Rediscovering Mathematics is an eclectic collection of mathematical topics and puzzles aimed at talented youngsters and inquisitive adults who want to expand their view of mathematics. By focusing on problem solving, and discouraging rote memorization, the book shows how to learn and teach mathematics through investigation, experimentation, and discovery. Rediscovering Mathematics is also an excellent text for training math teachers at all levels. Topics range in difficulty and cover a wide range of historical periods, with some examples demonstrating how to uncover mathematics in everyday life. Rediscovering Mathematics provides a fresh view of mathematics for those who already like the subject, and offers a second chance for those who think they don’t.

Front Matter Front Matter (pp. ivi) 
Acknowledgments Acknowledgments (pp. viiviii) 
Table of Contents Table of Contents (pp. ixxii) 
A Guide for the Reader A Guide for the Reader (pp. xiiixvi)SS 
Introduction: How to Read Mathematics Introduction: How to Read Mathematics (pp. xviixxxii)Areading protocolis a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than nonfiction. It would be ridiculous to read fiction and ask oneself what is the author’s source for the assertion that the hero is blond and tanned; it would be wrong to read nonfiction and not ask such a question. This reading protocol extends to aviewingorlisteningprotocol in art and music. Indeed, much of the introductory course material in literature,...

1 Mathematical Discovery in the Classroom 1 Mathematical Discovery in the Classroom (pp. 112)You know what arepeatingdecimal is—it’s a number such as 0.3333333333. . . that repeats forever. This particular repeating decimal equals 1/3. In contrast, aterminatingdecimal is a number that ends, such as 6.5 or 8.123. Although a calculator displays any number as a terminating decimal, a repeating decimal actually continues forever, eventually repeating the same sequence of digits. That sequence, however, does not have to be one digit long. The digits can repeat in cycles of length two, three, or more. For example, the number 12.314141414. . . is a repeating decimal that repeats in cycles...

2 Don’t Reach for Your Calculator (Yet) 2 Don’t Reach for Your Calculator (Yet) (pp. 1330)When I calculate the test average for a class of 20 students, I could add up the scores and divide by 20, but I never have a calculator when I need one, and if I did, I would likely mess up the data entry. Instead of a calculator, I use the following pencil and paper method, which is easier on the fingers.
First, I glance at the numbers and make a guess, usually a whole number. For example, consider the following class scores:
92, 86, 75, 69, 88, 98, 78, 86, 82, 84, 73, 71, 80, 70, 91, 95, 72,...

3 Have Another Piece of Pie, Zeno? 3 Have Another Piece of Pie, Zeno? (pp. 3140)If I crumple a piece of paper and throw it at the wastebasket, my students yell “miss” and they are almost always right. They are not prophets— they have, after all, seen me throw before, but missing the wastebasket gives me a chance to talk about Zeno and his paradoxes of motion.
Zeno (200 B.C.E.) explains that to reach the wastebasket, the ball must first travel half the distance from my hand to the basket, and then from there, half the remaining distance, and so on. Hence the ball must pass through an infinite number of locations and travel an...

4 Thinking Like a Mathematician—Lessons from a Medieval Rabbi 4 Thinking Like a Mathematician—Lessons from a Medieval Rabbi (pp. 4162)Each of the numbers 1² = 1, 2² = 4, 3² = 9, and 4² = 16 can be drawn as a square of dots as shown in Figure 4.1.
Notice that each succeeding square number can be drawn by adding a layer of dots around the right side and bottom of the previous square number. Each such layer is shown in bold in Figure 4.2. Do you recognize these odd numbers?
Figure 4.2 suggests that
1² = 1
2² = 1 + 3
3² = 1 + 3 + 5
4² = 1 + 3 + 5 + 7....

5 What is Mathematics Good For? 5 What is Mathematics Good For? (pp. 6376)People sometimes ask me if I can give them a few good illustrations of how math is used in thereal world. These requests are often posed as a challenge, as if knowing a good application might pique their interest in math, or somehow justify its study. However, even some of the greatest mathematicians had no interest in applications.
In 1940, toward the end of his career, the great British mathematician G. H. Hardy wrote an essay entitled “A Mathematician’s Apology.” The essay provides a look inside the mind of a working mathematician. It is less an apology than a...

