Manifesto on the Teaching of Mathematics (2015)

Why not? Did you never experienced the differences in learning something because you had to vs. learning something because you are interested in it? I found the latter to be far more effective and the first mostly a sad waste of time.

School math was mostly wasted on me. So many hours for nothing (even though I had somewhat good grades). But when I have a specific coding problem now, that I can solve with math - then I see a reason and then I enjoy doing math, as it now has a purpose. If people think, I will never need that crap, than their brain will resist learning it. The result is wasted time and energy for everyone involved.

I'm grateful now, and was not at the time, to my educator mother for drilling me and drilling me again on addition, subtraction, multiplication, and division. She drilled me until we were both so frustrated we couldn't see straight.

Her relentlessness gave me a basic numeracy that set me free to explore conceptual math and actually have fun doing it. She did it for my sibs too. It made us kiddos capable of playing car-trip games like "spot the prime number on the license plate".

She knew there was no magic pedagogy to learning that basic arithmetic, just drilling. Now she was no mathematician herself. Mention the central limit theorem to her and you'd get "huh"? She studied classics in school. But she sure knew how to to teach.

I think math instruction should take a cue from Pythagoras who, according to Iamblichus, taught by first paying his students a small amount for each successful learning accomplishment — until the students were older and wanted to pay him to learn more. It’s legendary, but the truth is that we shouldn’t expect kids to want to learn math.

Learning core skills should not rely on intrinsic motivation. It is so much easier to be intrinsically motivated once you have a base. Otherwise everything is just difficult and frustrating.

Modern formalism is because of calculus!… and the fact that contrary results were obtained regarding continuity and derivatives due to subtly different conceptions of the terms.

Enter the Weierstrass function:

> Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness.

https://en.wikipedia.org/wiki/Weierstrass_function

When it comes to math, it's actually used by two distinct categories: real engineers and mathematicians. There's also physicists but they share the same trait with engineers: math is a tool and not a purpose. A mathematician will become very anal when you skip some tiny detail in a demonstration, like go from Taylor series expansion to Ito lemma by approximating dt^2 =~ 0 (I had this happen to me). An engineer (that is, me ;) couldn't give a funk since Ito lemma is a well known and already proven fact and they only use the quick derivation from Taylor expansion as a way to mentally remember the former when they need it for some actual, real-life use case.