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.)
Marilyn Burns ( @mburnsmath ) posted an interesting example of how one child divided 56 by 4.
It just so happens that I’ve a whole collection of different ways pupils tackled this very question and got the correct answer. (There’s one incorrect answer, but it’s another interesting method.) ((PS Everyone has been far too polite to point out this is complete nonsense – Marilyn’s example is actually 56÷14 rather than 56÷4, but I should have spotted that some time ago. My apologies.))
There’s a brilliant animation of number patterns from Stephen Von Worley. You can find it at http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/
Try it straight away. It displays first a single dot, then two, then three in a triangle, four in a square, five in a pentagon. From 6 onwards, the number is likely to be shown as a pattern, so for 8 you get two squares of four.
As you explore, it becomes clear that the displays aren’t any old pattern, but are based logically upon the factors of each number.
I’ve not seen so much excited discussion in my classroom for ages. My Y6 children were transfixed. Words and descriptions tumbled out, ideas and predictions were offered, challenged, revised, replaced.
What would 9 look like? There were two opinions. One was we’d see a hollow triangle, the other was that we’d get a triangle of three small triangles. What delight to find both were correct, and the two suggestions were offering alternative descriptions of the same pattern.
If your pupils are anything like mine, one snag you often find when they’re solving a problem is the failure to build on evidence. Not here. Several times when trying to predict a number they asked to look at a relevant previous one. When thinking about 15 it was “Can we see 5 again?”, and used this to decide that 15 would show a pentagon with each vertex a triangle of three dots.
A week after the first session they were knocking the door down to take things further. Why were some numbers not in a pattern but arranged in a circle and labelled “Prime”? Why did we never get two of these in succession? Which numbers were made up of block of four dots in a square?
I had plenty of frustrations. It moves quite fast and the display changes every second, so we need to stop it each time to look at the pattern, and talk about it and discuss what the next one will look like. The control buttons are quite small, so I often miss. And I dearly wanted to be able to call up a number of my choice. But if we want to see what 243 looks like (and you probably will) we have to start again from the beginning, and what seemed fast now becomes rather slow. There is a faster speed option, which changes three times a second, but even that’s prohibitive when we want to explore larger numbers. My pupils were delighted to learn it would display up to 10 000, less so when we talked about how long it would take. By the way, to reset we have to tell it to count back all the way to 1.
I got round some of these problems by taking snapshots of the display for all the numbers up to 100. I’ve put them into a Powerpoint that gives me greater control, and made subsets with odd numbers, even numbers, and multiples of 3, 4 ,5, and 6. When we got back to the classroom after half-term I was pleased I’d done this; it worked really well and we spent a whole hour working through the first thirty or so counting numbers. Virtually nothing went on to paper, but a thousand diagrams were drawn in the air, and as the session went on – and the numbers and patterns increased – these were often dispensed with, so one person’s mental image was articulated and received and understood by their partner.
Yes, I do wish it offered a few more options, but make no mistake – I’m 100% sold on the animation. It’s brilliant, it’s free, and though I was using it with 10 yearolds it will entrance and stimulate any group of children and adults. If you haven’t tried it already, you should do so at once.
It was a great pleasure to be working with a couple of pupils I knew would accept any challenge I offered, so you won’t be surprised to know that we spent two or three sessions exploring all the ideas around Stars that I wrote about in several recent posts.
I may well have been the only teacher in the country disappointed that the end of the summer term was coming up fast, but there was still time for one further session. I really don’t think there’s any exploration more accessible and productive than the Tower of Hanoi. It’s intensely practical and visual and you need just two simple rules. I was using it with two very bright nine-yearolds, but I’ve used it both with teachers and with much younger children – one teacher used it with her Reception class “Baby Teddy can sit on Mummy Teddy’s lap or Daddy Teddy’s lap ….” and it worked a treat.
There’s so much to find that even now I’m still discovering new aspects, but it won’t take long to start wondering how many moves it takes to move a stack of 3, a stack of 4, a stack of 5, …., or to observe a dazzling array of patterns and movement rules.
If you need refreshing on the rules and background there must be hundreds of websites devoted to the problem, with diagrams, formulae, and animations. Many of them spoil the fun, but you’ll easily find all the information you could possibly want and much more besides.
In the spring I used it with a Masterclass group of Y6 children and we dealt with numbers up to quintillions, and derived a procedure to allow them to solve the puzzle for a stack of any size. We used boxes gleaned from the supermarket, and I was struck that for these children it’s probably rather rare that they get they chance to manipulate apparatus. It seems a little sad, but I suspect that one reason they enjoyed the session so much was that there was a strong element of play involved. There were 30 people in the group and next year the organiser has decided she wants to invite 90. Collecting enough boxes will be a massive task, and we’re hoping we can persuade IKEA to sponsor us with a few dozen sets of their toddlers’ stacking cups at £1.50 a set.
