I started by asking them what multiplication was.
I said, "OK. When I say 'multiplication', what comes to mind. There's no wrong answers. Throw some words out."
One student said, "Times tables?" Another said, "Skip counting." After a pause another offered, "That line with two dots around it."
I'm guessing she meant the division symbol.
Here's the thing - I don't think the answers in most 4th and 5th grade regular education classes would be much different. Kids think that math is about tricks we do with numbers, and that the key to being good at math is learning those tricks. Special ed students, and lots of other kids think that because they are bad at memorizing rules, they'll never be good at math. That's wrong.
Next, I told those kids that multiplication, and math, was not about numbers. They looked at me like I had three heads. At least I had their attention. I reiterated, "You've been lied to your entire life. Math is not about numbers. And I'm going to prove it to you."
They were hooked. Probably because they didn't believe me and they couldn't wait to see the head teacher crash and burn in a fiery mess. Figuratively, of course. Well, maybe literally for some of them.
So I told them, "Draw a picture of three chickens on your white boards. I'm going to draw chickens, too. Please don't laugh at my chickens. I'm a mathematician, not an art teacher."
|I'm a Mathematician, Not an Art Teacher|
I said, "OK, each chicken just laid 3 eggs. Go ahead and draw 3 eggs under each chicken. You see, multiplication isn't '3x3'. That's just the numbers and symbols we use to describe multiplication. Multiplication is all about groups. '3x3' just means that we have three groups of three."
The light bulbs started to go on. They stopped looking at me like I was crazy, even if they didn't totally understand yet.
We modeled groups of airplanes with passengers. Cookies with chocolate chips. Flying saucers with aliens. My flying saucers rocked. Much better than my poor attempt at chickens. Each time we talked about how there were repeated groups of the same number. I only showed the multiplication problem in number/symbol form after we had figured out the answer to "How many do we have in all the groups."
I was pretty sure they understood, but I wanted to make sure. So I told them, "Now, I'm going to give you a multiplication problem. I don't care what you draw, but I want to see you express the problem as groups." I gave them 7x4.
Some drew cookies, others drew flying saucers. One kid drew seven tornadoes with four cars being mangled in each cyclone. A bit graphic, but mathematically sound. All of them were able to model the problem without help.
For years I've talked about teaching math differently. I've talked about the need to solve real problems, that there needed to be relevance behind everything that we teach - even basic facts. I've talked about helping kids to understand before asking them to memorize. Some were receptive. Most thought I was crazy. Many told me that I'd think differently if I didn't only teach the high math group.
They were wrong. There's no reason that we shouldn't expect almost all students to have understanding of math the way we expect almost all students to learn to read. We just have to stop expecting kids that have trouble memorizing to have no problems memorizing stuff that has no relevance to them. In 40 minutes these special education kids went from thinking that multiplication was "the line with two dots around it" to being able to model multiplication problems.
Math isn't about numbers any more than writing is about the alphabet. Numbers and symbols are just our way of expressing the quantities in the world around us just as the letters we use to write are the symbols we use to express our thoughts. When we take away that context of that world around us, we take away students' understanding.
We'd never tell a student that they should do 40 letter manipulation problems for homework to get better at writing. Or at least we shouldn't. (I'm not a big fan of spelling workbooks.) That's not going to turn them into a writer. Heck, it's probably going to turn them into a kid who hates writing.
Math is no different. It shouldn't be about following rules to manipulate numbers. It should be more about drawing chickens.