Open peer commentary on the article “Enactive Metaphorizing in the Mathematical Experience” by Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes. Abstract: Welcoming their scholarly focus on metaphorizing, I critique Díaz-Rojas, Soto-Andrade and Videla-Reyes’s selection of the hypothetical constructs “conceptual metaphor” and “enactive metaphor” as guiding the epistemological positioning, educational design, and analytic interpretation of interactive mathematics education purporting to operationalize enactivist theory of cognition - both these constructs, I argue, are incompatible with enactivism. Instead, I draw on ecological dynamics to promote a view of metaphors as projected constraints on action, and I explain how mathematical concepts can be grounded in perceptual reorganization of motor coordination. I end with a note on how metaphors may take us astray and why that, too, is worthwhile.

Alsup J. (1993) Teaching probability to prospective elementary teachers using a constructivist model of instruction. In: Proceedings of the Third International Seminar on Misconceptions and Educational Strategies in Science and Mathematics. Cornell University, Ithaca, 1–4 August 1993. Misconceptions Trust, Ithaca NY: **MISSING PAGES**. https://cepa.info/7242

This paper is a report of a study conducted with preservice elementary teachers at the University of Wyoming during the summer of 1993. The study had two purposes: (1) to observe the effectiveness of using a constructivist approach in teaching mathematics to preservice elementary teachers, and (2) to focus on teaching probability using a constructivist approach. The study was conducted by one instructor in one class, The Theory of Arithmetic II, a required mathematics class for preservice elementary teachers.

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.

Dörfler W. (1987) Empirical investigations of the construction of cognitive schema from actions. In: Bergeron J. C., Herscovics N. & Kieran C. (eds.) Proceedings of the Eleventh Conference of the International Group for the Psychology of Mathematics Education, Volume 3. University of Montreal, Montreal: 3–9.

The theoretical basis of the interviews reported about here is a Piagetian-like approach to the origin and genesis of cognitive schemata representing mathematical concepts. Such schemata are postulated to reflect the abstract and general structure of material, imagined or mental actions and of relations induced by these actions. The main cognitive tools for the mental construction of such schemata are seen to be: Actions, symbolic representations, prototypes of objects, reflection and abstraction, schematization, generalization. The interviews were devised such that the subjects were guided appropriately in their individual cognitive constructions. The mathematical topics treated are: Place value system, divisibility, word problems, geometric sequence, Riemann integral. In general the results support the view that the individual construction of cognitive schemata is possible and effective in the proposed way.

Fidelman U. (1991) Experimental testing of constructivism and related theories. Behavioral Science 36(4): 274–297. https://cepa.info/7902

The purpose of this article is to show that experimental scientific methods can be applied to explain how the analytic mechanism of the left cerebral hemisphere and the synthetic mechanism of the right one create complex cognitive constructions like ontology and mathematics. Nominalism and ordinal mathematical concepts are related to the analytic left hemisphere while Platonism and cardinal mathematical concepts are related to the synthetic right one. Thus persons with a dominant left hemisphere tend to prefer nominalist ontology and have more aptitude for ordinal mathematics than for cardinal mathematics, while persons with a dominant right hemisphere tend to prefer platonist ontology and have more aptitude for cardinal mathematics than for ordinal mathematics. It is further explained how the Kantism temporal mode of perceiving experience can be related to the left hemisphere while the Kantian spatial mode of perceiving experience can be related to the right hemisphere. This relation can be tested experimentally, thus the Kantian source of constructivism, and through it constructivism itself, can be tested experimentally.

Howison M., Tminic D., Reinholz D. & Abrahamson D. (2011) The mathematical imagery trainer: From embodied interaction to conceptual teaming. In: Fitzpatrick G., Gutwin C., Begole B., Kellog W. A. & Tan D. (eds.) Proceedings of the annual meeting of The association for computer machinery special interest group on computer human interaction: “human factors in computing systems” (CHI 2011). CM Press, Vancouver: 1989–1998. https://cepa.info/8092

We introduce an embodied-interaction instructional design, the Mathematical Imagery Trainer (MIT), for helping young students develop grounded understanding of proportional equivalence (e.g., 2/3 = 4/6). Taking advantage of the low-cost availability of hand-motion tracking provided by the Nintendo Wii remote, the MIT applies cognitive-science findings that mathematical concepts are grounded in mental simulation of dynamic imagery, which is acquired through perceiving, planning, and performing actions with the body. We describe our rationale for and implementation of the MIT through a design-based research approach and report on clinical interviews with twenty-two 4th-6th grade students who engaged in problem-solving tasks with the MIT.

