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The Power of Mathematical Visualization

The Power of Mathematical Visualization

Professor James S. Tanton, Ph.D., Princeton University
The Mathematical Association of America

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The Power of Mathematical Visualization

Course No. 1443
Professor James S. Tanton, Ph.D., Princeton University
The Mathematical Association of America
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4.8 out of 5
25 Reviews
96% of reviewers would recommend this series
Course No. 1443
Video Streaming Included Free

What Will You Learn?

  • A new, visual way to think about math.
  • How similar patterns hold the key to astounding feats of mental calculation.
  • A completely new way to multiply that is graphical.
  • The world premiere of Professor Tanton’s amazing Fibonacci theorem.

Course Overview

Many people believe they simply aren’t good at math—that their brains aren’t wired to think mathematically. But just as there are multiple paths to mastering the arts and humanities, there are also alternate approaches to understanding mathematics. One of the most effective methods by far is visualization. If a picture speaks a thousand words, then in mathematics a picture can spawn a thousand ideas.

The Power of Mathematical Visualization teaches you these vital problem-solving skills in a math course unlike any you’ve ever taken. Taught by award-winning Professor James S. Tanton of the Mathematical Association of America (MAA), these 24 half-hour lectures cover topics in arithmetic, algebra, geometry, number theory, probability, statistics, topology, and other fields—all united by fascinating connections that you literally see in graphics and projects designed by Professor Tanton. In demand worldwide for his teacher and student workshops, Dr. Tanton is MAA’s Mathematician-at-Large—a globe-trotting advocate for teaching math “with beauty and joy and wonder and humanness,” as he was recently quoted in The New Yorker magazine.

As an example of Dr. Tanton’s approach, see the many applications of a simple game called dots-and-boxes, which is the gateway to a universe of mathematical concepts and operations– some of which might seem quite unrelated:

  • Long division: Elementary school students typically learn a traditional method of long division that works but can seem abstract. By contrast, the dots-and-boxes approach is more intuitive and actually explains why the traditional method works.
  • Binary arithmetic: The binary base system uses only 1’s and 0’s, which is how computers calculate with on/off switches. The game of dots-and-boxes makes arithmetic in binary and any other base system child’s play—even for fractional bases.
  • Polynomials: The study of polynomials in algebra is, astoundingly, mostly a repeat of grade-school arithmetic, done in base x rather than base 10. Dots-and-boxes comes to the rescue for intimidating-looking polynomial problems, and even for dividing polynomials.
  • Decimals: With dots-and-boxes, you can demonstrate that every fraction has an infinitely long decimal expansion with a repeating pattern. For example, 1/3 = 0.33333…; 1/4 = 0.25000… (the repeating pattern is zero); and 13/99 = 0.131313….

And that’s just the beginning. Once your mind is attuned to think about mathematical relationships in terms of visual models such as dots-and-boxes, the insights start to pile up. That’s when you are truly doing mathematics—not just mechanically following an algorithm or formula you memorized in school. Visual thinking lets you see the logical steps that lead to an answer and grasp the solution that must be true.

Throughout the course, Dr. Tanton often adjourns to his tabletop lab to illustrate mathematical ideas with activities that you can try at home, involving poker chips, marbles, strips of paper, and other props. Some seem positively magical, like the miraculous division of a pile of jelly beans in the last lecture, where your method is inspired by a simple folding pattern.

Do Math the Way the Pros Do

Visual thinking is not a trick or a crutch designed for beginners; it is a key technique often employed by professional mathematicians to achieve brilliant insights, forge new paths of discovery, and find deeper connections in the world of mathematics. For this reason, The Power of Mathematical Visualization is suitable for a wide audience, including:

  • math students at every level, who want to survey the subject from the refreshing heights of the visual perspective;
  • puzzle and math aficionados, who love the creative side of mathematics and the opportunity for endless exploration;
  • math teachers, who want an idea-filled 12-hour demonstration of joyous and effective teaching; and
  • parents, who can best help their children with math homework by fostering a playful, enquiring attitude—just like Dr. Tanton’s.

You start the course with pictures that go with grade-school arithmetic. As you study them, you see how they are springboards to more advanced ideas. For instance, visual thinking about multiplication can make perfect sense of why negative times negative is positive. Then you venture deeper, seeing how pictures that help you keep track of combinations of objects lead to Pascal’s triangle and from there to the concept of structure in randomness. And simple exercises in folding paper end up with exquisite fractal images and the consideration of a truly astronomical problem.

Next, the famous Fibonacci numbers come into focus thanks to a visual model that is a well-kept secret. You also learn how symmetry can save the day with quadratic equations. You play with balance points in statistics and the idea of a fixed point in a stirred cup of coffee. And there’s more!

