Eleven damn lies we tell our kids (about maths) - A bunch of opinions by David C, Aug 2011
1.You need to be good atmaths to get through life
Actually no. Most people never use algebra after they leave school. Thatincludes all the skills based on algebra, such as calculus, trigonometry,probability theory, complex numbers and matrices. Most people never needto interpret graphs, measure areas and volumes, calculate or interpretstatistics or bisect an angle. The trouble is that some of us – a very smallpercentage – do. And of those who do use these abstract skills, each ofthem uses it sparingly and probably uses just one or two of those skills andnot the whole menagerie.
The only maths you really, really, really need is the art of counting.Anything more sophisticated than that can be handed over to other peopleto do for you: accountants, social workers, probation officers, bankers andthe like. Counting is quite likely to be the only kind of maths you arerequired by circumstances to do frequently and quickly. After all, you’dfeel pretty silly if you had to ask a shop assistant to put eight apples into abag for you.
Of course, if you want to manage your affairs personally, you’ll need to beable to add, subtract, multiply (at least just a little bit) and just maybedivide. These form about the first half of arithmetic – the art of crunchingnumbers. The other half is fractions (which you’ll never need except for anunderstanding of halves and quarters), decimal numbers (which you’llnever need unless you count money or millimetres) and percentages (whichyou’ll never need unless you want to invest).Now, managing your own affairs is only an option. Some people prefer tomarry someone who can manage their affairs for them, or go on a benefit inwhich case someone else does the work for them.
So far I’ve been talking a little tongue-in-cheek, but from here on I’mserious. Most people learn the higher mathematical skills at high school,understand them to varying degrees and then promptly forget them whenthey leave school, only to confront them again twenty years later whentheir children go to high school. And when each generation who goesthrough this process asks the generation that came before, ‚Why do I needto learn this stuff?‛, we have no honest answers. We shake our heads andsay we don’t know, or we lie.Why are we doing this??? This is ridiculous and we all know it!Worldwide, the mistake is repeated generation after generation. Whose ideawas this? And why can’t we stop doing it?
The answer may be that underneath all our doubts about the utility ofmaths, we have in our hearts a secret desire to see our kids do better thanus. No-one knows which of our kids will be amongst the one percent ofalgebra-learners will go on to be algebra-users at university, and so wehappily push them into these classes to get a taste of it. That much is good,but from there, we go wrong, because none of us has the courage to marchinto an algebra class after our kids have had a year-long taste of it and say‚She’s tried it, doesn’t like it, can’t understand it, and wants to stop.‛ That,basically is what we need to be able to do.
But then if we allow that kind of thing, we have to be absolutely certainthat our decision to quit algebra at an early stage is the right one. I’m partof a cohort of kids from the 1970s who didn’t particularly like algebra atschool but didn’t dare to say so out loud, and so knuckled down and learntit anyway, only to find a few years later that I actually liked it. You see,learning algebra is like learning to play the piano: you have to put in a lotof dreary practice in the early days to become good at it, and you never seethe point of all that dreary learning until you’ve mastered a fair bit of it.
Now as an adult who has studied algebra for many years, I can use it toplay around with ideas. I can design a spaceship with it and fly to the Moonon a computer program I’ve designed myself. What I find particularlyremarkable is that this is only possible because I have a desktop computer,and the people who gave me this remarkable skill with algebra did so at atime when nobody’d even thought of putting computers into the home. Soalthough I can understand why I like algebra, I can’t fathom why myteachers liked it.
Whatever generation you’re born into, no-one needs algebra but some of us– a very small number of us – actually like algebra and use it in ourhobbies. And I’ve even heard that there are people out there, somewhere,who actually use algebra once in a while in their jobs! They must be as rareas hen’s teeth, but just imagine the kinds of things they must be doing: 3Dgraphics for Hollywood movies? Or maybe writing the code that enables acomputer to do 3D graphics. Or maybe designing a computer that can runthe code for 3D graphics. Or maybe designing a flight simulator that usesthat 3D graphics to teach pilots how to fly. Or maybe it’s a simulator forputting men on the Moon. Or maybe it’s the formulae used to land a roboton Mars. Or maybe “. Do I need to go on? It’s all fantastic, world-changing work.
We have to stop telling our kids that maths is necessary. Instead we have topresent it as an option that – for some – will be fun and possibly useful. Todo that, we have to stop believing ourselves that algebra is somehownecessary for our kids. We have to make dropping out of a high schoolmaths class permissible and not a sign that people who do so are stupid.There are plenty of people in high places who can’t even multiply andnobody calls them stupid.To balance all that, we have to make the learning of algebra a veryattractive option so that those who are at least a little curious about it willtry it. And from there, we - parents, teachers and students alike - will beable to identify those who really enjoy it and who want to go on to learn thedeeper levels of the subject.
I’m using the word ‘enjoy’ in place of the word ‘capable’ because I believethat in most cases (if not all cases) the one precedes the other. If you enjoyhigher maths, it will give you the stamina to go on to get good at it, in thesame way that a student who enjoys playing chopsticks on the piano willgo on to learn trickier stuff.Believe it or not, kids who enjoy algebra actually exist. They’re very rareand their experience of algebra is often hampered by the need to sitamongst kids who hate it. But imagine if we could find these kids early inlife and focus our teaching effort on them. Imagine where their lives couldgo. And imagine how much happier their teachers would be.
