematics classrooms led researchers to studying teachers’ practices (e.g., Hennessy et al., 2005; Haspekian, 2014), knowledge, and skills needed for effective use of digital technology (e.g., Niess et al., 2009; Rocha, 2013), obstacles to technology integration (e.g., Jones, 2004), and affective aspects such as beliefs or personal orientations (e.g., Thomas & Palmer, 2014). Considering both cognitive (knowledge) and non-cognitive dimensions (e.g., affective, motivational) underscores the need to focus on teachers’ competencies, as Carr (1993) points out:

It is argued that we need teachers who are not just knowledgeable or well-informed about education, but whose knowledge and understanding is expressed or exhibited in their abilities - teachers, in short, who are competent by virtue of the intelligent application of their knowledge and understanding in effective practice (p. 254, author’s emphasis).

Hence, the notion of digital competency emerged in the context of digital technology. Various frameworks are available that describe this competency, mostly in institutional documents such as the European Commission Digital Competence Framework for Educators (Redecker & Punie, 2017) or the UNESCO ICT Competency Framework for Teachers (Butcher, 2018) at the international level. These frameworksFootnote 1 are general and apply to any subject matter and school level.

In their recent literature review of teachers’ professional digital competence (TPDC), Skantz-Åberg et al. (2022) point to the scarcity of conceptualizations, stating that almost 75% of papers reviewed do not provide a clear definition of the terms they use. The authors express regret that the view of technological competence as “a set of basic technological skills” or “something that applies mainly to teachers’ handling of digital technology” (p. 10) is the most prominent pedagogical view and that its development is considered to be the responsibility of individual teachers. The authors conclude by claiming that

the conceptualisation of TPDC needs to be directed away from the strong focus on the technological competence and basic hands-on skills of individual teachers to a focus on a collective responsibility and accountability for TPDC (p. 16).

In a similar study that examines teachers’ digital competencies in higher education, Basilotta Gómez‑Pablos et al. (2022) note that the various conceptualizations in the literature converge in “the need for teachers to have didactic and technological knowledge that allow them to make use of digital technologies in their professional practice” (p. 2). The authors call attention to emerging lines of research, among them the necessity “to rely on frameworks and theoretical reference models that identify the dimensions and components of digital competence” (p. 8, authors’ emphasis) and to develop methods for assessing teachers’ digital competencies, since currently most methods are based on “teachers’ self-perception” (p. 9).

Gonscherowski and Rott (2023) also recently noted that self-assessment is a dominant instrument for assessing teachers’ digital competencies. Indeed, at CERME12 (Clark-Wilson et al., 2022) mathematics teachers’ digital competences emerged as one of the “hot” topics in the field of research on teaching mathematics with digital technology and resources. Conceptualising these competences requires considering relevant theoretical frameworks to provide an operational definition of their dimensions and components and to elaborate research methods for capturing and assessing them.

Recent mathematics education research includes initiatives toward defining teachers’ digital competencies that consider specificities related to teaching mathematics. Based on the assumption that the distinction between students’ mathematical and digital competencies tends to fade away, Geraniou and Jankvist (2020) contend that mathematical and digital competencies must be articulated for teaching as well. These authors use a network of four theoretical frameworks—the Danish KOM framework of mathematics teachers’ competencies (Niss & Højgaard, 2011), mathematics knowledge for teaching (Ball et al., 2008), TPACK (Mishra & Koehler, 2006), and instrumental orchestration (Trouche, 2004)—to conceptualise mathematical digital competencies for teaching (MDCT). According to Geraniou et al. (2022), MDCT are “the competencies teachers need (or have) to select and implement technology in their practice in pedagogically productive ways” (p. 167). In this issue, Geraniou et al. (2024) further elaborate on these competencies by drawing on the concept of mathematics digital competence for students (Geraniou & Jankvist, 2019). They rely on the theory of instrumental orchestration, the KOM framework, and the documentational approach to didactics (Gueudet & Trouche, 2009; Trouche et al., 2018), illustrating their arguments with the case of an experienced teacher.

Dilling et al. (2024, this issue) adopt a different approach. Extending the TPACK framework (Mishra & Koehler, 2006), these authors propose the media, pedagogy and content model (MPC) to describe mathematics teachers’ professional media competencies, which they define as “competencies that enable a teacher to evaluate and select digital technologies and use them in the mathematics classroom effectively”. Although these two conceptualizations—MDCT and MPC model—draw on different theoretical frameworks, they share the view that digital competencies allow teachers to identify the affordances and limits of digital tools and consequently select the appropriate ones to be used in support of students’ mathematics learning.

