Sawyer - Vision in Elementary Mathematics

HANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES, EDITED BY MILTON ABRAMOWITZ AND IRENE A. STEGUN. (0-486-61272-4)

ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, JIRI ADAMEK, HORST HERRLICH, GEORGE E. STRECKER. (0-486-46934-4)

NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, SERGIO ALBEVERIO, JENS ERIK FENSTAD, RAPHAEL HEGH-KROHN AND TOM LINDSTRM. (0-486-46899-2)

MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. ALEKSANDROV, A. N. KOLMOGOROV, AND M. A. LAVRENTEV. (0-486-40916-3)

COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THE CIRCLE, NATHAN ALTSHILLER-COURT. (0-486-45805-9)

THE WORKS OF ARCHIMEDES, ARCHIMEDES. TRANSLATED BY SIR THOMAS HEATH. (0-486-42084-1)

REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, ROBERT B. ASH. (0-486-47220-5)

INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, LOUIS AUSLANDER AND ROBERT E. MACKENZIE. (0-486-47172-1)

PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, BONNIE AVERBACH AND ORIN CHEIN. (0-486-40917-1)

THEORY OF LINEAR OPERATIONS, STEFAN BANACH. TRANSLATED BY F. JELLETT. (0-486-46983-2)

VECTOR CALCULUS, PETER BAXANDALL AND HANS LIEBECK. (0-486-46620-5)

INTRODUCTION TO VECTORS AND TENSORS: SECOND EDITION--TWO VOLUMES BOUND AS ONE, RAY M. BOWEN AND C.-C. WANG. (0-486-46914-X)

ADVANCED TRIGONOMETRY, C. V. DURELL AND A. ROBSON. (0-486-43229-7)

FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, LEON EHRENPREIS. (0-486-44975-0)

THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 1, EUCLID. EDITED BY THOMAS L. HEATH. (0-486-60088-2)

THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 2, EUCLID. (0-486-60089-0)

THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 3, EUCLID. EDITED BY THOMAS L. HEATH. (0-486-60090-4)

AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, STANLEY J. FARLOW. (0-486-44595-X)

PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, STANLEY J. FARLOW. (0-486-67620-X)

STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, AVNER FRIEDMAN. (0-486-45359-6)

ADVANCED CALCULUS, AVNER FRIEDMAN. (0-486-45795-8)

POINT SET TOPOLOGY, STEVEN A. GAAL. (0-486-47222-1)

DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. GARDINER. (0-486-45299-9)

LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, GEORGE GRTZER. (0-486-47173-X)

ORDINARY DIFFERENTIAL EQUATIONS, JACK K. HALE. (0-486-47211-6)

METHODS OF APPLIED MATHEMATICS, FRANCIS B. HILDEBRAND. (0-486-67002-3)

BASIC ALGEBRA I : SECOND EDITION, NATHAN JACOBSON. (0-486-47189-6)

BASIC ALGEBRA II: SECOND EDITION, NATHAN JACOBSON. (0-486-47187-X)

NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY THE FINITE ELEMENT METHOD, CLAES JOHNSON. (0-486-46900-X)

ADVANCED EUCLIDEAN GEOMETRY, ROGER A. ,IOHNSON. (0-486-46237-4)

GEOMETRY AND CONVEXITY: A STUDY IN MATHEMATICAL METHODS, PAUL J. KELLY AND MAX L. WEISS. (0-486-46980-8)

TRIGONOMETRY REFRESHER, A. ALBERT KLAF. (0-486-44227-6)

CALCULUS: AN INTUITIVE AND PHYSICAL APPROACH (SECOND EDITION), MORRIS KLINE. (0-486-40453-6)

THE PHILOSOPHY OF MATHEMATICS: AN INTRODUCTORY ESSAY, STEPHAN KRNER. (0-486-47185-3)

COMPANION TO CONCRETE MATHEMATICS: MATHEMATICAL TECHNIQUES AND VARIOUS APPLICATIONS, Z. A. MELZAK. (0-486-45781-8)

NUMBER SYSTEMS AND THE FOUNDATIONS OF ANALYSIS, ELLIOTT MENDELSON. (0-486-45792-3)

