The Dragonfly and the Pond

Life is how you prioritize your time. Some people manage to squeeze more into the 24-hours while other less. The other part of life is how deep you want to go into a subject. Some requires you to explore the furthest depth of the skill, such as music, while others require great spread of skills, general history class. Then, there are the ones in between, most notably for high school students, math and science.

The problem with math is that, some people get it, some people don't. For the people who get it, they want to learn more, but where? The answer to that question usually involves going up in the "math ladder" (id Pre-Calc, Calc BC and higher) but is that right? Is it really a good idea to have a wide swath of knowledge rather than deep insight? My answer is no.

Part of my reasoning is that, sometimes, it is just a waste of time. Why learn all the instruments of an orchestra if you can only play one at a time? There is no reason (besides just for kicks). The stuff for the USAMO doesn't even include calculus. If you're going to play tennis, why bother learning how to tackle a person? Sometimes, the time in life just won't let you spread out. In math, everything is dependent on one another. If you know the basics of functions, you can derive limits and derivative. If you know the basics of trig functions, you can find all the identities yourself. Why memorize when you can derive?

The deeper you learn something, the more transferable it will be. On one of my All-State band trips, my guest conductor (Dr. Richard Blatti from Ohio State) told us that music is about transferring ideas and thoughts. If the song repeats its theme, you should play it the same way unless otherwise noted. The same can go for any skill. A good clarinet player can easily switch to a saxophone; for that matter, any instrument involving wind. The deeper you go, the more skill you acquire that can be transferred. That is especially true of problem solving skills where it's not a "one-step solved" problem. In a problem that requires creativity, having depth of knowledge always helps.

The last part of the reasoning is that there are always something deeper, more insightful about a topic. The best analogy I could find is the Bible, where after every time you read it, you always learn more. There is no stopping that cycle. For example, think you know polynomials? Try to tackle this problem (Source: 1983 AIME):

Let equal . Determine the remainder upon dividing by .

If you never looked deeper at polynomials/number theory, it's pretty hard to figure out that problem. Rather than learning more tools, why not apply new uses to those old tools? Then later after learning the more advanced tools of math, applying them to even more complicated problems will come more easily.

The reason for the title is of a Chinese idiom. The story I learned about it was a conversation between a frog and a dragonfly. It goes some what like this:

Dragonfly: I have seen the WHOLE pond and know it like the back of my hand
Frog: Yes, but do you know where the temperature is the coldest?
Dragonfly: No, but I do know the whole pond.
Frog: Yes, but do you know where the fish like to gather?
Dragon: Uhm, no, but I did touch every part of the pond.
Frog: Do you know where the tastiest food lay in the pond?

After a few of those type of questions, the dragonfly left in frustration over its knowledge of the pond.

-runiteking1

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