Category Archives: Maths

How To See Into Their Heads

How To See Into Their Heads: Picturing a child’s own number line.

‘Miss, why we doing this?’ is something you hear from time to time. And however irritating it might be in tone, it’s a question that deserves an answer. After all, if we are going to take anyone’s time up teaching them anything, we should be able to say why that particular thing is worth the bother. Our reason doesn’t have to be of a narrow ‘you’ll need this to get a job’ type. It could be: ‘Understanding this will make you a better human being in countless ways’, but there must be a sense of purpose in education.  Familiarity with our curriculum can allow us to disregard fundamental questions that affect someone coming to the topic for the first time.

Let’s take an example: percentages. Why do children study them? Come to that, why do adults use them? What are they for?

This is a key question because we say nothing with percentages that can’t be said another way. So if 4% of people in my constituency voted UKIP, I could just as easily say that 0.04 of us voted UKIP. Or 4/100. Or I could (sticking with the raw data) say that it was 1971 out of 49 449.

Percentages are actually of course just fractions: they are hundredths. And at some point in the past, someone decided it would be useful to talk about parts of a whole in hundredths. Why? Why not just stick with ordinary fractions?

Well, percentages have one main advantage, which is that they are good for visualising and comparing. So let’s say I want to compare how UKIP did in my neighbouring consitituency – did they do better or worse than in mine? If I am told they got 6%, it is easy for me to compare. I can see immediately that they got more votes there than here, and a moment’s thought tells me that they got half as much again compared to here (4 + 2 = 6). But it is still less than 10%, so not a direct threat to the winner. And I can quickly conclude that even the winner of this seat would consider UKIP’s 4 % worth trying to win over to his own side, unlike the Socialist Party of Great Britain’s share, less than 1%.

All of that strikes me straight away, without me having to puzzle or calculate. Go back to the raw data of 1971 votes out of a 49 449 turnout, however. Is that better or worse than, say, 3707 from 73 788?! I can work it out, but it’s not immediately obvious in the same way.

Although percentages make comparison easier, there is one main disadvantage to them: they are not 100% (enjoy the pun) accurate. So usually when I use a percentage I will be rounding to make the figure into hundredths: 67%, 33%, 8% etc. Except… it’s not actually a disadvantage, it seems. In almost every ordinary life situation, (so, not including specialist financial data) a percentage is accurate enough for our purposes and makes the point we want to make. We simply don’t need the exact data.

How many children are taught that point when they are taught percentages?

It might not seem to matter. It might seem to be the pursuit of curiosity and trivia when there is real work to be done. But the whole procedure of converting data into percentages is meaningless without a reason for doing it.  They need to see that they are adding to their skills, understanding or wisdom.

So how do we go about proving to students that percentages make decisions and comparisons easier? The quickest way is to get them to put a set of unwieldy fractions in order of size:

4/7           5/12         16/22      4/9

They can put these in size order if they give every fraction the same denominator. But that’s a big ‘if’, and a big faff. Quicker and easier to divide the top by the bottom on a calculator and note the first two digits after the decimal point (ignore any digits after that). Like this:

0.57         0.42         0.73         0.44

It’s pretty easy to put them order of size now. Percentages, of course, are just these numbers written differently (57 % etc.)

OK, but that’s not the thing. Because I have still made a big assumption. I have assumed that kids can do what we do.  Assumed that when fractions are converted to numbers between 0 and 100, kids can now compare them easily, and immediately spot the proportions and relations between them. But can they?

One way to find out is to stretch a line of some kind along the classroom floor. You could make it one metre – there are some advantages to this – but it could be longer, which makes it easier for more people to see and participate. Mark one end 0 and the other 100. You then cut out some triangular pieces of paper with various numbers between and 1 and 99 on them. Do one each, and choose strategic numbers and a few random ones (so 25, 50, 75, 33, 66 and then random ones like 9, 42 etc).  Ask the child with the 50 triangle to place it along the line where it should go (if you’ve used triangles then you can use the point of the triangle to mark an exact point on the line). Hopefully, he’ll aim it smack in the middle. If he doesn’t you really have work to do, but the others in the class should be able to help get it to the right place. Then ask children who are confident they know where theirs goes to step forward and put their triangle point on the line.  Others can comment and suggest adjustments.

What you will see is the children’s own number lines – how it looks in their heads. These emerge as they make their attempts to divide the line visually and make an estimate of distance and proportion.

Now I guarantee that most basically educated adults, whatever their perceived ability at maths, would be able to divide the line into halves, quarters or thirds in their minds – perhaps tenths too. They would then place their triangle on the correct side of these points. For example, if you had the 40 triangle, you would know it goes on the left of the halfway point 50, because it’s lower. You might then imagine the line divided into tenths and judge one tenth left of 50. Or you imagine it divided into thirds and place your 40 slightly closer to the third point (because it’s 7 away from 33, which is a third) than the half (which is 10 away from 50).