To make algebra an attractive option, we have to bring in people fromindustry who use it as part of their jobs, who can really show kids thepower that algebra provides in certain very specialised careers.Finally, we need to associate algebra very conspicuously with theprofessions it serves: medicine, engineering, astronomy and economics (toname a few). Either we start teaching these professional courses a littleearlier in life, or entirely postpone the teaching of algebra to university.Only then will it rightly be seen for what it is: the language upon which allthose professional courses depend. It would NOT be seen as the last relic ofa dying brigade of subjects forced onto teenagers in less enlightened times,these being Latin, French, rhetoric, philosophy and the religious practicesof the day.
Nope. To see clear evidence for that, you only have to look at me and whatI do. I’m a maths teacher and I get peanuts compared to the rich folks incareers that never use more than the basic skill of counting.I met a bank CEO once who hired me to teach his daughter maths becausehe described himself as ‚no good with numbers.‛ Professional musicians,entertainers, tour guides, sports coaches, restaurateurs, “ c’mon, you don’tneed me to list these people because you know plenty of them yourself.They MAKE MORE MONEY than your typical maths teacher or scientist.They’re generally happier too because they work closely with people. Theyentertain people. People seek them out. That does not happen to scientistsvery often. Scientists spend a lot of time in solitude because that’s wherethey need to be to fit an equation to a chunk of data.
Consider the aeroplane test:Who would you rather sit next to for a ten-hour flight: a musician, amagician, a pro sportsman, an actor or a maths teacher? I don’t supposemany mathematicians get invited to cocktail parties except for thoseorganised by other mathematicians, and nobody goes to these anywaybecause even mathematicians can’t stand the company of mathematicians!Have you heard all the jokes about scientists on aeroplanes? A passengersits next to a cosmologist and starts asking questions about skin-careproducts. The cosmologist says, ‚No no, I do astronomy.‛ So the passengerthen asks for advice on her horoscope and the astronomer says ‚No, no, I’ma kind of physicist‛. So the passenger then starts talking about her backpain and wants to know what she should do about it. Now quiteexasperated, the physicist says ‚No, no, I’m a scientist,‛ after which thepassenger turns away and starts to read the newspaper.
I laugh at that too. Yet I’ve seen the TED talks online and I think scientistsare just about the most interesting people on Earth because of what theyproduce. These folks tinker away in obscurity for years and some of them –just some – create machines or theories or just insights that decide what thefuture will look like. Like painters, some of them get amply rewarded forwhat they do and others die before their work is recognised.Even so, it takes an awful lot of drudge learning to get to the point wherethey can do magic with mathematical modeling. And even while they’re atthe pinnacle of their careers, they’re still spending an awful lot of timelooking at mathematical procedures that 99 percent of the people in theirwhanau do not understand or even want to discuss at the dinner table.That’s the nature of science. It’s fascinating for some, but it’s certainly notteenage idol stuff and it’s certainly not well-paid. It just has the potential tomake the world a better place, that’s all.
One of the saddest things that’s happened to maths is that it has beenorganised into some kind of scholastic ladder that leads to jobs that don’teven use the material that is taught. A generation ago, school cert was allthat was needed to show you were smarter than half the population andtherefore smart enough to be trained in the basics of any profession, be itmathematical or not. Maths was used simply as an intelligence test. Today,the benchmark has moved up the scale to approximately the level of a basicuniversity degree. You need to have a degree to show that you’re smart,and most degrees require you to do at least stage 1 maths; differentialequations and the like. Pretty soon it will be PhD’s that separate the smartfrom the not-so-smart.
The problem is that the school qualification system is set up to identify andsupport those who are interested in becoming researchers. A degree inchemistry prepares a student for a life in academia, in which case she needsa background in calculus. But the rest of the world has piggybacked on thisacademic filtration system and used it simply as an intelligence test, so thatwe arrive at the bizarre situation where a person needs to know how to findthe determinant of a matrix in order to get a job as a secretary!
As a maths tutor, I was often asked ‚Do you teach NCEA level 2?‛ towhich I would usually reply ‚No, I teach maths.‛ And then if they askedme what kind of maths, I would list the options as probability, geometry,trigonometry, calculus, matrices, complex numbers and so on. The usualresponse from that would be ‚Is any of that in NCEA level 2?‛ and I’d say‚Some of it.‛ The conversation would then become a matter of determiningwhich parts of what I teach were in that exam so that we could decidewhether to go ahead with a course of tuition or not.
I get the feeling sometimes that if it were legal for me to dish out theanswers to this year’s exam questions, parents would actually ask me forthem, and pay me a good salary too.Of course I’m not doing myself any favours as a businessmen by being soelusive about what I teach, but I’m trying to make the point that maths isnot measured in exams, or at least shouldn’t be. Maths should be measuredin topic areas that are specific to a particular career path or even learnedsimply for the pleasure of it.
In a course on trigonometry, for example, I could go into the history of thesubject: who first invented it, how for a few years it was the secret maths ofastronomers, how later it came to be used in architecture, and then after thatit was used by navigators on ships to reach the New World. I could talkabout how it was once used as a way of turning multiplication problemsinto addition problems, and how today it is used in flight simulators andonscreen war-games. I would teach everything associated with the subjectfor as long (or as short) a time as the students were interested. It would notbe the bare bones formulae required for getting ‘achieved’ in this year’send-of-year exam.
The next saddest thing I often hear about maths is that ‚My daughter justwants to learn enough to get through the exam. Don’t teach her toomuch“‛ That to me is a very, very sad thing, and shows how out-of-touchthe teaching of maths has become as a preparation for adulthood. Imagineif we said the same things about learning to drive a car: ‚I want her to justlearn the basics. Don’t make her too good a driver“‛ Or maybe we couldsay the same thing about literacy: ‚I just want to be able to read road signs.Don’t get me reading newspapers“‛
It sounds ridiculous in those comparisons, but we have allowed this sort ofthinking to be sensible because maths is just about the only subject you willlearn at school that has absolutely no worldly application except for the fewpeople who go on into specialised professions. By dragging kids kickingand screaming through the process of learning algebra, all we are doing iscreating an unhappy industry for frustrated teachers and makingmathematics the jackass of all school subjects.
