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Exploring the Essential Features of “James Tanton – The Power of Mathematical Visualization”
The Power of Mathematical Visualization
Discover the advantages of seeing math from an entirely new angle, guided by a brilliant and engaging teacher.
LESSON
Trailer
01: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
02: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
03: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
04: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
05: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
06: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
07: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
08: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
09: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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
DETAILS
Overview
World-renowned math educator Dr. James Tanton shows you how to think visually in mathematics, solving problems in arithmetic, algebra, geometry, probability, and other fields with the help of imaginative graphics that he designed. Also featured are his fun do-it-yourself projects using poker chips, marbles, strips of paper, and other props, designed to give you many eureka moments of mathematical insight.
About
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.
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.
REVIEWS
JPM54
Fantastic!
This is really an amazing course; awesome and wonderful! Everyone should be exposed to the marvels of mathematical visualization from an early age. I enjoyed the class a great deal. Professor Tanton makes the class very enjoyable and engaging.
lah2022
excellent visuals
This is a great course. As someone who was uncertain as to why my teachers taught me the addition/subtraction/multiplication/division algorithms that they did, I love Dr. Tanton’s visual work and presentations.
I wonder how/whether I can share this course with my 7th grader without undermining her math teacher’s efforts.
Please see the full list of alternative group-buy courses available here: https://lunacourse.com/shop/
