Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: Herr’s target article outlines a teaching approach that illustrates and explains how radical constructivism can be used to teach architectural design principles to a large cohort of students. Herr’s approach consists of a hybrid set of instructional activities whose implementation was supported by her establishment of a social climate in the classroom that encouraged the contributions of individuals in the learning process. The activities included the integration of lectures with other teaching formats, the use of open‑ended problem solving tasks, and the use of small‑group projects. I will discuss each of these activities and offer a recommendation for future research.

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: Drawing from the ideas of Varela, Proulx and Maheux, I propose a theoretical framework to examine problem-posing and problem-solving and provide evidence for their ideas with examples of student work from their research studies. I will draw comparisons between the approach taken by the researchers to the constructivist approach I have taken in my studies of problem-solving and those conducted with collaborators. My intent with these comments is not to argue the merits of one perspective over the other; rather, I look to point out and elaborate on these differences and make some specific suggestions to the researchers.

Open peer commentary on the article “Problematizing: The Lived Journey of a Group of Students Doing Mathematics” by Robyn Gandell & Jean-François Maheux. Abstract: Gandell and Maheux present an interesting analysis of the problem posing (PP) and problem solving (PS) of a group of students playing a throwing game. The analysis builds on recent studies that explain how PP and PS co-evolve as students interpret and ultimately complete the tasks that we give them. I will make some preliminary comments on the researchers’ enactivist approach to studying PP and PS, draw some comparisons with constructivist approaches, and then comment on the analysis of the student work.

Open peer commentary on the article “Enactive Metaphorizing in the Mathematical Experience” by Daniela Díaz-Rojas, Jorge Soto-Andrade & Ronnie Videla-Reyes. Abstract: Díaz-Rojas, Soto-Andrade and Videla-Reyes advocate an approach to the teaching and learning of mathematics that emphasizes enaction, embodiment and metaphorization. I comment on their analysis of one of the illustrative examples, the handshake problem. First, I provide some historical context and rationale from mathematics education for how tasks such as the handshake problem have been used in studies of problem solving and why they also can be effective examples of rich problem-solving tasks that can be used in instructional settings. Then I comment on the robustness of the researchers’ analysis of the handshake problem by examining an extension problem, finding the number of diagonals in an n-sided polygon.

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

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.