The integration of digital technology into everyday school experience in mathematics lessons is still slow (Clark-Wilson et al., 2020). Our previous study highlights that, although mathematics teachers use digital technology to search for resources and to plan their lessons, they use it much less in the classroom (Tabach & Trgalová, 2020). Hence, we would like to investigate the affordances of digital technology likely to support teachers in their classroom teaching practices.

Norman (2013) used the term affordance in the context of human–computer interaction to refer to action possibilities that are readily perceivable by an actor. We further elaborate on this point below. There is a lack of studies that analyse affordances (and constrains) of digital platforms for mathematics teachers. Research studies focusing on teachers’ use of digital technology tend to show an increased complexity of teachers’ professional activity, requiring mastering the technology not only for doing mathematics but also and foremost for teaching mathematics (Haspekian, 2011). The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources and platforms specifically designed to help teaching and learning mathematics.

As was pointed out by Clark-Wilson et al. (2020), the term “digital technology” is very broad, and researchers need to be explicit when reporting about their research with respect to the technology used in their work. In our study, we refer to platforms specifically designed to allow for mathematical communication between multiple users: teacher–teacher, teacher–students, teacher–student and student–student. Considering this innovative technology, we explore affordances (and constraints) of several platforms specific for mathematics teaching and learning. We also consider the implications in terms of teachers’ digital competences that need to be developed in order to make use of the affordances.

This article is organised as follows. In the first section below, we present the conceptual and theoretical framework constituted from the concept of affordance, the instrumental approach and five practices framework. Then, based on a review of existing literature, we discuss the issues related to educational digital platforms and their use by teachers (the following section). In the third section, we present our research methodology, followed by the findings of our analyses of four digital platforms in the fourth section that are further discussed in the fifth concluding section.

Theoretical Framework

We start in the first sub-section by outlining the concept of affordance, followed in the second by the presentation of the core concepts of the instrumental approach (Rabardel, 2002). Instrumental approach offers a conceptual framework for analysing subject-tool interactions, shedding light on how affordances (and constraints) of the tool may shape the subject’s activity and how the subject may adapt the tool to her needs. We stress that our analysis is a theoretical one. As it focuses on affordances of platforms for teaching mathematics as tools that teachers can use, in the third sub-section, we refer also to the five practices pedagogical model (Stein et al., 2008) suggested for an efficient management of learning activities.

The Notion of Affordance

From the ecological perspective, Gibson (1979) defines affordance as follows: “The affordances of the environment are what it offers the animal, it provides or furnishes, either for good or ill” (p. 127). Heft (1989) discusses this definition and claims that:

Affordances, then, are properties of the environment taken with reference to an individual. As such, they have both objective and subjective qualities (p. 3).

According to the author, these qualities are objective in the sense that they are “facts of the environment” (p. 3). As an example, he takes a seat: “what constitutes, e.g. a seat, depends on the physical characteristics of an object” (p. 3). On the other hand, since what the environment affords must be perceived by an observer, its qualities may be considered as subjective. Heft further claims that “affordances are not subjective in the sense that they reside in mind […;] they are ecological facts […] they are relational in nature” (p. 4). From this point of view, affordances are seen as emerging from the interactions between an individual (animal) and the environment:

With respect to the environment-individual analysis, a relational concept refers to a property that emerges out of the interaction between an animal and the environment. Affordances are located at this boundary; they are synergetic properties of an environment-animal system (p. 4).

From a human–computer interaction perspective, Norman (2013) suggested that “An affordance is a relationship between the properties of an object and the capabilities of the agent that determine just how the object could possibly be used” (p. 11). He continued, “The presence of an affordance is jointly determined by the qualities of the object and the abilities of the agent that is interacting” (p. 11). Importantly, Norman stressed that “affordance is not a property. An affordance is a relationship. Whether an affordance exists depends upon the properties of both the object and the agent” (p. 11). Hence, while we are studying the affordances of platforms for teaching mathematics, we inquire into what the platforms afford and, at the same time, envision what the teacher needs to be able to do in order to recognise these affordances and to take benefit from them.

To be able to speak about the teacher as the main user of these affordances, we need to have relevant terminology. Considering the relational dimension of the concept of affordance leads us to choosing the instrumental approach as the theoretical framework: that is, we envision what kind of didactic instrument (in the sense of the instrumental approach, outlined in the next sub-section) the teacher can develop from the platform considered as a digital artefact. In addition, as we focus on didactic instrument for planning and enacting of mathematical activities in a classroom, we use the terminology suggested by the five practices pedagogical model (in the final sub-section).

Instrumental Approach

The instrumental approach (Rabardel, 2002) allows studying processes by which a user transforms a (digital) tool—an artefact, into an instrument enabling her to achieve her goals. The approach thus posits the idea of interaction between human and machine. While the artefact (material or symbolic) is available to the user, the instrument is a personal construct elaborated by the user during her activity with the artefact in the course of the so-called instrumental genesis. The process of instrumental genesis comprises two interrelated sub-processes: instrumentation leading to the constitution and the evolution of schemes of use of the artefact in the user and instrumentalization during which the user adapts and personalises the artefact according to her knowledge and beliefs. The development of schemes of use manifests itself in an invariant organisation of user’s activity in a given class of situations (Vergnaud, 1990).

The theoretical construct of double instrumental genesis (Haspekian, 2011) was developed in accordance with the instrumental approach. It encompasses both the personal and the professional instrumental geneses of teachers who use ICT. Whereas personal instrumental genesis is related to the development of a teacher’s personal instrument for a mathematical activity from a given artefact, professional instrumental genesis yields a professional instrument for a teacher’s didactic activity. This view is resonant with Krumsvik and Jones (2013), who claim that “digital competence of teachers is more complex than in other occupations” (p. 172), as it embeds two dimensions: (1) ability to use technology (personal use) and (2) ability to use technology in a pedagogical setting (professional use). That is, teachers must also “continually make pedagogic-didactic judgments which focus on how ICT can expand the learning possibilities for pupils in subjects” (Krumsvik, 2008, p. 283).

To avoid any confusion between teachers’ personal and professional activities, we use the term mathematical instrumental genesis to refer to teachers’ personal activities in relation with their teaching (transforming an artefact into a mathematical instrument, i.e. for doing mathematics with technology) and the term didactic instrumental genesis to refer to a teacher’s professional activities (transforming the same artefact into a didactic instrument, i.e. for teaching mathematics with technology) (Tabach & Trgalová, 2020; Trgalová & Tabach, 2018).

