# Approach «Radical Constructivism»

Purpose: One of my goals in the paper is to investigate why realists reject radical constructivism (RC) as well as social constructivism (SC) out of hand. I shall do this by means of commenting on Peter Slezak’s critical paper, Radical Constructivism: Epistemology, Education and Dynamite. My other goal is to explore why realists condemn the use of RC and SC in science and mathematics education for no stated reason, again by means of commenting on Slezak’s paper. Method: I restrict my comments to Slezak’s paper and leave it to the reader to judge which, if any, of the reasons that I advance for these two states of affairs are not specific to Slezak’s paper. Other readers might not agree with my interpretations of Slezak’s paper, including Slezak himself, but I offer them after having worked with von Glasersfeld in interdisciplinary research in mathematics education for over 25 years. Findings: My findings are that Slezak: (1) rejects RC and SC on the basis of unjustified criticisms, (2) does not explore basic tenets of RC nor of SC beyond the unjustified criticisms, (3) rejects how SC and RC have been used in science and mathematics education, based at least in part on the unjustified criticisms, (3) dislikes how SC has been used in science and mathematics education, a dislike that fuels his rejection of any constructivism, and (4) doesn’t explore how RC has been used in scientific investigations in mathematics education. On the basis of these findings, I conclude that how epistemological models of knowing might be used in science or mathematics education would be better left to the educators who use them in interdisciplinary work.

Purpose: My goals in this paper are to comment on some of the roles that Ernst von Glasersfeld played in our work in IRON (Interdisciplinary Research on Number) from circa 1975 up until the time of his death, and to relate certain events that revealed his character in very human terms. Method: Among my recollections of Ernst, I have chosen those that I felt would most adequately portray his impact on the field of mathematics education and his ethics in the field of human affairs. Findings: It has not often been said but, in his work in IRON, it was Ernst’s explicit intention to start a conceptual revolution in mathematics education and beyond. Other than serving as a revolutionary, Ernst also served as a mentor for many investigators in mathematics education, including myself. He excelled as a scientist as well an epistemologist, and his scientific work fueled his epistemological work. Ernst was a very ethical and wonderful human being who was dedicated to the betterment of humankind by means of his revolutionary ideas.

Number is presented as a uniting operation that can have collections of sensory items as material and a unit of units as the result. One thesis of the paper is that children who have constructed this uniting operation have not necessarily constructed number sequences. Problem situations are suggested for such children that might encourage the internalization of counting and the concomitant construction of the iterable unit of one. Situations are then suggested in which the child can use the number sequence that is based on the iterable unit of one to construct other iterable units and their corresponding number sequences. A second thesis of the paper is that for children who are yet to construct the uniting operation, counting is a sensory-motor scheme that should be coordinated with spatial, finger, and auditory patterns. Problem situations are suggested for these coordinations. While a teacher cannot give a child the uniting operation, the suggested situations can encourage its construction by the child.

In an earlier publication (Steffe, von Glasersfeld, Richards, & Cobb, 1983), we presented a model of the development of children’s counting schemes. This model specifies five distinct counting types, according to the most advanced type of unit items that the child counts at a given point in his or her development. The counting types indicate what children’s initial, informal numerical knowledge might be like, and reflect our contention that children see numerical situations in a variety of qualitatively different ways. These constructs constituted the initial theoretical basis of the teaching experiment and served as a catalyst for the first years work. Consequently, we provide an explanation of the counting types as we defined them in 1980.

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.

Radical Constructivism is currently a very influential view in mathematics education. This paper examines its philosophical roots through a review of the book Radical Constructivism by Ernst von Glasersfeld. It begins by describing the structure of the book, and then considers a number of philosophical issues raised by the book. The paper concludes with some reflections on the relationship between philosophy and mathematics education which the book provoked

Open peer commentary on the target article “From Objects to Processes: A Proposal to Rewrite Radical Constructivism” by Siegfried J. Schmidt. Upshot: My suggestion is that the shift from objects to processes can be seen as grounded in the processes of self-generation common to all living organisms. Specifically human cognition is a subsequent evolutionary emergence.

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