Why Teach Math?

Sometimes I sit in my classes, and wonder what I am getting out of the class that I couldn't get out of a book. One of my professors recently commented that faculty for math graduate students are really just guides. They have sat down, with all of the background knowledge they have about whatever subjects and possibly a few books to use as guidelines, and prepare for us a path through the theory. Perhaps sitting down at the start of the semester they think up a few key ideas or theorems they think we should know about some area, and then sit down to find a nice path between them all. Which I appreciate. Entering a new subject, I have no idea which of the multitude of books and resources are the 'best'. Which ones present the material in a manner that will give me insight and understanding? Surely, most of the definitions and results will be the same in any two books on the same topic. And I find it easy to forget that there's more to life than knowing definitions and theorems. Being able to look them up is easy (though a nice online database has potential to make it easier...), once you know basically what you want. But just staring at a list of new definitions and theorems isn't entirely helpful in itself. So the professors pick them out, and guide me through them. Which is awesome. With the understanding I get, I should be able to go back to those books and, even if I haven't picked out the 'best', be able to get much more out of it.

So that's nice. What about the math I teach? The stage I'm at now, I get to teach calculus. Should we focus on definitions and theorems for students who largely won't go into pure math? Should we just focus on computation and applications? How much should we let students have computers do the algebra and number crunching? What is the point of teaching the topic, if most students will forget it all after the semester and not use it again? Even for those that will re-use the information, I wonder if teaching the first 6 chapters of Stewart's Calculus book in a given semester is the best use of time. Shouldn't the student's be able to look up definitions and theorems and examples on their own, just like I can? If they can, what parts should I actually be teaching? And if they can't, how do I teach students how to learn math - how to process the information in a section of a textbook and be able to work similar problems, and moreover identify and understand the key ideas? In my setting, the courses are coordinated among several sections, with common exams. If I don't want my students to fail exams, I have to teach them all the facts they need, so how much of the time can I use teaching learning, versus teaching facts?

These problems get worse as technology enters the classroom. Students will literally have all of the definitions and results at their fingertips. A library of examples will be keystrokes away. And most of the examples we get them to do, they're exactly the sorts of things we made computers to do. Now I know that we want students to understand what's going on well enough that they can do some sanity check on whatever answers the computers return, and even be able to properly format a problem for a computer to be able to do it (a problem that I expect will decrease as computers become more intelligent).

Just recently I found this post on the BBC by Bill Thompson (whose posts I regularly enjoy) which discusses the changing face of teaching, focusing on the impact of technology. There he states that

... knowing facts provides a framework for understanding, a source of insight into problems and a way of boundary-checking solutions.

This bit about 'providing a framework for understanding' feels, somehow, like a lot of the answer I was looking for. Newton stood on the shoulders of giants, and we want out students to have a similar footing. By showing students the information that is there, and guiding them through it, we help show them our viewpoint, hopefully from somewhere near those great shoulders.

So anyway, what is my point? Where am I going with this? Well, as I want to be a math professor, it seems important to think about just what I teach my students, and with what goals in mind. I remain in a state of uncertainty about the best way to do this. Balancing definitions and results with examples, and mixing technology in, is certainly a great challenge. But it is one I look forward to working on. One of these days I'd like to start having answers to all the questions I post here.