Primes in Pre-K
This is a story of how my oldest boy discovered prime numbers and I made another step in my conversion to unschooling. I’m writing it down to help me think about what sorts of things kids can learn and what a parent can do to help. If you want to skip the details, here’s the moral I get out of the story: don’t talk down to kids, listen to them and think about what they say, and go where their questions take you. You may be surprised where you end up. I sure have been.
When G was 3, before he could reliably add numbers under ten or count much past 20, we started playing a game with his counting frogs: I give him some frogs and he tries to make a rectangle from them. I was just trying to plant some seeds for him to think about multiplying and dividing later on, but as we played and talked about it the conversation turned pretty interesting.
G noticed that for some numbers you just couldn’t make a good rectangle. For example, if I gave him 7 frogs, all he could make was a 1-by-7 rectangle. Other numbers, like 12 made a bunch of rectangles. We called the “good” numbers “rectangle numbers” and listed them up to about 20. Even though we didn’t really play the make-a-rectangle game much, G talked about this on and off for months. He taught his grandparents about rectangle numbers. He didn’t know how to find his rectangle numbers systematically, but he understood. And he was excited.
After a while, the talk died down and I forgot about it. Then one day, we were reading through a book we got at the library (Math Matters! series, ISBN: 0-7172-9294-0 published by Grolier — can’t find it online, but the series is great). It is very “above level”, and there was a section about prime numbers. I told him that prime numbers we just numbers that were not rectangle numbers, and soon the fireworks started. The next morning he was working at his desk in my office and announced “There are 4 prime numbers less than 10.”
“You’re right” I said
“How many are there less than 20?” G asked.
I told him we should make a list to find out. So we went through the numbers less than 20 and figured out whether each one was prime. We found there were 8.
“How many primes are there less than 100?” G asked.
I was a little taken aback. I didn’t expect this and I certainly wasn’t going to go through each number up to 100, checking if each one was prime. But I couldn’t let this line of questioning drop, so I was going to have to introduce a more efficient technique. It was time for the Sieve of Eratosthenes. Before pulling that out of my hat, I told him that he was asking a very interesting question that people have been thinking about for thousands of years. Even today nobody knows how to answer those questions as well as we think we should. I didn’t say the words “Riemann Hypothesis“, but that was what I was thinking.
He told me it was easy, you just count them. I told him to keep thinking it was easy, and to keep looking for patterns and structure in the prime numbers, because I’d really love to see it.
Then we moved on to the Sieve. Look at the wikipedia page for this — they have a great illustration of the Sieve in action. I started by writing down all of the numbers from 1 to 100 on the legal pad I was using for work.
Me: “Is 2 prime?”
Me: “What about numbers that are made out of twos?” (we say a number is made out of twos when you can get it by adding a bunch of twos)
G: “No that’s not prime.”
Me: “OK, let’s cross all of those numbers off” So I crossed every other number. The first number (after 1) that wasn’t crossed off was 3.
Me: “The first number left is 3. Is 3 prime?”
Me: “How about numbers made out of threes?”
G (laughing): “Not prime!”
So we circled 3 and crossed out every third number. We repeated this for 5 and 7. It was getting old, so I took a shortcut here and told him that if a number was made out of elevens but not twos, threes, fives, or sevens it had to be bigger than 100 because 11 times 11 is 121. So all of the numbers left on the page were prime. He circled them all and we counted 25.
G: “So how many are less than 1000?”
I should have seen this coming. It was time to teach about computer programming. If I were ambitious I would have done something in Squeak to make programming more accessible, but I’m not very familiar with it yet, so I just stuck with C and wrote a quick little prime listing and counting program. The answer was 168. Then he asked, and asked, and asked. At one point he asked for 64 million (there are 3,785,086). It took so long I was wishing I had written a program that was a bit more clever. But it was fun.
Now he will do the Sieve on his own for fun. I’ve seen him do it right up to 50, and make just a few mistakes up to 100. I even make mistakes going up to 100. I never dreamed that a 4-year-old who has never been asked to multiply two numbers would pursue this line of questioning and seem to really understand the answers. Despite the recommendations of the experts, this confirms my belief that limiting Math to place value and whole number arithmetic through the early elementary years would be a great loss. Even if you aren’t a Math geek like I am, you can still follow the sideroads with them. I know G seems most excited when he can tell I’m discovering something too.
Since then he seems to have taken up my challenge to find structure in the primes. He draws number lines on my notebooks all the time “to help me with my Math”. One day he was drawing 10 sets of 10 number lines to get to 100 (I still haven’t taught him to multiply!) and he when he was filling in the 20′s he looked and said: “I think 3 makes primes. 7 kind of makes primes. See, pattern!” For a second I thought he’d been to one of Terry Tao’s lectures… The number 33 sort of burst his bubble. Still, he keeps trying.
Two weeks ago, when we were talking about his Rainbow Arithmetic, I drew the colors of the rainbow around a circle and started writing numbers next to the colors just like he puts colors on his number line. I thought I was going to show him how you can wrap the number line around a circle. The picture looked something like this:
He looked at it and got a big smile on his face. He said “Aha, look at this!” He pointed to blue and said “Prime!” He pointed to red and said “Prime!”. He pointed to each of the other colors and said either “Not prime!” (green and purple) or “Sort of prime” (orange and yellow). Sure enough, every prime number bigger than 3 will be blue or red in this scheme. This is analogous to the statement that all primes bigger than 5 end in 1, 3, 7, or 9. But that only rules out 60% of numbers, where this red/blue rule filters out 67% of all numbers. After reflecting a bit I think it is pretty easy to generalize this, throw in an unjustified statistical model for the occurrence of primes, and make a good guess at the prime number theorem. I had never thought about that before.
My confidence in unschooling is growing quickly. Listen to your kids and take their ideas seriously. You never know where they will lead. You never know what you both will learn. Rigid curricula, checklists and timetables will just hold them back.