It is reasonable to assume that these two developmental processes, that is mathematical and didactic instrumental geneses, are interconnected. In this article, we focus on the didactic instrumental genesis. Although we do not study actual uses of the platforms by teachers, we deem that the instrumental approach and the notion of double instrumental genesis remain relevant theoretical frameworks. Indeed, they offer conceptual tools to envision which functionalities of the platforms can be recognised by teachers as affordances supporting their teaching activities—hence, to illuminate didactic instrumental geneses likely to occur while using the platforms.

Five Practices Framework

Being a mathematics teacher includes several professional activities, among them lesson planning, enacting planned learning situations, monitoring and assessing students’ activities and progress. Stein et al. (2008) aimed at helping teachers to manage instruction based on challenging tasks and students’ suggested solutions. To this end, the researchers brought to the fore five practices that should be mastered by teachers: anticipating, monitoring, selecting, sequencing and connecting. The innovative move of Stein and colleagues was in grouping together these practices as a sequence that together makes the enactment of such instruction more manageable for teachers. The first practice, anticipation, should be done before the lesson, as part of the lesson planning. The teacher is invited to anticipate what students might answer, what (mis-)conceptions might be expressed in students’ work and what the goals toward which the teacher is aiming during the lesson are.

The other four practices are enacted during the lesson, in the following sequence. Monitoring takes place while students work on the chosen mathematical task. The teacher observes their actions and as needed provides challenging questions or provides hints without giving away the procedure that might solve the task. Selecting is done in preparation for a whole-class discussion. It is based on the teacher’s observations of students’ actual work while monitoring it during the lesson. The teacher considers which of the solutions she noticed to present to the whole class. Next, the teacher needs to consider how to sequence these chosen solutions during the discussion, i.e. in which order to present the selected solutions in the whole-class discussion, so as to lead the discussion toward the lesson aims. Finally, as students are invited to present their thoughts, the teacher needs to connect the various solutions among them and also connect them with the lesson aim.

The anticipation practice is done before the lesson, without the pressure of the actual orchestration of the lesson. As such, it can set the ground and prepare the teacher to the monitoring practice, as she observes anticipated students’ thoughts and solutions, at least to some extent. Monitoring is the basis for the selecting practice and together with what was anticipated may help the sequencing practice. The sequencing of the presented solutions should also take the connections with the mathematical and pedagogical aims of the lesson into account, which were made explicit during the anticipating practice. Research findings show that connecting is the hardest to enact practice of the five (Boston & Smith, 2011).

Pre-service teachers’ learning, as well as practicing teachers’ professional development opportunities, used the five practices as a steppingstone leading toward change in teaching practices, also termed as ambitious instruction (Lampert et al., 2010). Yet, to the best of our knowledge, the five practices were not linked to teaching in a technological environment.

Literature Review

This section starts with presenting a particular kind of digital technology that we consider in this study, namely digital platforms (the first sub-section). Subsequently, we present existing studies on teachers’ use of such technology (the second sub-section). As digital platforms usually afford teachers with data gathered about students’ activity, the third sub-section is devoted to reviewing literature about data-based decision-making.

The Kind of Technology We Are Interested in

We focus on teaching mathematics using a digital environment. In order to highlight what mathematics teachers need to know and be able to do to teach efficiently within such an environment, we consider specific kinds of technology that provide both a learning environment for students and a system that affords support for teachers’ activity (planning, monitoring, assessing). We are therefore interested in virtual learning environments (VLE) or learning management systems (LMS) (Borba et al., 2016), defined as web-based platforms that:

could allow participants to be organised into groups; present resources, activities and interactions within a course structure; provide for the different stages of assessment; report on participation; and have some level of integration with other institutional systems (p. 600).

In the sequel, we use the term learning management systems (or LMSs) to refer to such environments.

Research on Educational Platforms

Learning management systems (LMSs) become prevalent at all education levels for various academic disciplines. These platforms are general in the sense that they are not specific to any academic subject. Indeed, researchers studied their general use. For example, it was found that instructors make use of the features of such platforms based on the communicative aspect, whether it allows teacher–students, student–student or student–teacher interactions (Derboven et al., 2017). Also, researchers found that instructors make use of general components of LMSs and adapt them to their needs, rather than use specific features with a limited scope of applicability. Less is known research-wise about instructors’ use of LMSs that are specific for teaching and learning mathematics. Next, we review studies that focus on LMSs dedicated to mathematics.

A few research studies (Cazes et al., 2006; Gueudet, 2008) pertain to e-exercise bases (EEBs) consisting of a repository of web-based exercises embedded in an environment that can include “suggestions, corrections, explanations, tools for the resolution of the exercise and score” (Cazes et al., 2006, p. 327). EEBs are therefore examples of LMSs. However, Gueudet (2008) points out a gap in research about the use of these LMSs: “The use of such tools has not been researched very much, perhaps because they seem inferior to, for example, microworlds” (p. 171). Moreover, existing studies focus rather on students’ learning with EEB, and little is known about teachers’ use of these tools.

Quaresma et al. (2018) report on the Web Laboratory for Geometry (WGL) platform, which is based on GeoGebra, but with some other features that make it an LMS. The teachers can assign students to classes and groups within the class, assign activities, communicate via private and collective channels with the students and collect students’ work and react to them. Each of the two basic working modes of the system—in-class and out-of-class—was implemented with one teacher. An in-class case study was done in Portugal, with 22 students, and an out-of-class case study was conducted in Serbia, with 69 students using the platform to develop homework. The two case studies show significantly better achievements for students who used the system as compared to their peers who did not use it (Santos et al., 2018). However, these studies focused on students and did not look at teachers’ use of the system.

Regarding teachers’ use of LMS, Kobylanski (2019) reports about the use of WIMS platform.Footnote 1 Her findings are based on a survey aimed at instructors using the platform for their teaching. The results show that the platform is mostly used out of school (by 69% of the respondents), but also in a school computer lab with one or two students per computer (53%). The use in a whole class with a beamer is much rarer (16%). The platform is mostly used for assigning exercises to students, either to practice in the classroom or as homework. Available resources are reused after modification by 35% of respondents, while 30% create new resources. Respondents appreciate the possibility of following-up their students’ work and achievement, of differentiating a same exercise according to students’ levels and the possibility of using the platform for assessment purposes thanks to affordances such as automatic correction, scores and marks.

Chorney (2022) studied challenges encountered by four mathematics teachers as they integrate DESMOSFootnote 2 platform in their practice. The results show that each teacher was facing somewhat different challenges, based on their previous expertise. The teachers’ craft knowledge was developed based on their ongoing work and trials of the platform rather than on official training.