2.2 Teachers as designers

Many researchers have acknowledged the importance of tasks in mathematics teaching and learning. Sierpinska (2004) considers “the design, analysis and empirical testing of mathematical tasks, whether for the purposes of research or teaching, as one of the most important responsibilities of mathematics education” (p. 10). Watson et al. (2013) claim that “task design is core to effective teaching” (p. 8). In relation to digital technology, tasks appear even more critical, as “some of the key affordances arising from technology use emanate from the tasks we use with it” (Thomas & Lin, 2013, p. 109). The introduction to the proceedings of the ICMI study 22, which is devoted to task design in mathematics education, define a task as follows:

anything that a teacher uses to demonstrate mathematics, to pursue interactively with students, or to ask students to do something. Task can also be anything that students decide to do for themselves in a particular situation. Tasks, therefore, are the mediating tools for teaching and learning mathematics (Watson et al., 2013, p. 10).

Teaching thus includes the selection, adaptation, design, enactment, and evaluation of tasks (ibid.). Acknowledging that interactions with tasks are an important facet of teachers’ professional activity led researchers to consider teachers as designers or partners in task design, rather than merely as task implementers (Brown, 2009; Jones & Pepin, 2016). Hence, teaching design is gaining increased interest in mathematics education research.

Two emerging research trends can be observed in recent research studies in relation to teachers as designers of their instruction. On the one hand, researchers’ attempts to understand and describe the requirements of designing and using (digital) tasks in terms of teachers’ competencies have led to a focus on teacher design capacity. On the other hand, the emergence of communities engaged in collective task design, such as communities of practice (Wenger, 1998), communities of inquiry (Jaworski, 2014), or teacher design teams (Handelzalts, 2009), has initiated a new research field in teachers’ collaborative work that may be accompanied and supported by facilitators.

Teacher design capacity. During the last two decades, researchers have begun exploring teachers’ interactions with curricular materials, for example by considering teaching as a design process (Brown, 2009) and by viewing curriculum materials as resources supporting such a process. Researchers agree on the existence of a mutual influence between teachers and resources, such that resources affect teacher practice and teachers modify resources to adapt them to their own context and needs. Examining such teacher-resource interactions necessitates understanding “pedagogical design capacity”, defined as a teacher’s “skill in perceiving the affordances of the materials and making decisions about how to use them to craft instructional episodes that achieve her goals” (Brown, 2009, p. 29). Pepin et al. (2017) further conceptualised Brown’s pedagogical design capacity as “teacher design capacity” that incorporates three main components: (1) an orientation or a goal for the design, (2) a set of design principles, and (3) “reflection-in-action” (p. 802). Building on existing conceptualisations of teacher design capacity, Trgalová and Tabach (2024, this issue) propose a framework that incorporates the specificities of mathematics and the use of digital technology, yielding a definition of teachers’ digital resource design capacity (DRDC). The authors then elaborate a conceptualisation of mathematics teachers’ DRDC, which they illustrate using two case studies from a course for pre-service mathematics teachers aimed at developing participants’ design capacity based on the DRDC conceptualization.

Teacher design teams. In their ICME13 survey paper, Robutti et al. (2016) claim that “the notion of mathematics teachers’ working and learning through collaboration… gains increasingly more attention in educational research and practice” (p. 651, authors’ emphasis). They point to “ever-increased interest in exploring and examining different activities, processes, and the nature of differing collaborations through which mathematics teachers work and learn” (p. 652). Their paper provides three emblematic examples to illustrate various forms of mathematics teachers’ collaboration: (1) a school-based collaborative lesson study in Japan that relies on iterative teacher processes of planning, teaching, lesson observation, and post-lesson discussion and reflection; (2) collaborative teacher projects in England involving schools and an “expert other” (p. 655), aimed at producing resources, planning, implementing, and evaluating an intervention, and providing professional development for other teachers; (3) mathematics learning communities in Norway involving teams of researchers and teacher educators in partnership with schools and based on “three layers of inquiry” (p. 656): in students’ mathematical activity, in mathematics teaching, and in the research process during the collaboration.

The 25th ICMI study devoted to teachers of mathematics working and learning in collaborative groups (Borko & Potari, 2020) confirms the growing interest in this topic. The emergence of new theoretical frameworks during the last decade has allowed researchers to address collective aspects in teachers’ professional activities and to consider how these aspects have affected teachers’ learning and professional development. Examples include the documentational approach to didactics (Gueudet et al., 2013) and the meta-didactical transposition theory (Arzarello et al., 2014) of collaboration between two communities: teachers and researchers or teacher educators. Most research studies focus on in-service mathematics teachers engaged in collaborative work. Other examples include the collective design of e-textbooks by an association of teachers (Sabra, 2016), the documentation workmates initiative in which a pair of teachers designed new resources for teaching algorithms and programming (Wang, 2018), and collaborative work on scenario design to consider possible enactments of classroom activities within a professional development programme (Cusi et al., 2020). Yet very little research has considered collaboration among pre-service and in-service teachers. To fill this gap, Dilling et al. (2024, this issue) report on a collaborative project between pairs of teachers made up of an in-service teacher and a pre-service teacher trained to assist the in-service teacher in integrating digital technology in the classroom. The paper highlights the importance of a sound distribution of responsibilities within the teacher pairs so as to divide the workload.