Watch any show, any movie that features schoolteachers. Usually they’retwo-dimensional characters who wear funny clothes and talk in high-pitchvoices and don’t seem to have a sense of humour. Look a little moreclosely. If we have movies that distinguish the good teachers from thelousy ones, the good ones are always the arts teachers, the music teachers,the history and social science teachers, maybe even the English teachers.HEL-LO-O!! Who’s missing in this picture?! “.
“..Where are the really cool, funny, inspirational math teachers whochanged our perspective forever when we were growing up? Somebodytaught calculus to Neil Armstrong. I know this because like all otherastronauts of the time, he has a degree in aeronautical engineering. He gotthat way back before anyone talked to him about going to the Moon. NASAdoesn’t go up to people on the street and say ‚We’d like to send you to theMoon,‛ and hear them say in response ‚Okay. I’d better go and learn somecalculus then.‛ Somehow these guys who flew spacecraft to the Moon wereinspired by maths teachers and physics teachers enough to study it topostgraduate level. That couldn’t have been an accident.
The funny thing is that in their private lives, math teachers are about assmart and funny and inspirational and cool as any other kind of teacher,which is about as funny and cool and smart and inspirational as most otherpeople in society. But when a math teacher is at the board in front of aroomful of bored kids, teaching material he knows no-one wants to learn,he (and it’s usually a male teacher) is about as unhappy as any human beingcan ever get. It needn’t be this way.
I’ve been lucky in my teaching life to catch kids at a very young age andteach them something that they haven’t yet learned to dislike. This is thebest kind of teaching. Take a ten year old, and show him how to add a longlist of numbers in a few simple steps. Straight away you can see in his facethe joy of mastery, the pleasure that comes from having increased one’sown mathematical power. These kids go home and show their mums anddads, they even show their school teachers and classmates. They get to beknown as ‘good at maths’.
At the other extreme, I come across kids well after the school system hasruined them. They’ve learned to hate maths for all the obvious reasons:(a) It’s forced upon them(b) it was taught to them while they were engaged in a critically important conversation at the back of the room(c) they missed a few critical lessons when they were off school on a basketball trip, and never caught up(d) it’s taught too fast because the teacher has to finish the syllabus before the end of year(e) it has no connection with the real world so kids can’t remember it two weeks after they mastered it(f) the teacher himself doesn’t understand the subject that well and therefore can’t field questions from the class.It’s the same old thing, generation after generation.
So a subset of bored kids grow up to become the next cohort of schoolteachers. They got into teaching because they had an inspirational artteacher or maybe they’re good at music or story-writing, or maybe theysimply like the idea of igniting chemicals in a laboratory. Along the waythey are told that they will have to teach a maths course from time to time.So they brush up on their calculus, just enough to get them through thethree weeks they are required to teach it. Then they forget it for a wholeyear and have to brush up again just in time for the next wretched season.
I met one of these fellows a few years ago. He was a trainee teacher whohad been told that before the end of the year he would be going ‘onassignment’ to a primary school and that part of his duties would beteaching fractions. So, could I teach him fractions, he asked. He’d of courseforgotten how to do fractions half a lifetime ago because he’d never hadany reason to use them as an adult. He said that he would be required toteach this subject to kids for a whole term, and then he asked me for one - Irepeat one – lesson in fractions so he would know how to teach it. I toldhim that this was not going to be enough and finally persuaded him to signup for six lessons to cover the basics of adding fractions, subtractingfractions, multiplying fractions, dividing fractions, turning fractions intodecimals and percentages and getting them back from decimals andpercentages. Like anything else, mastery at a subject requires over-learningit, and you have to be a master if you are going to teach somebody else. Hesigned up for six classes, then dropped out after two. I don’t know if heever got to be a schoolteacher.
In the same way that kids are forced to learn maths at school, schoolteachers are also forced to teach maths, It is the common complaint ofschool-teaching; ‚Maths is not my forte‛, ‚I hate teaching something thatthe kids are never going to use‛, ‚I hate being asked what it’s used for‛, ‚Ihate struggling for new ways to make it interesting.‛ On that last point I’veoften seen teachers try to jazz up a maths class by introducing music orgames or silly mnemonics, or switching on a computer game whichfeatures aliens shooting each other down with numbers. Some of theseteachers don’t trust themselves to be able to remember how to do mathsand end up simply handing out pieces of paper, and measuring the successof the lesson by counting how many of these pages came back withanything written on them.
Whereas I can see the value of having generalist teachers in schools, I’dalso like to see specialist teachers in schools to handle the moresophisticated and narrow skills, such as music or languages or maths. Tohave a person who hates maths teaching maths is an absurdity. I see Yodatalking to Luke Skywalker all of a sudden: ‚Do or do not. There is no try.‛Perhaps he meant it in a different way, but it seems to me that if you don’tfind something really easy and fun to do, then no amount of ‘trying’ willever turn you into a good teacher for that particular subject.
The trouble is that enthusiastic math teachers are few and hard to find.Most folks who are good at maths are in industry somewhere doing betterthings than teaching. Well, we have to find a way to turn this around. Wehave to restructure our society so that all those wonderful meteorologistsand architects and carpenters can come into school for a couple of hours aweek each, and teach their particular specialty in maths. AND, we shouldhope that this would be the fun part of their week. It’s a day away from theoffice, the boss, the loser in the cubicle next door. It’s a day with cute kidswho ask intelligent questions.