Likewise, Fahlgren and Brunström (2020, 2021) studied four high school teachers’ use of DESMOS platform to monitor, select and sequence students’ responses while working in pairs and during the subsequent whole-class discussion. The teachers used research-based designed activities. The findings of these studies show that “it was challenging for them to orchestrate the pair-work stage, i.e. both to provide help to students and to prepare for whole-class discussion” (Fahlgren & Brunström, 2021, p. 19) and difficulties were also reported for these teachers to conduct a connected whole-class discussion (Fahlgren & Brunström, 2020).

Systems with Learning Analytics Supporting Teachers’ Decision-Making

LMSs are often provided with features that afford instructors to monitor learners’ activities. The analysis of the four platforms conducted in this article (in the penultimate section) takes learning analytics features into account. We focus in particular on the kind of data collected and their visualisation as affordances, so as to foresee possible exploitation of these data to inform teachers’ instruction. Based on these analyses, we attempt to infer components of teachers’ didactic instrument (described in the previous section). This section is therefore devoted to reviewing literature on technology-supported decision-making often called data-based decision-making (DBDM). DBDM has emerged and evolved as a key field in education for nearly two decades.

According to Mandinach and Schildkamp (2021):

DBDM has become important, in part, because policymakers have stressed the need for education to become an evidence-based field, causing educators to rely more on data and research evidence, and not just experience and intuition (p. 1).

Referring to Hamilton et al. (2009), the authors define DBDM as “the systematic collection and analysis of different kinds of data to inform educational decisions. [They point out that] data use is a complex and interpretive process, in which goals have to be set, data have to be identified, collected, analysed, and interpreted, and used to improve teaching and learning” (p. 1).

Data-informed instruction is recognised as an essential practice for improving students’ achievement (Massell, 2001). However, Curry et al. (2016) bring obstacles to the fore successfully using data that include “difficulty in accessing relevant data despite technological advances, inability to decipher meaning from data, and data that is too far removed from students to be useful” (p. 91).

Technology empowered with learning analytics usually includes dashboard applications aiming at supporting actors (learners or teachers) in their decision-making by visualising traces of their activity (Verbert et al., 2014). Dashboard applications “capture data about learner activities and visualise these data” (p. 1500) to support DBDM. Regarding the dashboards and the data they provide, one of the biggest criticism is that the developers of assessment systems produce reports that summarise results into red, yellow and green categories that indicate to the user which students are failing, borderline or passing (Mandinach & Gummer, 2018). According to Penuel and Shepard (2016), this problematic form of presentation, called “stop light”, oversimplifies the results and fails to provide a roadmap for instructional steps. However, these categories may provide a starting point for further analysis. Yet, new technological opportunities make it possible to go beyond the stop light approach.

Regarding the use of learning analytics empowered technology, research study conducted by Molenaar and Knoop-van Campen (2018) with (primary school) mathematics teachers shows that:

The data drove reflection and sense making and teachers used their existing pedagogical knowledge to come to new understandings, which in turn lead to pedagogical actions. [Pedagogical knowledge activated was mostly] at the individual student level, such as knowledge of the student, progress of the student and error analysis of the student’s work (p. 353).

These considerations highlight several issues related to DBDM that will be taken into account in our analysis of the platforms: nature of the data collected and the learning analytics embedded in the platforms (ranging from stop light to more elaborated data analysis), information provided to teachers (ranging from information about individual students and the whole class to possibilities to obtain profiles of groups of students) and support afforded to teachers about how to exploit this information.

Based on the literature review and the theoretical framework exposed above, and keeping in mind our position that teachers need to plan and enact mathematical activities with digital platforms in class, the research question that guides our study is: What are the affordances and constraints of the platforms likely to support instrumentation and instrumentalization processes leading to the development of the teacher’s didactic instrument?

Methodology

To address the abovementioned research question, we perform an a priori analysis of four platforms specifically designed to support mathematics teaching and learning. In other words, we do not observe how teachers or students use the platforms, but we rather attempt to identify their affordances in order to understand their potentials and limitations in supporting teachers’ didactic instrumental geneses in relation with the five practices on the one hand, and also to infer competences that mathematics teachers need to develop in order to recognise and take benefit from these affordances. This analysis will enable us to foresee the kinds of didactic instruments for mathematics teaching teachers can develop while utilising the platforms. To perform the platform analysis, we elaborate a method described in the first sub-section. In the second sub-section, we provide a rationale for the choice of the four platforms that are analysed in the subsequent section.

Method of Analysis of Virtual Learning Environments

Following Berthelsen and Tannert (2020), we assume that platforms as digital artefacts have affordances in the sense that they offer to the user possibilities for action and interaction. We are only interested in affordances from the point of view of teachers. Gueudet et al. (2021) claim that digital education platforms foster specific instrumentation and instrumentalization processes in teachers. Indeed, the fact that the platforms “allow the teacher to design according to his/her pre-existing schemes” (p. 88) is directly linked to instrumentalization. On the other hand,

a platform can structure and support teachers’ design practices: through the mathematical content it offers, how the content can be sequenced and through particular features that are offered for the lesson designs. […] this corresponds to instrumentation processes; its outcome is a modification of the teachers’ schemes

(p. 88).

Following these authors, we first analyse affordances the platforms offer in terms of potential instrumentation and instrumentalization processes (macro-level analysis). In particular, we are interested in platforms affordances allowing teachers to design their own resources (instrumentalization) and supporting their professional practices (instrumentation).

Since we focus more particularly on teachers’ planning and enacting a technology-supported mathematical activity in a classroom, referring to the five practices framework leads us to look for a support the platforms offer to teachers in:

  • Anticipating students’ answers and (mis-)conceptions that these answers might reveal. Such a support can take different forms. The simplest form could be allowing teachers to preview the proposed mathematical activities in a student’s mode, in order to get awareness of the potentialities and constraints of the digital environment in which they will be solving them. Among the most elaborated supports we can think of suggesting possible students’ answers and describing (mis)conceptions they reveal;

  • Monitoring students’ instrumented actions and understanding mathematical ideas at stake. Our analysis focuses on the information that can be obtained from the platform about individual students, groups or the whole class, in particular via dashboard;

  • Selecting particular students’ work to be shared with the rest of the class. We look for suggestions (if any) of possible answers, correct or not, that would be deemed by the designers worthwhile to be addressed in the class;

  • Sequencing the students’ responses that will be displayed, in order to make a discussion mathematically more coherent and make it lead to the lesson aim. We explore whether the designers suggest particular ordering of possible answers and provide a rationale for this order;

  • Connecting students’ solutions with each other and with the lesson aim. We look for hints, possibly provided by the designers, helping teachers link the “mathematical ideas that are reflected in the strategies and representations that they use” (Stein et al., 2008, p. 330).

