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.

Anoraks of the world unite!

What is it with maths teachers and the sheer, unfettered joy of keeping lists or ordering things? Or is it just me?

Och, we complain about paperwork - and rightly so - but, you know what, give us a list of marks to write up, or papers to put in order, or a good bit of solid filing to do... and we're in heaven. Maybe it's to do with always looking for order and reason, or maybe we're just control freaks. Mind you, I bet the truly gifted mathematically are probably less inclined to get out the coloured A4 dividers and go crazy filing old Higher homework solutions....

What's all this leading to, you ask? Well, jings, crivvens and help ma boab, have I not found the perfect website for... sorting out your library!!

What's that? "Why would I want to bother?" Well, see what you think: have a look at www.librarything.com and see if you can resist temptation. I'm in there somewhere, happily cataloguing and rating books. Feel free to drop by and say hello, if you can find me!

And yes, it's free.

We can call him Al

Caught a bit of BBC News 24 today (hey, I have to have something to watch while doing the Sunday ironing): they were reporting live from the Edinburgh TV Festival and had an interview with Al Gore. Al's in town plugging his film "An Inconvenient Truth", which is in the Film Festival, and he seems to be doing all the other festivals he can too.

Anyway, I only saw the end of the interview but I did hear a superb soundbite from Mr Gore. Asked if he had any concerns about how unpopular his message about impending global warming and climate change was, he said this: "The truth will find its own constituency."

I like it. I really like it.

Dear maths teacher: your questions answered

Those letters just keep coming in. First up, a query from Miss Markitt, an NQT in Scotland-shire. She writes: "Dear maths teacher, do you have any advice on the best way to get through the mountain of marking that comes with the job? Night after night I sit down to mark more and more jotters, and it's getting me down. Help!"

Well, Miss Markitt, you're right to point out that maths teachers the world over are busier with the red pen than most of their counterparts, but it sounds to me like no-one has yet told you the best way to get through the long dark night of the ticks. I'm referring, of course, to the use of performance-enhancing substances. Yes, I'll say it if no-one else will: just as no-one can seriously expect a Tour de France cyclist to go the distance without the occasional furtive visit to Superdrug, so too does your maths teacher need a little extra to get him or her through the evening. Heaven forbid that you should think I'm referring to illegal drugs, mind, because that would hardly fit our image. No, rather we say hello to The Glass of Red Wine. No maths teacher would think of marking at home without it.

Now, let me stress here, I'm not advocating you should mark jotters while squiffed/steaming/legless - though it might be fun to see the results, you have to admit. But one or two glasses of a well-chosen red, and trust me, that red pen will start to flow more freely. And, when you see someone forgetting that a negative times a negative makes a positive for the twelfth time in as many jotters, it won't seem quite so bad as it might otherwise.

Experienced teachers know that it's best to choose the wine to suit the class. For example, marking a Higher Ink Exercise is to become part of an age-old tradition, so obviously a decent Cotes du Rhone brings sufficient gravitas and helps you feel you belong. Marking a first year test, by contrast, is a lighter affair, well-suited to a cheeky Beaujolais Nouveau. (There being no famous Australian mathematician to speak of, by the way, maths teachers tend to reject New World wines as young upstarts that should be left to the Media Studies department, or PE.)

If, however, you ever face the prospect of having to read your School's Improvement Plan... well, get yourself a decent Islay Malt Whisky, a large tumbler and a pillow, and the pain will recede eventually. Failing that, call for Doctor Macallan...

Who's counting?

First up, thanks to liturgybuff and covenanter who have both recently posted comments; as user-friendly as Blogger seems to be, I can't quite work out yet how to show recent comments on the sidebar, though I'd like to. Bear with me...

