
Young Children Continue to Reinvent Arithmetic, 2nd Grade: Implications of Piaget's Theoryreviewed 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 Search for book at Amazon.com 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
selfreflection 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 baseten
blocks represent the baseten 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 socialcultural 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.


