TTCVideo – Paul Zeitz – Art and Craft of Mathematical Proble

TTCVideo – Paul Zeitz – Art and Craft of Mathematical Problem Solving DVD no guide [course 1483] L24 [NEw RIP][HuntR][PDU]

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One of life’s most exhilarating experiences is the “aha!” moment that comes from pondering a mathematical problem and then seeing the way to an elegant solution. Try your hand at these:

# Can you cut a sheet of typing paper with scissors into only one piece so that an elephant can step through it?

# What is the first time after noon at which the hour and minute hands meet?

# Seventeen pennies are on a table. Two players take turns removing 1, 2, 3, or 4 pennies. The last to make a legal move wins. Is there a winning strategy?

# How fast can you find the sum of the numbers 1 + 2 + 3 up to 100?

Each of these problems can be solved relatively quickly with the right strategy. For example, the last question was famously answered in the late 1700s by the 10-year-old Carl Friedrich Gauss, later to become one of history’s greatest mathematicians. Young Gauss noticed that by starting at opposite ends of the string of numbers from 1 to 100, each successive pair adds up to 101:

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

and so on through the 50th pair,

50 + 51 = 101

Gauss was already thinking like a good problem solver: The sum of the numbers from 1 to 100 is 50 � 101, or 5,050�obtained in seconds and without a calculator!

In 24 mind-enriching lectures, The Art and Craft of Mathematical Problem Solving conducts you through scores of problems�at all levels of difficulty�under the inspiring guidance of award-winning Professor Paul Zeitz of the University of San Francisco, a former champion “mathlete” in national and international math competitions and a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.

These are not mathematical exercises, which Professor Zeitz defines as questions that you know how to answer by applying a specific procedure. Instead, problems are questions that you initially have no idea how to answer. A problem by its very nature requires exploration, resourcefulness, and adventure�and a rigorous proof is less important than no-holds-barred investigation.

Think More Lucidly, Logically, Creatively

Not only is solving such problems fun, but the techniques you learn come in handy whenever you are presented with an unfamiliar problem in mathematics, giving you the confidence to try different approaches until you make a breakthrough. Also, by learning a range of different problem-solving approaches in algebra, geometry, combinatorics, number theory, and other fields, you see how all of mathematics is tied together, and how techniques in one area can be used to solve problems in another.

Furthermore, entertaining math problems sharpen the mind, stimulating you to think more lucidly, logically, and creatively and allowing you to tackle intellectual challenges you might never have imagined.

And for those in high school or college, this course serves as an enriching mathematical experience, equal to anything available in the top schools. Professor Zeitz is a masterful coach of math teams at every level of competition, from beginners through international champions, and he knows how to inspire, encourage, and instruct.

Strategies, Tactics, and Tools of Math Masters

The Art and Craft of Mathematical Problem Solving is more than a bag of math tricks. Instead, Professor Zeitz has designed a series of lessons that take you through increasingly more challenging problems, illustrating a variety of strategies, tactics, and tools that you can use to overcome difficult math obstacles. His goal is to give you the persistence and creativity to turn over a problem in your mind for however long it takes to reach a solution.

The first step is to come up with a strategy�an overall plan of attack. Among the many strategies that Professor Zeitz discusses are these:

# Get your hands dirty: Dive in! Plug in numbers and see what happens. This is a superb starting strategy because it almost always shows a way to keep on investigating. You’ll be surprised at how often a pattern emerges that takes you to the next step.

# Think outside the box: Break the bounds of conventional thinking. Professor Zeitz shows you the original think-outside-the-box problem, in which the key idea is to disregard the boundaries of an implied box. He also explains why he prefers to call this strategy “chainsaw the giraffe.”

# Wishful thinking: Turn a hard problem into an easy one by removing the hard part. For example, substitute small numbers for big ones. This is a confidence-builder that often gives you a partial solution that shows you how to solve the original problem.

# Change your point of view: Every problem has a natural point of view, such as a time or place where something is happening. Step back and try a different point of view. This could mean recasting an algebra problem as one in geometry, or vice versa.