Anyway, here's the thing: how often do any of us have to do basic arithmetic these days? It's hard to find examples from real-life - and the old answer of "when you're shopping" doesn't really hold up under scrutiny in these modern times. Do you see people keeping a running total as they shove the trolley round? No. Do people work out a rough total? No. Does the shop assistant? No. The computer on the till does the work, and maybe - just maybe - you have to count out some notes, but more likely you hand over some plastic and cross your fingers. (And if you need change, the assistant doesn't work that out either.) Does anyone check the computer's calculations? Hardly ever.

Y'see, speaking as a teacher, we do teach kids the basics, honest. And things are a whole lot better now that the main exams have parts where you are not allowed to use a calculator, unlike in the 80's and 90's, when kids were welded to their calculators. So in theory at least kids should be getting better at the basics.

But, but, but... surely the important difference nowadays is that there's hardly any contexts where they then have to use these skills. Whereas for those of us who lived in a time before computers could do all the work, it was pretty important to be able to do the maths, or rather arithmetic, and you weren't short of opportunities so to do.

And this throws up a question of precisely what we should teach these days under the banner of mathematics. If it's just for personal finance, then as far as I can see kids need to be able to do the four basic operations, understand percentages and maybe just a tiny bit of fractions, though you could maybe get away without that. Under current guidelines this content would be covered and understood by a notionally "average" 14 year old. Meanwhile, maths is compulsory roughly to age 16.

Should we let kids get away from the subject early? Am I talking myself out of a job?

Trust me... this stinks

Courtesy of the wonders of digital TV (and a subscription fee, I might add), I caught up with a TV prog yesterday which was originally broadcast on Friday night, 7pm, BBC2. "Trust me... I'm an economist" purports to demonstrate how the science (?) of economics can show us how to live more successful lives. It's presented by I-can't-even-be-bothered-to-look-it-up in a manner both smug and incompetent - always a winning combination. And what do you know, programme one was on "love", and did indeed try to use the language of economics to tackle such problems as how to meet the right partner/how to be sure marriage is the right step/how to decide whether or not to have a child.

The audacity of this programme was breathtaking. And disturbing. Quite the worst thing is, it struck me as a programme almost entirely devoid of any sense of morality. I shudder at the memory.

Naturally the programme maker had occasional equations flitting across the screen, cos hey! this is a science, right? Which makes me wonder what any of my pupils would make of this stuff. I think I trust them to call it for what it is.

Let's hear it for Howie

Big splash in today's Guardian about government plans to beef up the content and difficulty of GCSEs in Maths and English - which doesn't apply north of the border, of course, but may end up being an indication of the way the wind is blowing, metaphorically speaking.

Anyhoo, there's a comment by Baroness Warnock which you can find here, and I've got to say, the old girl could be talking a lot of sense. I know that a lot of teachers and educational thinkers were disappointed when the government turned down the recommendation of the Tomlinson Report, and it does get me vaguely nostalgic for the Howie Report which came out in Scotland... ooh, ages ago. Basically Professor Howie - the guy's a mathematician, which has to count for something - said we needed to do something about the Higher exams, which are taken mainly by fifth years, who then get a university place based on their Highers, provided they do well enough. Which means they then do... er, what, exactly, in sixth year? "P*ss about!" I can hear the teachers saying in frustration. Well, the Prof suggested we go for a Baccalaureate thingy, which more academic pupils could get by the end of sixth year, which would (a) keep them studying (b) be a nice European way of doing things and (c) insist on students studying a reasonably wide range of disciplines. Less academic/more vocational pupils meanwhile would study for a vocational qualification, with exit points at the end of fourth, fifth or sixth year.

Not a bad idea, I thought at the time - though no-one asked me. But this was not great news to a couple of groups of people. Firstly, to the universities, who insisted that they wanted the Highers (fifth year, remember) to remain the benchmark for uni entry. And why would that be, exactly? Anyone? Well, that way you get more people through the door, of course. (Well, duh!) Which has to be a good thing, surely? See Baroness Warnock for thoughts on that argument.

