This article reports on students’ construction of fraction composition schemes. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme.

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Cifarelli & Sevim outline the distinction between “representation” and “re-presentation” in von Glasersfeld’s thinking. After making this distinction, they identify how a student’s problem solving activity initially involved recognition, then re-presentation, and finally reflective abstraction. I use my commentary about the Cifarelli & Sevim article to identify two ways they could extend their current line of research.

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: Borg et al. provide a framework that contributes to a growing body of research on how radical constructivism can help teachers and researchers to understand the complexity of classroom interactions. The bulk of my commentary is written to clarify theoretical points that I think are important to the endeavor that Borg et al. set out for themselves. My points come in part from thinking about how non-constructivist readers may interpret what Borg et al. outline in their article.

In this paper, we engage in a thought experiment about how students might notate their reasoning for taking a fraction of a fraction and determining its size in relation to the whole. We situate this discussion within a radical constructivist framework for learning in order to articulate how developing systems of notation with students can contribute to their learning. In particular, we posit that developing systems of notation with students is likely to contribute to what Piaget called reflected abstractions – a retroactive thematization of one’s reasoning.

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