11 damn lies we tell our kids (about maths)Presentation Transcript
Eleven damn lies we tell our kids (about maths) - A bunch of opinions by David C, Aug 2011
1. You need to be good at maths to get through life
Actually no. Most people never use algebra after they leave school. That includes all the skills based on algebra, such as calculus, trigonometry, probability theory, complex numbers and matrices. Most people never need to interpret graphs, measure areas and volumes, calculate or interpret statistics or bisect an angle. The trouble is that some of us – a very small percentage – do. And of those who do use these abstract skills, each of them uses it sparingly and probably uses just one or two of those skills and not 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 people to do for you: accountants, social workers, probation officers, bankers and the like. Counting is quite likely to be the only kind of maths you are required by circumstances to do frequently and quickly. After all, you’d feel pretty silly if you had to ask a shop assistant to put eight apples into a bag for you.
Of course, if you want to manage your affairs personally, you’ll need to be able to add, subtract, multiply (at least just a little bit) and just maybe divide. These form about the first half of arithmetic – the art of crunching numbers. The other half is fractions (which you’ll never need except for an understanding of halves and quarters), decimal numbers (which you’ll never need unless you count money or millimetres) and percentages (which you’ll never need unless you want to invest). Now, managing your own affairs is only an option. Some people prefer to marry someone who can manage their affairs for them, or go on a benefit in which case someone else does the work for them.
So far I’ve been talking a little tongue-in-cheek, but from here on I’m serious. Most people learn the higher mathematical skills at high school, understand them to varying degrees and then promptly forget them when they leave school, only to confront them again twenty years later when their children go to high school. And when each generation who goes through this process asks the generation that came before, “Why do I need to learn this stuff?”, we have no honest answers. We shake our heads and say 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 idea was this? And why can’t we stop doing it?
The answer may be that underneath all our doubts about the utility of maths, we have in our hearts a secret desire to see our kids do better than us. No-one knows which of our kids will be amongst the one percent of algebra-learners will go on to be algebra-users at university, and so we happily 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 march into 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 certain that our decision to quit algebra at an early stage is the right one. I’m part of a cohort of kids from the 1970s who didn’t particularly like algebra at school but didn’t dare to say so out loud, and so knuckled down and learnt it 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 lot of dreary practice in the early days to become good at it, and you never see the 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 to play around with ideas. I can design a spaceship with it and fly to the Moon on a computer program I’ve designed myself. What I find particularly remarkable 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 a time when nobody’d even thought of putting computers into the home. So although I can understand why I like algebra, I can’t fathom why my teachers 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 our hobbies. 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 rare as hen’s teeth, but just imagine the kinds of things they must be doing: 3D graphics for Hollywood movies? Or maybe writing the code that enables a computer to do 3D graphics. Or maybe designing a computer that can run the code for 3D graphics. Or maybe designing a flight simulator that uses that 3D graphics to teach pilots how to fly. Or maybe it’s a simulator for putting men on the Moon. Or maybe it’s the formulae used to land a robot on 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 to present it as an option that – for some – will be fun and possibly useful. To do that, we have to stop believing ourselves that algebra is somehow necessary for our kids. We have to make dropping out of a high school maths 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 and nobody calls them stupid. To balance all that, we have to make the learning of algebra a very attractive option so that those who are at least a little curious about it will try it. And from there, we - parents, teachers and students alike - will be able to identify those who really enjoy it and who want to go on to learn the deeper levels of the subject.
I’m using the word ‘enjoy’ in place of the word ‘capable’ because I believe that in most cases (if not all cases) the one precedes the other. If you enjoy higher maths, it will give you the stamina to go on to get good at it, in the same way that a student who enjoys playing chopsticks on the piano will go on to learn trickier stuff. Believe it or not, kids who enjoy algebra actually exist. They’re very rare and their experience of algebra is often hampered by the need to sit amongst kids who hate it. But imagine if we could find these kids early in life and focus our teaching effort on them. Imagine where their lives could go. And imagine how much happier their teachers would be. To make algebra an attractive option, we have to bring in people from industry who use it as part of their jobs, who can really show kids the power that algebra provides in certain very specialised careers.
Finally, we need to associate algebra very conspicuously with the professions it serves: medicine, engineering, astronomy and economics (to name a few). Either we start teaching these professional courses a little earlier 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 all those professional courses depend. It would NOT be seen as the last relic of a dying brigade of subjects forced onto teenagers in less enlightened times, these being Latin, French, rhetoric, philosophy and the religious practices of the day.