I used to work in one of these office environments. If my boss ever talkedto me at all, it was usually to inquire why something was behind scheduleand he was usually mad. When I became a tutor, most of my conversationswere with really happy parents and genuinely likeable kids. That’s what gotme hooked on teaching, and has kept me (loosely) in the profession for halfof my life.
When I was a kid, these pieces of paper were bound in a book. These days,the pieces of paper seem to be downloads from the internet. The font isalways too small for kids to want to read, and the words are jammedtogether as tightly as possible to cut down on the number of pages. Theworking space, if it’s there at all, is about as big as a postage stamp. I hearstories about teachers who hand these awful things out at the beginning of aclass and then go sit down for an hour. Now I know that these are not ALLmath teachers and probably aren’t even half of the math-teaching fraternity,but the fact that they are there at all is a bad sign. Maths is increasinglyabout pieces of paper!
NO! Maths is about people. I have written articles on how to do variousthings in maths. Nobody reads them. I’ve put talking lessons on theinternet. Nobody uses them. It’s not that my material is bad, it’s that peopleinherently want to learn from people.Years ago I used to use interactive software as part of my repertoire.Though I’d heard that kids love playing learning games, I always found thatas soon as I walked away from the computer, the kid lost interest.Obviously I was more fun than the computer. I could flatter myself intobelieving that I am fun to be around, but I think the same thing wouldhappen even if I was having a real bad day and was as dull as ditchwater.When a kid is learning anything, she wants to be able to ask questions asthey come up because no two kids learn precisely the same way. Nocomputer program can anticipate all the things a kid might want to ask.
But more importantly, kids like to chat. Hasn’t anyone noticed that? MyGod, we hear kids chattering in class all the time. Why can’t we see thatthat is a way of holding a kid’s attention!? I chat with kids while they learn,and they chat with me. They tell me about their dogs and sisters and idiotclassmates. They chat so much they can’t even hear the advice that is beingrecited to them from the recorded voice on the computer. They chat to meabout the funny characters on the computer screen and we make a jokeabout why it was drawn the way it was. And sometimes – wow, sometimes– they even chat about what they are learning. It’s that simple. Let kidshave a teacher to chat with while they learn and they will learn.
Kids are much better than us at multi-tasking, mainly because theirattention to any one thing is never all that deep. They need to occupy thedormant part of their head with conversation. Otherwise they’ll fill it withsomething else, like fidgeting with a ruler and poking the kid next to them.If you chat with a kid, you can keep the rest of him engaged in the learningactivity, whether it’s a paper exercise or a computer game.
Islam has it pretty well nailed. One of its teachings is that there are naturalroles for people to fill, and teaching is one of them. If that role is leftempty, then the role will be filled by the Devil, which means to say thatsomething will go wrong. Maybe there will be mischief or maybe she willlearn the wrong thing. It needn’t be conscious or willful mischief. Maybethe kid is spending all her time learning how to make a particular characteron the computer screen squawk rather than learning anything mathematical;in which case the sight of kids hunched over their keyboards is not anindicator of their enthusiasm for maths. Sometimes all you need to do is sitnext to them and they feel your presence. They sense that you care and areavailable for help. And it could happen that they never need to ask you aquestion, but your presence focuses them on the task at hand.
I knew a teacher once who thought that teaching was about handing outpieces of paper with notes and pictures on them, and asking the kids to readthese notes out loud. Following that the teacher would try to get the kids todiscuss the material and then get them to write ‚in their own words‛ theanswers to some questions and then hand the sheets back in so that shecould mark them. Now I have put the phrase ‚in their own words‛ betweenquotation marks because inevitably this would become ‚the teacher told thekids what to write‛. The whole lesson, each time I observed her in action,required clean pieces of paper going out and bescribbled pieces of papercoming back in.
Now of course the teacher has to have material to browse through later todetermine who is actually learning and who is not. That is a consequence ofhaving as many as thirty kids in one place, supposedly learning a topic inmaths simultaneously for the first time. That’s the nature of the job. But weall know that being busy is not necessarily being productive. Sometimes I’dask that teacher how the kids were progressing and she’d tell me all thethings SHE was doing. That’s not the same thing. I know she’s busy and Iknow the kids are busy, but I want to know if they are getting anyknowledge from all that busy-ness. Somehow that question wouldn’tregister and I’d get a rephrased description of how busy she was.
Now I don’t want to get drawn into a big debate about what constitutesteaching and learning, but I do want to say that when I teach maths – and Ihave the freedom to teach what I want when I want and to whom I want –the only pieces of paper I use are completely blank. And when I amfinished teaching someone something, I don’t collect the papers back in,nor do I mark them. If I think the kid has mastered something – shall wesay, a part of the times table – I will get her to recite those numbers to herenthusiastic parent. I don’t need assessment in the formal sense because Ihave the privilege of working with kids one-on-one or in very tiny, selectgroups. I can see in an instant whether a student has made any progress ornot.
Einstein once wrote ‘Real education is what is left over once everythingelse you learned at school has been forgotten’ (or words to that effect). Iwant the knowledge to be in the kid’s head, not on some piece of paper.Our society has gone bonkers over pieces of paper. Most people, when theycomplain along these lines, complain about too much assessment in school.But that’s not the same thing. Assessment can be achieved verbally andbehaviourally. And if the kid allows herself to forget what you have taughther over a period of time, is that really a sin? Most learning, in my ownpersonal experience, has involved repeated visits to a particular topic,forgetting two bits for every three that I have absorbed, staggering up theprogress ladder until a time comes when I can say I really know the subjectwell.
