Abrahamson D. (2021) Grasp actually: An evolutionist argument for enactivist mathematics education. Human Development, online first. https://cepa.info/7084

What evolutionary account explains our capacity to reason mathematically? Identifying the biological provenance of mathematical thinking would bear on education, because we could then design learning environments that simulate ecologically authentic conditions for leveraging this universal phylogenetic inclination. The ancient mechanism coopted for mathematical activity, I propose, is our fundamental organismic capacity to improve our sensorimotor engagement with the environment by detecting, generating, and maintaining goal-oriented perceptual structures regulating action, whether actual or imaginary. As such, the phenomenology of grasping a mathematical notion is literally that – gripping the environment in a new way that promotes interaction. To argue for the plausibility of my thesis, I first survey embodiment literature to implicate cognition as constituted in perceptuomotor engagement. Then, I summarize findings from a design-based research project investigating relations between learning to move in new ways and learning to reason mathematically about these conceptual choreographies. As such, the project proposes educational implications of enactivist evolutionary biology.

Abrahamson D., Dutton E. & Bakker A. (2021) Towards an enactivist mathematics pedagogy. In: Stolz S. A. (ed.) The body, embodiment, and education: An interdisciplinary approach. Routledge, New York: in press.

Enactivism theorizes thinking as situated doing. Mathematical thinking, specifically, is handling imaginary objects, and learning is coming to perceive objects and reflecting on this activity. Putting theory to practice, Abrahamson’s embodied-design collaborative interdisciplinary research program has been designing and evaluating interactive tablet applications centered on motor-control tasks whose perceptual solutions then form the basis for understanding mathematical ideas (e.g., proportion). Analysis of multimodal data of students’ hand- and eye- movement as well as their linguistic and gestural expressions has pointed to the key role of emergent perceptual structures that form the developmental interface between motor coordination and conceptual articulation. Through timely tutorial intervention or peer interaction, these perceptual structures rise to the students’ discursive consciousness as “things” they can describe, measure, analyze, model, and symbolize with culturally accepted words, diagrams, and signs – they become mathematical entities with enactive meanings. We explain the theoretical background of enactivist mathematics pedagogy, demonstrate its technological implementation, list its principles, and then present a case study of a mathematics teacher who applied her graduate-school experiences in enactivist inquiry to create spontaneous classroom activities promoting student insight into challenging concepts. Students’ enactment of coordinated movement forms gave rise to new perceptual structures modeled as mathematical content.

Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. https://cepa.info/3731

Research in mathematics education is a discursive process: It entails the analysis and production of texts, whether in the analysis of what learners say, the use of transcripts, or the publication of research reports. Much research in mathematics education is concerned with various aspects of mathematical thinking, including mathematical knowing, understanding and learning. In this paper, using ideas from discursive psychology, I examine the discursive construction of mathematical thinking in the research process. I focus, in particular, on the role of researchers’ descriptions. Specifically, I examine discursive features of two well-known research papers on mathematical thinking. These features include the use of contrast structures, categorisation and the construction of facts. Based on this analysis, I argue that researchers’ descriptions of learners’ or researchers’ behaviour and interaction make possible subsequent accounts of mathematical thinking.

Open peer commentary on the article “Designing Constructionist E-Books: New Mediations for Creative Mathematical Thinking?” by Chronis Kynigos. Upshot: Chronis Kynigos’s article invites us to explore how to make familiar objects for learning — namely, books — more constructionist. In my response, I ask questions about the affordances and potential limitations of books as central objects, particularly about the role of the learner in relation to the objects.

Brown T. (1994) Creating and knowing mathematics through language and experience. Educational Studies in Mathematics 27: 79–100.

The radical constructivist assertion that the student constructs his or her own knowledge as opposed to receiving it “ready made” echoes the classical debate as to whether the human subject constitutes the world or is constituted by it. This paper shows how the philosophical traditions of post-structuralism and hermeneutic phenomenology offer approaches to effacing this dichotomy and how this forces a re-assertion of the teacher’s role in the student’s constructing of mathematical knowledge. It is also shown how hermeneutic phenomenology provides an opportunity to ground constructivist mathematical thinking in the material qualities of the world.

Corcoran D. (2015) Thoughts on Developing Theory in Designing C-Books. Constructivist Foundations 10(3): 316–317. https://cepa.info/2139

Open peer commentary on the article “Designing Constructionist E-Books: New Mediations for Creative Mathematical Thinking?” by Chronis Kynigos. Upshot: As a mathematics teacher educator and “digital tourist,” I focus my response to the many questions posed by Kynigos from three perspectives. First, I outline the theories he uses to frame the reporting of the research into the design of constructionist e-books. Second, I compare his theoretical tools with design-based research as an organising framework for a research project of this nature. Third, I propose the possible contribution of further theory-testing to the work.

