Tuesday, August 26, 2008

The power of distinction

There are lots of jokes that start out, "There are two kinds of people in the world." My favorite goes, "There are two kinds of people in the world: those who think there are two kinds of people in the world, and those who don't. I'm one of those who don't."

Making simple distinctions is a powerful tool for thinking. A favorite is the distinction between concepts that are necessary and those that are arbitrary.

I think this should be taught in kindergarden. In many domains there can be differences of opinion that could cause troubles. But mathematics is a good example, and one where there shouldn't be a lot of room for opinion.

Consider the concept of number. We teach our children to distinguish between groups that contain two items and groups that contain three items, and four, and so on. That distinction, in and of itself, and depending on your point of view might be either necessary or arbitrary. For example, some might think it more important what color the objects are (or what shape, etc.), rather than their number.

However, once we introduce the concept of number, things get interesting. It is necessary and not arbitrary that if you combine two groups that each contain two elements, that the new, larger group contains four elements. On the other hand, it is arbitrary and not necessary that we use the numeral "2" to label groups with two items and the numeral "4" to label the new combined group.

This use of the graphics "2" and "4" is a social convention and nothing more. We might just as well have used "2" and "9". The fact that we agree on the former, and that everyone uses it, is, of course, helpful for communication. But which symbols we use is arbitrary. In the other system, it would still be true that "2" combined with "2" yields a new group of "9".  That is necessarily so.

Sorry that I don't have the reference, but I have seen an educational study in which a boy who could not "get" the symbols we use was helped by a researcher. The researcher got several matchboxes and put different numbers of pebbles in each box. Then the researcher invited the boy to look in each box and draw something on each box to help him know how many pebbles were in each one. This technique helped the boy past his difficulty with arithmetic, and once he realized the numerals were arbitrary, he could accept and learn the conventional ones.

Another distinction that I have been thinking about is that between things that work whether you believe in them or not, and things that only work if you believe.

Every time that I fly in an airplane, I find myself disbelieving that something so large and heavy can actually get off the ground. Despite my disbelief, every one--so far at least--has actually taken off.

As a lad, I watched Peter Pan with my sisters and they were dismayed when I didn't join in the clapping to save poor Tinkerbell's life. The fact is that the film had already been made and there was nothing that could change the outcome.

I don't have anything profound to offer about this last distinction, except to urge my audience to graciously accept the things that happen in spite of their disbelief, and to believe for all they are worth in situations where that can make a difference.


Myrna said...

Well, now I understand why you didn't clap. The funny thing about me was that, as often as I listened to that record and therefore should have known the outcome, everytime I listened I clapped and clapped, with tears streaming down my cheeks, like it was a new experience every time. I don't know how that kind of reaction fits in with all your theorizing.

Abra said...

I think, after reading both Uncle Bruce, and Mom, that there are at least 2 kinds of people in the world... and many more I'm sure.

I look at my kids, Malachi is an immediate disbeliever. You could show him over and over again that, something can likely happen and he will choose to not believe. He is not a big risk taker, and tends to be timid around new experiences. He's not a clapper.

Piper, is a do first, ask questions later kind of a kid. If you told her she could fly with a blanket off the balcony, she'd give it a try. She, claps too mom... just so you know.

Deklan, is a thinker. If I promise him he can dive off the diving board, he doesn't go right on out and test it out. He thinks about it, tries it off the edge of the pool, thinks some more, then finally gathers the courage to try. He probably would clap the first time, realize that either way the faerie lives and laugh at Piper clapping over and over again.

I guess my theory is, you can choose to distinguish between certain things or not, but I think a lot of the outcome depends on the person. No matter how hard you clap, or how much you want to disbelieve it is what it is.

Myrna said...

Abra, you are getting to be so erudite. All the writing you are doing shows! Keep it up! Sorry to talk to my daughter on your blog, my brother...thanks for the platform! You are both good writers and I like to read what you have to say.

Canadian Dave said...

There are three kinds of people in the world. Those who can count and those that can't.