3^5 * 3^2 = 3^7for example. And I think the students really got it.
3^5 / 3^2 = 3^3
(3^2)^5 = 3^10
I broke them into groups of threes and fours and gave them three different problems that, in being worked out, hint at the above listed laws. It was great to see them working together, debating their answers, enthusiastically sharing their discoveries with each other, and doing all those synergistic things that you can generate in student-led, group-structured activities. (I'm also starting to see a lot of that in their work with integer operations, and especially on last Friday's lesson on the various effects of negative signs on exponents.) Being early into my second year teaching, I know I'm still pretty much a novice at the whole thing, but it's good to see that I'm on the right track to really mature pedagogy.
I've been reading a lot of Jack lately and I will be for the next couple months. It's really refreshing in a "getting back to the basics" sort of way. Finished a really worthwhile biography on Mr. Lewis last weekend, and I've started reading through Mere Christianity for probably my fourth time. It's interesting, on a personal level, to see how I've matured intellectually over the past six years since my first taste of Lewis. Whereas I once honestly (innocently) considered whether or not C.S. Lewis' work might be inspired in a way not much different than the Bible, I find now I have a healthy critical distance between Lewis' beliefs and my own. Much more, I'm able to map out the weaknesses and strengths of Lewis' arguments in ways that previously escaped the weaker reasoning of my freshman/sophomore years.
I'm finding some new angles and questions for Lewis, too. For example, What is the compatibility of Lewis' logic with the twenty-first century man in light of the linguistic turn in contemporary philosophy to which he was rather self-consciously immune? And, Might it be valuable for Covenant Theologians to import Lewis' description of the "Law of Nature" (as he calls it) and use such language in elaborating the principals of the Covenant of Works? Further, Lewis' dealings with Christian dogma, especially with regards to theories of atonement, share a curious similarity to remarks made my men of FV/NPP sorts. Is this something we can attribute to the common influence of Anglicanism? Or is it a coincidence?
Anyway, I'll be (re-)reading through a lot of the Lewissian corpus over the next few months, and honestly, I'm pretty happy about it. Lewis, Wright, and Leithart being my three favorite authors, being forced to spend time with works I cherish and have long cherished is hardly a school assignment I'll be complaining about.
Unrelated: That last post wasn't intended to be about anyone but me, nor should it be read as about anything but my own shameful spiritual struggles. Just in case there was any question about that. Also, it was written (with great intention) rather vaguely. Be careful about assuming you get the point. There's none but myself who should take offense to what was written there.
Additionally unrelated: I've been meaning to make it a comment somewhere that all reductio ad absurdem arguments are (strictly speaking) inherently "reductionistic." Hence, the "reductio." So let's all be much more careful about what we mean by pointing out an interlocutor's "reductionistic" reasoning as if it were clearly a flaw. To illustrate that, when you strip away (i.e., "reduce") the pomp and trite argumentative garb of another's position to its plain and simple absurdity, you are not failing to be reasonable. Your opponent is simply failing to be honest.