Okay, so let me introduce myself. My name is Dave Coulson and I’m agraduate in mathematics as well as a graduate in psychology. I’ve beentutoring kids in maths off and on since 1983, which means I’ve been doingthis a looooonnnng time and have seen a lot of interesting characters.Putting these three ingredients together, I’ve learnt a lot, first of all aboutmaths, secondly about how people learn and don’t learn maths, and thirdly alot about how kids learn and don’t learn maths.
Most recently I’ve been teacher-aiding at a primary school that my son usedto attend. I’m teacher-aiding rather than teaching because I don’t have ateaching qualification despite the quarter century of experience I’ve pickedup from teaching and tutoring in many different places, sometimes in classeswith as many as 45 students in them.Rather than see this as an obstacle, I see this as a means of escaping most ofthe drudge and frustration that most teachers have to live with. It means thatall of my time on the job is spent actually teaching rather than accounting forwhat I’m doing.
What were those three numbers I showed you earlier?
What were those three numbers I showed you earlier? Do you remember?
17, 34 and 51Are you curious as to why I’m repeating these numbers? See if you can guess.
One thing we have to recognise if we are to be any good as teachers is thatkids are social learners.A lot has been said over the last couple of decades about different learningstyles, and most often we’re told that there are auditory learners (who like tohear stories) and visual learners (who like to see diagrams) and kinestheticlearners (who like to pick the thing up and play with it).
When I started tutoring, this was the kind of thing I tried to see in the kids Iworked with. But even though it appeared that these kids were often more ofone thing than the others, it was never entirely clear which definition Ishould shove them into. They generally seemed to be a mix of all three, andthough I was faithful to the principle of learning styles, I could never reallyget the idea to work for me.What I learned instead, just through interacting with these kids one-on-oneas most tutors do, is that kids are almost universally social learners, andsome are more intense about it than others.
Social learners learn by interacting with someone whose company they enjoy;mentors, parents, friends, even teachers. They will listen to them if they likethem, and chat back to them if they like them, and play learning games withthem if they like them; anything, so long as they can get that half hour’sworth of solid , locked-on attention from that person they like.More than anything else, kids want to have your undivided attention, and ifyou can give them that while you are sharing whatever it is you want to teachthem, then half your job is done.
I’m sure that this is a consequence of something I learned about in mypsychology days, which is called attachment theory. You might have heardof it. It says that kids are programmed before they are born to attachthemselves to whoever looks into their eyes and talks to them after they areborn. Furthermore, their basic programming compels them to copy thebehaviour of the people they see, particularly those individuals they haveattached themselves to by a bond of liking.
These instincts serve young creatures well because it keeps them out oftrouble before they are old enough to understand what trouble is. When theherd runs, it is probably a good idea to run too, and run in the same direction.Copying others when they graze teaches the newborn how to graze. Thereforecopycat behaviour is pretty useful to all young creatures, including humans,who get a lot out of following Mum and Dad and doing what they do.
But I think we forget that these instincts follow us through to adulthood. Ofcourse others learning strategies slowly take over, but the instinct to copy thebehaviour of others is still recognisable in kids at school and still (by myreckoning) the most useful way of passing a skill down to young people.As long as the monkeys like each other, it’s monkey see and monkey do.You just have to get their attention by giving them all of yours.
So what does this say about the role of rote learning in the classroom and athome?Lately, rote-learning has taken a lot of flack from people who set educationalpolicy, who like to portray rote-learning as a painful process in which kids areforced to memorise meaningless facts and figures under the threat of tortureif they don’t comply.
I want to point out that kids rote learn all the time, whether you want them toor not. They can’t stop themselves, and neither can we as adults. People rotelearn telephone numbers, names of friends and neighbours, the route toschool, the location of shops and the roles played by family members. Noneof this has any understandability to it. You can’t understand a telephonenumber, you just memorise it. People have to go to school for years beforeanyone tells them why darkness comes at the end of each day, yet even littlechildren know that there is a thing such as nighttime. Why? It just IS.