People who have to do this for practical purposes, like builders, may well have better judgment. Perhaps artists would too.

Some children can do this kind of a thing a bit. Others barely at all. They will see this for themselves when you measure out the line.   By the way, this is where it is good to have a metre-long line after all because you don’t have to convert the distances into hundredths, it’s already there in the cm markings.  Alternatively, you could make a long line that has the correct markings on the underside that can be revealed when you flip the line over at the end.

There is certainly a big difference between what children of average ability manage and what adults of any ability at all can do. But if most of the students can’t do it well, then a lot of the purpose of percentages is lost on them. Knowing, for example, that a rise of 18 to 26% takes you past the 25% mark is the whole point. Without a grasp of these milestones, percentages don’t help nearly as much in appreciating the significance of data.

What should we do? Try to help them develop their mental number line, perhaps.  Her are some some suggestions…

  1. Get children estimating all kinds of distances under 1m and checking their accuracy until they develop a feel for where numbers between 1-100 are on the scale.  Some children will do this competitively in their breaks.
  2. Get children practising questions like ‘Is 67 closer to 55 or 75?’. See how they visualise it in their own drawings and help them to settle on strategies that help.
  3. Get children to choose which way to represent parts of a whole (common fractions, decimal fractions or percentages) when doing a task – make sure it’s not always decided in advance by the rubric of the question.
  4. So that they can succeed at point 3, make sure children experience the practical value of the three different ways of talking about amounts between 0 and 1. When I say ‘practical’ I include practical for completing calculations as well as problems in everyday life.
  5. Always let children show you how they see numbers fitting together. Don’t be in a rush to straighten out the wonky bits. Instead, help the child build a better map of the ‘numberverse’ in a way that they understand.

The first part of this blog is an abridged version of Go Compare, my chapter on percentages in The Numberverse.  The second part is something I did in a classroom once with results that surprised me.


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Maths In Singapore: Why You Don’t ALWAYS Want To Start With A Concrete Example

Question: How do you introduce new concepts in Maths?
One Answer: You demonstrate and explore them in a concrete way, then get students to represent the concept pictorially, then record it numerically – from the concrete to the abstract, in other words. So if you were introducing fractions, you’d get students to cut up a cake, then draw or arrange pictures of cakes cut up, then use digits to record the process.
That all makes sense to me, and apparently it’s the main principle behind Singapore Maths, the curriculum and methods that started in Singapore and have been followed by schools around the world attracted by the country’s performance in Maths teaching.

However, I spent last week at the impressive UWCSEA international school in Singapore. They don’t use ‘Singapore’ Maths, though, because they are an international school and so outside the Singapore system. I was there to work with the teachers on Enquiry Maths, an approach I came to through doing philosophy in schools.

The central insight of my book The Numberverse – the thunderbolt that hit me when I first got into this area – is that children will explore numbers in the abstract. So yes, they are helped enormously if they can proceed from concrete to abstract, and see how concepts are applied first. But they also, at times, can make strides by pursuing their own curiosity about numbers in the abstract. I’d like to give an example of how.

One of the many challenges the teachers at UWCSEA set me was how to use enquiry to introduce the multiplication of fractions by whole numbers, e.g. 3/4 x 7. When I learned this topic, in about 1980, it certainly wasn’t by enquiry. I had a good teacher that year but she was the sort that just showed you how to do stuff: I was told to multiply the top number (numerator) by the whole number, e.g 3 x 7 and keep the bottom number (denominator) the same, e.g. 21/4. Bingo. This gives you a correct answer, though you may need to simplify the fraction, e.g. 5 and 1/4.

I learned to do this mechanically, and as the arithmetic involved is pretty simple even for someone like me, successfully. It was some time – perhaps years – later that I twigged that 3/4 multiplied by 7 is exactly the same as three-quarters OF 7.  My confidence with the procedure wasn’t matched by a confidence with the concept, the problem lying in wait being that when I needed to apply the knowledge (whether in practical situations or abstract ones, like algebra) I was hesitant.

So I wanted the group I worked with last week to ground the new concept in the knowledge they already had. I followed the principle of starting with the known as the gateway to the unknown. I wrote this on the board:

6 x 6 = 36

6 x 5 = 30

Without saying anything about what I was doing, I asked if anyone could continue it. They wrote the 6x table backwards down to 6 x 1 = 6. I asked if we had finished. Someone added 6 x 0 = 0. Have we finished now, I asked. After discussion in pairs the children said that you could continue by ‘doing minus numbers’. I agreed that you could. Then I asked:

‘Does anything go in between these?’ and pointed to two lines in the list. The room buzzed with activity, and after a few minutes each pair had suggested another entry to the list, where 6 was multiplied by a fraction or mixed number. They wrote their ideas in the gaps between the lines. Although they used the word ‘fraction’ when they discussed it out loud, they mostly switched to decimals when writing. The children attempted:

6 x 1.5
6 x 1.75

… and so on. Some of their calculations weren’t correct, but two of the children thought through how to test the calculations practically: by imagining six people all with one and a half cakes each, for example, and counting how many cakes there are in total.

