Watch The Power of Mathematical Visualization

Name: The Power of Mathematical Visualization
Released: 2016-10-21
Cast: James S. Tanton

2016
1 Season

The Power of Mathematical Visualization is a fascinating and enlightening course presented by James S. Tanton, a renowned mathematician and educator. The course is part of The Great Courses Signature Collection and explores the power of visualization in mathematics, showing how it can be a powerful tool for understanding complex concepts and solving difficult problems. Throughout the course, Tanton leads viewers through a series of engaging and challenging mathematical exercises, using colorful visual aids to bring complex concepts to life. His approach is both practical and intuitive, allowing viewers to explore some of the most fascinating mathematical ideas and concepts in a fun and accessible way. One of the key themes of the course is the idea of visualizing mathematics, and Tanton introduces viewers to a range of powerful tools and techniques for doing so. Whether it's using diagrams, graphs, or other visual aids, Tanton shows how visualization can help to make abstract concepts more concrete, and make complex problems easier to understand. Another important theme running throughout the course is the idea of problem-solving. Tanton introduces viewers to a range of different problem-solving strategies, including visualizing, pattern-finding, and exploring different solutions. Whether it's tackling a simple arithmetic problem or a complex algebraic equation, Tanton shows how having a strong problem-solving mindset can help you to overcome obstacles and find creative solutions to challenging problems. The course covers a wide range of topics, including geometry, algebra, probability theory, and more. Tanton uses a variety of examples and exercises to explore each of these areas, showing how visualization can help to make even the most complex ideas more accessible. One of the most interesting aspects of the course is the way that Tanton encourages viewers to think creatively about mathematics. He shows how even simple math problems can have multiple solutions, and how exploring different approaches and perspectives can lead to new insights and discoveries. Throughout the course, Tanton's enthusiasm for mathematics is infectious. He has a real passion for the subject, and he does an excellent job of conveying that passion to his viewers. Whether you're a seasoned mathematician or just starting out, this course is sure to inspire you to think differently about mathematics and to approach it with fresh curiosity and excitement. Overall, The Power of Mathematical Visualization is a fascinating and engaging course that is sure to be of interest to anyone who is curious about mathematics. With its clear, accessible style and practical approach to problem-solving, it is a great resource for anyone looking to improve their math skills and gain a deeper understanding of the subject.

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Seasons

24. Bringing Visual Mathematics Together

October 21, 2016

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!

23. Visualizing Fixed Points

October 21, 2016

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.

22. Visualizing Balance Points in Statistics

October 21, 2016

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.

21. Symmetry: Revitalizing Quadratics Algebra

January 1, 1970

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.

20. Symmetry: Revitalizing Quadratics Graphing

October 21, 2016

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.

19. The Visuals of Graphs

October 21, 2016

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.

18. Visualizing the Fibonacci Numbers

October 21, 2016

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!

17. Visualizing Orderly Movement, Random Effect

January 1, 1970

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.

16. Visualizing Random Movement, Orderly Effect

October 21, 2016

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.

15. Visualizing Pascal's Triangle

October 21, 2016

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.

14. Visualizing Combinatorics: Art of Counting

October 21, 2016

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.

13. Visualizing Probability

October 21, 2016

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.

12. Surprise! The Fractions Take Up No Space

October 21, 2016

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.

11. Visualizing Mathematical Infinities

October 21, 2016

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!

10. Pushing the Picture of Fractions

October 21, 2016

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.

9. Visualizing Decimals

October 21, 2016

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!

8. Pushing Long Division to Infinity

January 1, 1970

"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.

7. Pushing Long Division to New Heights

October 21, 2016

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.

6. The Power of Place Value

October 21, 2016

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.

5. Visualizing Area Formulas

October 21, 2016

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.

4. Visualizing Extraordinary Ways to Multiply

October 21, 2016

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.

3. Visualizing Ratio Word Problems

October 21, 2016

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.

2. Visualizing Negative Numbers

October 21, 2016

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.

1. The Power of a Mathematical Picture

October 21, 2016

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.

Description

Throughout the course, Tanton leads viewers through a series of engaging and challenging mathematical exercises, using colorful visual aids to bring complex concepts to life. His approach is both practical and intuitive, allowing viewers to explore some of the most fascinating mathematical ideas and concepts in a fun and accessible way.

One of the key themes of the course is the idea of visualizing mathematics, and Tanton introduces viewers to a range of powerful tools and techniques for doing so. Whether it's using diagrams, graphs, or other visual aids, Tanton shows how visualization can help to make abstract concepts more concrete, and make complex problems easier to understand.

Another important theme running throughout the course is the idea of problem-solving. Tanton introduces viewers to a range of different problem-solving strategies, including visualizing, pattern-finding, and exploring different solutions. Whether it's tackling a simple arithmetic problem or a complex algebraic equation, Tanton shows how having a strong problem-solving mindset can help you to overcome obstacles and find creative solutions to challenging problems.

The course covers a wide range of topics, including geometry, algebra, probability theory, and more. Tanton uses a variety of examples and exercises to explore each of these areas, showing how visualization can help to make even the most complex ideas more accessible.

One of the most interesting aspects of the course is the way that Tanton encourages viewers to think creatively about mathematics. He shows how even simple math problems can have multiple solutions, and how exploring different approaches and perspectives can lead to new insights and discoveries.

Throughout the course, Tanton's enthusiasm for mathematics is infectious. He has a real passion for the subject, and he does an excellent job of conveying that passion to his viewers. Whether you're a seasoned mathematician or just starting out, this course is sure to inspire you to think differently about mathematics and to approach it with fresh curiosity and excitement.

Overall, The Power of Mathematical Visualization is a fascinating and engaging course that is sure to be of interest to anyone who is curious about mathematics. With its clear, accessible style and practical approach to problem-solving, it is a great resource for anyone looking to improve their math skills and gain a deeper understanding of the subject.