The next step after choosing a strategy is to find a suitable tactic. For example, suppose you live in a cabin that is two miles north of a river that runs east and west, and your grandma’s cabin is 12 miles west and 1 mile north of your cabin. Every day you go to visit grandma, but first you stop by the river to get fresh water for her. What is the length of the route that has the minimum distance?

You start with the “draw a picture” strategy. Once you have something to look at, you realize that the “symmetry” tactic will give you the shortest distance. Here’s how it works: Imagine an alternate you on the same errand but on the south side of the river, in a mirror image of the situation on the north side. By drawing a line connecting your real cabin with the alternate grandma’s cabin, and another line connecting the real grandma’s cabin and the one belonging to the alternate you, you find an intersecting point at the river that is the perfect place to stop.

On some problems you also need a special-purpose technique�a tool. For example, the 10-year-old Gauss’s trick of pairing numbers in the earlier example is a tool whose underlying idea�symmetry�can be applied to a wide range of problems. You learn the strengths, as well as possible pitfalls, of such tools.

Prepare for an Exhilarating Experience

Professor Zeitz compares this systematic approach to problem solving�in which you deploy strategies, tactics, and tools�to the mountaineer’s quest to reach the top of a high peak. The mountain may seem insurmountable, but there is always a way to conquer it by proceeding one step at a time.

Looking at an impressive mountain, you can’t but feel a sense of awe at the prospect of climbing it. Math problems, too, can produce this same reaction. But don’t be daunted: You are more ready than you think. So sharpen your pencil, get some paper, and prepare for the exhilarating experience of The Art and Craft of Mathematical Problem Solving.

About Your Professor

Dr. Paul Zeitz is Professor of Mathematics at the University of San Francisco. He majored in history at Harvard and received a Ph.D. in Mathematics from the University of California, Berkeley, in 1992, specializing in ergodic theory.

One of his greatest interests is mathematical problem solving. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO. He has helped train several American IMO teams, most notably the 1994 “Dream Team,” which, for the first time in history achieved a perfect score. He founded the San Francisco Bay Area Math Meet in 1994 and cofounded the Bay Area Mathematical Olympiad in 1999. These and other experiences led him to write The Art and Craft of Problem Solving (1999; second edition, 2007).

He was honored in March 2002 with the Award for Distinguished College or University Teaching of Mathematics by the Northern California Section of the Mathematical Association of America (MAA), and in January 2003, he received the MAA’s national teaching award, the Deborah and Franklin Tepper Haimo Award.

Available Exclusively on DVD

This course features hundreds of visual elements to aid in your understanding, including animations, graphics, and on-screen text.

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Art and Craft of Mathematical Problem Solving

by Paul Zeitz (Biography)

The following materials are provided to enhance your learning experience. Click the links below for free information including a professor-authored course summary, recommended web links, and a condensed bibliography.

# Course Summary – Professor’s written description of the course.

# Professor Recommended Links

# Condensed Bibliography – Prepared by the professor for this course.

Course Summary

This is a course about mathematical problem solving. The phrase “problem solving” has become quite popular lately, so before we proceed, it is important that you understand how I define this term.

I contrast problems with exercises. The latter are mathematical questions that one knows how to answer immediately: for example, “What is 3 + 8?” or “What is 3874?” Both of these are simple arithmetic exercises, although the second one is rather difficult, and the chance of getting the correct answer is nil. Nevertheless, there is no question about how to proceed.

In contrast, a problem is a question that one does not know, at the outset, how to approach. This is what makes mathematical problem solving so important, and not just for mathematicians. Arguably, all pure mathematical research is just problem solving, at a rather high level. But the problem-solving mind-set is important for all who take learning seriously, especially lifelong learners. Much of the current craze in brain strengthening focuses merely on exercises. These are not without merit�indeed, mental exercise is essential for everyone�but they miss out on a crucial dimension of intellectual life. Our brains are not just for doing crosswords or sudoku�they also can and should help us with intensive contemplation, openended experimentation, long wild goose chases, and moments of hard-earned triumph. That is what problem solving is all about.