And the second group? That's the group of people who go wild-eyed crazy when they see someone venture anywhere near the phrase "more academic pupils" like I did above; who complain that there can be no division between "vocational" and "academic" study, and who think that suddenly we're talking about reintroducing the 11-plus (which we never had here anyway). These people are more politicians than they are teachers, believe me.

It's weird: most of the pupils who leave for university see it as part of the process of training for a better-paid job, which is what the polytechnics and colleges used to do so well. So maybe that's Mrs Thatcher's legacy (always blame Mrs T, remember, though by jings Tony and chums don't seem keen to change things): polytechnics get to call themselves universities, and universities get to do the job of polytechnics. Orwell would be amused, at least.

In-service days and how to survive them

In-service days, or ISIS days ("In-School In-Service" apparently)... ah, the joys of days in school with no kids around.

You'd think.

Time to tidy up the room, do that filing you meant to do at the end of last session, organise the jotters and books and folders and rulers and protractors and etc etc before the weans arrive fresh-faced the next day.

You'd think.

Well, OK, let's be fair, there is a certain amount of time available for this sort of stuff. But! there are also Meetings. Lots of Meetings. And these Meetings are Important because they are organised by our managers, and because... er, hang on, I'll get back to you on this.

Don't get me wrong, we do need to meet with colleagues, whether that be in departments or as a whole school. But (whisper it quietly) do so many of them have to be so boring, I wonder? What is it about standing in front of staff that makes a teacher - a teacher, for goodness' sake - go off the scale in the bore-o-meter? I've heard people deliver the most bahookie-numbingly dull talks, about how inspiring we should be as teachers. I'd like to say they were being ironic, but no.

So, this session, let's be radical. Here's to creative meetings; to a chance to discuss teaching and learning with our chums; to actually talk about how to teach maths. It could happen... but! we need to learn how to talk the talk first. Don't say "we're going to have a chat about how best to teach factorising quadratics"; rather, say "we'll be discussing proactive ways in which we can enhance the provision for accessing key algebraic skills and concepts, with opportunities provided for both individual and group learning and discussion". And keep a straight face while you're saying it...

Be careful out there

Cruel as it may seem, tomorrow sees the start of schools returning for the new session, in some poor authorities at least. Some others get to wait until towards the end of the week, while others yet get to wait another week... but wherever we may be, we can hear the bell tolling (or maybe the bleeper bleeping, in newer schools) and we know the holidays are not long for this world.

So, wherever you are, whenever it is you're back out there, let me wish you good luck and - in the never-bettered words of Hill Street Blues - "be careful out there".

Good luck especially to all the newbies (sorry, NQTs), and to any student teachers out there - won't be long now.

Och, I'm getting all emotional...

(Those of you not in the teaching profession may, of course, wish at this point to mutter astonished comments along the lines of "six weeks holiday - what're they moaning about?!" Go on, you know you want to...)

Do the maths...

Well, the SQA results duly came out on Tuesday and I have to say there didn't seem to be too much of a fuss made about them by the media. Most went mainly for the human interest angle, following some pupil somewhere as they opened their envelope to find out their results - though I do wonder: have the schools in question "spilled the beans" in advance to the press, guaranteeing a decent set of results for the kid in question? It can be done... and if it is, what's the legality - to say nothing of the morality - of such practice?

Anyway, the only other thing they fixed on was that the pass rate for Higher English has fallen by a couple of percent (can't recall exact figures offhand, but somewhere in the 60%s). This was put down to a tricky exam, but it does strike me as odd that no-one ever takes a more considered view of such a statistic. For a start: how many people actually sat the exam? Was it more than last year, in which case in theory it's possible for more people to have passed Higher English than before? After all, many pupils do sit the exam against all professional advice, at the express wish of their parents. How many? Difficult to say, but at a rough guess, I'd say that easily two-thirds of those who failed will have been expected so to do.