The previous post used an activity where cards have different numbers on each side, and the possible totals are found.
I followed that by using another set of numbered cards. There were six in the set, each with a different number on each side, i.e. there were twelve different numbers in all. I gave Jenna and Den a free choice of which number should be face up on each of the cards, and gave them an opportunity to revise their choice. I asked them to add the six numbers on display – and then produced a sealed envelope which they opened to find I’d predicted the correct total in advance!
You don’t have to work very hard on presentation for your audience to be wholly baffled. I performed the trick with a set of eight cards in South Africa, and the audience included Toni Beardon, who’s the founder of NRICH and a very clever person indeed. I treasure the look of complete amazement on her face when the sealed envelope was opened and the prediction displayed.
I encouraged Jenna and Den to inspect the cards carefully. Their first observation was that each had an odd number one side and an even number the other. Secondly, on each card the even number was the lower one. Thirdly, on each, the odd number was 17 more than the even number.
So the total of the numbers on display would always be the sum of the six even numbers, plus 17 for however many odds were visible.
From the work earlier in the session the children told me there would be 64 possible arrangements. A significant number of these – twenty – show three odds and three evens, and that’s the situation I need to see for the trick to work. It doesn’t matter which three odds /three evens they are, and on about one occasion in three this will happen anyway, but Jenna and Den’s original selection showed four odds and two evens. So I invited them to “do a further randomisation” and turn over one of the six, and not surprisingly they turned one of the four odds. So we now had my desired situation of three odds and three evens, and the total had to be 51 more than the total of the six even numbers and it was safe to open the envelope.
Of course, it’s possible the “further randomisation” doesn’t do what you want, and you’re now looking at five odds and one even, at which point you need to request a final randomisation of two cards – but whenever I’ve done it one randomisation has been sufficient, and frequently the initial arrangement does the trick and you can open the envelope immediately.
Performing has always been part of teaching, and hamming up the amount of choice you’re giving the children not only disguises the fact that you’re actually controlling the situation, but should make the opening of the envelope both dramatic and amazing.
For my second session with Jenna and Den I used another of my long-time favourite number activities. It another one that’s very accessible but can make people think quite hard.
On each side of a card square write a number. You don’t actually have to use a different number on each side, and they don’t actually have to be whole numbers, but that’s what most people do. And on another piece of card do things similarly – again, you don’t have to use whole numbers, and they don’t actually have to be different from the ones on the first card, but that’s what usually happens.
Now toss the two cards as if they were coins, and add the two numbers you see. Do it again, and record the totals you see; do this until you’re satisfied you’re not going to get any new totals.
If you do this with a class some groups are likely to find they’ve made three different totals, and some will find they’ve got four. If they have two new pieces of card and number these, do they still get three (or four) totals? Can they discover how you ensure you always get three different totals, or four different totals?
With only a small number of children I may steer it in a different direction. Here’s the account I wrote up for school of what happened with Jenna and Den, including some false starts and blind alleys:
Today I asked them to devise two double-sided cards with different numbers on each face, so that the four possible totals they could display would give a set of consecutive numbers. Before long they found – not quite by accident, but not completely by design (Den had first decided that one card should be 0/1, and suggested 3/4 for the other) – the cards 0/1 and 2/4, which generate 0+2, 1+2, 0+4, 1+4 (i.e. 2,3,4,5).
I asked them to find a second set and Jenna offered 2/3 and 4/6, making 6,7,8,9.
I asked them to generalise from this and they suggested one card had to be even / odd and the other even / even, but it didn’t take long to find a counter-example, before Den came up with the correct suggestion that the numbers on one card should have a difference of 1, and on the other a difference of 2.
I asked them to explore the situation with three cards. They thought there would be six combinations, and Jenna suggested the cards would need differences of 1, 2, and 3. They used a logical process to derive each combination in turn, and both contributed equally. Having reached six they realised there would be eight possibilities, and they observed that each number appeared in four combinations and were able to use this to check they had a complete set. However, one of the totals in their set was repeated, and Jenna then suggested the cards needed to show differences of 1, 2, and 4 (rather than 1, 2, and 3).
They wanted to explore four cards, which Jenna suggested would need to display differences of 1, 2, 4, and 8. They quickly devised the set 2/3, 7/5, 4/8, and 9/1. Den thought there would be 12 combinations, but they again used their logical strategy for generating every combination, and so decided there would be 16. They found these with no slips, and found the 16 showed every total from 12 to 27, once each.
