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The Calculus: a Genetic Approach / Otto Toeplitz ; with a New Foreword by David M

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THE CALCULUS THE CALCULUS A Genetic Approach OTTO TOEPLITZ New Foreword by David Bressoud Published in Association with the Mathematical Association of America The University of Chicago Press Chicago · London The present book is a translation, edited after the author's death by Gottfried Kothe and translated into English by LuiseLange. The German edition, DieEntwicklungder Infinitesimalrechnung,was published by Springer-Verlag. The University of Chicago Press,Chicago 60637 The University of Chicago Press,ltd., London © 1963 by The University of Chicago Foreword © 2007 by The University of Chicago All rights reserved. Published 2007 Printed in the United States of America 16 15 14 13 12 11 10 09 08 07 2 3 4 5 ISBN-13: 978-0-226-80668-6 (paper) ISBN-10: 0-226-80668-5 (paper) Library of Congress Cataloging-in-Publication Data Toeplitz, Otto, 1881-1940. [Entwicklung der Infinitesimalrechnung. English] The calculus: a genetic approach / Otto Toeplitz ; with a new foreword by David M. Bressoud. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-80668-6 (pbk. : alk. paper) ISBN-10: 0-226-80668-5 (pbk. : alk. paper) 1. Calculus. 2. Processes, Infinite. I. Title. QA303.T64152007 515-dc22 2006034201 § The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences-Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992. FOREWORDTO THECALCULUS: A GENETICAPPROACHBYOTTO TOEPLITZ September 30, 2006 Otto Toeplitz is best known for his contributions to mathematics, but he was also an avid student of its history. He understood how useful this history could be in in­ forming and shaping the pedagogy of mathematics. This book, the first part of an uncompleted manuscript, presents his vision of an historically informed pedagogy for the teaching of calculus. Though written in the 1930s, it has much to tell us to­ day about how we might-even how we should-teach calculus. We live in an age of a great democratization of calculus. A course once reserved for an elite few is now moving into the standard college preparatory curriculum. This began in the 1950s, but the movement has accelerated in the past few decades as knowledge of calculus has come to be viewed as a prerequisite for admission to the best colleges and universities, almost irrespective of the field that will be stud­ ied. The pressures and opportunities created by this popularization have resulted in two significant movements that have shaped our current calculus curriculum, the New Math of the 1950s and '60s, and the Calculus Reform movement of the 1980s and '90s. These movements took the curriculum in very different directions. The New Math was created in response to the explosion in demand for scientists and engineers in the years following World War II. To prepare these students for ad­ vanced mathematics, the curriculum shifted to focus on abstraction and rigor. This is the period in which Riemann's definition of the integral entered the mainstream calculus curriculum, a curriculum that adopted many of the standards of rigor that had been developed in the nineteenth century as mathematicians extricated them­ selves from the morass of apparent contradictions revealed by the introduction of Fourier series. One of the more reasoned responses to the New Math was a collective statement by Lipman Bers, Morris Kline, George P6lya, and Max Schiffer, cosigned by many others, that was published in The American Mathematical Monthly in 1962.1 In this letter, they called for the use of the "genetic method:" "The best way to guide the mental development of the individual is to let him retrace the mental development IThe American Mathematical Monthly, 1962,69:189-93. v vi FOREWORD of the race-retrace its great lines, of course, and not the thousand errors of detail." I cannot believe it was a coincidence that one year later the University of Chicago Press published the first American edition of The Calculus: A Genetic Approach. The Calculus Reform movement of the 1980s was born from the observation that too many students were confused and overwhelmed by an approach to calcu­ lus that was still rooted in the rigor of the 1950s and '60s. In my experience, most calculus students genuinely want to understand the subject. But as students en­ counter concepts that do not make sense to them and as they become confused, they fall back on memorization. These students then emerge from the study of calculus with nothing more than a capacity to handle its procedures and algorithms, with lit­ tle awareness of its ideas or the range of its uses. In the 1980s, departments of math­ ematics were facing criticism from other departments, especially departments in en­ gineering, that we were failing too many of their students, and those we certified as knowing calculus in fact had no idea how to apply its concepts in other classes. The Calculus Reform movement tried to achieve two goals: to create student awareness of and ability to work directly with the concepts of calculus, and to increase the ac­ cessibility of calculus, to make it easier for more students to learn what they would need as they moved into subsequent coursework and careers. It created its own backlash. The argument commonly given against its innovations was that it weak­ ened the teaching of calculus, but much of the resistance came from the fact that it required more effort to teach calculus in ways that improve both accessibility and understanding. Today, the battles over how to teach calculus have receded. Most of the innova­ tive curricula created in the late 1980s have either disappeared or mutated into something that looks suspiciously like the competition. The movement did change what and how we teach: more opportunities for exploration, greater emphasis on the interpretation of graphical and tabular information, a recognition that the abil­ ity to read and communicate mathematical ideas is something that must be devel­ oped, more varied and interesting problems, a recognition of when and how com­ puting technology can aid in the transmission of ideas and insights. At the same time, we still use Riemann's definition of the integral, and there is a lingering long­ ing for the rigor of epsilons and deltas. The problems that initiated both the New Math and the Calculus Reform are still with us. We still have too few students pre­ pared for the advanced mathematics that is needed for many of today's technical fields. Too few of the students who attempt calculus will succeed in it. Too few of those who complete the calculus sequence understand how to transfer this knowl­ edge to other disciplines. Toeplitz's The Calculus: A Genetic Approach is not a panacea now any more than it was over forty years ago. But it brings back to the fore an approach that has received too little attention: to look at the origins of the subject for pedagogical in­ spiration. As Alfred Putnam wrote in the Preface to the first American edition, this is not a textbook. It is also not a history. Though Toeplitz knew the history, he is not attempting to explain the historical development of calculus. What he has created is a distillation of key concepts of calculus illustrated through many of the problems by which they arose. FOREWORD vii I agree with Putnam that there are three important audiences for this book, though I would no longer group them quite as he did in 1963. The first audience con­ sists of students, especially those who are not challenged by the common calculus curriculum, who can turn to this book to supplement their learning of calculus. One of the penalties one pays for accessibility is a leveling of the curriculum. There are too few opportunities in our present calculus curricula for students with talent to wrestle with difficult ideas. I fear that we lose too many talented students to other disciplines because they are introduced to calculus too early in their academic ca­ reers and are never given the opportunity to explore its complexities. For the high school teacher who wonders what to do with a student who has "finished" calculus as a sophomore or junior, for the undergraduate director of mathematics who won­ ders what kind of a course to offer those students who enter college with credit for calculus but without the foundation for higher study that one would wish, this book offers at least a partial solution. Its ideas and problems are difficult and enticing. Those who have worked through it will emerge with a deep appreciation of the na­ ture and power of calculus. Putnam's second and third audiences, those who teach calculus and those preparing to teach secondary mathematics, have largely merged. Prospective sec­ ondary mathematics teachers must be prepared to teach calculus, which means they must have a depth of understanding that goes beyond the ability to pass a course of calculus. Toeplitz's book can provide that depth. The final audience consists of those who write the texts and struggle to create meaningful curricula for the study of calculus. This book can help us break free of the current duality that limits our view of the calculus curriculum, the belief that calculus can be either rigorous or accessible, but not both. It suggests an alternate route through the historical development of difficult ideas, gradually building the pieces so that they make sense. Too many authors think they are using history when they insert potted accounts that attempt to personalize the topic under discussion. Real reliance on history should throw students into the midst of the confusion and exhilaration of the moment of discovery.
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