Math Through the Ages: A Gentle History for Teachers and Others (Dover Books on Mathematics) - Softcover

Berlinghoff, WilliamP

Softcover

ISBN 10: 0486832848 ISBN 13: 9780486832845

Verlag: Dover Publications Inc., 2019

Inhaltsangabe

"This is a beautiful, important book, a pleasure to read, in which the history recounted truly illuminates the mathematical ideas, and the ideas themselves are superbly explained; a wonderful accomplishment." — Barry Mazur, Harvard University

Designed for students just beginning their study of the discipline, this concise introductory history of mathematics is supplemented by brief but in-depth sketches of the more important individual topics. Covering such subjects as algebra symbols, negative numbers, the metric system, quadratic equations, and much more, this widely adopted work invites and encourages further study of mathematics.

"Math Through the Ages is a treasure, one of the best history of math books at its level ever written. Somehow, it manages to stay true to a surprisingly sophisticated story, while respecting the needs of its audience. Its overview of the subject captures most of what one needs to know, and the 30 sketches are small gems of exposition that stimulate further exploration." — Glen Van Brummelen, Quest University 

Über diesen Titel

Über die Autorin bzw. den Autor

William P. Berlinghoff received his Ph.D. in mathematics from Wesleyan University and was a Professor of Mathematics at Southern Connecticut State University and a Visiting Professor at Colby College.
Fernando Q. Gouvea received his Ph.D. from Harvard University and is the Carter Professor of Mathematics at Colby College.

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Math Through the Ages

A Gentle History for Teachers and Others

By William P. Berlinghoff, Fernando Q. Gouvêa

Dover Publications, Inc.

Copyright © 2014 William P. Berlinghoff and Fernando Q. Gouvêa
All rights reserved.

ISBN: 978-0-486-83284-5

Contents

Preface, 
to the Second Edition, v, 
to the First Edition, vi, 
History in the Mathematics Classroom, 1, 
The History of Mathematics in a Large Nutshell, 5, 
Beginnings, 6, 
Greek Mathematics, 15, 
Meanwhile, in India, 25, 
Arabic Mathematics, 29, 
Medieval Europe, 33, 
The 15th and 16th Centuries, 35, 
Algebra Comes of Age, 37, 
Calculus and Applied Mathematics, 43, 
Rigor and Professionalism, 48, 
Abstraction, Computers, and New Applications, 54, 
Mathematics Today, 59, 
Sketches, 
1. Keeping Count: Writing Whole Numbers, 63, 
2. Reading and Writing Arithmetic: The Basic Symbols, 69, 
3. Nothing Becomes a Number: The Story of Zero, 73, 
4. Broken Numbers: Writing Fractions, 77, 
5. Less Than Nothing: Negative Numbers, 83, 
6. By Tens and Tenths: Metric: Measurement, 89, 
7. Measuring the Circle: The Story of π, 93, 
8. The Cossic Art: Writing Algebra with Symbols, 97, 
9. Linear Thinking: Solving First Degree Equations, 103, 
10. A Square and Things: Quadratic Equations, 107, 
11. Intrigue in Renaissance Italy: Solving Cubic Equations, 111, 
12. A Cheerful Fact: The Pythagorean Theorem, 115, 
13. A Marvelous Proof: Fermat 's Last Theorem, 121, 
14. On Beauty Bare: Euclid's Plane Geometry, 127, 
15. In Perfect Shape: The Platonic Solid s, 133, 
16. Shapes by the Numbers: Coordinate Geometry, 137, 
17. Impossible, Imaginary. Useful: Complex Numbers, 143, 
18. Half Is Better: Sine and Cosine, 149, 
19. Strange New Worlds: The Non-Euclidean Geometries, 155, 
20. In the Eye of the Beholder: Projective Geometry, 161, 
21. What's in a Game?: The Start of Probability Theory, 165, 
22. Making Sense of Data: Statistics Becomes a Science, 171, 
23. Machines that Think?: Electronic Computers, 177, 
24. The Arithmetic of Reasoning: Boo lean Algebra, 183, 
25. Beyond Counting: Infinity and the Theory of Sets, 187, 
26. Out of the Shadows: The Tangent Function, 193, 
27. Counting Ratios: Logarithms, 199, 
28. Any Way You Slice It: Conic Sections, 205, 
29. Beyond the Pale: Irrational Numbers, 211, 
30. Barely Touching: From Tangents to Derivatives, 217, 
What to Read Next, 223, 
The Reference Shelf, 223, 
Twelve Historical Books You Ought to Read, 226, 
History Online, 228, 
When They Lived, 231, 
Bibliography, 237, 
Index, 251, 

CHAPTER 1

History in the Mathematics Classroom

Where did mathematics come from? Has arithmetic always worked the way you learned it in school? Could it work any other way? Who thought up all those rules of algebra, and why did they do it? What about the facts and proofs of geometry?