2. Maths will get you a good job
Nope. To see clear evidence for that, you only have to look at me and what I do. I’m a maths teacher and I get peanuts compared to the rich folks in careers that never use more than the basic skill of counting. I met a bank CEO once who hired me to teach his daughter maths because he described himself as “no good with numbers.” Professional musicians, entertainers, tour guides, sports coaches, restaurateurs, … c’mon, you don’t need 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. They entertain people. People seek them out. That does not happen to scientists very often. Scientists spend a lot of time in solitude because that’s where they 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, a magician, a pro sportsman, an actor or a maths teacher? I don’t suppose many mathematicians get invited to cocktail parties except for those organised by other mathematicians, and nobody goes to these anyway because even mathematicians can’t stand the company of mathematicians! Have you heard all the jokes about scientists on aeroplanes? A passenger sits next to a cosmologist and starts asking questions about skin-care products. The cosmologist says, “No no, I do astronomy.” So the passenger then asks for advice on her horoscope and the astronomer says “No, no, I’m a kind of physicist”. So the passenger then starts talking about her back pain and wants to know what she should do about it. Now quite exasperated, the physicist says “No, no, I’m a scientist,” after which the passenger turns away and starts to read the newspaper.
I laugh at that too. Yet I’ve seen the TED talks online and I think scientists are just about the most interesting people on Earth because of what they produce. These folks tinker away in obscurity for years and some of them – just some – create machines or theories or just insights that decide what the future will look like. Like painters, some of them get amply rewarded for what 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 where they can do magic with mathematical modeling. And even while they’re at the pinnacle of their careers, they’re still spending an awful lot of time looking at mathematical procedures that 99 percent of the people in their whanau 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 not teenage idol stuff and it’s certainly not well-paid. It just has the potential to make the world a better place, that’s all.
3. Maths is measured in exam credits
One of the saddest things that’s happened to maths is that it has been organised into some kind of scholastic ladder that leads to jobs that don’t even use the material that is taught. A generation ago, school cert was all that was needed to show you were smarter than half the population and therefore smart enough to be trained in the basics of any profession, be it mathematical or not. Maths was used simply as an intelligence test. Today, the benchmark has moved up the scale to approximately the level of a basic university 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; differential equations and the like. Pretty soon it will be PhD’s that separate the smart from the not-so-smart.
The problem is that the school qualification system is set up to identify and support those who are interested in becoming researchers. A degree in chemistry prepares a student for a life in academia, in which case she needs a background in calculus. But the rest of the world has piggybacked on this academic filtration system and used it simply as an intelligence test, so that we arrive at the bizarre situation where a person needs to know how to find the 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?” to which I would usually reply “No, I teach maths.” And then if they asked me what kind of maths, I would list the options as probability, geometry, trigonometry, calculus, matrices, complex numbers and so on. The usual response 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 determining which parts of what I teach were in that exam so that we could decide whether 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 the answers to this year’s exam questions, parents would actually ask me for them, and pay me a good salary too. Of course I’m not doing myself any favours as a businessmen by being so elusive about what I teach, but I’m trying to make the point that maths is not measured in exams, or at least shouldn’t be. Maths should be measured in topic areas that are specific to a particular career path or even learned simply for the pleasure of it.
In a course on trigonometry, for example, I could go into the history of the subject: who first invented it, how for a few years it was the secret maths of astronomers, how later it came to be used in architecture, and then after that it used by navigators on ships to reach the New World. I could talk about how it was once used as a way of turning multiplication problems into addition problems, and how today it is used in flight simulators and onscreen war-games. I would teach everything associated with the subject for as long (or as short) a time as the students were interested. It would not be the bare bones formulae required for getting ‘achieved’ in this year’s end-of-year exam.
The next saddest thing I often hear about maths is that “My daughter just wants to learn enough to get through the exam. Don’t teach her too much…” That to me is a very, very sad thing, and shows how out-of-touch the teaching of maths has become as a preparation for adulthood. Imagine if we said the same things about learning to drive a car: “I want her to just learn the basics. Don’t make her too good a driver…” Or maybe we could say 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, be we have allowed this sort of thinking to be sensible because maths is just about the only subject you will learn at school that has absolutely no worldly application except for the few people who go on into specialised professions. By dragging kids kicking and screaming through the process of learning algebra, all we are doing is creating an unhappy industry for frustrated teachers and making mathematics the jackass of all school subjects.