Coming back to maths, some kids will genuinely want to know in advancewhy it is that the interior angles of a triangle add up to 180 degrees, but theyare the rare ones. Most kids are just happy to memorise that attrociouslyuseless bit of information without proof if they see that a well-liked teacherhas also memorised it. They don’t have to memorise it forever, but if they seeit enough times - even unintentionally - they will remember it.
Remember those three numbers I gave you at the start? 17 34 51Those are the first three numbers of the seventeen times table. I didn’texplain to you what they were. I just asked you to memorise them. What’smore, you willingly complied. I didn’t coerce you with a clip around the ear ora threat of detention. You just went along with it because you were curious.
Now I’m going to ask you NOT to memorise something.Whatever happens, do not remember that 17 x 5 is 85.Have you got that? No! Don’t memorise it!I said DON’T REMEMBER THAT 17 x 5 is 85.What is the matter with you!? I will punish you if you memorise that 17 x 5 is 85. Stop!!!!
So you remembered it, huh?I couldn’t stop you rote learning even when I asked you not to. You just did.That’s what I think of rote learning. It’s human nature to do it and it isnonsense to say it is not a valuable way of acquiring knowledge.Understanding usually comes a little later in life.
By the way, when I gave you the seventeen times table, I missed a numberout. Gosh darn, I went 17 34 51 ….… and then skipped over a number to 85.Damn! Can you work it out for me?My brain’s gone to putty.I can’t think anymore.The number is 17 bigger than 51,or twice the size of 34. What is it???? I can’t work it out.
See? I didn’t even have to teach you that one.You worked it out because someone put the question in your head.It’s a useless piece of information that you’ll probably never need to know inthe whole of your future life, but if someone ever does ask you what 17 x 4is, you’re probably going to reach the answer faster than if I’d never asked youto work it out here. Welcome to copycat learning.
Okay, so now I want to talk a little about the growth of a child’s brain. Most ofyou will probably know this stuff, but I want to be sure because I think this isimportant.The human brain continues to grow (so I was told at university) right up untilthe age of nearly thirty. I find that interesting because most of the people inmy class at the time were probably about 21 or 22 and therefore still had a bitof brain growing to do.But maybe this explains why some of us at the age of thirty find ourselvesreflecting on our younger days and saying “Oh Cripes, that’s why my life wassuch a mess. Now I understand.”
Be that as it may, most of the growth of the brain happens in two bursts earlyon in life, one soon after birth and the other associated with puberty.The precise details are not important. But what you should be aware of is thatthe part of the brain which develops later in life is the part responsible forabstract thinking, that part of the brain that handles imaginary times andplaces, things that are not actually happening right here and now, things thatdon’t and possibly can’t even exist.Like numbers. What’s a number? What does it taste like? What does it looklike? What about quadratic equations? Do they exist? Where? And if theydon’t exist, why should I waste my mental resources learning about them?This is the kind of thinking that goes on subconsciously in a child’s head.
Children live in the here and now. They don’t have an awful lot to reflect withbecause that part of the brain is still developing (probably behind a ring oforange cones somewhere).That means most kids (in my experience) pay shallow attention to what I amshowing them. As a teacher, I accept this and don’t get upset when kids twitcharound and seem to be looking over my shoulder sometimes, and yawn a bitwhen I’ve been talking too long, and at the end of it all only remember thescantiest little bit of what I’ve been showing them. In essence, I’m teachingthem now so that they will forget so that another teacher later in life will havean easier time teaching them the same material. After all, do you rememberwho taught you how to spell, or add or multiply? I don’t, and yet it went insomehow, probably while I was looking out the window.
The point is that kids pay shallow attention to anything. Put more correctly,children pay shallow attention to everything. They live very much in a sensoryworld, much more so than we do. Kids are aware of everything – at a shallowlevel – because they haven’t yet mastered the ability to block out whatever isnot important. That is why a kid can listen to you and listen to the TV at thesame time and still understand you. It would drive me mad because I canonly focus on one thing at a time, but that is because I’ve learned how tofocus and forgotten how to multitask.So if kids are looking around at everything, it doesn’t necessarily mean thatthey are not listening to you. They are listening to you as much as they arelistening to everything, which is to say not an awful lot but enough to make adifference.