2.3 Innovative teaching practices leveraged by digital resources

In the current digital age, mathematics teachers are provided with a profusion of digital resources, mainly available through the Internet. Consequently, much research in mathematics education has focused on teachers’ interactions with resources. Indeed, these interactions are thought to form the core of their professional activity (Remillard, 2005; Gueudet & Trouche, 2009). As noted above, the consensus is that teachers’ interactions with resources have the potential to affect both the resources themselves as they are modified and combined with other resources and the teachers’ practices that are shaped by the resource use. Hence, it is legitimate to expect that innovative digital resources will lead to innovative teaching practices. Indeed, as Clark-Wilson (2017) notes, digital resources have the potential to offer learners an environment in which they can explore mathematical ideas:

The advent of dynamic mathematical digital resources in the early 1990s promised a transformation of the teaching and learning mathematics as the technology enabled teachers and learners to experience and explore difficult mathematical ideas in more tangible ways.

Yet Clark-Wilson also notes that despite their potential, such technologies remain underused, mostly because of the complexity of integrating them into teachers’ practices. Research projects examining teachers’ professional development have sought to help teachers overcome barriers to technology use. One of these is the UK Cornerstone Maths project, which aimed to exploit the dynamic and visual affordances of digital technology and provide students an environment in which “to explore and solve problems within structured activities” (Clark-Wilson et al., 2013, p. 13). Teachers’ appropriation of specifically designed resources was accompanied by professional development programs. Likewise, Naftaliev and Barabash (2024, this issue) orchestrate teachers’ interactions with a specific type of digital resources known as technology-based interactive resources to foster the evolution of in-service teachers’ practices toward teaching experimental mathematics.

Digital technology has also been explored as a vehicle for the development of creativity, which is considered a 21st century skill, for example in the book Creativity and Technology in Mathematics Education (Freiman & Tassel, 2018). The European MC2-Mathematical creativity squared project demonstrated that digital technology can support social creativity in the design of digital resources (Kynigos et al., 2020) and at the same time can foster students’ creative mathematical thinking (Trgalová et al., 2018b; Kolovou & Kynigos, 2017). Ye et al. (2024, this issue) observe and analyse the mathematical creative actions of in-service teachers engaged in problem solving activities with Scratch, alongside their development of mathematics and programming knowledge.

To summarize, the above survey demonstrates several emerging trends and research evolutions in teaching mathematics with digital technology, which are further discussed in the concluding section: a shift from reporting on the complexity of digital technology integration in mathematics classrooms to exploring what teachers need to use digital tools effectively in terms of mathematical and digital competencies; viewing teachers as designers of (digital) resources rather mere resource implementers; development of innovative teaching practices enhanced by digital technologies; and the need for long(er) term support for teachers in integrating digital technology in their classrooms.

3 Learning with technology

In her paper “Solid Findings in Mathematics Education: The Influence of the Use of Digital Technology on the Teaching and Learning of Mathematics in Schools”, Hoyles (2014) stated: “Digital tools have the potential for transforming teaching and learning mathematics in ways not possible with other tools” (p. 50). Indeed, according to Drijvers (2019), learning mathematics with technology was found to be beneficial for student learning, though the average effect size was small to moderate. At the same time, Hoyles stated:

In order for the potential for transforming mathematical practice through the use of digital technologies for the benefit of all learners to be realised, teachers, teachers’ practice and their beliefs about learning must form part of the process (2014, p. 50).

Taken together, these quotations tell the story of integrating technology in the mathematics classroom and student learning: on the one hand, a clear vision of new opportunities that are realised to some extent, and on the other hand acknowledgment of the practical complexity involved in “making good use” of these opportunities for the benefit of all students.

The issue of student learning with technology raises many questions. For example, what do we know about learning with technology in different mathematical domains? Does learning algebra differ from learning geometry? How does learning algebra in a CAS environment differ from learning algebra in a graphical environment? What is the influence of tasks provided to students which use the same technological tools? How does learning mathematics in technological environments in elementary school differ from high school? Some of these questions are discussed in depth in the chapters of the newly published handbook titled Digital Resources in Mathematics Education (Pepin et al., 2023). For example, Leung et al. (2023) explore how digital technology enhances geometric skills, Haspekian et al. (2023) study algebra education with digital resources, and Biehler et al. (2023) question the impact of digitalization on content and goals of statistics education.

In the following concise review, we were guided by the issues emerging from the papers in this issue. We chose two themes that on the one hand go beyond mathematical topics and on the other hand are linked to recent developments in our field. We begin by discussing the use of multiple resources while learning mathematics—a conceptualization that may shed new light on our understanding of learning with resources. We then refer to a new adoption of technologies for learning mathematics that is influenced by developments in hardware, software, and accompanying pedagogy. In each subsection we first describe the topic based on the literature and then refer to the papers in this issue.