Watch any show, any movie that features schoolteachers. Usually they’re two-dimensional characters who wear funny clothes and talk in high-pitch voices and don’t seem to have a sense of humour. Look a little more closely. If we have movies that distinguish the good teachers from the lousy 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 who changed our perspective forever when we were growing up? Somebody taught calculus to Neil Armstrong. I know this because like all other astronauts of the time, he has a degree in aeronautical engineering. He got that way back before anyone talked to him about going to the Moon. NASA doesn’t go up to people on the street and say “We’d like to send you to the Moon,” and hear them say in response “Okay. I’d better go and learn some calculus then.” Somehow these guys who flew spacecraft to the Moon were inspired by maths teachers and physics teachers enough to study it to postgraduate level. That couldn’t have been an accident.
The funny thing is that in their private lives, math teachers are about as smart 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 other people in society. But when a math teacher is at the board in front of a roomful 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 being can ever get. It needn’t be this way.
4. Maths is awful
I’ve been lucky in my teaching life to catch kids at a very young age and teach them something that they haven’t yet learned to dislike. This is the best kind of teaching. Take a ten year old, and show him how to add a long list of numbers in a few simple steps. Straight away you can see in his face the joy of mastery, the pleasure that comes from having increased one’s own mathematical power. These kids go home and show their mums and dads, they even show their school teachers and classmates. They get to be known as ‘good at maths’.
At the other extreme, I come across kids well after the school system has ruined them. They’ve learned to hate maths for all the obvious reasons: 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 it’s taught too fast because the teacher has to finish the syllabus before the end of year it has no connection with the real world so kids can’t remember it two weeks after they mastered it 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 school teachers. They got into teaching because they had an inspirational art teacher or maybe they’re good at music or story-writing, or maybe they simply like the idea of igniting chemicals in a laboratory. Along the way they 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 the three weeks they are required to teach it. Then they forget it for a whole year 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 who had been told that before the end of the year he would be going ‘on assignment’ to a primary school and that part of his duties would be teaching fractions. So, could I teach him fractions, he asked. He’d of course forgotten how to do fractions half a lifetime ago because he’d never had any reason to use them as an adult. He said that he would be required to teach this subject to kids for a whole term, and then he asked me for one - I repeat one – lesson in fractions so he would know how to teach it. I told him that this was not going to be enough and finally persuaded him to sign up for six lessons to cover the basics of adding fractions, subtracting fractions, multiplying fractions, dividing fractions, turning fractions into decimals and percentages and getting them back from decimals and percentages. Like anything else, mastery at a subject requires over-learning it, and you have to be a master if you are going to teach somebody else. He signed up for six classes, then dropped out after two. I don’t know if he ever got to be a schoolteacher.
In the same way that kids are forced to learn maths at school, school teachers are also forced to teach maths, It is the common complaint of school-teaching; “Maths is not my forte”, “I hate teaching something that the kids are never going to use”, “I hate being asked what it’s used for”, “I hate struggling for new ways to make it interesting.” On that last point I’ve often seen teachers try to jazz up a maths class by introducing music or games or silly mnemonics, or switching on a computer game which features aliens shooting each other down with numbers. Some of these teachers don’t trust themselves to be able to remember how to do maths and end up simply handing out pieces of paper, and measuring the success of the lesson by counting how many of these pages came back with anything written on them.
Whereas I can see the value of having generalist teachers in schools, I’d also like to see specialist teachers in schools to handle the more sophisticated and narrow skills, such as music or languages or maths. To have a person who hates maths teaching maths is an absurdity. I see Yoda talking 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’t find something really easy and fun to do, then no amount of ‘trying’ will ever 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 better things than teaching. Well, we have to find a way to turn this around. We have to restructure our society so that all those wonderful meteorologists and architects and carpenters can come into school for a couple of hours a week each, and teach their particular specialty in maths. AND, we should hope that this would be the fun part of their week. It’s a day away from the office, the boss, the loser in the cubicle next door. It’s a day with cute kids who ask intelligent questions.
I used to work in one of these office environments. If my boss ever talked to me at all, it was usually to inquire why something was behind schedule and he was usually mad. When I became a tutor, most of my conversations were with really happy parents and genuinely likeable kids. That’s what got me hooked on teaching, and has kept me (loosely) in the profession for half of my life.
