Representation Reading ECI 314 Early Childhoodhood Mathematics

Representation Reading ECI 314 Early Childhoodhood Mathematics

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The use of copyrighted materials in all formats, including the creation, online delivery, and use of digital copies of copyrighted materials, must be in compliance with U.S. Copyright Law ( http://www.copyright.gov/title17/ ). Materials may not be reproduced in any form without permission from the publisher, except as permitted under U.S. copyright law. Copyrighted works are provided under Fair Use Guidelines only to serve personal study, scholarship, research, or teaching needs. From: Young /Children Reinvent Arithmetic, 2nd ED Author/Editor: Kamii, Constance Teachers College Press (2000) ISBN 0807739049 Item Title: Chapter 2 - Representation
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CHAPTER 2 Representation In Chapter 1, we explained how children construct number concepts. The rep­ resentation of these concepts will be discussed in this chapter. Piaget’s theory about representation is different from the traditional, empiricist assumptions on which mathematics education has been based. These differences give rise to classroom practices that diverge from traditional instruction. Workbooks for kindergarten and first-grade math have many pictures. These pictures are there on the assumption that young children go from the “concrete” (objects) to the “semiconcrete” (pictures), and then to the “abstract” (written numerals). We will argue in the first part of this chapter in light of Piaget’s theory that this assumption is erroneous and that children do not need any of those pictures. Teachers often ask, “Why do you give playing cards to kindergartners? I thought young children needed concrete objects to manipulate.” The issue of “manipulatives” and representation will be discussed in the second section of this chapter. The third section will deal with equations. More specifically, prob­ lems such as 4 + _____ = 6 and “equations” such as “5 + 5 = 10 + 5 = 15” will be discussed. PICTURES IN WORKBOOKS Authors of workbooks assume that numerals and mathematical symbols (e.g., “+”) are too abstract for first graders at the beginning of the school year and that pictures are halfway between concrete objects and mathematical sym­ bols. According to Piaget (1945/1951), pictures and mathematical symbols have different sources, and working with pictures is not necessarily a step toward becoming able to deal with mathematical symbols. Symbols and Signs Piaget (1945/1951) distinguished between symbols, such as pictures and tally marks (the left-hand side of Figure 2.1), and signs, such as words and writ­ ten numerals (the right-hand side). His terminology is confusing at first because he used the term symbol differently from common parlance. 19
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Figure 2.1 . The representation of "eight* in Piagets theory. 2 q Theoretical Foundation In Piaget’s theory, symbols such as pictures bear a resemblance to the ob­ jects represented and can be invented by each child. In other words, the source of symbols is children’s thinking. For example, children can think about “eight" (the rectangle in Figure 2.1 labeled “The child’s idea of‘eight’”) and draw 8 apples or 8 people without any instruction. Likewise, they can use 8 fingers, 8 counters,
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