There’s a story I tell whenever I get the opportunity. You must know it too. It’s the story of how the young Carl Friedrich Gauss, who in 1785 or so was aged about eight, was set the task of adding the whole numbers from 1 to 100. Rather than adding each number in turn, he promptly wrote the answer on his slate and placed it on the teacher’s desk.
It’s a great story, and it offers probably the only piece of genius mathematics which we can all grasp. I’ll invite children – and indeed teachers – to consider how he might have been able to give the answer so quickly. He never did explain his method, but presumably recognised that you can take the highest and lowest numbers, 1 and 100, and add them to make 101. Then the next highest and the next lowest, 99 and 2, making 101 again, and so on. Then all he had to do was notice that there will be 50 pairs totalling 101, so giving a total of 101×50, equalling 5050.
One of the things I love about this is the immense power it gives us. We’re not restricted to adding the integers from 1 to 100; adding the whole numbers from 1 to 1000 is little more work. Your set of numbers doesn’t have to start with 1, and as long as they increase by the same amount each time they don’t have to be whole numbers either. Once you’ve understood the method you can find the total of sets which include fractions, decimals, and negatives – there’s a formula you can use for summing such series, but learning it becomes wholly redundant.
Another reason the story’s so popular is its great human interest and it’s been told time and time again; there’s a website with well over a hundred versions (http://bit-player.org/wp-content/extras/gaussfiles/gauss-snippets.html ). Many of them are very fanciful, but it’s easy to pull out the basis – the task itself, the little boy, and the school-master Johann Georg Büttner.
Many of the versions have incorporated details which are distinctly fanciful – that Büttner was idle, or a sadistic bully, who was scornful and disbelieving of his young pupil. Often there’s a David and Goliath slant – the ingenious pupil defeating the hulking teacher. Now in the last couple of years I’ve done a large amount of reading about mathematics teaching and I’d like to offer a different interpretation which I think is far more accurate.
It’s lucky Gauss was born in Germany. If he’d been English it’s likely the world would never have heard of him. It’s frequently said England was the worst educated country in Europe; in England it’s unlikely there would have been a school for him to go to, and there was no great desire from anyone to do much about it. The church and the gentry didn’t want their peasants to be too well educated, and parents were happy to put their children out to work – most English eight-yearolds would already have been working and earning for a couple of years.
And where there was provision it was often scarcely deserving of being called a school, with the teacher someone looking to top up his main income, or an older person no longer able to earn a living in other ways. England was so slow developing an educational system that Gauss was middle-aged by the time the first tentative steps towards a national English system of schools were taken, and the first generation who’d studied and trained to be teachers didn’t emerge until he was an old man. Indeed, it’s scarcely believable, but when Gauss died in 1855 there were hundreds of English teachers who were illiterate and couldn’t sign their name to documents.
So Carl was indeed fortunate to have been born in one of the German states. Prussia, for example, had established teacher training programmes before 1750 (virtually a century before England), and had compulsory state education to 13 before 1800. In England attendance didn’t become compulsory until 1880 and it was only at the very end of the century that the leaving age was raised even to 11, and then 12. But even in 1898 attendance was still nowhere near 100% and there were still cases of 5 and 6-yearolds working 12 or 15 hours a week.
Far from being an ignorant oaf Büttner was a trained professional. Rather than ridicule Carl’s achievement, he created an individual programme specially for him. His assistant Johann Martin Bartels lived on the same street as Carl, and Büttner arranged for him to give Gauss individual tuition. Bartels may well have been the most remarkable teaching assistant of all time – indeed, he became a university mathematics professor himself, numbering Lobachevsky among his students. His relationship with Gauss was so productive that they were still corresponding forty years later. What an amazing piece of good fortune that a tiny school should have such a tutor available!
The help Büttner and Bartels gave Carl didn’t end there. From his own purse Büttner bought Carl the best mathematics texts available, and he had the contacts to ensure that Carl’s education didn’t end at the elementary stage but continued into secondary school; from there he and Bartels arranged for the Duke of Brunswick to provide for a university fellowship which set him on the path to become the “Prince of mathematicians”.
Few of us will have the good fortune to number a genius among our pupils – the closest I’ve got is to have known Dick Tahta, who Stephen Hawking has always acknowledged as his inspiration. Johann Georg Büttner appreciated a pupil with exceptional ability, and deserves a far better reputation than he’s been given. He recognised and nurtured one of the greatest mathematical geniuses of all time and rather than traduce his memory all teachers should be proud of the example he set us nearly 250 years ago.
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).
