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Jacinto, H.; Carreira, S. Techno-Mathematical Fluency. Encyclopedia. Available online: https://encyclopedia.pub/entry/59748 (accessed on 20 May 2026).
Jacinto H, Carreira S. Techno-Mathematical Fluency. Encyclopedia. Available at: https://encyclopedia.pub/entry/59748. Accessed May 20, 2026.
Jacinto, Hélia, Susana Carreira. "Techno-Mathematical Fluency" Encyclopedia, https://encyclopedia.pub/entry/59748 (accessed May 20, 2026).
Jacinto, H., & Carreira, S. (2026, May 19). Techno-Mathematical Fluency. In Encyclopedia. https://encyclopedia.pub/entry/59748
Jacinto, Hélia and Susana Carreira. "Techno-Mathematical Fluency." Encyclopedia. Web. 19 May, 2026.
Peer Reviewed
Techno-Mathematical Fluency
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia6050101

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.

mathematics education mathematical problem solving digital technologies teaching and learning fluency affordances
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.

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.

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.

References

  1. Borba, M.; Villarreal, M. Humans-with-Media and the Reorganization of Mathematical Thinking; Springer: New York, NY, USA, 2005.
  2. Goos, M.; Galbraith, P.; Renshaw, P.; Geiger, V. Perspectives on Technology Mediated Learning in Secondary School Mathematics Classrooms. J. Math. Behav. 2003, 22, 73–89.
  3. Hoyles, C. Transforming the Mathematical Practices of Learners and Teachers through Digital Technology. Res. Math. Educ. 2018, 20, 209–228.
  4. Santos-Trigo, M. Problem solving in mathematics education: Tracing its foundations and current research-practice trends. ZDM Math. Educ. 2024, 56, 211–222.
  5. Moreno-Armella, L.; Hegedus, S. Co-Action with Digital Technologies. ZDM Math. Educ. 2009, 41, 505–519.
  6. Hegedus, S.; Laborde, C.; Brady, C.; Dalton, S.; Siller, H.-S.; Tabach, M.; Trgalova, J.; Moreno-Armella, L. Uses of Technology in Upper Secondary Mathematics Education; ICME-13 Topical Surveys; Springer: Cham, Switzerland, 2017.
  7. Sinclair, N. On Teaching and Learning Mathematics–Technologies. In STEM Teachers and Teaching in the Digital Era; Kolikant, Y., Martinovic, D., Milner-Bolotin, M., Eds.; Springer: New York, NY, USA, 2020; pp. 91–107.
  8. Gibson, J.J. The Theory of Affordances. In Perceiving, Acting, and Knowing: Toward an Ecological Psychology; Shaw, R., Bransford, J., Eds.; Erlbaum: Hillsdale, NJ, USA, 1979; pp. 67–82.
  9. Martinovic, D.; Freiman, V.; Karadag, Z. Visual Mathematics and Cyberlearning in View of Affordance and Activity Theories. In Visual Mathematics and Cyberlearning; Martinovic, D., Freiman, V., Karadag, Z., Eds.; Springer: New York, NY, USA, 2013; pp. 209–238.
  10. Artigue, M. Digital Technologies: A Window on Theoretical Issues in Mathematics Education. In Proceedings of CERME 5; Pitta-Pantazi, D., Philippou, G., Eds.; Cyprus University Editions: Nicosia, Cyprus, 2007; pp. 68–82.
  11. Noss, R. For a Learnable Mathematics in the Digital Cultures. Educ. Stud. Math. 2001, 48, 21–46.
  12. Brown, J.; Stillman, G.; Herbert, S. Can the Notion of Affordances Be of Use in the Design of a Technology Enriched Mathematics Curriculum? In Mathematics Education for the Third Millennium: Towards 2010. Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia; Putt, I., Faragher, R., McLean, M., Eds.; MERGA: Sydney, Australia, 2004; Volume 1, pp. 119–126.
  13. Drijvers, P.; Godino, J.D.; Font, V.; Trouche, L. One Episode, Two Lenses: A Reflective Analysis of Student Learning with Computer Algebra from Instrumental and Onto-Semiotic Perspectives. Educ. Stud. Math. 2013, 82, 23–49.
  14. Trouche, L.; Drijvers, P.; Gueudet, G.; Sacristan, A.I. Technology-Driven Developments and Policy Implications for Mathematics Education. In Third International Handbook of Mathematics Education; Bishop, A.J., Clements, M.A., Keitel, C., Kilpatrick, J., Leung, F.K.S., Eds.; Springer: New York, NY, USA, 2013; pp. 753–789.
  15. Chemero, A. What We Perceive When We Perceive Affordances. Ecol. Psychol. 2001, 13, 111–116.
  16. Chemero, A. An Outline of a Theory of Affordances. Ecol. Psychol. 2003, 15, 181–195.
  17. Stoffregen, T.A. Affordances as Properties of the Animal–Environment System. Ecol. Psychol. 2003, 15, 115–134.
  18. Greeno, J.G. Gibson’s Affordances. Psychol. Rev. 1994, 101, 336–342.
  19. Norman, D.A. The Design of Everyday Things; Doubleday: New York, NY, USA, 1990.
  20. Norman, D.A. The Design of Everyday Things, Rev. and expanded ed.; Basic Books: New York, NY, USA, 2013.
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