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=”https://www.youtube.com/watch?v=BRRolKTlF6Q”>

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:http://www.resourceaholic.com/2014/08/fractions.html ]

If you’d like to take a peek at The Numberverse, try:
http://www.amazon.co.uk/The-Philosophy-Foundation-Numberverse-everything/dp/1845908899
<a href=”http://www.philosophy-foundation.org/resources/philosophy-foundation-publications/the-numberverse&#8221; target=”_blank”>http://www.philosophy-foundation.org/resources/philosophy-foundation-publications/the-numberverse