Young Children Continue to Reinvent Arithmetic, 2nd Grade: Implications of Piaget's Theory
reviewed by David Kuschner - 2004
Title: Young Children Continue to Reinvent Arithmetic, 2nd Grade: Implications of Piaget's Theory
Author(s): Constance Kamii with Linda Leslie Joseph
Publisher: Teachers College Press, New York
ISBN: 0807744034, Pages: 194, Year: 2004
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To fully understand the ideas expressed in this book by Constance Kamii and her colleague Linda Leslie Joseph, it is important to take note of the word reinvent in the title. The phrase, reinvent arithmetic, suggests that although there may be fixed products for arithmetic operations, e.g., 5 + 12 will always equal 17, the understanding of the operations is constructed anew by each individual child. This book, therefore, is not about the teaching of arithmetic to second graders. Based on the Piagetian concept that children must construct knowledge and meaningful understanding of intellectual concepts for themselves, Kamii and Joseph suggest that the purpose of the arithmetic curriculum in second grade, and along with it the role of teacher, is to provide the right context for this construction and reinvention to take place. The ultimate goal for the curriculum, furthermore, is not the mastery of the multiplication tables or the ability to solve problems as quickly as possible but rather the development of children’s intellectual autonomy.
What is the right context for this reinvention of arithmetic to take place? According to the authors it is one that encourages children’s development of their own strategies for thinking about and arriving at solutions to arithmetic problems. As children think about these problems, they are encouraged to share their strategies with others, thus providing the opportunity for self-reflection and exposure to alternative perspectives. It is also a context in which wrong answers are valued as the products of intellectual activity and are seen as important components of the ongoing process of constructing knowledge. Kamii and Joseph believe that if children are encouraged to develop their own strategies for thinking about the problems, arriving at the correct answer is inevitable.
In arithmetic, a major objective of traditional instruction is to get children to learn correct techniques of producing right answers. In the Piagetian approach, by contrast, the objectives are conceived in terms of children’s ability to think, that is, their ability to invent various ways of solving problems and to judge whether a procedure makes logical sense. We do not stress the correctness of the answer because if children can think, they sooner or later will get the correct answer. (p. 157)
The book itself is divided into four sections. The first two sections outline the theoretical foundations, and the goals and objectives for this approach to arithmetic education. The remaining two sections offer suggestions for activities that would foster children’s reinvention of arithmetic. The activities are built around computational and story problems, situations from daily living, and group games.
In general, the theoretical discussion is clear and persuasive. The first chapters provide the reader with a solid introduction to the constructivist perspective as it applies to arithmetic education. There are also interesting examples of children’s efforts at figuring out problems that powerfully illustrate and amplify the theoretical points. The chapters discussing activities relate well to the earlier chapters and offer the reader a good sense of how the theoretical concepts can be translated into practice.
I do have one criticism of the book. Even though the subtitle of the book is, “Implications of Piaget’s theory,” I found it curious that there was lack of any reference in the book to the social constructivist ideas of Vygotsky. The authors do emphasize children’s collaborative strategy building when it comes to solving arithmetic problems and have an entire chapter devoted to “The Importance of Social Interaction.” Their theoretical perspectives and activity suggestions, I believe, relate to such Vygotskian concepts as scaffolding and the zone of proximal development and the book would have been strengthened if those connections were acknowledged.
The following is an example of how Kamii and Joseph tend to emphasize the role of the individual in the construction of knowledge and ignore the contributions of the social world to the process.
In empiricist thinking, it is correct to say that the symbol ‘+’ represents addition, that the ‘2’ in ‘23’ represents ‘twenty,’ and that base-ten blocks represent the base-ten system. In Piaget’s theory, however, all the previous statements are incorrect because representation is what a human being does. Symbols do not represent; it is always a human being who uses a symbol to represent an idea (p. 17).
I do agree that representation and the use of symbols involve the transformation of meaning on the part of the individual knower, i.e., representing meaning by use of a symbol and then interpretation of the symbol back into some sort of meaning. And I also agree that the meaning is personal on both ends of the transformation. But to say that the ‘+’ symbol doesn’t represent addition and that the ‘2’ in ‘23’ doesn’t represent ‘twenty’ is ignoring the fact that the problem exists in a social context. It is true that a real understanding of ‘+’ or ‘2’ requires an individual construction of meaning and that this meaning can’t be simply transmitted from the culture to the child. There is, however, the fact that cultures “agree” to let a particular mark or symbol stand for a specific concept and this agreement then allows for communal understanding and communication. I have no argument if Kamii and Joseph are making the point that the meaning of a symbol cannot simply be transmitted from the culture to the child. But the origin of the symbol does lie in the social-cultural world. We don’t ask children to construct their own personal symbols to represent the concepts of ‘addition’ and ‘twenty’; we ask them to construct an understanding of the culturally agreed upon symbols, i.e., symbols that originate in the social world.
I’ll end this review with a personal anecdote. I doubt that she would remember, but in 1975 Contstance Kamii interviewed me for a faculty position when she was teaching at the University of Illinois, Chicago Circle. The interview took place over dinner at a restaurant, and at some point during the interview I took a napkin and diagrammed my idea for an activity that would help children develop their sense of number and counting. Kamii was quite attentive as she listened to my description of the activity, but when I was through, she summed up her evaluation of my idea by simply saying: "It's a trick." She used this expression to capture the essential problem with my suggested activity: the material would lead children to a superficial representation of number awareness without their having to actually think about number. She was right then, and her work almost thirty years later is still focused on that distinction: it is one thing to be able to arrive at the correct answer, and it is quite another thing to be able to think about the problem and construct your own understanding of the solution.