I came across this collection of linear algebra lectures, recommended some time ago on the mollyrocket forums, and found I liked them:
It’s interesting though — I feel like linear algebra is a perfect example of a problem in the structure of our educations. I had the experience which is apparently quite common of taking linear algebra: doing fine at it, but being completely puzzled as to the applications of half the stuff I learned. Everyone comes out knowing what an eigenvalue is, but no one knows why they care. Then, years later, I learn why I care, and suddenly linear algebra finds a place in my mind — I see how it comes together, and how to use it. But the foundation I had built up in class years ago … is all gone. I have to relearn pretty much everything. And I relearn it all out of order, and not very well — I learn eigenvalues again, but don’t know basic terms like ‘null space’ until a year later, when someone uses the term and confuses me. And every time I come across something I ‘should’ know, that people are using with ease, it’s intimidating.
So, what went wrong? I took the pre-requisite, did well, took the class that used it … this is like the ideal case for education. But it mismatches the way the brain works, I think. Our minds are all about finding connections — this information is connected to that, and that and that. We remember things and understand them by their relation to other things. So, when you put a whole random structure into the mind, and don’t connect it up properly? Obviously it vanishes. Sure, when you are forced to re-learn it there are traces of the original structure there, but largely it’s gone.
Most of education is structured in this “logical” way — you learn a whole bunch of things because ‘they’ll be useful later’. And it’s a lot of work, getting that stuff to stick to your mind. You have to over-ride that lack of connections, find ‘tricks’ to make it stick when by all rights it should be forgotten. And ultimately, it doesn’t work — the knowledge doesn’t really stay long after the final exam.
It would work better, I think, to flip the whole thing on its head, and learn the prerequisites AFTER the things that require them. But, of course, that has its own problems …