[skhunted]

Your comment is interesting to me. A few days ago someone asked why math classes don’t the how and why and rather tend to just present the formulas for the operations and tell people to compute. Here you are focusing on what the operations do rather than on why they work. It’s an interesting contrast and one that teachers of mathematics have to balance. The two questions,

How does it work?

Why does it work?

Can’t always both be answered well in a given course.

[aleph_minus_one]

> A few days ago someone asked why math classes don’t the how and why and rather tend to just present the formulas for the operations and tell people to compute.

The lecturer typically do know quite well the how and why, but teaching these points takes a lot of time (you also have to explain a lot about the practical application until you are able to explain why this mathematical structure is helpful for the problem).

Since lecturers are typically very short of time in lectures, they teach the mathematics in a concise way and trust the students to be adults, capable of going to the library, and reading textbooks about the how and why by themselves if they are interested in this. At least in Germany, there is the clear mentality that if you are not capable of doing this, a university is a wrong place for you; you should better get vocational training instead.

[rhelz]

// focusing on what the operations do rather than why //

YMMV, of course, but in general, I always found it easier to understand the why once I understood the what and the how, rather than the other way around.

[skhunted]

Yes. Usually, though, when someone complains about the teaching of mathematics they say we focus too much on how to do the operations and not enough on why they work the way they do. I agree it is much easier to understand why after knowing how.

[ajkjk]

I think there's a big difference between "presenting a formula" and "presenting rules for doing calculations". People are very good at extrapolating complicated results from a few simple rules: that's why taking derivatives and doing (elementary) integrals is fairly easy and also fairly easy to remember a long time after you've taken a calculus course. On the other hand, a bunch of literal miscellaneous formulas is very hard to hold on to --- for instance that's how introductory physics is taught, a bunch of disjoint relationships that you have to make sense of in your mind to make any use of.

In fact all anyone really wants for vector calculus is a bunch of "tools" they can use that will generally give the right answer if applied mindlessly. I think that's why GA is relatively popular, because it says how to do basic geometric operations (rotations, reflections, etc) without any thought.

[ducttapecrown]

This is because "how does it work" and "why does it work" are Fourier transforms of each other!

[Nevermark]

> How does it work?

> Why does it work?

Where can I use it?

[aleph_minus_one]

> Where can I use it?

Relevant:

https://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-...