Lately there have been a number of posts discussing the state of contemporary elementary maths teaching. While I certainly agree that Reform Math is bound to be a complete failure I’m not sure back to basics is the best alternative. One thing I’ve certainly despised about school is the endless hours we’ve spent wasted on practising long division. Of course I can still do long division but I highly doubt that it made me a better mathematician. Talking of old school algorithms my father was taught to take square roots using paper and pencil. Years ago I learned that algorithm on a whim and used it to calculate square roots mentally to keep me busy on my way to work.
Anyway, the point of this post is that while looking for pencil and paper algorithms for taking square roots (as I have since forgotten how to do this) I came across a paper entitled: Let’s Abolish Pencil-and-Paper Arithmetic which makes a point that mental arithmetic should be taught rather than pencil and paper methods and that there’s nothing wrong with using calculators.
Talking of people who dislike the use of calculators reminds me of a substitute teacher we once had for A-level maths who seemed almost violently opposed to calculators but loved slide rules so much that he wasted parts of a lesson to show us what they were and how to use them (as our generation had been so unfortunate to grow up without them). I mean, seriously, both calculators and slide rules are tools to facilitate computation so grow up and embrace the enemy.
Besides calculators are so old fashioned already. The new way to corrupt our youth are computer algebra systems. One of my first year computing lecturers might have been right as he said that at some point in the future these systems would become as commonplace as calculators nowadays and it could mean that the necessity to practice taking derivatives, etc. would vanish. I doubt it will happen soon but he’s got a point. Let’s face it, so far I’ve used Maple, Matlab and GAP and I have no intention to forgo their use in the future. Often when working on something or playing with a particular problem one may want to check a number of cases or just collect some data to get a feeling for what is happening. In such cases where one looks for patterns in the answers the actual computations are rather unimportant and it is only sensible to automate this task.
Then again, don’t take anything I say too seriously. After all I’m the kind of guy who’d favour getting rid of a number of compulsory mathematical methods courses and introduce at least one decent set theory and mathematical logic course in the standard undergraduate curriculum (and I would place it in the first year).
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April 16, 2007 at 10:35 pm
Foxy
Don’t forget Mathematica : )
All good points. The one thing that interest me about long division is the way it’s much more of a notation based algorithm than value based, and I think that has prejudiced me when I analyze things like factoring and multiplication.
I do wish I knew how to use a slide rule, though.
April 17, 2007 at 10:49 am
Jean-Noël
I didn’t really forget Mathematica, it’s just that I’ve never used it and I figured I’d only mention those computer algebra systems I’ve actually used myself.
If you really want to know how to use a slide rule Good Math, Bad Math has two posts explaining slide rules:
Using a slide rule (part 1)
Using a slide rule (part 2)
May 9, 2007 at 8:25 am
mike3
I wouldn’t just abolish pen-and-paper math methods like that. What happens if you DON’T HAVE A CALCULATOR? Gasp! You’re out of luck! You talk about doing “mental arithmetic”, but what if you need to add 595 + 723… You’re not seriously suggesting we have our kids rote memorize 1,000 x 1,000 addition and multiplication tables?! Even to do that in your head you still need to use a few PNP-like steps. For example, if you only know that 5+3 = 8, 9+2 = 11, and 5+7 = 12, then to get the sum you still need to be able to carry, etc. which is what the gist of the PNP addition algorithm is! If you want to teach mental arithmetic something there still needs to resemble PNP in it’s basic concept. Then you talk about phasing in computer algebra systems, now what happens if you don’t have access to a computer, much less one with a CAS on it? And you need to do something as simple as 12x = 9? Oh my!
I think that overdependence on technology is not good. Technology is useful, but to become so dependent on it is just insane.
May 9, 2007 at 8:50 am
Jean-Noël
1318. No pen, no paper. (No fancy technology either)
You should read the paper I linked to.
x = 9/12 = 3/4. (btw I’d love to see your PPA for that one)
I mean, seriously, it’s a great paper.
I never said anything about dependence, let alone overdependence, on technology. There’s a difference between using technology and being dependent on it.
Most children (and adults) do not know the basic concept of most (if not all) PPA algorithms – they might just be able to use them (yes, I said might).
Using CAS is not insane dependence on technology. I’d conjecture that most people who can use these systems proficiently can do your calculations easily enough (in several different ways, too). CAS give you the power to take things significantly further and investigate maths in ways that using traditional methods would take ages. It’s not insane to use the best tools available, it’s the smartest choice.