What these two children were doing was working the opposite way to the concrete-to-abstract method I mentioned at the beginning. I’d got them to think about it purely as a matter of logic: i.e. there must be something in between 6 x 0 and 6 x 1, so what could it be? Now they were testing that abstract reasoning by applying it to a concrete scenario. Some felt more inclined to do that testing than others, and that’s fine.

So my point is that you can start at either end – concrete or abstract. Different topics, different students, different teachers and resources – all of these may influence a teacher’s decision about which angle to come from.

Where this comes into its own is if you extend this from multiplication to division of fractions. So now you might write:

6 divided by 3 = 2

6 divided by 2 = 3

6 divided by 1 = 6

Now if you just stop a moment you might spot something new this time. Whereas last time moving on to 6 x 0 was quite straightforward, 6 divided by 0 is far from that. In fact I once set this to a class as a starter and asked them to figure it out, having stumbled across the problem myself and got very confused. I initially thought the answer must be 6 or 1. But neither makes sense: 6 by 1 is 6 and 6 by 6 is 1, so neither of those can have the same answer as 6 by 0… surely?

If you try the concrete approach of physically trying to share 6 oranges, say, between zero people you find that you are in fact left with 6 oranges as you have no-one to share them with. But then the whole point of sharing is that I should share out all that I’ve got till I have nothing – not have everything still left.

Working at an abstract level was something I found more helpful on this occasion. It took me a while to figure this, but I remember one boy who came up with it within about 10 seconds of being presented with the problem:

‘You can’t do it. Because you can’t do it backwards. If 6 divided by zero had an answer, that would mean something multiplied by 0 equalled 6, but it can’t.’

This is a perfect reductio ad absurdum argument. It shows that if we allow that 6 divided by 0 is possible then we have to allow also that something multiplied by 0 equals 6, which is absurd. This boy was thought not to be very academic, by the way, but OK because he was good at sport. I’ve got a feeling he’ll do just fine in life.

If you’re interested, you can see Matt Parker prove it more mathematically and entertainingly here <a href=””>

But that’s all a digression! The point is that by running the same enquiry for dividing by fractions as you did for multiplying them you might get the children filling in values like this:

6 divided by 3 = 2

6 divided by 2 = 3

6 divided by 1 = 6

6 divided by 1/2 = ???

If children can spot a pattern in what comes above they can make a conjecture about what comes next. For example, they might say that the answer must be higher than 6.  Good start.  How much higher…?  Tricky to say.  Or they could see that the inverse operation works in each line, so they ask themselves: ‘what do you multiply by a half to get six?’, or ‘how many halves make six?’. Then the answer is quite straightforward: 12.

Were you, on the other hand, to try and start from a concrete scenario, then what? You imagine yourself giving six oranges to half a person?! I would be genuinely interested to hear from anyone who has managed this, as I’d like to be able to come at it both ways. Until I hear different, though, I’ll go on believing that sometimes the concrete-to-abstract is perfect, but that the purely abstract sometimes blows it away.

[Since posting this I’ve seen an excellent round up on the x & ÷ fractions issue: ]

If you’d like to take a peek at The Numberverse, try:
<a href=”; target=”_blank”>

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INSET day yesterday. Hello teachers, I’m one of those people that come to your school on the first day back from holidays and interrupt your preparation for the coming term with power points of wisdom on how to teach.

I like to start by finding out something about what the teachers want, and what their beliefs about education are. Yesterday, one of the ideas that was mentioned – and generally agreed with – was ‘risk-taking’; the staff wanted their pupils to be willing to experiment and explore, and not to fear making mistakes, particularly in Maths. I agree with this aim, but… it’s ironic to hear it coming from teachers.

Why is it ironic? Because a lot of teachers are scared of making mistakes. I suppose there is more than one cause of this. But one of the causes is the current pressures of the profession. (This piece ends on a positive note, by the way, so stick with it!).

The problem is that of the teacher who is running scared of negative judgments. This person might be an NQT or an old-timer, but they are threatened by new ideas because they are worried that they won’t be ‘able to do it’. They are not hostile, and aim to please, but fear assessment, and fear failure.

In some cases, they may simply be an unconfident sort of person. But often, they have acquired this blanket professional anxiety because their training and their CPD have often involved nerve-wracking observations, crushing comments or clumsy feedback. Things have not been explained to them properly. They have been told things that contradict other things they were told. They are not supported by interested experts who believe they can succeed.

