An Historical Reflection on Adapting Piaget’s Work for Ongoing Mathematics Education Research

My reflection on adapting Piaget’s work in mathematics education starts with the national efforts during the 1960s to upgrade the mathematics curriculum. From there, after recounting my introduction to Piaget’s work by Henry Van Engen at the University of Wisconsin and my efforts to carry on his research program at the University of Georgia, I describe how my conception of Piaget’s work changed after meeting Ernst von Glasersfeld and how we launched the research program, Interdisciplinary Research on Number (IRON). I then convey how Glasersfeld was introduced to Piaget’s work after his arrival at the University of Georgia and cite a critical excerpt from his original paper on radical constructivism where he argued that radical constructivism constituted a legitimate interpretation of Piaget’s genetic epistemology. From that point, I make a distinction between first-order and second-order models and explain how we used Glasersfeld’s first-order model of attentional moments in the analysis of video-records of children’s counting in the construction of second-order models of the units children construct and use in counting. I then provide an interpretation of children’s counting using Glasersfeld’s model of scheme and go on to illustrate children’s construction of the first numerical counting scheme in an attempt to operationalize Piaget’s concept of reflective abstraction. In this context, I found that reflective abstraction usually works on its own time and the role that experience plays remains an unsolved problem. I then turn to explaining learning stages in children’s construction of counting schemes, the perceptual stage, the figurative stage, the stage of the initial number sequence, the stage of the tacitly nested number sequence, the stage of the explicitly nested number sequence, and the stage of three levels of units. I end the paper with a conceptual analysis of the percent of children across five of the six grades of the elementary school in each learning stage.