Izmirli I. M. (2014) Wittengstein’s language games and forms of life from a social constructivist point of view. American Journal of Educational Research 2(5): 291–298. https://cepa.info/2949

In this paper our main objective is to interpret the major concepts in Wittgenstein’s philosophy of mathematics, in particular, language games and forms of life, from a social constructivist point of view in an attempt to show that this philosophy is still very relevant in the way mathematics is being taught and practiced today. We start out with a brief discussion of radical constructivism followed by a rudimentary analysis of the basic tenets of social constructivism, the final blow against absolutism – the soulless landmark of mathematics as often construed by the uninitiated. We observe that, the social constructivist epistemology of mathematics reinstates mathematics, and rightfully so, as “…a branch of knowledge which is indissolubly connected with other knowledge, through the web of language” (Ernest 1999), and portrays mathematical knowledge as a process that should be considered in conjunction with its historical origins and within a social context. Consequently, like any other form of knowledge based on human opinion or judgment, mathematical knowledge has the possibility of losing its truth or necessity, as well. In the third section we discuss the main points expounded in Wittgenstein’s two books, Tractatus Logico-Philosophicus and Philosophical Investigations, as well as in his “middle period” that is characterized by such works as Philosophical Remarks, Philosophical Grammar, and Remarks on the Foundations of Mathematics. We then briefly introduce the two main concepts in Wittgenstein’s philosophy that will be used in this paper: forms of life and language games. In the fifth and final section, we emphasize the connections between social constructivism and Wittgenstein’s philosophy of mathematics. Indeed, we argue that the apparent certainty and objectivity of mathematical knowledge, to paraphrase Ernest (Ernest 1998), rest on natural language. Moreover, mathematical symbolism is a refinement and extension of written language: the rules of logic which permeate the use of natural language afford the foundation upon which the objectivity of mathematics rests. Mathematical truths arise from the definitional truths of natural language, and are acquired by social interaction. Mathematical certainty rests on socially accepted rules of discourse embedded in our forms of life, a concept introduced by Wittgenstein (Wittgenstein, 1956). We argue that the social constructivist epistemology draws on Wittgenstein’s (1956) account of mathematical certainty as based on linguistic rules of use and forms of life, and Lakatos’ (1976) account of the social negotiation of mathematical concepts, results, and theories.

James K. H. (2022) The embodiment of letter perception: The importance of handwriting in early childhood. In: Macrine S. L. & Fugate J. M. B. (eds.) Movement matters: How embodied cognition informs teaching and learning. MIT Press, Cambridge MA: 55–76. https://cepa.info/7990

Excerpt: Action has long been known to play a strong role in perceptual development. A large number of studies have shown the importance of action, and specifically self-generated action, in visual perceptual development and many different domains of cognitive development (e.g., Bertenthal & Campos, 1987; Bushnell & Boudreau, 1993; Gibson, 1969; Needham et al., 2002). In childhood, we learn to associate self-generated actions with percepts to construct representations of objects. Active interaction with the world facilitates learning about three-dimensional objects (Deloache, 1989; James & Swain, 2011; Piaget, 1953), depth perception (Richards & Rader, 1981; Wexler & van Boxtel, 2005), various types of spatial processing (Christou & Bülthoff, 1999; Held & Hein, 1963; Wohlschläger & Wohlschläger, 1998), eye-hand coordination (Needham et al., 2002), and mathematical concepts (Alibali & Nathan, 2012; Marquardt Donovan & Alibali, chapter 10 in this volume). Visual perception also uses information gained through action – locomotion, handling objects, head movements. Thus, we perceive in order to act, and we act in order to perceive (Gibson, 1979). All these coupled experiences of perception and action sculpt connections among sensorimotor brain systems that support typical cognitive development. For this knowledge to be useful for educators, we must address how active interaction with the environment has specific effects on learning in a school setting. Of the many educational competencies that are positively affected by self-generated action, one that is often not considered is learning to read. In what follows, I will review the importance of letter recognition for learning to read, how learning letters is affected by self-generated action – specifically handwriting – and review how brain imaging can help us understand why handwriting is important for letter learning. For educators, this chapter is intended to provide information regarding how we can improve letter knowledge (and subsequent literacy) through self-generated action, and importantly why handwriting has these positive effects on letter learning.

Koch L. C. (1989) Constructivism: A model for relearning mathematics. In: Maher C. A., Goldin G. A. & Davis R. B. (eds.) Proceedings of the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 11), Volume 1: Research reports. PME-NA, New Brunswick: 334–340. https://cepa.info/6882

A large number of students enter college each year not knowing basic mathematical concepts such as fractions, decimals and percents. Although many students take mathematics for twelve years in elementary school, junior high school and high school they lack the fundamental processes necessary to be successful in college level mathematics courses. In this paper a teaching model is outlined that will benefit those students in the learning of mathematics and mathematical processes. This model is based on the tenets of constructivism as put forth by von Glasersfeld (1983).

Open peer commentary on the article “Roles and Demands in Constructionist Teaching of Computational Thinking in University Mathematics” by Chantal Buteau, Ana Isabel Sacristán & Eric Muller. Abstract: Buteau, Sacristán and Muller’s target article raises the pertinent issue of how to describe a sustained undergraduate course on computational thinking and programming for mathematical learning. There is so little work on this issue that it is worthwhile to reflect on this study and to raise questions regarding the options and tools available or needed to understand sustained constructionist activity. Here, I focus on questions regarding how to understand the instructor’s craft knowledge in practice and what constructionist mathematical learning may look like when programming is at the service of engagement with mathematical concepts.