5. Maths comes off a piece of paper
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 is always too small for kids to want to read, and the words are jammed together as tightly as possible to cut down on the number of pages. The working space, if it’s there at all, is about as big as a postage stamp. I hear stories about teachers who hand these awful things out at the beginning of a class and then go sit down for an hour. Now I know that these are not ALL math 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 increasingly about pieces of paper!
NO! Maths is about people. I have written articles on how to do various things in maths. Nobody reads them. I’ve put talking lessons on the internet. Nobody uses them. It’s not that my material is bad, it’s that people inherently 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 that as soon as I walked away from the computer, the kid lost interest. Obviously I was more fun than the computer. I could flatter myself into believing that I am fun to be around, but I think the same thing would happen 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 as they come up because no two kids learn precisely the same way. No computer program can anticipate all the things a kid might want to ask.
But more importantly, kids like to chat. Hasn’t anyone noticed that? My God, we hear kids chattering in class all the time. Why can’t we see that that 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 idiot classmates. They chat so much they can’t even hear the advice that is being recited to them from the recorded voice on the computer. They chat to me about the funny characters on the computer screen and we make a joke about 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 kids have a teacher to chat with while they learn and they will learn. Kids are much better than us at multi-tasking, mainly because their attention to any one thing is never all that deep. They need to occupy the dormant part of their head with conversation. Otherwise they’ll fill it with something 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 learning activity, whether it’s a paper exercise or a computer game.
Islam has it pretty well nailed. One of its teachings is that there are natural roles for people to fill, and teaching is one of them. If that role is left empty, then the role will be filled by the Devil, which means to say that something will go wrong. Maybe there will be mischief or maybe she will learn the wrong thing. It needn’t be conscious or willful mischief. Maybe the kid is spending all her time learning how to make a particular character on the computer screen squawk than learning anything mathematical; in which case the sight of kids hunched over their keyboards is not an indicator of their enthusiasm for maths. Sometimes all you need to do is sit next to them and they feel your presence. They sense that you care and are available for help. And it could happen that they never need to ask you a question, but your presence focuses them on the task at hand.
6. Maths goes onto a piece of paper
I knew a teacher once who thought that teaching was about handing out pieces of paper with notes and pictures on them, and asking the kids to read these notes out loud. Following that the teacher would try to get the kids to discuss the material and then get them to write “in their own words” the answers to some questions and then hand the sheets back in so that she could mark them. Now I have put the phrase “in their own words” between quotation marks because inevitably this would become “the teacher told the kids what to write”. The whole lesson, each time I observed her in action, required clean pieces of paper going out and bescribbled pieces of paper coming back in.
Now of course the teacher has to have material to browse through later to determine who is actually learning and who is not. That is a consequence of having as many as thirty kids in one place, supposedly learning a topic in maths simultaneously for the first time. That’s the nature of the job. But we all know that being busy is not necessarily being productive. Sometimes I’d ask that teacher how the kids were progressing and she’d tell me all the things she was doing. That’s not the same thing. I know she’s busy and I know the kids are busy, but I want to know if they are getting any knowledge from all that busy-ness. Somehow that question wouldn’t register 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 constitutes teaching and learning, but I do want to say that when I teach maths – and I have 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 am finished 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 we say, a part of the times table – I will get her to recite those numbers to her enthusiastic parent. I don’t need assessment in the formal sense because I have the privilege of working with kids one-on-one or in very tiny, select groups. I can see in an instant whether a student has made any progress or not. Einstein once wrote ‘Real education is what is left over once everything else you learned at school has been forgotten’ (or words to that effect). I want 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 they complain along these lines, they complain about too much assessment. But that’s not the same thing. Assessment can be achieved verbally and behaviourally. And if the kid allows herself to forget what you have taught her over a period of time, is that really a sin? Most learning, in my own personal experience, has involved repeated visits to a particular topic, forgetting two bits for every three that I have absorbed, staggering up the progress ladder until a time comes when I can say I really know the subject well.
So wouldn’t it be nice if we could admit that by teaching a kid today, we are sometimes planting the seeds for something that will not produce fruit until much later in that kid’s life. Furthermore, can we admit that it’s okay to 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 issued on any particular day? I had maths exams in my childhood that were so poorly answered I should have been weeded out by natural selection and gone the way of the monkeys. In theory I should never have ended up as a maths teacher. And yet through the chaos of life, here I am! Should exams ever have been used to separate ‘the men from the boys’? In earlier days, maybe. But not now, not in our generation. Our classes are much smaller, and we know that not everyone wants to be a university professor. Our kids have diverse natures and ambitions. Let the proof of their learning be seen out there in the field.
