For addition and subtraction there is no better tool than a number line and moving right for adding and left for subtracting. Anything else is pointless, and Common Core excels at pointlessness. I've seen this in my daughter's homework:
Instead of memorizing familiar steps, students now practice even stranger rituals, like drawing dots only to count them or breaking simple addition problems into complicated forms (62+26, for example, must become 60+2+20+6) without understanding why. This can make for even poorer math students. “In the hands of unprepared teachers,” Lampert says, “alternative algorithms are worse than just teaching them standard algorithms.”
No wonder parents and some mathematicians denigrate the reforms as “fuzzy math.” In the warped way untrained teachers interpret them, they are fuzzy.
Common Core sucks and traditional methods and no "approximate math" is the best way.
I don't have a strong view about the exact methods of Common Core, but I agree with a few ideas that I think factored into the initiative:
(1) I believe in coming up with a single model curriculum for each subject on a national basis. Lots of people move from district to district and state to state, and so if every area is following its own curriculum, those who move will be getting all sorts of redundant instruction, while also having stuff thrown at them for which they haven't yet got the foundational material. A fifth grade teacher should be able to rely on the idea that her students have had a particular set of instruction, already, regardless of whether they spent fourth grade in the local school district or elsewhere.
(2) I believe in coming up with the curriculum and the guidance on methods using the expertise of top people who have actually made teaching kids their life's work -- the finest educational experts in the country. I don't believe these things should be come up with by the whim of local school board members, who often have no relevant training or experience, and who can be counted on to abuse their positions to deal with their own pet peeves rather than trying to give the kids the best education they can (e.g., mandating the teaching of intelligent design in science class).
(3) I believe in studying what works and what doesn't, in practice, and emulating the successes. That means both studying successful school systems in the US (e.g., Massachusetts) and around the world (Finland, Canada, South Korea, etc.), and seeing what lessons we can derive from them.
(4) I believe we should be willing to try out new things, rather than just assuming the old ways were better for no reason other than that was how we were taught.
(5) I believe that what we were doing before wasn't working terribly well, since we spend fairly lavishly on education with mediocre results, relative to our peers.
As for "approximate math," I think that has a place, too. I can't tell you how many times I've seen people get a math problem off by an order of magnitude or two, because of a little carelessness about the decimal place, or because they multiplied when they should have added, or something like that. They get so focused on marching mindlessly through the rote steps they were taught, that they lose a sense of generally what is it they're trying to figure out, and they produce results that would be easily spotted as nonsense by anyone who took a step back from the trees, for just a moment, to observe the forest.
For example, take a classic word problem asking where two trains will meet. They're 100 miles apart, heading towards each other, and Train A is moving 60% faster than Train B. You'll get students dutifully walking through the algorithm they were taught and making some dumb mistake in the process and telling you that the trains will meet at mile 615.4, without even stopping to realize that's way, way past the far end of the track. And then they'll promptly forget the algorithm, once the test is over, because they never really grasped the idea behind it, anyway. Teaching them to approximate the result, first, will serve as a reality check on their answer, and will also help to cement the algorithm in their heads, because they get what it's about, and could derive it themselves if they forgot it.
I suspect most adults solved problems just like that as kids, but couldn't solve it today, because they only memorized the steps to the solution as if it were an abstract procedure, rather than an intuitive concept.
To give a specific example I remember from science class: We were taught the formulas to convert among Celsius, Fahrenheit, and Kelvin. On one test, we got an extra-credit question asking us to convert to an invented temperature scale called the Kolias scale. We were told it was a linear scale where water froze at 73 Kolias and boiled at 401 Kolias. What is 64 degrees Fahrenheit on the Kolias scale? For me, it was a painfully easy question. I'd never even bothered to memorize the conversion formulas, since if you knew where water froze and boiled in a scale you could derive the formulas in three seconds, anyway. So, I just did the same thing and got the answer as easily as I would have if converting between Fahrenheit and Celsius. I assumed everyone else got the answer easily, too. Then I found out that in two classes, with 25 students each, only two of us had gotten the answer right.
Some students had come up with completely absurd answers. They hadn't even located the converted temperature less than halfway between freezing and boiling, which even a first approximation should have told them to do. They'd been taught to apply their algorithms by rote, and never even understood the concept behind them, so all they were able to do is act as mindless calculators, and when asked to think conceptually, even in an absurdly simple fashion, they were at a total loss. I think what Common Core proponents hope to do is to give the kids the conceptual building blocks they need to be something other than mindless calculators.