3.1 Learning with multiple resources

It is clear today that digital resources do not replace paper and pencil resources, but rather are used alongside them. Acknowledging the fact that learning mathematics takes place both with analogue and with digital resources, Geraniou et al. (2023) asked

how students (and subsequently teachers) conceptualise mathematics as they make transitions when using digital resources, involving different mathematical representations and different semiotic systems that may be using ‘new’ representations of mathematical concepts? (p. 1).

According to these authors, the use of graphs, symbolic expressions and numerical data can be considered as dynamic realisations of the same mathematical idea or as a dynamic and connected means of bridging between virtual concrete-like phenomena and their abstract counterparts. The authors further define transitions while working with digital and other resources using the words “within, beyond and across” (ibid., p. 1).

A transition is considered “within” a resource if several different representations of the same mathematical object or concept are available for students, who need to make sense of them. Bach (2023), for example, examines pairs of students who interacted with a dynamic geometry environment and were also asked to report individually in a paper environment. Bach noted:

‘Transitions within a digital tool’ concern the coordination of representations presented in a digital tool, which may be related to different content areas. ‘Transitions beyond a digital tool’ include the coordination of digital tool representations to paper-based representations. Such a transition may be evident even when it is not undertaken (p. 344).

Thus, observing learners as they interact with resources can shed light on transitions, as can researchers’ expectations regarding whether such transitions will take place, which may be absent in students’ work. For example, while working with a spreadsheet, students need to write expressions using such Excel-cell notations as 2*A2, whereas in a paper environment they would use 2*x, 2x or 2*A.

A transition is considered to be “across” resources when the opportunity is provided to navigate between two or more digital resources. In such cases, mathematical knowledge about a concept or an object is learned with one resource and then used with a different resource, possibly also with different representations. For example, Panorkou et al. (2023) studied students’ covariation thinking while using several resources (simulation, table, and graph) and described their transition across the resources as “messy”. Specifically, the researchers “provide an insight into the nature of the synergy of artefacts that offers a constructive space for students to shape and reorganise their meanings about covarying quantities” (p. 131).

Four papers in this issue describe cases of transitions within, across, and beyond resources. Radmehr and Turgut (2024, this issue) analyse online video resources for studying the topic of derivation. The authors look for representations [realisations in their terminology] of mathematical objects in the video. These representations provide learners opportunities to make transitions within the video resource for the same mathematical object.

Jaber et al. (2024, this issue) studied students working in an augmented reality environment in the mathematical context of covariational change. The authors offer learners the possibility to make transitions across carefully designed physical phenomena and their mathematical representations (tables and graphs).

Brnic et al. (2024, this issue) studied students’ transitions across resources such as digital textbooks, dynamic visualisations, feedback formats, and digital tools, while they learned about conditional probability. The authors also studied students’ transition beyond resources by analysing students’ work on a paper and pencil test. They compared the achievements of students who learned with non-digital resources to those who learned with the digital resources and also compared gender differences.

Finally, Bach et al. (2024, this issue) studied students’ transition beyond resources, moving from working with dynamic geometry resources to oral and written communication concerning the dynamic aspects of the mathematical objects observed.

3.2 Technological developments adopted for school mathematics

A different line of research examining students’ learning with digital resources is related to technological hardware and software developments and their influence on the uses of innovative technologies for teaching and learning mathematics. For example, augmented reality (AR) was first introduced by the US Air force in the early 1990s. AR can be defined as a system that incorporates three basic features: real and virtual worlds, real-time interaction, and accurate 3D registration of virtual and real objects (Wu et al., 2013). For mathematics education, AR is “a mixed reality technology that contains virtual objects that are implemented or ‘augmented’ with the real world” (Ahmad & Junaini, 2020, p. 107). That is, AR technology facilitates design in which virtual elements can be added for the viewer while looking at real world objects such as tables or graphs.

In a systematic review of the use of AR in mathematics education, Palancı and Turan (2021) found an increased number of reports since 2010 that mainly used qualitative research methods. The researchers note that the use of AR

supported learning and motivation and enhanced the spatial abilities of students. Additionally, the most frequent disadvantages of AR in mathematics education were that it caused technical inconveniences and it is difficult to develop materials through AR (p. 89).

As this AR example shows, it is less accurate to speak about “new” and “old” technologies. Rather, we look for technological developments accompanied by new ways of exploiting hardware and software that have the potential to benefit students’ mathematics learning (Hoyles & Lagrange, 2010).

