When Numbers Won’t Behave

Some people like Maths because it feels like a land of certainty. Where other subjects – indeed life in general – teem with doubts and contradictions, numbers are cool, hard and permanent. You know where you are with them.

But is this how numbers really are, or just how we want them to be?

Once we know that two plus two equals four it can be quite comforting. We feel confident that there will never be a time that teachers, the government, our parents or anyone else turn out to have misled us about this simple fact. George Orwell, in 1984, floated the idea that ‘freedom is the freedom to say that two plus two make four’. So for him, or the protagonist of his novel at least, it was the perfect example of objective truth.

That may well be so, and certainly I myself can’t think of a better example. But despite that, students of maths can find what they are being told vague, or illogical. And some of the most confusing moments come with the most basic concepts.

I once got into a discussion with a class about whether 7 was a number or a digit. One child said ‘both’. We then started to discuss whether a digit and a number are the same thing. Some children said ‘yes’, some ‘no’. Their teacher sighed and crinkled her brow, having gone to some pains to teach them recently that a digit and a number are not the same thing.

Except that sometimes they sort of are…

Think of it like this:
51 is a number made of two digits.
5 is a number made of one digit

In the first example, 51 is a number and I have used two digits to write it. A computer working in binary, on the other hand, might communicate it as 110011, using six digits to express the same number. So it is clear to me that one number can be expressed with different digits.

I could, of course, communicate it another way altogether: ‘fifty-one’. That’s two words. Is it two digits? Maybe. What if I make up a single word to express the number… let’s say, ‘flimp’. If my new word caught on then people would be referring to the number 51 without using any digits. So this proves that numbers and digits are not the same thing. Doesn’t it?

But what about a single-digit number? The 5 in 51 was a digit, we said, but what about the 5 in the number 5? It’s a digit because it’s in a column where it means ‘5 units’. But isn’t it a number as well? And if it isn’t… what is a number?

Let’s widen this out a bit. Because ‘5’ (and I am talking about that bendy line on the page here) is also a numeral. A numeral is a mark, and can vary between languages. In Arabic, for example, 5 is written with a circle much like our sign for zero.

So 5 (European etc) and 0 (Arabic) are two different numerals, but they stand for the same digit. It helps to compare numbers with words here. The Arabic for ‘book’ is ‘kitab’, apparently. So ‘kitab’ and ‘book’ are two different words for one thing – a book.

The problem is that numbers aren’t quite like books. I have no doubt what both ‘book’ and ‘kitab’ stand for because there are some books on the shelf two metres away from me, and I could pick one up and drop it down on the table to illustrate. However, if I look at the numerals ‘5’ and ‘0’, there is nothing I can drop on the table to show what they refer to. So what do they refer to? If what they refer to is a number, how can I demonstrate what that number is without going back to the numerals all over again?

This may sound as if I am looking for a problem where there isn’t one – or at least not if you don’t go looking. But in my experience, people learning new concepts or procedures in maths come up against precisely these weird questions as they try to bend their brains around the new idea. This can happen when they are introduced to double-digit figures. Or decimal fractions.

Wittgenstein, a philosopher who was fascinated by the fundamentals of mathematics, wrote that as a philosopher he was trying to ‘show the fly the way out of the fly-bottle’. I always wondered what a fly bottle was but apparently it’s something they used to use to trap flies. Anyway, you can imagine the predicament of a fly inside a bottle. It buzzes around, ever more frantically, blocked by an invisible wall and unable to locate the exit.

That is a fitting metaphor for intellectual confusion. And you don’t have to be an intellectual to suffer from it. An ordinary 8-year-old can be unable to progress because a new concept is, basically, doing her head in. At that moment, numbers are doing things she thought they weren’t meant to do. They are behaving irrationally.

What can a teacher do? What a philosopher does. That is to:

• listen to the confusion
• try to understand the confusion rationally (i.e. how it stems naturally from the student’s current knowledge and beliefs)
• sympathise with the confusion (e.g. by saying ‘I see why you’re confused – that must be annoying’)
• consider why you the teacher believe what you believe instead, and try to explain it

Remember, we all have an upper limit of understanding in any subject, a point where the next stage doesn’t make sense to us. Reaching that point is anxious and frustrating. A guide who takes those feelings of ours seriously rather than brushing them aside is more likely to boost us up to that next level of understanding.

**Check out my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
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Posted by Lubos Remplik on 14th August 2014 at 12:00am