Table 1 summarises the method of analysis of selected LMSs.

Table 1 Features of LMS
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Selection of Learning Management Systems

Platform affordances that we aim at studying are related to teachers’ planning and enacting technology-supported mathematical activities. Therefore, the main criteria that guided our choice of platforms to analyse are the following:

  • Platforms providing a virtual learning environment for students, i.e. students can work on interactive activities within the platform digital environment;

  • Platforms providing some kind of support for teachers’ management of students’ work, such as the possibility to adapt or create resources and to follow-up students’ activity;

  • Platforms designed by various design groups (researchers, teachers) and underpinned by different principles so as to obtain a wider range of possibilities offered to teachers in terms of affordances.

We have chosen four platforms that satisfy these criteria, namely STEP, DESMOS, WIMS and Labomep. In the following section, we analyse each of these platforms. Through their analysis, we attempt to highlight affordances likely to support teachers’ activity when planning and enacting technology-supported activities in their classes.

Analysis of the Platforms: Findings

We start by presenting briefly the four platforms that we have chosen for our analysis (the first sub-section). We then present their macro-analysis (the second sub-section) and micro-analysis (the third sub-section) that were carried out following the method outline above (in the first sub-section of the previous section). We provide illustrative examples of some affordances taken within algebra and function domain.

The Four Platforms

In what follows, we present the four platforms we chose for analysis.

STEP Platform

STEP (Seeing The Entire Picture),Footnote 3 designed by mathematics education researchers from Haifa University and developed by Carmel-Haifa University Economic Corporation Ltd. (Israel) is.

an automatic formative assessment platform in mathematics that helps teachers and students make use of rich and interactive assignments in the classroom in order to empower the teacher’s decision making in real time – during the actual course of the lesson.Footnote 4

The platform provides teachers with interactive exercises they can assign to their students and with an automatic analysis of students’ answers to these exercises, thus helping teachers take the students’ answers in their teaching into account.

DESMOS Platform

DESMOSFootnote 5 was developed in the USA by a team of researchers, teachers, software engineers and developers and is available in 18 languages. Besides mathematics software tools such as graphing calculator, scientific calculator or geometry tool to support students’ mathematical activity, the platform offers “free digital classroom activities, thoughtfully designed by teachers for teachers to support and celebrate the different ways students come to know mathematics”.Footnote 6 These activities are guided by DESMOS “pedagogical philosophy”,Footnote 7 including, for example creating objects that may promote mathematical conversations between teachers and students that may have impact on teachers’ practices. We therefore consider this philosophy in the platform analysis in the next sub-section.

WIMS Platform

WIMSFootnote 8 (WWW Interactive Multipurpose Server) is developed by the Côte d’Azur University, Nice, France, and is available in eight languages. Initially designed for university mathematics students to offer them a wide range of exercises, it is nowadays used also by secondary mathematics teachers and students (Kobylanski, 2019). WIMS is a web-based exercise repository hosting “online, interactive, random, self-correcting exercises [designed to] support the development of students’ competency by providing them with the opportunity to practice and test their knowledge in a wide range of exercises” (p. 128).

Labomep Platform

Labomep is developed by Sésamath, an association of French mathematics teachersFootnote 9 aiming at promoting the use of digital technology in teaching mathematics, cooperation between teachers and peer-supported professional development.

LabomepFootnote 10 is presented at the home page as follows:

Labomep is a platform for teachers to offer their students exercises for discovery, learning, training, deepening and monitoring the results, as well as summaries of animated courses, and many more resources (mental calculation, supervised manipulation of geometry software, etc.).

Its design allows for differentiated pedagogy; it is indeed possible to adapt the educational paths to each student, by creating sub-groups (from one individual), by using a large number of resources (constantly evolving) and by creating new ones, as well as by structuring these resources according to personalised order and logical criteria (Our translation).

This presentation highlights the designers’ intentions to support teachers’ activity by offering a variety of resources to be used in all phases of students’ learning (discovery, training, deepening), by supporting teachers’ monitoring of the students’ results and by fostering differentiation strategies.

This brief description of the four platforms highlights the intentions of the platform designers to support specific, and different, teaching practices: formative assessment (STEP), student-centred pedagogy (DESMOS), students’ autonomous learning through practicing (WIMS) and differentiation (Labomep).

Macro-level Analysis of the Four Platforms

All four platforms embed both an interactive environment for students and affordances for teachers. They have in common some similar affordances, which define them as learning management systems. As we mention above, a learning management system provides instructors with a way to create and deliver content, monitor student progress and participation and assess student performance (Borba et al., 2016; Pilli, 2014). First, in terms of organising, teachers can create a class, assign students into a class and assign activities to the class, and students can submit their solutions to the system. Second, in terms of creating activities, the particularities of what information the system needs to be provided with are different, but in all four platforms, the teacher can create an activity.

Third, all platforms include activities to be used and the teacher can search among the existing activities: all platforms afford searching based on mathematical topic and on grade level. Each platform might have other unique search fields, but these two basic search criteria are common to all. Fourth, in terms of modifications, all platforms allow to choose an activity and duplicate it to a particular use and modify the activity or parts of it. The specific modification options vary between the platforms. Fifth, in terms of following students’ progress along the assigned tasks, the platforms provide the teacher with a dashboard on which she can see for each student at least which tasks she has already done. Becoming familiar with the interface of each platform and mastering its use are part of the didactic instrumental genesis of the teachers.

Subsequently, we analyse each of the platforms to highlight their unique affordances.

STEP Platform

The platform may be used by the teacher for creating assessment activities from scratch. In this case, as the STEP platform is built on GeoGebra, the teacher needs to create the mathematical situation via the software. The teacher needs to make several decisions along the process of creating a new activity. This is part of the instrumentalization process the teacher is undergoing while creating an activity. The teacher needs to decide on the type of task that is the most relevant to the aims of the assessment. Several types of tasks can be created: multiple selection items; yes–no questions; provide up to ten examples of …; provide three examples which fulfil a given set of conditions.

While creating an activity, the system guides the teacher along a sequence of screens. These screens support the design of the activity by the teacher and are part of the instrumentation process the teacher undergoes. The information needed by the system asks the teacher to write the name of the activity and its description and instructions for students, as well as choosing among several options as of the way that the activity is going to be displayed to the students. In addition, the teacher can choose from a set of given filters that she will enact as the students will work and submit their solutions, to help her design the summary phase of the lesson. To choose the filters, the teacher needs to know how students tend to think about the task at stake, what kind of examples their lines of thought will result in and base on this the selection of the filters that will help externalise students’ understanding.