So wouldn’t it be nice if we could admit that by teaching a kid today, weare sometimes planting the seeds for something that will not produce fruituntil much later in that kid’s life. Furthermore, can we admit that it’s okayto learn something today to forget it tomorrow, to relearn it next week?And if we can do all that, what meaning can there be to a written test issuedon any particular day? I had maths exams in my childhood that were sopoorly answered I should have been weeded out by natural selection andgone the way of the monkeys. In theory I should never have ended up as amaths teacher. And yet through the chaos of life, here I am!
Should exams ever have been used to separate ‘the men from the boys’? Inearlier days, maybe. But not now, not in our generation. Our classes aremuch smaller, and we know that not everyone wants to be a universityprofessor. Our kids have diverse natures and ambitions. Let the proof oftheir learning be seen out there in the field.If we want to see how good kids are at basic arithmetic, let’s see if theystart using it in their daily lives. Let’s see if kids can compare prices in thesupermarket while pushing a trolley down the aisle; more important, let’ssee if they are naturally inclined to do this without any prompting from us.And finally, let’s see if they choose to use arithmetic even when no-one iswatching.
As for trigonometry, let’s see how well they can measure the distances andheights of buildings out there on the horizon, or maybe design a bridgeacross a river, maybe even build it. And let’s see them do it without use ofany calculators or trig tables, because if you can estimate these sines andcosines then you really do understand the subject. Perhaps the ultimate testof real learning is not the A on a test paper but the tree hut in the back yard.But I can go one better than that; I think the real test of learning is whethera student will care enough about the subject to go off and teach it tosomeone else.
Some years ago I came across a bunch of guys who were flying remotecontrol model aeroplanes from a steep hillside. I was so fascinated by thesight of these things that I hiked up the hill and sat amongst them andeventually began talking with them. One of them made the observation,‚The funny thing is that we fly these planes all the time, and yet we neverknow how high they actually fly,‛ which to me was like a red rag to a bull.Before I knew it I was showing a couple of these guys how to squintbetween your fingers at a plane in flight and use some really basicgeometry to estimate the height of the plane, a number which could bededuced in less than a minute. These guys looked at me as if I was fromMars, but estimating distances and heights is something I do just forentertainment. I used to sit on a bench on the coast and try to estimate,using this same procedure, how far out to sea the big ships were. Maybesome day my life will depend upon it. Probably not. The point is, it’s mathsand I’m using it outside of school.
For a lot of people, maths begins and ends at school. My high schoolFrench was like that. I’ve never spoken in French to anybody, and Iremember so little of what I learnt now that it’s not worth talking about.Had I gone to France, I’d have had to brush up using some phrase bookperhaps. And the funny thing is that all those tired old neurons would’vestarted firing again, and I’d very quickly be able to speak the subject aswell (ahem) as I did when I was fifteen.
Too often what we teach at school is streamlined for passing an exam. Isuspect that very few math teachers ever talk about maths as anything otherthan a way to get brownie points in an exam. At the same time that studentsare trying to spot what particular types of question will be in the exam,teachers too are trying to spot these same questions and gear their lessonsto focus especially on these skills. And if that is insufficient, parents will goout into the community to find math tutors who will use their expertise toprep kids for what they think will be in the exam. IT’S ALL ABOUT THEEXAM, FOLKS!
This sort of thinking is not going to help us. All we are doing is training ourkids in the mystical art of question-spotting. My wife is brilliant at this. Sheslept through many of her engineering classes at university and passed thecourse anyway because she knew how to spot questions, memorise keyformulae and regurgitate answers in a particular way. She and I bothstudied mechanics, and while she passed it in one year and I failed and hadto repeat it, I am the one who actually remembers those formulae andknows how to use them. People who are especially good at memorisingcrap for exams are also very good at forgetting it so that their minds areclear for the next exam.
My wife has a degree in engineering that she never used. I never finishedmy engineering degree. I was taking too long trying to see how formulaewould work in different situations. I was also trying to comprehend how ithappened that people in pre-computer times could come up with suchimmensely powerful procedures for designing buildings. I was using all thistime trying to connect engineering with everything else that I knew aboutthe world, so that I was never getting work done on time and washed out ofthe course. Strangely, half a lifetime later, I still hang around libraries andread engineering books. Why? Because the subject is so interesting! WouldI want to be an engineer? Absolutely not! But I’d love to teach the stuff,and I could probably write a really good book or two on the subject.
Isaac Asimov taught himself astronomy. Never in his life did he study thesubject in a classroom. He did apply once to enroll in an astronomy coursebut was told by the course coordinator that he lacked the pre-requisites andtherefore would not be admitted to the course. During this meeting, Asimovhad the pleasure of discovering that the recommended textbook for thecourse was a book he had written some years earlier, and he pointed thisfact out to the coordinator, who blushed and conceded that Asimov wascertainly knowledgeable enough to enroll on the course after all. But bythat stage it was clear to both men that Asimov had nothing to gain fromjoining the class since he had already written the material that would betaught.
Stories like these pamper our egos a little and give us the feeling that wedon’t really need to go to school and don’t really need to take much noticeof people who do. But that’s wrong. People like Asimov are rare; most ofus need a bit of a push to learn anything substantially new, and school isgenerally the place in which to do it. It’s just that schools have become soswept up in the maelstrom of paper-based assessment that there is littlescope left for anything else. Exams and assignments drive us to get throughthe material as quickly as we can. In a sense, we aren’t exploring a subjectso much as we are driving through it on a learning expressway. The examsare the tollgates that we pass through. Little else is seen.
