YOUNG CHILDREN CONTINUE TO REINVENT ARITHMETIC 3rd Grade IMPLICATIONS OF PIAGET'S THEORY

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1 YOUNG CHILDREN CONTINUE TO REINVENT ARITHMETIC 3rd Grade IMPLICATIONS OF PIAGET'S THEORY Constance Kaii with Sally Jones Livingston

2 Published by Teachers College Press, 1234 Asterda Avenue New York, NY Copyright 1994 by Teachers College, Colubia University Figure 6.3 fro Look Into the Facts: Multiplication and Division, by Carol A. Thornton & Cathy Noxon is reprinted by perission of the publisher, Creative Publications, Inc, of Palo Alto, California. Copyright 1977 by Carol A. Thornton & Cathy Noxon. Quotations fro "The Developent of Children's Place-Value Nueration Concepts in Grades Two through Five" by Sharon H. Ross, a paper presented at the annual eeting of the Aerican Educational Research Association, San Francisco, in April 1986, are reprinted by perission of the author. ERIC Docuentary Reproduction Service No. ED Algoriths fro "Suing It All Up: Pre-1900 Algoriths" by Eleanor S. Pearson, an article appearing in Arithetic Teacher, 33 (March, 1986), are reprinted by perission of the publisher, National Council of Teachers of Matheatics, Reston, Virginia. Figures 9.1 and 9.3 are adapted fro Figures 6.2 and 6.3 of the chapter "Thinking Strategies: Teaching Arithetic through Proble Solving" by Paul Cobb and Graceann Merkel in New Directions for Eleentary School Matheatics (1989 NCTM Yearbook), edited by Paul R. Trafton and are adapted and reprinted by perission of the publisher. National Council of Teachers of Matheatics, Reston, Virginia. Quotations fro Jean Piaget, The Moral Judgent of the Child, translated by Marjorie Gabain. New York: The Free Press Reprinted with perission of the publisher. Quotations fro Jean Piaget, The Psychology of Intelligence (1963) are reprinted by perission of the publisher. Paterson, NJ: Littlefield, Adas, & Co. Quotations fro La Foration des Raisonneents Recurrentiels by P. Greco, B. Inhelder, B. Matalon, and J. Piaget, Copyright 1963 by P. Greco et al., are translated and reprinted by perission of the publisher, Presses Universitaire de France, Paris, France. Quotations fro A Survey of Matheatics: Eleentary Concepts and Their Historical Developent, by Vivian Shaw Groza, Copyright 1968 by Vivian Shaw Groza, are reprinted by perission of the author, and of the publisher, Holt, Rinehart, and Winston, Inc., Fort Worth, Texas. Excerpts of historical atheatical ethods of different ethnic-geographic groups are adapted fro History of Matheatics, Vol II, by David Eugene Sith, Copyright 1925, by D. E. Sith, published by Cinn and Copany. Used by perission of Silver Burdett Ginn, Inc. All rights reserved. No part of this publication ay be reproduced or transitted in any for or by any eans, electronic or echanical, including photocopy, or any inforation storage and retrieval syste, without perission fro the publisher.. Library of Congress Cataloging-in-Publication Data Kaii, Constance. Young children continue to reinvent arithetic 3rd grade : iplications of Piaget's theory / Constance Kaii; with Sally Jones Livingston, p. c. Includes bibliographical references and index. ISBN ISBN (pbk.) 1. Arithetic Study and teaching (Eleentary) 2. Nuber concept Study and teaching (Eleentary) 3. Piaget, Jean, I. Jones Livingston, Sally. II. Title. QA135.5K '2044 dc Printed on acid-free paper Manufactured in the United States of Aerica