It was fun working with these two All I Can Throwers, but in some ways I prefer using Two Cards with students whose thinking isn’t so streamlined. Jenna and Den did offer a couple of suggestions which didn’t work out, but they immediately corrected them and got back on track, so they missed out on a lot of the red herrings that most people might experience. Incidentally, I was intrigued that in both the explorations they’d done so far neither of them had shown the slightest inclination to make notes or do any recording on paper.
Actually, this exploration was only half of what we did in the session, and I’ll tell you about the other activity in my next post.
Enough people have noticed the Consecutive Numbers piece to justify a follow-up.
Discovering just which numbers can’t be made by summing a set of consecutive positive whole numbers is such an elegant and surprising result it brings a smile to the face. On the way – and worthy of being an enquiry in its own right – is the key observation that while any set of three consecutive numbers sums to a multiple of 3, four consecutive numbers do not sum to a multiple of 4.
There are other spinoffs as well.
* Those sets which start with 1, e.g. 1, 2, 3, 4 sum to give the triangular numbers.
* What if you do the Consecutive Numbers enquiry with just the consecutive odd numbers? In this case, of course, those sets which start with 1 (e.g. 1, 3, 5) sum to give the square numbers.
* Which rather suggests that if you use numbers from the series 1, 4, 7, 10, 13, … you ought to find something interesting.
And that leads me to Roof Numbers, which you probably won’t have heard of. I was asked to work with some B.Ed. students, and in every respect except one I was given a totally free hand. So I was able to create a course built around exploratory maths, and a quite wonderful term it was too (I was asked to give them a second course the following year, and they insisted I was a guest to their graduation).
The fly in the ointment was the university’s requirement that I set a timed unseen written examination. But at least I got to set it, and Roof Numbers were my response. If the Consecutive Numbers question is a highly open problem, then Roof Numbers are hyper-open.
Here’s the problem:
Start with a bottom row of dots.
Above it, add a row which is three shorter than the bottom row.
Keep going till you feel like stopping, or until it’s impossible to carry on.
You have made a Roof Number.
o o o o o
o o o o o o o o
o o o o o o o o o o o
So 24 is a (level 3) Roof Number.
11 is a level 2 Roof Number:
o o o o
o o o o o o o
What can you find out about Roof Numbers?
I promised anyone who was desperate could buy a hint, but I knew perfectly well that anyone who’d spent a term doing problem-solving investigative mathematics would be able to spend their hour finding out interesting things, such as:
* are there numbers which are roof numbers in more than one way?
* are there numbers which cannot be made as roof numbers?
* what can you find out about level 3 (for example) roof numbers?
And since the step size of 3 is wholly arbitrary, you could just as well have roof numbers where the step size is 4, or 2 – and if the step size is 1, then you get the original Consecutive Numbers enquiry as simply a special example of Roof Numbers.
I think most of us are pretty uncomfortable with labels like “slow learners”, “less able” etc. And terms like “gifted”, “quick”, “high ability” aren’t much better. Apart from anything else, many are highly pejorative and they all suggest that everyone has a fixed level of learning.
All the same, I do from time to time meet children who do have the experience and background that lets me know they can take all I can throw at them, and recently I had the chance to work with a couple of them for a few weeks. The challenge was not to accelerate them through the syllabus but to give them the opportunity to explore some ideas at greater depth than is normally possible.
So what did I do with my All I Can Throwers? In looking for ideas I had many of the same criteria I’d use for any group, but basically I wanted themes that were accessible, intriguing, offered scope for asking questions, and lots of things to find out.
My first almost chose itself. I was asked to go to Cambridge a while back to talk to NRICH about my favourite activities and we both agreed that number 1 on our list would be the Consecutive Numbers question.
- I can write 12 as the sum of three consecutive numbers; 12 = 3 + 4 + 5
- Another example: 9 = 4 + 5
- Another: 14 = 2 + 3 + 4 + 5
- (and some numbers can be made in more than one way: 21 = 10 + 11, and also 21 = 5 + 6 + 7).
So the question is whether all numbers can be made in this way.
In truth, it’s the perfect enquiry for almost anyone. It’s immediately accessible using the simplest arithmetic, and offers scope to explore in your own way. There’s lots of scope for formulating questions, making observations, and reasoning and generalising. Perhaps best of all, it gives up its secrets gradually. It won’t take long to make the first observation, which explains half of all numbers, but others may take a little longer. There are further generalisations on the way, each adding a little more to the understanding.
