Math G2: We don't need no stinking correct answers.

["Arthur Hu" <arthurhu@comcast.net>]

2ND GRADE MATH: WE DO NOT STRESS CORRECTNESS OF ANSWER IF CHILDREN CAN THINK
z75\clip\2004\02\cons2nd.txt
http://www.tcrecord.org/Content.asp?ContentID=11260
Young Children Continue to Reinvent Arithmetic, 2nd Grade:
Implications of Piaget's Theory. Constance Kamii with Linda Leslie
Joseph. New York: Teachers College Press, 2004, ISBN: 0807744034, 194
pp. (review by)  David  Kuschner University of Cincinnati
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...  . 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)


Beyond Piaget: A Philosophical Psychology  [1986]
Young Children Reinvent Arithmetic: Implications of Piaget's Theory
[1986]
Jean Piaget and Rudolf Steiner: Stages of Child Development and
Implications for Pedagogy [1982]
What Does Piaget's Theory Describe?  [1982]
Beyond Piaget: A Philosophical Psychology
Education and Psychology: Plato, Piaget, and Scientific Psychology
Play and Development: A Symposium with Contributions by Jean Piaget,
Peter Wolff, Rene A. Spitz, Konrad Lorenz, Lois Barclay Murphy, and
Erik H. Erikson

 Young Children Continue to Reinvent Arithmetic, 2nd Grade:
Implications of Piaget's Theory

Young Children Continue to Reinvent Arithmetic, 2nd Grade:
Implications of Piaget's Theory. Constance Kamii with Linda Leslie
Joseph. New York: Teachers College Press, 2004, ISBN: 0807744034, 194
pp.

Browse for this title at Amazon.com®

  David Kuschner University of Cincinnati Author Bio | E-mail Author

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.



----------------------------------------------------------------------------
----
Comment on this article
----------------------------------------------------------------------------
----


----------------------------------------------------------------------------
----
Teachers College Record Volume 106 Number 5, 2004, p. -
http://www.tcrecord.org ID Number: 11260, Date Accessed: 2/7/2004
----------------------------------------------------------------------------
----


Content Tools
----------------------------------------------------------------------------
----
  E-mail this article

  Printer-ready version

  Comment on this article


...Recently posted comments are displayed below...

Psychobabble
By: Marsh Kaminsky  2/6/2004



----------------------------------------------------------------------------
----
View all reader comments
----------------------------------------------------------------------------
----











----------------------------------------------------------------------------
----
Copyright 2002 Teachers College, Columbia University. All rights reserved.
Privacy Policy | Terms of Use | Copyright Agreement | Contact TCRecord.org |
Awards and Distinctions