[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.