Let us illustrate this on an example related to linear functions (Fig. 1). Students can be given a point on the Cartesian plane—both graphically and with the numerical values of (x, y) coordinates—and are asked to write a symbolic representation of two linear functions that pass through this point. Students may click to change the given point and are to submit three examples of such pairs of functions. The system can filter students’ submissions based on, for example, whether the two functions are increasing, decreasing, one is increasing and the other decreasing or if one of the functions is constant. Such information, which is rarely collected by the teacher, may inform her about her students’ example space in this respect and may guide her further instruction.

Fig. 1
figure 1

Teaching guide from the STEP platform

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In addition, the platform offers activities that can be adopted and modified. An activity consists of a sequence of several tasks. Modifications can be done by removing one or more tasks from the suggested sequence of activities or modifying one or more tasks. Finally, while students are working on a particular activity and submitting parts of it, the platform affords the teacher to follow the submissions using a dashboard, to analyse these submissions online, in order to make decisions on the summary phase of the lesson.

DESMOS Platform

The activities provided in the DESMOS platform are guided by DESMOS “pedagogical philosophy”Footnote 11 reflected in the following recommendations: Incorporate a variety of verbs (e.g. not only calculating but also arguing, predicting, comparing, validating) and nouns (e.g. not only produce numbers but also represent them on a number line and write sentences about those numbers); ask for informal analysis before formal analysis, e.g. ask estimation before calculation, sketch before graph, conjecture before proof; create an intellectual need for new mathematical skills; create problematic activities; give students opportunities to be right and wrong in different, interesting ways; delay feedback for reflection, especially during concept development activities; connect representations; create objects that promote mathematical conversations between teachers and students; create cognitive conflict; keep expository screens short, focused, and connected to existing student thinking; integrate strategy and practice; create activities that are easy to start and difficult to finish; ask proxy questions.

This set of principles helps us understand the nature of the activities in this platform. There is an attempt to move away from drill-and-practice activities toward encouraging opportunities for conceptual understanding and students’ engagement. From the teacher’s perspective, it seems that the platform encourages teachers to adopt instruction based on students’ mathematical solutions. These solutions are expected to be a starting point for the teacher to make sense of the ways that students are thinking.

There is a special websiteFootnote 12 which is devoted to help a teacher learn how to build an activity from the beginning—DESMOS activity builder. An activity can be created by a single teacher or by more than one author. The site is user-friendly, with short videos and many demonstrations. An important “rule of thumb” is that each screen within the activity will be devoted to one mathematical goal. To build an activity, the platform leads the teacher along several screens in which she can choose components from the collection of possible components the platform offers: graph, table, sketch, media, notes, inputs, choice and checkboxes, graphing calculator, marble slides and card sort.

For each component, the site provides several screens taken from existing activities with variations of the component that can be implemented as part of mathematical activity. These screens are examples of different ways of using the component. A basic option is to copy a feature from an already existing activity. Finally, the teacher can send a message to the help desk and get feedback from the DESMOS team. Creating an activity is part of the instrumentalization process a teacher may undergo while working with the platform.

WIMS Platform

Among the affordances of the WIMS platform from the teachers’ point of view are those allowing creating own resources, either from scratch or by modifying the available ones (instrumentalization). The platform provides several tools for designing various types of resources. The most basic type of WIMS resource that any user can create is an interactive exercise, whereas other types as modules are more complex resources whose design is reserved for registered users only. Like other platforms, in order to support exercise design, the platform provides a guided mode in which the user either creates an exercise step-by-step by filling in a pre-established form or follows one of the available models of customisable exercises. Besides the guided mode, an experienced user can write directly online a source code of the exercise or upload a source file created with a text editor. The interface of the exercise design tool guides the teacher first to set the number of answers for the exercise, either open or multiple-choice. Next, and differently from STEP or DESMOS, the teacher needs to:

  • Define parameters if she wishes to. These parameters can then be used in the text of the exercise, the answers, the hint and the solution. The possibility to set up parameters prompts the teacher to reflect on mathematically and didactically relevant values (instrumentation). Indeed, as Kobylanski (2019) says, taking an example of solving second-degree equations:

    • Programming WIMS exercises may require a good mastery of randomness and didactic variables. Indeed, some exercises may not be random enough, e.g. solve the equation ax2 + bx + c = 0 only in the case where a = 1 and b and c are integers such that b > 2c is not a sufficient frame in terms of exploration of the possibilities. The even more particular case where b2 – 4c is the square of an integer is also not sufficient. It is nevertheless interesting because it allows the setting up of fast procedures, finding, for example a particular solution and deducing the other one. At the other end of the spectrum, proposing only to solve ax2 + bx + c = 0 for a, b, c decimal numbers presents a technical difficulty that is not necessary. […] Moreover, if only this variability is proposed, it will almost never allow the student to confront the case b = 0 or c = 0. But not knowing how to treat these falsely simple cases is detrimental to the control of the resolution of a second-degree equation. A student will be exposed to these cases only if the teacher pays attention to them in the setting of the worksheet (pp. 133–134);

  • Write the text of the exercise;

  • Define the responses by indicating their name, correct solution and type that is chosen from 30 or so predefined types, such as a number, an algebraic expression, a matrix or a free text. Such a variety of response types opens a way to in-depth didactic reflections;

  • Provide hints and a solution (optional). Reflecting on relevant hints triggers a teacher’s anticipating of students’ possible actions and answers.

Like other platforms, all along the design process, explanations with examples are provided at request.

Another outstanding affordance of the WIMS platform, when creating an exercise, is the possibility offered to the teacher to also programme feedback. Besides automatic feedback in terms of the correctness of the provided answer to the exercise, which is based on the comparison of the latter with the correct solution indicated by the author, the student can receive other types of feedback, for example hints to refer to the definition of the notion at stake. According to Kobylanski (2019), “(a) more ambitious and perhaps sometimes more useful type of feedback would depend on the question asked and the student’s answer” (p. 133). The author mentions feedback in the form of an additional question in case of a correct answer that aims at deepening the student’s understanding of the notion at stake or in the form of recalling a definition of the notion at stake and explaining a provided counterexample in the case of an incorrect answer.

Finally, and yet unique, part of the exercise design is defining grades students will obtain. The teacher has the possibility to programme grading for a single exercise or for an exercise string, which might be a series of random versions of the same exercise to avoid trial and error strategies. Exercises can be organised into sheets of exercises to be assigned to students. Teachers can define the weight of an exercise in an exercise sheet, as well as the weight of exercise sheets in the global average. Decisions teachers are supposed to take when creating exercises, organising them into sheets, setting up feedback and marking and managing groups of students require in-depth didactic reflections, thus fostering teachers’ instrumentation processes.

