This paper addresses the learning paradox, which obliges radical constructivists to explain how cognition can advance from a lower level of reasoning to a higher one. Although the question is at least as old as Plato, two major flaws have inhibited progress in developing solutions: the assumption that learning is an inductive process, and the conflation of epistemological and ontological questions. I adopt a radical constructivist perspective and present a few related solutions from previous mathematics education literature. I then provide a new solution that relies on Peirce’s theory of abduction and Piaget’s theory of operational schemes. However, with the learning paradox resolved, an ontological paradox remains: If individuals construct their mathematical realities based on their personal actions and experiences, how can we explain the predictive power of scientific hypotheses that are based on this mathematics?

This commentary responds to a criticism of constructivism by Wolff-Michael Roth, published in For the Learning of Mathematics 30(2). At times, Roth oversimplifies and mischaracterizes constructivist perspectives on learning while promoting embodied cognition as an alternative. I argue that a simple transposition of terms largely aligns his description of embodied mathematical objects with the constructivist conception of mathematical objects as interiorized action.

This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs – especially reflective abstraction – to build explanatory models for teaching and learning. LTA attempts to provide the teacher-researcher with a greater degree of determination in student learning trajectories, but then the teacher-researcher must address constraints in the students’ available ways of operating. These issues are exemplified in the case of teaching students about multiplying fractions. Additional theoretical issues arise in explaining logical necessity in students’ ways of operating and the role of reflective abstraction in organizing new ways of operating.

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: Research from various perspectives has converged on the idea that mathematics arises from a coordination of actions. This commentary examines the approach taken in the target article and other approaches to modeling such coordinations, raising three related questions.

This opening chapter provides an introduction to the book. It also introduces a theme that integrates many of the contributions from the remaining chapters: we adopt Kant’s perspective for merging rationalist and empiricist philosophies on the construction of knowledge. In particular, we focus attention on ways that biologically based abilities and experience in the world (coordinations of sensorimotor activity) each contribute to the construction of number. Additional themes arise within the content chapters and the commentaries on them.

Key words: embodied cognition, epistemology, numerical development, radical constructivism, sensorimotor activity.

Open peer commentary on the article “Enactive Metaphorizing in the Mathematical Experience” by Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes. Abstract: Constructivist and embodied theories of learning each focus on action as the basis for cognition. However, in restricting action to sensorimotor activity, some embodied perspectives eschew abstraction and replace it with metaphor. Here, we argue that metaphors, and mathematical development itself, depend upon coordinations of actions, within structures that we call abstract.

This article reports on results from a study that quantitatively tested hypotheses arising from Les Steffe and John Olive’s Fractions Project. It affirms their work and scheme theory in general. For example, the study showed that additional mental operations are necessary for middle school students to generalize their partitive conceptions from unit fractions to other proper fractions.

Piagetian theory describes mathematical development as the construction and organization of mental operation within psychological structures. Research on student learning has identified the vital roles two particular operations – splitting and units coordination – play in students’ development of advanced fractions knowledge. Whereas Steffe and colleagues describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students’ psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students’ development with a structure that is isomorphic to the positive rational numbers under multiplication – “the splitting group.” We use a quantitative analysis of written assessments from 59 eighth grade students in order to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students’ constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students’ constructions of three levels of units coordination. Because three levels of units coordination also plays a vital role in other mathematical domains, such as algebraic reasoning, implications from the study extend beyond fractions teaching and research. Relevance: The paper uses constructivist theories of learning, including scheme theory and Piaget’s structuralism, to study how students construct mature conceptions of fractions.

Upshot: In reading the commentaries, we were struck by the fact that all of them were in some capacity related to what we consider a core principle of radical constructivism - interaction. We characterize interaction from a radical constructivist perspective, and then discuss how the authors of the commentaries address one kind of interaction.

Context: This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. Problem: The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. Method: The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. Results: The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. Implications: This paper has the implication that radical constructivism has the potential to impact practice.

Key words: Second-order model, radical constructivism, teaching, mathematics education