When I set up as a maths tutor many years after getting out of school, Ifound to my dismay that most parents felt that tuition should only happenduring the weeks that their kids were at school. This surprised me. Schooloccupies forty weeks of the year, which means there are 12 weeks in whichkids can come for tuition in the daytime instead of at night or directly afterschool. There was even time on the weekends. But who in their right mindwants to learn maths on Saturday or Sunday? Somehow kids do sports onweekends and schoolwork during the week. And holidays, they dosomething else.This seems so silly. Imagine a toddler learning how to talk for 40 weeks outof every 52. Imagine him losing interest in speech for two days out of everyseven. Imagine him only interested in talking between the hours of 9:00amand 3:00pm. I just don’t get it. ‚Do or do not. There is no try.‛
In more recent times I have volunteered my expertise in a primary schoolthat claims with fair pride to be a school that challenges old assumptionsand wants to explore new ways of learning. As much as it talks to parentsabout learning occurring anywhere and at any time, it is really, really hardto get parents interested in having their kids learn maths in front of thefireplace at Dave’s house on a weekend, even if there is no money involvedin the process. In a sense I’m offering to take one or two hours out of theMonday-to-Friday learning cycle and transplant them onto the weekend, inan environment that is safe, peaceful and pleasant. In ten years only threekids from this school have ever been to my place for lessons.
This 9-to-3 mindset is not exclusively in the minds of the parents. Teacherstoo are very territorial and do not like to see slices of their teaching weekhacked into by an unorthodox outsider. Once faced with the task of tryingto teach four kids in a very noisy and distracting school, I took these kidsout to the car-park and taught them maths in my old camper-van. The kidssat at the back, my whiteboard was in the boot and I stood in the drivewaybehind the van with the back door of the van raised up like a sunshade. Thekids had a great time. “.
“.And while this breaks just about every rule in the school’s guidelines forsafe teaching practice, it worked and it was fun. It could not have happenedin most schools and was only possible in this particular school because ofthe foundation of trust that existed between all the parties involved. One ofthose four boys was my son and the other three were friends that hadvisited our place more times than I can remember, for birthday parties andthe like. I knew their parents, the teachers, the school principal. Everyoneunderstood that for Dave there is no partition between working life and therest of life – as I would like much more of society to be.
Now I’m going to have a go at those people who write maths textbooks,especially the books meant for university students. These are the worstwritten books on the planet and are that bad because the author has tried sohard to make it ‘good’. The professorial version of ‘good’ means that thelogic that joins one page to another is precise and correct. The booktherefore becomes a logical ladder that proceeds from what is universallyknown to .. well, wherever the author decides to take us. There will beabout ten chapters in the book, and each chapter consists of thirty pages ofvery densely-packed theorems and lemmas and corollaries.
The problem with this style of writing is that the only people who will havethe patience to read the book in the intended sequence are people whoalready know the subject well are and reading the book only to fill in a fewgaps in their understanding. The rest of us browse back and forth, skippingsome sections entirely and re-reading some sections many times. The bookmay have been written in a logical manner, but very few of us are reading itthat way.
Learning is not a logical process. We come across something that takes ourfancy and we look a little closer at it. This means we want to see what itdoes and where it is useful. We respond well to humorous and friendlyexplanations as long as they are informative. We don’t necessarily need toknow where the material came from, or how it works, but we know that ifwe stick around to the very end, someone will show us all that stuff too.So it is with teaching. It would be neat to have a teacher show us somethingfabulous and convenient and clever about maths, and then later – for thosewho are keen – show how the material can be derived from simpler stuff.The best math teachers I ever had taught in this fashion. The rest werelogical.
Logic has been pestering maths since at least the time of Euclid but gaineda stranglehold on it in the nineteenth century when it was found that somemathematical processes didn’t work as well as they should in allcircumstances. Someone needed to go back and verify the assumptions onwhich a mathematical technique worked, and in so doing plug up the holesthat people sometimes fell through when using the technique improperly.That’s fine for writing a PhD thesis but no good at all for writing aninspiring maths book and worse still for teaching kids in the classroom. Ifany part of maths-learning is going to turn a student’s curiosity off, it’salmost always a very logical teacher.
The better we get at a subject, the worse we get. We may understand thelogical connections that hold a mathematical procedure together, but welose the ability to present it in ways that the students can follow, which is tosay that we lose the ability to teach it illogically.Consider Calculus. This is a fascinating area of maths. It’s the language ofphysics and astronomy. It’s the procedure that makes it possible to workout the flow of air around an aircraft’s wings, or to determine which part ofan overloaded bridge will crumble first. Calculus told Isaac Newton thatgravity drove the Moon around the sun. It also told Einstein that time slowsthe faster you go. Usually the first taste a kid ever gets of calculus isderiving it from first principles. She learns how to differentiate polynomialsand trig functions, and then goes on to integration. If the teacher is good, hewill explain why people do this, but some teachers are not so good. And sothe student sees calculus as bizarre and pointless.
Imagine if we taught carpentry in this manner. Let’s say we gave a kid ahammer and told him how it was made and what it was made from. Let’ssay he spent six months swinging this hammer back and forth in the airbefore we introduced him to a nail. Hammers were NCEA level 1 and nailswere NCEA level 2. We put them together at Level 3.We would do well to think more like children than like professors when wesit down to create a lesson plan.