6 Three Averages 6 Three Averages (pp. 7792)If you ever drive over the speed limit on the highway without being stopped by a police officer, you might be very surprised to still receive a speeding ticket in the mail. The ticket states that you entered the highway at 9:00 AM at mile marker 10, and exited at 11:00 AM at mile marker 160, for an average speed of 75 miles per hour. Ouch! The rest of the mail is no better, as your 401K statement indicates that your investments have dropped an average of 10% over the last three years.
Whether it’s miles per hour, home prices,...

7 Algorithms—The Unexpected Role of Pure Mathematics 7 Algorithms—The Unexpected Role of Pure Mathematics (pp. 93112)How does Google return the most appropriate matches when you search for something on the Internet? How are the directions of the shortest route from your home to your destination determined? How is your credit card number encrypted so that it is safe from cyberthieves? Each one of these tasks is performed using analgorithm.
An algorithm is a procedure, a method, or a list of instructions for solving a problem. Much of mathematics education in the early years focuses on memorizing and applying algorithms for various problems from long division to adding fractions. By the time you enter high...

8 Pythagoras’ Theorem and Math by Pictures 8 Pythagoras’ Theorem and Math by Pictures (pp. 113128)Pythagoras’ theorem is probably the most famous mathematical theorem in the world. Every child can recite “a² +b² =c²,” but alone, and without context, these symbols do not capture the essence of the theorem. What area,b, andc? As a young child, the phrase “asquared plusbsquared equalscsquared” utterly confused me. I mistakenly thought that this theorem had something specifically to do with the first three letters of the English alphabet. How was I supposed to know?
Of course the three letters mean nothing out of context. The theorem applies toright...

9 Memorizing Versus Understanding 9 Memorizing Versus Understanding (pp. 129142)Memorizing mathematics without comprehension is often harmful. If you memorize a poem that you don’t understand, there is still the chance that the flow of the words may have an effect on you. When you memorize dates of historical events, at least you know the chronological order of those events, even if you may not know their significance. When you memorize mathematics without understanding, you delude yourself into thinking that you know something, when in fact you do not. This delusion compounds the lack of understanding. Your ability to apply the knowledge, generalize it, or even question its truth is...

10 Games and Gambling 10 Games and Gambling (pp. 143158)It’s a hot summer day, and you are off to the local carnival with your niece and nephew to have some fun. You find the usual collection of carnies hawking their games, and your nephew is begging you to play. One welldressed fast talking gentleman waves to you and asks if you have ever played Monopoly? It’s too late for you to sneak away since your niece has already run towards him bragging about how you are the greatest Monopoly player in the world.
You quickly learn that the game has almost nothing to do with Monopoly, except for rolling...

11 Soccer Balls and Counting Tricks 11 Soccer Balls and Counting Tricks (pp. 159168)Sometimes, the easiest way to count something is to count it incorrectly. Yes, count it incorrectly. It can be right to do something wrong, as long as the wrong way is easy andyou know exactly how wrong you are.
For example, in my classes, I often choose people to help me with something or other—carrying books, making copies, whatever. When my middle school students hear me say, “I need two people to. . . ” I usually see every hand in the class go up. It doesn’t matter how I finish the sentence—a break from mathematics seems...

12 Pizza Pi and Area 12 Pizza Pi and Area (pp. 169184)Most people know how to calculate the area of a square, a rectangle, and even a triangle. Indeed, the formula that computes the area of a rectangle is so easy to discover, that nobody really knows who first discovered it. It is quite intuitive to draw a picture of a 5by7 grid, and understand that it takes 5 × 7 squares to fill it. That the area of a triangle is one half its base times height is less obvious, but the formula can be rediscovered by drawing appropriate rectangles. The details are left as a challenge at the end...

13 Back to the Classroom 13 Back to the Classroom (pp. 185194)Putting the themes of this book into practice requires courage and skill. Just as an English teacher leads a discussion of a novel, a math teacher should flexibly guide a class’s discovery, sometimes straying from a lesson plan in order to follow the unexpected suggestions of the class. The spontaneous interaction between students and teacher, as they together discover mathematical truths, is worth the detour.
Throughout this book, the challenges and solutions are outlines of much deeper investigations. The right way to approach each challenge in the classroom is to allow open discovery filled with dead ends and backtracking until,...

Resources for Rediscovering Mathematics Resources for Rediscovering Mathematics (pp. 195198) 
Further Reading Further Reading (pp. 199200) 
Index Index (pp. 201206) 
About the Author About the Author (pp. 207207)