Techno-Mathematical Fluency

Definition

Techno-mathematical fluency (TmF) is the ability to coordinate mathematical knowledge with technological means—digital and non-digital—to solve mathematical problems and express solutions, by recognising affordances, selecting appropriate tools and data, and integrating them with mathematical ideas in iterative cycles of exploration and integration. It goes beyond instrumental tool use to encompass reasoning, modelling, representation, and communication mediated by technologies, and functions as a form of expertise important for both students’ learning and teachers’ professional practice.

1. Origins and Theoretical Roots

Our purpose is to outline the essential features of the concept of techno-mathematical fluency (TmF). Rather than emerging from a single empirical study, this discussion traces the genealogy of the concept and develops its meaning and structure. By mapping its intellectual origins and clarifying its boundaries, we aim to establish TmF as a distinct and analytically productive lens for understanding students and teachers’ problem-solving-based mathematical activity in digital environments.

1.1. The Transformative Role of Digital Technologies

The role and impact of digital tools in mathematics learning and teaching have stimulated fertile discussions over the past decades within Mathematics Education. Research on the use of computers in mathematics learning traditionally conceives these tools as pedagogical aids. There are, however, challenging views in academia. Borba and Villarreal [1] claim that, although digital tools may be considered as substitutes, aids or complementary to human activity, those are only minor roles. Instead, they propose that the (mathematical) processes mediated by technologies produce a reorganisation of the human mind and that knowing stems from the interactions between individuals, technology and the surrounding media. Technology not only gives rise to innovative ways of accessing information, but it also affords new styles of thinking and knowing. Humans-with-media conceptualises the transformational power of the media with which one thinks and acts. Goos and colleagues [2] have also approached such transformational power by stressing the notion of cognitive reorganisation in a learning community mediated by technological tools. They emphasise that such cognitive reorganisation occurs precisely when the interaction of learners with technology qualitatively transforms their mathematical thinking; they suggest, for example, that one of the effects of using spreadsheets and graphical software is to bring graphic and numerical reasoning to the forefront, relegating algebraic reasoning to the background. This means that the very process of learning mathematical ideas and concepts involves a productive appropriation of tools that alter the ways in which individuals formulate and solve problems. Therefore, the subject is the indivisible unit of human-and-tool.

1.2. The Subject–Tool as an Integrated Unit

This fundamental argument was clearly expressed by Hoyles [3] when stating that mathematical knowledge is inextricably linked to the tools in which the knowledge is expressed. Her work, standing from a constructionist perspective, led to the identification of different didactical roles that digital technologies can play in students’ mathematical activity. The argument is further supported by the conceptualization of the interaction between the tool and the individual in devising a problem-solving strategy or investigating mathematical properties in a problem situation [4]. That form of interaction is regarded as a co-action between the individual and the tool [5]. An illustrative metaphor of the concept is offered by [6], based on a virtuoso musical performance. The authors point out that the musician and the instrument come to function as a single integrated unit, with the music being co-produced by both. Through years of reflective practice, this seamless human–artefact integration, termed co-action, becomes a distinctive and creative interplay between the person and their tool or symbol system.

Along similar lines, Sinclair [7] discusses the possibilities of interaction between the learners and the mathematical concepts that technology has the power to change, thus suggesting that tools change both the concepts and the learning process. So, the focus must be on the student/tool as a unit, since it is not possible to establish a clear boundary between the students’ actions and the mathematical thinking triggered by the tools. In this view, different collectives (student + tool) can yield different ways of knowing, even when tackling the same problem with the same tool, because the technological medium reorganises the reasoning pathways and the representational moves available to the user.

In assuming that digital technologies can function as extensions of an individual’s cognitive activity in the production of mathematical thinking [2], it is important to recognise that different forms of knowledge may emerge from technology-mediated activity. Effective use of such tools relies on an adequate understanding of their affordances—the possibilities for action they offer [8]. Originally formulated within ecological psychology, the concept refers to the properties of an object that invite or enable particular actions, simultaneously grounded in the environment and in the perceiver’s engagement with it [8,9]. The notion has been widely adopted in mathematics education to examine human–technology interactions [10,11,12,13,14].

Several authors have expanded the relational character of affordances, arguing that they emerge from the interplay between the individual and the environment rather than from intrinsic properties of objects alone. Chemero [15,16] proposes that perceiving an affordance involves recognising the possibility for action in a given situation, while [17] emphasises that affordances describe what is possible to do, not what must be done. Greeno [18] further argues that understanding affordances requires considering the agent’s “abilities”, thus reinforcing the inseparability of agent and environment. In the field of human–computer interaction, Norman [19,20] highlights the interpretive dimension of affordances, shaped by the users’ prior experiences, goals, and knowledge.

When taken together, these perspectives converge on the understanding that affordances represent possibilities for action that arise from the indivisible relationship between individuals and their environment. This relational view, aligned with ecological perspectives and activity theory, has proven particularly valuable in mathematics education research, especially in examining the role of digital tools in shaping learners’ mathematical activity and the development of mathematical knowledge.

A unifying thread of those theoretical roots is a consistent shift: from viewing technology as a neutral aid to thinking of it as co-constitutive of mathematical thinking. Across the perspectives reviewed, digital tools are not mere add-ons to pre-existing cognition, they are participants in the very formation and reorganisation of mathematical activity. Humans-with-media, co-action, and related notions coincide on the idea that the unit of analysis is the human–tool collective, within which cognition, representation, and reasoning are dynamically reorganised.

Within this view, learning mathematics becomes a process of appropriating technological tools in ways that transform how problems are formulated, explored, and solved, and what counts as a legitimate mathematical pathway. Concepts such as affordances cement the inherently relational nature of these transformations: what a tool “affords” depends on the evolving interplay between its properties, the user’s goals, experiences, and abilities, and the surrounding activity system.

2. From Mathematical Problem Solving with Technology to Techno-Mathematical Fluency

The Problem@Web research project marked the beginning of our efforts, and for over two decades we have continued to promote, collect, and analyse a variety of scenarios in which students—primarily in middle grades (6th to 9th grade)—have engaged in solving non-routine mathematical problems by using the technological tools of their choice. These problems have covered a range of mathematical topics and concepts, including algebra, numbers, geometry, counting, statistics, and analytical thinking [21]. In a subsequent phase, we also explored how veteran mathematics teachers interact with technology when solving non-routine problems, focusing on their reasons for selecting specific tools and how these choices influence their strategies and thinking, both mathematically and pedagogically. More recently, our attention has turned to the study of individuals solving real-world modelling problems with the support of technology.

A substantial body of empirical data, gathered across diverse settings—including problem-solving competitions, classroom activities, home-based student contexts, online math clubs, and task-based interviews—has enabled the development of a theoretical framework describing problem-solving processes with technology. This work has provided the foundation for investigating how mathematical problem-solving activity is shaped by the use of technological tools among both students and teachers.

2.1. A Model of Mathematical Problem Solving with Technology: The MPST Model

Building on a socio-cultural perspective of learning, especially relying on the concepts of activity system and humans-with-media, and combining established models of mathematical problem solving and digital problem-solving frameworks, we developed the descriptive and analytical model known as mathematical problem solving with technology (MPST) [22,23], which comprises ten processes (

Table 1. Mathematical problem solving with technology model, adapted from Jacinto and Carreira [23].

