Psst! Rules!

PDS is a progressive school. right? Right. So it doesn't bother with silly things like rules, right?

Well--no. We do bother with rules. As well we should. The first rule of the school is, or used to be, "No one may interfere with the learning of another." A good rule, and a sensible one, which recognizes the important distinction between rights and responsibilities, even if it lacks the pizazz of what used to be the school's second rule, "Windows are not doors."

We have plenty of other rules regarding behavior, too, many of them developed by the children themselves (most of these begin with the word No, as in No hitting/No kicking/No biting/No poking people with sharp sticks/No knocking over other people's block buildings) or by the children in conjunction with a teacher (these are stated much more positively--Walk in the halls/Be nice to people/Share classroom materials/Treat other people's block buildings with respect). My favorite of these from my classroom-teacher days was Eat regularly, which had nothing to do with having frequent snacks but was one second grader's attempt to condense No burping at lunch and No opening your mouth while you're eating to show people what's inside into one relatively positive statement...

But the rules I want to discuss in this post are math rules. Yes! Math rules, as I use the term, are statements about math that are always true. We encourage children to come up with these rules as they work on problems and as they talk about math. Here are 3 examples, named for the children who first suggested them:
These may seem basic to you, and of course they are. None of these will make the pages of American Mathematical Monthly. They've all been discovered before. On the other hand, let's keep in mind that these are first and second grade children who are just beginning to make sense of the number system. The idea that numbers will always behave in a certain way is by no means obvious. Most things about school aren't that predictable. Think about reading: the letter o can stand for several different sounds, some words add an s in the plural while others add es, a word like set or right can have multiple meanings. Math is different. Math is a little more--reliable.

So we ask children to look for these always situations in math. We ask them to think carefully about how numbers work, about whether a particular result will happen no matter what or whether it will happen only sometimes. The benefits, I think, are clear. A child who recognizes that an odd number plus an even number will always be an odd number is thinking hard about numbers and their properties. She's doing some basic number theory; she's acting like a scientist, making and then testing a conjecture. A child who sees that adding or subtracting zero never changes the original number is finding a pattern, generalizing from special cases, and boldly going where (to the best of his knowledge, anyway) no man has gone before.

Yes, we could tell children that odd + even is always odd, and very often we do just that. But it's more powerful, and the effects are longer-lasting, if children come to discover these rules themselves. Now if you'll excuse me, I need to go determine whether every even number greater than 2 can be written as the sum of two primes...