Every now and then a child says something that really makes you sit up and go Wow! See what you think about this Wow! moment.
I’ve borrowed a vast number of ideas from other people, but I have had one or two good ones of my own, and Envelope puzzles are up there with the best of them. I’ve written about them before (April 2015) but I’ve no hesitation in doing so again. They do give a hugely accessible way for children to develop a chain of rigorously justified reasoning.
I gave Amy and her partner this set of envelopes. They knew each envelope contained two cards from a 0 to 9 set of digits and that the product of the two digits was displayed on each envelope. Their job of course was to identify the cards in each envelope.
Amy’s partner and I agreed it would be sensible to leave the 0 envelope till last, since though we could be sure it contained the 0 we wouldn’t know which the other digit was until we’d eliminated all the other possibilities.
“No”, said Amy, “you can say immediately that the 0 envelope must have the 0 and the 1”.
“Why’s that?” I said. I rather assumed Amy was a bit unclear about the multiplicative properties of 0 and 1.
“Well”, she said, “if the 1 is in any other envelope then it must have a single-digit number as its partner. That would mean that one envelope would have a single-digit number written on it, but none does. So 1 cannot be in any other envelope, and so it must be in the 0 envelope.”
Wow! indeed. What a terrific and totally water-tight chain of reasoning that had never occurred to me when I devised the set. With a National Curriculum which aims that we focus upon problem solving, reasoning and fluency I reckon Amy’s pretty much on the right lines.
A footnote: I was almost as flabbergasted at the end of the afternoon when I eagerly buttonholed a couple of teachers. “Can I tell you about Amy?”, I said. “Ah, Amy”, they said ruefully, “she’s always had problems with maths!”
(Don’t get me wrong – I’m not saying this to show how brilliant I am; these are experienced and committed expert teachers who spend every moment every day devoted to thirty pupils, many very challenging. I, on the other hand, merely swan in for the afternoon and have no other responsibility than to work with two or three children on aspects of their mathematics. My point is rather that locked away in Amy’s head was potential and insight and I was lucky enough to find the right key to bring some of this out into the light of day.)
I can’t believe I’ve never posted this tale before – everyone should have their own Christmas story, and this one’s mine. And I promise you that every word is true.
Part 1 It was Christmas Eve five or six years ago. It was a proper Christmas Eve, cold and with three inches of snow. And on every Christmas Eve my son and I are encouraged to get out of the house for the afternoon, so we set off for our traditional trip to see a film. Halfway to the bus stop I saw an object lying in the snow. It was a combined purse and wallet.
The wallet opened to show a student identity card – a rather attractive student, I had to admit. Like you and me, she also had a whole collection of cards – bank cards, store cards, library cards – so I soon knew she was called Katarina and quite a bit more about where she shopped, but what I didn’t know was her address, nor phone number, nor email. The poor girl, losing her wallet on Christmas Eve! She’d be devastated and if I couldn’t do something about it she’d face the most awful Christmas ever.
Part 2 So we spent most of the afternoon contacting all the organisations we could think of. Banks, libraries, clubs, stores. Many had closed, plenty were suspicious, but eventually I managed to get an address for her, just a mile or so away. So, like Good King Wenceslas and his page, Simon and I trudged through the falling snow, knowing how thrilled and delighted she’d be and how she would after all be able to get every enjoyment from her Christmas. “My hero!”, she’d cry, as she planted a warm kiss upon my frozen cheek. “Come in, come in, sit by the fire and have a mince pie and glass of mulled wine!”.
Part 3 We knocked on the door, and waited. I knocked again, and waited again. Eventually we heard sounds of movement upstairs, and finally the front door opened. Now I guess any adult male can fabricate plenty of scenarios built on a young woman in her night attire opening her front door. But it’s fair to say that none of my fantasy scenarios had got even close to this one. Yes, she was just about recognisable, but 5.30pm on Christmas Eve was obviously all too early in the day for her and it would take a lot of work on her make-up, hair, complexion, clothing, and above all her demeanour to become the agreeable and attractive student in the photo. It wasn’t quite a snarl, but it wasn’t far off: “Oo’r’yu, ‘n’whaddya wan’?”
“Katarina?”, I enquired mildly, “I think we’ve found your wallet”.
“’Ow ja geddis? Whad’ja doin’ wivvit?”
Slightly bemused, we went through the whole story, and she became more hostile rather than less. She started by denying she’d ever been at our end of Tring and not even knowing her wallet was missing, and was swiftly moving towards us having stolen it in the first place. I suppose it’s possible she’d ingested some chemical which had affected her manner, but it now looked perfectly possible she was going to make a scene and even call the police. She seemed quite capable of accusing me of helping myself to the contents of the wallet or calling upon her neighbours to sort us out.
