No doubt–and I’m inclined to agree that Common Core is a good idea from a structural perspective. The distinction I’m drawing is between areas of legitimate disagreement (e.g., this kind of structural concern) and foolish conspiracy-theory arguments (Common Core is creeping socialism! Obama is trying to indoctrinate our children into cryptoMuslim atheist Marxism!)
I have stayed out of this debate because I don’t really care that much about it. I have two kids that are in 5th and 7th grade and I am happy with their education mostly because my wife and I are very involved. I have thus ignored all the debate about the new math and common core math. I was however in a waiting room today and read the article you linked to kill time. Now I find I need to chime in.
I was really excited when (~ 4 years ago or so) my children started doing math the way that is talked about in the article and the way it is talked about in this this thread. I am a huge supporter of this methodology and I have been encouraging these kinds of methods actively. This is exactly the way I have been doing math since college and it was responsible for a great leap in my understanding and ability to do math.
When I was in college (majoring in physics and applied mathematics), I learned that these were the methods my professors used to solve problems. I was constantly stunned by how quickly they could solve math problems in their head and over time I learned their methods through osmosis. By graduate school (at a different university) all of my peers were doing math using the same methods. When you have a number line in your head and can recognize that 5*2=10 says exactly the same thing as 10/2=5 or even (15-5)/5=2, it makes it easy to do all sorts of complicated math in your head by breaking it down. The way that common core teaches math is IMHO much better than the rote memorization and brute force techniques I learned in high school. YMMV.
Today I taught the distributive property of multiplication to my 8-year-olds. This is something I didn’t learn in school till like eighth or ninth grade. And when I learned it, it was something I was expected to memorize and to use with variables, not something I was expected to internalize and use in everyday life.
I was showing my students a basic word problem. Dorkness has six shirts, each shirt has six buttons, how many buttons in all? 6x6=36, right?
But I wanted to show ways of solving it ranging from basic-but-tedious to sneaky. The basic way involved drawing all the shirts and buttons and then counting them. Then I showed that you could be lazy, draw six circles instead of six shirts, write “6” in each circle instead of drawing six buttons. Then you’re just adding six sixes.
Someone pointed out that you could pair the shirts into groups of 12 buttons, you’d get 3 sets of 12. Cool! I showed that halving one factor and doubling the other gave the same product (sort of the associative property).
And then I baffled them: I drew a big lumpy circle around five of the shirts. They had no idea why I would do that.
I showed them that the problem was now (5 x 6) + (1 x 6). The first factor had been changed from 6 into 5+1. And they can all flip 5 x 6 into 6 x 5 (commutative property, but they don’t know that), then count by fives six times; they all got to 30 really fast. And 30 + 6 is 36. There was a chorus of “Woah!”
Now, for 6 x 6, that may seem like a lot of work. But it serves three purposes:
- A lot of them don’t have their facts memorized yet. This gives them a way of deriving a fact. Even something like 7 x 8 can be solved if you can solve (5 x 8) + (2 x 8), much faster than having to count by 8 seven times.
- It’s really useful for multiplying bigger numbers in your head. 24 x 7 is a snap if you can do (20 x 7) + (4 x 7).
- Later, when they’re doing algebra, they’ll already be familiar with this aspect of math.
I didn’t mention that it was the distributive property today. I’ll mention that in passing at some point, but I won’t require them to know that. It’s the opposite of how I learned it: instead of expecting them to memorize the definition of the distributive property and apply it to multivariable algebra, I’ll just expect them to be able to fiddle around with it when they’re solving problems.
It’s a tool in their box. And like you say, it’s a tool that real mathematicians use. Sure, they’ll memorize the facts at some point; but even once they have, it’ll stay useful.