He passed on 8 years ago this February 28th. I miss him, and I miss co-discovering who we are, as I grow and change. I find that I can still mine the recollections of long ago for new lessons, however.
For example, in First or Second grade we were working on our "minusses" and I could see that 3-2 is 1, and 3-1 is 2, so there ought to be an answer to 2-3. I could feel it. Something that "lived," for numbers were alive to me then, on the other side of Zero. Dad was working in the garage, on a car or hanging the garage door or something.
"What's 2 minus 3?" I asked.
"It doesn't have an answer," he said.
"It must have an answer!"
"Nope. You can't take a bigger number from a smaller number," he explained.
"Well, I think there needs to be an answer, so I'll make one up," I told him.
I went away, and came up with a cumbersome system not unlike Roman numerals for these numbers on the other side of Zero. 2 minus 3 was 00 (the first zero is of course zero, acting as a gatekeeper and a sentry, to let us know we've crossed over; the second zero indicates the quantity), 2 minus 4 would be 000, and so on. I wanted something better, but this was the best I could think of on my own. I showed Dad. I don't have any recollection of what he thought about it. That suggests he wasn't terrifically impressed.
I do know that a few years later, when I discovered negative numbers, I was pretty upset with him. He knew about them all along, of course. Why didn't he go into the "teachable moment" with me when I was 8? Why force me to plod along with the dullards? I had a glimpse of something more than the spoonfed information, and he didn't take the time to peel back the curtain a little bit. Grrrr.
Tonight son Nicholas was working on Least Common Multiples, and he asked me if a positive and negative number could have a LCM. I asked if "least" meant closest to zero, or if it meant "lowest." He decided it meant closest to zero. He worked out then, that the LCM for 3 and -4 has two answers, 12 and -12. He was quite pleased that even though it was a two-answer problem, he'd been able to extend his learning into some new territory. I told him that I thought this new idea was pretty cool.
This then, is what this nearly 40 year old exchange between Dad and me teaches me now. To take the time to explore seemingly insignificant questions between a father and son. To help my son think his question through until he comes up with a satisfying answer, and to support him as his mind extends the concepts he's learning.
I'm sure, 40 odd years ago, I'd have wanted the answer about negative numbers more than this lesson. But would I have carried it for so long?