Somewhere, somehow, in recent times, rote learning has been given a verybad and undeserved reputation. Somehow rote learning is stigmatised assome sort of punishment, as if kids were being nailed to the walls andforced to recite mindless facts and figures to earn their release. Poppycock!Kids memorise all the time. They memorise their Mums and Dads, theymemorise their toys and telephone numbers. They memorise the way toschool and they memorise their favourite TV shows. We can’t stop kidsfrom memorising; they’re programmed by millions of years of evolution tocopy what they see. At a young age, when the cognitive centres of the brainare not fully developed, memorisation by mere mimicry is in some casesthe only way they learn. Understanding generally comes a little while later.
They don’t understand why everyone has a seven-digit telephone number,but they memorise these numbers anyway. They don’t understand whyMum is Mum but they memorise her anyway. They memorise what theywant, and believe it or not some kids actually want to memorise the times-table. So aren’t we obstructing them by telling them they have tounderstand it before they memorise it?I’ve been teaching kids for years and never saw a kid back away from thechallenge of memorising a string of numbers. They love it! If they canmemorise five numbers on the times table in ten minutes, they feel proudenough to try the next five. The point is to desist from telling them thatmemorising numbers is bad, and similarly desist from forcing them tounderstand things when they’re not ready.
Speaking now as a person who uses maths quite a lot, I can tell you thatperformance in maths is a mixture of analysis and memory. The moreconnections you memorise, the less you have to analyse.For example, when you multiply 7 by 8, are you using memory or analysis?If you are using memory, the number 56 flashes into your mind withouteffort. It just does. If on the other hand you are counting on your fingers ordoing some dumb cluck strategy that requires you to multiply 8 by 5 andthen add on another couple of 8s, then you are using a lot of analysis andthe process is tiresome. The more you memorise, the less work you have todo.
Try finding two-thirds of 57. If you just happen to know that 57 is three19s, then the task is easy. Did I memorise that? I guess I did, but I didn’t doit consciously. Somehow I must have seen that connection enough times inmy life that it stuck. The more arithmetic you do, the more you(unconsciously) memorise, which means you will have less work to do thenext time you see these numbers.Just imagine what a student can do if he memorises some basic decimalnumbers. Imagine what YOU can do if you memorise some basic decimalnumbers.
Let me test you out. I’ll bet you never knew that three 17s make 51. Whyshould you know this? No reason at all, but I’ll bet you’re going toremember it for the next few seconds, simply because I drew your attentionto it. If I was to write 51 down somewhere later on in this document, a weelight would flick on and go ‚oh! That’s 3 x 17‛. HAH! Got you again!You’ve seen it twice now so you really will remember it the next time yousee it. See, I don’t actually have to tell the truth about whether or not youwill remember that 3 x 17 = 51, I just have to draw your attention to it afew times, and you will remember it. You can’t stop yourself. Even if I sayDON’T MEMORISE THAT 3 x 17 = 51, you will go ahead and memoriseit anyway.
Rote learning is as easy as that. You don’t understand why multiplicationworks when you first meet it, no more than you understand how a carworks when you first drive it. No kid in her first decade of life wants tounderstand how things work, she just wants to be able to do what thegrown-ups are doing, and if enough grown-ups are multiplying numbers,she will see that as something worth learning.Kids are social creatures. They learn in the company of people they like.They copy them, they feed motivation from them. We’re fools if we don’tuse this as a pathway to learning.
I put this one in here to get folks to realise that teachers are everywhere onthis planet and not all of them stand in classrooms and not all of them arecertified as teachers. It’s a topic I sometimes hear about on radio debates:does a teacher need to be trained as a teacher to be a good teacher? And canit happen that an untrained teacher can teach as well if not better than acertified teacher?I honestly believe that there are people in this world who have a natural,intuitive feel for teaching. Teaching is a calling, a yearning that somepeople have that motivates them to train as a teacher, even though there arebetter career paths to follow. But school is hard on teachers these days, sonot everyone who has a yearning to teach dares to be a schoolteacher.
We could ask the reverse question, which would be, Does granting acertificate to a trainee make them a competent teacher? Chances are theanswer is yes, but it’s not guaranteed, and we know this to be true becausewe have all had bad teachers in our lives. And I’ll bet that once you left theclassroom, there was someone you happened to meet who knew a fair bitabout something, probably to do with his or her job, and that personchanged your life.It’s the same old story of lifting a pebble off the ground and seeing auniverse beneath it. Sometimes we think something is small andinsignificant, until someone draws our attention to it and we see itscomplexity for the first time.
So am I dissing the teacher-training process? Not at all. But I do wonderhow many of those excellent teachers who came out of Teacher’s Collegewere excellent teachers even before they were trained.My point is that a good relationship between your child and your child’steacher is not always guaranteed by a diploma. Be vigilant and supportyour child and if you can, support your child’s teacher too. But if therelationship between your child and his teacher ever diminishes to the pointof no return, be prepared to find a new teacher.
Teaching is all about relationships. We are coloured by many shades ofpersonality and that means we will click with certain people, and not withothers. Some kids love teacher A and hate teacher B. Why? Plenty ofreasons. But suppose as many kids love teacher B and hate teacher A? Whoare we to say that one is better and one is worse? It’s all in thepsychological chemistry when two minds meet.So when your kid says she doesn’t like maths, she’s probably saying shedoesn’t like her maths teacher. Go back to your own school days anddecide whether or not your interest in some subject suddenly ceased orbegan with a particular teacher. I’ll bet it happened that way for a lot ofyou.
