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Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas

Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas

Taught By Multiple Professors
Course No.  1423
Course No.  1423
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Course Overview

About This Course

24 lectures  |  30 minutes per lecture

Discover mathematics as an artistic and creative realm that contains some of the greatest ideas of human history. This course explores infinity, the fourth dimension, probability, chaos, fractals, and other fantastic themes.

Is it worth Bill Gates's time to pick up a $100 bill if he sees it on the sidewalk? Amidst the frenzied screaming from the audience on television's Let's Make a Deal, is there sound advice to give the contestant trying to decide whether to swi

The world of mathematics contains some of the greatest ideas of humankind—ideas comparable to the works of Shakespeare, Plato, and Michelangelo. These mathematical ideas can add texture, beauty, and wonder to your life. Most importantly, you don't have to be a mathematician to have access to this world.

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Discover mathematics as an artistic and creative realm that contains some of the greatest ideas of human history. This course explores infinity, the fourth dimension, probability, chaos, fractals, and other fantastic themes.

Is it worth Bill Gates's time to pick up a $100 bill if he sees it on the sidewalk? Amidst the frenzied screaming from the audience on television's Let's Make a Deal, is there sound advice to give the contestant trying to decide whether to swi

The world of mathematics contains some of the greatest ideas of humankind—ideas comparable to the works of Shakespeare, Plato, and Michelangelo. These mathematical ideas can add texture, beauty, and wonder to your life. Most importantly, you don't have to be a mathematician to have access to this world.

A Mathematical Journey

The Joy of Thinking is a course about fun, aesthetics, and mystery—about great mathematical ideas that arise from puzzles, observations of everyday life, and habits of curiosity and effective thinking. It is as much about learning to think abstractly as it is about what we traditionally think of as mathematics.

You explore the fourth dimension, coincidences, fractals, the allure of number, and geometry, and bring these weighty notions back down to earth to see how they apply to your own life.

Rather than focusing on adding figures or creating equations (in fact, there are fewer numbers than you might expect), this course enables you to uncover and grasp insightful strategies for approaching, enjoying, and understanding the world around you.

"Wonderful ... the Best"

Taught by Professors Edward B. Burger of Williams College and Michael Starbird of the University of Texas at Austin, this course is based on their innovative textbook, The Heart of Mathematics: An invitation to effective thinking, which a reviewer for The American Mathematical Monthly called "wonderful ... possibly the best 'mathematics for the non-mathematician' book that I have seen."

Paradoxical Phenomena

Consider these examples:

  • The game show Let's Make a Deal® entertained viewers with Monty Hall urging contestants to pick a door. The choice involves a question of chance that has been the source of many heated arguments. You explore the mathematics that prepares you for future game-show stardom and explains a paradoxical example of probability.
  • Coincidences are striking because any particular one is extremely improbable. However, what is even more improbable is that no coincidence will occur. You see that finding two people having the same birthday in a room of 45 is extremely likely, by chance alone, even though the probability that any particular two people will have the same birthday is extremely low.
  • One of the most famous illustrations of randomness is the scenario of monkeys randomly typing Hamlet. Another, called "Buffon's needle," shows how random behavior can be used to estimate numbers such as pi. Physicists discovered that a similar needle-dropping model accurately predicts certain atomic phenomena.

The Fourth Dimension

Mathematical thinking leads not only to insights about our everyday lives and everyday world but also points us to worlds far beyond our own. Take the fourth dimension. The very phrase conjures up notions of science fiction or the supernatural.

Because the fourth dimension lies beyond our daily experience, visualizing, exploring, and understanding it requires us to develop an intuition about a world that we cannot see. Nevertheless, that understanding is within our reach.

You learn how to construct a four-dimensional cube and why a four-dimensional surgeon could remove your appendix without making an incision in your skin.


Or take a world that we can see: the two-dimensional realm. It can be just as rich with surprises. You learn how the simple exercise of repeatedly folding a sheet of paper introduces the concept of fractals—a geometric pattern that is infinitely complex—repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry.

You discover that the paper-folding sequence offers an example of the classical computational theory of "automata," developed by Alan Turing—the father of modern computing. Fractal construction processes may also relate to the behavior of the stock market and even to your heart rate.

