Abrahamson D., Nathan M. J., Williams-Pierce C., Walkington C., Ottmar E. R., Soto H. & Alibali M. W. (2020) The future of embodied design for mathematics teaching and learning. Frontiers in Education 5: 147. https://cepa.info/7086

A rising epistemological paradigm in the cognitive sciences – embodied cognition – has been stimulating innovative approaches, among educational researchers, to the design and analysis of STEM teaching and learning. The paradigm promotes theorizations of cognitive activity as grounded, or even constituted, in goal-oriented multimodal sensorimotor phenomenology. Conceptual learning, per these theories, could emanate from, or be triggered by, experiences of enacting or witnessing particular movement forms, even before these movements are explicitly signified as illustrating target content. Putting these theories to practice, new types of learning environments are being explored that utilize interactive technologies to initially foster student enactment of conceptually oriented movement forms and only then formalize these gestures and actions in disciplinary formats and language. In turn, new research instruments, such as multimodal learning analytics, now enable researchers to aggregate, integrate, model, and represent students’ physical movements, eye-gaze paths, and verbal–gestural utterance so as to track and evaluate emerging conceptual capacity. We – a cohort of cognitive scientists and design-based researchers of embodied mathematics – survey a set of empirically validated frameworks and principles for enhancing mathematics teaching and learning as dialogic multimodal activity, and we synthetize a set of principles for educational practice.

Borg P., Hewitt D. & Jones I. (2016) Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom. Constructivist Foundations 12(1): 59–69. https://cepa.info/3810

Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we constructed a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L (Mathematics-Negotiation-Learner) as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” (CT) during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings.

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.

Ellerton N. F. & Clements M. A. (1992) Some pluses and minuses of radical constructivism in mathematics education. Mathematics Education Research Journal 4(2): 1–22. https://cepa.info/2950

A brief overview of the radical constructivist movement in mathematics education, as it has expressed itself around the world over the past ten years, is provided. Three benefits of the movement are then identified, namely: (a) its emphasis on the need for learners to construct their own mathematical meanings, and thereby come to believe that they “own” the mathematics they are learning; (b) its recognition and advocacy of quality social interaction as the basis for quality mathematics learning; and (c) its identification, clarification, and advocacy of principles for improving mathematics teaching and learning. Three weaknesses that might be associated with the radical constructivist movement are also outlined: (a) the missionary zeal of some radical constructivists who tend to accuse mathematics educators outside their ranks of advocating and practising transmission modes of education; (b) the downplaying of the role of linguistic activity in the development of abstract thought by many radical constructivists; and (c) the tendency of radical constructivists to provide oversimplified answers to the ontological question “What is mathematical knowledge? "

Goldin G. A. (1990) Epistemology, constructivism, and discovery learning of mathematics. In: Davis R. B., Maher C. A. & Noddings N. (eds.) Constructivist views on the teaching and learning of mathematics. National Council of Teachers of Mathematics, Reston VA: 31–47. https://cepa.info/2976

Excerpt: What is the best way to characterize the body of knowledge that we call mathematics? How do children and adults learn mathematics most effectively? How can we best study their learning processes, and assess the outcomes of learning? Can meaningful learning be consistently distinguished from nonmeaningful or rote learning? What constitutes effective mathematics teaching, and how can elementary and secondary school teachers be enabled to provide it?

Hardy M. (1997) Von Glasersfeld’s radical constructivism: A critical review. Science & Education 6(1-2): 135–150. https://cepa.info/2983

We explore Ernst von Glaserfeld‘s radical constructivism, its criticisms, and our own thoughts on what it promises for the reform of science and mathematics teaching. Our investigation reveals that many criticisms of radical constructivism are unwarranted; nevertheless, in its current cognitivist form radical constructivism may be insufficient to empower teachers to overcome objectivist cultural traditions. Teachers need to be empowered with rich understandings of philosophies of science and mathematics that endorse relativist epistemologies; for without such they are unlikely to be prepared to reconstruct their pedagogical practices. More importantly, however, is a need for a powerful social epistemology to serve as a referent for regenerating the culture of science education. We recommend blending radical constructivism with Habermas‘ ’theory of communicative action‘ to provide science teachers with a moral imperative for adopting a constructivist epistemology.

Lesh R., Doerr H. M., Carmona G. & Hjalmarson M. (2003) Beyond constructivism. Mathematical Thinking and Learning 5(2–3): 211–233.