If we want to see how good kids are at basic arithmetic, let’s see if they start using it in their daily lives. Let’s see if kids can compare prices in the supermarket while pushing a trolley down the aisle; more important, let’s see 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 is watching. As for trigonometry, let’s see how well they can measure the distances and heights of buildings out there on the horizon, or maybe design a bridge across a river, maybe even build it. And let’s see them do it without use of any calculators or trig tables, because if you can estimate these sines and cosines then you really do understand the subject. Perhaps the ultimate test of 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 whether a student will care enough about the subject to go off and teach it to someone else.
7. Maths only happens at school
Some years ago I came across a bunch of guys who were flying remote control model aeroplanes from a steep hillside. I was so fascinated by the sight of these things that I hiked up the hill and sat amongst them and eventually 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 never know 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 squint between your fingers at a plane in flight and use some really basic geometry to estimate the height of the plane, a number which could be deduced in less than a minute. These guys looked at me as if I was from Mars, but estimating distances and heights is something I do just for entertainment. 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. Maybe some day my life will depend upon it. Probably not. The point is, it’s maths and I’m using it outside of school.
For a lot of people, maths begins and ends at school. My high school French was like that. I’ve never spoken in French to anybody, and I remember 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 book perhaps. And the funny thing is that all those tired old neurons would’ve started firing again, and I’d very quickly be able to speak the subject as well (ahem) as I did when I was fifteen. Too often what we teach at school is streamlined for passing an exam. I suspect that very few math teachers ever talk about maths as anything other than a way to get brownie points in an exam. At the same time that students are 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 lessons to focus especially on these skills. And if that is insufficient, parents will go out into the community to find math tutors who will use their expertise to prep kids for what they think will be in the exam. IT’S ALL ABOUT THE EXAM, FOLKS!
This sort of thinking is not going to help us. All we are doing is training our kids in the mystical art of question-spotting. My wife is brilliant at this. She slept through many of her engineering classes at university and passed the course anyway because she knew how to spot questions, memorise key formulae and regurgitate answers in a particular way. She and I both studied mechanics, and while she passed it in one year and I failed and had to repeat it, I am the one who actually remembers those formulae and know how to use them. People who are especially good at memorising crap for exams are also very good at forgetting it so that their minds are clear for the next exam.
My wife has a degree in engineering that she never used. I never finished my engineering degree. I was taking too long trying to see how formulae would work in different situations. I was also trying to comprehend how it happened that people in pre-computer times could come up with such immensely powerful procedures for designing buildings. I was using all this time trying to connect engineering with everything else that I knew about the world that I was never getting work done on time, and washed out of the course. Strangely, half a lifetime later, I still hang around libraries and read engineering books. Why? Because the subject is so interesting! Would I 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 the subject in a classroom. He did apply once to enroll in an astronomy course but was told by the course coordinator that he lacked the pre-requisites and therefore would not be admitted to the course. During this meeting, Asimov had the pleasure of discovering that the recommended textbook for the course was a book he had written some years earlier, and he pointed this fact out to the coordinator, who blushed and conceded that Asimov was certainly knowledgeable enough to enroll on the course after all. But by that stage it was clear to both men that Asimov had nothing to gain from joining the class since he had already written the material that would be taught.