Another example is the use of video technology to capture live images by electronic means. This technology was first developed for commercial television in the early 1950s to capture live images by electronic means. As is the case with other technical tools, these videos were adopted for educational purposes. Specifically, video recordings were adopted for professional teacher development in that the instruction of actual teachers in the classroom was recorded as a basis for discussions and reflections at a professional development meeting (Jaworski, 1990). Teacher educators who are also mathematics education researchers used recordings of professional development meetings as an opportunity to reflect on their own practice and research (Coles, 2014; Tirosh et al., 2014). Video technology was also adopted as part of learning while engaging with (serious) video games (e.g., dragonbox algebraFootnote 2 for learning to solve linear equations, see Kapon et al., 2019). Indeed, it is hard to imagine flipped classrooms or MOOCs without the use of videos (Lo et al., 2017).

Kay (2012) identified two main factors that facilitated a tremendous increase in the use of instructional videos for learning aims: one is the YouTube platform, launched in 2005, that allowed users to upload and watch videos with ease; the other is the increasing size and availability of bandwidth, influencing the quality of what one might learn by watching such media. Kay and Kletskin (2012) identify two kinds of videos for use in mathematics higher education—algorithm presentations and problem-based demonstrations—claiming that problem-based demonstrations were less prevalent in our field. There is still no consensus regarding how to design an instructional video or how to measure its effectiveness (Netzer & Tabach, 2023). Some researchers use indirect methods such as students’ course grades, course dropout rate, and student feedback, while others examine the actual way in which learners use the video (Kim et al., 2014).

Three papers in this issue describe cases of using AR or video technology.

As noted above, Jaber et al. (2024, this issue) designed special software that students could use to experiment with a physical phenomenon: while a cube slid down an inclined plane (Galileo experiment) students observed the covariation of time-distance in a table and in graph representations. They also observed the elongation of a physical spring under various masses, where the stiffness of the spring served as a parameter (Hooke’s Law). The AR technology enabled juxtaposing the view of the physical experiment and the mathematical representations, allowing students to discuss and make sense of these covariation phenomena.

Radmehr and Turgut (2024, this volume) took advantage of the YouTube platform to identify online instructional video on derivation by using the number of views for the video they chose to analyse as an inclusion criteria (more than 3.2 million views!). The starting point of the analysis was that this highly observed resource provides learning opportunities for the students who watch it.

Finally, Wirth and Greefrath (2024, this issue) designed a video resource that students worked on in pairs to learn how to handle modelling problems. The authors made use of our knowledge as a research community regarding the use of written examples and implemented this knowledge in their instructional video. This video was shown to 18 pairs of students without prior experience in modelling; each pair worked individually with the video to make sense of the modelling process. The authors examine the students’ perceived advantages and pitfalls following this experience.

In summary, although learning with technology has already been examined for several decades, the above survey provides evidence that the field is far from reaching stability. It is still characterised by new conceptualizations that researchers have raised to further our understanding of this complex issue. It is interesting to note how much time it is taking the education system to adopt technological innovations for research purposes, not to mention how much time it is taking for these innovations to be implemented in mathematics classrooms as an everyday reality.

4 Assessment with technology

Digital assessment also offers new types of affordances for assessment, such as automated scoring, feedback, and adaptivity. These affordances provide new opportunities with respect to developed tasks and assessment problems as well as styles of performing tasks and means of solving problems (e.g., see Drijvers & Sinclair, 2023). Interactive software such as Computer Algebra Systems (CAS), Dynamic Geometry Systems (DGS), and spreadsheets offer opportunities for carrying out interactive and dynamic tasks that are not possible using paper and pencil means, as do films, animations, and simulations. Assessment can be individualised in that feedback can be adapted to individual learners. Nevertheless, problems and difficulties associated with digital assessment must be considered. One such problem is the possibility that problems will become more complex due to a greater variety in representations. Another is the need to assess conceptual knowledge in addition to procedural knowledge, a form of assessment that is generally thought to be more difficult. Moreover, media use in the classroom and on examinations can change many or even all aspects of assessment.

There are new possibilities for the ways in which tasks are selected for use in assessments, in the way they are presented to students, in the ways that students operate while responding to the task, in the ways in which evidence generated by students is identified, and how evidence is accumulated across tasks. (Stacey & Wiliam, 2013, p. 722)

In the following section, we begin with some general observations on the use of digital technologies in formative and summative assessment. We then discuss specific aspects we expect to be more important in the coming years: automatic feedback and adaptive assessment, assessment of conceptual understanding, and task development for digital assessment.

4.1 Digital technologies in formative and summative assessment

Assessment in mathematics education is characterised by a well-accepted duality between large-scale or summative assessment on the one hand and classroom or formative assessment on the other. Summative assessment refers to the assessment of learning that is associated with evaluation of student knowledge and learning at the end of a course or teaching unit, quite often with the aim of assessing students’ proficiency level. In contrast, formative assessment is assessment for learning that entails gathering information about a student’s current state of knowledge and learning to guide teachers and students in subsequent teaching and learning (e.g., Stacey & Wiliam, 2013). Digital technology has been and will continue to be used to enhance both summative and formative assessments.