So we decided we’d completed our Christmas errand, quickly said farewell, and set off down the path. Belatedly she remembered some of the lessons her mummy had taught her as a little girl. “Oh yeah”, she muttered, “I spose – ‘Appy Chrismuss”, and slammed the door hard enough to dislodge the snow from the rooftops.
It’s now a few years ago, so I guess my son won’t mind if I disclose mathematics wasn’t the high point of his school career. To some extent this was personal taste – history and languages were more his forte – but a succession of teachers didn’t do much to light any fires.
The one occasion which needed any originality and problem-solving techniques was when he wanted to find the surface area of a hemisphere. He and his friends had decided to while away a particularly boring series of lessons by establishing the amount of material needed to make a pair of trousers for the generously portioned backside of Mrs F.
Mrs F was certainly his maths teacher I remember best. All the evidence suggested that not a spark of creativity or enquiry made even a fleeting appearance in her lessons. As Parents’ Evening approached I put a lot of thought into how I might gently and tactfully comment. “I’m afraid he doesn’t come to your lessons for intellectual stimulus”, I began.
“Oh, thank you, thank you”, she gushed, “you’re very kind”!
I recently wrote about a pupil who uses a subtraction technique he invented for himself; this is a rather similar situation.
There are two reasons I particularly remember the inspection I did at Hamphill Middle School. The first happened when I sat down at the back of the classroom. Now, like most of us, I’m pretty experienced at sitting down, so I wasn’t concentrating on the process very hard, but I became aware that something strange was happening. I couldn’t work out what it was, and it continued to happen very much in slow motion until I found myself on the floor – one of the back legs of the chair had given way. I was hugely impressed by the reaction of the children, who were most solicitous and greatly concerned for my welfare even though no inspector has ever made a more undignified and hilarious spectacle.
After this dramatic start the lesson continued with the children doing some division examples. Partway through, I noticed a girl with her back to me, behaving in a manner so furtive it would have resulted in instant arrest in the outside world. She looked around to check the teacher wasn’t watching, hunched herself up to conceal what she was doing, and scribbled something on a piece of paper. She wrote something in her exercise book and scrunched up the paper and stuffed it into her pocket.
The calculation she was doing was 95÷5 and her method is totally transparent. She’s seeing it as a sharing, with 95 to be distributed into five packages. She makes an initial distribution of 10 into each package and follows this with a further 5. She’s keeping track of the amount distributed, and decides she can make another distribution of 5 into each package. You can almost hear her exclamation of frustration as she realises that will be too many and the 5 needs to be downgraded to 4.
At that point she knows she’s distributed all 95; she adds the 10, 5, and 4 in any package and has the answer of 19. She transfers this to her exercise book in the approved format, and the piece of paper has now served its purpose and goes into her pocket.
I talked in detail with her, and what she said went along these lines:
(1) My teacher’s taught us how to do these, but I don’t understand her method, so I’m not a very good pupil.
(2) However, I can do them using my own method, which of course is really cheating, because it can’t be as good as the correct method.
(3) I love my teacher, and if she thinks I can’t understand her method she’ll be disappointed because she’ll think she’s let me down.
(4) So I’ll work them out my way and write them up as if I’ve used her method.
Poor Justine! Her emotions involved her perception she was a failure who could only use a method she saw as inferior, coupled with a very real concern that she didn’t want her teacher to be disappointed either in herself or in Justine.
My reaction was of course exactly the opposite. As far as I was concerned, Justine was actually doing better maths than many others who were simply following a rule without insight or understanding. The example shown is pretty low-level, but it’s extensible. As understanding grows, you can divide – should you need to do so – both larger and smaller numbers using the same starting point.
Tim and his friend were both happy to spend their first few minutes with me chatting about themselves. The friend had a pretty good feeling about himself and said he felt he was good at mathematics. Tim, on the other hand, said “I’m pretty bad at maths” – so even though school feels they’re of pretty similar abilities they’ve clearly got very different self-images.
A few minutes later the calculation 74 – 46 came up. Tim gave it some thought and gave the correct answer, 28. He’d done it mentally, so I asked him to explain what he’d done. “6 minus 4 is 2” he said.
“Erm, aren’t you doing 4 minus 6?”, I asked.
“6 minus 4 is 2”, he repeated. “That tells me how many I’ve got to take away from 70. So I get 68, and then I take off the 40, and 28 is the answer.”
He did several more the same way and some three-digit subtractions as well, using the same method and getting each of them correct.
