> I have noted that perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus, because it impedes development of a more general and powerful concept of vector. I call it the coordinate virus!
It's hard to tell what the author is proposing, but from the surrounding pages it sure sounds like he's talking about teaching vectors as geometric objects with various manipulation rules and avoiding coordinate descriptions.
This is actually the sort of angle that may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system.
But for physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction. In my teaching experience as a graduate student, students adopt the geometric picture of a vector as soon as they are able to, because it's vastly easier and more pleasant to work with. (That's why the abstraction was originally developed.) The students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.
It's hard to tell what the author is proposing, but from the surrounding pages it sure sounds like he's talking about teaching vectors as geometric objects with various manipulation rules and avoiding coordinate descriptions.
This is actually the sort of angle that may be possible for mathematicians (although I doubt it's sensible) but is not for physicists. In math, you're studying the logical implications of various axioms, and the abstract ideas are the goal. It's perfectly possible for a mathematician to learn and master an abstraction that was historically inspired by a physical system without personally ever actually connecting the abstraction to the physical system.
But for physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction. In my teaching experience as a graduate student, students adopt the geometric picture of a vector as soon as they are able to, because it's vastly easier and more pleasant to work with. (That's why the abstraction was originally developed.) The students who don't fail to do so because they struggle to build the proper abstract machinery in their brains, not because they have been "infected" with the coordinate picture. The pictures aren't exclusionary, they are both necessary.