Life Lessons from Math

Dr. Tanton is a charming teacher who makes math both easy and enjoyable with his playful approach. As you might expect from the winner of Raytheon’s Math Hero Award (plus other prizes) and the author of acclaimed books on the delights of math, he has plenty of problem-solving tips, among them:

  • Make it happen: “If there’s something in life you want,” he says more than once, “then make it happen!” If an additional five on the left side of an equation makes the calculation easier, just tack it on—along with five on the right side to keep things balanced.
  • Take an easier way out: Instead of learning formulas and procedures by rote, simply follow your nose and common sense. Once you discover the deeper reason for a rule, such as the quadratic formula in algebra, then you won’t need to memorize anything.
  • Don’t stop: Nothing in mathematics leads to just one place. Think of a mathematical picture as a doorway to many destinations. One of the big lessons from this course is that an image can be interpreted in multiple ways, which is a powerful technique in mathematics.
  • Mull it over: When perplexed, don’t be intimidated. Do a lot of staring and mulling. Play with the problem. Then take a break. Go for a walk. More often than not, your brain will surprise you! This is actually a good approach to many of life’s muddles.

Mulling comes naturally to Professor Tanton. As a research mathematician, he can’t help sharing a recent discovery he made by contemplating a simple diagram related to the Fibonacci series, turning it over and over in his mind. In Lecture 18, he walks you through the result, which is the world premiere of a brand new theorem. He discusses it exactly as he would with his colleagues over dinner—with barely contained excitement that you will find infectious.

Discover the advantages of seeing math from an entirely new angle, guided by a brilliant and engaging teacher in The Power of Mathematical Visualization.

