If you do not know how to explain what you are doing, then you are useless.
I know that sounds like a harsh statement, but bear in mind that when it comes to mathematics, literally the only thing that makes you better than a computer is your ability to understand what all of these numbers and symbols mean, and therefore use some creativity, logic, and proper judgement to get out appropriate conclusions. In every other aspect, computers are better than you: they are faster, they never make mistakes, they never file for worker compensation, they never generate complaints with human resources, they never take sick days, etc., etc. If your only mathematical skill is to be able to take in some inputs and run some algorithms that you know to get out an answer, then there is no point in having you—a computer would do that job literally hundreds of times better than you can ever hope to do it.
If this is your only skill, you aren’t just obsolete; you are a liability. I offer two case studies toward this point: Verizon Doesn’t Know How To Count and Apparently, Calculus Was Invented In 1994. The first story is from 2006, when Verizon representatives misquoted their data plan as being 0.002¢/kilobyte when it should have been $0.002/kilobyte. This then generated a massive amount of negative publicity when their customer service representatives could not figure out what was wrong—listening to the transcript of the call, it is obvious that they were simply plugging numbers into a calculator, without any understanding of what those numbers mean. The second story is thematically similar, but with a very different setting: a biologist published a paper in 1994 explaining her method of computing the area under the curve. The only problem is that her method was just the Trapezoid Rule, known to mathematicians since the 17th century at the very latest.
Considering that biologists generally do have to take a calculus course, it is clear that she learned how to crunch formulas and write down integrals, but what it all meant never actually sunk in. (Even more frustratingly, this same paper has since been cited by other biologists, who refer to it as “Tai’s method.”)
Considering that biologists generally do have to take a calculus course, it is clear that she learned how to crunch formulas and write down integrals, but what it all meant never actually sunk in. (Even more frustratingly, this same paper has since been cited by other biologists, who refer to it as “Tai’s method.”)
As such, it is crucial that teachers be able to assess not just that you are able to push symbols around, but that you actually have an understanding of what it all means. Furthermore, the skill of being able to explain your reasoning is supremely valuable—in the business world, it will help you tremendously if your boss ever asks you to explain why your proposed solution is the right way to go. In academia, if you do not understand how to explain your reasoning, you will not get published. If nobody gets what you are doing, they are more likely to conclude that you are crank than that you are an inscrutable genius. (With good reason. Inscrutable geniuses are only a few a generation, while cranks are as abundant as stars in the sky.) One of my best friends in graduate school wrote a paper, and then his adviser forced him to edit and rewrite that paper for a year before he finally concluded that it was in a fit state to be published. I have known many a budding mathematician who was very clever and very bright, but was absolutely awful at expressing their thoughts—this is something that has to be outgrown.
-Senia Sheydvasser
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