Process	Description
Grasp	First engagement with the task (reading or stating it), making sense of the problem situation and conditions, and forming initial ideas about what it involves.
Communicate	Interactions with others of relevance while dealing with the problem.
Notice	Initial attempt to discern the key aspects of the problem, that is, what mathematics might be relevant and which digital tools may be useful.
Interpret	Consider the affordances of the technological resources to explore possible mathematical approaches to the solution.
Integrate	Combine technological and mathematical resources as part of an exploratory approach.
Explore	Draw on technological and mathematical resources to explore and analyse conceptual models that may lead to a solution.
Plan	Formulate an approach to obtain the solution on the basis of analysing the conjectures explored.
Create	Implement the planned approach by recombining resources in new ways to support the solution and generating new knowledge objects that contribute to solving-and-expressing the problem.
Verify	Engage in explaining and justifying the obtained solution, drawing on the available mathematical and technological resources.
Disseminate	Share the solution or outputs with relevant others and reflect on the effectiveness of the problem-solving process.

).

Under this lens, problem solving is not merely about generating an answer but involves a process of solving-and-expressing—a synchronous activity in which explanation, representation, and communication of thinking are integral to the solution itself and are closely intertwined with the media used to articulate and express them. In this re-specification of mathematical problem solving for technology-rich environments, classic models of mathematical problem solving (MPS), originally designed for paper-and-pencil approaches, were extended to include processes that are triggered and supported by technology. The MPST highlights how technological knowledge and actions (such as using graphing tools, dynamic geometry software, or spreadsheets), interweave with mathematical knowledge and actions (like formulating conjectures, testing hypotheses, or analysing patterns). The model also reveals how solutions evolve progressively, through successive micro-cycles of exploration and integration, into conceptual models that not only represent the problem situation but also serve as foundations for more formal mathematical reasoning.

Resulting from the application of the MPST model to the analysis of individuals’ problem-solving activity in which they resort to mathematical and expressive technological tools, the inseparability between the tool and the mathematical thinking of the solvers becomes evident and adds to the concept of humans-with-media. Thus, when students or teachers freely select digital tools to solve non-routine mathematical problems, such as dynamic geometry software or a spreadsheet, their solutions become self-explanatory artefacts that externalise the conceptual models they have built, and the quality/diversity of those models depends on the solver’s ability to coordinate mathematical ideas with the tool affordances. A clear conclusion is the following: the mathematics produced by humans-with-paper-and-pencil is, for instance, qualitatively different from that produced by humans-with-spreadsheets or that produced by humans-with-GeoGebra [22].

2.2. Techno-Mathematical Fluency and Problem-Solving Activity

Techno-mathematical fluency emerges from a sociocultural view of mathematical activity in which problem solving is conceived as action mediated by cultural tools. Drawing on activity-theoretical ideas, Jacinto [22] frames mathematical problem solving as an activity system whose structure and outcomes are reorganised when technological artefacts are present; the subject and tool form a functional unit that reshapes the kinds of mathematical actions that are possible and valued in the activity (e.g., exploration, observation, conjecture, proof). The results gathered from the actual analysis of individuals solving problems with diverse technological tools have further supported and refined the construct of techno-mathematical fluency, highlighting how learners’ interactions with technology foster new forms of mathematical reasoning and expression. In particular, the following findings are relevant to understanding TmF:

• Solving mathematical problems with technology involves multiple sequences of processes that repeat iteratively (micro-cycles).
• The micro-cycles emerging in the activity of solving mathematical problems with technology involve the process of Integrate.
• Typically, the activity includes several back-and-forth movements between the processes of Integrate and Explore.
• The process of Integrate generates an exploratory activity, which in turn leads to a new stage of integrating technological and mathematical resources, and so on, until a conceptual model with potential for realisation is obtained, allowing for the solution to be reached.
• The construction of the solution to the problem depends on the relationship between mathematics and technology; that is, these micro-cycles are characterised by an effective use of technology to overcome the challenges posed by the problem.
• The process of Integrate, when associated with the process of Explore, demonstrates a conscious search for affordances to support mathematical thinking, which in turn triggers new exploratory activity.

3. Conceptualising Techno-Mathematical Fluency

The previous results differ from some proposed conceptualizations of the path taken by problem solvers, particularly concerning the perception and integration of the affordances of technology-rich environments. For example, in her study related to the solution of tasks about functions, Brown [24] identifies the complexity of the activity involving perceiving and enacting affordances. As she states, part of the complexity arises from “doing mathematics” in a technology-rich teaching and learning environment (TRTLE) that substantially changes the scope and complexity of the tasks that can be given to students in a certain school grade. The task is no longer limited to one that can be solved manually within a reasonable time, nor restricted to mathematical ideas that can only be explored through manual methods.

Her account of the process of enacting affordances unfolds through several stages, beginning with a “situation” in which particular affordances can be utilised, and progressing toward a “resolved situation”, encompassing the actions required to move from one stage to the next. The model consists of seven stages—situation, strategy identified, affordances perceived, affordance bearers selected, affordance bearer(s) operationalized, affordance bearer(s) output interpreted, and situation resolved—together with six actions—translating, perceiving, choosing, using, interpreting technological output, and interpreting mathematical output with respect to the situation. Such representation of the process suggests a cognitive activity where successive steps occur sequentially, in a direct and unidirectional manner, regarding perceiving and enactment of affordances.

An important divergence between this account and the MPST model is that instead of a sequential progress, in successive steps, the MPST model reveals the complexity in the form of several small loops at multiple points in the activity, namely the back and forth between the integration and exploration processes, involving both the mathematical situation and the technological environment. This means that the subject thinks, solves, and expresses through the technological tool, giving rise to the co-action between the individual and the artefact. Such mathematical–technological thinking is a defining feature of the subject–tool unit, and it is precisely this distinctive capacity that gives rise to TmF.

Therefore, techno-mathematical fluency is seen as enclosing:

• the individual’s ability to engage in productive mathematical thinking using technological tools to reformulate or create new knowledge and to express that thinking effectively;
• the interdependent relationship between mathematical knowledge and technological or expressive tools while solving a mathematical problem or situation;
• technological tools as extensions of self, depending on one’s proficiency in aligning mathematical ideas with the tool affordances.

This last attribute of TmF is connected to the notion of task-technology fit, which is well established in the field of information systems research [25]. Rooted in the work of Goodhue and Thompson [26], the task-technology fit (TTF) suggests that information technologies are effective only when there is a close match between what the task demands and what the technology can do. They argue that, for a given technology to enhance individual performance, it must be employed in a way that is well aligned with the specific tasks it is designed to support.

Clearly, the TTF construct is not the only variable responsible for realising techno-mathematical fluency; rather, it is one among several conditions that jointly support how learners think and act mathematically with technological tools. So, how is techno-mathematical fluency revealed? It manifests in four interrelated ways:

• In the identification of an alignment or potential matching between the technological affordances and a given mathematical problem.
• In the capacity to perceive a problem in a dual way (mathematically and technologically), in a sort of (almost) bilingualism.
• In the quality of the interaction between the user (the subject) and the technology (the object or artefact) for a specific purpose.
• In the way integration and exploration are carried out, that is, in the ability to move cyclically between these two processes.

Techno-mathematical fluency is therefore evidenced when learners (a) recognise (mathematical) affordances in the tools, (b) select and configure resources purposefully, and (c) cycle iteratively between exploratory actions and justificatory explanations until a mathematically warranted solution is articulated in and through the chosen media. This construct, theorised from beyond-school contexts and later connected to classroom and professional knowledge, offers a conceptually robust, empirically grounded account of how technology and mathematics jointly shape cognition in the digital age.

Broadly, techno-mathematical fluency can be understood as a learners’ capacity to recognise, interpret and strategically mobilise the affordances of digital tools in order to formulate, explore and refine mathematical ideas. Our use of the term therefore emphasises not only what technologies make possible, but also how learners appropriate these possibilities in purposeful and conceptually grounded mathematical activity. As highlighted, TmF is situated within a theoretical landscape where mathematical knowledge, technological media, and the learners’ actions are inseparable. In this framework, different human–tool configurations can lead to qualitatively distinct forms of mathematical knowing and doing.

