Understanding how a person feels when she or he cannot grasp a matter that everyone else finds so easy to understand that it is “obvious” is vital. Because if the whole subject matter is outside our comprehension, it is hard to the point of impossible to teach.
For example, if we were faced with a child who believed that the sun went around the earth, then we could at once see that, because the child sees the sun travelling across the sky each day, from the child’s perspective the sun obviously goes around the earth.
That would give us a starting point. We would know how the child sees the issue, and from that see where the problem lies, and as teachers we would be able to start at that point.
But supposing we meet a child who simply doesn’t understand numbers – how can we relate to that? How can some people not understand that six take away four leaves two? It is, after all, obvious.
When we are in this position, it is clear that something has to be done to help us understand the child’s perception (or lack of it) of numbers. But this is difficult since quite obviously we as teachers cannot unlearn the maths we know and which we use as a fundamental part of our everyday life. And indeed, even if we could unlearn the maths we know, we could quickly relearn maths, because for most of us, maths is something we immediately get. It is out there, it is obvious, it is central to our everyday existence.
That gives us a problem when we come to working with dyscalculic pupils. But fortunately there are two ways of trying to understand what it is like to be dyscalculic.
First we can take the issue of number itself. Imagine you are asked to define “six”, you might first do this by reference to external objects such as “six cars” or “six people.”
But supposing I was being rather difficult, and said to you, “I don’t want you to tell me about six cars or six people, I want to know what six is,” then you might find it harder. You could try and say that six is one more than five – but that is no good if I don’t know what five is. And if you find that comment frustrating, that’s good, because this is what it is like to work with a dyscalculic person. It is very frustrating simply because most of us simply “get” numbers. They are out there. They are obvious.
So, if our first attempt to unravel dyscalculia by relating numbers to other things fails to get to the heart of the matter, we might try a different route. And it is this second approach that I will describe below.
We use, in the west, a numbering system that is generally known as Hindu-Arabic numerals, and which we might call today the “decimal number system”. The system originated in the 6th century in India, was then adopted in the Middle East, and got to Europe around the 12th century.
And you will probably realise that this raises a really interesting question: how did people count before that?
In fact that turns out to be a very important question, because before the 12th century the west had already had two great civilisations of its own: the Roman Republic, and then from the time of Julius Caesar onwards, the Roman Empire.
These civilisations installed order and governance across vast parts of Europe and north Africa, and they did it without the Indo-Arabic system of counting. But, of course, they had their own system: Roman numerals.
Now I am labouring this point, because there is a fundamental difference between Roman numerals and the Indo-Arabic system of numbers in that the Indo-Arabic system has a number that does not appear in the Roman system. You might care to close your eyes and think what that number is, before reading on.
And that number is…
In Roman numerals there was no zero. So if you had five oranges and gave them to your friends, the calculation of five minus five could not and would not be undertaken. It was not a part of reality. The answer was nothing, it was nowhere, it was like asking today, “what lies outside the universe?” The question is meaningless.
Now this lack of zero might strike you as a bit surprising given that the Roman Republic on its own lasted almost 500 years (or “D” if you want to go native), and the Roman Empire lasted for 1000 years beyond that (“M”). 1500 (MD) years, and not a zero in sight.
We would find it impossible today to have any conversations about anything involving numbers without zero, and yet for well over 1000 years the accountants of the Roman Republic and Empire (and believe me there were a lot of accountants in the Roman Republic and Empire) did exactly that. They ran European civilisation without the zero.
I go through all this because most people brought up with our numbering system find this rather odd. And then they often find it odder if asked to explain what zero is without simply using another word for zero (such as nothing).
And again this gets us a bit closer to understanding how a dyscalculic person feels. It is not that they are not very good about numbers but rather that they don’t get numbers at all, in the way that most of us today can’t get the idea of a system that simply fails to acknowledge zero.
But to help our understanding of this oddity, we can take this matter further.
Everywhere in our universe there is something. Even in the depths of space there is something; as any physicist will tell you, there is no such thing as a total vacuum. There is always something. Nothing does not and cannot exist.
So in that sense “zero” or “nothing” is a construct – something we have invented to signify an impossible state, but it is also a notion that is quite handy when we have five oranges, take five away and then want to say what is left.
Not surprisingly, this totally artificial construct of “zero” turns out to be not a proper number at all. If you try to multiply by it, you get zero. If you try to divide by it you get zero. If you add or subtract it, nothing happens. Three minus two equals one. Zero minus zero equals zero. How can that be?
Now as I have said, we all accept this, because it is a way of doing maths. And with a lot of help the dyscalculic person can come to accept it too. But initially for that person zero, and all the other numbers that can exist, are all odd, because the gene that allows most of us to understand numbers and accept the oddity of zero without thinking about it, is faulty. It is not their fault, that is just how it is.
It does us no harm to puzzle over the oddity of expressing nothingness as “0” because it does remind us how odd all maths seems to a dyscalculic person, and helps us understand why conventional ways of teaching maths, which just assume all this oddness is, well, normal, make all sorts of assumptions.
When we recall that the Romans had no concept of “zero” and yet could raise massive armies, count the men, and take over much of the civilised world, imposing law, order and civilisation as they went, it reminds us that maybe maths is not quite as natural as we might want to assume in order to make our everyday lives easier.
If we want to teach dyscalculic pupils and students we need to take ourselves back to the very basics and realise that if we walked into a class of students who were learning maths in the days of the Roman Republic and started talking about zero, the whole class – indeed the whole civilisation – would appear to be dyscalculic to us.
Dyscalculic pupils and students need a different way of teaching. One that moves slowly through the physical world and relates that physical world to numbers in a way that makes no assumptions. It is a completely different approach from that which we can use with the 97% of the population that is not dyscalculic. Such teaching can be done, but it needs to be done slowly, with patience, and with a different approach from the norm.
Tony Attwood C.Ed., B.A., M.Phil (Lond) F.Inst.A.M
Head of the Dyscalculic Centre