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.

This reminds me of the ‘demonstration’ in Plato’s Meno where Socrates questions an uneducated slave boy in order to show that knowledge ‘comes from within’. In that the boy starts off trying to answer the question of what line length would give a square of 8 square feet until he gets stuck and comes to a dead-end. Socrates realises that the boy is in need of the concept of ‘area’ before he can proceed. And again, all this reminds me, Andy, of what you once said at a talk you gave about your maths book at The Sunday Times Festival of Education. You said something like ‘we need to get the children asking the questions to the answers we are teaching them.’ In other words we need to provide a circumstance where the children realise that they need pi or area or symmetry and this is exactly what this kind of teaching approach results in. Great stuff!

What a fascinating discussion from an interesting prompt. I love the way the children’s thinking diverged from the prepared questions. The skill of a great teacher is seen in the ability to follow the children’s interest and help them discover what they wish to know.