I haven't seen any comment on Maths results (perhaps tomorrow's TES will oblige?), so I don't yet have a fix on how my school's results compare with elsewhere. Everyone knows that you get "good" and "bad" year groups, of course, but that doesn't count as an excuse to the powers that be, and newspapers like The Scotsman are still publishing "league tables" based on results.

Still, at least as a mathematician you can get away with waffling on about means, medians, sample sizes, confidence intervals etc etc and hope that you don't get found out. It's worked so far...

(At this point I can't help but repeat an anecdote I once heard about a Headteacher demanding to know of a particular PT why more pupils hadn't got above the average mark...!)

Search for a red pen: update

Out browsing today and came across a model of red pen I hadn't seen before, so before you could say "stationery fetish" I bought myself one. After rigorous roadtesting, here are my thoughts:

Make: Faber Castell
FC are well know in the art world but this is the first time I've seen them venture for a wider audience.

Model: Finepen 1511 Document

Price: £1.70 ish

Colour of ink: red
Quite a bright red, actually. There 's no colour stated on the pen, but I think it is what we call in the teacher trade "scarlet displeasure", perhaps with a hint of "vague encouragement".

Appearance:
Reasonably thick without being chubby; large pen lid. The body of the pen is a dark green (with a red end indicating colour), so I worry that after a heavy day I may inadvertently grab it thinking it's green rather than red. Time will tell.

Taste:
The pen top is chunky so should survive a good chewing, if you're given to such nervy behaviour. Phenol tones dominate on the palate, with an aftertaste of burnt sugar, lavender, pomegranate and wellies.

Nib width:
Not written on the pen, but a width of 0.3mm is apparently claimed. Electron micron measurement reveals a precise width of 0.324mm, which is fine by me, as the extra 0.024 always comes in handy for hefty ticking.

Bleed through:
Reasonable on a standard 0.5 square centimetre jotter, though comments written in anger ("NO!", "ugh!", "Show your WORKING!!!" etc) fared poorly - perhaps one to use with a more able class?

Performance:
The pen was road tested by first computer generating 100 basic arithmetic questions and answers, which I then marked under standard teacher conditions, ie at 11.30pm after consumption of two-thirds of a bottle of cheap red wine. The pen felt good on the ticks, with a free-flowing action, but on the crosses there was a tendency for deposit build up on the second stroke. To be fair, this is a well-recognised problem in assessment, and to the pen's credit, it did manage to write reasonably well on wine-stains.

Underwater performance:
Fair, but the jotters did get soggy.

Performance at altitude:
In order to check the pen's ability I took a long-haul flight (for greater altitude); the pen performed reasonably well but did smudge some jotters marked in economy class. To be fair, this may be because I'd dropped my mini-bread roll on the page earlier.

Works in zero gravity?
A delay to the space shuttle launch leaves this question pending. For wider space travel, however, it's worth mentioning that the pen worked well in a Type 40 Time Travel machine, though it didn't unlock any doors, so a sonic device remains a safer bet in such an environment.

Overall score: 6.83/10.
Not a classic by any means, but shows promise and may age well. Use now to end of term two.

A plea from the heart

Only about another week to go now, and it'll be back to school. So what better way to spend the time remaining, than to sort out the maths teacher's wardrobe - we wouldn't want to be out of step with fashion, would we? Time, surely, to rummage through those drawers, cupboards, wardrobes and make sure everything's OK. Failing that, a quick visit to Matalan and all shall be well.

I love how, as soon as the holidays have started, shops like BHS or M&S have "back to school" displays... not after a couple of weeks of holiday, no, but straight away. Thanks for reminding us, guys! But these displays are always of weans wearing vaguely schoolish clobber; why not push the boat out and have a display of teacher wear instead? They could display the latest Harris tweed bullet-proof number, complete with leather elbow patches. Mmm - very Jude Law.