They and their mentors are not helped by the incessant tides of new initiative, new curriculum, restructuring, ideology, overcooked controversy and daft fashion that erode the kind of stable foundations across the industry that help professionals to grow. Confusion messes with confidence.

If I am right that these problems exist (though the causes and extent may be arguable), how can teachers expect pupils to take risks? We must remember that the best way to teach children values and habits (as opposed to facts and practical skills) is to model them ourselves. And that is where so many teachers fall down. So many will not risk being wrong in front of their class, their colleagues, or their managers.

In Maths, the subject in which I have a particular interest, the need to create a fear-free environment is most vital, as so many children will only engage partially in class discussions or activities out of fear of looking foolish. There is a risk involved in venturing a daring or untested idea. Safety lies in passivity and caution.

So the teacher needs to model the behaviour he wants to see. That means he has to work things out in front of the class, to correct his working, rub out, re-try, re-think and see the funny side of his own errors. Unless he does this, he can’t model risk-taking. And if he can’t model it, he is probably only preaching it, not teaching it.

It follows logically that for teachers to do this, it requires head teachers (as teachers of teachers) to do it too.

And that is why I was so pleased that in yesterday’s INSET, the head teacher was in the room throughout. He put forward some answers in an experimental spirit and wasn’t worried if they weren’t adopted by the rest of the group – or me, the facilitator. He certainly came across as an intelligent and interesting guy, but it was partly because he risked voicing half-formed ideas to see what other people thought.

Not only was he there all the way through, he volunteered to try out the techniques we were introducing in front of the rest of the group. He wasn’t fazed at all when he found some unfamiliar aspects quite tricky, but described how it felt and what he thought, for everyone’s benefit. He really was prepared to ‘take one for the team’ in that respect!

Don’t get me wrong. I’ve worked with some brilliant head teachers who haven’t sat in on INSETS. And I know they have a mountain of other things to do. So I’m not advocating it as standard procedure across the board. But it was fantastic that someone was willing to demonstrate such a willingness to learn, and to learn by taking risks, when his own staff had cited that as one of their objectives.

So thanks, Mark. I hope you and your team – and the children – benefit from what we brought along.

For more thoughts on Maths teaching take a look at my book, The Numberverse, available on The Philosophy Foundation website and, of course, via Amazon.

For more on teaching risk-taking see Peter Worley’s article on the 9 Dot Problem on Innovate my School.

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Is 3 The Same Upside-Down?

This is a session that started as general philosophy, but led into a Mathematics focus. (It’s Year 6, mixed ability in a 1-form entry primary school.) Here is the story/scenario I began with:

“There was once a man who had been working for 50 years, and it came to the day of his retirement. People asked him ‘What will you do all day now that you don’t have a job to go to?’. And the man thought about this for a while and said ‘Something I’ve never had the time to do before’.

The next day he woke up and said to his wife ‘I’m going to invent something. I’ve always wanted to invent something, and now’s my chance’. So he went down to the shed at the bottom of the garden, where he kept all his tools and bits and bobs for making things.

After a week, he came back up to the house after another long day in the shed. He made his wife a cup of tea and as he gave it to her with a sigh. ‘How’s it going?’ she enquired, sympathetically.

‘Well… I have all these ideas for things I’d like to invent. But I don’t know how to invent them. I’ll just have to keep trying.’

His wife nodded, and said gently: ‘To invent something, you’d need to be a bit more up-do-date with technology. And you’re not really that good with technology. Your phone looks like something out of a museum. And that time you tried to take a picture of your grandchildren with my phone, you took a selfie by accident. Technology is pretty complicated these days.’

The man thought about this for a while. Then he jumped up out of his chair, and said: ‘You’re right. I’ve been thinking about this all the wrong way’ and off he trotted, back to the shed.

The next morning, at breakfast he looked very pleased with himself. ‘I’ve worked out what to do,’ he explained to his wife. ‘I’ve got to invent something that doesn’t need technology. So I’ve narrowed it down to three possibilities. I’m going to invent either… a new word, a new shape, or a new number.’ “

I didn’t give the class a question at this point. I just asked them to discuss their reactions to the story in pairs, saying:

There is no question at the moment. Just tell your partner what you think.

I had questions up my sleeve, in case this initial prompt came to nothing. These were the questions:

If you were his wife what would you say?
Are all three possible, and what makes you say so?
Which would be easiest, and why?

One of the first responses was:

‘He can’t invent a new number, because numbers go on for ever.’

The next answer was:
‘He can’t invent a new number but he can invent a new rule. Like he could say that a number written upside down means a minus number, so an upside down 5 would mean minus 5.’