Assessment is further marked by another duality: assessment with digital technologies and assessment through digital technologies (Stacey & Wiliam, 2013).Footnote 3 Assessment with digital technologies is characterised by traditional paper-based learning methods in which learners have access to digital learning opportunities such as software and videos (Drijvers et al., 2016; Fahlgren et al., 2021). In many countries today, especially in the West, handheld technology is allowed to be used or is even mandatory in examinations. In contrast, in assessment through technology or technology-based assessment (TBA), technology serves as a testing environment.

Computer-based tests and (final) examinations have been in use for many years at the university level, e.g., in medicine and engineering studies (Iannone, 2020). Recently, computer-based examinations have begun to be introduced in school mathematics. Computer-based versions of the PISA test have been offered since 2012; by 2015, PISA’s primary mode of assessment of students’ skills was computer-based and tasks were coded automatically (Jerrim, 2016). One part of the national assessment of 18-year-old high school students in Finland is a computer-based test using the Abitti system. In this system, students work in an environment isolated from the digital world yet still have access to mathematical tools like GeoGebra, Maxima, Casio ClassPad or TI-Nspire (Drijvers, 2018). The Dutch Ministry of Education developed a diagnostic test for 15-year-old students used on final examinations in lower secondary mathematics education (ibid.).

In addition to summative assessment, in recent years formative assessment has gained importance in assessing classroom work and supporting the learning process, especially while using digital technologies (Aldon et al., 2017; Dalby & Swan, 2019; Olsher, 2019; Cusi & Morselli, 2024). In 2009, Black and Wiliam (2009) provided a widely accepted specification of formative assessment:

Formative assessment (FA) is conceived as a method of teaching in which evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited (p. 7).

In addition to the key areas and moments in which formative assessment is used, the European Formative Assessment in Science and Mathematics Education project (FaSMEd, Aldon et al., 2017) emphasises three main elements or technology functionalities of formative assessment: sending and displaying; processing and analysing; and providing an interactive environment (Cusi, 2022). These functionalities should be integrated into the whole teaching and learning process. They are especially represented in classroom communication (e.g., with systems like MOODLE or ILIAS), in feedback (e.g., with systems like STACK), and in interactive work with digital technology (e.g., with systems like GeoGebra). In this ZDM-issue, two contributions refer to specific questions concerning formative assessment. Cusi and Morselli (2024, this issue) investigate the specific roles played by teachers when making conscious in-the-moment decisions during classroom discussions to foster the development of effective formative assessment processes. Hershkovitz et al. (2024, this issue) evaluate the use of feedback on the topic of reflective symmetry in digital learning environments among 9- to 12-year-old elementary school students.

4.2 Automatic feedback and adaptive assessment

Feedback is perhaps the most important aspect of teaching and learning (Hattie & Clarke, 2018). Feedback with digital technologies, and particularly computer algebra systems like GeoGebra-CAS, Mathematica or STACK, are best suited for algorithmic calculations. When these technologies are used for learning, two types of use-software interactions are important: system input and system output. In the case of input, users must be confident with the question to be answered and its representation in the technical input scripts. In the case of output, users must to be able to interpret the system feedback and validate it with respect to the posed problem. Even though the calculations are done automatically, the feedback system developer, the task designer or the teacher must construct the kind of feedback required in advance. For example, STACK generates automatic differentiated feedback from student solutions. For each task, the designer must define relevant and possible outputs in advance, which will then be checked vis-à-vis the task or problem solution. Various forms of feedback can provide verbal, numerical, graphical or algebraic information about students’ answers (Pinkernell et al., 2019). Students in technical and science fields use the interactive adaptive assessment system OPTES (or its follow-up system DigikoS) as supplementary materials to repeat school mathematics in the introductory phase of their studies (see Roos et al., 2019). In OPTES, STACK tasks are used to develop an adaptive learning and training system (Wankerl et al., 2019), making it possible to select appropriate exercises for individual learners. These kinds of adaptive systems will become increasingly meaningful in the coming years.

An important issue in current research is how to help teachers make productive use of the results of digital learning assessments in the classroom (Dalby & Swan, 2019). Klingbeil et al. (2024, this ZDM-issue) describe the development and evaluation of the online formative assessment system SMART, which is designed improve mathematics learning among upper primary and junior secondary school students. The system provides teachers an informative diagnosis of students’ conceptual understanding (Stacey et al., 2009; Haspekian, 2020; Fahlgren et al., 2021). Each SMART test is individually evaluated and identifies students’ understanding, misconceptions, and knowledge gaps. Further, it provides teachers useful information about possible future steps, tasks to solve, or instructions to improve. In SMART, only the teacher receives the results of the test because the teacher is considered the initiator of the developmental process.

