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.

Key words: idealism, experiential reality, constraints, interaction, adaptation, first-order models, second-order models, scheme, self-reflexivity

In this paper, I discuss my experiences with collaborative research and how interdisciplinary research has always been a major part of the collaborations in which I have engaged. Collaborative research has much to recommend interdisciplinary research, but the concept has been distorted in recent documents. For example, to enhance research culture, the College of Education at the University of Georgia has published a strategic plan in which a basic action item is the initiation of strategic research collaborations and partnerships with faculty and organizations inside the College and outside the College, where the Associate Dean for Research is the person primarily responsible1. Why I consider this action item a distortion of the concept of collaborative research will become clear as I develop the concept. Initially, suffice it to say that in my experience, it is individuals. not institutions, who initiate and do collaborative research. Although it cannot be institutionalized, collaborative research can and must be supported by institutions when it might occur.

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.

Key words: paradigm shift, constructivism, mathematics education, conceptual analysis

Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically with von Glasersfeld for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise.

Open peer commentary on the article “Religion: A Radical-Constructivist Perspective” by Andreas Quale. Upshot: The first of my three main goals in this commentary is to demonstrate that Quale’s radical separation between cognitive and non-cognitive knowledge is not viable. The second is to establish Quale’s assertion that a radical constructivist cannot be genuinely religious is a result of taking radical constructivism and religion as abstracted first-order models and is a result of comparing and contrasting elements of these models. The third goal is to establish how religious radical constructivists establish relations between their religious beliefs and their radical constructivist beliefs. To accomplish the first goal, I appeal to the work of Damasio to establish that what Quale refers to as “non-cognitive knowledge” is inextricably cognitive. I also appeal to the work of both Damasio and von Glasersfeld to demonstrate that the Cartesian duality between mind and body is not viable. To accomplish the last two goals, I make a distinction between first- and second-order knowledge and contrast Quale’s argument that a radical constructivist cannot be religious with the relation between religious beliefs and radical constructivism from the perspective of actual religious radical constructivists.

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some basic terms. Second, I explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics.

Starting with Woodworth and Thorndike’s classical experiment published in 1901, major periods in mathematics education throughout 20th century and on into the current century are reviewed in terms of competing epistemological and psychological paradigms that were operating within as well as across the major periods. The periods were marked by attempts to make changes in school mathematics by adherents of the dominant paradigm. Regardless of what paradigm was dominant, the attempts essentially led to major disappointments or failures. What has been common across these attempts is the practice of basing mathematics curricula for children on the first-order mathematical knowledge of adults. I argue that rather than repeat such attempts to make wholesale changes, what is needed is to construct mathematics curricula for children that is based on the mathematics of children. Toward that end, I present several crucial radical constructivist research programs.

Key words: learning trajectories, research methods, cognition, curriculum.

Open peer commentary on the article “I Can’t Yet and Growth Mindset” by Fiona Murphy & Hugh Gash. Abstract: Murphy and Gash interpret Bateson’s deutero-learning as constructivist, but the only rationale for their interpretation is that deutero-learning is adaptive to the environment. Their limited rationale opens the possibility that their interpretation and use of deutero-learning is behavioristic. My goals in this commentary are to (a) interpret Bateson’s deutero-learning, as well as his Learning I, using radical constructivism, and (b) critique the authors’ interpretation and use of deutero-learning.