Labomep Platform

Labomep platform offers a variety of freely available resources that teachers can use as they are or modify them according to their needs. Different types of resources are available (Fig. 2, left): animated dynamic constructions (“Constructions instrumentpoche”, organised by mathematical topic), series of interactive exercises for lower and higher secondary school (“J3P collège” and “J3P lycée”,Footnote 13 organised by school level and then by mathematical domain, see Fig. 2, right), mental calculation exercises (“Calcul@TICE”, organised by school level and then by specific calculation skills), and e-textbooks for Grades 10 and 11 (“Manuel 2nde” and “Manuel 1ère”Footnote 14) written by Sésamath association.

Fig. 2
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Screenshot showing available resources in Labomep (left) and the organisation of interactive exercises for lower secondary school (“J3P collège”, right—in red, our translation)

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When a teacher wants to create a new resource, she needs first to provide the following information: title; language (to be chosen among ten available languages); technical type (to be chosen from among seven types: tree [hierarchical list], Calcul@TICE exercise [calculation], J3P activity, dynamic geometry animation, Mathgraph figure [dynamic geometry with calculus engine], external page, multiple choice questions); restriction (to be chosen from none, teacher, group(s), author(s)); abstract, description, comments (reserved for the instructor; it is possible, for example to provide information about which parameters in the resource can be changed); category (to be chosen among eight types: static activity, animated activity, static lesson, lesson with animation, static exercise, exercise with animation, interactive exercise, list of resources); school level (ranging from pre-school to upper secondary). Like in the other platforms, for each type of resource the teacher is proposed a specific design interface to guide the creation of the resource thus supporting instrumentation processes. To foster differentiated pedagogy in teachers using the platform, the “J3P activity” type affords a relatively easy way for teachers to prepare personalised paths from existing exercises adapted to their students’ needs.

The macro-analysis of the platforms highlights several affordances. Among those that are common to all platform is the support provided to teachers when creating of modifying resources (instrumentalization). Other affordances are unique, e.g. the possibility afforded to teachers to choose filters to be applied on students’ solutions enabling the platform to provide the teacher with a very specific information about their students’ example space related to a mathematical concept at stake (STEP); affordances supporting teachers’ decision making during resource design by providing hints about possible mathematical discussions to organise with students (DESMOS); affordances prompting teachers to reflect on exercise parameters, feedback and scores (WIMS); affordances supporting easy way of preparing personalised paths through exercises thus fostering differentiated pedagogy (Labomep). These affordances are likely to impact teachers’ practices and therefore support instrumentation processes.

Micro-level Analysis of the Four Platforms

STEP Platform

Support is embedded in the STEP platform while the teacher plans and enacts a lesson. Here, we refer to each of the five practices—anticipation, monitoring, selecting, sequencing and connecting. In fact, this support is part of the instrumentation process a teacher undergoes while developing her didactic instrument.

Anticipation—for each task within an activity, the platform provides a short teacher guide. Specifically, the guide informs the teacher about the stage of learning the particular formative assessment activity is suitable for, and also what prior knowledge is needed by the students in order to enact the tasks. Taking this information into account, together with what the teacher knows about her students’ learning process, may allow her to anticipate students’ ways of thinking around the specific tasks. Also, the platform provides examples of submissions by past students. Each example is accompanied by a concise description of the main mathematical emphasis that the example highlights. Observing these examples may help the teacher anticipate what examples her students will come up with, what lines of thought lead to these examples and how to suggest some guiding questions to foster students’ thinking.

Monitoring—while students are working on the tasks, the teacher on her screen can enter individual students’ submissions. In addition, the teacher can see on the dashboard in a tabular form the general progress of the class as a whole with indications which students hand in each task. The teacher can also follow students’ submissions for a particular task in a “carpet” form, where each solution is a small figure on a collection of all submissions. Moreover, the teacher can ask the platform to filter the submissions according to specific criteria associated with the particular task. These criteria are mathematical properties relevant to the task at hand. For example, in the case of linear functions, a possible filter may be whether the functions increase, decrease or are constant. Choosing filters results in a Venn diagram that displays the submitted solutions based on the chosen mathematical properties (see Fig. 3). This unique mode may guide the teacher while selecting which of the solutions may be the basis for a whole-class discussion.

Fig. 3
figure 3

Venn diagram view of a task based on filtering submissions (STEP platform)

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Selecting—as mentioned, the selection can be supported by the filtering option of the platform that supports the analysis of students’ submissions, based on the mathematical properties of the examples the students produced. Also, in the teacher’s guide, there are some hints such as example submissions and possible ways of students’ thinking that led to them. In this way, the platform may support the selecting of examples to be discussed, but the platform cannot do the selection for the teacher, who needs to take her didactic goals into account while selecting these examples.

Sequencing—the teacher needs to consider based on her selected solutions how to sequence them so as to promote the learning goals set. Here, the platform provides minimal support.

Connecting—again, some hints for connecting possible students’ submissions and the didactic aims are provided in the teacher guide. These include possible questions for discussions and their possible sequence. The teacher needs to combine the information she receives from the platform while selecting and sequencing the examples to be discussed with the information provided in the teachers’ guide and her own aims of the lesson.

DESMOS Platform

We refer to the support embedded in the DESMOS teacher platform for planning and enacting a lesson for each of the five practices, as part of the instrumentation process, a teacher might undergo while developing her didactic instrument.

Anticipating—a typical activity includes several screens that reflect a collection of several tasks. The teacher can view each of the tasks in the “student preview” mode (Fig. 4). The main feature that the DESMOS teacher platform offers with respect to preparing a lesson is a checklist for teachers to consider before enacting an activity in class. The list includes the following: Complete the activity using student previews; identify your learning targets for the activity; determine the screens you will bring to the whole class using the following function: “Teacher pacing and pause class. What will you discuss on those screens?”; anticipate screens that will cause students to struggle and then plan your responses; plan a challenge for students who finish the activity quickly and successfully; during the activity, when appropriate make yourself available to students who need individual help or have questions; write a summary of the activity main ideas: How can you incorporate student work in that summary? What parts of the activity can you skip to ensure there is sufficient time for the summary? In fact, the checklist encourages the teacher to anticipate students’ actions on the various screens, as well as to plan her responses for intervention during the enactment. This is a different mechanism than the one provided in STEP.

Fig. 4
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Example of one screen of an activity in the DESMOS teacher platform

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Monitoring—like in the STEP platform, while enacting the activity, the teacher can view the students’ progress on a dashboard, see whether their submissions are correct and view their answers by using a mode similar to the STEP carpet mode. Different from the STEP, the teacher can also ask the system to provide an aggregated view of students’ responses and can select anonymous responses to share with the whole class. In addition, for each task, the platform provides a short message for the teacher on the student preview screen: a brief description of the purpose of the particular task and hints for how to facilitate students’ thinking.