4. Empirical Evidence on Students’ Techno-Mathematical Fluency

In her thesis, Jacinto [22] conceptualises techno-mathematical fluency (TmF) in young learners as the capacity to solve and express mathematical problems with digital technologies, through a concomitant mobilisation of (i) knowledge of the technology, (ii) mathematical knowledge that can be enacted with that technology, and (iii) productive ways of coordinating both to create new techno-mathematical objects of knowledge (e.g., inscriptions, models, and digitally mediated arguments) that support understanding, obtaining a solution, and communicating it effectively.

The thesis further argues that TmF has distinct varieties, because the mathematical thinking that flows during solving-and-expressing (for example, geometric, visual, or covariational thinking) is moulded by the technologies learners choose and by how those technologies are used to develop mathematical work. Across three empirical cases, TmF is characterised through the specific techno-mathematical ways each student integrates digital tools into developing a conceptual model of the situation, conjecturing, generalising, and (to varying degrees) justifying. In Jessica’s case [23], TmF is grounded in dynamic geometry work with GeoGebra: the student uses robust constructions and dynamic variation to explore conditions, generate conjectures, and support explanation/proof, with conceptual development that shows traits of vertical mathematization. In Marco’s case [27], TmF is centred on visual thinking supported by combining tools such as spreadsheets, GeoGebra and image-editing software to construct and transform figures that function as visual arguments, tending to sustain more horizontal mathematisation through context-near inscriptions. In Beatriz’s case [28], TmF is strongly oriented towards expressivity, with PowerPoint operating as a tool for expressing (and thereby further developing) mathematical ideas: by composing and manipulating digital objects (images, colour, symbols, text and simple calculations), the student assigns mathematical meaning to artefacts and refines conceptual models, with indications of progression towards more sophisticated mathematisation, including vertical mathematisation supported by visual and covariational thinking. In this way, TmF is not treated as technological skills added to mathematics skills, but as a form of thinking-mathematically-with-the-tool, emerging from learners’ recognition of a tool’s affordances for mathematical work and from intentional appropriation shaped by prior experiences and the broader activity system in which techno-mathematical actions take place. When TmF is prominent, the learner–tool relationship [7] approaches symbiosis: the technology becomes “absorbed” into the activity, and the subject of such activity becomes the student-with-media [1].

In the work of [29], techno-mathematical fluency is used as a theoretical anchor—via the notion of mathematical digital competency [30] and building on earlier conceptualizations of TmF [22]—with a particular focus on techno-mathematical discourse at the intersection of computational thinking (CT) and mathematics. The study examines how such discourse develops when the technological medium is a programming environment.

Based on classroom episodes with 6th-grade students working with Scratch and GeoGebra, the authors show how students’ mathematical meanings and programming actions become intertwined but also potentially ambiguous. For instance, students initially accept a “forever” loop drawing repeated squares because it does not visibly “harm” the drawing, until the teacher reframes the issue as a need to communicate a non-redundant algorithm; later, when experimenting with small turning angles (e.g., 15°), students shift from “polygon” to “circle-arc” reasoning and are guided towards the idea that repeated turns must total 360°. A key breakdown occurs when students compare Scratch and GeoGebra: in Scratch, the “angle” parameter refers to a turning (external) angle, whereas in GeoGebra it aligns with the internal angle of a polygon—an overlap in language that can cause confusion but also creates opportunities for conceptual clarification. The authors argue that, when programming is involved, students need a specific form of techno-mathematical discourse: they must coordinate and often explicitly disambiguate mathematical and computational meanings. This discourse is closely linked to the artefact–instrument duality (instrumental genesis), since learners must appropriate the programming environment as an instrument for testing and communicating mathematical ideas through code and the representations it produces.

This study contributes conceptually by showing that TmF in programming contexts involves specific tensions and possibilities, such as semantic overlaps between environments, differences in modelling conventions, and the need to express mathematical objects algorithmically. On this basis, they argue for a more fine-grained theorisation of techno-mathematical discourse when the “technology” is not a dedicated mathematical instrument, such as in the case discussed by [28] where presentation software becomes a medium for producing and communicating mathematical meanings.

Techno-mathematical fluency is endorsed by [31] as a conceptual reference point for theorising what it means for young students to learn geometry with a digital artefact—in this case, GGBot (a drawing robot) programmed through SNAP!. The paper explicitly builds on Jacinto’s view of TmF as the activity-based symbiosis between mathematical and technological knowledge shaped by perceived tool affordances [22], and it proposes to take a step forward by framing this symbiosis through an embodied, socio-semiotic lens at the intersection of the Theory of Objectification [32] and Humans-with-Media [1].

The study develops a theoretical account of how and why a drawing robot supports geometry learning, focusing on how students’ perception, signs, activity, and knowledge are reconfigured through its use. Based on classroom episodes with 8-year-olds using GGBot—exploring movement and trace, figure-to-code reasoning, and code-to-figure prediction tasks—it identifies four key building blocks: (i) objectification processes and semiotic means, (ii) sensuous cognition and “domestication of the eye,” (iii) emerging thinking collectives (humans-with-paper-and-marker vs. humans-with-paper-and-marker-and-GGBot), and (iv) learning as transitions between activity domains.

The authors argue that the GGBot is effective when students’ perception is progressively “domesticated” across three domains: drawing with paper/marker before the robot, drawing with the robot (where actions must be expressed via step/turn commands), and drawing after the robot (predicting robot drawings without the robot present). Crucially, it is the transitions between these domains—not merely “adding” a digital tool—that enable students to “see geometry with the eye of the GGBot” and become “information-technology actors” who can think with the tool even in its absence. Empirical episodes illustrate both progress and struggle: while one student achieves a stable shift (e.g., interpreting turns as rotations), others exhibit partial domestication and difficulties, particularly in coordinating reference frames and handling turning direction.

This work links TmF to embodied semiotics and perceptual transformation, offering “transitions between domains of activity” as a theoretical tool for analysing why certain technology-rich learning trajectories work (or fail) with young learners, and for identifying where difficulties originate (e.g., angle-as-rotation; reference-frame coordination).

In [33] techno-mathematical fluency is treated as the students’ growing capacity to “speak the language” of a digital interactive mathematics learning environment (DIMLE) to construct and communicate mathematical generalisations. Becoming fluent in a DIMLE’s language is seen as a mechanism through which interaction with the environment can “plant seeds” of algebraic thinking. The study aims (i) to capture, in fine-grained detail, how a student’s learning of algebraic generalisation develops through interaction with the DIMLE eXpresser, and (ii) to advance a methodological approach for inferring students’ instrumented schemes from their visible actions. The authors build an analytic method grounded in the theory of instrumental genesis (TIG) (artefact/instrument; instrumentation/instrumentalization; scheme/technique) and focus on the recurring “colouring task” in eXpresser—constructing figural patterns so they remain correctly coloured as an “unlocked” number (a variable) changes. Empirically, they introduced eXpresser to six 7th graders and present a case study of one student (“Molly”), using session recordings and transcribed think-aloud excerpts (re-enacted in a new web-based version of eXpresser) organised into vignettes and coded as successive instantiations of the task.

The student’s progress is described as a development of instrumented schemes that makes her TmF increasingly visible. She moves from giving a single numerical answer for a fixed case to constructing and using a general expression, treating it as a mathematical object within the digital environment. This shift is driven by the DIMLE’s feedback, which prompts her to connect the pattern’s structure with symbolic representation and to coordinate different tools in order to maintain invariance. The paper also highlights that TmF should not be reduced to procedural use of tools, but understood as involving sense-making about numbers, operations, and generality. Overall, the study shows how TmF can be analysed as the co-development of tool use and algebraic thinking in a digital environment, and it proposes an analytic approach that may be extended to other DIMLEs and to future AI-supported learning systems.

