The Power of Mathematical Visualization

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.


01 The Power of a Mathematical Picture
02 Visualizing Negative Numbers
03 Visualizing Ratio Word Problems
04 Visualizing Extraordinary Ways to Multiply
05 Visualizing Area Formulas
06 The Power of Place Value
07 Pushing Long Division to New Heights
08 Pushing Long Division to Infinity
09 Visualizing Decimals
10 Pushing the Picture of Fractions
11 Visualizing Mathematical Infinities
12 Surprise! The Fractions Take Up No Space
13 Visualizing Probability
14 Visualizing Combinatorics: Art of Counting
15 Visualizing Pascal's Triangle
16 Visualizing Random Movement, Orderly Effect
17 Visualizing Orderly Movement, Random Effect
18 Visualizing the Fibonacci Numbers
19 The Visuals of Graphs
20 Symmetry: Revitalizing Quadratics Graphing
21 Symmetry: Revitalizing Quadratics Algebra
22 Visualizing Balance Points in Statistics
23 Visualizing Fixed Points
24 Bringing Visual Mathematics Together

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