3 CHAPTER 5 Autonoy: The Ai of Education for Piaget H 1 II Piaget's greatest direct contribution to education ay well be a short chapter he wrote in a sall, siple book entitled To Understand Is to Invent (1948/ 1973). In the fourth chapter of this book, he argued that education ust ai at autonoy rather than at obedience and confority. Since traditional education unwittingly ais at fostering obedience and confority, changing the goal to autonoy entails changes in the way teachers ake every decision fro one oent to the next. This is why we believe that Piaget's conceptualization of autonoy as the ai of education ay be his greatest direct contribution to education. As can be anticipated fro the previous chapter, autonoy has two aspects for Piaget, the oral and the intellectual. Since autonoy in Piaget's theory eans soething different fro its usual eaning, we first clarify "oral autonoy" and "intellectual autonoy." We then discuss autonoy as the ai of education and show in what ways this goal changes what teachers do during every inute of the school day. We also argue that autonoy as the ai of education would drastically change the efforts being ade in the nae of education refor. WHAT PIAGET MEANT BY AUTONOMY In coon parlance, autonoy eans the right of an individual or group to be self-governing. For exaple, when we speak of Palestinian autonoy, we are referring to this kind of political right. In Piaget's theory, however, autonoy refers not to the right but to the ability to be self-governing. Autonoy is the ability to think for oneself and to decide between right and wrong in the oral real, and between truth and untruth in the intellectual real, by taking all relevant factors into account, independently of reward and punishent. A characteristic of autonoy is the ability to cooperate, which has a unique eaning in Piaget's theory, as we saw in Chapter 4. For Piaget, there is no autonoy without cooperation. Autonoy is the opposite of heteronoy. Heteronoous people are governed by soeone else, as they are unable to think for theselves. Below is a discussion of the oral and the intellectual aspects of this difficult concept. II IWffl H i 59

4 60 Goals and Objectives Moral Autonoy A clear exaple of oral autonoy is Martin Luther King's struggle for civil rights. King was autonoous enough to take relevant factors into account and to conclude that the laws discriinating against African Aericans were unjust and ioral. Convinced of the need to ake justice a reality, he fought to end the discriinatory laws, in spite of the police, jails, dogs, water hoses, and threats of assassination used to stop hi. Morally autonoous people are not governed by reward and punishent. An exaple of extree oral heteronoy is found in the Watergate coverup. The en under President Nixon were governed by hi and went along with what they knew to be orally wrong, reaping the rewards the president dispensed to those who helped hi in the cover-up attept. In The Moral Judgent of the Child, Piaget (1932/1965b) gave ore coonplace exaples of autonoy and heteronoy. For exaple, he asked children 6 to 14 years of age whether it was worse to he to an adult or to another child. Young children tended to reply that it was worse to fie to an adult. When Piaget inquired why, young children revealed their heteronoy by explaining, "Because adults can tell if soething is not true and punish you." Piaget went on to ask, "Would it be OK to tell lies if you were not punished for the?" and young children answered "Yes." Piaget (1932/1965b) also ade up any pairs of stories and asked children which one of the two children in the stories was the worse. Following is an exaple of such a pair: A little boy (or a little girl) goes for a walk in the street and eets a big dog who frightens hi very uch. So then he goes hoe and tells his other he has seen a dog that was as big as a cow. A child coes hoe fro school and tells his other that the teacher had given hi good arks, but it was not true; the teacher had given hi no arks at all, either good or bad. Then his other was very pleased and rewarded hi. (p. 148) Young children systeatically anifested the orality of heteronoy by saying that it was worse to say "I saw a dog as big as a cow." Why was it worse? Because dogs are never as big as cows and adults do not believe such stories. Older, ore autonoous children, however, tended to say that it was worse to say "The teacher gave e good arks" because this lie was ore believable. For ore autonoous children, a believable lie is worse than one that is so outlandish as to not deceive people. The iportant question for parents and teachers is: What causes certain children to becoe ore autonoous than others? Piaget's answer to this question,

