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Number in Preschool and Kindergarten: Educational Implications of Piaget's Theory by Constance Kamii University of Illinois at Chicago Circle and University of Geneva (Switzerland) National Association for the Education of Young Children Washington, D.C. \ l.tt 1,...·. ~ •. : -~·. c :· . ' Photographs: Constan~~ _K~mji a~d RfletCVL'Il or eight, he has the possibility of representing this idea either with symbols or with signs. In Piaget's theory, symbols are different from signs in that symbols bear a figurative resemblance to the objects being represented and are created by the child. An example of symbols is "o o o o o o o o" or" 11111111 ."Examples of signs are the spoken word eight and the written numeral 8. ·Unlike symbols, signs are created by convention and do not bear any resemblance to the objects being represented. The reader may recognize signs as belonging to social knowledge. Representation with signs is overemphasized in early childhood _gducation_and I prefer to put it in the background. Teachers too often Cteach children to count and to read and write numerals, believing that ( they are thereby teaching number concepts. It is good for children to \ learn to count and to read and write numerals, but a more important \objective is for the child to construct the mental structure of number. If a child fias constructed this structure, he will be able to assimi!Jte signs into it with the greatest of ease. If he has not constructed it, all the counting and reading and writing of numerals can only be by rote. While I prefer to deemphasize the teaching of signs, l also feel that it is good to teach them if children are genuinely interested in learning them. In reading, there must be things to read in the environment if Kamii 26 the child is to become interested in reading. When he becomes interested in reading at whatever age, it is best to satisfy his curiosity and pride in acquiring new knowledge. Counting is likewise a joy for most preschool and kindergarten children, and if children want to learn to count, there is no reason to refuse this knowledge. The teacher must, however, know the difference between counting by rote and counting with numerical meaning. The numerical meaning can come only from the logico-mathematical structure constructed by the child in his head. All the spoken and written signs in the world are only surface knowledge. While there must be spokf:>n and written numbers in the environment for the child to become interested in them, understanding them can come only from the mental structure that he constructs from within. 16 ----- .............. In conclusion, the objective for "teaching" number-is_th~ ~hild's construction of the mental structure of numbe.r~CSince this structure cannot be taught directly, the teacher must focus on encouraging the child to think actively and autonomously in all kinds of situations. A child who thinks actively in his own way about airkina·s-ofobje"cts and events, including quantities, will inevitably construct number. "'fhe task of the teacher is to encourage the child's thinking in his own way, which is very difficult because most of us were trained to get children to produce "right" answers. Some principles of teaching to achieve this goal will be discussed in Chapter 3. 16 F. Siegrist, A. Sinclair, and H. Sinclair are conducting research in Geneva on the ideas young children have about numerals at ages four to six, before first grade. Numerals are everywhere in the environment-on houses, buses, cans and boxes, price tags, football players' uniforms, license plates, etc. 3 Principles of teaching In the following discussion, I will speak of "teaching number" even though number is not directly teachable. fhe reason for the use of this term is that the e