Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed.

Key words: Mathematics education, mental representation, problem solving, mathematics learning

Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: In the target article, Christiane Herr offers an insightful characterization of how von Glasersfeld’s radical constructivism can be implemented in structural design education in architecture. In this commentary, I articulate possible connections between research on problem solving and problem posing in mathematics education and design processes in structural design education as described in the target article.

Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: The significance of Proulx and Maheux’s target article lies in their thorough grounding of some of the ideas of the mathematical problem-posing and problem-solving literature in a strong theoretical framework. They direct our attention to two distinct epistemological assumptions that underlie explanations of problem-solving: the so-called “selection-then-execution hypothesis” and Varela’s problem-posing perspective. In this commentary, I will offer two ways their line of research could be extended.

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: In this commentary, I point to a few ways that Gandell and Maheux’s important work on problem posing|solving could be further extended. I also offer some pedagogical insights that tie the authors’ ideas to my own experiences as an educator and researcher.

Open peer commentary on the article “Enactive Metaphorizing in the Mathematical Experience” by Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes. Abstract: I offer some connections between Díaz-Rojas, Soto-Andrade and Videla-Reyes’s work on the role of students’ enactive conceptual metaphorizing experiences in mathematics, and existing research on learning and problem posing in mathematics education.

Upshot: In this response to the open peer commentaries on our target article, we address two emerging themes: the need to explicate further the nature of learning processes from a radical constructivist perspective, and the need to investigate further the implications of our research for classroom teaching.