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Eleanor Duckworth

1 Jean Piaget, conference at Cornell University, Ithaca, N.Y., 1964.

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Arthur F. Coxford Jr

For approximately forty years the French psychologist Jean Piaget and his various collaborators have been producing volumes dealing with the formation of concepts in children. He has intensively studied the concepts of number, geometry, physical causality, space, the world, and reality; also the moral judgment, reasoning, language, and thought of the child. He has published a total of twenty-five books on these subjects, four teen of which are now in English.

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Daiyo Sawada

Over the past ten years, and with increasing frequency, the name of Jean Piaget has become a familiar sight to the eyes of mathematics teachers, especially at the elementary school level. Numerous books have appeared (Almy 1966; Athey and Rubadeau 1970; Beard 1969; Copeland 1970; Furth 1970), along with a large number of journal publications, th at address themselves to the topic of Piagetian implications for teaching.

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John Richards and Ernst Von Glasersfeld

The challenging title of Brian Rotman's book neatly formulates the conclusion the author draws from his investigation of some of Piaget's writings. Piaget, Rotman argues, is a realist. External reality is the constant, everpresent guide that leads Piaget to a progress-oriented, accretionist theory of evolution, cognition, knowledge, and finally, mathematics. Rotman brings out the deficiencies of the progress-oriented approach and applies them in his critique of Piaget.

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Raymond J. Duquette

Perhaps because his findings have been published mostly in French and he used French for his few American appearances, it has not been until recently that Jean Piaget has become known in America. Although some of his works date back to the 1920s-for instance, The Language and Thought of the Child—his findings have had very little effect on American education. However, his four stages of development can play an important role in curriculum considerations in our schools.

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J. Fred Weaver

Based on remarks that I have made on more than one occasion, some persons may view me as being anti-Piaget. I am not.

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Constance Kamii and Kelly A. Russell

Based on Piaget's theory of logico-mathematical knowledge, 126 students in grades 2–5 were asked 6 questions about elapsed time. The main reason found for difficulty with elapsed time is children's inability to coordinate hierarchical units (hours and minutes). For example, many students answered that the duration between 8:30 and 11:00 was 3 hours 30 minutes (because from 8:00 to 11:00 is 3 hours, and 30 more minutes is 3 hours 30 minutes). Coordination was found to begin among logicomathematically advanced students, through reflective (constructive) abstraction from within. The educational implications drawn are that students must be encouraged to think about durations in daily living and to do their own thinking rather than being taught procedures for producing correct answers to elapsed-time questions.

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Arthur F. Coxford Jr.

In his book, The Child's Conceplion of Number, Jean Piaget stated that the concept of number has three basic aspects: cardinal number, ordinal number, and unit. He has given criteria for determining when a child understands each of the basic concepts. A child understands cardinal number when he is able' to construct a one-to-one correspondence between two sets of objects and to conserve this corrspondcnce when it is no longer perceptually obvious.

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Anderson Norton and Jesse L. M. Wilkins

Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.

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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.