Stories like these pamper our egos a little and give us the feeling that we don’t really need to go to school and don’t really need to take much notice of people who do. But that’s wrong. People like Asimov are rare; most of us need a bit of a push to learn anything substantially new, and school is generally the place in which to do it. It’s just that schools have become so swept up in the maelstrom of paper-based assessment that there is little scope left for anything else. Exams and assignments drive us to get through the material as quickly as we can. In a sense, we aren’t exploring a subject so much as we are driving through it on a learning expressway. The exams are 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, I found to my dismay that most parents felt that tuition should only happen during the weeks that their kids were at school. This surprised me. School occupies forty weeks of the year, which means there are 12 weeks in which kids can come for tuition in the daytime instead of at night or directly after school. There was even time on the weekends. But who in their right mind wants to learn maths on Saturday or Sunday? Somehow kids do sports on weekends and schoolwork during the week. And holidays, they do something else. This seems so silly. Imagine a toddler learning how to talk for 40 weeks out of every 52. Imagine him losing interest in speech for two days out of every seven. Imagine him only interested in talking between the hours of 9:00am and 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 school that claims with fair pride to be a school that challenges old assumptions and wants to explore new ways of learning. As much as it talks to parents about learning occurring anywhere and at any time, it is really, really hard to get parents interested in having their kids learn maths in front of the fireplace at Dave’s house on a weekend, even if there is no money involved in the process. In a sense I’m offering to take one or two hours out of the Monday-to-Friday learning cycle and transplant them onto the weekend, in an environment that is safe, peaceful and pleasant. In ten years only three kids 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. Teachers too are very territorial and do not like to see slices of their teaching week hacked into by an unorthodox outsider. Once faced with the task of trying to teach four kids in a very noisy and distracting school, I took these kids out to the car-park and taught them maths in my old camper-van. The kids sat at the back, my whiteboard was in the boot and I stood in the driveway behind the van with the back door of the van raised up like a sunshade. The kids had a great time. And while this breaks just about every rule in the school’s guidelines for safe teaching practice, it worked and it was fun. It could not have happened in most schools and was only possible in this particular school because of the foundation of trust that existed between all the parties involved. One of those four boys was my son and the other three were friends that had visited our place more times than I can remember, for birthday parties and the like. I knew their parents, the teachers, the school principal. Everyone understood that for Dave there is no partition between working life and the rest of life – as I would like much more of society to be.
8. Maths is logical
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 worst written books on the planet and are that bad because the author has tried so hard to make it ‘good’. The professorial version of ‘good’ means that the logic that joins one page to another is precise and correct. The book therefore becomes a logical ladder that proceeds from what is universally known to .. well, wherever the author decides to take us. There will be about ten chapters in the book, and each chapter consists of thirty pages of very densely-packed theorems and lemmas and corollaries. The problem with this style of writing is that the only people who will have the patience to read the book in the intended sequence are people who already know the subject well are and reading the book only to fill in a few gaps in their understanding. The rest of us browse back and forth, skipping some sections entirely and re-reading some sections many times. The book may have been written in a logical manner, but very few of us are reading it that way.
Learning is not a logical process. We come across something that takes our fancy and we look a little closer at it. This means we want to see what it does and where it is useful. We respond well to humorous and friendly explanations as long as they are informative. We don’t necessarily need to know where the material came from, or how it works, but we know that if we 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 something fabulous and convenient and clever about maths, and then later – for those who 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 were logical.
Logic has been pestering maths since at least the time of Euclid but gained a stranglehold on it in the nineteenth century when it was found that some mathematical processes didn’t work as well as they should in all circumstances. Someone needed to go back and verify the assumptions on which a mathematical technique worked, and in so doing plug up the holes that people sometimes fell through when using the technique improperly. That’s fine for writing a PhD thesis but no good at all for writing an inspiring maths book and worse still for teaching kids in the classroom. If any part of maths-learning is going to turn a student’s curiosity off, it’s almost always a very logical teacher. The better we get at a subject, the worse we get. We may understand the logical connections that hold a mathematical procedure together, but we lose the ability to present it in ways that the students can follow, which is to say that we lose the ability to teach it illogically.
Consider calculus. This is a fascinating area of maths. It’s the language of physics and astronomy. It’s the procedure that makes it possible to work out the flow of air around an aircraft’s wings, or to determine which part of an overloaded bridge will crumble first. Calculus told Isaac Newton that gravity drove the around the sun. It also told Einstein that time slows the faster you go. Usually the first taste a kid ever gets of calculus is deriving it from first principles. She learns how to differentiate polynomials and trig functions, and then goes on to integration. If the teacher is good, he will explain why people do this, but some teachers are not so good. And so the student sees calculus as bizarre and pointless.
Imagine if we taught carpentry in this manner. Let’s say we gave a kid a hammer and told him how it was made and what it was made from. Let’s say he spent six months swinging this hammer back and forth in the air before we introduced him to a nail. Hammers were NCEA level 1 and nails were NCEA level 2. We put them together at Level 3. We would do well to think more like children than like professors when we sit down to create a lesson plan.
