François K. (2014) Convergences between Radical Constructivism and Critical Learning Theory. Constructivist Foundations 9(3): 377–379. https://cepa.info/1098

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: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process.

Glasersfeld E. von (1981) An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education 12(2): 83–94. https://cepa.info/1356

A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget’s postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.

Goodson-Espy T. (2014) Reflective Abstraction as an Individual and Collective Learning Mechanism. Constructivist Foundations 9(3): 381–383. https://cepa.info/1100

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 and Sevim discuss the development of individual students’ abstract conceptual structures while problem solving, using constructs for analysis that are consistent with von Glasersfeld’s radical constructivism: re-presentation and reflective abstraction. This commentary discusses the on-going contributions of reflective abstraction to individual and collective learning.

Kenny V. (2012) Continuous Dialogues III: Processes of Construction Ernst von Glasersfeld’s Answers to a Wide Variety of Questioners on the Oikos Web Site 1997–2010. Constructivist Foundations 7(3): 208-221. https://cepa.info/508

Context: Up to the time of his death in 2010, Ernst von Glasersfeld had, for the previous thirteen years, directly answered a wide variety of questions posed to him on the Oikos web site. Purpose: This is the third article in a series of four that is based on a selection from all of the questions posed in the thirteen-year period and is aimed at highlighting key aspects of radical constructivism. Method: The question-answer pairs are grouped into eight categories, and each article deals with two of these categories. This third article deals with the issue of “constructing,” split into the two main themes of (i) the personal construing system and (ii) the threat that the radical constructivist view of the personal constructing system holds for realism. Results: In the first part of this article, von Glasersfeld emphasises his view of the individual’s construction of personal meaning as a phenomenon of “private language.” The second part outlines some of the implications of radical constructivism for “realism.” Implications: Von Glasersfeld’s answers offer an easily accessible resource for people of all levels of understanding who are seeking to deepen their appreciation of his work.

Steffe L. P. (1991) The constructivist teaching experiment: Illustrations and implications. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer, Dordrecht: 177–194. https://cepa.info/2098

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge and how it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.

Steffe L. P. & Ulrich C. (2013) Constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 102–109. https://cepa.info/2959

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.

Open peer commentary on the article “A Computational Constructivist Model as an Anticipatory Learning Mechanism for Coupled Agent–Environment Systems” by Filipo Studzinski Perotto. Upshot: The CALM cognitive agent with its learning mechanism, as presented by the author, can be described as “trivially constructivist.” Probably, at best, it can be seen as a model of the empirical abstraction but not of the reflective abstraction. The “intrinsic motivations” in the simulated agent presented as “evaluative signals” sent from the agent’s “body” to its “mind” can be seen as low-level physiological drives. They cannot account for far more sophisticated intrinsic motivations such as curiosity.

Tillema E. S. (2014) Reflecting on a Radical Constructivist Approach to Problem Solving. Constructivist Foundations 9(3): 383–385. https://cepa.info/1101

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.