Now, for something like 70 years whenever I see something like 74 – 46 a little voice in my head says “4 minus 6 you can’t do, so you borrow a 10 ….” But the voice in Tim’s head is saying “4 minus 6 of course you can do, it’s negative 2”.
So Tim’s method is actually not a convenient trick, it’s actually better than the method I was taught. It’s based on clarity and understanding rather than confusing and misleading terms like “borrow” and “pay back”.
I’ve heard anecdotal accounts of children who’ve used this method before, but Tim is the first of my pupils who’s discovered it and articulated it to me. As you’d expect, I took some pleasure in telling Tim that that far from being bad at maths, his method represents much more insight and achievement than is needed by those pupils (like me in 1950 or thereabouts) who simply follow a rule given by the teacher.
(I recall with considerable embarrassment the first talk I ever gave at a parents’ evening. It was a new school and the hall was full with parents who wanted to know about about teaching. I demonstrated a subtraction example – and with 150 people watching I got in a total mess. Never in my life had I needed to think about what I was doing, I knew the rule and applied it automatically – until now. As I floundered around, I knew what every person in the audience was thinking, and I knew it wasn’t complimentary.)
I asked Tim when he’d devised his method – probably about 7, he thought. What did his teacher say about it, I asked. As I rather suspected, Tim had never previously disclosed it to any teacher, believing they wanted him to use the traditional written method – so Tim has always worked his subtractions mentally, and then writes out the sum with borrowing and carrying figures so the teacher won’t suspect he’s done anything unusual.
PS: I’ve just discovered this in my files. I can’t remember anything about Shelley – she may have been someone I met or someone who a colleague alerted me to – but clearly she’d used the same reasoning as Tim but recorded her working in a different way:
I’m not too well up in these things, but I read that a winter-themed Lottery had to be withdrawn. Purchasers were offered a prize if their ticket showed a temperature less than -8°C. People got very upset, complained to Trading Standards, threatened lawsuits, etc when their -6° ticket didn’t win a prize. “6 is lower than 8, so -6° must be lower than -8°” was a typical response.
A few days after reading this I came across a similar story from 15 years ago. I’d written it up for ‘Mathematics Teaching’ at the time and completely forgotten about it. By the way, just to avoid possible confusion, I suppose I’d better mention that in the UK ‘vet’ refers to a veterinary surgeon rather than someone who’s seen service with the armed forces.
I’d been waiting for ages behind the woman in front of me at the vet’s. At least at the doctor’s people tend to be telling the receptionist of their own problems; at the vet’s they need to talk about a couple of dogs, the cat, some rabbits, the budgie and the guinea pig as well!
Eventually she said, “There’s just one more thing … I’ve got to inject doses from 0.5 ml decreasing to 0.13 ml. But I don’t understand: 13 is more than 5, so the doses will get bigger! The instructions must be wrong.”
The receptionist peered at the instructions. “Yes, you’re right,” she said. “Hmm. I know, I’ll get the nurse and we’ll ask her.”
The nurse arrived and had the problem explained. “Yes, you’re right,” she said. “Hmm. I know, I’ll get the vet.”
Now, our cat has been undergoing a whole range of treatments in recent weeks, so I was getting pretty worried about the way the conversation was going.
I’ve always said that one of the great skills in teaching is not intervening and letting people sort their own problems out, but knowing when to stop not intervening is perhaps even more important. So I decided it was time that I stopped being a part of the furniture and gave three people an impromptu lesson in decimals – after all, the graduations on a syringe are simply another number line. I’m not sure how much understanding there was, but everyone was not only surprised but grateful and I left behind an assortment of pieces of paper and syringes with new lines on them.
Not surprisingly, I left the vet’s deep in thought. There was also some relief. We’ve lived in Tring for 35 years now and I regularly bump into former pupils. At least none of the people at the vet’s said, “Hello, Mr Parr. You were my maths teacher!”
I’ve been lucky enough to meet quite a lot of people on my personal list of heroes. In almost every case my heroes have not just been inspiring, but they’ve been prepared to give me their time and encouragement and they’ve always given a good impression of being pleased to know me.
But there is a saying that you should never meet your heroes, and on just a couple of occasions I found myself rather wishing the meeting hadn’t taken place at all. One of those disappointments was EB.
Early on in my career I had a class who with deliberate malice would make their teachers’ lives miserable, and they were old enough and clever enough to make a pretty decent job of it. I was unlucky enough to have to teach them three years running, and it wasn’t until the final year that they decided I was good enough to be allowed to teach them in an agreeable manner.