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24 lectures
 |  30 minutes each
  • 1
    The Power of a Mathematical Picture
    Professor Tanton reminisces about his childhood home, where the pattern on the ceiling tiles inspired his career in mathematics. He unlocks the mystery of those tiles, demonstrating the power of visual thinking. Then he shows how similar patterns hold the key to astounding feats of mental calculation. x
  • 2
    Visualizing Negative Numbers
    Negative numbers are often confusing, especially negative parenthetical expressions in algebra problems. Discover a simple visual model that makes it easy to keep track of what's negative and what's not, allowing you to tackle long strings of negatives and positives-with parentheses galore. x
  • 3
    Visualizing Ratio Word Problems
    Word problems. Does that phrase strike fear into your heart? Relax with Professor Tanton's tips on cutting through the confusing details about groups and objects, particularly when ratios and proportions are involved. Your handy visual devices include blocks, paper strips, and poker chips. x
  • 4
    Visualizing Extraordinary Ways to Multiply
    Consider the oddity of the long-multiplication algorithm most of us learned in school. Discover a completely new way to multiply that is graphical-and just as strange! Then analyze how these two systems work. Finally, solve the mystery of why negative times negative is always positive. x
  • 5
    Visualizing Area Formulas
    Never memorize an area formula again after you see these simple visual proofs for computing areas of rectangles, parallelograms, triangles, polygons in general, and circles. Then prove that for two polygons of the same area, you can dissect one into pieces that can be rearranged to form the other. x
  • 6
    The Power of Place Value
    Probe the computational miracle of place value-where a digit's position in a number determines its value. Use this powerful idea to create a dots-and-boxes machine capable of performing any arithmetical operation in any base system-including decimal, binary, ternary, and even fractional bases. x
  • 7
    Pushing Long Division to New Heights
    Put your dots-and-boxes machine to work solving long-division problems, making them easy while shedding light on the rationale behind the confusing long-division algorithm taught in school. Then watch how the machine quickly handles scary-looking division problems in polynomial algebra. x
  • 8
    Pushing Long Division to Infinity
    If there is something in life you want, then just make it happen!" Following this advice, learn to solve polynomial division problems that have negative terms. Use your new strategy to explore infinite series and Mersenne primes. Then compute infinite sums with the visual approach." x
  • 9
    Visualizing Decimals
    Expand into the realm of decimals by probing the connection between decimals and fractions, focusing on decimals that repeat. Can they all be expressed as fractions? If so, is there a straightforward way to convert repeating decimals to fractions using the dots-and-boxes method? Of course there is! x
  • 10
    Pushing the Picture of Fractions
    Delve into irrational numbers-those that can't be expressed as the ratio of two whole numbers (i.e., as fractions) and therefore don't repeat. But how can we be sure they don't repeat? Prove that a famous irrational number, the square root of two, can't possibly be a fraction. x
  • 11
    Visualizing Mathematical Infinities
    Ponder a question posed by mathematician Georg Cantor: what makes two sets the same size? Start by matching the infinite counting numbers with other infinite sets, proving they're the same size. Then discover an infinite set that's infinitely larger than the counting numbers. In fact, find an infinite number of them! x
  • 12
    Surprise! The Fractions Take Up No Space
    Drawing on the bizarre conclusions from the previous lecture, reach even more peculiar results by mapping all of the fractions (i.e., rational numbers) onto the number line, discovering that they take up no space at all! And this is just the start of the weirdness. x
  • 13
    Visualizing Probability
    Probability problems can be confusing as you try to decide what to multiply and what to divide. But visual models come to the rescue, letting you solve a series of riddles involving coins, dice, medical tests, and the granddaddy of probability problems that was posed to French mathematician Blaise Pascal in the 17th century. x
  • 14
    Visualizing Combinatorics: Art of Counting
    Combinatorics deals with counting combinations of things. Discover that many such problems are really one problem: how many ways are there to arrange the letters in a word? Use this strategy and the factorial operation to make combinatorics questions a piece of cake. x
  • 15
    Visualizing Pascal's Triangle
    Keep playing with the approach from the previous lecture, applying it to algebra problems, counting paths in a grid, and Pascal's triangle. Then explore some of the beautiful patterns in Pascal's triangle, including its connection to the powers of eleven and the binomial theorem. x
  • 16
    Visualizing Random Movement, Orderly Effect
    Discover that Pascal's triangle encodes the behavior of random walks, which are randomly taken steps characteristic of the particles in diffusing gases and other random phenomena. Focus on the inevitability of returning to the starting point. Also consider how random walks are linked to the gambler's ruin" theorem." x
  • 17
    Visualizing Orderly Movement, Random Effect
    Start with a simulation called Langton's ant, which follows simple rules that produce seemingly chaotic results. Then watch how repeated folds in a strip of paper lead to the famous dragon fractal. Also ask how many times you must fold a strip of paper for its width to equal the Earth-Moon distance. x
  • 18
    Visualizing the Fibonacci Numbers
    Learn how a rabbit-breeding question in the 13th century led to the celebrated Fibonacci numbers. Investigate the properties of this sequence by focusing on the single picture that explains it all. Then hear the world premiere of Professor Tanton's amazing Fibonacci theorem! x
  • 19
    The Visuals of Graphs
    Inspired by a question about the Fibonacci numbers, probe the power of graphs. First, experiment with scatter plots. Then see how plotting data is like graphing functions in algebra. Use graphs to prove the fixed-point theorem and answer the Fibonacci question that opened the lecture. x
  • 20
    Symmetry: Revitalizing Quadratics Graphing
    Throw away the quadratic formula you learned in algebra class. Instead, use the power of symmetry to graph quadratic functions with surprising ease. Try a succession of increasingly scary-looking quadratic problems. Then see something totally magical not to be found in textbooks. x
  • 21
    Symmetry: Revitalizing Quadratics Algebra
    Learn why quadratic equations have quad" in their name, even though they don't involve anything to the 4th power. Then try increasingly challenging examples, finding the solutions by sketching a square. Finally, derive the quadratic formula, which you've been using all along without realizing it." x
  • 22
    Visualizing Balance Points in Statistics
    Venture into statistics to see how Archimedes' law of the lever lets you calculate data averages on a scatter plot. Also discover how to use the method of least squares to find the line of best fit on a graph. x
  • 23
    Visualizing Fixed Points
    One sheet of paper lying directly atop another has all its points aligned with the bottom sheet. But what if the top sheet is crumpled? Do any of its points still lie directly over the corresponding point on the bottom sheet? See a marvelous visual proof of this fixed-point theorem. x
  • 24
    Bringing Visual Mathematics Together
    By repeatedly folding a sheet of paper using a simple pattern, you bring together many of the ideas from previous lectures. Finish the course with a challenge question that reinterprets the folding exercise as a problem in sharing jelly beans. But don't panic! This is a test that practically takes itself! x

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  • Illustrations and tables
  • Recommended reading
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Your professor

James S. Tanton

About Your Professor

James S. Tanton, Ph.D., Princeton University
The Mathematical Association of America
Dr. James Tanton is the Mathematician in Residence at The Mathematical Association of America (MAA). He earned a Ph.D. in Mathematics from Princeton University. A former high school teacher at St. Mark's School in Southborough and a lifelong educator, he is the recipient of the Beckenbach Book Prize from the MAA, the George Howell Kidder Faculty Prize from St. Mark's School, and a Raytheon Math Hero Award for...
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Reviews