Individualisation of the learning process requires more than individual feedback on solutions and solution processes. Indeed, the problems of the next learning step must also be individualised and individual learning trajectories must be adaptive. Since the time teaching and learning machines were first introduced, adaptive assessment was always an important aspect of individual learning (Olsher et al., 2023). OPTES, for example, categorises a special mathematical field, e.g., algebra or calculus, based on a didactical reference model concerned with particular aspects of knowledge. Problems in one category are more closely related than problems from different categories. All problems in this field are arranged in a graph with special transformation probabilities. If a learner does not solve a problem, the probability of getting a similar problem is higher than if the problem is solved correctly. The transformation probabilities are calculated in a training process with a large number of users (Wankerl et al., 2019).

4.3 Digital assessment and conceptual understanding

Competency assessment is a central but demanding task for which digital technologies are increasingly being used in different contexts. In assessing procedural knowledge—i.e., knowing how to use procedures like transformations of terms, how to solve equations with formulas or how to calculate square roots with the Heron algorithm—computer algebra systems such Mathematica or STACK are adequate tools that allow the development of fully automated examination evaluation and grading. The assessment of conceptual understanding is far more complex for it entails assessing the understanding of mathematical concepts and the adequate application of mathematical concepts in internal and external mathematical problem-solving situations. This assessment is best accomplished by oral tests, interviews or any kind of project work. Concerning digital assessment, Olsher et al. (2023) note the risk that this kind of assessment “is not commonly testing competencies that match what it means to ‘do mathematics’ in the 21st century” (p. 21). Hoogland and Tout (2018) even see “mathematics education… at risk of focusing too much on assessment of lower order goals, such as the reproduction of procedural, calculation based, knowledge and skills” (p. 675). Nevertheless, in the context of the increasing importance of digital assessment, means must also be developed for assessing conceptual understanding.

The so-called Basic Mental Models (in German “Grundvorstellungen”) (vom Hofe & Blum, 2016) can serve as a good basis for the development of conceptual understanding. A Basic Mental Model (BMM) of a mathematical concept is a conceptual interpretation that gives it meaning. Yet BMMs still need to be developed and empirically validated (Greefrath et al., 2021), as do appropriate assessment tasks.

Some empirically based suggestions for the use of digital technologies in developing conceptual understanding have already been proposed. Weigand and Günster (2022) used digital tools (GeoGebra) to develop the dynamic view of functions on these levels of understanding. Ruchniewicz and Barzel (2019) developed a digital tool for student self-diagnostics as they work with different representations of functions. Yerushalmy and Olsher (2020) and Popper and Yerushalmy (2022) developed a digital tool for the development of reasoning while working with and classifying quadrilaterals.

4.4 Task design for digital assessment

Task design is a crucial element in mathematics education (Watson et al., 2013; Watson & Ohtani, 2021). Digital assessment raises a number of questions concerning task design: What new kinds of questions do digital tests allow (e.g., how to integrate films, simulations or interactive applets into the questions)? How do working styles in digital tests differ from those of paper-and-pencil tests (e.g., doing preliminary considerations and sketches on paper alongside the digital test, using interactive programs like GeoGebra or hand-held-technology in addition to the digital test)? How do the test questions influence the general classroom work (e.g., does the use laptops, hand-held devices or computer programs in examinations influence the need for hand calculations with paper-and-pencil)?

Concerning the development of tasks in digital assessment systems, four aspects need to be taken into consideration: (1) the task representation; (2) the way students or learners should work (in relation to software functionality); (3) the evaluation of students’ work and answers, and (4) the feedback on students’ solutions. A major theory has evolved regarding principles of designing assessment tasks for both summative and formative assessment. Suurtamm et al. (2016) believe that the principles underlying these two forms of assessment are similar because “large-scale and classroom assessment interact with one another in different ways” (p. 25). Still of major importance is the emphasis in the NCTM assessment standards (1995) “that both large-scale and classroom assessment should take into account not only content but also mathematical practices, processes, proficiencies, or competencies” (p. 5). Yet many questions concerning task design in digital environments still remain: Who designs the problems? Who has control over the assessment platforms (e.g., to what extent can individual teachers write their own assessments)? Which tools are allowed? (e.g., Nortvedt & Buchholtz, 2018).

Questions with verbal answers are clearly better suited for assessing higher order mathematics skills than multiple choice tests, as these questions enable students to express themselves more freely. Yet assessing handwritten questions is often challenging. Moons et al. (2024, this issue) developed a “checkbox grading system” that (human) assessors can use to assess handwritten mathematics exams as “atomic feedback”. The idea is to combine pre-defined answers for test problems by using a list of checkboxes for classification. The system then automatically calculates the grade and provides individualised feedback to students.

A great deal of experience has accumulated regarding assessment with technology, especially using CAS-systems. For example, CAS-supported examinations have been in use in Great Britain, France, Germany and the Scandinavian countries for many years (Bach, 2023). Leigh-Lancaster and Stacey (2022) give an overview of the 20 years in which CAS-calculators were used in Australia, showing that the availability of sophisticated digital technologies does not necessarily make assessment easy.