The platform does not provide additional support for selecting or sequencing.

Connecting—the platform provides some support for connecting, by the checklist mentioned and by encouraging the teachers to anticipate students’ actions. The teacher may link her anticipation with students’ actual activity to plan her moves during the discussion phase of the lesson toward connecting with the general aims of the lesson.

WIMS Platform

The system does not offer any explicit support for anticipating students’ possible actions or answers. However, there is the unique possibility to parameter didactic variables and the feedback prompts teachers’ in-depth didactic reflections and anticipations.

Regarding monitoring, as described by Kobylanski (2019), the teacher can see each student’s performance (see Fig. 5), first by a global average (Fig. 5a), then by an average on each sheet or a detail exercise by exercise in a given sheet (Fig. 5b). Several statistical data are provided, among which “the difficulty index of an exercise”, indicating “the average number of times necessary to complete the exercise” (As the author explains, “if this indicator is between 1 and 2, the exercise is not difficult. Experience shows that when this indicator is above 3, the teacher should consider explaining the solution to the exercise to the class” (p. 137)). This unique feature might help the teacher identify students’ difficulties and consider remedial activities.

Fig. 5
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WIMS dashboard (Kobylanski, 2019, p. 138)

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Selecting—the system does not provide any specific support for selecting students’ answers. However, the difficulty index of an exercise is an interesting feature of the platform that a teacher can consider to identify exercises that posed some difficulties to the students and make decisions accordingly.

Sequencing, connecting—the platform does not provide any specific support for these practices.

Labomep Platform

Anticipating—the system does not support the teacher in anticipating students’ answers.

Monitoring—during a session, the teacher can visualise students’ results in real time (Fig. 6). After the session, she can edit a synthesis with statistics about the ratio of success of each student or to see the recorded answers and thus identify encountered difficulties.Footnote 15

Fig. 6
figure 6

Labomep dashboard showing individual student’s performance on six series of ten exercises each (in rows). Green rectangle means that the student answered correctly at the first attempt, dark green one that she answered correctly at the second attempt, red one that she was mistaken at all proposed attempts and blue one that she did not answer (gave up the exercise)

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Green rectangle means that the student answered correctly at the first attempt, dark green one that she answered correctly at the second attempt, red one that she was mistaken at all proposed attempts and blue one that she did not answer (gave up the exercise).

These data help the teacher see which exercises pose difficulties to the student and take decisions accordingly. The visualisation of the data for the whole class is very “basic” according to the developers,Footnote 16 they advise rather to export the data to treat them according to the teachers’ questions. Such a treatment of the data does not support timely feedback to students but helps adjusting teaching strategies.

The platform does not provide any specific support for selecting, sequencing or connecting practices. Besides the abovementioned affordances, Labomep provides unique, specific resources that may help teachers prepare their lessons. These resources called “instrumenpoche constructions” are kind of animated lessons featuring methods related to a given mathematical topic. For example, in relation with affine functions, Labomep offers animated methods of how to draw a graphical representation of a linear function given its algebraic representation (see screenshot in Fig. 7), how to draw a graphical representation of an affine function given its algebraic representation, how to read an image or an antecedent by an affine function from its graphical representation and how to determine an algebraic representation of an affine function given by its graphical representation.

Fig. 7
figure 7

Screenshot of the animated method of drawing a graphical representation of a linear function defined by f(x) =  − 5/3x. The method shows step by step how (1) to find the image of a well-chosen pre-image, for example 3; (2) to plot the point A(3, - 5) and (3) to draw a line passing through this point and the origin

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The method shows step by step how (1) to find the image of a well-chosen pre-image, for example 3; (2) to plot the point A(3, − 5) and (3) to draw a line passing through this point and the origin.

This kind of resource can be used in various ways, e.g. by the teacher in the institutionalisation phase as part of the lesson or by the students who need to revise the lesson.

Discussion and Conclusion

In this final section, we respond to our research question: What are the affordances and constraints of the platforms likely to support instrumentation and instrumentalization processes leading to the development of teacher’s didactic instrument? We first highlight affordances of the platforms likely to support instrumentation and instrumentalization processes (macro-level) and then focus on affordances offering support of the five practices (micro-level), likely to lead to the development of a didactic instrument.

Potential Instrumentation Processes

The potential instrumentation processes that we consider are related to possible influences of the platforms on teachers’ teaching practices in general.

Clearly, all four platforms have been designed to influence teachers’ practices, though in different ways. STEP platform intends to afford teachers to integrate formative assessment in their practices, DESMOS platform makes explicit the underlying pedagogical principles that may influence teachers’ practices and WIMS and Labomep offer affordances toward differentiated teaching strategies.

Clear orientation toward generating predefined instrumentation processes is visible in the support offered to teachers when they design their own resources. All platforms provide a template guiding teachers through a series of items to fill in (e.g. learning goal or type of activity), which strongly impacts the ways teachers design their teaching. Some platforms (e.g. WIMS) require filling in aspects related to students’ activity, such as feedback or grading, which prompts teachers to anticipate students’ behaviour and leads to deep reflections on the mathematical and didactic aspects of their activity.

Nevertheless, these latter affordances can also be considered as constraints to which the teachers need to conform. If the design guides are too distant from teachers’ design practices, the creation of a resource may be perceived as requiring too much effort, and the corresponding affordance would not be recognised as such.

Potential Instrumentalization Processes

All four platforms afford teachers to modify, adapt and create their own resources, which, according to Gueudet et al. (2021), “can foster instrumentalization processes: the teachers will use the tools and/or the contents proposed by the platform according to their pre-existing schemes” (p. 96). Beside general affordances offered by all platforms toward creating new resources, some platforms offer additional support when modifying existing resources. For example, when creating a resource in Labomep, the author can provide indications about parameters that can be changed, which helps other teachers identify variables they can modify.

Affordances Supporting Planning and Enactment of Mathematical Activities

With respect to the micro-level analyses, in Table 2, we summarise affordances platforms offer to teachers in terms of supporting five practices to plan and enact mathematical activities.

Table 2 Platforms affordances supporting the five practices
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Table 2 demonstrates that there is a great variety in the affordances that each platform offers to support teacher’s work along the five practices. It seems that monitoring is a clear affordance in all four platforms. Indeed, the four platforms provide the teacher with information regarding the correctness of students’ answers. The minimum support is in the form of indication, mainly colour coded, about not attempted/correct/incorrect response. As one of the roles of teachers is to verify the correctness of student work, this affordance of the platforms takes some of the load off the teacher, allows her to make a better use of her time in class and supports the monitoring practice.