Díaz-Rojas D. & Soto-Andrade J. (2017) Enactive metaphors in mathematical problem solving. In: Doole T. & Gueudet G. (eds.) Proceedings of CERME10. Dublin, Ireland: 3904–3911. https://cepa.info/6173

We are interested in exploring the role of enactive metaphoring in mathematical thinking, especially in the context of problem posing and solving, not only as a means to foster and enhance the learner’s ability to think mathematically but also as a mean to alleviate the cognitive abuse that the teaching of mathematics has turned out to be for most children and adolescents in the world. We present some illustrative examples to this end besides describing our theoretical framework.

Díaz-Rojas D., Soto-Andrade J. & Videla-Reyes R. (2021) Enactive Metaphorizing in the Mathematical Experience. Constructivist Foundations 16(3): 265–274. https://cepa.info/7155

Context: How can an enactive approach to the teaching and learning of mathematics be implemented, which fosters mathematical thinking, making intensive use of metaphorizing and taking into account the learner’s experience? Method: Using in-person and remote ethnographic participant observation, we observe students engaged in mathematical activities suggested by our theoretical approach. We focus on their idiosyncratic metaphorizing and affective reactions while tackling mathematical problems, which we interpret from our theoretical perspective. We use these observations to illustrate our theoretical approach. Results: Our didactic examples show that alternative pathways are possible to access mathematical thinking, which bifurcate from the metaphors prevailing in most of our classrooms, like teaching as “transmission of knowledge” and learning as “climbing a staircase.” Our participant observations suggest that enacting and metaphorizing may indeed afford a new and more meaningful kind of experience for mathematics learners. Implications: Our observations highlight the relevance of leaving the learners room to ask questions, co-construct their problems, explore, and so on, instead of just learning in a prescriptive way the method to solve each type of problem. Consequently, one kind of solution to the current grim situation regarding mathematics teaching and learning would be to aim at relaxing the prevailing didactic contract that thwarts natural sense-making mechanisms of our species. Our conclusions suggest a possible re-shaping of traditional teaching practice, although we refrain from trying to implement this in a prescriptive way. A limitation of our didactic experience might be that it exhibits just a couple of illustrative examples of the application of our theoretical perspective, which show that some non-traditional learning pathways are possible. A full fledged ethnomethodological and micro-phenomenological study would be commendable. Constructivist content: We adhere to the enactive approach to cognition initiated by Francisco Varela, and to the embodied perspective as developed by Shaun Gallagher. We emphasize the cognitive role of metaphorization as a key neural mechanism evolved in humans, deeply intertwined with enaction and most relevant in our “hallucinatory construction of reality,” in the sense of Anil Seth.

The unsolved problem of induction is closely linked to “the unreasonable effectiveness of mathematics in the natural sciences” (Wigner 1960) and to the question “why the universe is algorthmicly compressible” (Davies 1960). The problem of induction is approached here by means of a constructivist version of the Evolutionary Epistemology (CEE) considering both, the perceived regularities we condense to the laws of nature and the mathematical structures we condense to axioms, as invariants of inborn cognitive and mental operators. A phylogenetic relationship between the mental operators generating the perceived and the mathematical regularities respectively may explain the high suitability of mathematical tools to extrapolate observed data. The extension of perceptional operators by means of experimental operators, i.e., by means of measurement devices) would lead to the completion of the classical world picture if both the cognitive and the physical operators are commutable in the sense of operator algebra (quantitative extensions). Otherwise the physical operators will have invariants which no longer can be described in classical terms, and, therefore, would require the formation of non-classical theories (qualitative extension), exceeding the classical world picture. The mathematical analogon would be the algorithmic extension of elementary mathematical thinking exceeding the axiomatic basis previously established according to Gödel’s incompleteness theorem. As a consequence there will be neither a definitive set of axioms in mathematics, nor will be there a definitive theory of everything in physics.

Diettrich O. (1997) Kann es eine ontologiefreie evolutionäre Erkenntnistheorie geben? Philosophia naturalis 34(1): 71–105. https://cepa.info/3914

Most of what nowadays is called evolutionary epistemology tries to explain the phylogenetic acquisition of inborn ‘knowledge’ and the evolution of the mental instruments concerned – mostly in terms of adaptation to external conditions. These conditions, however, cannot be described but in terms of what is provided by the mental instruments which are said to be brought about just by these conditions themselves. So they cannot be defined in an objective and non-circular way. This problem is approached here by what is called the mathematical thinking and their algorithmic extensions it follows that there will be no definitive set of axioms, i. e. it would explain Gödel’s incompleteness theorem. The ontological prerequisites being the basis of the various epistemologies discussed in the philosophy of science, are replaced by the requirement of consistency: our cognitive phenotype has to bring about a world picture within which the cognitive phenotype itself can be explained as resulting from an abiotic, then biotic, organic, cognitive and eventually scientific evolution. Any cognitive phenotype reproducing in this sense (together with its organic phenotype) represents a possible and consistent world together with its interpretation and mastery – and none of them is ontologically privileged.