Anyway, on to the plea from the heart. Now I'm no fashion guru, but there are quite definitely things that should not be done. So, to all maths teachers everywhere, here's my wish list for the coming session... follow this carefully and thou shalt not be too tragically unhip:

1. Stop wearing the following ties: Wallace & Gromit; the Simpsons; Doctor Who, or basically any character based paraphernalia. Do you really want the world to say "Hello, Grandad"?
2. Give up the comb-over
   Yes, early onset male pattern baldness is a terrible thing, but think Sean Connery, think Jean Luc Picard... or think Gregor Fisher in the Hamlet cigar advert. Your choice.
3. Put those pants away, madam
   Your class does not want to see them, trust me. Or if they do, they won't be getting much work done.
4. "Non-iron" shirts and trousers
   still need ironing.
5. Jeans
   don't need a crease.
6. A smart suit
   indicates a desire to join senior management. Before you know it, you could find yourself supervising lunches - is that really what you want?

Always remember that, historically, maths teachers have tended to operate within a narrow band of the fashion spectrum, unlike (say) art teachers who can wear almost anything in the name of creativity; though you could always go for the ever-popular "mad professor" look, but it's hard to pull this off because the moment you start trying to dress eccentrically, you're doomed.

One final tip: some pupils apparently have difficulty seeing red or green colours when used at the "blackboard". Theoretically this should mean that if you dress in red or green, you will become invisible to such pupils. It's worth a try, surely?

The calm before the storm

You wouldn't know it to listen, watch or read the national media, but up here in hairy Scotland the examination results are due early this coming week. On Tuesday morning, the SQA will deliver the official verdicts on pupil abilities via the traditional letter through the post. (Though apparently there are plans to send them via text messaging next session - can you imagine it: "Maths not gr8"?)

I'd say good luck to one and all, but it's all decided now. It's a strange time for a teacher, but let me also say that it's one of the real joys of the job if you are able to go back to work in the new session and meet pupils in your care who have done well. And, of course, to sympathise with those who haven't - well, provided they worked hard, that is, otherwise you can let the unspoken "if only..." hang in the air.

Of course this also means that - in the Scottish press, at least - on Tuesday we the teachers get some form of verdict delivered unto us as well, in that we'll waken up to headlines either along the "standards falling: what can be done?" or "exams get too easy: what can be done?" variety.

I wonder which one it will be?

Aaarrrrggggghhhhhh!

I'll say it again: aaaarrrrggggghhhhhh!

Yes, tonight I enjoyed that fabulous bit of entertainment, so loved by so many of us: kids misbehaving while their parents do nothing at all to stop them.

I'll spare you the gory details, but basically for the best part of an hour I had to put up with two wee brats running more or less riot in a train carriage, while parental figures (one per child) looked indulgently on.

Did I say anything? Did anyone else? Of course not. Who does, these days? We just did our best to ignore them. But since I was sitting directly opposite them, this was no mean feat, and I eventually sought refuge in the Proclaimers, played loudly through the headphones of my mp3 player. This almost worked.

A question well worth asking is, is enduring such behaviour harder for teachers than for the general public? I get frustrated for two reasons: firstly, that most teachers can get 30 kids behaving better than such parents can manage with 1; secondly, I can't help but wonder how these kids then behave when they get to school, if this is the measure of structure and control they are used to.

Or, to be blunt: I blame the parents.

OK, this is what teachers always say, so of course it can't be the whole truth, but by jings we're talking high percentages here. I've got pie charts and graphs and everything to prove it. Somewhere...

So, what's to be done?

Heck, who knows? We certainly seem to have reached a tipping point, where it's just not the done thing for anyone to comment on kids' behaviour when the parents are around (for fear of causing offence) or even when they're not (for fear of getting abuse from the kid in question, if not actual physical violence). How do you get back to the good old days, assuming they ever existed? How do you "untip"?

