By Leslie P. Steffe

The stories provided during this ebook can be of curiosity to anyone fascinated by the instructing of mathematics to youngsters or with cognitive improvement in most cases. The 'eaching scan· was once conducted with part a dozen kids coming into first grade over years in biweekly periods. Methodologically the authors' examine is unique. it's a longitudinal yet now not a naturalistic examine, because the experimenter-teachers directed their interplay with each one person baby with a purpose to his or her attainable growth. it truly is experimental within the experience that teams of topics have been chosen in accordance with standards derived from an past learn (Steffe, von Glasersfeld, Richards & Cobb, 1983) and that the issues proposed have been similar, although faraway from exact around the topics; yet in contrast to extra inflexible and shorter "learning" or ''training" experiences it doesn't contain pre-and posttests, or predetermined strategies. Theoretically, the authors sign up for Piagefs constructivism: numbers are made by way of teenagers, now not came across (as they might locate a few lovely rocks, for instance) or authorized from adults (as they could settle for and use a toy). The authors interpret alterations within the kid's counting behaviors when it comes to constructivist techniques comparable to assimilation, lodging, and reflective abstraction, and sure excerpts from protocols supply online examples of such tactics at paintings. in addition they sign up for Vygotsky's notion for academics '0 make the most of the sector of proximal improvement and to guide the kid to what he (can) now not but do· (1965, p. 104).

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Two. (Places all the marbles into one cup) How many are altogether? (Sequentially puts up the fingers of his left hand and one of his right hand) 1-2-3-4-5-6. (Pauses, and then puts up one more finger of his right hand) Seven. He stopped counting after he put up the seventh finger, completing a visually separated finger pattern for two in a manner analogous to that of Brenda (cf. 06). 08. The substitution of finger patterns for hidden collections occurred when he wanted to find how many marbles were hidden in a container.

It does not matter under what circumstances the connection between the experience of a pattern called three and the experience of counting the three components of the pattern is made. Once made, this connection will provide the first and most immediate opportunity for the revelation that the number word refers both to a unitary thing (the constellation) and to a collection of units (the elements of the constellation). Meaning Theory The preceding conceptual constructs are, in our opinion, crucial in any consideration of children's early numerical meanings.

One principal contributor to the structural school believed that "meaning is to be sought in the structure, the organization, the relationships of the subject itselr (Brownell, 1945, p. 81). The origins of the second school, the operational school, can be traced to Percy Bridgman's operational analysis of the fundamental concepts of modem physics (Bridgman, 1927). Van Engen (1949), a principal contributor to the operational school, believed that the meaning of a mathematical symbol such as "4" is "an intention to act and .

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