The study by [34] reports on the first teaching unit of a project designed to introduce grade 10 students to Operations Research (OR) through authentic optimisation problems, collaborative group work, and digital tools (GeoGebra and Excel with Solver). The study set out to examine whether OR problems are suitable for regular secondary mathematics lessons, whether collaboration and digital technologies can foster students’ modelling and problem-solving processes, and how this experience influences students’ appreciation of OR. Their theoretical framing explicitly connects learning with digital technologies to TmF, emphasising the entanglement of mathematical and technological knowledge and skills in solving-and-expressing problems with technology, and highlighting the central role of recognising tool affordances.

Techno-mathematical fluency is used as an analytic lens for interpreting students’ success and difficulties when moving between modelling decisions and technological implementation, especially with Excel Solver. For instance, one group’s difficulties are explained by limited TmF: they failed to notice/interpret Solver affordances such as the ‘bin’ setting for binary variables, entered constraints inconsistently with the model, and even ignored Solver alerts that could have helped revision. In contrast, progress across tasks is documented through increasingly effective integration of mathematical structure and spreadsheet functions (e.g., SUMPRODUCT) and more successful use of Solver, with the authors arguing that recognising affordances is a “main generator” of TmF and that sustained work with Excel/Solver can generate a symbiotic relationship between mathematical concepts and digital representations.

In terms of advancing research on TmF, the study contributes fine-grained, classroom-based evidence of what it can look like in secondary students’ work on OR modelling: it becomes visible in how learners coordinate conceptual modelling choices with specific technological actions (variable types, constraint encoding, interpreting solver output), and in how breakdowns often stem from missed affordances rather than only from lack of mathematical knowledge. This extends the use of TmF from more general technology-based problem solving to the particular demands of optimisation and modelling with a spreadsheet, while also underscoring the need for instructional designs that explicitly support students in learning to see and use key affordances when new digital tools enter the modelling process.

Simões [35] analysed the impact of a teaching experiment, grounded in the development of students’ techno-mathematical fluency, on 5th graders’ ability to solve mathematical problems with technology. Based on the MPST model [22], she designed a classroom teaching experiment in a Portuguese public school to examine students’ (i) appropriation of the problem situation (ability to interpret the task statement and select relevant data), (ii) implementation of a techno-mathematical approach (adopt appropriate strategies, integrate technological resources, and explore the results of such integration), and (iii) explanation of the solution (obtaining, explaining and justifying it).

As reported by [36] the study followed a quasi-experimental pre-test/post-test design with a control group. After an initial “adaptation phase” to familiarise students with several digital tools (Excel, Scratch, GeoGebra), the intervention consisted of four 100 min inquiry-based lessons over roughly three months. Each lesson centred on a different problem type aligned with the curriculum and designed to “fit” [25,26] particular digital tools (spreadsheets, dynamic geometry, visual programming). Students’ performance was assessed through parallel pre/post-tests scored with a rubric across the criteria appropriation, implementation, and explanation, followed by descriptive and inferential analyses (t-tests and non-parametric comparisons). Results showed that groups were similar at baseline, but the Experimental Group significantly outperformed the control group at post-test, with significant gains in appropriation and implementation, and improvement in explanation mainly for obtaining the solution. However, justifying the obtained solution showed weaker improvement, indicating that supporting students’ justification may require additional instructional emphasis. After the teaching experiment, the Experimental Group showed significantly higher performance in indicators that reflect techno-mathematical fluency such as interpreting the problem statement, selecting relevant mathematical and digital tools, adopting and implementing strategies, integrating mathematical and technological resources, exploring what that integration produces, and obtaining a solution. This pattern suggests students became better at mobilising technology as part of mathematical problem solving—choosing and using tools to support reasoning and reach solutions.

Main Contributions from Research with Students to the Theoretical Framing of TmF

For a structured overview of the ways in which recent research contributes to the understanding of students’ techno-mathematical fluency,

Table 2. Summary of empirical studies on students’ techno-mathematical fluency.

Study	Participants	Context and Digital Tools	How TmF Is Framed	What Becomes Demanding for Students	Main Contribution to TmF Research
[22,23,27,28]	Individual young learners	GeoGebra Spreadsheet Image-editing software PowerPoint	TmF as thinking-mathematically-with-the-tool Productive integration of mathematical and technological knowledge	Coordinating tool affordances to devise conceptual models Generating conjectures Moving between horizontal and vertical mathematisation	Establishes distinct varieties of TmF shaped by tool choice and use Introduces learner–tool symbiosis and “student-with-media”
[29]	Grade 6 students	Scratch GeoGebra	TmF as techno-mathematical discourse at the intersection of computational thinking (CT) and mathematics	Disambiguating overlapping meanings Coordinating mathematical and computational interpretations Expressing mathematical ideas algorithmically	Refines TmF for programming contexts Highlights semantic tensions and instrumental genesis in code-based environments
[31]	8-year-old students	GGBot SNAP!	TmF framed through embodied and socio-semiotic perspectives Symbiosis between perception, action, and digital artefact	Coordinating reference frames; interpreting angle as rotation Transitioning between activity domains (before/with/after robot)	Introduces “transitions between domains of activity” as mechanism for developing TmF Links fluency to perceptual transformation
[33]	Grade 7 students	eXpresser	TmF as developing instrumented schemes; learning to “speak the language” of a DIMLE	Moving from fixed-case reasoning to invariant generalisation Assigning meaning to symbolic expressions Coordinating multiple digital artefacts	Provides TIG-based analytic method for tracing TmF Co-development of algebraic thinking and tool-mediated techniques
[34]	Grade 10 students	GeoGebra Excel with Solver	TmF as coordination of modelling decisions and technological implementation	Recognising and exploiting affordances (e.g., variable types, constraint encoding); interpreting solver outputs and alerts	Breakdowns often stem from missed affordances rather than lack of mathematics Extends TmF to optimization and modelling contexts
[35,36]	Grade 5 students	Excel Scratch GeoGebra	TmF aligned with MPST model	Integrating mathematical and digital tools Explaining/justifying solutions	Intervention-based evidence that TmF can be developed instructionally Multi-dimensional nature of TmF

offers a synthesises in terms of participants, context and digital tools, how TmF is framed, what becomes demanding for students, and each paper’s main contribution to TmF research; this is followed by a more detailed discussion of their relevance and main inputs to the advancement of the construct of TmF.

Across the studies reviewed, techno-mathematical fluency is consistently framed beyond students’ technical proficiency with digital tools: it refers to students’ efficient and meaningful coordination of mathematical reasoning with the affordances of digital media, in order to solve-and-express mathematical problems. While the foundational account is [22], more recent work extends TmF research by specifying what becomes demanding for students’ fluency in different digital environments and by sharpening theoretical and methodological lenses. One strand foregrounds techno-mathematical discourse, showing that when the technological medium is programming, fluency depends on coordinating and disambiguating mathematical and computational meanings (e.g., the meaning of “angle” across environments) and on the processes of instrumental genesis as students learn to communicate mathematically through code [29]. Another strand links TmF to embodied and socio-semiotic perspectives in robotics, arguing that fluency can be fostered through transitions between activity domains (before/with/after the robot), which progressively “domesticate” perception and support the objectification of key concepts [31]. A further strand, based on a designed interactive environment (eXpresser), frames fluency as learning to “speak the language” of a DIMLE to express generality: students’ efficiency is evidenced in the development of instrumented schemes, moving from fixed-case numerical answers to algebraic expressions with variables that preserve invariance under change, while also highlighting that fluent tool use may remain procedural unless coupled with structural sense-making [33].