In a recent book titled Beyond Constructivism: A Models & Modeling Perspective on Mathematics Problem Solving, Learning & Teaching (Lesh & Doerr, 2003a), the concluding chapter describes a number of specific ways that a models and modeling perspective moves significantly beyond the implications that can be drawn from constructivist theories in the context of issues that are priorities to address for teachers, curriculum developers, or program designers. In that chapter (Lesh & Doerr, 2003b), the following topics were treated as cross-cutting themes: (a) the nature of reality, (b) the nature of mathematical knowledge, (c) the nature of the development of children’s knowledge, (d) the mechanisms that drive that development, (e) the relationship of context and generalizability, (f) problem solving, and (g) teachers’ knowledge and the kinds of teaching and learning situations that contribute to the development of children’s knowledge. In this article, we organize our comments directly around the preceding topics and describe how a models and modeling perspective provides alternative ways of thinking about mathematics teaching and learning that enable teachers, researchers and others to produce useful and sharable conceptual tools that have powerful implications in the context of decision-making issues that are of priority to practitioners.

Proulx J., Simmt E. & Towers J. (2009) The enactivist theory of cognition and mathematics education research: Issues of the past, current questions and future directions. In: Tzekaki M., Kaldrimidou M. & Sakonidis H. (eds.) Proceedings of the 33rd conference of the international group for the psychology of mathematics education. Volume 1. P. M. E., Thessaloniki: 249–278. https://cepa.info/6863

Excerpt: A number of intentions triggered this research forum on enactivism and mathematics education research, and those are significant to highlight as they have in return structured the content and form that this forum takes. First, there has been and continues to be a substantial amount of research and writing on issues of enactivism undertaken by mathematics education researchers; thus we wanted to highlight and synthesize this body of research. At the same time, although much research has been conducted within the enactivist perspective, many of those contributions, and their authors, are not always well known and have often been seen merely as “interesting” orientations or “alternative” perspectives – but clearly not mainstream. Because we believe enactivism offers an insightful orientation which shows promise for enhancing our understanding of mathematics teaching and learning, we wanted to bring forth the nature and wide spectrum of enactivist contributions in order to share and create dialogue with the PME community about significant issues raised through this orientation. A third intention is in reaction to what might be thought of as a hegemony of constructivism in the mathematics education literature. We believe that enactivism, as a theory of cognition, offers a more encompassing and enlightening perspective on learning, teaching, and epistemology. Therefore, the following concerns will orient and be continuously present in the research forum unfoldings: retrospectives (as well as perspectives and prospectives) on research studies and writing done on enactivism in mathematics education will be shared; contributors will focus on insightful features that enactivism offers us; particularities of enactivism as a theory of cognition will permeate all discussions and presentations; and finally, but not least, interactions and discussions will take place about the ideas put forward.

Riegler A. & Steffe L. P. (2014) “What Is the Teacher Trying to Teach Students if They Are All Busy Constructing Their Own Private Worlds?”: Introduction to the Special Issue. Constructivist Foundations 9(3): 297–301. https://constructivist.info/9/3/297

Context: Ernst von Glasersfeld introduced radical constructivism in 1974 as a new interpretation of Jean Piaget’s constructivism to give new meanings to the notions of knowledge, communication, and reality. He also claimed that RC would affect traditional theories of education. Problem: After 40 years it has become necessary to review and evaluate von Glasersfeld’s claim. Also, has RC been successful in taking the “social turn” in educational research, or is it unable to go beyond “private worlds? Method: We provide an overview of contributed articles that were written with the aim of showing whether RC has an impact on educational research, and we discuss three core issues: Can RC account for inter-individual aspects? Is RC a theory of learning? And should Piaget be regarded as a radical constructivist? Results: We argue that the contributed papers demonstrate the efficiency of the application of RC to educational research and practice. Our argumentation also shows that in RC it would be misleading to claim a dichotomy between cognition and social interaction (rather, social constructivism is a radical constructivism), that RC does not contain a theory of mathematics learning any more or less than it contains a theory of mathematics teaching, and that Piaget should not be considered a mere trivial constructivist. Implications: Still one of the most challenging influences on educational research and practice, RC is ready to embark on many further questions, including its relationship with other constructivist paradigms, and to make progress in the social dimension.

Scott B. (2011) Thinking mathematically. In: B. Scott, (ed.) Explorations in Second Order Cybernetics: Reflections on Cybernetics, Psychology and Education. Vienna, echoraum. https://cepa.info/1814

This paper discusses the nature of mathematical thinking from the viewpoint of cognitive psychology. This is followed by a discussion of the practical needs and problems associated with mathematics teaching in Primary Schools. Some necessarily cautious and speculative recommendations are made. Where appropriate, reference is made to the literature on mathematics teaching and associated methods and materials. The author also has opportunity to refer to his own experiences as a teacher of mathematics, in a Primary School. Finally, there is a discussion of the emerging media in education and their likely impact on a new generation of teachers and pupils.