You’ll often find that kids have to talk in order to think. They live in theirsenses because that abstract cave we slink into as adults hasn’t been built yet.So if you want a kid to learn something, let him or her talk about it with you.And while they’re at it, let them talk about everything else that is passingthrough their minds at the time. Let them gossip to you about their friends orwhat they can see out the window, as long as it is interleaved appropriatelywith whatever it is you are trying to teach them. Kids minds are all over theplace, but they are still sometimes where you want them to be, as long as theylike you and like doing what you are doing.
For a child, talking and fiddling is the same as thinking. They live in theirsenses. So as far as possible let them pick up and fiddle with what you areteaching; let them go off on detours while they come to grips with the subjectmatter. Let them take their own time over a concept. Let them hear it and sayit and fidget with it and write it and see it written in front of them – anythingso long as it appeals to the senses and keeps them in mental motion. Amotionless child is (for all intents and purposes) asleep.
Another way to characterise children’s thinking is to associate it withchildren’s feeling. In the same way as it is impossible to separate a child’sconcentration from the physical senses, it’s impossible to separate a child’sconcentration from emotional sensation.Have you ever asked a kid to give you a number at random? Adults have noproblem offering random numbers but a kid will hesitate a while beforesuggesting a number. That’s because asking a kid to suggest a randomnumber is like asking him or her ‘which is your favourite number?’ They can’toffer a number without having to like it in some way.
That suggests that a good element of any successful teaching is a transmissionof emotion. In the past this might have meant scaring the kid half to deathwith the threat of punishment if a block of information wasn’t learned ontime, but it could just as easily mean appealing to the kid’s sense of humourand having a good laugh while something is being taught. I know which pathI’d rather take.
I think this is part of how I have succeeded with children. I look them in theeyes to engage their attention at a very primitive level. I ask them about theirlives and listen when they tell me. We go outside and count things; notalways, but even if I’m indoors on a rainy day, we’ll still move around a lot. I’mkeeping those inattentive parts of the brain engaged while I’m talking to theattentive part. This is a skill I learned while teaching a few ADHD kids earlyin my career: keep the body active and the mind will generally follow you.So we walk around a lot. It keeps me fresh as much as anything else, and aslong as I am having a good time, the kids pick up on that and have a goodtime too. Again it’s monkey see and monkey do. If I let them make me boredthen a vicious cycle sets in and no-one benefits. I may as well go home.
That’s the next thing I want to say about teaching kids: there’s no useflogging a dead horse – and by dead horse I’m not talking about the kid buttalking about the task. If for some reason a lesson is not working, don’t persistwith it. Try something else or regroup on another day when the child is in abetter state of mind. By doggedly persisting with a bad lesson, all you’re doingis ruining the relationship you have with that student, and that relationship isthe only key you have to that kid’s head. You’re going to have good classes farmore often than bad classes, so who cares if you don’t succeed once in a while.Let it go.
Now, what were those numbers again?Oh yes, 17 34 51 and (that number I asked you to work out) … and then 85.
Okay, comes time to talk about the times table.One of my better-known routines is putting the numbers of the times tableonto your fingers. This comes from the Tony Buzan / Kevin Trudeau school ofmemorisation. I first came across this sort of thing – known as pegging – in apsychology class many years ago. The idea is that you use your imaginationand your senses to ‘peg’ information onto items around you, such as thefingers of your hand. You could put the number seventeen on your first finger,the 34 on the second and the 51 on the third.
As you ‘write’ these numbers on your fingertips, say the numbers to yourselfand imagine the numbers being burned or carved into your skin in somesensational way, such as using a laser beam. See the shapes of the numbers.What do the digits look like as you inscribe them? How do the digits interactwith one another? Is the 7 part of the 17 poking the 1 in the backside? Does ithurt? What does a 1 say to a 7 when it’s poked in the backside?
