If you’re a parent, teacher, or anybody concerned with kids’ abilities in basic arithmetic in the fourth or fifth grades, watch this video. It’s 10 minutes long, but the first half is enough to get the idea.
These alternative methods that are being presented here aren’t exactly wrong from a mathematical standpoint — they all calculate the product or quotient successfully if carried out correctly — but they are no better as algorithms than the standard algorithm and in some cases are worse.
Take the lattice method of calculating the product. This is exactly the same process as the standard algorithm. But because it’s exactly the same process, what’s the basis for teaching it versus something like the standard algorithm that’s well-known? If it ain’t broke, why should we fix it?
And as for the “cluster” algorithms, I found myself recognizing this method instantly because this is how I do mental math when multiplying or dividing. I can see the idea behind teaching this method — students might be more likely to do several easy multiplication problems quickly and put the results together than they would to try the standard algorithm. But the catch is in the complexity of the algorithm. There is much more computation needed for the same problem done by clustering than if it were done the standard way. And just like with computers, the more computations that are involved the greater the probability of error.
Also with the cluster algorithm, it takes a pretty good amount of mathematical sophistication to see those clusters. Where does that sophistication come from? Apparently the curriculum authors think this is hardwired into kids’ brains — they “just know” that 6 goes into 133 twenty times. In general, I’m struck by just how much these alternative methods rely on “mental math”, “just knowing” certain things about arithmetic, and use of a calculator. But where does proficiency with mental math come from? Or the knowledge you have when you “just know” something? Isn’t it from mastery of basic algorithms and a deep personal pool of experience with working problems?
I think somebody ought to do an O(n) analysis of each of the algorithms presented here and see which one is the most efficient from a computational complexity standpoint.
It’s as if the education specialists who designed these methods found something that works for them, and decided it’ll work for everybody else too, regardless of whether that’s the case.
Update: Isabel has some related thoughts on the computational complexity of arithmetic at God Plays Dice, with a classic Foxtrot comic as backup.
[Hat tip: Math Blog]