There followed a spirited and very flowing response from a youngster called Tabatha. Two of the things she said were:

1. It would be easier to invent a new word. Because in Mary Poppins they did it with Super-calli-fragilistic-expi-alidocious.
2. Turning numbers upside down to make them into minus numbers wouldn’t work because if you turn 1000 upside down, it’s still 1000.

Now, I love this second point, and I would like to put it to you, the reader, to ponder. Is 1000 the same upside down? Plump for an answer before you read on!

The children discussed it for a while. One of the answers was:

‘Numbers are the same upside down if they join up. 1, 8, and 0 are the same upside down. It’s because they join up. 4 and 2 don’t join up.’

By this time, several pupils have come up to the board to try to illustrate their points. I switched the term ‘number’ to ‘digit’ at one point, saying ‘Which digits are the same upside-down?’. Although it was obvious to me that the ‘joined up’ theory was fallacious, I waited for it to be superseded by something else.

Quite soon, a long-winded and rather confused answer contained the word ‘symmetry’ at some point. At the end, I said: ‘He mentioned the word ‘symmetry’. Does anyone else think this is anything to do with symmetry or not?’

The discussion went off at another tangent at this point, though in an interesting way, as the children debated whether the numeral ‘3’ is the same upside down. This brings into focus the very same issue that complicates whether 1000 is the same upside down: it sort of depends how you make it go upside-down.

By this time, I and most of the class were periodically turning our heads upside down to read the numbers on the board. If you do this, you will find that 1000 is not the same, because it reads as 0001, because all the digits look the same, but appear in a different order (this assumes that we are writing 1 with a single vertical stroke, though the children noticed that you might not). The numeral 3 is also different upside-down because it appears ‘backwards’ with its open side facing right instead of left. So…. is that the answer?

What about if we use a mirror? If I place a mirror under the numeral 3, what do I see in the mirror? Another 3, facing the correct way. The number 1000 will also read correctly. Which is weird because I always thought that if you look at something in a mirror then left is right and right is left.

Huh? This is a nice example of how a simple question can lead down an interesting and unexpected route. With this class it led away from the concept of whether numbers are invented or discovered into the practicality of whether something looks the same upside-down. But this then led into a discussion where the class needed the concept of symmetry to explain a phenomenon that they could all observe – which is a great way of consolidating that concept in the minds of the children. They would also needed to refine their application of the concept of symmetry to explain why 3 is the same upside-down if you use a mirror, but not if you turn your head upside down.

We didn’t quite get there. Can you explain it?

By the way, we did also have fun talking about how you invent a new word, but this blog series is about maths. Next time I do it, we might end up on shapes. I just don’t know. If we do, I’ll report back.



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Mine Do It Already: Nought To Reasoning In 60 Seconds

Are your children reasoning in the lesson? Not sure? Or maybe you want to prove that they are?

Here is a simple activity that is fun for the children and shows you – or anyone else – how they are reasoning. Before I go on to explain it, it may be useful to give a definition of reasoning that we can use here – just so we know what we are talking about.

Reasoning is described this way by ACARA, the Australian curriculum authority:
‘Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.’

A simpler, neater, definition is that when we reason, we use information that we already have to prove information that we don’t have. So, for example, we use DNA evidence to reason that someone must be guilty. We use the lengths of a polygon’s sides to calculate its area.

It also helps to define something negatively – in this case, to say what is NOT reasoning. And here are some examples:


Not that those things are bad in themselves. Sometimes we have to guess to get started. Sometime the best thing to do in a given situation is just copy someone who seems to know what to do. And sometimes it is great to realise that you know the answer because you remember it from before. That’s all fine. It is just that where we rely on these strategies we are not, at that point, reasoning.

Here is the activity…

The simplest way to explain is to say that it’s 20 Questions, but with numbers. So 20 Questions goes like this:
• One individual thinks of a type of object at random – such as ‘chair’.
• The rest of the group can ask him/her questions. The individual will only answer Yes or No.
• If the group can guess the word in 20 questions or less they defeat the individual.

For maths, the individual thinks of a number instead of an object. Usually I say it has to be between 0 and 100. With Year 2 or lower, you might want to set it at 1-20. Also, instead of a limit to the number of questions, our goal is to get the answer in the lowest number of questions, improving our score with practice.

If you play this game you will be able to see the reasoning of your class and the people in it. At primary level, the person who finally guesses the answer often wants to claim maximum credit from the others even though he or she may have contributed very little to the hunt. That is a sign that reasoning – using reasons to move from one question to the next – is not taking place, at least with that person.

Between each game, I ask the children to say which questions were good and why (so it’s a good idea to write the questions, or short forms of them, on the board as you go). At the beginning, some may say that ‘Is it 17?’ was the best question because the answer was 17. But gradually the children will start to spot that the final shot was the easy one as all the other numbers had been eliminated. And if not enough numbers have been eliminated then glory-seeking stabs in the dark like ‘Is it 17?’ are a waste of a question (unless the questioner happens to get lucky). The class often takes a while to articulate the idea that a good question ‘narrows it down’. In other words, it reduces the possibilities to a narrower range.

