What should we learn?
In a recent discussion about education, this YouTube video came up: Math Education: An Inconvenient Truth. It’s a look at some of the curricula used in American schools, and asks the question: what should students be expected to know at the end of Year 5? In particular, should we teach the “standard” methods for doing multiplication and division (the usual multiplication (”carry the 3″ etc) and the usual long division) or are students would be better off learning more “intuitive” methods? In some sense it’s a battle between speed, complexity and understanding.
(At this point, you really should watch the video, but read on regardless!)
The argument (according to the curricula authors) against the tried-and-true methods is that although most of us can “do” them (eventually) it can take a while to learn and not everyone understands why these methods work - the mathematical insight may be lost to the algorithm.
Instead, they suggest using more “intuitive” methods. One option for multiplication is the way most people do maths in their head, except on paper. Using their example, to multiply 26 x 31, you write it as 20×31+5×31+1×31. Then you can quickly work out 10×31=310, and double and halve it to get 20x and 5x. Finally, you add it all together and you’re done. It’s good because it’s very clear what you’re actually doing - what multiplication actually means, something which can easily get obscured in the traditional method. But I would have to think it would also be slower on paper - and probably more prone to making mistakes.
A compromise is the “partial products” method, where you do keep track of what you’re multiplying (e.g., in 26×31 you don’t multiply 2 by 3, but 20 by 30.) Finally, there’s the “lattice method” (which I read about when I was much younger in a fantastic coffee table book “Oddities”), which is very cool from a geek perspective, and useless otherwise.
So what should be taught? Certainly, some of these methods are ridiculous, but the basic idea isn’t. One Big Issue is calculators (I think there’s a whole separate post there). Most mathematicians go apoplectic when this comes up, but I think there’s points on both sides. You could argue, why bother teaching multiplication at all if a calculator can Just Do It? Realistically, I don’t often multiply by hand. Aside from arguments like “you won’t always have a calculator” (which is still important (think shopping!) but I don’t feel key here) in maths you need to develop your skills and, if you like, intuition for how things go together, and how to approach questions. Problems arise when students assume every problem can be solved just by sticking it into the calculator (and this gets “worse” with graphics calculators!) - as soon as a problem is out of the ordinary, I’ve watched students spend more time hunting for a possible equation solver in their calculator than it would have taken to solve by hand!
I can’t provide data, only anecdotes, but I’d think that if you never learned how to multiply, you’d find the more complex skills which build on the basic ideas much harder. And for just the sheer number of times you need to multiply, and for how much of other maths it underlies, it’s just such a valuable skill.
What about intuitive vs algorithmic approaches? I think the word “intuitive” is a bad choice, actually - really what they’re advocating is “guess and check” methods (”10 13’s fit into 200, subtract them off and we’re left with 70. At least one 13 fits in 70, etc) It helps build that deep understanding of how maths works, but I’d argue the method itself really isn’t “intuitive” - just frustrating after you understand the principles. Interestingly, when I tutor I always emphasise understanding rather than doing - but at the same time, I encourage students to follow procedures (e.g., for differentiation) at the expense of quicker “guess and check” shortcuts, because it drastically reduces errors. (Again, another post I think).
Again, I think multiplication is important enough to justify the proper algorithm. Division, though, I’m less certain on. I’ve learned long division, only ever used short division, and even that I use only rarely these days. I’d be tempted to say the guess-and-check understanding of division might be sufficient.
And, of course, all these arguments extend to other areas of school as well. I don’t remember most of what I learned in primary school - names of explorers, poems, etc. But was the “learning how to learn” more important than the content itself? (I generally have little interest in games like Trivial Pursuit, preferring games that require skill than can be developed rather than knowledge that you either have or you don’t. I guess I feel the same about school 
)
So I ask again, what do you think should be taught in primary school?