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Mathematical Pedagogy Analogy

I've ranted before about the fallacy of arithmetic skills equating to mathematical skills, but it is probably no better illustrated than in discussions about timestables. There is a school of thought, and I believe it is slowly dying, that memorising timestables is worthwhile mathematical pedagogy. There are two situations I am regularly presented with that demonstrate this illusion: either someone shows that they too, are good at mathematics by rattling off a series of timestable entries; or, someone complains to me that maths sucked or they were not good at it, because they never could remember their timestables.

Both situations demonstrate a failure on behalf of those people's eduactors. While familiarity with a few timestable entries can be a useful life skill, the disconnect between memorising timestables and possessing mathematics skills should be made clear in mathematics education. I'll pass this time on presenting a thorough explanation on why this is the case, and instead present a cute analogy that just occurred to me.

Assuming you're a native and competent English speaker, can you voice the 5 vowels, and their corresponding long and short soundings in the English language? With some thinking, most English speakers will manage this. They may discover that they did not realise, prior to hearing the question, that the relationships between the vowel pronounciations and their long and short soundings are not at all obvious. For example, while the short 'o' sound sounds much like the letter's pronounciation, the same is not true for 'a'. Nonetheless, by thinking of simple words and working backwards, most competent English speakers will figure it out.

Consider for moment what would happen if vowel soundings were taught like timestables. "The long 'e' sound is 'ee', the long 'a' sound is 'ah'..." and so on. Imagine the trouble you would have applying the information! Every time you came across another word you would have to refer back to your vowel sounding mnemonic and find the relevant sound. Imagine the time that would be wasted memorising relationships that any Joe Blow English speaker would be able to figure out, or if necessary, simply look up in a reference.

Fortunately we are not taught English in that way. Generally it is assumed that people can consult a reference if necessary, and instead the depth, application and beauty of the language is presented and explored. Less mechanical, but more readily and widely applicable skills are developed along the way, allowing further exploration of the subject.

Now you can probably guess where the analogy is going: imagine if mathematics were taught in the manner I've just described for English. Ignore for now the fact that the subjects require very different skills, and consider that any time spent memorising timestables may well be time wasted, and could be better spent exploring the depth, application and beauty of mathematics. A natural familiarisation with multiplication will develop over time and constructing a timestable will become relatively simple, just as constructing a table of vowel sounds would be to the average English speaker. Further, thanks to the human tendency to forget that which is not used, the average student may end up with a similar recall of the basic timestable anyway. The tradeoff however, may very well be mathematics students who have a diverse mathematical toolkit, an appreciation of the attractive qualities of mathematical reasoning and the ability to apply mathematical skills to new problems.

Perhaps then, the practice of developing a timestable or a vowelstable should carry the same stigma of irrelevancy.

Note this post has been edited for minor grammatical improvements in anticipation of submission in the Carnival of Mathematics. (18/2/2008)


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Holy cow!

Next you'll be telling us that teachers make kids memorize something called an alphabet before teaching them how to read.

We better put a stop to that, as well.

Great counter argument mr. k. I realise you're being hyperbolic (pun not intended), but the simple explanation would be to equate "memorisation" of the alphabet as being on par with "memorisation" of the integers. Knowing the letters of the alphabet off heart is as critical to literacy as knowledge of the numbers is to numeracy.

By the way, I think the importance of number sense is particularly relevant to this issue. I also think an intuitive ability to estimate and "feel" the magnitude of numbers is critical for mathematicians and socially intelligent people alike. That said, I don't think de-emphasis on timetables contradicts that goal.

I'm looking forward to more comments as people work their way through the Carnival of Mathematics, particularly if those closer to actual education than I can speak from their point of view.

I don't think much needs to be memorized in mathematics.

Unfortunately, the times tables are one of those things that really needs to be down solid, because it will impair students greatly to have to stop at *every* juncture they need to, say, pull out all the possible ways to multiply two numbers to make 20 because they need to factor a binomial.

It's certainly possible to make it to Pre-Calculus without knowing times tables -- I have one student who doesn't -- but that student will take 3 times as long as the others to work out some assignments.

The unit circle, on the other hand, is something that I believe doesn't have to be memorized cold. I give a memorization test to my Trig students but I teach them how to recreate it, not neccesarily to know cold in 2 seconds what the sine of 210 degrees is. I've never seen in real mathematics a call for that; angles tend to be less book-contrived things like 22.534 degrees anyway.

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