My current curriculum uses some pieces from new math. My basic take on it is that there are two competing schools of thought–traditionalists and constructivists–and the folks who ascribe to both theories are ideological idiots. You gotta combine both approaches, just like you gotta combine phonics teaching with reading whole books.
So I teach (following the fairly good curriculum) the addition facts, such that by the end of the year my second-graders should know everything from 0+0 to 10+10 instantly. But I don’t teach them all at once. For example, tomorrow we’ll start learning the facts with a sum of 10, and in doing so we’ll examine the commutative property. We’ll use 2x5 tables in this discussion, make it clear that the table has 10 cells, and put dots in some cells. How many cells have dots? (This also helps kids decompose numbers, seeing, for example, that 7=5+2, a really useful skill for flexible addition strategies). How many cells don’t have dots? If 7 cells have dots and 3 don’t, what do you need to add to 7 to get 10?
That kind of thing.
Next we’ll work on facts that add 1 to a number, then facts that add 2 to a number, then ones that are doubles, then ones that add 10 to a number, then ones that are doubles-plus-one, then ones that add 9 to a number–and when we’ve done all those clusters, we’ll have something like 16 facts left in 8 pairs, instead of having 121 facts to learn. Throughout the process we’ll use games and manipulatives (I have a fun little magic trick I use to help kids visualize doubling a number).
One of the big controversies in math education is whether kids can use algorithms. Quick, y’all: add these numbers
347
+538
Most of y’all are adding the 7 and 8, carrying the 1, adding 1+4+3, adding 3+5, and arranging the digits from right to left, yes? And that works fine. Plenty of folks when challenged on this can’t explain why it works, however. And plenty of kids who learn this method screw it up: they’ll write 8715 as the answer, or 876, or 985. And they’ll have a reason for each of these specific answers related to an imperfect understanding of the algorithm. (Bonus points: figure out what misunderstanding leads to each of these answers).
The constructivists don’t want me to teach the algorithm. The traditionalists want me only to teach the algorithm. I think they’re both kind of idiots. I model the equation for the kids, and I pound place value into their heads over the entire year, and I force them to build the equation using base-10 blocks (a 1 block is a tiny cube, a 10-strip is 10 1-blocks put together, a 100-flat is 10 10-strips put together, and a 1000 cube is 10 100-flats put together). I give them my patented “Build a Borg” game to play. I compare place value to pennies, dimes, and dollar bills, and remind them how you could make equal trades between these money amounts.
And when they understand addition, including regrouping (or borrowing, or trading, or carrying, or whatever you want to call it) from the ground up–when they can explain how the system works–then I’ll show them the algorithm. I don’t emphasize it, and indeed I explain that it’s often not the fastest way to solve a problem: you can solve 495+342 much faster if you do 342+500-5, for example. But I let 'em use it if they can explain why it works.
You gotta combine approaches.
Seriously, there are records of the Whole Word approach dating back to pre-Roman Greece.