I wish I had seen some of this beauty from a younger age.
It goes to show how much the right teacher, especially early on, can have a profound impact on your life.
But I don't know that my younger self had the maturity to see the deeper beauty. It was only starting college that I had the necessary complexity of thought (combined with wonderful professors).
It takes a while to develop a taste refined enough to perceive these things. In the meantime, rote memorizing your multiplication tables and polynomial expansions might be a necessary evil to have foundations strong enough to later support that kind of more abstract, artistic reasoning.
This goes for other fields too. You have to practice your chords to play Bach, and learn your conjugations to write a novel.
I disagree on both ideological and technical terms. Won't students lose the "flow" as soon as they start memorizing? It seems like an anti-intellectual activity to memorize data or particular steps (e.g. (a+b)^2 = a^2 + 2ab + b^2). I think the further we stay from memorization the better the learner's experience will be.
Now for the technical objection. You said some degree of memorization might be a "necessary" evil, but I have a counter example: me. I've been doing math for the past 15+ years, but to this day I never learned the multiplication table. People are often surprised when I need 78 and I have to do 74 and add the result twice. "I thought you were a math person, and you don't know the multiplication table?" some will say... and I'm, like, yeah.
> Won't students lose the "flow" as soon as they start memorizing?
The core of my argument is that memorization will help them better find and stay in the flow in the first place, over the long term.
People wouldn't argue that having to learn your scales, chords, etc. gets in the way of creative piano playing, for instance; how and why is math different?
My view is that pursuing fluency in the basic arithmetic skills as an intermediate goal towards higher understanding might not be a necessary step. Assuming the person learns a lot of math (like five years' worth) then they will be forced to develop fluency in the process of using the basic skills as building blocks for the more advanced topics. You're right that having useful "chunks" of memorized procedures would make learning the advanced stuff easier in the first place, but I think students could also develop arithmetic/algebra fluency "just in time" while learning the more advanced levels.
Unfortunately, it takes much more teacher skill (e.g. to assess individual gaps in student understanding), doesn’t scale as well to large class sizes (ideally problems should to be catered to the ability level of each individual student), and isn’t as easy to assess with standardized tests.
who else think it could be better to explain such identities with geometric combinatorial diagrams ? at least at first step toward formal memorization.
This is a method which was known <50 BCE to the Babylonians, and is taught to every high school calculus student as the “trapezoid rule”.
Thx for posting this excellent article.
While true, I think that this is still a bit abstract. Rather than "a toolbelt to understand the universe", I would say "a toolbelt to formally characterize problems in order to analyze and solve them".
Let me add in addition that in order to learn to use the fun tools in the toolbelt, one must first grasp the basic ones. This (in my limited experience teaching lower level math courses) is another obstacle in teaching math: (continuing with the tool analogy) it would be quite boring to spend 3 months studying screwdrivers, another 3 months on wrenches, etc. without applying these tools to any "fun" problems.