One reason for digital assessment is pragmatic: it offers an effective means of designing and implementing tests and examinations. Moreover, the use of digital technologies in tests and examinations is known for allowing enhanced experimental and heuristic work in problem-solving phases and for providing opportunities for realistic modelling tasks (Fahlgren et al., 2021). Yet this experimentation requires sufficient time to try out different methods and pursue them thoroughly, which is very difficult to implement within the confines of the time-limited tests that have been the predominant form of examination up to now. Yet, aligning learning activities in the instructional learning process with assessment activities is a crucial part of the learning process: “Any course needs to be designed so that the learning activities and assessment tasks are aligned with the learning outcomes that are intended in the course. This means that the system is consistent” (Hattie, 2009, p. 6). The issue of the use of digital media in examinations must therefore also be accompanied by a discussion of alternative forms of examination, such as portfolios, individual studies, and project work (Ball et al., 2018).

In summary, in the coming years assessment with and through technologies will play an important and increasing role in formative and summative assessment. During the pandemic, many teachers were exposed to new possibilities for digital assessment (e.g., online quizzes or digital interactive learning trajectories) and later adapted some of these in their traditional teaching (Cusi et al., 2023; Engelbrecht & Borba, 2023). Some current developments will surely influence assessment in the coming years. Inputting symbolic expressions using a formula editor is still not very convenient and “limits the expressivity of ideas due to the efforts of communicating them. Here technological developments such as optical character recognition (OCR) may provide useful solutions” (Olsher et al., 2023, p. 22). Furthermore, the issue of how to use Virtual and Augmented Reality not only in the classroom but also in assessment situations remains an open question (Wu, 2013; Palancı & Turan, 2021).

5 Conclusion

Teaching, learning, and assessment are closely related in learning environments, especially in mathematics classrooms. Hence, they must be oriented towards goals in terms of mathematical knowledge and general competencies. On the one hand, digital technologies and resources must be selected with a view to achieving these goals, while on the other hand, digital technologies and resources also influence the way in which these goals are achieved. This interrelationship, along with new developments in digital resources such as artificial intelligence products (e.g., adaptive digital learning systems or generative artificial intelligence systems like chatbots) make planning for the future very difficult or even impossible. In the digital age, all stakeholders in the educational system—politicians, school administrators, teachers, and students alike—must be flexible and prepared for continuous adaptations and corrections in the educational process. Moreover, in view of the diversity of educational systems, together with different national and regional peculiarities and traditions and diverse convictions and emphases, it is unlikely that generally accepted strategies for dealing with digital technologies and resources in mathematics education will emerge.

Nevertheless, based on the experience of the last decades and on the discussion in the above sections about teaching, learning, and assessment with technologies, we can still make some comments, suggestions, and recommendations regarding the use of digital technologies and resources in the digital age:

  • Despite major expectations on the part of the public, the use of digital technologies is not an end in itself. Rather, these technologies must be critically evaluated in the context of the intended goals of mathematics learning.

  • Digital technologies for new curriculum developments, task design, and the development of new problems and tasks for teaching, learning, and assessing are continually being explored.

  • Current digital technologies should be integrated into all levels of pre- and in-service teacher education. Nevertheless, this integration must be flexible and open to further technical developments and must not focus on special hardware or software.

  • As we show in the section devoted to teaching with digital technology, the mathematics education research community has demonstrated growing interest in conceptualising mathematics teachers’ professional competencies, with the goal of using digital artefacts in classrooms effectively. The current lack of consensual conceptualization highlights the diverse opinions regarding what makes mathematics teachers competent users of digital artefacts in their instruction.

  • As discussed in the section on assessment with digital technology, formative assessment could benefit from the potential offered by digital technologies, especially concerning individual feedback as a crucial element of mathematics learning and teaching.

  • The development of digital technologies for tests, examinations, and summative assessment should be intensified, especially concerning conceptual understanding. The fact that examination problems set standards for learning and teaching in the classroom is well-known.

  • Most recent research studies, including those in this volume, highlight the need for networking theories to address complex issues related to the use of digital artefacts in mathematics teaching, learning, and assessment. Likewise, the field requires large-scale studies using quantitative methods to complement the qualitative research methodologies dominating the research field.

  • Embodiment and embodied cognition, especially in connection with virtual and augmented reality, will become increasingly important. Hence, research on this topic as well as adequate classroom examples that go beyond the equipment debate need to be developed.

  • Finally, it is reasonable to assume that Artificial Intelligence will exert a major influence on educational systems. For the time being, the topic of how language generation models such as ChatGPT will influence teaching, learning, and assessment remains an open question.

Despite the ubiquity of digital technology and resources, this issue demonstrates that old but still relevant themes require further theoretical and methodological investigation.