Rolfe Schmidt

A few months ago G (my 4-year-old) developed a fascination with rainbows, and it has led to a surprising number of fun math games and activities. I’ll describe how it all evolved here.

I didn’t really know the order of the colors in the rainbow until G set out five of his cups in a line on the floor of my office and announced that we must be missing one. I looked more carefully, and noticed that the cups were in this order: Red, Yellow, Green, Blue, Purple. There was a bit of a gap between the red and yellow.

“We are missing orange” G said.

Of course: red, orange, yellow, green, blue, purple. The order of the colors in the rainbow. Once I figured out what was going on I told him how impressed I was that he figured that out, and that unfortunately we never had an orange cup. After that we both started noticing how many places you see colors in the “rainbow order”, and G is sure to complain whenever colors are in the wrong order. It became a common topic of conversation.

About a month ago, he came out of the blue and asked “Did you know that you can plus colors?”

“No I don’t, how do you do it?”

“Red plus red is orange, and red plus orange is yellow, and orange plus orange is green, see?”

I saw. Red was like 1, orange like 2, yellow like 3, etc. I was curious what he would do when we “went past purple”, so I asked him:

“OK, what is purple plus red?”

He paused, then said “It is red.”

“And blue plus yellow?”

“Is orange.”

Neat, he is adding colors as if they wrap around a circle. This has nothing to do with mixing colors, even though that’s what I thought he was talking about at first. Here is the addition table for his rule:

He is basically doing modular arithmetic — something that is usually kept safely away from students who don’t go looking for it. It turns out that this has been a great source of fun for our math classes. I actually think it might be a better way for me to teach pre-arithmetic than using numbers

• It puts us on a level playing field. If someone asks me “what is yellow plus green?” I really have to think about it. My brain does not associate colors with quanitities, so I have to think pretty hard about it. I’m sure young kids have the same problem with numerals that I have with colors. Using colors, we both have trouble.
• It is simpler. There is no carrying or place value.
• It is fun. G likes to quiz me, and to be quizzed, when we go for walks. He talks about it all the time. And it lets him draw some nice looking pictures.
• It gives us the opportunity to talk about some concepts that don’t show up so readily with regular arithmetic.

After a couple of weeks of playing with this idea, I started asking him a few more questions:

• Can you make all of the colors from red (adding red again and again)?
• What colors can you make from orange?
• What colors can you make from yellow?
• Is there a number that acts like zero?

So basically I was trying to plant some group theory concepts in his head, and he seemed receptive. I was pleasantly surprised. Then last weekend he drew this picture on his easel, associating numbers with colors:

And this led to an interesting set of questions:

• If you are adding blue and yellow, does it matter which blue you use? Will blue number 5 give you the same answer as blue number 11?
• What if you change the yellow too?

At first he thought the answer should depend on which particular number you picked for the color. But we worked out a lot of examples and he changed his mind. He was surprised and excited.

G has also realized that we can add letters this way, and that we can add numbers but “start over at 10″ to produce similar systems. So we can talk about more than just colors now.

We do still have some stumbling blocks with our rainbow math. The most common problem is that sometimes G thinks zero should be nothing instead of purple, and that trips him up. We talked about changing the rules so that there is a “nothing color” for zero, but then we figure out that red plus purple is nothing and he doesn’t like it.

In a few weeks I’ll start talking about how we multiply colors by numbers (what is red 3 times?), and maybe multiply colors by colors, and draw a rainbow multiplication table.

I’m still trying to sort out the morals of this story. I’ve been surprised at how much kids can understand, so why do we spend years only teaching them plus, minus, times, and divide? Do we really need to crank out a bunch of little 4-operation computers?

I thought it was great that we found an arithmetic system that was just as new to me as it was to G . It made me realize how unnatural a question like “what is 2 + 3?” must be to a young child. It also made G feel comfortable making mistakes, after all his dad was making them. And we’ve both had fun doing it. We never would have spent so much time and energy thinking about an addition drill sheet.

I have been thinking for a while about how to balance drills with exploration, and this experience is driving me farther away from the drills. I will need to teach proper arithmetic one day, but I think after a few years exploring numbers and arithmetic operations like we’re doing now arithmetic should be fairly easy. I’m not in a rush.