[xg15]

> What a lot of math learners fail to understand is that grinding through concrete examples imbues you with intuition that you will not get if you jump directly to studying the most abstract ideas.

I feel that's more a lesson for a lot of math teachers to understand. I remember some frustrating linear algebra, calculus and computational complexity courses where the lector basically threw some formulas onto the blackboard, went line-by-line through a formal proof of their correctness and then called it a day. Giving actual examples of the application of the formula was an afterthought left to the student aides. Giving examples that could explain the derivation of the formula was not even considered as an idea.

It always reminded me of someone teaching an "introduction to vi" course but then just scrolling through vi's source code without any further explanation - and in the end expecting the students to be able to fluently use vi.

[JadeNB]

> > What a lot of math learners fail to understand is that grinding through concrete examples imbues you with intuition that you will not get if you jump directly to studying the most abstract ideas.

> I feel that's more a lesson for a lot of math teachers to understand.

That's certainly true, but the teachers who teach that way were probably once students who tried to learn that way (I was one on both counts, though I got better), and it'll be better for them as teachers if they learn the lesson as students.

[dehrmann]

Adding, not disagreeing, but at some point, those abstract concepts like dot products become concrete on their own when you get into things like SIMD programming.

[xg15]

Yeah, but to arrive at that point, you'd have to have understood dot products already.

It's one thing to take two float arrays, multiply them componentwise and sum the results. It's another thing to understand why this operation constitutes the dot product in R^n vector spaces.

[imtringued]

You also have to understand that a^T b is a popular way of writing the dot product. <a,b> is for dumb high schoolers.

Then there is the fun part in German that the dot product and inner product are both called Skalarprodukt. There are many inner products of which the dot product is only one.

[gowld]

Does it matter whether the professor or the teaching assistant is the one giving the examples?

The professor is in an awkward position, because the professor at the front of of the large-group lecture hall doesn't have anything to do to add value. Watch videos, read book, work exercise, and then go to recitation or office hourse for interactive tutoring.

[tom_]

Yeah. It's kind of ironic given how unhappy the author sounds about people being unable to figure out what's going on after seeing the same thing over and over again.