The Power of Mathematical Visualization is rated 4.8 out of 5 by 25.
Rated 5 out of 5 by from So simple yet so clever This is a marvelously entertaining and enlightening course. I shall use it to teach my grandchildren.
Date published: 2017-07-19
Rated 5 out of 5 by from Astonishingly Wonderful. Take this course! There are few courses I recommend for everyone, regardless of your background and interests. This is one of them. If you love math, you will love this course. If you think you hate math, all the more reason to take this course - it will change your mind. By way of background, I took college math through multivariable calculus and linear algebra (aren't I wonderful?). Given the fact that the math treated here is high school level, I assumed I might be bored, but thought I would see what the visual approach had to offer. I was truly astounded at how fascinating I found (most of) the course, despite the fact that almost none of the math was new to me. I lost track of how many times I found myself thinking, this is really, really cool. And for those who are math-phobic, the visualizations give a very different perspective on the mathematics from what you were exposed to in school. They are often fascinating and beautiful in themselves, even apart from the mathematical ideas, and are an ideal way to put your fear of math behind you. Professor Tanton, as others have noted, is outstanding. His enthusiasm and energy keep things going (although, professor, I could have done without the various explosive sound effects - "Ka-boom!" - for changing one ten into ten ones), and the clarity of his explanations is exemplary. Of course, the visuals are wonderfully done. Now, the course isn't perfect. There was an occasional topic which I found just too simple to hold my interest. And while I greatly appreciated the lectures on quadratics, these, in contrast, might be a bit too complex for those not already familiar with the subject. Plus, I do wish TGC would teach its professors how to pronounce "voilà." But I am very sincere in saying this is a magnificent course which everyone should take, regardless of your prior knowledge or level of interest. You can thank me later. Enjoy.
Date published: 2017-07-13
Rated 5 out of 5 by from Fantastic lecturer I could not take a break from this course. I wish I could have had him as a professor. This course brings math to a whole new light. Even if you think you know math, you will absolutely learn something new.
Date published: 2017-07-06
Rated 5 out of 5 by from Best if Viewed right before bed :) I just wanted to give this course 5 stars. While many of the examples were quite fundamental, they show subtle truths that apply to much more advanced topics. I felt I got the most benefit from watching just before bed and ruminating over the ideas presented while drifting off to sleep. As an aside, I had three semesters of calc and a semester of diff eq many years ago. I got through the courses by memorizing formulas and regurgitating them on tests... didn't learn a lick of actual math. When trying to dust off the cobwebs for a new new project, I realized I need to build a whole new foundation.
Date published: 2017-06-24
Rated 2 out of 5 by from Disappointing His visual approach is far from efficient. He is often hard to follow. I believe visual mode to be the most efficient way to learn, but his approach is lacking. I didn't get value for my money.
Date published: 2017-06-18
Rated 5 out of 5 by from Wow, what a course! TGC has produced many great courses, but I think they have outdone themselves with this one. There are so many good things I could say about it that I hardly know where to begin. Dr. Tanton has an enthusiasm and flair for presenting the material that is simply amazing. As some other reviewers have pointed out, most of the math concepts presented (arithmetic, algebraic equations, fractions, statistics, the Fibonacci series, etc) are likely to be familiar to many of us, the way in which they are presented is refreshingly original and intuitive. Having said that, there were several surprising results presented. For instance, I had never realized that non-integer numbers in the binary or other non-decimal systems could be expressed using figures to the right of the "decimal" point, which as Dr. Tanton points out should not really be called the decimal point in these instances. If his methods are not the way that math is currently being taught in schools, they certainly should be. His use of visual aids, such as on-screen graphics and demonstrations on the tabletop, is outstanding. I highly recommend this course to anyone with an interest in general mathematics, and I especially recommend it to those who are involved in the teaching of basic mathematical concepts.
Date published: 2017-06-13
Rated 5 out of 5 by from I found Math Visualization to be Revealing Just as a single example of many found in the course, I had never seen the link between a geometrical square and a quadratic equation. Visualization of the equation in terms of a square can be powerful in doing the algebra. If one has a child bogged down in numbers (positive and negative), algebra, arithmetic, geometry, etc., the examples in this course might be the approach that powers on the light bulb. The professor's example of paper folding I found boring until the last example. That example is a wonderful practical application of the math described by the folding that can be used over and over. I enjoyed the Professor's style of delivery and enthusiasm.
Date published: 2017-05-07
Rated 5 out of 5 by from The Power of Mathematical Visualization I wish I had this when I was teaching math to give to students who need to be challenged. I believe visualization could also help slower students that have trouble with our older methods of rote teaching.
Date published: 2017-03-26
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