The mysterious hero bowed. “I ask only this: please give me a single coin today. Tomorrow, give me double that amount. The day after that, pay me double what you paid me yesterday. Continue to do this for a month. This is what I wish to be my reward.”
The king snorted. “Don’t be ridiculous. You won’t be compensated nearly enough.”
Shrugging, the hero replied, “Trust me: it is definitely enough.”
That was how my eighth grade math teacher taught us about exponents. He gave us this scenario and asked the class if the hero was right. Then he devoted two or three classes to allowing us to figure out the answer to the question, “Would the hero make more money by being given a large upfront fee, or by being paid twice as much as the day before every day for a month?” We had to calculate the amount of money the hero would have by the end, which took quite a bit of trial and error--multiplying new numbers for every day of the month, then adding those to the numbers that had already been calculated for each previous day. It turned out that the hero was, in fact, correct, and would be extremely rich by the end of the month.
I now understand exponents very well.
Paul Lockhart, the author of A Mathematician's Lament, probably would have liked my eighth grade math teacher. The idea of presenting students with a problem and asking them to solve it, rather than telling them how in a step-by-step process, is one that the two both seem to view as a ‘real’ strategy for teaching math. I agree with them both. As someone who has always thought of herself as an ‘English person’ rather than a ‘math person,’ this way of teaching math appeals to me because it aligns very well with the skills that I consider my strengths. Analysis, problem-solving, looking at a problem from a number of different angles--this is what I do best. I enjoy debate. I spend my free time reading about solutions to world issues (which are rather similar to math problems in the sense that they both involve trying to answer questions). Maybe I am a ‘math person.’
Of course, I’m only a ‘math person’ if Lockhart (and my eighth grade math teacher) are correct. It’s very convenient for me if that’s the case, but is it a universal truth? The answer lies in the very definition of mathematics (because isn’t that really the issue here? whether the definition of math is about numbers or about exploration, about formulas or about discovery?). In hope that Merriam-Webster or Encyclopedia Britannica could provide some aid, I did some reading about the meaning of the word ‘mathematics.’ I found that math is, above all, a study concerning logic and relationships between numbers. Like mathematics itself, these definitions tell us what mathematics is, but do not inform us as to how it is best learned. So from this research, I’ve deduced that the definition of math is an end product. Each person must decide for him- or herself how to get from the question “What is math?” to the answer provided by a dictionary. The ‘how’ is subjective. Just like math.
This means that Lockhart is right. If everyone explores this question, even if some people decide that math is formulaic, then everyone is fulfilling Lockhart’s idea that math is something that one must define for oneself. What matters is not the conclusion that is reached; it is that people have reached this conclusion themselves rather than accepting and parroting back what they have been taught.
As for me, I’m very excited. I’m still marvelling at the fact that I’m a ‘math person.'