Life Lessons

As Professors Burger and Starbird lead you through these and other examples, you pick up some valuable life lessons:

  • Just do it. If you're faced with a problem and you don't know how to solve it, begin by taking some action.
  • Make mistakes and fail but never give up. Mathematicians are supremely gifted at making mistakes. The key is to use the insight from your mistakes to identify the features of a correct solution to your problem.
  • Keep an open mind. If we are never willing to consider new ideas, then we can never hope to increase our understanding of the world around us.
  • Explore the consequences of new ideas. This strategy pushes us to see where an idea leads and in this way to discover new ideas and insights.
  • Seek the essential. One of the biggest obstacles in solving real-world problems is the noise and clutter of irrelevant issues that surround them.
  • Understand the issue. Identifying and clarifying the problem to be solved in a situation is often a significant step in reaching a solution.
  • Understand simple things deeply. We can never understand unknown situations without an intense focus on those aspects of the unknown that are familiar. The familiar, in other words, serves as the best guide to the unfamiliar.
  • Break a difficult problem into easier ones. This strategy is fundamental to mathematics and, indeed, applicable in everyday life.
  • Examine issues from several points of view. We can, for example, gain new insights by looking at the construction of an object, rather than the object itself.
  • Look for patterns. Similarities among situations and objects that are different on the surface should be viewed as flashing lights urging us to look for explanations. Patterns help us to structure our understanding of the world, and similarities are what we use to bring order and meaning to chaos.

The Un-Math Math

This is probably not like the mathematics you had at school. Some people might not even want to call it math, but you experience a way of thinking that opens doors, opens minds, and leaves you smiling while pondering some of the greatest concepts ever conceived.

One of the great features about mathematics is that it has an endless frontier. The farther you travel, the more you see over the emerging horizon. The more you discover, the more you understand what you've already seen, and the more you see ahead. Deep ideas truly are within the reach of us all. How many more ideas are there for you to explore and enjoy? Well, how long is your life?

tch his choice to Door Number 2? How can we see the fourth dimension in a Salvador Dali painting?

These certainly aren't the kinds of questions you would normally ask in typical lectures about mathematics. But then again, this isn't an ordinary math course.

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24 Lectures
  • 1
    Great Ideas that Bring Our World into Focus
    A way to refine our worldview is to become more precise in describing what we see. Each of the classical theories of numbers, geometry, topology, fractals, and probability offer tools. x
  • 2
    How Many? Counting Surprises
    Numbers accompany us throughout our lives and play a fundamental role in the realm of mathematics. By counting and quantifying, we understand our world with more refinement. x
  • 3
    Fermat’s Last Theorem and the Allure of Number
    To a mathematician, numbers have their own personalities. This lecture explores the ways they have been used and understood—and have captivated humankind—through the ages. x
  • 4
    Pining for Nature’s Numbers
    We see how a hidden order of numbers actually underlies much of nature's beauty, and explore the remarkable insights available in the pattern known as Fibonacci numbers. x
  • 5
    Sizing up the Fibonacci Numbers
    A potent method for discovering new insights is to isolate and examine patterns—a process that leads us to the most pleasing proportion in art and architecture: the Golden Mean. x
  • 6
    The Sexiest Rectangle
    We investigate our newly honed sense of mathematical aesthetics to see how it illuminates the structure behind aesthetically pleasing art and architecture to arrive at a new appreciation for what is known as the Golden Rectangle. x
  • 7
    The Hidden Beauty of the Golden Rectangle
    Why, exactly, is the Golden Rectangle so visually appealing? A surprising property may hold the answer. x
  • 8
    The Pythagorean Theorem and Geometry of Ellipses
    The Pythagorean Theorem perhaps best represents all of mathematics, and we examine some of its most elegant proofs, along with the alluring relationship between the conic section and the ellipse. x
  • 9
    Not-so-Platonic Relationships in the Platonic Solids
    Symmetry and regularity lie at the heart of classical beauty. The five regular, or Platonic, solids embody not only elegant symmetry but also an elegant duality in their nature. x
  • 10
    Hunting for a Sixth Platonic Solid
    For millennia, the five Platonic solids inspired thinkers with a mystical allure. Kepler mistakenly thought they explained the orbits of the then-known planets. But planets aren't involved, as we see when we discover why there are only five Platonic solids. x
  • 11
    Is There a Fourth Dimension? Can We See It?
    Though the fourth dimension lies beyond our daily experience, understanding is within our reach, and we can visualize and explore it by using our knowledge of familiar realms and arguing by analogy. x
  • 12
    The Invisible Art of the Fourth Dimension
    We consider the geometry of the fourth dimension, beginning with artistic works inspired by dimension, then building and visualizing our own four-dimensional cube. x
  • 13
    A Twisted Idea—The Möbius Band
    Must every surface have two sides? Surprisingly, the answer is no. We explore a remarkable surface known as a Möbius band. x
  • 14
    A One-Sided, Sealed Surface—The Klein Bottle
    Though a single-sided surface with no edge at all cannot be constructed entirely in three-dimensional space, it can be effectively described and modeled, as illustrated by the elegant surface of the Klein bottle. x
  • 15
    Ordinary Origami—Creating Beautiful Patterns
    Even the act of folding a piece of paper can be the gateway to a rich trove of nuance, introducing us to the idea of fractals and showing how patterns and structure can emerge from seemingly unpredictable "randomness." x
  • 16
    Unfolding Paper to Reveal a Fiery Fractal
    Our simple paper-folding sequence leads us not only to the secrets of the dragon curve fractal, but to an example of the classic computational theory of automata developed by Alan Turing, the father of modern computing. x
  • 17
    Fractals—Infinitely Complex Creations
    What does it mean to speak of an infinitely detailed image? We look at what is possible by repeating a simple process infinitely and then reasoning about the result, producing images that illustrate the ideas of self-similarity and symmetry. x
  • 18
    Fractal Frauds of Nature
    We examine how chance, with some simple rules, leads us to an infinitely intricate world of fractals, which quite possibly overlaps with our own physical world. x
  • 19
    Chance Surprises—Measuring Uncertainty
    The uncertain and unknown are not forbidding territories into which we dare not tread. Instead, they can be organized and understood as we construct a means to measure the possibilities for an undetermined future. x
  • 20
    Door Number Two or Door Number Three?
    The game show Let's Make a Deal® involved a question of chance that surprises people to this day, and leads us to an exploration of probability and the ways we measure it. x
  • 21
    Great Expectations—Weighing the Uncertain Future
    This lecture shows us how to put a number to the possibilities of the unknowable future as it examines the quantitative measure known as expected value and how it can be used. x
  • 22
    Random Thoughts—Randomness in Our World
    Coincidences and random behavior do occur, often with predictable frequency. This lecture takes a look at randomness and how the principles of probability help us to understand it better. x
  • 23
    How Surprising are Surprising Coincidences?
    Coincidences are so striking because any particular one is extremely improbable. But what is even more improbable is that no coincidences will occur. We examine why. x
  • 24
    Life Lessons Learned from Mathematical Thinking
    This final lecture looks at 10 "lessons for life" that can be drawn from a range of mathematical themes and concepts. x