Finally, classroom-based intervention studies make TmF visible as a set of conditions for success and sources of difficulty in students’ work with digital tools. In optimisation modelling tasks using Excel/Solver, efficiency is shown to depend strongly on students’ ability to notice and exploit key affordances (e.g., choosing variable types, encoding constraints, interpreting outputs), with difficulties often stemming from missed affordances rather than from “lack of mathematics” alone—supporting TmF as a distinct explanatory construct for tool-mediated modelling [34]. Complementary, quasi-experimental evidence indicates that targeted instruction can produce significant gains in dimensions aligned with “solving with technology” (problem appropriation and techno-mathematical implementation), while progress in explaining/justifying solutions is more limited, suggesting that TmF-related efficiency is multidimensional and that the expressing component may require more explicit instructional attention [35,36].

Taken together, these studies suggest that students’ efficient use of digital tools in mathematics learning is best understood through TmF as a tool-ecology-sensitive competence, with distinct varieties shaped by the mathematical thinking and technologies involved (including general-purpose tools), and as a learnable set of practices in which students come to think-mathematically-with-the-tool, approaching a “symbiotic” learner–tool relationship, which can be strengthened through purposeful teaching.

5. Empirical Evidence on Teachers’ Techno-Mathematical Fluency

Hernández, Perdomo-Díaz and Camacho-Machín [37] set out to analyse the kinds of classroom situations or events that future secondary mathematics teachers anticipate, based on their own experience of solving problems with GeoGebra. This exploratory study draws conceptually on the MUST framework (Mathematical Understanding for Secondary Teaching), particularly the perspectives of mathematical activity and mathematical context. In this study, techno-mathematical fluency is used as an analytic category to capture a particular kind of teacher competence that becomes relevant when students solve problems with GeoGebra.

The authors identify three types of prompts anticipated by the future teachers: prompts aimed at explaining the mathematical functioning of GeoGebra tools, prompts requiring the justification of properties observed in dynamic constructions, and prompts involving the formulation and testing of conjectures through construction. A key finding is that only one pair of students anticipated prompts across all three types, suggesting that recognising and working with techno-mathematical contingencies is not automatic for future teachers and should be explicitly addressed in teacher education.

The study by Hernández et al. [37] contributes to shifting the discussion of techno-mathematical constructs from being mainly about learners’ tool-mediated problem solving to also addressing teacher preparation, by showing how anticipation of classroom events can be analysed through a techno-mathematical lens. It positions techno-mathematical fluency as a specific and teachable component of professional noticing: recognising when a digital tool’s behaviour becomes mathematically consequential, and turning that contingency into an instructional opportunity, which is an aspect that complements broader frameworks of mathematical understanding for teaching in technology-rich problem-solving contexts.

In the work of Weinhandl et al. [38], techno-mathematical ability appears as an implicit requirement of what the authors call a “techno-mathematical learning environment”—a modern, IT-based environment in which teachers and students are expected to be fluent in the “language” of mathematical inputs and outputs to technologies and able to interpret and communicate with them. They explicitly add that, to solve problems in such an environment (with a focus on GeoGebra), students need the combined skills of problem solving with mathematics and technologies.

The goal of the empirical study is to identify secondary mathematics teachers’ anticipated concerns and benefits before the nationwide introduction of student digital devices at the start of lower secondary schooling in Austria. Based on interviews with 14 teachers from six schools, the authors identify four main categories: concerns about student discrimination through technology and the loss of basic mathematical knowledge and skills, and perceived benefits related to teachers’ development of technological competence and the use of technology to support differentiation and individualization. A notable finding is that most teachers felt sufficiently capable of integrating technology and did not report a strong need for further technical training.

The study shows that teachers anticipate the conditions under which students’ fluency would (or would not) develop in a techno-mathematical learning environment. For instance, their concern that students will arrive with very unequal experiences and skills in purposeful technology use directly signals an expected unequal distribution of techno-mathematical ability (some students “only play”, others can act meaningfully), with implications for equity and participation. At the same time, the concern that technologies may undermine “basic skills” suggests that many teachers still frame the relationship between technology and mathematics through a substitution lens, for example by seeing calculators as replacing mental or written calculation, rather than in line with the paper’s definition of a techno-mathematical learning environment as one requiring fluency in interpreting and communicating mathematical meanings through digital inputs/outputs. This article contributes a teacher-belief, pre-implementation baseline at a rare moment of large-scale digitalisation, clarifying the perceived obstacles and enablers that will likely shape whether techno-mathematical fluency becomes an educational reality—and pointing to the need for follow-up studies that examine how such fluency is actually cultivated once devices and digital resources become embedded in classroom practice.

The work by Santos-Trigo, Barrera-Mora and Camacho-Machín [39] documents the extent to which high-school mathematics teachers enhance and extend their problem-solving strategies when they systematically rely on GeoGebra, as a dynamic geometry system, and other online developments (e.g., Wikipedia) to contextualise tasks, review concepts, and construct dynamic models. The data analysis a representative task and traces how teachers’ reasoning evolves through problem-solving episodes supported by tool use. The authors show that teachers used online sources to contextualise the problem historically and mathematically, and—crucially—to motivate a classical heuristic: reducing complexity by exploring simpler/limiting cases. Within GeoGebra, this heuristic is amplified: teachers construct dynamic models, drag objects to observe invariants, measure attributes to test conjectures, and trace loci to identify key structures (e.g., conic sections) that become central to solution pathways. Thus, the results show the teachers’ increasing capacity to coordinate mathematical reasoning with the affordances of digital tools in order to represent, explore, justify, and extend problem solutions. In other words, the study conceptualises technology use as a factor that reconfigures the repertoire and scope of problem-solving strategies through dynamic modelling, exploratory activity, and communication, core features that techno-mathematical fluency research typically foregrounds. Furthermore, this work makes visible a progression from empirical/visual exploration towards more explicit geometric justification, suggesting that the technology-mediated exploration supports, but does not replace, mathematical argumentation.

Although in their study, Santos-Trigo et al. [39] do not frame their analysis explicitly in terms of techno-mathematical fluency, the construct is compatible with what the authors document. This study offers evidence of teachers’ developing techno-mathematical fluency as tool appropriation for mathematical inquiry: teachers learn to see mathematical structure through the affordances of dynamic modelling, and to treat digital representations as legitimate for conjecturing, validating, and communicating solutions. It also strengthens a key research claim in this field: when teachers move beyond “paper-and-pencil substitution” and exploit affordances such as dragging, measuring, and locus generation, technology can extend the domain of classical problem-solving strategies and reshape what counts as a productive route to mathematical reasoning.

Through an in-depth continuation of their earlier work, Hernández et al. [40] focused on the analysis of what prospective teachers actually do while solving a problem with GeoGebra in pairs, capturing how technology shapes their mathematical activity in real time. Within this study, techno-mathematical fluency appears as an explanatory construct for understanding differences in how pairs of prospective secondary mathematics teachers build and exploit dynamic models, and how this conditions their reasoning routes. The study aims to analyse the mathematical activity (noticing, reasoning, creation) that emerges as the pairs of prospective teachers solve an optimisation problem with GeoGebra, structured through three problem-solving episodes (understanding, exploration, search for multiple approaches).

The results show that the quality of the dynamic construction, and the capacity to use GeoGebra’s affordances mathematically, strongly influences what the pair can notice, conjecture, verify and justify. For example, some pairs struggled to construct a triangle with the given side lengths, revealing difficulties related both to mathematical knowledge and to the use of GeoGebra, which the authors interpret as reflecting the influence of prospective teachers’ techno-mathematical fluency on their approach. The chapter then develops a more detailed case-like account of the activity of pair of prospective teachers working on another problem, illustrating techno-mathematical fluency as the ability to coordinate tool-driven exploration (dragging, quantification, dynamic invariants), mathematical conjecturing, and validation/justification, including knowing when to extend the toolset (e.g., consulting external digital resources) to advance the solution.

