Calculus: Big Ideas, and a visit from Mr. Feynman

We’ve been cruising along through various applications of derivatives. As an interesting optimization application, we talked about the refraction of light by a change of media (like a pencil appearing bent in a glass of water) and derived Snell’s law, using Fermat’s principle, that light takes the path requiring the least time (hence optimization). And a student asked, “How does the light know which path will take the least time?” My response: “That is an awesome question! I promise to do right by it, but I am going to need to do a little homework, and we are out of time for today anyway. But thank you so much for asking that!”

So I had to dust off the modern physics books and really get to thinking how to answer that question. My training is in math, not physics, but I took this one amazing course in college that was an intensive physics class based on the Feynman lectures. I had a vague memory that the path of least time was most probable, but I wasn’t prepared to give a full answer, by any means.

My homework led me to the videos of Feynman speaking in New Zealand on the topic of QED, which are truly marvelous. I recommend every teacher watch them, just to see how he communicates with the audience. One interesting thing I discovered is that Fermat invented the principle of least time to fit the experimental findings of Snell; of course it makes sense that the model was created to fit the data, but we obscure this when we use Fermat’s principle to derive Snell’s law!

In the end, I decided to talk with the class about the history of our understanding of light, from Newton’s particle model to Maxwell’s wave theory, and explain why neither model completely worked. I introduced the twentieth-century quantum model where “waves” of probabilities rule particles. I tried to tell them just enough to let them know there’s a big, amazing world of knowledge out there waiting for them. Then I played a clip of Feynman from the first NZ lecture (“Corpuscles of Light”); on the YouTube version it’s from about 20:00 to 26:00. In this part of the lecture he speaks eloquently about not understanding. He describes the laws of nature as hard to accept and calls them “screwy,” then announces, “If you don’t like it, go somewhere else! Another universe, where the laws are simpler!” The kids were losing it; it was so fun to watch their reactions. Then I told them that links to all the videos were posted on the class website, and invited them to explore more on their own.

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We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have FUN.
-Paul Lockhart

In theory, theory and practice are the same. In practice, they are not.

-Yogi Berra


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