[erikerikson]

This question seems to bring a frame that implies there is an example that would apply to everyone. I'm claiming such an example doesn't exist but I will try (please note these are overly simplified).

Consider addition...

"We're going to learn addition today"

One approach: "it is combining two quantities into one. For example, 1 + 1 = 2, 2 + 2 = 4"

Another: "1 + 1 = 2, 2 + 2 = 4, you see, we are combining the quantities"

For some of us, getting the abstraction first gives us the mental framing that allows the examples to teach us the most and the alternative feels like noise.

For others, they're not sure what we're going on about until we have given them enough examples so that they can understand the abstraction. They want to generalize off of the examples they've seen.

Neither is inherently right but they have trade offs and we have preferences on this dimension that end up being strong and usually outside of our awareness. Frequently it can not matter if we jump back and forth between concreteness and abstraction.

[seeknotfind]

I don't think this is the distinction being made here. Abstractions may be useful framing, but the author contrasts, after hearing what addition is, are you going to start studying multiplication, or are you going to look at examples of addition. We'd never think about teaching a child multiplication before they know their addition tables. However, in higher level math, and I saw this in uni, it's common to study structures without knowing many examples of them. You can do proofs about groups without connecting any of these proofs to actual examples that show their utility.

[ndriscoll]

Ironically properties of groups were one of the first things to come to mind where working on an abstract level makes sense. You can convince yourself groups are an interesting interface/definition (both with a couple concrete examples plus a fuzzy argument that anything you want to call "symmetries" ought to be a group), but then if you want to understand something like commutators and abelianization, I don't really see any insight from working with concrete groups. The point is more abstract: [G,G] is normal (easy to show), so you can mod it out, and doing so gives you "G but everything commutes" (because you quotiented away all the commutators).

Similarly I'm not sure there's a lot of understanding to be gained from writing down cosets (as in the actual list of set members). It's still necessary to know how to calculate, but the understanding of what you're trying to do comes from the fundamental theorem on homomorphisms/first isomorphism theorem. You'll never get it from looking at cosets, and in some way it's actually a bit of a distraction IMO. They're often kind of just there to prove quotients exist.

Tensor products feel similar I think: actually constructing the tensor product is not terribly interesting and mostly just demonstrates that it exists. Focusing on the concrete definition makes it easy to miss the forest for the trees.

Maybe having given a couple examples, we can extend it to the abstract idea that universal properties are usually more interesting than the associated concrete constructions. :-)

[erikerikson]

In fact, I taught my child multiplication and negative numbers without running him through a lot examples. He publicly demonstrated the ability to multiply (by multiplying numbers during conversations) before his school ever taught it via drilling (in their case the Singapore method).

I agree about the shift in university. I really enjoyed what felt like a removal of the noise. Of course, the interconnection of the pieces and learning to instantiate the concepts into concrete are expected work of students. That all said, abstraction (or models) are lossy compressions and noticing their failures is essential to avoid becoming detached and/or delusional.