I woke early on a Wednesday school morning and couldn’t fall back asleep, so I studied a little and took a shower, and have an hour left to kill before I go and so I’m writing, and what I write here is only really a draft. No, do not ask when I woke.
It seems there’s always those smart kids in class, aren’t there, who seem to understand math how you don’t? It must be that these students can memorize. These are people that will contribute nothing mathematically interesting to the field.
Paul Lockart writes in Mathematician’s Lament:Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.
Honestly, I could quote his entire 25-page essay but it can be read here.
I will acknowledge that to the some that maths comes easily to are those birthed with the ability of being good reasoners, those that recognize patterns. But it’s a skill to be honed and developed. Height is an advantage when it comes to basketball, but that too is a skill to be honed and developed. You would not say, perhaps, well, I could never hope to play sudoku. I wasn’t born with those abilities— see I’m not a sudoku person.
At times, mathematics is claimed to “actually be useful”. I suppose so, but we all do carry with us calculators in our pockets nowadays. Even if it was mathematics was not useful, it would have value as all arts do. Literature classes too have value, do they not? But beyond that, I’d argue the true value of mathematics for the layperson that may never intent to pursue anything beyond high school is logical reasoning.
Formal logic is a field both in philosophy and mathematics; and very personally, I interpret mathematics as philosophy beyond abstraction. Mathematicians and latin philosophers will conclude their arguments; their proofs with this simple abbreviation at times, “QED”, meaning “that which was to be demonstrated”.
I do believe in schools, proofs must be taught and so should some further discrete mathematics. Proofs though are introduced in the gradeschool curriculum through their horrid manner; geometry, and usually only to prove congruence or similarity— and then are their reasonings spoken of? Criticized? Asked to be improved upon, of course not. There is a problem and there is the specific points one must make. It’s simply a fill-in-the-blanks game. An argument is an art, and of course mathematical arguments are never criticized to be improved upon— all the art of it has been stripped (Lockhart, 2002).
Mathematical arguments are not like those of debate classes concerning politics, but of abstractions. Like I said, mathematics is about logical thinking. The students most impressive to me have been those that visualize the situation and apply what they have already known rather than learning a formula and going, “This is the situation in which the formula is used. Remember the formula if the question is asked with these keywords.” Really, it’s just looking at it, perhaps there is a pattern, and thinking on it a little— logically reasoning why a thing is the case. Perhaps afterward, comparing your original process with a method that may be faster, you can derive a formula. And that’s your creation!