EXPERIMENTAL STATISTICS, MARY GIBBONS NATRELLA. (0-486-43937-2)

AN INTRODUCTION TO IDENTIFICATION, J. P. NORTON. (0-486-46935-2)

BEYOND GEOMETRY: CLASSIC PAPERS FROM RIEMANN TO EINSTEIN, EDITED WITH AN INTRODUCTION AND NOTES BY PETER PESIC. (0-486-45350-2)

THE STANFORD MATHEMATICS PROBLEM BOOK: WITH HINTS AND SOLUTIONS, G. POLYA AND J. KILPATRICK. (0-486-46924-7)

SPLINES AND VARIATIONAL METHODS, P. M. PRENTER. (0-486-46902-6)

PROBABILITY THEORY, A. RENYI. (0-486-45867-9)

LOGIC FOR MATHEMATICIANS, J. BARKLEY ROSSER. (0-486-46898-4)

PARTIAL DIFFERENTIAL EQUATIONS: SOURCES AND SOLUTIONS, ARTHUR DAVID SNIDER. (0-486-45340-5)

INTRODUCTION TO BIOSTATISTICS: SECOND EDITION, ROBERT R. SOKAL AND F. JAMES ROHLF. (0-486-46961-1)

MATHEMATICAL PROGRAMMING, STEVEN VAJDA. (0-486-47213-2)

THE LOGIC OF CHANCE, JOHN VENN. (0-486-45055-4)

THE CONCEPT OF A RIEMANN SURFACE, HERMANN WEYL. (0-486-47004-0)

INTRODUCTION TO PROJECTIVE GEOMETRY, C. R. WYLIE, JR. (0-486-46895-X)

FOUNDATIONS OF GEOMETRY, C. R. WYLIE, JR. (0-486-47214-0)

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Copyright

Copyright 1964 by W. W. Sawyer All rights reserved.

Bibliographical Note

This Dover edition, first published in 2003, is an unabridged reprint of the work originally published by Penguin Books Ltd., Hammondsworth, Middlesex, England, in 1964.

Library of Congress Cataloging-in-Publication Data

Sawyer, W. W. (Walter Warwick), 1911

Vision in elementary mathematics / W. W. Sawyer.

p. cm.

Originally published: Hammondsworth, Middlesex, England; Baltimore, Md. : Penguin Books, 1964, in series: Introducing mathematics; 1.

9780486143620

1. MathematicsStudy and teaching (Elementary) I. Title.

QA135.6 .S39 2002

372.7dc21

2002034825

Manufactured in the United States by Courier Corporation
42555X02
www.doverpublications.com

Introduction

VISION IN ARITHMETIC AND ALGEBRA

THIS is a book about mathematics at a very low level. In recent years there seems to have been a sudden increase in the worlds demand for mathematicians. A consequence of this has been that many people who do not feel themselves particularly qualified in mathematics find themselves called upon to teach it, either in an official or an informal capacity. A schools regular mathematics teacher leaves for a post in industry, and the French teacher has to take over his classes. A teacher of young children, who never liked arithmetic much and never did very well at it, finds that public attention is being focused on how she teaches it. Students at teacher training colleges have a similar experience. Parents find themselves unofficially involved in a similar dilemma. They want to help their children with mathematics but they fear that little good will come of the blind leading the blind.

LEARNING MATHEMATICS AND LEARNING CHINESE

The difficulty of learning a subject depends enormously on the way in which the subject is presented. A comparison may be helpful. Suppose that, instead of mathematics, we were setting out to teach Chinese. This seems a good comparison, because to many people mathematics seems much like Chinese. We will try to make the task sound as depressing as possible. To be a Chinese scholar you need to know 20,000 characters. In your first lesson you learn some of these. You go on until you know the lot. However depressing this may be for the pupil, the teachers job seems straightforward enough. He must give the pupil a certain number of new characters each day to learn, and make sure the pupil does not forget what he learnt on former days. The list for the first lesson might be something like . The pupil begins to learn these 14 signs. On other days he adds others, and in due course reaches his 20,000.