This one’s a classic. In it’s usual form, the question goes, ‚Why should Ilearn this if I can do it on the calculator?‛ The answer is that it all dependson whether you really can do it on the calculator, and in my experience ofworking with hundreds of kids, those who can’t do the math on paperusually can’t do it by calculator either. Pen-and-paper maths requirespractice which requires patience which requires interest. So does using acalculator. Those who are good at one are usually good at both becausethey know how to learn. You’ll generally find that they are good at othersubjects too: music and languages, for example, which also requirepractise, patience and interest, and draw upon the same symbol-processingparts of the brain as maths.
Buying a calculator for a kid who is poor at maths is like buying amultiplication chart. The kid will look at it and go ‚uh-huh‛ and go back towhatever else he was doing at the time. It may give a kid a bit of comfort tofiddle with the buttons in the middle of a calculus problem, but that’s justabout all the good it will do him if he hasn’t done the paperwork.You see, it’s all about learning to do a process and practising it. It takesabout as much time to learn to operate a calculator as it takes to learnarithmetic. Of course, arithmetic won’t save you from the super-nastycalculations, but you’ll find that the person who can multiply 7 x 8 in hishead can also multiply 7.13 x 8.024 on a calculator, whereas the guy whojumps onto his calculator to multiply 7 x 8 generally has to do it three timesto make sure he’s pressed the buttons in the right order, and even then isnever sure.
I’ve met teenagers who use a calculator to multiply by ten or to doublenumbers. As well as being a sad sight – much like watching an alcoholictelling you he doesn’t need any help – it’s also tiresome because by thetime he’s actually done the calculation three times in a row to verify hisanswer, I’ve gone to sleep.Doing maths in this way is like forcing numbers through a keyhole. It takesso much attention to do the merest, simplest calculations that by the timemy student has multiplied by ten successfully, he can’t remember what hewas supposed to use the answer for. Could you expect such a student toplod his way through the dozen or so arithmetic operations required to findthe area under a curve?
The point is that it takes as much (or as little) effort to memorise the timestable as it takes to memorise the keystrokes of a calculator, and you have topractise both regularly to become reliable. Most calculator-dependent kidsspend more time remembering how to put a formula into a calculator thanactually putting the formula in. After some trial-and-error, they get asuitable answer out that suggests they entered the formula properly, butcan’t remember exactly how they entered it and don’t seem to care anyway.That’s kind of like using guesswork to work out if the antifreeze goes in theradiator or the petrol tank. You only need to do it once in a long while, sowhy memorise it?
Now it sounds like I might be telling kids they have to practise using theircalculators every day, even if they don’t have any homework to do. No!Not my point. My point is that if maths is useful, fun and interesting, kidswill learn it. They’ll learn to use the calculator for tough problems and usetheir head for easy problems. They’ll get good at both because they’rethinking about maths frequently. It’s not a chore.
So should all kids be excited about the prospect of coming home with mathhomework? Of course not. I don’t like homework. Never liked it as a kidand like it even less as a teacher. Some other time I’ll give you a long rantabout homework, but not today. No, my point once again is that some kidslike maths and will learn to use the calculator as an adjunct to using theirheads. Any other kid who is dragging his heels through the process istelling you he shouldn’t be there, wasting your energy as much as his. Letthe poor suffering fellow go; his life won’t be ruined by not knowing howto solve integrals. Think about how you want him to speak to you, ten yearsfrom now when you pass him on the street with his wife and kids in tow.He might introduce you to his family and tell you all the great things he’sdone in his life, none of it using maths. Or he might turn his head down ashe walks past and softly call you a bastard.
I’m 51 years old as I write. I was thirteen when I saw a calculator for thefirst time, and nineteen when I saw a home computer for the first time. Isuppose I stand on the cusp between two generations, one that is old andhas no interest in computers, and another that is young and strugglessometimes to find anything interesting that’s not on a computer.Before calculators and spreadsheets, hand-calculations were tedious andwere done only when absolutely necessary. Therefore older peoplegenerally did not play with numbers in their youth, and still don’t today.Calculators and (especially) spreadsheets are so easy to use that I’mactually encouraged by them to do calculations even when I don’t need to.Having solved one important problem, I might wonder, ‚What if I changedthis parameter“‛ and go off to explore other possible outcomes. From thisI can get clarity over which factors drive a process, and how they drive it.
But I think I might have been lucky. I had the times table drilled into mewhen I was a kid. I had to learn fractions and decimals and percentages,and forgot them all later in life. But when I needed to relearn them, theycame back quickly. And with those basic number skills working so wellthat they are nearly subconscious, I can concentrate on the truly enormousand interesting challenges, using a calculator, or a spreadsheet or even afull-blown mainframe computer. That’s probably why maths is a hell of alot more fun for me than it is for you.
When I was a teenager, I turned my brother’s programmable calculator intoa Moonlanding simulator. I didn’t have to do that. It was fun and it waseasy because I knew the basics of both maths and programming. A fewyears later I wrote a sophisticated Moonlanding simulator on my desktopcomputer. It took me weeks to create, and each time it was activated, itcrunched through thousands of calculations a second to make sure thatwhat I saw on the screen was accurate. I didn’t have to do it. Again, it wasfun and it was easy.
There’s only ever one good reason for learning maths and it is the samereason for learning music, as my son does. It’s the same reason that drives aperson to study languages or ancient history, or yugioh. In each there isstrategy and variation, and links that are unseen at first but then surpriseyou after a while when they pop up. And the funny thing is that the masterof any of these skills is the only one who would be fascinated by thesesurprises because no-one else cares or understands. You have to haveexplored a subject for a while before you really can see what’s interestingabout it.