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.

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.

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.

Panorkou N. & Maloney A. (2015) Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment. Constructivist Foundations 10(3): 338–347. https://cepa.info/2146

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies.

Steffe L. P. (2014) Constructing Models of Ethical Knowledge: A Scientific Enterprise. Constructivist Foundations 9(2): 262–264. https://constructivist.info/9/2/262

Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically with von Glasersfeld for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise.

Steffe L. P. & Thompson P. W. (2000) Teaching experiment methodology: Underlying principles and essential elements. In: Lesh R. & Kelly A. E. (eds.) Research design in mathematics and science education. Lawrence Erlbaum, Hillsdale NJ: 267–307. https://cepa.info/2110

A primary purpose for using teaching experiment methodology is for researchers to experience, firsthand, students’2 mathematical learning and reasoning. Without the experiences afforded by teaching, there would be no basis for coming to understand the powerful mathematical concepts and operations students construct or even for suspecting that these concepts and operations may be distinctly different from those of researchers. The constraints that researchers experience in teaching constitute a basis for understanding students’ mathematical constructions. As we, the authors, use it, “constraint” has a dual meaning. Researchers’ imputations to students of mathematical understandings and operations are constrained by the language and actions they are able to bring forth in students. They also are constrained by students’ mistakes, especially those mistakes that are essential; that is, mistakes that persist despite researchers’ best efforts to eliminate them. Sources of essential mistakes reside in students’ current mathematical knowledge. To experience constraints in these two senses is our primary reason for doing teaching experiments. The first type of constraint serves in building up a “mathematics of students” and the second type serves in circumscribing such a mathematics within conceptual boundaries.