The Missing Puzzle Piece

Recently, I decided to go back to basics, and learn mathematics from the ground up. You will find that post here. Calculus was first on the list. I’m taking the MIT course in single variable calculus, it’s available for free online.

It’s tempting to try and jump a few rungs. We understandably want to learn the fancy applications, the impressive stuff. Instead, I’ve chosen a more mundane route. Getting a strong handle on the basics. Understanding deeply the foundations is underrated. Josh waitzkin, a chess “prodigy” and martial artist, puts it brilliantly in his book:

“[We] began our study with a barren chessboard. We took on positions of reduced complexity and clear principles. Our first focus was king and pawn against king—just three pieces on the table… Layer by layer we built up my knowledge and my understanding of how to transform axioms into fuel for creative insight… This method of study gave me a feeling for the beautiful subtleties of each chess piece, because in relatively clear-cut positions I could focus on what was essential. I was also gradually internalizing a marvellous methodology of learning—the play between knowledge, intuition, and creativity.”  

Josh Waitzkin

Today I began with calculus, a subject that I’ve often touched on but never deeply understood. What is a derivative? Well (on the geometric interpretation) it’s the general formula for slope of the tangents to a curve. What is a tangent to a curve? Well if you take two points on a curve, and take the slope of that line, then push the points arbitrarily close together (take the limit as the difference between the points approaches 0), that’s a tangent line. I was beginning to get into the flow of the lecture. I already know what a derivative is, but I never had such a clear handle on why the formula is how it is. Then we started to solve the standard formula for taking the derivative of y=x^n. Although common place, I’ve never derived it myself. I’m feeling excited by now.

Suddenly, the flow stops. “…by the binomial theorem we have such and such”, the lecturer continues. I couldn’t remember the binomial formula. Worse, I didn’t understand it in this context. I didn’t know why the terms were exactly that way, and how the lecturer had found them. And a “whole load of junk” was a mystery to me. What was it? and Why was it junk?

I paused the lecture and jumped on to another website. Time to find the root of this problem. I watched some videos on the binomial theorem. I mostly understood, but not quite. There was a funny combinatoric thing, which was familiar previous study, but I couldn’t exactly remember it. I went back further, building up my understanding. First permutations, then combinatorics, then back to binomials. Then to pascals triangle and binomials, and linking all three.

When I switched back to the lecture, I understood.

The once formidable formula looked mundane. I knew what the pieces were, and why. I understood what the “junk” was, and how the junk would disappear in the final solution. When I got to the end, I quickly finished off my notes, replicating the proof from memory. My two hour journey into beginner calculus had begun.

When something doesn’t make sense, more often than not, we are just missing a piece of the puzzle. As someone who never took math seriously in high school, I’m missing a lot of the early puzzle pieces. It’s rewarding to find the missing pieces and bring them together.

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