Children also fail to realise, to begin with, that a ‘No’ answer is not worse than a ‘Yes’ answer, necessarily. If you ask ‘Is it an odd number?’, then either a Yes or No answer gives you exactly the same amount of help. And there are some Yes answers that tell you very little. For example, if you ask ‘Is it a two-digit number?’ the answer will probably be ‘Yes’ but it doesn’t get you far because you still have so many possibilities left. Now, you may still feel that ‘Is it a two-digit number?’ is a pleasing answer for you the teacher, because it shows the children recruiting prior knowledge to apply to the task (i.e. they’ve been learning about one/two/three-digit numbers, have remembered it, and are trying to apply that knowledge here). I couldn’t agree more, so you may want to praise some questions while preparing to nudge the questioner towards a more effective strategy.

After a few runs through, I ask the children if they can come up with a strategy that will always get them the answer in a set number of questions – so, can they guarantee to get the answer in 10 questions, or 5? Children then explain their strategies and we try them out. This is crucial because they are now thinking about their reasoning.

Most classes latch onto 0-50 as the first question before long. Either that or ‘Is it even/odd?’. However, it can go in two ways from there. Usually, you have two different strategies being used within the class. For example, after ‘Is it between 1 and 50?’ with the answer No, we might get:

Is it odd? Yes
Is it between 50 and 75? Yes
Is it in the 3-times table? No

Now it is quite hard from here to work out which numbers that leaves. Try it yourself! It’s better to stick to one strategy. So:

‘Is it between 0-50?’ halves the possibilities should be followed by a question that halves what’s left in a predictable, memorable, way. So if the answer was No, then the next question should be either 50-75 or 75-100, and so on, halving each time. For example:

0-50? No
50-75? Yes
50-62? Yes
50-56? No
56-59? Yes
56-57? No
58? No
The only remaining number is 59.

Using the ‘halving’ method outlined above, the class should be able to guarantee to find any number within 7 or 8 questions. You may able to refine this further. I’ve only had a couple of classes who got that far (I generally teach primary).

When I’m playing the game, though, I’m content for the children to circle gradually closer to a strategy like this, and don’t worry if they never quite nail it. What I’m interested in is them looking at a task and saying to themselves: ‘How do I make this simple? How do I work steadily towards the answer?’.

One side issue that comes up here is a problem with the edge of the range. For example: is 50 itself between 0 and 50? You might be sure that it is. But imagine if you have a group of children sitting in a row in this order:

Floriana, Luke, Chester, Adibola, Polly.

We wouldn’t say that Floriana is sitting ‘between’ Floriana and Polly (we’d say that Luke, Chester and Steven were). So the word ‘between’ can be applied differently. You can introduce the word ‘inclusive’ here to help the children, (‘Is it between 0 and 50 inclusive?’) and they will have learned a valuable lesson about the definition of a range. In a similar way, by the way, the question ‘Is it below 50?’ doesn’t make it clear (for some children) whether 50 itself should be judged above or below! You could argue that ‘below’ is not ambiguous at all – 49 is below 50; 50 isn’t – and that’s exactly the kind of precision in the use of terms that we want the children to learn.

If, instead of the ‘halving’ method, children try to extend the ‘Is it odd?’ question into a strategy, another set of problems is thrown up. Because to extend that strategy means using times tables:

• Is it odd?
• Is it in the 3xTable?
• Is it in the 4xTable?

This is much harder to operate, and soon challenges the reasoning of the child. For example, if the answer to ‘Is it odd?’ was Yes, then there is no point in then asking if the number is in the 4xTable (or the 6, 8, or 10xTable, come to that) as odd numbers won’t feature in any of those times tables.

Another issue is that it is very hard to know what numbers are left if you eliminate them through timestables. You could do it with a number line or square, crossing out the eliminated numbers, but few of us could do it in our heads.

And finally, and most fascinating, is that if the person choosing has chosen a prime number, then it is not in any times tables – except of course its own. So you’d have to wait until you got to ‘Is it in the 97xTable?’ to eliminate 97.

All of these knotty problems are rich pickings for reasoning. Lead the children again and again back to a discussion of their strategies. Do that by asking ‘What did the answer to that question tell you?’ to develop the children’s logical thinking. Try not to jump in and tell them – stick to questions. If you feel they’re falling short of what you’d hope, just keep encouraging them – that’s more valuable than getting them to the best strategy fast. After all, it’s only a game – not a SAT.

