(I'm going to post a followup to the Snow Crash post either tonight or tomorrow, but I just needed to make this post tonight.)
So tonight was the first night of Multivariable Calculus. Now, I don't really expect to learn anything on the first day of pretty much any class, but tonight was (with a few exceptions) dreadfully boring. The reason being that we spent about half the class time doing an introduction to vectors.
Vectors are important. This I know. Being able to break up quantities into their components is vital, especially in physics. And the connection between geometry and algebra that vectors allow for is also very useful. This I know, too, because I've been introduced to vectors about a dozen times in my life. That might be an exaggeration, but I'm not sure.
I was tempted (not really -- this isn't the type of thing I would ever do) to raise my hand during class and ask if there was anyone in the class who hadn't gotten a vector intro before. And if no one raised their hand, could we please maybe move on to new stuff? I could not show up to class until we started covering something I didn't know ten years ago, but apparently this is a very popular class/professor and I'd be dropped and replaced if I did that. (This, too, isn't really something I would do.)
Perhaps colleges could offer modules -- short classes that introduce/refresh a topic that will be used in other classes. That way those classes wouldn't be bogged down going over stuff people have already learned elsewhere. Vectors could be one. Polar geometry could be another.
I understand that the remedial math classes at my college are kind of like this. They're modular, and students just take the sections they need so that they can move on to whatever's next.
That said, I did get something useful out of tonight's class. The professor also introduced 3-space, which I've gotten the basics of before as well, but she framed it in a way that was quite illuminating for me. I have a lot of trouble with visualizing objects in more than two dimensions, and anything to help that process is good. She discussed 3-space in terms of the room we were in. One bottom corner of the room was the origin. The front wall was the yz-plane, the side wall the xz-plane, and the floor the familiar xy-plane.
Literally being inside the three-dimensional coordinate system she was describing was immensely helpful for me. Planes made sense; surfaces made sense. And I think I can start to look at the world through the lens of a coordinate system, something that should help the process of abstraction that is necessary for creating models of reality.
Anywho, that's all for today. I should probably think of something to post about yesterday's physics class as well.