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. (shrink)

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. (shrink)

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.