9. Rote learning is bad
Somewhere, somehow, in recent times, rote learning has been given a very bad and undeserved reputation. Somehow rote learning is stigmatised as some sort of punishment, as if kids were nailed to the walls and forced to recite mindless facts and figures to earn their release. Poppycock! Kids memorise all the time. They memorise their Mums and Dads, they memorise their toys and telephone numbers. They memorise the way to school and they memorise their favourite TV shows. We can’t stop kids from memorising; they’re programmed by millions of years of evolution to copy what they see. At a young age, when the cognitive centres of the brain are not fully developed, memorisation by mere mimicry is in some cases the 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 why Mum is Mum but they memorise her anyway. They memorise what they want, 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 to understand it before they memorise it? I’ve been teaching kids for years and never saw a kid back away from the challenge of memorising a string of numbers. They love it! If they can memorise five digits on the times table in ten minutes, they feel proud enough to try the next five. The point is to desist from telling them that memorising numbers is bad, and similarly desist from forcing them to understand things when they’re not ready.
Speaking now as a person who uses maths quite a lot, I can tell you that performance in maths is a mixture of analysis and memory. The more connections 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 without effort. It just does. If on the other hand you are counting on your fingers or doing some dumb cluck strategy that requires you to multiply 8 by 5 and then add on another couple of 8s, then you are using a lot of analysis and the process is tiresome. The more you memorise the less work you have to do.
Try finding two-thirds of 57. If you just happen to know that 57 is three 19s, then the task is easy. Did I memorise that? I guess I did, but I didn’t do it consciously. Somehow I must have seen that connection enough times in my life that it stuck. The more arithmetic you do, the more you (unconsciously) memorise, which means you will have less work to do the next time you see these numbers. Just imagine what a student can do if he memorises some basic decimal numbers. Imagine what YOU can do if you memorise some basic decimal numbers.
Let me test you out. I’ll bet you never knew that three 17s make 51. Why should you know this? No reason at all, but I’ll bet you’re going to remember it for the next few seconds, simply because I drew your attention to it. If I was to write 51 down somewhere later on in this document, a wee light 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 you see it. See, I don’t actually have to tell the truth about whether or not you will remember that 3 x 17 = 51, I just have to draw your attention to it a few times, and you will remember it. You can’t stop yourself. Even if I say DON’T MEMORISE THAT 3 x 17 = 51, you will go ahead and memorise it anyway.
Rote learning is as easy as that. You don’t understand why multiplication works when you first meet it, no more than you understand how a car works when you first drive it. No kid in her first decade of life wants to understand how things work, she just wants to be able to do what the grown-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’t use this as a pathway to learning.
10. Teacher knows best
I put this one in here to get folks to realise that teachers are everywhere on this planet and not all of them stand in classrooms and not all of them are certified 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 can it happen that an untrained teacher can teach as well if not better than a certified 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 some people have that motivates them to train as a teacher, even though there are better career paths to follow. But school is hard on teachers these days, so not everyone who has a yearning to teach dares to be a schoolteacher.
We could ask the reverse question, which would be, Does granting a certificate to a trainee make them a competent teacher? Chances are the answer is yes, but it’s not guaranteed, and we know this to be true because we have all had bad teachers in our lives. And I’ll bet that once you left the classroom, there was someone you happened to meet who knew a fair bit about something, probably to do with his or her job, and that person changed your life. It’s the same old story of lifting a pebble off the ground and seeing a universe beneath it. Sometimes we think something is small and insignificant, until someone draws our attention to it and we see its complexity for the first time. So am I dissing the teacher-training process? Not at all. But I do wonder how many of those excellent teachers who come out of Teacher’s College were excellent teachers even before they were trained.
My point is that a good relationship between your child and your child’s teacher is not always guaranteed by a diploma. Be vigilant and support your child and if you can, support your child’s teacher too. But if the relationship between your child and his teacher ever diminishes to the point of no return, be prepared to find a new teacher. Teaching is all about relationships. We are coloured by many shades of personality and that means we will click with certain people, and not with others. Some kids love teacher A and hate teacher B. Why? Plenty of reasons. But suppose as many kids love teacher B and hate teacher A? Who are we to say that one is better and one is worse? It’s all in the psychological chemistry when two minds meet. So when your kid says she doesn’t like maths, she’s probably saying she doesn’t like her maths teacher. Go back to your own school days and decide whether or not your interest in some subject suddenly ceased or began with a particular teacher. I’ll bet it happened that way for a lot of you.