This is spatial separation, reinforced a bit by rehearsal, humour, and vision,audition and tactile sensation (which is why I will touch the tips of thefingers as we attach the numbers).Then we’ll go outside and write the numbers on the wall using a fingerinstead of a marker pen. We say the numbers as we go. And then I ask theseendless, banal questions as if I am so stupid I can’t remember the numbersmyself. What was that number after the 51? The one before the 85? Oh yes,that one. Which finger is that on? Oh that must mean that four timesseventeen makes … (go on, work it out. You know what it is).
You’ll find after a few repetitions that the child can recall which finger anumber is on just as easily as recall which number is on which finger. Theycan access the numbers in any random order and confidently tell you whichnumbers are NOT on the fingertips. It takes about 45 minutes.
As you’re attaching the numbers, keep the audition as minimal as possible.The auditory working memory (as I was told in psychology) has a limit of (Ithink) about fifteen seconds or so, which means it’s like the looped bit ofrecording tape on an aeroplane that captures what was said in the last fifteenseconds but then is over-recorded by new information after that.So if you waste time saying “seventeen times four is” then you are going toclog up the memory with unnecessary sounds. Just simply say“17…34….51….(etc)”. The location of the number on your hand is enough totell the student which part of the sequence it belongs to.
Over the course of time, many repetitions later, the numbers will develop alife of their own in the child’s memory and (s)he will no longer need to referto the hands to ‘see’ where they are. The hands therefore are a kind of inputdevice that can be recycled for other lists of numbers at a later date.Once the numbers are attached to the hands, ask questions. Say “I’ll say thefirst half of a number and you say the back half. I say fifty and you say…(one).I say thirty, you say…(four).” Then write the numbers on the whiteboard butmake some mistakes so that the child can correct you. Every parent knowsthis gag. “NO!!” They shout. “It’s not 81 it’s 85!!!”
Whatever you do, keep the child physically busy and well jazzed up withhumour and interaction, and keep asking them to do things for you. Rub outsome of the numbers and ask the child to write the numbers back in. It’sbanal stuff to an adult learner, but the kids just seem to love it… which is whyI do it.
The point of committing chunks of knowledge to memory is that it’s a trade-off. The more you memorise, the less you have to compute.I can do a lot of calculations in my head not because I have a big head butbecause I’ve memorised a lot of things. I’ve memorised that 1/ is 110.090909… and that 1/ is 0.11111... and lots of other useless things 9besides, and the funny thing is that I’ve never actually set out to memorisethem. I’ve just come across them enough times in my life that I’ve learned torecognise them.
The important thing, though, is that once memorised, you no longer have towaste energy re-computing all these things. If I had to recalculate the seventimes table every time I needed to multiply by seven, then I would never haveenough mental space left over to handle the more challenging stuff likecalculus and trigonometry.I’ve met a lot of teenagers in this predicament; 15 year olds who pull out acalculator to multiply by ten and still get the answer wrong because theyhaven’t even memorised the right key strokes on their calculator. If you don’tremember the little stuff then your mind will be clogged up with calculationevery time you try to solve a problem. It’s a trade-off: memorise more,calculate less.
Here’s a classic situation: What’s 17 and 3? If you’ve memorised (or simplylearned to recognise) that 7 and 3 add to ten then the answer rolls out of yourhead without much effort. But if you’re still counting on your fingers (someadults count on imaginary fingers so they don’t look dumb) then it will takeyou a bit longer and require your full attention.
These are called number bonds. Kids should learn that 7 goes with a 3 and 8goes with a 2 and 6 goes with a 4 to make 10 because it makes the rest ofaddition and subtraction so much easier. Do you know what you get whenyou subtract 13 from 100? You might not know it immediately but you onlyhave to do half as much work if you know straightaway that the last digit hasto be a 7. If you say that 10 subtracted from 100 is 90 and therefore the answershould be 97, you’ll still be wrong but at least you’ve used number bonds toget a plausible answer (which you can then correct) in a fraction of the time itwould take you to count backwards 13 steps from 100.