Lecture Titles

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2 Professors
Michael Starbird Edward B. Burger
Ph.D. Michael Starbird
The University of Texas at Austin

Dr. Michael Starbird is Professor of Mathematics and University Distinguished Teaching Professor at The University of Texas at Austin, where he has been teaching since 1974. He received his B.A. from Pomona College in 1970 and his Ph.D. in Mathematics from the University of Wisconsin-Madison in 1974. Professor Starbird's textbook, The Heart of Mathematics: An Invitation to Effective Thinking, coauthored with Edward B. Burger, won a 2001 Robert W. Hamilton Book Award. Professors Starbird and Burger also collaborated on Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, published in 2005. Professor Starbird has won many teaching awards, including the Mathematical Association of America's 2007 Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics, which is the association's most prestigious teaching award. It is awarded nationally to 3 people from its membership of 27,000. Professor Starbird is interested in bringing authentic understanding of significant ideas in mathematics to people who are not necessarily mathematically oriented. He has developed and taught an acclaimed class that presents higher-level mathematics to liberal arts students.

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Ph.D. Edward B. Burger
Southwestern University

Dr. Edward B. Burger is President of Southwestern University in Georgetown, Texas. Previously, he was Francis Christopher Oakley Third Century Professor of Mathematics at Williams College. He graduated summa cum laude from Connecticut College, where he earned a B.A. with distinction in Mathematics. He earned his Ph.D. in Mathematics from The University of Texas at Austin. Professor Burger is the recipient of many teaching awards and accolades. He was named by Baylor University as the 2010 recipient of the Robert Foster Cherry Award for Great Teaching for his proven record as an extraordinary teacher and distinguished scholar. Baylor University lauded Dr. Burger as truly one of our nation's most outstanding, passionate, and creative mathematics professors. His other teaching awards include the Nelson Bushnell Prize for Scholarship and Teaching from Williams College, the Distinguished Achievement Award for Educational Video Technology from the Association of Educational Publishers, and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America. In 2006 Reader's Digest honored him in its annual 100 Best of America special issue as Best Math Teacher. Professor Burger is the author of more than 40 scholarly papers and books, including Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, which has been translated into seven languages. He was honored with the Robert W. Hamilton Book Award for his coauthored work with Michael Starbird, The Heart of Mathematics: An invitation to effective thinking. He served for three years as mathematics advisor for the educational series NUMB3RS, produced by CBS, Paramount Studios, and Texas Instruments.

