Proofs without Words : Exercises in Visual Thinking (Classroom Resource Material)

Rating:
Summary: Engaging exercises to train your intuition
Review: Famous mathematicians have often emphasized the role of visual intuition; e.g., Hilbert: "Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?" (from his famous address at the 1900 International Congress). This book is a collection of well over 100 one-page proofs, collected from various sources. The topics range from number theory to calculus, and most of them require no advanced mathematics. Typically there is a statement of a result, with a labelled diagram showing how it is "proved"; in some cases there are a few auxiliary equations along with the picture. These are not simple, often requiring quite a bit of thought before the "Aha!" moment. Working through them is a valuable exercise for the student of mathematics--having seen, e.g., six different visual proofs of the Pythagorean theorem, one comes to really *understand* the result, not just "follow the logic". I have not encountered any better way than this book to "see" how mathematical truth is discovered and proved. It can be valuable as a supplement to courses through precalculus and elementary calculus. Perhaps one of its best uses is to inspire teachers to present results in a more lively way then "definition-theorem-proof" or "just memorize it".

Rating:
Summary: Engaging exercises to train your intuition
Review: Famous mathematicians have often emphasized the role of visual intuition; e.g., Hilbert: "Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?" (from his famous address at the 1900 International Congress). This book is a collection of well over 100 one-page proofs, collected from various sources. The topics range from number theory to calculus, and most of them require no advanced mathematics. Typically there is a statement of a result, with a labelled diagram showing how it is "proved"; in some cases there are a few auxiliary equations along with the picture. These are not simple, often requiring quite a bit of thought before the "Aha!" moment. Working through them is a valuable exercise for the student of mathematics--having seen, e.g., six different visual proofs of the Pythagorean theorem, one comes to really *understand* the result, not just "follow the logic". I have not encountered any better way than this book to "see" how mathematical truth is discovered and proved. It can be valuable as a supplement to courses through precalculus and elementary calculus. Perhaps one of its best uses is to inspire teachers to present results in a more lively way then "definition-theorem-proof" or "just memorize it".

Rating:
Summary: Visual justification has a role in mathematics
Review: The first mathematical proofs were no doubt primarily diagrammatic in structure, and we all should appreciate the role they have played in the development of mathematics. Unfortunately, the figure is now somewhat maligned as a tool in mathematics. A symbol used in a proof is a representative of an abstract concept, and if a diagram is also considered in that way, then it should be just as acceptable. The proofs in this book are not truly without words, as most of the time there is a formula as well. However, they are easy to understand and cannot fail to be appreciated.
Proof by diagram does have a place in the mathematical educational experience as well. After all, the point of a proof is to convince us of the validity and also explain why the result must hold. Students who struggle their way through abstract formulas and symbols can be exposed to proofs like this and learn there is a place for visual thinking in mathematics.
Mathematics teachers face a difficult task and should use every tool that is available to present the wonder and greatness of mathematics as a form of human endeavor. Proofs without words will not work everywhere, but when they do, it can be the difference that makes the light bulb of understanding burn bright. This book should be read by all teachers of mathematics.