Trust me, you can spend a whole hour on trying to crack this. Alternatively, you can use it to warm up/down at the beginnings or endings of lessons or weeks. Just remember that its main value as a teaching tool – rather than a mere time-filler – is in developing awareness of reasons, and how a chain of reasoning can solve a problem.

Once they have exhausted the possibilities of this game, you could try some of the games on the NRICH website. I particularly like Strike It Out
…and Got It – where the whole class can play against the computer

I am indebted to Peter Worley at The Philosophy Foundation for showing me this game, and Andy West (also TPF) who reminded me of its value.

For more on how to introduce Enquiry into your set of teaching skills, try my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.

You can also buy it here:

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Reasoning & Problem-solving – The New Black? The New Brain Gym? Or Just The New Curriculum?

Reasoning and Problem Solving

When the new Australian curriculum goes live in 2015, teachers across the country are being asked to – among other things – get students thinking mathematically. Two of the four proficiency strands are Problem Solving and Reasoning, (the others being Understanding and Fluency).

There is a great deal of overlap between Problem Solving and Reasoning. After all, it would be hard to solve problems without reasoning about them and it would be hard to demonstrate reasoning ability without some sort of problem to get our teeth into.

At the foot of this post is a link to the helpful ACARA curriculum website and I have also pasted in the clear definitions they give there.

In England the new KS3 curriculum says students should:
• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

To help any teachers trying to incorporate these aims into their practice, here are some thoughts about Problem Solving and Reasoning, their difference and similarities, and what to do…

We can begin with problem-solving. People use certain skills, or develop certain habits, to solve problems. Take a jigsaw puzzle, for example. There are certain things that a skilled puzzler might do, such as: search for corners and edges, group pieces of the same colour, stand the box lid up to compare its picture to the one being formed, use both piece-shape and picture detail to match pieces, and so on.

However, it is possible to use some of these skills without reasoning all that much. So someone may ‘do the edges first’ because that’s the way they were taught to do it, but they would be unable to explain why it is better to start with the edges. So when pupils do exam questions, they may be solving problems quite effectively because they have encountered problems of that type before (like the puzzler who’s done lots of jigsaws) and ‘know the rules’.

Problem-solving skills are very important. An electrician uses them to identify a fault, logically testing until the cause of a problem emerges. A doctor uses them to diagnose a single condition from a set of symptoms and test results. And although I have given a very narrow picture of problem-solving above, with someone knowing the rules and being familiar with the problem, it is fair to say that really good problem-solvers attack unfamiliar problems effectively too. According to ACARA ‘Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations’.

So how does reasoning differ? Basically, it is more abstract. Whereas we can judge problem-solving according to outcome, to a large extent, (so electricians or doctors who fix things – or have good ways of fixing things – are good problem-solvers), someone could possibly demonstrate good reasoning without managing to find a solution. A scientist might observe a phenomenon and show that it needs explaining, but have no answer. He/she may observe ‘Sunrises look different to sunsets: sunsets have a warmer, redder light and are more likely to cast spectacular colours across the sky. This must be because of a difference in the air the light is passing through or a difference in the objects it strikes, as the sun’s rays themselves would not be affected by the movement of the Earth.’ There is no problem to solve here, as such. And the scientist may be wrong about the reasons too, or even the observation itself, but he or she is reasoning: noticing, asking why, and attempting to work out where explanations might be found. Think of that great reasoner Isaac Newton: when he saw the apple fall out of the tree and wondered why, it wasn’t because gravity was a problem!

In my opinion, though, there is no need to ponder at length over the difference between problem-solving and reasoning, as so much of it is the same – especially in practice. The one useful distinction for me is that reasoning ability is a far more open thing, and problem-solving might not be enough. Reasoning leans more towards independence and creativity.

Can teachers teach these kind of skills? Those that say No have a point when they say that a lesson on ‘problem-solving’ is unlikely to be effective. I agree with that, but you could say the same about things like discipline, imagination, initiative and conscience. None of these things are taught as topics in themselves but… and here is the key thing… children learn them from adults or peers and they learn them (if indeed they do!) the whole time. The adult can teach by being a role model or by guiding children through experiences in such a way that these skills, or virtues, are acquired almost imperceptibly.

And so I believe that I have, on occasion, taught both problem-solving and reasoning through enquiry. An enquiry is an attempt by a group of learners to explore a topic. The exact question or problem to be explored is not automatically provided by the teacher. The teacher is more likely to present a stimulus that is rich in possibility and controversy. The question the pupils will be trying to answer will emerge from their own reactions and reflections.