For most of those first couple of years they’d have me in near despair. I didn’t have many weapons in my armoury. Just two really; one was a dogged refusal to be beaten by a gang of kids when my self-respect was on the line; the other was a couple of books by EB. The books presented an unglamorised picture of a young teacher in the toughest of London secondary schools, inch-by-inch moving from regular humiliation to something like comfort. If he could do it in the most demanding of environments, then there ought to be some hope for me; I probably read the books half a dozen times, not really looking for tips, but using them like a comfort blanket.
Fifteen years later we’d both moved on. I had indeed become soundly established, and EB had long since left teaching to become a full-time writer and broadcaster. Our English department invited him to speak to our top year and I couldn’t have been more excited at the thought of meeting the person who’d meant so much to me, and whose books are still on my shelves today.
And what a disappointment it was! I couldn’t blame him for being an old-age pensioner rather than a dynamic young teacher, but I could resent that there were no signs of the spark that had won over his pupils a generation earlier. What was left was just another visiting speaker with nothing of interest to say to young people, and – worst of all – not for one moment did he sound like anyone who’d ever met a group of schoolchildren before. Afterwards I introduced myself and attempted to say how much he’d helped me, but he clearly felt he’d left those days far behind, and the conversation didn’t last long.
EB died nearly twenty years ago and an obituary said of his classroom books “EB portrays himself as a martyr rather than the boastful messiah of other autobiographical classroom accounts published around that time. But behind the initial panic he never lost sight of the essential good-humour of the young tearaways he was in charge of. Gradually teacher and taught came to an accommodation satisfying to both. His account of those years is still the best book ever about life in the classroom. Lessons that did not work are described with a rueful honesty that makes descriptions of the more successful times to come all the more convincing.”
I suspect that he and his books are no longer particularly well-known, though anyone interested will be able to identify him easily enough. He was a decent, civilised man and I’ll never forget how much his books meant to me – but I really do wish I’d never met him.
I was helping out in the class of a friend, and she asked me to have a word with Susie, sitting at the back of the class. It’s a long time ago now, and I can’t begin to recall what the problem was, but I do know our conversation went pretty much like this:
Susie: Can you help me please? I can’t do this one.
Alan: Let’s have a look at it. What’s the problem?
S: I can’t do any of it.
A: OK. Well, how do you think you might start?
S: (Pause, then) Well, I suppose I could ….
A: That seems a good idea.
S: But what do I do next?
A: What do you think you could try?
S: (Pause, then) Well, I might do ….
S: But what then?
A: Well –
S (Pause, then huge, radiant, smile): Oh, I see! It’s OK! I’ve got it now! Thank you so much!
Now I’d no wish to destroy Susie’s conviction that I’m the greatest maths teacher that ever lived, but I’d not said one insightful word about the problem. I’d not even needed to look at it – all I’d needed to do was lend an ear and a little encouragement. Any teacher, or any parent, could have done the same.
And the message? Well, an obvious one is that Susie got far more satisfaction from solving the problem herself than if I’d said “First you do this, then you do that, …”. But the bigger one for teachers and pupils is that seeing maths as a subject that has to be done at a hundred miles an hour does nobody any favours. The biggest factor in Susie’s success was being able to think her own way through the question without pressure and in her own time.
It was Denise Gaskins in her blog Let’s Play Math ( http://letsplaymath.net/about/ ) who reminded me about Susie. Denise posted a famous clip of a couple of primary pupils tackling a problem about fractions, and if you’ve got 6½ minutes to spare you can see the video at https://www.youtube.com/watch?v=Q-yichde66s
It’s a very similar message – that by giving the girl and boy the time to tackle the problem (and the resources to do so practically) followed by the opportunity to talk about it at their own pace you get a depth of learning that goes far beyond giving them a batch of mechanical rules.
Here’s what Denise had to say: http://letsplaymath.net/2014/08/13/fractions-15-110-180-1/
Children are creatures with a wonderfully evolved talent for getting adults to do things for them. A pitiful weak smile, a blunt “I can’t do this”, a puzzled frown, …. They’re all strategies to avoid actually getting stuck into a problem and children have enormous success with them – and my response is often to smile nicely and go and talk to someone else.
So I treasured it as a very significant compliment when Amanda said to her friend “Don’t bother asking Mr Parr for help, it’s a total waste of time!”
You don’t get monsoons in Chesham, but this was quite close enough for me. The rain drenched you on the way down and then had another go as it bounced up again. As morning break approached the downpour intensified even further.
A group of transfixed Y6 girls looked at the deluge in horror. At last they could stand it no more, and a deputation approached the teacher: “Please miss, you’re not going to let a few drops of rain interfere with our rugby practice, are you?”