The Power of Mathematical Visualization

Course Overview

Course No. 1443

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

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.

Video Lectures

01: The Power of a Mathematical Picture (34 min)

Professor Tanton shares how childhood ceiling tile patterns inspired his math career, demonstrating visual thinking’s power through mental calculation patterns.

02: Visualizing Negative Numbers (29 min)

Master negative numbers and parenthetical expressions using a simple visual model that clarifies long strings of positives and negatives.

03: Visualizing Ratio Word Problems (29 min)

Conquer word problems involving ratios and proportions using visual tools like blocks, paper strips, and poker chips.

04: Visualizing Extraordinary Ways to Multiply (30 min)

Explore a revolutionary graphical multiplication method and solve the mystery of why negative × negative = positive.

05: Visualizing Area Formulas (30 min)

Never memorize area formulas again with visual proofs for rectangles, triangles, circles, and polygon dissections.

06: The Power of Place Value (33 min)

Create a “dots-and-boxes” machine that performs arithmetic in any base system (decimal, binary, even fractional).

07: Pushing Long Division to New Heights (29 min)

Make long-division intuitive using the dots-and-boxes machine, then apply it to polynomial algebra.

08: Pushing Long Division to Infinity (30 min)

Solve polynomial division with negative terms, exploring infinite series and Mersenne primes visually.

09: Visualizing Decimals (32 min)

Connect decimals to fractions using dots-and-boxes, converting repeating decimals to fractions effortlessly.

10: Pushing the Picture of Fractions (30 min)

Prove √2 is irrational by showing it can’t be expressed as a fraction, using visual reasoning.

11: Visualizing Mathematical Infinities (30 min)

Compare infinite set sizes à la Georg Cantor, discovering some infinities are infinitely larger than others.

12: Fractions Take Up No Space (29 min)

Map all fractions onto the number line to reveal they occupy zero space—just the start of infinite weirdness.

13: Visualizing Probability (31 min)

Solve classic probability riddles (coins, dice, medical tests) and Pascal’s 17th-century problem using visual models.

14: Visualizing Combinatorics (34 min)

Master counting combinations by rearranging letters, using factorial operations to simplify combinatorics.

15: Visualizing Pascal’s Triangle (32 min)

Uncover Pascal’s triangle patterns linking to algebra, grid paths, powers of eleven, and the binomial theorem.

16: Random Movement, Orderly Effect (31 min)

See how Pascal’s triangle models random walks, diffusion, and the “gambler’s ruin” theorem.

17: Orderly Movement, Random Effect (31 min)

From Langton’s ant to paper-folding fractals, explore simple rules that create chaotic-seeming results.

18: Visualizing Fibonacci Numbers (34 min)

Trace Fibonacci’s rabbit-breeding origins to discover sequence properties through a single unifying image.

19: The Visuals of Graphs (30 min)

Use scatter plots and function graphs to prove the fixed-point theorem and solve Fibonacci puzzles.

20: Symmetry in Quadratics Graphing (31 min)

Graph quadratic functions effortlessly using symmetry, bypassing the quadratic formula.

21: Symmetry in Quadratics Algebra (28 min)

Solve quadratics by “completing the square” visually, then derive the quadratic formula from this approach.

22: Balance Points in Statistics (30 min)

Apply Archimedes’ lever law to calculate averages and use least squares for best-fit lines on scatter plots.

23: Visualizing Fixed Points (33 min)

Prove that crumpling a paper always leaves at least one point directly above its original position.

24: Bringing Visual Math Together (32 min)

Synthesize course concepts through paper-folding patterns and a jelly-bean sharing challenge.