5 Autonoy: The Ai of Education for Piaget 61 1 i found in The Moral Judgent of the Child, was that adults reinforce children's heteronoy when they use rewards and punishent, thereby hindering the developent of autonoy. By refraining fro using rewards and punishent, and by exchanging points of view with children instead, we can foster the developent of autonoy, he said. For exaple, if a child tells a lie, an adult could punish hi or her by saying, "No dessert tonight." Alternatively, the adult can look the child straight in the eye with affection and skepticis and say, "I really can't believe what you are saying because (and give the reason). And when you tell e soething next tie, I a not sure I'll be able to believe you. I want you to go to your roo (or seat) and think about what you ight do to be believed." Children want to be believed, and when they are confronted with this kind of stateent, they are likely, over tie, to coe to the conclusion that it is best for people to deal honestly with one another. ' In general, punishent leads to three possible outcoes. The first outcoe is calculation of risks. Children who are punished will learn to calculate their chances of getting caught the next tie and the price they ight have to pay if they are caught. The second possible outcoe is, interestingly, the opposite of the first one blind obedience. Sensitive children will do anything to avoid being punished, thereby giving the ipression that punishent works. The third outcoe of punishent is a derivative of the second, naely, revolt. Many "good," odel children surprise us eventually by beginning to cut classes, take drugs, and engage in other acts characteristic of delinquency. Their reason for switching to these behaviors is that they are tired of living for their parents and teachers and think that the tie has coe for the to start living for theselves. Piaget was realistic enough to say that it is soeties necessary to ipose restrictions on children. However, he ade an iportant distinction between punishent and sanctions by reciprocity. Depriving the child of dessert for telling a lie is an exaple of a punishent, as the relationship between a lie and dessert is copletely arbitrary. Telling children that we cannot believe what they said is an exaple of a sanction by reciprocity. Sanctions by reciprocity are directly related to the act we want to sanction and to the adult's point of view. They have the effect of otivating the child to construct rules of conduct fro within, through the coordination of viewpoints. Other exaples of sanctions by reciprocity, such as exclusion fro the group, depriving the child of the thing he or she has isused, and restitution, can be found in The Moral Judgent of the Child and in Kaii (1982, 1985). When adults exchange viewpoints with children, this fosters the developent of autonoy by enabling children to decenter and consider relevant factors, such as other perspectives. When children can take relevant factors into account, especially other people's rights and feelings, they construct fro within

6 62 Goals and Objectives J" the rule of treating others as they wish to be treated by the. A person who has constructed this conviction fro within cannot he in situations such as the Watergate affair, no atter what reward is offered. Many behaviorists and others believe that punishent is bad because it is negative, but that rewards are positive and good. However, rewards do not ake children any ore autonoous than does punishent. Children who help their parents only to get oney, and those who fill out worksheets only to get a sticker, are governed by soeone else just as uch as those who are "good" only to avoid being punished. Intellectual Autonoy In the intellectual real, too, autonoy eans the ability to govern oneself by being able to take relevant factors into account, and heteronoy eans being governed by soebody else. An exaple of outstanding intellectual autonoy is Copernicus or the inventor of any other revolutionary theory in the history of science. Copernicus invented the heliocentric theory when everybody else believed that the sun revolved around the earth. But although ridiculed, he was autonoous enough to reain convinced of his own idea. An intellectually heteronoous person, by contrast, unquestioningly believes what he or she is told, including illogical conclusions, slogans, and propaganda. A ore coon exaple of intellectual autonoy is a child who used to believe in Santa Claus. When she was about 6, she surprised her other one day by asking, "How coe Santa Claus uses the sae wrapping paper as we do?" Her other's "explanation" satisfied her for a few inutes, but she soon cae up with the next question: "How coe Santa Claus has the sae handwriting as Daddy?" This child had her own way of thinking about Santa Claus, which was different fro what she had been told. Unfortunately, in school, children are not encouraged to think autonoously. Teachers use reward and punishent in the intellectual real, too, to get children to give "correct" responses. An exaple of this practice is the use of worksheets. In first-grade arithetic, for exaple, if a child writes "4 + 4 = 7," ost teachers ark this answer as being wrong. The result of this kind of teaching can be seen when we walk around a first-grade classroo while children are working on worksheets and stop to ask individual children how they got particular answers. They typically react by grabbing their erasers, even when their answer is perfectly correct! Already in first grade, any children have learned to distrust their own thinking. Children who are thus discouraged fro thinking critically and autonoously will construct less knowledge than those who are confident and do their own thinking. In the next section, on autonoy as the ai of education, we argue further that we ust replace an education that unwittingly ais at obedience and con-