I’m as agnostic as anyone about contemporary art, but if one of an artist’s intentions is to make you look at something with a fresh approach, then I had something of a Road to Damascus moment earlier this year.
It’s nearly six months ago now, but as part of the golden wedding celebrations in April we took ourselves off to Blenheim Palace for a slap-up meal. It was the most beautiful day, and we strolled around the grounds – or at least some of them, for Blenheim isn’t your common or garden palace. The building itself is reputed to cover seven acres; it contains a library 55m long. It was designed by Vanbrugh in 1705 and the landscaping was done by Capability Brown.
It’s the home of the Duke of Marlborough, and Winston Churchill was born there.
When we visited, there were a number of pieces of contemporary art both inside and art, but the one that took my eye consisted of a square array of blue porcelain sort-of-three-quarter-spheres. Neither Jill, nor our son, were greatly impressed, but each time I looked I saw something more in the layout.
Looking down a corner I saw a diagonal of six near-hemispheres. To each side of the diagonal there were further diagonals, of 5, 4, 3, 2, 1. So 6² = 1+2+3+4+5+6+5+4+3+2+1, and I could write a similar statement for a square of any size.
But another look brought out that the 5+4+3+2+1 grouping is the fifth triangular number, so it’s also true that 6² = T5 + 6 + T5 So this is another way of looking at square numbers.
Another one is to count the long diagonal in with one of the sets of 5+4+3+2+1. So this time we have (6+5+4+3+2+1) + (5+4+3+2+1) – in other words T6 + T5
And then I could use the near-hemisphere at one corner as the central point of an L-shape, in which nestled a series of smaller and smaller L-shapes. Numerically, 6² = 11+9+7+5+3+1
I don’t think any of these visualisations were actually new to me, but the array certainly brought them to life in a way that was original and powerful. To me, at any rate – neither Jill nor Simon was ever convinced.
You’ll gather that we don’t usually move in such circles, but we discovered that quite by accident we’d visited an exhibition of the work of the Chinese artist Ai Weiwei. Though confined to his home and studio in Beijing, he created the largest exhibition of his work in the UK via 3D computer models. The exhibit with the near-hemispheres is called Bubbles, and took him two years to create as he experimented over and over again to find the precise details of the shade and glaze of the porcelain ceramic.
For nearly 35 years I’ve run my own games magazine, originally by post and now of course by email. Some really expert gamesplayers have taken part, and one game we played with great success was, believe it or not, Snakes and Ladders.
Here’s an introductory version we’ll call Invisible Snakes, and Ladders. As the Gamesmaster, I’ll publish a board showing only the Ladders. I am the only person who knows the positions of the snakes, but I will tell you they’re not positioned randomly (for example, their heads may be on multiples of 11 with each snake sending you back 6 spaces). Furthermore, each player may choose their own die-roll each turn, with the proviso that in your first six turns you must use each number 1,2,3,4,5,6 exactly once. For the next set of turns you again have to use each number once – this is easily managed by giving each player a set of 1 to 6 cards.
So as the game goes on players can deduce something about the nature of the snakes and plan how to make the most of their knowledge by using their die rolls to the best effect. It’s a game full of observation, hypothesising, and strategy.
Of course, I can make things a little more interesting by making the Ladders invisible as well as the Snakes.
And when you play with really expert gamesplayers they’ll want things to be as challenging as possible. So if you’ve got invisible snakes and invisible ladders, the next step is to make the board invisible as well!
So now we really do have something close to Ultimate Snakes and Ladders. As well as having to deduce the positioning and effects of the snakes and the ladders, players have to find how the board is laid out. Is it a normal S&L board or a conventional Hundred Square? Does the numbering run from left to right or in another direction? Is the board in ten columns or six, or eight, or eleven? In fact, is the board rectangular at all (I recall we used triangular layouts and perhaps even circular ones)?
I recently heard an eminent maths educator say Snakes and Ladders held no interest because it was purely random, so that’s why I’ve done this series of four posts. And I haven’t even mentioned shrinking ladders (which wear down as they’re used), or growing snakes (which expand as they feed), let alone boards with Trap Doors!
Our new National Curriculum in England demands that mathematics should incorporate Fluency, Reasoning, and Problem-Solving, and there’s no reason why Snakes and Ladders shouldn’t stimulate all three of these aims.
We’re used to the regular patterns multiples display on a conventional Hundred Square. Multiples of 2, for example, fill the even-numbered columns, and you can probably visualise without much difficulty the different, but equally regular pattern they display on a Snakes and Ladders Square.
As a mental exercise, try a couple more. Start with multiples of 10, and that should give you a start to see if you can predict what happens with multiples of 5.