Science & Tech

Transforming Problems

Terence Tao

Lesson time 09:20 min

Visualization, thinking about a problem in a different way—sometimes even in a physical way—may help lead you to solutions. This process of transforming your thinking can help you move beyond the limitations of intuition and preconceived ideas.

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Topics include: Transforming Problems Get Physical Disrupt Your Thought Patterns Playing With Transformations


[MUSIC PLAYING] - One of the great insights of mathematics is that there are different frameworks, different ways of thinking about the same mathematical problem, which may look quite different but are mathematically related. One of the great advantages is that we can then transform the problem and think of it in a new way, which has maybe nothing to do with the original context. The human brain has got many different modes of thinking. So we have visual modes, we have symbolic modes, we have modes where we are trying to fight some sort of adversary. And by changing the language of your problem, you are activating different areas of your brain. So, for instance, if you are transforming the problem into a geometric problem, then you are activating the visual centers of your brain. So transformation is a way of swapping your thought patterns. For some problems, actual physical sensation can actually be useful. Many mathematicians, you will find, they wave their hands, or gesture somehow when thinking about a problem. And that act of making the thoughts physical is often quite, quite useful. I know people who, they work on a pen and paper for a while, on a problem. If they get stuck, they take a walk. They force themselves to think about the question while walking without the pen and paper. And that's a lot harder, but it forces you to somehow only focus on the essence of the problem, concepts are simple enough to keep in your head at one time without having to write down any computations. And that can sometimes lead to a better way of thinking about the problem. Talking about the problem to other people can help, even if they're not mathematicians. And the process of verbalizing the problem can often lead them to actually realize what the problem is, when previously they were only just thinking about it internally in their head. I have occasionally used my own physical location as a way to transform. There was one time when I was trying to understand a very complicated geometric transformation in my head involving-- I was rotating a lot of spheres at the same time. And the way I actually ended up visualizing this was actually lying down on the floor, closing my eyes, and rolling around. And I was staying at my aunt's place at the time. And she found me rolling on the floor with my eyes closed. And she asked me what I was doing. And I said, I was thinking about a math problem, and she didn't believe me. You find whatever analogies, physical, or mental, or whatever, that work for you. And sometimes it makes you look silly, but that's an occupational hazard. [MUSIC PLAYING] I sometimes find myself solving little mathematical problems on the fly. To give one example, I was once at an airport. And I had to make a connection. And I was at one end of the airport, and I had to race to the other end to catch my flight. But my shoelace was undone. I didn't know when I should stop to tie the shoelace. ...

About the Instructor

A MacArthur Fellow and Fields Medal winner, Terence Tao, PhD, was studying university-level math by age 9. Now the “Mozart of Math” is breaking down his approach to everyday problem-solving—without complex equations or formulas. Learn how to deconstruct challenges, use storytelling as a tool, and discover solutions, whether you’re trying to level up in a computer game or just catch your plane on time.

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Terence Tao

World-renowned mathematician Dr. Terence Tao teaches you his approach to everyday problem-solving—without complex equations or formulas.

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