Comment by omnicognate

It's trivial to provide motivating examples for vector spaces, and there's no reason you can't do so while explaining what they actually are, which is also very simple for anyone who understands the basic concepts of set, function, associativity and commutativity. The notion of a basis falls out very quickly and allows you to talk about lists of numbers as much as you like without ever implying any particular basis is special.

I hesitate to call anything pedagogically "wrong" as people think and learn in different ways, but I think the coyness some teachers display about the vector space concept hampers and delays a lot of students' understanding.

Edit: Actually, I think the "start with 'concrete' lists of numbers and move to 'abstract' vector spaces" approach is misguided as it is based on the idea that the vector space is an abstraction of the lists of numbers, which I think is wrong.

The vector space and the lists of numbers are two equivalent, related abstractions of some underlying thing, eg. movements in Euclidean space, investment portfolios, pixel colours, etc. The difference is that one of the abstractions is more useful for performing numerical calculations and one better expresses the mathematical structure and properties of the entities under consideration. They're not different levels of abstraction but different abstractions with different uses.

I'd be inclined to introduce the one best suited to understanding first, or at least alongside the one used for computations. Otherwise students are just memorising algorithms without understanding, which isn't what maths education should be about, IMO. (The properties of those algorithms can of course be proved without the vector space concept, but such proofs are opaque and magical, often using determinants which are introduced with no better justification than that they allow these things to be proved.)