Here are three possible stimuli, or starting points, for enquiries:
1. The children make a number line on the floor with each number on a separate piece of paper. Questions that might arise are:
Where will the line stop? How do we know?
Is there anything before zero? How do we know?
Is there anything in between the numbers? What?
2. Give the children a 10×10 number square. Challenge the children to find patterns in it. The class vote for which patterns they find most interesting. Questions or observations arising from the class or teacher might be:
If you draw a straight line through any group of numbers can you always say what the pattern is? (e.g. if you draw a vertical line, you always add 10 to get the number on the next line; if you draw a horizontal line, you add 1 – a diagonal line… then what?)
3. Show the children a set of different shaped drinking glasses. See what emerges. If nothing, then prompt them by asking which glass is ‘the biggest’. Then ask if the biggest one will take the most water, and how we can find out.
The process of facilitating an enquiry mainly involves reducing the length and frequency of teacher interventions to the minimum, and – more importantly – withholding judgment of the content of the children’s discussion.

Teaching Maths – or anything else – through enquiry gives a teacher a chance to nurture reasoning in the children. It treats mathematics as a field of discovery – as it once was for earlier mathematicians in history and still is for those high-fliers at the cutting edge of maths today. It allows students to pursue their own thoughts rather than rely on the teacher always to present them with a problem, still less a ready-made solution.

This is not to say that they wander aimlessly; the mere fact that children have freedom would not guarantee useful learning. Nor is it to say that teaching styles that are not enquiry-driven are ineffective or wrong. It is more that if children never have freedom to follow their own curiosity, then learning – if it is taking place – is more limited in scope, and less likely to allow young minds to fully flourish.

For more on how to introduce Enquiry into your set of teaching skills, try my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.

The ACARA definitions:
Problem Solving
Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

The above are excerpts from:


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What’s Philosophy Got To Do With It? Or… We’re Mates With Maths

When I told people that my work with philosophy for children had moved me into mathematics for children, and how we teach it, a lot of them were surprised. They are still surprised when I insist that philosophy and maths are closely related. For many, those two subjects would seem opposite ends of the spectrum: at one end is cold hard mathematics with its truths set in stone, and the other is philosophy, as vague and elusive as a puff of smoke. But this is to misunderstand them both.

Historically, mathematics and philosophy went hand-in-hand for centuries. The mathematician whose name is probably known to the most people, Pythagoras, was a Greek philosopher. And one of the most famous of all philosophers, Descartes, was also a mathematician.

These days, both disciplines have come so far that it would be asking a lot for even a genius to master both. So there probably won’t be another Descartes or Pythagoras, with a foot in both camps. But the two subjects are still linked. One reason why is that they both combine the mental abilities of logic and imagination.

As most of us know from school, we need logic to solve the mathematical problems we are set. When we say to ourselves, ‘The answer to this multiplication must be an even number, because we are multiplying two even numbers together’, that is pure, simple logic. But where does the imagination come in? Well, if you speak to mathematicians who work on the research side, trying to map out parts of the universe of numbers that have not been fully explored, they all maintain that imagination is essential. They mean the imagination to ask ‘What if…?’ and follow a train of thought onto new ground. Or the imagination to conceive of shapes and forms that lesser brains simply boggle at.

The same is true of philosophy. In philosophy, the illogical is not allowed. There are, of course, schools of philosophy which claim that the illogical is a necessary part of thinking. But even this discussion over whether logic is the be-all-and-end-all still puts logic at the heart of things. And as for imagination, yes, the great philosophers have all been hypothetical thinkers, able to picture the world in a multitude of ways.

This is important for the study of maths. Every single school child studies it. To get the maximum from each one, we need to make sure that the imaginative, curious children have the chance to explore maths in that way. Yes, they too need to be drilled and practised on calculation methods. But they need more if they are going to take maths to their hearts.

In the UK, and the US too, taking maths to your heart is quite a comical notion for many. But that’s not the case elsewhere. Many children in East Asia, for example (China, Japan, Korea) claim to love maths. There is no social stigma to liking it, or being good at it. And that is a fact about the whole society, much more than its teaching methods.

Until we in the West take curiosity, imagination, wonder and mystery as essential parts of mathematics, we will lag behind.

We can do this by seeking out questions in maths that are difficult and starting to think about them. And the difficult questions are not only in the difficult areas of maths. Just asking yourself if zero is a number, and trying to prove your answer is enough.

Here are some others to get you started:

What happens if you divide a number by zero?
Does Pi go on for ever? How do we know?
Are there more fractions or more integers (whole numbers)?
How many shapes are there?
How many lines of symmetry does a circle have?

Some of these big questions we can answer. Some we can’t. Some no-one ever will but we don’t know which ones those are.

Philosophy is the missing link that makes maths meaningful, which is why philosophers are very good at sniffing out questions like this. But do you know what? They are not as good as children. Given the chance, children will bamboozle you with queries that strike right at the heart of what numbers actually are. And you won’t be able to answer them sometimes. And that’s great, because it keeps the flame of curiosity alive – for them and you.

**Check out my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
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You can also buy it here:


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