A lot of teachers pick up on the value of teaching kids number bonds to tenbut I haven’t met the teacher yet who will teach kids number bonds to fifteen.Fifteen is a useful number because it’s exactly halfway between the tens andallows you to put big numbers together as effortlessly as little numbers. I’llshow you what I mean.Say you have to add a long list of single-digit numbers. 13457694236782156796784342356
If you group them together to make tens, you are already doing the tasksmarter than the person who simply starts at the left and grinds rightwardsthrough the list. But the time will come when all the 1s and 2s and 3s aregone and you’re left with the task of adding 7s to 8s and 9s.If you group these numbers into pairs that add up to fifteen – 9+6 and 7+8,for example – then there are no longer ‘big’ numbers and ‘small’ numbers.They’re all the same. 13457694236782156796784342356
Personally, I’d add all the nines to sixes, and all the sevens to eights first tohave a bunch of fifteens. Then I’d add pairs of fifteens to make thirties, andpairs of thirties to make sixties. Chances are there’ll be a few numbers leftover that could be added to make ten. That’s fine, add them to the total. Itdoesn’t take any effort. Instead it feels more like a treasure hunt, which Iguess is why kids like adding this way. 13457694236782156796784342356
You see, addition doesn’t have to be strenuous. And if addition isn’t strenuousthen it could conceivably be fun and if it’s fun you might get good at it and beable to show off to your friends. And if you get good at it you might findyourself doing it even when you don’t need to, like a cook who goes exploringnew recipes even when the old recipes work fine. (If we taught cooking thesame way we teach maths, everyone would eat the same damn thing everynight until the cook got it right and moved on to the next recipe).
Learning what adds to make fifteen sheds light on what adds to make 14 and16. This is what I call the lighthouse effect. Without really trying, thestudent learns that one less than 7+8 makes one less than 15, and that onemore than 7+8 makes one more than 15. So the knowledge spreads outlaterally from 7+8 to 6+8 or even as far as 5+8.And then there is another kind of spread. Kids who learn that 7+8 makes 15sometimes spot for themselves that 17+8 makes 25. What do you think 17+18will be?
Then there’s the halo effect that I’ve read about in psychology books andseen in real life with the kids I’ve tutored. Parents come up to me sometimes,after a few weeks. They’re mystified but want to share with me the peculiarobservation that little Johnny is suddenly taking an interest in things henever used to be bothered with at school. It’s as if getting good at maths hassomehow made it possible for him to get good at music or reading. I don’tknow how this works any more than you do, but I’ve seen it enough times tobelieve there’s a strange connection somehow.
We make the mistake sometimes of telling our kids that the only reason forlearning maths is to get a good job someday as an adult. Most of the time thisis well-intended crap that even we don’t believe as we say it. But we say itanyway because we don’t want to deny our children the chance to become thenext big thing in science. That’s fine, but you’ll get a lot more traction if youtell your kids that math is easy and let them see you fooling around with it.Use it in the supermarket to compare prices. Better still, use maths in thegames that you play at home. (er.. You still do this, don’t you?)
I learned a lot of maths playing a card game at home called Cribbage. Myparents taught me and I taught my son and then taught lots of kids throughmy tuition work. Some kids take the idea home and end up buying cribboards to play with Mum and Dad on holidays, but a lot of parents are justtoo busy for trivia like this and instead buy computer games for their kids.
Computerised math games generally miss the point, though they can beuseful if an adult is sitting at the computer with them. That’s what I said atthe start of this essay: kids are social creatures and learn from people, notfrom machines. People laugh at their corny jokes and make corny jokes too. Amachine doesn’t know your little darling is there, and if the machine doesn’tlisten to the child then you’ll usually find that the child doesn’t listen to themachine. Maybe a recorded voice is explaining to the student how it is thattwo negatives make a positive, but if the machine doesn’t know that the childhas turned around to look out the window and keeps on blabbering, then themachine is just blabbering into thin air. Kids know this and therefore don’tlisten.
That’s pretty much it.Oh yeah. What were those numbers again? The ones I got you to memorise? 17 37 51 69 85Wait a minute! I got a couple of those numbers wrong. Which ones?