The Power of Mathematical Visualization

Rated 4 out of 5 by from Thought Provoking It's been decades since I studied math, algebra, and geometry and appreciated re-learning something about them in this course. I found the visualization techniques thought provoking in that it had never occurred to me to think of things that way, especially the 10:1 and 2:1 machines. I found the presentation style too dramatic for my taste - too many kapows, and zzsshhoooms which is why I rated it 4 instead of 5. I also found Dr Tanton spoke too quickly at times and found it easier to follow if I downloaded the video file and played it at 90% speed. However the content is enjoyable and enlightening. I'll have to watch it again to get the most out of it.
Date published: 2020-08-05
Rated 1 out of 5 by from Disappointing As several other reviewers have stated this is aimed at a very low level of understanding, junior high to high school at best. The continual exhortations to have an epiphany, even when I have easily worked out answers in my head, are an irritant. The constant claims that certain features of elementary arithmetic are scary or mysterious are equally irritating and, for many of us I am sure, completely false. I gave up after the third lecture--I tried looking ahead at some of the further lectures, but found them the same--no new information, no new insights.
Date published: 2020-06-25
Rated 5 out of 5 by from Mathematical Visualization I am a math teacher and I really enjoyed this program.
Date published: 2020-05-10
Rated 5 out of 5 by from Intriguing. I have a degrees in math and computer science, but was intrigued by the course description. After listening to 8 lectures,I was sufficiently impressed that I bought a copy of if for my 14 year old nephew who has aspirations of becoming an engineer.
Date published: 2020-05-03
Rated 5 out of 5 by from Making Math Interesting I love how the patterns are found and the new ways of solving problems is demonstrated. I especially loved how negative numbers were represented using playing in the sand. The enthusiasm makes me want to learn more and the graphics are very helpful to make it more clear.
Date published: 2020-05-02
Rated 5 out of 5 by from A Great Help To Struggling Students I purchased this course to help me with tutoring middle and high school students. All of my students struggle with fractions and negative numbers. The visualization exercises presented by Professor Tanton will be a great help. This is an excellent alternative to the purely numerical approach to teaching mathematical concepts. I highly recommend this course. I recommended it to my daughter who is home schooling three children.
Date published: 2020-04-23
Rated 5 out of 5 by from Captivating G'day!! Prof Tanton does an amazing job of explaining difficult sounding problems in a very engaging and clear manner. Well done. He wants you do understand how to solve the problem - not to mechanically operate a formula. Take this course - you'll get a lot out of it. Highly recommended.
Date published: 2020-01-21
Rated 5 out of 5 by from Life changing This is the first math course I’ve ever taken that has made sense and been enjoyable. I no longer fear math, but find it fascinating!
Date published: 2019-12-13
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The Power of Mathematical Visualization
Course Trailer
The Power of a Mathematical Picture
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....

34 min
Visualizing Negative Numbers
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....

29 min
Visualizing Ratio Word Problems
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....

29 min
Visualizing Extraordinary Ways to Multiply
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....

30 min
Visualizing Area Formulas
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....

30 min
The Power of Place Value
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....

33 min
Pushing Long Division to New Heights
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....

29 min
Pushing Long Division to Infinity
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....

30 min
Visualizing Decimals
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!...

32 min
Pushing the Picture of Fractions
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....

30 min
Visualizing Mathematical Infinities
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!...

30 min
Surprise! The Fractions Take Up No Space
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....

29 min
Visualizing Probability
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....

31 min
Visualizing Combinatorics: Art of Counting
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....

34 min
Visualizing Pascal's Triangle
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....

32 min
Visualizing Random Movement, Orderly Effect
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....

31 min
Visualizing Orderly Movement, Random Effect
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....

31 min
Visualizing the Fibonacci Numbers
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!...

34 min
The Visuals of Graphs
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....

30 min
Symmetry: Revitalizing Quadratics Graphing
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....

31 min
Symmetry: Revitalizing Quadratics Algebra
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....

28 min
Visualizing Balance Points in Statistics
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....

30 min
Visualizing Fixed Points
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....

33 min
Bringing Visual Mathematics Together
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!...

32 min
James Tanton

Our complex society demands not only mastery of quantitative skills, but also the confidence to ask new questions, to explore, wonder, flail, to rely on ones wits, and to innovate. Let's teach joyous and successful thinking.


Princeton University


The Mathematical Association of America

About James Tanton

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 excellence in math teaching. Professor Tanton is the author of a number of books on mathematics including Solve This: Math Activities for Students and Clubs, The Encyclopedia of Mathematics, and Mathematics Galore! Professor Tanton founded the St. Mark's Institute of Mathematics, an outreach program promoting joyful and effective mathematics education. He also conducts the professional development program for Math for America in Washington, D.C.

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