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Rated 4.4 out of 5 by 35 reviewers.
Rated 5 out of 5 by Superb course for the neophyte mathematician This course has touched my life as the two professors hoped the course would do to its participants. It has opened the doors of mathematics for me in a surprisingly accessible way.. Both profeeors are superb instructors. I appreciated the pace of delivery, the clarity and the careful presentation of material, The review lecture at the end -- the recapitulation of the course and a reminder of its application to life-- is absolutely top notch. Both professors are also very charming and delightful persons Thank you both. August 26, 2012
Rated 4 out of 5 by A Fun and Interesting Exploration This 24 lecture course presented by Professors Starbird and Burger explores many of the somewhat eccentric and quirky mathematical ideas of history. It held my interest throughout as it explored aspects of number theory, geometry, and probability. The mathematics is not complex. However, the ideas and the basic principles behind them open up some new ways of thinking. The professors focus attention on the principles of mathematical thinking as a guiding theme throughout the course. Professors Starbird and Burger are both excellent teachers and clearly enjoy presenting this material. If you are an aficionado of Teaching Company mathematics courses you will find some of the material covered in this course a bit repetitious. Professor Benjamin’s "Joy of Mathematics", Professor Burger’s "Zero to Infinity: A History of Numbers", and Professor Starbird’s "What are the Chances? Probability Made Clear" are courses where similar material has appeared. Forays into the 4th dimension and probability issues that defy our intuition and expectations were favorite sections of the course for me. I recommend this course for those with an interest in things mathematical and for those with little previous mathematics experience. It has something to offer to everyone. June 17, 2014
Rated 5 out of 5 by Informative and Entertaining Mathematics can be entertaining? Professors Burger and Starbird introduce complex concepts by looking at simple things deeply. By looking at simple things deeply, cultivating an open mind, approaching an unknown from multiple perspectives, some amazing and unexpected insights can be gained. The course is divided into three sections, numbers and counting, shapes and forms, and uncertainty and the unknown. They explore the conceptual elegance, the aesthetics of mathematics and show and draw lessons applicable to any endeavor. This course is perfect for the non-math person to approach, understand and appreciate mathematics as more than a tool relegated to calculators when absolutely necessary. It requires no advanced math knowledge, just curiosity and a willingness to explore. I recommend their 2005 book, "Coincidences, Chaos, and All That Math Jazz" if you want to explore the themes of this course while viewing or to reenforce your learning. The answer to my question at the beginning of this review, is an unqualified 'Yes,' mathematics can be entertaining and learning at the same time. June 26, 2013
Rated 5 out of 5 by An aptly named course I enjoyed this course in no small part due to the presentation styles of Professors Starbird and Burger. It would be really, really hard to find two professors who express more “joy” and enthusiasm for their course material. The professors appear together in the first and last lectures and split the remainder between them. If I had to pick a favorite, it would be Burger, especially in his coverage of the Fibonacci numbers. However, they are both excellent lecturers offering detailed explanations which employ excellent graphics and models. The professors make a strong effort to drive a point home. If you have a good math background you may find some lectures to be a bit repetitious. However, if your math is more limited, you’ll likely find that sometimes repetitious teaching technique to be a positive. Their technique is as it should be in order to have broad appeal. This course is dependent on mathematics up through basic algebra. Math is utilized to provide insight into the world we live in…in sometimes surprising ways. Examples here would include lectures on the Fibonacci numbers, fractals and probability. In a few of the courses, mathematical and geometric concepts are explored for the sheer “Joy of Thinking” as per the course title. Lectures on the Mobius Band and the Klein bottle would fit into this category. The final lecture is philosophical in providing an approach to life based on mathematical thinking. I have a background in mathematics. It was the title of the course that intrigued me….I was interested in discovering what justified it. I didn’t expect to learn any math. However, I truly enjoyed the way these professors explored the world through math and showed math as a thing of beauty in its own right. The purchase was worth it for those reasons alone. Yes, I could quibble about the content of a few of the lectures…the two platonic solids lectures seemed to degrade into an endless discussion of vertices, edges, faces, and lines until my eyes glazed over. The lecture on the Klein bottle pushed the limits of my geometric interest. However, I felt most lectures were outstanding in both subject and content. There’s something in here for everyone. At least some interest and ability in basic math is a prerequisite to taking this course. Without it, there would be less “joy of thinking”. However, if you have any mathematical interest at all, this course will likely go a long way towards expanding upon it. I highly recommend it to those having that interest. It definitely does make you think and the course is aptly named. April 5, 2013
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