An analogy that I frequently use compares an exerciser to a gym rat and a problem solver to a mountaineer. The latter’s experience is riskier, messier, dirtier, less constrained, less certain, but much more fun. For those of you who prefer more civilized pursuits, consider 2 ways to learn Italian. One involves toiling over grammar exercises and translations of texts. The other method is to spend a few months, perhaps after a short bit of preparation, in a small town in Italy where no one else speaks English. Again, the latter approach is messier but fundamentally richer.

Becoming a good problem solver requires new skills (mathematical as well as psychological) and patient effort. My pedagogical philosophy is both experiential and analytic. In other words, you cannot learn problem solving without working hard at lots of problems. But I also want you to understand what you are doing at as high a level as possible. We will break down the process of solving a problem into investigation, strategy, tactics, and finer-grained tools, and we will often step back to discuss not just how we solved a problem but why our methods worked.

Problems, by definition, are hard to solve. Solving problems requires investigation, and successful investigations need strategies and tactics. Strategies are broad ideas, often not just mathematical, that facilitate investigation. Some strategies are psychological, others organizational, and others simply commonsense ideas that apply to problems in any field. Tactics are more narrowly focused, mostly mathematical ideas that help solve many problems that have been softened by good strategy. Additionally, there are very specialized techniques, called tricks by some, that I call tools.

This course is devoted to the systematic development of investigation methods, strategies, and tactics. Besides this “problemsolvingology,” I will introduce you to mathematical folklore: classic problems as well as mathematical disciplines that play an important role in the problem-solving world. For example, no course on problem solving is complete without some discussion of graph theory, which is an important branch of math on its own but is also a very accessible laboratory for exploring problem-solving themes. Many of the lectures will include small amounts of new mathematics that we will build up and stitch together as the course progresses. The topics are largely drawn from discrete mathematics (graph theory, integer sequences, number theory, and combinatorics), because this branch of math does not require advanced skills such as calculus. That does not mean it is easy, but we will move slowly and develop new ideas carefully.

A small but important part of the course explores the culture of problem solving. I will draw on my experience as a competitor, coach, and problem writer for various regional, national, and international math contests, to make the little-known world of math Olympiads come to life. And I will discuss the recent educational reform movement (in which I am a key player) to bring Eastern European�inspired mathematical circles to the United States.

Problem solving is not a vertically organized discipline; it is not something that one learns in a linear fashion. Thus the overall organization of this course has a recursive, spiral nature. The first few lectures introduce the main ideas of strategy and tactics, which then are revisited and illuminated by different examples. We will often return to and refine previously introduced ideas. Overall, the topics get more complex toward the end of the course, but the underlying concepts do not really change. An analogy is a theme and variations musical piece, where the main theme is introduced with a slow, stately rhythm and later ends in complex avant-garde interpretations. By the end of the course, you should understand the main theme (the basic and powerful strategies and tactics of problem solving) quite well because you had to struggle with the complex interpretations (the advanced folklore problems that used the basic strategies in novel ways).

Problem solving is not just solving math problems. It is a mental discipline; successful investigations demand concentration and patient contemplation that few of us can do, at least at first. Also, problem solving is an aesthetic discipline�in other words, an art� where we create and contemplate objects of elegance and beauty. I hope that you enjoy learning about this wonderful subject as much as I have!

Professor Recommend Links

Art of Problem Solving

Welcome to The On-Line Encyclopedia of Integer Sequences? (OEIS?)

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Paul Zeitz

University of San Francisco

Ph.D., University of California, Berkeley

Dr. Paul Zeitz is Professor of Mathematics at the University of San Francisco. He majored in history at Harvard and received a Ph.D. in Mathematics from the University of California, Berkeley, in 1992, specializing in ergodic theory.

One of his greatest interests is mathematical problem solving. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO. He has helped train several American IMO teams, most notably the 1994 “Dream Team,” which, for the first time in history achieved a perfect score. He founded the San Francisco Bay Area Math Meet in 1994 and cofounded the Bay Area Mathematical Olympiad in 1999. These and other experiences led him to write The Art and Craft of Problem Solving (1999; second edition, 2007).

He was honored in March 2002 with the Award for Distinguished College or University Teaching of Mathematics by the Northern California Section of the Mathematical Association of America (MAA), and in January 2003, he received the MAA’s national teaching award, the Deborah and Franklin Tepper Haimo Award.



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