Shortcuts and dead-ends

Time, surely, for some mathematical musings.

Here's one: to what extent to teachers follow teaching methods prescribed in a maths textbook, just because they are there?

I don't mean to knock those who write textbooks, because by and large the quality isn't too bad. But sometimes you come across a prescribed method and think, "wait a minute, this is pants!". Or at least I do.

Now I'm not going to name the textbook, but here's a famous example: I wonder just how many pupils doing Credit maths in Scotland have been taught to expand brackets using the "FOIL" mnemonic just because it's in the pages of a very popular textbook? (Technical alert: you might want to look away for a bit...) Basically the book suggests that to expand, say, (x+2)(x+3), you multiply the first terms (x multilpied by x to give x squared), then the outsides (x multiplied by 3 to give 3x), then the insides (gives 2x) and finally the last terms (gives 6). F O I L - geddit?

Well, yes, but why do I need it? Is expanding using the distributive law - to give x(x+3) + 2(x+3) - really so tricky that we need a mnemonic? Strangely enough, the book starts off with the distributive method before offering the FOIL shortcut. But surely the shortcut is a dead-end, not a shortcut? Because all the student ends up remembering is "FOIL", which means they get well-stumped when subsequently faced with expanding a linear and a quadratic factor, eg (x+2)(xsquared + 3x +2), having forgotten all about the distributive law. Worse yet, the FOIL method seems to hide rather than highlight the distributive law. And yet this is still a very popular method, as far as I can tell.

So, stuff this FOIL business, I say! And let's start questioning more closely the methods suggested in textbooks, and chatting about them with colleagues. You see, having ranted to fellow teachers I've discovered they feel the same way - but none of us has ever said. (Mind you, that's probably because we're too busy scoffing chocolate biscuits...)

Of course, this all raises a fairly key question in maths teaching: to what extent does a student have to understand a process, in order to be able to use it? Does understanding always precede the ability to carry out, say, a successful calculation? Or can you do something without really understanding what's going on? And if you can, is that a good or a bad thing?

More of this to follow.