Comment by arjie
8 hours ago
Haha, this works if you already know what a vector space is. But I think pedagogy needs to provide motivating examples. I'll quote one section of a text by Poincaré (translated by an LLM since most here do not speak French).
> We are in a geometry class. The teacher dictates: “A circle is the locus of points in the plane that are at the same distance from an interior point called the center.” The good student writes this sentence in his notebook; the bad student draws little stick figures in it; but neither one has understood. So the teacher takes the chalk and draws a circle on the board. “Ah!” think the students, “why didn’t he say right away: a circle is a round shape — we would have understood.”
> No doubt, it is the teacher who is right. The students’ definition would have been worthless, since it could not have served for any demonstration, and above all because it would not have given them the salutary habit of analyzing their conceptions. But they should be shown that they do not understand what they think they understand, and led to recognize the crudeness of their primitive notion, to desire on their own that it be refined and improved.
The learning comes from making the mistake and being corrected, not from being taught the definition, I think.
Anyway, it's from Science and Method, Book 2 https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre...
There's more to the section that talks about the subject. I just find this particular paragraph amusingly germane.
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.)
I have nothing against starting out with motivating examples, obviously they are needed for understanding. But they should motivate the definition of a vector space. Not the definition of vectors as mappings of indices.
Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.
Thinking about this specific case, I think you are right. The manner of describing actually confuses the concept more than if it never tried to introduce the index-mapping.