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?
Don't Panic Says:
I found the video infuriatingly one-sided and on the verge of being deliberately obtuse as well as a bit reminiscent of the "This is the way I learned it so it must be the only acceptable way; anyone who learns/thinks differently is just broken" attitude.
I think the best response is the video: http://www.youtube.com/watch?v=9skRrnN2_HU&feature=related. I think he makes some useful points.
I think this touches on a pet peeve of mine for the last [mumble] years — the unrealistic expectation that everyone should be able to do these "simple" math problems quickly and without aids. I don’t have a good short term memory — I was even pulled out of 4th-5th grade for extra training on this — so I can’t add two 2-digit numbers in my head. And until I turned 42 a few years ago, I could not tell you what 6 times 7 was without doing it as 6 times 6 is 36, add 6 makes 42. But you can’t imagine the disapproval I’ve gotten over the years for demonstrating these traits. On the other hand, I managed to get a PhD in Physics (not a math-phobic field) because while I wasn’t adept at simple calculations I understood the underlying concepts in math. I’ve loved word problems since I was a child. So what has served me better in life: skills or understanding? Well, definitely in terms of career it’s been understanding the underlying concepts. In terms of living everyday life — well, being better at some of those skills might have smoothed things over a bit, made things a little faster and less awkward but I don’t think I’ve been seriously impaired.
Sonic Charmer Says:
I think you’re biasing the question a little when you paint it as “traditional” vs. “intuitive”. Part of the problem with the “intuitive” method is that it’s only “intuitive” *if you already understand what multiplication is*. But to gain that intuition, you have to…learn multiplication. You have to dig into it and examine its nooks and crannies. This can be tedious but it’s part of learning. Tell a kid who knows nothing about multiplication “just split it up into 20×31 + 5×31 + 1×31″ and he’ll be lost. He’ll be looking for rules and patterns he can understand. If you don’t convey those patterns, he’ll get frustrated and lose interest. (I would!) Or worse, he’ll invent a bunch of rules and patterns that seem to fit the “intuitive” methods you’re teaching him, and make mistakes.
This isn’t really a question of speed vs. understanding, either. The “traditional” methods can be slow. And the teachers who stand up there and say “Okay kids, just pick an intuitive way and do it that way, however you feel like” aren’t promoting any understanding. They are assuming the understanding is already there. Rather, it’s a question of methods that consistently work and methods that aren’t really methods because they don’t tell the kid how to approach any given problem. Once a kid understands multiplication, of *course* he can split up 26×31 any way he feels like.
The one thing “traditional” methods don’t have going for them is that they aren’t that *fun*. They aren’t exciting. They aren’t the result of modern research or theories. And as a result, I think a lot of elementary-school math teachers *don’t enjoy teaching them*. So they look for excuses not to use them.
But the one thing I hope everyone can agree on is that schools are not there for the enjoyment of the teachers.
S :-| Says:
I think more emphasis should be given to teaching kids core level maths as well as spelling etc (one of my personal flaws). The airy fairy approach that is used in schools is just not working. When my little sister and a cousin were in grade 8, they were having trouble with their math & they asked if I could help them out. The first thing that completely shocked me was their dificulty in multiplying single digits let alone double digits. Their response “Just let me find my calculator”. OK so I’ll admit that neither of them were overly interested their achedemic development but I found it hard to believe that they got that far without anyone picking them up. My sister in particular is quite smart when she wants to be (heaven forbid people notice it, she may be called a geek like her big brother). I started teaching her using the more traditional methods and she picked up a lot of stuff in a very short time frame.
I’m not saying I’m an advocate of winding back the clock and so everything is taught as it was 50 years ago, I’m just saying that more attention should be paid to these core principles of education in the early years.
As for using the “traditional” math principles in day to day life, well if its a single multiplication I’ll probably use a calculator but if I’m working out a complex problem a work, i’ll do most of it on paper.
Trubby Says:
I agree with S:-/, Though computers are dominating our lives it is still an important skill to write and do maths on paper and these skills are losing importance in our education. The good thing with the ‘traditional’ method though is you can teach how the algorithm works and thus you know how the numbers work. I think that partial algorithm method is good for initial teaching and understanding. Then you could compress it into the ‘traditional’ method with the students understanding how it works. It is important, when teaching the ‘traditional’ method, that teachers know to teach the student how the algorithm works. By the way I am a high school student in grade 10 so I remeber more about grade 5
You are more likely to have a cell phone on you (that has a calculator) than a pencil-and-paper at any give random moment, I would guess. So the value algorithms derive by being a simple path to the right answer (lattice method, for example) has rapidly decreased.
On the other hand, manual computation is good mental exercise. So how do we choose? Stick to algorithms that display the patterns needed in higher math. For instance, multiplication algorithms that make explicit use of the distributive property (in their process, they show that 21*36 = 20*36 + 1*36, and so on) teach principles that apply also to polynomials: (x+5)*y = x*y+5*y. In other words, teach the stuff that builds mathematical continuity with what’s needed for higher level thinking. The goal should be to get as many kids to a higher level as fast as you can.