[GuiA]

I feel the same way as you.

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.

[ivansavz]

> In the meantime, rote memorizing your multiplication tables and polynomial expansions might be a necessary evil...

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.

[GuiA]

I've been doing math for the past 15+ years as well, and the nature of my education (French system) forced me to learn a number of things by rote. I'm very glad I now have these automatisms built in my brain and basic things come instinctively to me.

> 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?

[ivansavz]

I can't argue with the music analogy—it's a very strong point.

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.

[jacobolus]

Indeed. It’s much better to practice basic arithmetic in the context of solving some harder problem than to just drill on worksheets of completely repetitive trivial problems for years on end.

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.

[agumonkey]

> (a+b)^2 = a^2 + 2ab + b^2

who else think it could be better to explain such identities with geometric combinatorial diagrams ? at least at first step toward formal memorization.