I’ve learned that, when presenting content, the more concrete, the better. This is something I thought I knew quite well by the time I started my first (and only, so far) regular job as a secondary math teacher, but I found otherwise! For example, many of my Algebra II Honors students were having trouble understanding negative exponents. To make the relationships involved more concrete, I drew the following table on the greenboard, showing how positive integer exponents relate to repeated multiplication:
x^{1} | x^{2} | x^{3} | x^{4} | |||
x | x· x | x· x· x | x· x· x· x |
Then I observed that each time you add one to the exponent, you multiply by x one more time. I asked the class how this could be extended to non-positive exponents. Not getting an answer from them, I answered myself by pointing out that that rule is equivalent to the rule that each time you subtract one from the exponent, you divide by x one more time. I added a few columns to the left:
x^{–2} | x^{–1} | x^{0} | x^{1} | x^{2} | x^{3} | x^{4} |
1/( x· x) | 1/x | 1 | x | x· x | x· x· x | x· x· x· x |
I commented further that defining powers of x below the 1^{st} that way is the only way to be consistent with positive powers.
That helped some, but not as much as I expected. It was only much later that I realized I could easily have made things really concrete simply by substituting, say, 3 for x ! (It might also have helped if I’d used a variable other than x, since x looks a lot like a multiplication sign.) I’ve made this mistake of not being as concrete as possible many times. I think the main reason is that I tend to assume that expressions like the ones above are already so concrete, there’s no need to get even more so. But that’s not a safe assumption, to put it mildly—not even with an honors class like mine.