I started my spring semester's classes in the last few days, including two introductory algebra classes. It's possibly the best and most powerful start to a semester I've ever had; I got an extraordinarily good vibe from all my classes. So that's a good thing.
So here's a quick observation I've said aloud several times. For me, teaching college math is a continual exploration of the things people don't know. If I listen really carefully, I continually discover that the most basic, fundamental ideas imaginable are things that a lot of people in a community college simply never encountered. Whenever I start teaching a class and think "oh, christ, that's so basic it'll bore everyone to tears, skip over that quickly," I discover at some later point that a good portion of people have never heard of it in their life.
That's actually a good thing. It keeps me interested with this ongoing detective work I do to see exactly how far the unknowns go. And it provides the opportunity to share in the never-ending amazement, through the eyes of a student who never saw something really fundamental.
Here's one from this week (which I got to run in each of two introductory algebra classes). We're going to want to simplify expressions, with variables, even if we don't know what the variables are. ("Simplifying Algebra", I write on the board.) To do that, we can use a few tricks based on the overall global structure of numbers and their operations. I'm about to give 3 separate properties of numbers, here's the first. ("Commutative Property", I write on the board.)
Let's think about addition, say we take two numbers, like 4 and 5. 4 plus 5 is what? ("9", everyone says.) Now, if I reverse the order, and do 5 plus 4, I get what? ("9", everyone says.) Same number. Now, do you think that will work with any two numbers? With complete confidence, almost everyone in a room full of 25 people all say at once, "No, absolutely not."
So of course, I'm sort of thunder-struck by this response. Okay, I say, I gave you an example where it does work out, if you say "no" you need to give me an example where it doesn't work out. One student raises his hand and says "if one is positive and one negative". Okay I say, let's check 1 plus -8 ("-7"). Let's check -8 plus 1 ("-7"). So it does work out. Now do you think it works for any two numbers? At this point I get a split-vote, about half "yes" and half "no".
Okay, what else do you think it won't work for? One student raises her hand and says (and I bless her deeply for this), "if it's the same number". Uh, okay, let's check 5 plus 5 ("10"). And if I flip that around I again have 5 plus 5 ("10"). So now do you think it works for any two numbers?
At rather great length I finally get everyone agreeing "yes" to the Commutative Property of Addition. And it's obviously a point that no one in the class had ever realized before, that addition is perfectly symmetric for all types of numbers. You can sort of see a bit of a stunned look on some people's faces that they hadn't realized that before. Isn't that just incredibly amazing?
After that, we get to think about the Commutative Property of Multiplication (pointing out that commutativity does not work for subtraction or division), look at the Associative and Distributive Properties, do some simplifying exercises with association and distribution, and so on and so forth. But the thing I can't get over this week is that something so simple as flipping around an addition and automatically getting the same answer can, all by itself, be an enormous revelation if you listen closely enough.