Yet, as noted by Penuel and Shepard (2016), this stop light presentation might be an oversimplification of students’ learning. However, three of the platforms provide more information than the correctness of students’ responses. Two platforms allow for an aggregated view of students’ responses, which provides the teacher with additional valuable information on their students’ actions. The STEP platform in addition allows the teacher an interactive way to inquire into the mathematical properties of their students’ submissions.

The four platforms vary in their affordances for anticipating students’ answers. Again, these affordances take a different form. While STEP provides examples of possible submissions by students, DESMOS invites the teacher to respond to tasks in the activity using the students’ preview, in the form of a checklist to do before the lesson, bringing the practice of anticipation into one’s mind. The way WIMS indirectly provides affordance to the anticipation practice is on the one hand, by allowing the teacher to control the feedback provided to students, which might lead the teacher to anticipate students’ answers. On the other hand, as WIMS allows the teacher to set parameters by which the actual tasks to the students will be determined, again there is an indirect invitation to anticipate students’ activity.

The platforms indirectly afford selecting, again enacting it in a different way. For the STEP platform, it is the possibility of filtering students’ submissions that might support the teacher in determining which examples to select as a basis for the whole-class discussions. DESMOS invites the teacher to think about which productions could be discussed in the whole class while planning. For the WIMS platform, the system identifies the level of difficulty of a particular task based on the difficulty index of tasks which is calculated based on the percentage of incorrect students’ responses. This unique affordance supports another kind of selection—a selection of tasks from among a list of tasks in an exercise sheet that seem to be difficult for the students. Such a selection may be considered as a first step toward selecting students’ productions related to these tasks.

Sequencing is not supported by the platforms. This practice has to do with the actual submissions of the students in the particular class, and hence, this is up to the teacher to consider them in a way that leads to the lesson aims. We see the question of how an automatic system can afford help to this important teachers’ practice as a challenge for future developments of LMSs.

Finally, connecting is indirectly afforded by two platforms, though in different ways. STEP provides hints for possible connecting actions based on the examples provided as possible answers and a minimal mathematical analysis for each example, together with the filtering option that may allow the teacher to see at a glance the mathematical features of her student’s submissions. The checklist provided by DESMOS, specifically the following—“Write a summary of the activity’s main ideas: How can you incorporate student work in that summary? What parts of the activity can you skip to ensure there is sufficient time for the summary?”—may support teachers in connecting students’ answers and the lesson aims.

Concluding Remarks

Sinclair and Robutti (2020) brought to the fore two main functions of the use of digital technology, namely “(a) as a support for the organisation of the teacher’s work (producing work sheets, keeping grades) and (b) as a support for new ways of doing and representing mathematics” (p. 845). One of the results of our study is the identification of a third function brought about by LMS: as a support provided to the teacher while enacting technology-based mathematics activities.

Indeed, our analyses of the four platforms highlight that teachers’ practices related to planning and enacting a technology-based mathematical activity in a classroom can be supported by digital tools to some extent. Yet, the teachers need to be able to recognise the platform affordances and turn them into instruments for their teaching (didactic instrument).

Monitoring practice, for which all the four platforms analysed provide the strongest affordances, can allow for data-based decision-making (DBDM) to a much greater extent compared with the traditional paper-and-pencil environment. Verbert et al. (2014) propose a learning analytics model (Fig. 8) that brings to the fore four stages in DBDM: (1) awareness that is only concerned with data, which can be visualised in different ways (e.g. as activity streams or tabular overviews), (2) reflection that focuses on users asking questions, (3) sensemaking that concerns users answering the questions identified in the reflection stage and getting new insights and (4) impact aiming at inducing new meaning or changing behaviour.

Fig. 8
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Learning analytics process model (Verbert et al., 2014)

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Research studies report teachers’ difficulties regarding DBDM, in particular related to the last two stages, namely making sense of the data and taking data-driven instructional decisions:

Although teachers can read the data (i.e. find specific information), they struggle to read beyond the data and to make interpretations that link the data to instructional decisions (Hebbecker et al., 2022, p. 1697).

The authors conclude on benefits of teacher training to help develop teachers’ DBDM practices:

The results demonstrate that even shorter and, thus, more scalable and resource-saving teacher trainings combined with instructional recommendations and prepared teaching material can have the potential to improve DBDM practice in general education (p. 1793).

In line with the abovementioned conclusion, we claim that teacher educators should consider the need to help teachers getting acquainted with dashboard information (awareness stage) and recognising it as an affordance for their teaching (self-reflection, sense-making and eventually impact stages).

Anticipating practice, which is the second one mostly afforded by the digital platforms, intends to help teachers prepare the enactment of technology-based mathematical activities in a classroom. Such affordances range from providing examples of task solutions to prompting teachers’ didactic reflections when planning or designing tasks for students. The quality of anticipation impacts the other three practices, namely selecting, sequencing and connecting that are only indirectly, if at all, supported by the platforms.

These four practices are obviously not specific to digital environments. However, since we consider that the students’ activity takes place within a digital platform, teachers need to handle issues related to students’ instrumental geneses and take these into account when selecting, ordering and connecting. This ability is therefore part of teachers’ didactic instrumental genesis as well.

Our findings, therefore, highlight several components of didactic instrumental genesis that mathematics teachers need to develop in order to take benefit from digital platform affordances. These components include the ability to base decision-making on data gathered and visualised in dashboard and the decisions pertaining to the five practices. In accordance with Hamilton et al.’s (2009) claim that “making sense of data requires concepts, theories, and interpretative frames of reference” (p. 5), we argue that the development of these components needs to be supported in teacher education or professional development programmes. These should aim at helping teachers analyse platform functionalities, resulting in recognising their affordances, which is a condition for sparking off instrumental geneses.

Moreover, the findings reported in this article point out possible new developments of digital technology supporting teachers’ teaching mathematics, especially selecting, sequencing and connecting practices. Such developments would require synergy between mathematics education researchers and technology developers.

In our study, we have focused on five practices and their relation to teachers’ didactic instrumental genesis. Nevertheless, our findings show that beyond the five practices, the platforms provide other affordances fostering didactic instrumental geneses. Such affordances aim, for example at supporting differentiated pedagogical strategies within a digital environment by allowing grouping students according to their performance (Labomep and WIMS platforms), technology-supported formative assessment (STEP) or supporting teaching interventions based on students’ responses (DESMOS). Further studies are needed to get a more comprehensive view of the processes of didactic instrumental genesis of mathematics teachers using digital platforms.