11. The calculator will do it for you
This one’s a classic. In it’s usual form, the question goes, “Why should I learn this if I can do it on the calculator?” The answer is that it all depends on whether you really can do it on the calculator, and in my experience of working with hundreds of kids, those who can’t do the math on paper usually can’t do it by calculator either. Pen-and-paper maths requires practice which requires patience which requires interest. So does using a calculator. Those who are good at one are usually good at both because they know how to learn. You’ll generally find that they are good at other subjects too: music and languages, for example, which also require practise, patience and interest, and draw upon the same symbol-processing parts of the brain as maths. Buying a calculator for a kid who is poor at maths is like buying a multiplication chart. The kid will look at it and go “uh-huh” and go back to whatever else he was doing at the time. It may give a kid a bit of comfort to fiddle with the buttons in the middle of a calculus problem, but that’s just about 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 takes about as much time to learn to operate a calculator as it takes to learn arithmetic. Of course, arithmetic won’t save you from the super-nasty calculations, but you’ll find that the person who can multiply 7 x 8 in his head can also multiply 7.13 x 8.024 on a calculator, whereas the guy who jumps onto his calculator to multiply 7 x 8 generally has to do it three times to make sure he’s pressed the buttons in the right order, and even then is never sure. I’ve met teenagers who use a calculator to multiply by ten or to double numbers. As well as being a sad sight – much like watching an alcoholic telling you he doesn’t need any help – it’s also tiresome because by the time he’s actually done the calculation three times in a row to verify his answer, I’ve gone to sleep.
Doing maths in this way is like forcing numbers through a keyhole. It takes so much attention to do the merest, simplest calculations that by the time my student has multiplied by ten successfully, he can’t remember what he was multiplying by ten for. Could you expect such a student to plod his way through the dozen or so arithmetic operations required to find the area under a curve? The point is that it takes as much (or as little) effort to memorise the times table as it takes to memorise the keystrokes of a calculator, and you have to practise both regularly to become reliable. Most calculator-dependent kids spend more time remembering how to put a formula into a calculator than actually putting the formula in. After some trial-and-error, they get a suitable answer out that suggests they entered the formula properly, but can’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 the radiator or the petrol tank. You only need to do it once in a long while, so why memorise it?
Now it sounds like I might be telling kids they have to practise using their calculators 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, kids will learn it. They’ll learn to use the calculator for tough problems and use their head for easy problems. They’ll get good at both because they’re thinking about maths frequently. It’s not a chore.
So should all kids be excited about the prospect of coming home with math homework? Of course not. I don’t like homework. Never liked it as a kid and like it even less as a teacher. Some other time I’ll give you a long rant about homework, but not today. No, my point once again is that some kids like maths and will learn to use the calculator as an adjunct to using their heads. Any other kid who is dragging his heels through the process is telling you he shouldn’t be there, wasting your energy as much as his. Let the poor suffering fellow go; his life won’t be ruined by not knowing how to solve integrals. Think about how you want him to speak to you, ten years from 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’s done in his life, none of it using maths. Or he might turn his head down as he 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 the first time, and nineteen when I saw a home computer for the first time. I suppose I stand on the cusp between two generations, one that is old and has no interest in computers, and another that is young and struggles sometimes to find anything interesting that’s not on a computer. Before calculators and spreadsheets, hand-calculations were tedious and were done only when absolutely necessary. Therefore older people generally did not play with numbers in their youth, and still don’t today. Calculators and (especially) spreadsheets are so easy to use that I’m actually encouraged by them to do calculations even when I don’t need to. Having solved one important problem, I might wonder, “What if I changed this parameter…” and go off to explore other possible outcomes. From this I 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 me when 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, they came back quickly. And with those basic number skills working so well that they are nearly subconscious, I can concentrate on the truly enormous and interesting challenges, using a calculator, or a spreadsheet or even a full-blown mainframe computer. That’s probably why maths is a hell of a lot more fun for me than it is for you. When I was a teenager, I turned my brother’s programmable calculator into a Moonlanding simulator. I didn’t have to do that. It was fun and it was easy because I knew the basics of both maths and programming. A few years later I wrote a sophisticated Moonlanding simulator on my desktop computer. It took me weeks to create, and each time it was activated, it crunched through thousands of calculations a second to make sure that what I saw on the screen was accurate. I didn’t have to do it. Again, it was fun and it was easy.
There’s only ever one good reason for learning maths and it is the same reason for learning music, as my son does. It’s the same reason that drives a person to study languages or ancient history, or yugioh. In each there is strategy and variation, and links that are unseen at first but then surprise you after a while when they pop up. And the funny thing is that the master of any of these skills is the only one who would be fascinated by these surprises because no-one else cares or understands. You have to have explored a subject for a while before you really can see what’s interesting about it.