In terms of advancing research, the work by [40] strengthens the field by showing, in fine grain, where techno-mathematical fluency becomes visible in prospective teachers’ activity: besides in operating tools, in how tool affordances reorganise mathematical noticing, drive conjecture refinement, and open (or constrain) pathways towards explanation. This complements the authors previous study focus on anticipating classroom prompts [37], by documenting the actual techno-mathematical work that underpins such prompts: for instance, the very need to explain “why this construction/tool output makes sense mathematically” becomes meaningful only when pairs experience mismatches, partial constructions, failed conjectures, or the necessity to redesign representations during exploration.

Through an exploratory study of a veteran secondary mathematics teacher, Jacinto and Carreira [41] use techno-mathematical fluency as a lens for characterising teachers’ proficiency in solving-and-expressing non-routine mathematical problems with digital tools. The authors draw on Chapman’s [42] framework of Mathematical Problem-Solving Knowledge for Teaching (MPSKT) as a theoretical basis for examining teachers’ knowledge for teaching mathematical problem solving (in this case, with technology). The study’s goal is to examine the role technology plays in an expert teacher’s problem-solving process and in the way it supports (and shapes) his mathematical thinking, with the broader purpose of clarifying what counts as knowledge for teaching mathematical problem solving with technology.

The results show that expert problem solving with technology is cyclic, unfolding through micro-cycles in which the teacher repeatedly integrates mathematical and technological resources and then explores the outcomes to advance (or abandon) emerging approaches. Tools played major but different roles at different moments: GeoGebra supported early interpretation/modelling attempts, while the spreadsheet enabled systematic exploration and eventually the construction of a stronger conceptual model leading to the solution; importantly, the artefacts produced (e.g., spreadsheet outputs) served both as problem-solving resources and as expressive resources in the final explanation. The study argues that this is precisely where TmF becomes visible: the teacher’s proficiency is marked by his ability to recognise and exploit tool affordances to (a) interpret the situation techno-mathematically, (b) explore and refine a conceptual model, and (c) produce a coherent techno-mathematical solution and explanation.

This study contributes by making TmF a theoretically grounded component of teachers’ knowledge for teaching problem-solving with technology—extending discussions of teachers’ problem solving knowledge by foregrounding proficiency in solving-and-expressing with digital tools as a distinctive and necessary capability. Empirically, it provides a fine-grained account of how mathematical and technological knowledge intertwine in expert activity, while also pointing to future research on further conceptualising techno-mathematical thinking and examining how it develops across teachers with different backgrounds and in contexts such as mathematical modelling.

Ovadiya [43] examines how in-service teacher-researchers can develop techno-mathematical fluency by designing and teaching mathematical problems with digital tools. The study draws on Jacinto and Carreira’s [41] view of teachers’ TmF as an ability that bridges the didactic gap between paper-and-pencil teaching and reflective, concept-oriented uses of technology. In particular, TmF is understood as the teachers’ developing ability to pose and teach problems in ways that integrate mathematical understanding for teaching, knowledge and skills with digital tools, and communication and reflection about students’ interactions with digitally represented mathematics. The analysis highlights that this ability develops especially when teachers face unforeseen (contingent) classroom events that require them to depart from their original plan to respond to students’ reasoning, prompting in-the-moment adaptation and subsequent systematic reflection, and supporting a shift from using technology as a mere substitute to exploiting its affordances for mathematical exploration.

Findings point to four macro-triggers that show where and how in-service teachers’ techno-mathematical fluency develops: moving from a mainly procedural to a more ontological understanding through dynamic representations; learning to represent the essence of concepts and theorems visually and dynamically through the tool; realising that modifying dynamic constructions can generate new mathematical relationships and support task design; and recognising that fluency also depends on teaching the posed problem, which can reveal ambiguities and prompt redesign. Overall, TmF is used as an analytic construct to characterise teachers’ growing ability to coordinate tool affordances, mathematical goals, task design, and in-the-moment pedagogical decisions through iterative design–teach–reflect cycles.

Main Contributions from Research with Teachers to the Theoretical Framing of TmF

Empirical work on mathematics teachers’ and prospective teachers’ techno-mathematical fluency, as summarised in

Table 3. Summary of empirical studies on teachers’ techno-mathematical fluency.

Study	Participants	Context and Digital Tools	How TmF Is Framed	Where TmF Becomes Visible	Main Contribution to Teacher TmF Research
[37]	Prospective secondary mathematics teachers	Anticipating classroom events after solving problems with GeoGebra	TmF as a component of professional noticing within the MUST framework	In the anticipation of three types of prompts: explaining tool operation mathematically; justifying observed properties; fostering conjecture-and-test cycles	Positions TmF as a teachable professional competence in teacher education, showing that anticipating techno-mathematical contingencies is not automatic
[38]	In-service secondary teachers	Technology-rich “techno-mathematical learning environment” (GeoGebra and student devices)	TmF as mathematical–technological fluency required for participation in IT-based environments	In teachers’ expectations and concerns regarding students’ preparedness, equity, and preservation of “basic skills”	Provides a belief-based baseline of conditions shaping students’ future TmF development during large-scale digitalisation
[39]	High-school mathematics teachers	GeoGebra and online resources (Wikipedia) for problem solving	Implicitly aligned with TmF as tool appropriation for mathematical inquiry	In dynamic modelling, heuristic extension (e.g., exploring limiting cases), dragging, measuring, and locus generation	Shows how technology extends classical problem-solving strategies and supports progression from empirical exploration to explicit justification
[40]	Prospective secondary teachers	Real-time problem solving with GeoGebra	TmF as explanatory construct for differences in dynamic modelling and reasoning	In the quality of constructions, affordance perception, conjecture refinement, and validation during problem-solving episodes	Documents how tool affordances reorganise mathematical noticing and open or constrain explanatory pathways
[41]	Expert secondary mathematics teacher	Non-routine problem solving with GeoGebra and spreadsheets	TmF as a component of Mathematical Problem-Solving Knowledge for Teaching (MPSKT)	In cyclical micro-cycles integrating mathematical and technological resources; artefacts functioning as reasoning and expressive tools	Establishes TmF as a theoretically grounded dimension of teachers’ knowledge for teaching problem solving with technology
[43]	In-service teacher-researchers	Designing and teaching mathematical problems with digital tools	TmF as developing ability to bridge paper-and-pencil teaching and reflective technology integration	In “macro-triggers” arising from contingent classroom events requiring adaptation and redesign	Shows how TmF develops through iterative design–teach–reflect cycles and in-the-moment pedagogical decision-making

, positions it as a key condition for students’ efficient participation in technology-rich mathematics: teachers need to recognise and exploit tool affordances mathematically, orchestrate task design and implementation, and make sense of (and respond to) students’ digitally mediated reasoning. TmF is not treated as a “general digital skill”, but as a profession-specific ability visible in teachers’ capacity to (i) anticipate when tool outputs will be mathematically significant, (ii) guide students from tool outputs to mathematical meanings, and (iii) support justification and communication in digital environments [37,38,39].

Prospective teachers’ readiness to support students’ efficient use of digital tools can be analysed through the prompts they anticipate in technology-based problem solving, ranging from explaining the mathematics embedded in tool operations (explicitly linked to techno-mathematical fluency), to supporting proof and fostering conjecture-testing through dynamic constructions, anticipating this full range; however, this is not automatic and appears to require explicit attention in teacher education [37]. The quality of dynamic constructions and the ability to perceive mathematical affordances in the tools strongly shape what can be noticed, conjectured, verified, and justified, showing how TmF becomes visible as tool affordances reorganise mathematical activity and open or constrain paths towards mathematical explanation [40].