Some of these discoveries will be made by any pupil who tackles the question, but there’s a final gorgeous climax in store for those who, like my pupils Den and Jenna, are able to dig deep and make generalisations. It had been a great way to spend my first hour with them, and with just five minutes to go they realised, and were able to explain, just which numbers cannot be made as the sum of consecutive numbers. But you don’t have to have a Den or Jenna in your group – it’s a great topic to explore (and makes for a fine display for the first parents’ evening of the year).
Mr Nick Gibb, Minister of State at the Department for Education, objects to certain methods of tackling questions because he claims they’re inefficient. He’s making it plain that he wants to see children using traditional calculation algorithms, and he’s particularly had the grid multiplication method and the chunking process for division in his sights. Now I see how more children tackle these processes than most, and all the evidence that I have points to grid multiplication and chunking having brought about something close to a revolution in children’s understanding and fluency. Throwing away these valuable gains in favour of a purely notional increase in efficiency seems to me little short of crazy.
A dozen years ago I’d mark literally hundreds of children every summer who found it was the traditional long multiplication algorithm that was totally inefficient – because they couldn’t remember where or why they were supposed to be parachuting a zero or two into their operation. By contrast, the grid method allows each separate sub-product to be given the correct magnitude.
Exactly this situation occurred in yesterday’s lesson; the pupil is attempting to multiply 543 by 12. He uses the traditional algorithm and makes the classic error of multiplying first by 2 and then by 1 rather than by 10. He checked using the grid method and all the place value problems disappeared and the correct answer was reached. Which of the two methods was the inefficient one?
Today, every one of my average-attainment pupils will use the grid method with total efficiency and understanding. What’s more, unlike the standard algorithm, they use it as a basic and natural part of their toolkit to call upon when faced with new challenges. They can use the same method for every situation – three-digit by three-digit? No problem. Decimals? Lots of zeroes? And even fractions – for which the standard method is little help – they’ll use the same method for all of these. It’s not just my few pupils, either – today when I mark hundreds of pupils in their SATs papers I see the grid method used naturally and effectively as a general and reliable multiplication method.
A few weeks after Mr Gibb spoke, Freda and Joe, who were both on the level 4/5 borderline (i.e. a bit above average but by no means outstanding) got deep into triangular numbers. Effectively they were summing arithmetical progressions. They set themselves bigger and bigger challenges and surely even Mr Gibb would have been impressed how effectively they used the grid method. At one point we were walking down the corridor and Joe set himself the challenge of working out the 79th triangular number, which involved multiplying by 39½, but the grid method was so well established that he did the whole thing in his head. Go on, you have a go.
Here’s the board they filled with their calculations as the ideas and the challenges just flooded out. See how understanding the grid method is so complete that it’s become a tool that allows them to do their workings speedily without a second’s delay. All the work you see – along with more which got rubbed out – took place in a hectic burst of about 10 to 15 minutes. (One of yesterday’s pupils said “I haven’t got the time to rule a grid, I’m too busy working”.) Not much sign of inefficiency here, Mr Gibb.
With long division it’s much the same story. In this year’s Key Stage 2 Tests the long division question was 936÷36. In the conventional algorithm you first decide that you’re going to remove 720 (20 lots of 36); this leaves you with 216 which is 6 further lots of 36. This, I guess, is Mr Gibb’s preferred method – a similar calculation is done this way in a document National Curriculum Mathematics_Appendix_1 published in 2013.
Since this method involves removing first a batch of 20 lots of 36 and followed by subtracting a batch of 6 further lots of 36, someone will need to explain to me how it differs from an “inefficient” chunking method. I’m guessing – and it is only a guess – that it’s deemed as efficient because just two chunkings are used and the first of these is a multiple of 10.
I asked the statisticians for the information about the 600 000 responses to the 936÷36 question and they told me that 96% of pupils attempted the question, and 53% got the correct answer (another 9% used a valid method but made an error). These are figures that are far, far higher than anything we’ve ever seen before, and it was the chunking technique that gave children the confidence and the ability to handle the question. I can claim something like expert knowledge here, because this summer I marked no fewer than 3000 of those responses to this very question – which is a little more than 1 in every 200 children in the age-group.
Rather than a weakness, it seems a great strength of the chunking technique that you can do your chunkings however feels right to you – for example, I marked several cases where there were three chunkings, of 10, 10, and 6 lots of 36. Perhaps Mr Gibb believes using three chunkings is inefficient, though it doesn’t seem to me to devalue the method. By the way, many pupils preferred to chunk up to 936 and not down to 0. Again, that option seems a strength rather than a weakness.
