Multiple methods

Glenn Beck is criticizing the new Common Core standards. I have some interest in politics. But this is not about that. I have 25 years of my life invested into math education. So far what I have seen of Beck's criticism is wildly off base. Here is one correction. There may be more to follow.

Beck states that addition with regrouping is a new-new-math that is dangerous and that dumbs down our children. I'll explain what regrouping is and why it is actually awesome.

I'll use Beck's example: 29 + 17. How would you do this problem? (Without a calculator)

Most of you would want to grab pencil and paper and commence the adding algorithm. This involves "carrying a 1", takes a few seconds and is reasonably accurate.

However, do you know why you "carried a 1"? Do you realize that 9 + 7 = 16 and that 10 of the 16 can be transformed into a 6 in units column and a 1 in the tens column?

Regrouping emphasizes student understanding of why "carrying the 1" makes sense.

Most math folk don't do mental math from right to left. It doesn't work very well or is as quick as regrouping. If I did this problem I would think

20 + 10 is 30

9 + 7 = 16

30 and 16 make 46.

Beck ridicules this very method. When you write it on paper, it looks longer. In your mind, it is remarkably nimble. You might not trust my mental tricks. But perhaps you would trust Arthur Benjamin's. I've heard Dr. Benjamin talk about mental arithmetic. Like most mental arithmetic experts, one of the first things he will do is explain to you that you don't work mentally like you do on paper. And that you often work with regroupings and from left to right.

But its not just mental arithmetic that motivates regrouping. It is an understanding of place value. Let me move to a second example. How do you multiply 33 x 21? If you follow the typical American procedure, you will multiply by 1 x 33 and write down 33. Then you write a zero in the next row. STOP! 

Why? Why a zero to start row two? Because you're not about to multiply by 2. You're about to multiply by 20. That's right, TWENTY. (I have a friend who is a heart surgeon. When I explained this to him he said he never understood why the zero is there. And he's a smart guy. But he had never been taught this fundamental fact with any understanding!)

So if we regroup, we get something like this:

33 x 21 = (33 x 20) + (33 x 1)

And 660 + 33 is pretty easy to do in your head. It's 693.

Mr Beck might complain that this is longer (and he's sure to write it on his nifty chalk board as if it is much worse. I wonder if he knows that white-boards have been invented?) but if he paused for a second, he might realize that it's looking kind of familiar.  It's looking like the distributive property! The same property that's so handy in Algebra 1! And pretty much the rest of advanced math. In fact, a strong multiplication unit (in the common core) will distribute the problem out all the way:

33 x 21 = (30 + 3) x (20 + 1) = 30x20 + 30x1 + 3x20 + 3x1 = ...

You might recognize this as FOIL'ing (I'll have to correct the atrocity of that mnemonic on another post) and rest of the world introduces the distributive property in this manner pretty early on. It builds algebraic thinking from a young age.

More to follow. Comments welcome.