I will say this much: that a lot of the problems we have now are caused by the rise in the cult of the individual; the idea that there's no such thing as society. So, who am I to complain about how your kids behave? They're your kids and that's your call. Apparently. And who is the school to tell you that your kids are wee monsters? You know your kids, and they're lovely. And if you know one thing, it's that they never lie. Apparently.

I suppose, to be fair to parents (it won't last), teachers have access to a "big" picture at school - the behaviour of a class, or a year group - while they don't. We need to appreciate their point of view more, I can see that. But if we collectively decide that parenthood gives us a right to say "stuff the bigger picture", well that worries me. That way lies parents lying about their place of residence in order to get their kid into a "good" school, and then justifying their behaviour on the grounds that they're doing it for their child so that makes it OK.

We're already there, let's face it. And I don't like the look of it.

Cheery stuff, eh? Still, for any new teachers out there, remember: it's the parents fault. Failing that, blame Mrs Thatcher. Works for me.

[One final point: I have experienced contact with parents who, when faced with evidence of inappropriate behaviour on the part of their child, then go on to ask me if I have children myself. I have always refused to answer that question, and I'd suggest you do the same. It's entirely beside the point, whatever the answer.]

The campaign starts here...

OK, bear with me 'cos I haven't thought this through fully yet, but all the same, enough's enough.

I've been mulling over my earlier posting on maths in the media generally, and I think I've located a central problem. Carol Vorderman. Now don't get me wrong, there was a time when our Carol brought a general sense of happiness and well-being to many maths teachers, back when Countdown was in its infancy. But by heck that time has long passed. Now it's all 30-day detox and Carol's book of Sudoku and put them away woman and and and those flippin' loan adverts. Enough already! No wonder people aren't studying maths much anymore...

Now, leaving aside the issue of the current campaign against Ms Vorderman and these loan adverts (see the MoneySavingExpert website here - I encourage you to sign the petition, seriously), I'm here to say right, that's it, CV's place as the unofficial mathsy media mascot type person is over. Finito. I'm calling it. Maybe it was fun while it lasted, but her reign is over.

So, the next question has to be, who takes her place? Who should the public think of, when they think "maths", now that Johnny Ball is enjoying his retirement? Who can we turn to? Call on? Who will save us?

Here's where I reckon we have to think outside the box. Oh yeah, we could look around and find someone who's done a bit of maths - the Irish comedian Dara O'Briain, for example, as I read today - and try to get them to do maths work in the media, but that's what they're expecting us to do. It would be too easy. Why not come at the problem from another angle entirely - a sort of proof by contradiction, if you will?

Yes folks, the campaign starts here. My nomination for the UK's maths spokesperson is, has got to be...

... Boris Johnson.

Admit it, you're intrigued.

OK, I'll grant there may be arguments offered against this idea, so let's deal with them, one by one.

1. He's busy
   Well, yes, to an extent, but we all know Boris is game for a laugh. Surely he'd fit us in somewhere.
   
2. He's a bit of a buffoon
   Well, again, yes, but don't we say as maths teachers that we need to encourage pupils to believe that anyone can do the subject? If we can show 'em old BJ differentiating y with respect to x... well, can you imagine?
3. He has no idea whatsoever about maths in any way shape or form
   OK, so we're getting to the meat of the argument. The crux of the matter. Or, as Boris would put it, the ummmm ahh yes well ah you see the ummm crux yes indeed the crux, crux! of the emmmmmmmmmmmmmm... matter.
   But, be honest, since when has a politician ever let a lack of knowledge in a subject deter them from speaking as if they were an absolute expert? Aha! Y'see? I have absolute confidence that Boris can sound as knowledgeable about maths as he does about anything else.
4. He's Boris Johnson
   I'll get back to you on that one.

For me, it has to be Boris. It's that rare thing... that moment of clarity when you realise you've finally arrived at a proof, and that proof is so simple, so obvious, that you wonder how you could miss it. And answer me this: would you go for a loan company on his recommendation?