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The Evidential Basis for Knowledge Claims in Mathematics Education Research

Frank K. Lester Jr. and Dylan Wiliam

During the past few years, diverse individuals and groups have begun to promote a variety of old and new instructional approaches, programs, and policies for mathematics education (Dixon, Carnine, Lee, Wallin, & Chard, 1998; Jacob, 1997; Kilpatrick, 1997; Wu, 1997). Researchers are being exhorted to gather and analyze data for evaluating the efficacy of various instructional approaches and curricula. Moreover, individuals both within and outside of the mathematics education research community have offered “evidence” to support specific agendas. In view of this turbulent state of affairs, it seems especially timely for researchers and those interested in research to engage in discussions of the notions that are at the heart of all educational research activity. This essay addresses one of these fundamental notions—evidence.

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A Case of Interpretations of Social: A Response to Steffe and Thompson

Stephen Lerman

In their response to my (1996) article, Steffe and Thompson argued that I have taken an early position of Vygotsky's and that his later work is subsumed in and developed by von Glasersfeld. I argue that the two theories, Vygotsky's and radical constructivism, are, on the contrary, quite distinct and that this distinction, when seen as a dichotomy, is productive. I suggest that radical constructivists draw on a weak image of the role of social life. I argue that a thick notion of social leads to a complexity of sociocultural theories concerning the teaching and learning of mathematics, a perspective that is firmly located in the debates surrounding cultural theory of the last 2 decades.

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Interaction or Intersubjectivity? A Reply to Lerman

Leslie P. Steffe and Patrick W. Thompson

Lerman, in his challenge to radical constructivism, presented Vygotsky as an irreconcilable opponent to Piaget's genetic epistemology and thus to von Glasersfeld's radical constructivism. We argue that Lerman's stance does not reflect von Glasersfeld's opinion of Vygotsky's work, nor does it reflect Vygotsky's opinion of Piaget's work. We question Lerman's interpretation of radical constructivism and explain how the ideas of interaction, intersubjectivity, and social goals make sense in it. We then establish compatibility between the analytic units in Vygotsky's and von Glasersfeld's models and contrast them with Lerman's analytic unit. Consequently, we question Lerman's interpretation of Vygotsky. Finally, we question Lerman's use of Vygotsky's work in mathematics education, and we contrast that use with how we use von Glasersfeld's radical constructivism.

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Playing Mathematical Instruments: Emerging Perceptuomotor Integration With an Interactive Mathematics Exhibit

Ricardo Nemirovsky, Molly L. Kelton, and Bohdan Rhodehamel

Research in experimental and developmental psychology, cognitive science, and neuroscience suggests that tool fluency depends on the merging of perceptual and motor aspects of its use, an achievement we call perceptuomotor integration. We investigate the development of perceptuomotor integration and its role in mathematical thinking and learning. Just as expertise in playing a piano relies on the interanimation of finger movements and perceived sounds, we argue that mathematical expertise involves the systematic interpenetration of perceptual and motor aspects of playing mathematical instruments. Through 2 microethnographic case studies of visitors who engaged with an interactive mathematics exhibit in a science museum, we explore the real-time emergence of perceptuomotor integration and the ways in which it supports mathematical imagination.