7 H^H ^^^K Vr^B A Autonoy: The Ai of Education for Piaget 63 FIGURE 5.1 Autonoy as the ai of education in relation to the goals of ost educators and the public, Autonoy fority with one that ephasizes the honest, critical exchange of viewpoints aong peers. AUTONOMY AS THE AIM OF EDUCATION Figure 5.1 is our interpretation of autonoy as the ai of education in relation to the goals of ost educators and the public. In the shaded part of the circle labeled "the goals of ost educators and the public," we include those that resulted in our eorizing words, just to pass one test after another. All of us who succeeded in school achieved this success by eorizing an enorous nuber of words without understanding the or caring about the. The shaded part also includes the oral heteronoy that schools generally reinforce by using reward and punishent to enforce readyade rules. In the intersection with the circle labeled "autonoy," we list things we did not forget after each test. The ability to read and write, to do arithetic, to read ' aps and charts, and to situate events in history are exaples of what we learned in school that we did not forget after craing for tests. When oral and intellectual autonoy becoes our ai, educators work hard to increase the area of overlap between the two circles. Piaget chose atheatics as an exaple to argue that the atosphere of a classroo either enhances or squelches the developent of autonoy in ways that are inseparably oral and intellectual. If the classroo is governed by the authority of the teacher, children will be encouraged to confor to the teacher's wishes, without separating sociooral issues fro purely intellectual ones. If, on the other hand, the class is governed by the group, and the teacher reduces

8 64 Goals and Objectives his or her power as uch as possible, children will exchange points of view freely to ake decisions in both the sociooral and intellectual reals. If a teacher is coercive in one area, it is ipossible for children to feel free to ake decisions in the other. Just as exchanges of viewpoints are indispensable for children to overcoe their egocentricity in the intellectual real, exchanges of points of view are essential for children to decenter in the sociooral real. For exaple, if one child coplains that Johnny always goes first in a ath gae and Johnny disagrees, the teacher could ipose a solution such as "The person who rolls the biggest nuber goes first." However, the teacher who keeps autonoy constantly in ind as the ai of education would suggest cooperation and say, "Can you two think of a solution that would be fair to both of you?" The children ay end up aking the sae rule that the teacher could have iposed. Fro the standpoint of children's developent of sociooral autonoy, however, who akes rules akes an enorous difference. Children ust coordinate points of view and ake rules that ake sense to the. They are uch ore likely to respect a rule they ade than the sae rule iposed by the teacher. The essential principle for the developent of autonoy is to ask children "What do you think is the right thing to do?" rather than trying to anipulate the with reward and punishent. Reward and punishent at best leads only to confority. Children who are encouraged to debate possible solutions to sociooral probles think about the fro any perspectives and coordinate points of view. This is how, over tie, these children becoe able to ake decisions on the basis of what is best for all concerned. Autonoous people are governed by what is orally right and what is intellectually true, rather than by reward and punishent. CONCLUSION The reader can probably see that the teaching of algoriths reinforces children's heteronoy. By teaching readyade rules and using reward and punishent, albeit in ild fors, schools are unwittingly teaching confority, blind obedience, and dependence on adults. By fourth grade, if we ask children to explain the steps they follow in long division, they all say, "I don't know why [I brought down this nuber], but y teacher said to do it this way." The nations schools are having enorous probles with drugs, violence, AIDS, teenage pregnancies, suicides, and so on. Most educators and the public, including the National Governors' Association (U.S. Departent of Education, 1991), view these probles as being separate phenoena that are unrelated to ath education. However, autonoy as the ai of education enables us to see that if children learn atheatics through blind obedience, we cannot expect

9 Autonoy: The Ai of Education for Piaget 65 the to have the judgent and autonoy to say "no" to drugs, peer pressure, and sex. Children who can take relevant factors into account do not take drugs or confor autoatically to peer pressure. Those who can exchange points of view and negotiate solutions to probles do not resort to violence either. The probles of drugs, violence, and teenage pregnancies are all syptos of heteronoy. Autonoy as the ai of education is also iportant for children's selfdirectedness and otivation fro within during the ath hour. We have been aazed by the serene atosphere in soe classroos and children's respectfulness toward one another. When soe children take a long tie to write all over the blackboard, for exaple, their classates have surprised us by reaining keenly attentive. Such interest and respect for other students' ideas can coe only fro within. Teachers who genuinely respect children and ake decisions with the see to create such an atosphere of respect. If every school in the nation adopted autonoy as the ai of education, the work force would becoe ore copetent, racial and ethnic conflicts would be reduced, and the nation's prisons would becoe less crowded. It behooves us to conceptualize goals for atheatics in the context of overall long-ter goals for education if we want school refor to produce ore than higher test scores..at 1 ;nl