Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials Series, No.1) / Edition 1

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Brand new. We distribute directly for the publisher. Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may ... be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought.The proofs in this collection are arranged by topic into six chapters: Geometry and Algebra; Trigonometry, Calculus and Analytic Geometry; Inequalities; Integer Sums; Sequences and Series; and Miscellaneous. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics. Read more Show Less

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Overview

Just what are "proofs without words?" First of all, most mathematicians would agree that they certainly are not "proofs" in the formal sense. Indeed, the question does not have a simple answer. But, as you will see in this book, proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement is true, and also to see how one could begin to go about proving it true. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. Proofs without words bear witness to the observation that often in the English language to see means to understand, and in "to see the point of an argument."

Proofs without words have a long history. In this collection you will find modern rendition of proofs without words from ancient China, classical Greece, twelth-century India- even one based on a published proof by a former President of the United States! However, most of the proofs are relatively more recent creations, and many are taken from the pages of MAA journals.

The proofs in this collection are arranged by topic into six chapters: Geometry and Algebra, Trigonometry, Calculus and Analytic Geometry; Inequalities; Integer Sums; Sequences and Series, and Miscellaneous. Teachers will find that many of the proofs without words in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Read More Show Less

Editorial Reviews

Choice
"By acquiring a knack for explaining in words what they already understand, students will be better prepared to seek their own explanations.... The book offers models of the sort of visual thinking that is often woefully neglected in the classroom."
Crux Mathematicorum
"This book is a nice collection of results which certainly will enhance one's geometric sense and stimulate mathematical thought."
The Matheamtics Teacher
"Teachers who desire to give their students visual mathematical insights will find a great deal of value in this book... I found this book to be a great source for both teachers and their students at the high school level. For those teachers who are striving to satisfy the NCTM's call to reach all students, including those with different writing styles, this book will be an outstanding tool to help students who are visual learners."
The Mathematical Gazette
"Every time I have shown this book to a colleague, I have been asked how to get hold of it. It is an absolute delight.... Buy a copy today."
Booknews
Although not considered proper proofs, this collection of pictures helps students to visualize why a particular theorem is true and how one might go about proving it. Dr. Nelsen has gathered his pictures from many sources, including former President James Garfield, and Nicomachus of Gerasa (A.D. 100). Theorems are taken from geometry and algebra, trigonometry, calculus and analytical geometry, inequalities, integer sums, and sequences and series. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780883857007
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/1993
  • Series: Classroom Resource Materials Series , #1
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 152
  • Age range: 15 - 17 Years
  • Product dimensions: 6.90 (w) x 9.90 (h) x 0.40 (d)

Table of Contents

Introduction
Geometry & Algebra
Trigonometry, Calculus & Analytic Geometry
Inequalities
Integer Sums
Sequences & Series
Miscellaneous
Sources
Index of Names
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Introduction

See (se) v. saw, seen, seeing. - v.t.
5. to perceive (things) mentally; discern; understand: to see the point of an argument.
- The Random House Dictionary of the English Language (2nd Ed.) Unabridged

"Proofs without words" (PWWs) have become regular features in the journals published by the Mathematical Association of America- notably Mathematics Magazine and the College Mathematics Journal. PWWs began to appear in Mathematics Magazine about 1975, and, in an editor's note in the January 1976 issue of the Magazine, J. Arthur Seebach and Lynn Arthur Steen encouraged further contributions of PWWs to the Magazine. Although originally solicited for "use as end-of-article fillers," the editors went on to ask "What could be better for this purpose than a pleasing illustration that made an important mathematical point?"

 A few years earlier Martin Gardner, in his popular "Mathematical Games" column in the October 1973 issue of the Scientific American, discussed PWWs as "look-see" diagrams. Gardner points out that "in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." This dramatically illustrates the dictionary quote above: in English "to see" is often "to understand."

 In the same vein, the editorial policy of The College Mathematics Journal throughout most of the 1980s stated that, in addition to expository articles, "The Journal also invites other types of contributions, most notably: proofs without words, mathematical poetry, quotes,..." (their italics). But PWWs are not recent innovations- they have long history. Indeed, in this volume you will find modern renditions of proofs without words from ancient China, classical Greece, and India of the twelfth century.

 Of course, "proofs without words" are not really proofs. As Theodore Eisenberg and Tommy Dreyfus note in their paper "On the Reluctance to Visualize in Mathematics" [in Visualization in Teaching and Learning Mathematics, MAA Notes Number 19], some consider such visual arguments to be of little value, and "that there is one and only one way to communicate mathematics, and 'proofs without words' are not acceptable." But to counter the viewpoint, Eisenberg and Dreyfus go on to give us some quotes on the subject:

[Paul] Halmos, speaking of Solomon Lefshetz (editor of Annals), stated: "He saw mathematics not as logic but as pictures." Speaking of what it takes to be a mathematician, he stated: "To be a scholar of mathematics you must be born with... the ability to visualize" and most teachers try to develop this ability in their students. [George] Polya's "Draw a figure..." is classic pedagogic advice, and Einstein and Poincare's views that we should use our visual intuitions are well known.

So, if "proofs without words" are not proofs, what are they? As you will see from this collection, this question does not have a simple, concise answer. But generally, PWWs are pictures or diagrams that help the observer see why a particular statement may be true, and also to see how one might begin to go about proving it true. In some an equation or two may appear in order to guide the observer in this process. But the emphasis is clearly on providing visual clues to the observer to stimulate mathematical thought.

 I should note that this collection is not intended to be complete. It does not include all PWWs which have appeared in print, but is rather a sample representative of the genre. In addition, as readers of the Association's journals are well aware, new PWWs appear in print rather frequently, and I anticipate that this will continue. Perhaps some day a second volume of PWWs will appear!

 I hope that the readers of this collection will find enjoyment inn discovering or rediscovering some elegant visual demonstrations of certain mathematical ideas; that teachers will want to share many of them with their students; and that all will find stimulation and encouragement to try to create new "proofs without words."

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