(1) Stark Raving Calm
(2) I know, put my earmuffs on the cookie.
(3) My dog is smarter than me.
(4) He's my brother, but he's still heavy.
My first week of classes is over. (I only have a seminar on Fridays, but it's just an organization meeting tomorrow, and I don't think I have anything to contribute. I may go anyway, as a prelude to T.G.I.F. at the B.R.B.. It was really good. I'm sitting in on a lot of classes, and so far all of them are very interesting. One of them is something is Teaching Secondary Mathematics: Theories and Practices, and I've been very pleasantly surprised. I figured, quite correctly, that a lot of it would be discussion, but I'm happy to report that we've been discussing some issues that I haven't really thought about before, and that definitely deserve some thought.
Here are some that come to mind. Consider the "fact" that:
Obviously something is wrong here, because otherwise math is broken. But what exactly is wrong, and how on earth would you explain it to a high school student?
Something we talked about today is the difference between 2/0 and 0/0. They're both mathematical bad eggs, but one of the instructors of the class, Dave Bock, suggested that the two are, in fact, rather different. He called the first undefined, and the second indeterminate. He cited some pretty decent reasons for this distinction, and he did so without resorting to limits or advanced algebra. He did it based on the idea that a fraction a/b is the number which solves the equation b x = a. Using this logic, 2/0 can't be a number (because 0 times anything is 0), but 0/0 could be any number (again because 0 times anything is 0). We also talked about 0 raised to the 0 power, which is sometimes considered to be 1, and is sometimes considered undefined.
What I'm realizing now is that although we talked a little about the pure mathematical, foundational type reasons for defining things one way or another, the discussion leaders always steered things back to questions about how teachers can and do present these things to students. How can teachers help students make sense of these seemingly (and sometimes actually) arbitrary rules? How can we help them learn this stuff in a way that they'll remember it because they absorbed it into their structure of understanding, and not just their structure of random memory?
Here's one more issue that I found extremely interesting. It started because we were talking about the FOIL method of multiplying two binomials, and how much it sucks in regards to student understanding and extension to more difficult situations. We talked about learning to use the distributive property, and some nice graphical representations of that.
This led to Dave Bock talking about how he's always found it absolutely amazing how algebra, specifically the distributive property, inexplicably captures the very geometric idea of multiplying complex numbers. I was very curious. I said something like, "What is this geometric idea of multiplication? I mean, I know the geometric interpretation of multiplying complex numbers, but to me, this was always something that comes after the algebraic definition."
Dave Bock replied that he thought this was very sad. I decided not to take offense right away, and hear him out. (Yes, I am actually getting somewhat smarter as I get older.)
He explained that when he looks at a complex number a+bi, he sees a single number. The plus sign is kind of artificial, or just algebraically convenient. It is the same sort of thing as the hidden plus sign in the single number "1 and 2/3" = 1 2/3 = 1 + 2/3. When we think of complex numbers sitting in the complex plane, there's no reason for the plus sign to be there. In a way, the plus sign is there exactly because the multiplication of complex numbers, defined in the standard geometric way, obeys these nice algebraic rules when we put the plus sign in there.
(Thinking about it now, I realize that a standard way of defining the complex numbers in certain circles is as the algebraic closure of the real numbers, obtained by adjoining the number i, and that this field extension is isomorphic to the set of all polynomials in i with real coefficients modulo the relation i^2 = -1. This is very much not the way I think of the complex numbers, of course. I think of them of sneaky little bastards that are often uncooperative, and sometimes far too rigid. I love them anyway.)
At the time, I was only preliminarily convinced, but Dave Bock wasn't done messing with my mind. He pointed out the globally recognized difficulty of explaining to a high school student, or elementary student, or college student, or your average adult, why the product of two negative numbers is positive. He posited that this explanation is much like the geometric explanation of multiplication of complex numbers.
I immediately agreed, and was completely convinced. I also wasn't exactly sure why I was so convinced.
I think that the explanation of why (-2)(-2)=4 is that the best way to think of negation is as a reflection. In terms of the real number line, the number -x is the reflection of the number x about 0. When we do two such reflections, we don't change anything. Since negation is multiplication by -1, and not doing anything is multiplication by 1, this means (-1)(-1)=1. Since multiplication ought to be commutative, this means (-2)(-2)=4.
I believe this kind of thinking is exactly how one makes sense, geometrically, of multiplying complex numbers.
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