Whoever is going to have to write the rules that define an approved method deserves some sympathy. When you’ve got 600 000 pupils tackling a question not everyone wants to use anything resembling the standard algorithm in the first place. For example, it’s easy to see that 936 is divisible by both 9 and by 4, so what about the children who chose to tackle division by 36 as a two-stage short-division process, dividing by 9 and then by 4? Or dividing by 6 twice? They’re not using Mr Gibb’s method, so are they being inefficient – or are they being insightful and creative?
And the response I found most memorable was the child who first subtracted 36 from 936 to leave 900, and then called upon their recognition that 900 is 25×36, so immediately getting the answer of 1+25, or 26. It’s hard to see anyone could be more insightful or efficient than that. But wait a minute, hasn’t the child has actually done a chunking, which Mr Gibb says is inefficient? Even worse, they’ve taken the smallest chunk off first, which seems to be against Mr Gibb’s procedure.
And of the 600 000 children in the cohort, some will be seriously clever and will work the question out mentally. When you can work out 936÷36 in your head, how can anyone possibly have the nerve to criticise you for being inefficient because you didn’t use a standard method?
None of this is intended as a criticism of traditional algorithms. If teachers and children see them as their preferred choice and can use them effectively and with the fluency and understanding that the National Curriculum requires, then that’s fine with me. But my observations about the grid method and chunking don’t point that way. We subject well over half a million 11-year-olds to the Key Stage 2 Tests each year, and we use the results to grade children and their schools. I’d be far more comfortable if we used the huge mass of data to tell us about the mathematics children can do and the methods they use.
I’ve heard it said that Nick Gibb does some long division practice every day before breakfast. That may or may not be true, but it may be more relevant than you think, because Mr Gibb is Minister of State at the Department for Education and it’s certainly in keeping with many of his views. Politicians do seem to attach almost mystical importance to long division, though perhaps it’s rather like doing the laundry. Everyone knows the traditional way is to go down to the riverbank and bash your washing with a rock, but I bet it’s a long time since Mr Gibb attacked his shirts with a large stone.
It might be rather nice if Mr Gibb and those in his circle – and the Education Secretary Ms Nicky Morgan certainly seems to be one of them – could be persuaded that there might be other ways of tackling mathematics in the 21st century than automatically applying a mechanical procedure. After all, I’m at the end of a very long line of people who’ve pointed out that if you want a mechanical procedure it makes a lot more sense to get a machine to do it.
I’d guess one reason Mr Gibb enjoys long division is because division feels different from the other operations. In addition, subtraction, and multiplication you have your two numbers, you do things with them (conventionally beginning with the digits of least value) and out pops the answer.
With all but the simplest divisions it works differently. You do something (starting this time with the digits of greatest value rather than the least) to find part of the answer; then you do another something and find the next part of the answer, and so on. And since finding each of these numbers may involve trying one value, and then another and perhaps another, it’s all akin to a trial and improvement process and yes, it does feel a different kettle of fish entirely.
There’s a parallel to another process. In the sixth form I learned an algorithm that was reminiscent of a long division algorithm on steroids. You thought of a number, did several things with it, did it all several times more, and eventually you found the square root (or more likely, an approximation of the square root) of the number you started with. This process was even more arcane than long division – I recall very little apart from the fact that at one point you needed to double the number you first thought of. In fact, and I may be unique here, I actually used the even more complicated procedure to find cube roots (though I’d certainly appreciate it if you didn’t ask me for any details).
Even in 1958 this was the most pointless activity you can imagine. At no time in my life did I ever need to employ these algorithms and they’ve long since passed into history. Fortunately politicians have never heard of them, so I’ve never heard even the most diehard traditionalist demand that extracting square roots ought to be a fundamental part of the mathematics curriculum. Otherwise they might still be asking students to find square roots by hand, and indeed do so in the examination room.
Don’t get me wrong; I do actually have some sneaking regard for Mr Gibb, Ms Morgan, and others in their profession. They have qualities which I totally lack, they’ve got to the top of a hugely competitive profession, they’re prepared to tackle big problems, and I don’t doubt they’re motivated by a desire to make things better. But I do wonder how Mr Gibb’s advisers (or indeed the audience) could allow him to speak dismissively of “… inefficient methods such as the grid method for multiplication and chunking for long division …” (London Thames Maths Hub Primary Conference, 27th March).
Now I’m prepared to assume we’re going to devote some curriculum time to the processes of long multiplication and long division and I want to write about these “inefficient methods” in part (ii).