In the context of large-scale technology implementation in schools, teachers’ expectations highlight key conditions for the development of students’ TmF: while technology is associated with opportunities for differentiation and teachers’ own learning, concerns about unequal student preparedness and the possible weakening of “basic skills” suggest that TmF may develop unevenly unless it becomes an explicit focus of classroom work and support [38].

Another contribution characterises teachers’ developing techno-mathematical fluency through tool appropriation for mathematical inquiry, showing how dynamic modelling and online resources can extend classical problem-solving heuristics and support a progression from empirical exploration towards more explicit justification, features that are directly relevant to enable students to use digital tools efficiently [39].

Finally, two studies refine TmF as part of teachers’ professional knowledge by focusing on teaching and problem solving as intertwined practices. A case of an expert teacher makes visible “micro-cycles” of integrating and exploring with multiple tools, where digital artefacts function both as reasoning and as expressive resources, thereby strengthening the claim that TmF is a distinctive component of knowledge for teaching problem solving with technology [41]. In parallel, an action-research design shows that in-service teachers’ techno-mathematical fluency develops through contingent classroom events that force in-the-moment adaptation and later reflection; these “macro-triggers” clarify where teacher TmF grows (e.g., shifting from procedural to ontological understandings, making concepts visible dynamically, redesigning tasks after enactment), and why such growth matters for students’ efficient engagement in digital environments [43]. Together, these studies reinforce techno-mathematical fluency as a productive lens for mathematics education research.

6. Setting the Difference to Neighbouring Constructs

As is easily recognisable, the concept of techno-mathematical fluency is close to other established theoretical constructs in the field. At this point, we intend to clarify important distinctions between some of these constructs and the notion of techno-mathematical fluency in the context of solving non-routine mathematical problems using digital tools.

6.1. TmF in Contrast to Instrumental Genesis

The first distinction lies between TmF and the concept of instrumental genesis, as elaborated by key researchers such as Guin and Trouche [44], Trouche [45,46], Artigue [47], and Drijvers et al. [48]. Within this perspective, the use of a technological tool is understood as involving a process of instrumental genesis, through which the artefact is progressively transformed into an instrument. This instrument is conceived as a psychological entity that brings together the artefact itself and the schemes that the user develops to employ it in particular kinds of tasks. In these instrumentation schemes, technical knowledge of the artefact and domain-specific knowledge are closely interwoven. Instrumental genesis can thus be seen as the joint emergence of schemes and techniques for acting with the artefact. From this standpoint, interaction between the subject and the artefact is a crucial element. In this specific form of interaction, the subject develops efficient procedures to manipulate the artefact, thus acquiring knowledge that enables new or alternative uses of it. The process shapes instrumented activity through three specific features: (1) the artefact constraints that the subject must identify, understand, and manage; (2) the resources that artefacts afford for action; and (3) the procedures linked to the artefact use. As [46] explains, two essential ideas underlie this approach: (i) the distinction between an artefact and an instrument (resulting from an appropriation process); (ii) the distinction between a process of instrumentation, directed from the artefact towards the subject, and a process of instrumentalization, directed from the subject towards the artefact. Overall, the focus of instrumental genesis is on the process through which a learner appropriates an artefact (e.g., CAS, DGS, spreadsheet) and turns it into an instrument, through intertwined processes of instrumentation and instrumentalization.

Otherwise, TmF focuses on the resulting capabilities of the subject–tool unit, namely on how productively and flexibly a problem solver can mathematically think, reason, and express a solution to a mathematical situation using tools as extensions of self. As such, when addressing techno-mathematical fluency, technology is already taken as part of an integrated human–tool system (humans-with-media); the focus is less on “appropriating” the tool and more on the interdependence between mathematical ideas and technological affordances within problem-solving activity. Furthermore, attention is focused on how rich, flexible, and conceptually grounded is the mathematical thinking that the subject–tool unit can produce as well as on how affordances are strategically mobilised to formulate, explore, and refine mathematical ideas. In short, instrumental genesis helps to describe how learners come to work with a tool; techno-mathematical fluency helps to describe how well and in what ways they think and act mathematically with that tool in technology-rich problem-solving contexts. In fact, to a certain extent, one can argue that successful instrumental genesis is often a precondition for techno-mathematical fluency.

In essence, techno-mathematical fluency—as an activity theory-informed construct—rests on these core principles:

• alignment between task demands and technical affordances (task–tool fit),
• dual conceptualization of problems (mathematically and digitally),
• cyclical integration–exploration activity as a cornerstone of human–tool interaction.

6.2. TmF in Contrast to Mathematical Digital Competency

The second distinction involves the separation between techno-mathematical fluency and mathematical digital competency (MDC), the latter proposed by Geraniou and Jankvist [30]. MDC is conceived as a bridge between two domains: disciplinary mathematical competencies and generic digital competencies. To explain how these domains interact, the authors draw on instrumental genesis and the theory of conceptual fields. They acknowledge that TmF overlaps with aspects of MDC but argue that TmF alone is insufficient for analysing how digital tools shape students’ problem-solving activities, particularly the developmental interplay between their digital and mathematical knowledge.

The framing of MDC as a combination of two separate skill sets carries an implicit assumption: it suggests that digital proficiency and mathematical proficiency can be developed independently and later merged. However, our empirical studies show that when students engage in technology-rich mathematical problem solving, digital tools and mathematical thinking do not simply blend into one another, they transform each other, producing a qualitatively new, context-specific expertise that could not exist in isolation. Furthermore, empirical research on the interplay between digital literacy and domain-specific knowledge acquisition highlights the difficulty in disentangling their mutual influences. For instance, Govender [49] led a systematic literature review on the relationship between digital literacy and STEM skills, which supports the view of mutual constitution rather than simple combination. This bidirectional relationship indicates that digital and disciplinary competencies co-evolve: each of them shapes and is shaped by the other during learning.

If digital and mathematical competencies are interdependent and co-constitutive, then a construct that treats them as separable domains to be “blended” (like MDC) requires additional theoretical machinery—such as instrumental genesis—to explain their ongoing interaction. By contrast, TmF starts from the premise that mathematics and technology are already inseparable in mathematical problem-solving activity. It does not model their fusion; it describes the fluent, integrated performance that emerges when a learner thinks and acts mathematically through digital tools in non-routine problem-solving contexts. Thus, while MDC and TmF appear related, they remain conceptually distinct.

Finally, another distinction worth highlighting concerns the differentiation between digital competency and digital fluency, as currently advocated by researchers in the field of Educational Technology and, more globally, within Information Science.

According to Ilomäki et al. [50] digital competency is a loose and boundary concept. From their descriptive review of school-related empirical studies the authors concluded that digital literacy is commonly used as a synonym for digital competency. Otherwise, several scholars have conceptualised digital capability as a continuum encompassing different proficiency levels [51,52]. On this continuum, some models suggest a scale starting at digital foundations, progressing to digital literacy, and culminating in digital fluency. According to the model proposed by Cain and Coldwell-Neilson [51], the following stages are established: digital foundations consisting of access to digital tools or technologies and some technical proficiency; digital literacy consisting of consciously and competently applying digital skills; and digital fluency consisting of knowledge, skills, attributes and identity formation that enable adaptive inquiry, communication, problem solving and creation in digital worlds. A similar perspective is shared by Sparrow [53], who contends that digital literacy is not the same as digital fluency. The author explains the difference by drawing on the case of learning a foreign language. In that case, literacy enables reading, speaking, and listening for comprehension, while fluency allows creating original content like stories, poems, plays, or conversations. Likewise, digital literacy involves understanding how to use digital tools, whereas digital fluency empowers the creation of novel outputs with those tools. It may be described as harnessing technology to generate new knowledge, and solutions to challenges and problems, while integrating critical thinking, complex problem solving, and social intelligence to address them.