Mathematics is an ongoing human endeavor, like literature, physics, art, economics, or music. It has a past and a future, as well as a present. The mathematics we learn and use today is in many ways very different from the mathematics of 1000, or 500, or even 100 years ago. In the 21st century it will no doubt evolve further. Learning about math is like getting to know another person. The more you know of someone's past, the better able you are to understand and interact with him or her now and in the future.

To learn mathematics well at any level, you need to understand the relevant questions before you can expect the answers to make sense. Understanding a question often depends on knowing the history of an idea. Where did it come from? Why is or was it important? Who wanted the answer and what did t hey want it for? Each stage in the development of mathematics builds on what has come before. Each contributor to that development was (or is) a person with a past and a point of view. How and why they thought about what they did is often a critical ingredient in understanding their contribution.

To teach mathematics well at any level, you need to help your students see the underlying questions and thought patterns that knit the details together. This attention to such questions and patterns is a hallmark of the best curricula for school mathematics. It is the driving force behind the Standards for Mathematical Practice, a major component in all levels of the Common Core State Standards. It is also reflected in the National Research Council's "Framework for K-12 Science Education" section of their 2013 report, The Mathematical Sciences in 2025. Most students, especially in the early grades, are naturally curious about where things come from. With your help, that curiosity can lead them to make sense of the mathematical processes they need to know.

So what's a good way to use history in the math classroom? The first answer that comes to mind is "storytelling" — historical anecdotes, or, more generally, biographical information. Here's a typical scene. When introducing the idea of how to sum an arithmetic progression, the teacher tells a story about Carl Fried rich Gauss.

When he was about 10 years old (some versions of the story say 7), Gauss's teacher gave the class a long assignment, apparently to carve out some peace and quiet for him self. The assignment was to add all the numbers from 1 to 100. The class star ted working away on t heir slates, but young Gauss simply wrote 5050 on his slate and said "There it is." The astonished teacher assumed Gauss h ad simply guessed, and, not knowing the right answer him self, told Gauss to keep quiet until the others were done, and then they would see who was right. To his surprise, the answer that the others got was also 5050, showing that young Gauss was correct. How had he done it?

Telling such a story achieves some useful things. It is, after all, an interesting story in which a student is the hero and outwits his teacher. That in itself will probably interest students, and perhaps they will remember it. Being fixed in their memory, the story can serve as a peg on which a mathematical idea can hang (in this case the method for summing arithmetic progressions). Like most biographical comments, t he story also reminds students t hat there are real people behind t he mathematics that they learn, that someone had to discover the formulas and come up with the ideas. Finally, especially when told as above, the story can lead the class towards discovering the formula for themselves.

But this example also raises some questions. That story appears in many different sources, with all sorts of variations. The sum is sometimes another, more complicated, arithmetic progression. The foolishness of the teacher is sometimes accentuated by including elaborate accounts of his reaction to Gauss's display of attitude. Many, but not all, versions include an account of Gauss's method. Such variations raise doubts about the story. Did it really happen? How do we know? Does it matter?

To some extent, it doesn't matter, but one might feel a little queasy telling students something that might not quite be true. In the case of our example, it's actually not hard to settle at least some of these quest ions. The story was told to his friends by Gauss himself when he was older. There is no particular reason to doubt its truth, though it's possible that it grew in t he telling, as stories that old men tell about themselves...

Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2019
• Sprache: Englisch
• ISBN 10: 0486832848
• ISBN 13: 9780486832845
• Einband: Taschenbuch
• Anzahl der Seiten: 272
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Soc. Agr. La Raia Ss; http://www.la-raia.it; ; Strada Monterotondo, 79; Novi Ligure; 15067; Italien