Logarithms!

 I maintain that as an unschooling parent, I don’t teach, I facilitate. I try really hard to live by those words. One of the reason is the fringe benefits I reap by not ‘teaching’.

Let me stop here  for a moment to summarize ahead of time. The point I’d like to make is that you don’t have to know the things to help someone else learn the things. Even better, frequently I find myself learning very cool new things I didn’t know before. Unschooling works, and it benefits everyone! Everything else below is rambling about math. Here we go!

The kids and I have been talking about number bases for a few years now. Starting off in base two arithmetic—binary. It was an easy way to look at concepts without worrying about memorization. The addition and multiplication tables for that base have only four entries a piece. There’s not too much you have to memorize when the only numbers you have to work with are one and zero.

A few years into this odyssey, the kids and I started looking into raising numbers to powers, and what this meant with respect to binary. We quickly arrived at the idea, (a new idea to me also), that shifting numbers in a base was the same as raising the number of that base by powers. So, in other words, working in base 2 where we have the numbers 0 - 1 to work with, we saw that,

1 << 0  = 1 (one shifted to the left by 0 is 1, corresponding to the 0th power of 2)

1 << 1  = 10 (one shifted to the left by 1 is 2, corresponding to the 1st power of 2)

1 << 2  = 100 (one shifted to the left by 2 is 4, corresponding to the 2nd power of 2)

That was cool. It immediately made sense of raising something to the 0th power, you just don’t shift the number that forms the powers of the base, the number one. I’d never understood it as a kid. I’d been told that raising something to the zeroth power will always result in the answer one, and that this was a definition I should simply memorize. 

The kids and  I  also got to make sense of terms I came across in graduate school, namely, generators, operators, and sets. In our example above, using the generator 1 and the operator (shift left), we could create all the members of the set of powers of the label of a base, (where 2 would be the label of base 2, and the powers we can generate are 1, 2, 4, and so on, (see above).

I think I may be using the term generator incorrectly, but that’s part of the process. We can all, (the gang and I), use the terms, roll the sound and the feel of the words around in our mouths, as well as our conception of them around in our minds, and then evolve those meanings over time as we learn more. What we’re learning isn’t perfect, but that's ok, it's fluid. It’ll all evolve to the full story over time.

There are two real points here besides my current fascination with exponentiation, generators, and logarithms. First, I don’t have to be an expert in a subject to learn it with the kids. This is one of the unschooling ways around the old questions, “But how will you teach them everything?’ (The other way around is that they teach themselves, I help them find resources.) The second point is that because I’m not an expert, I get to participate in all this stuff, and it’s fun!

OK, so I said I was going to talk about logarithms. I’ve taken a few different swipes at them. The logarithm is the inverse of raising a number to a power. So, if I raised 10 to the 2nd power to get 100, (in whatever base), and then took the logarithm in that same base, I’d get two. For a bit, this got me to realize that two was the number of zeroes in 100. From that point of view, logarithm is a good way to count the digits in a given number in a given base. That's sort of cool. But for the last week or so, I’ve been thinking of logarithm as returning the power the number was raised to itself, rather than the number of digits. Both interpretations give the same answer, but they present different perspectives. For me, this is awesome. I’m learning what powers, logarithms, and bases really mean, as opposed to the definitions—devoid of thought—I was taught in school. I’m getting to do that because there are unschooling kids around me who asks questions. It’s all pretty nice!