In general, the idea of digital progression places digital fluency above digital literacy or digital competency, with digital fluency being marked by individuals creating something via technology. Digital fluency is manifested when individuals exercise agency to leverage appropriate digital tools and strategies as part of problem solving (such as learners’ productions in problem-solving-and-expressing) and generate new knowledge in response to needs [52]. This particular argument strongly foregrounds the needs driving the activity and reinforces the distinction between MDC and TmF. The core elements of activity theory—subjects (individuals or groups) pursuing an object toward a goal, mediated by tools, yielding results that often diverge from the intent and loop back as feedback—frame this dynamic. Techno-mathematical fluency emerges as the subject–tool unit targeting an object (e.g., solving-and-expressing mathematical problems), with cycles of integration and exploration generating productive, mathematically and technologically informed interpretations of feedback. Thus, it ties directly to technology-mediated problem solving, as outlined by the MPST model.

6.3. Synthesis of Fundamental Distinctions

The following core ideas structure the distinctions that make techno-mathematical fluency unique compared to neighbouring constructs:

(1)

TmF vis-avis Instrumental Genesis

• Different focus:
  • While instrumental genesis explains how a learner appropriates a tool, turning an artefact into an instrument (focus on the process), TmF describes what the learner can do—the quality of thinking and problem solving with the tool (focus on the outcome).

• Tool status:
  • Instrumental genesis treats the tool as something being learned and shaped, while TmF assumes the tool is already integrated into a human–tool system (“humans-with-media”).

• Type of knowledge emphasized:
  • Instrumental genesis focuses on developing schemes and techniques for using tools; instead TmF focuses on rich, flexible, and conceptually grounded mathematical thinking using those tools.

• Role of interaction:
  • While instrumental genesis states that interaction leads to understanding of constraints, affordances, and procedures, TmF sees interaction as strategically leveraged to explore, express, and refine mathematical ideas.

• Relationship between the two:
  • Although instrumental genesis is often a precondition for TmF, the latter goes further by emphasizing productive mathematical activity in complex, non-routine problem-solving.

(2)

TmF vis-avis Mathematical Digital Competency (MDC)

• Integration or combination:
  • MDC intends to combine digital and mathematical competencies (as two domains); instead TmF treats them as inseparable and co-constitutive in practice.

• Nature of capability:
  • MDC suggests competencies can exist separately and then be merged whereas TmF argues that in mathematical problem solving they form a new, unified capability that cannot be reduced to parts.

• Developmental perspective:
  • MDC relies on frameworks like instrumental genesis to explain development while TmF directly captures the enacted, situated performance of thinking-with-tools.

• View of learning:
  • MDC aligns with the idea of parallel development of digital and mathematical skills whereas TmF emphasizes co-evolution and mutual shaping during activity.

• Relationship between the two:
  • MDC describes a blended profile of digital and mathematical competencies but is not oriented toward mathematical problem-solving-and-expressing as a core activity. On the other hand, TmF aligns more with digital fluency but is domain-specific (mathematics). TmF frames fluency within goal-directed, tool-mediated activity driven by problem-solving needs.

7. Implications and Future Research Directions

The evidence reviewed in this entry positions techno-mathematical fluency as a construct with direct implications for how mathematics is taught and learned in technology-rich environments. Conceptualising mathematical problem solving as solving-and-expressing with technological tools, implies that instruction needs to legitimise and cultivate the iterative micro-cycles of integrating and exploring that characterise productive work with digital media [22,23]. This has curricular consequences: tasks should be designed so that students are expected (and supported) to interpret tool outputs mathematically, to refine conceptual models through successive cycles, and to communicate solutions through the chosen media rather than positioning technology as ancillary to the core mathematical work. At the level of classroom practice, a TmF perspective suggests shifting attention from “using a tool correctly” to recognising and mobilising affordances strategically, including the ability to perceive a problem “bilingually” (mathematically and technologically) and to sustain exploration until a mathematically warranted solution can be articulated [22,25,33].

Beyond classroom-level analyses of tool-mediated mathematical problem solving, studies situated in large-scale digitalisation contexts highlight enabling and constraining conditions for the development of TmF: teachers anticipate benefits, such as differentiation and richer representations, alongside risks such as unequal student preparedness and a perceived weakening of basic skills. From a TmF perspective, these findings foreground that students’ TmF is likely to become unevenly distributed unless classroom practices (and educational policies) deliberately address access, consider the students’ prior experiences and the quality of mathematical work expected with particular digital tools. They also suggest that teachers may need support to connect emerging technologies (e.g., generative Artificial Intelligence or Extended Reality tools) to concrete classroom activity rather than viewing them as peripheral to mathematics learning.

For teacher education and professional learning, the reviewed research reinforces that students’ opportunities to develop techno-mathematical fluency are strongly shaped by teachers’ own TmF, as reflected in how they work through technologies when tackling mathematically challenging tasks and how they respond to unforeseen events arising from students’ tool-mediated activity. TmF becomes visible in teachers’ capacity to predict how tool use may shape the mathematical work, to connect digital outputs to mathematical meanings, and to leverage emergent “teachable moments” in technology-based problem solving [37,40,43]. This has implications for teacher education and professional development: preparing teachers for technology-rich classrooms involves more than technical training, since a central demand concerns managing tool-mediated contingency and the interpretive openness of digital representations [43]. Work with prospective teachers further suggests that anticipating such events can be cultivated explicitly, strengthening the link between personal tool use and pedagogical decision-making in technology-rich problem solving [37,40]. In addition, teachers’ expectations in contexts of rapid digitalisation foreground the need to address unequal preparedness and participation: when students bring very different prior experiences and familiarity with digital tools, the development of TmF is likely to be uneven unless it becomes an explicit focus of classroom work and support [38]. This positions TmF as an equity-relevant educational aim, requiring attention to how access, prior experience, and classroom norms shape meaningful participation in techno-mathematical activity.

Future research can build on this body of work by addressing several priorities. First, there is a need to broaden and sharpen theorisation of TmF across a range of digital tool environments and representational systems, particularly where “technology” is not (always) a dedicated mathematical tool, demanding close attention to techno-mathematical discourse. Existing work illustrates both fine-grained models and intervention outcomes that separate “doing with tools” from “explaining/justifying”; hence a key direction is to develop assessment approaches that capture TmF as a coordination of processes.

Second, more longitudinal designs are warranted to study how fluency develops over time, and across domains, particularly the conditions under which students’ progress from tool-enabled exploration to robust justification and communication (a recurrent challenge across findings), and to trace how the earlier practices of noticing affordances support later, more complex tool-mediated reasoning.

Third, research should continue to expand beyond a narrow set of “mathematical software” by analysing TmF across broader tool ecologies—including general-purpose tools used for mathematical expressivity and hybrid environments where programming, robotics, and interactive systems reshape what it means to model, generalise, and explain. This opens a research agenda on how traces of students’ tool use might support timely feedback, personalised scaffolding, and potentially AI-enabled support, while also requiring caution that the interaction with the tool does not mask limited mathematical understanding, that is, apparent “fluency” should not be taken as evidence of robust structural sense-making when the underlying mathematical ideas have not been secured. In this sense, this expansion is not merely a change of tools, it requires theorising how tool actions, representations, discourse, justification, and meaning-making differ across media, and how these differences create new demands on students’ and teachers’ techno-mathematical fluency.

Across these directions, techno-mathematical fluency offers a coherent framework for studying efficiency with digital tools as the capacity to build, refine, and communicate mathematical meaning through the technologies that increasingly mediate mathematical work in school and beyond.