In the Guild of Florentine Bankers forbade the use Arabic numerals, demanding instead that numbers be written out using letters. Yet despite all the obstacles in its path, the study of mathematics has flourished. Our second book is by Brian Butterworth who's in Melbourne this week from Britain.
I'll tell you his title in a minute. At the moment he's working on where numbers go in the brain and it really matters.
Brian Butterworth: Well, some people like, for example, Piaget, argued that really mathematics was no more than an extension of logic and the mathematician Keith Devlin in a recent book called the "Maths Gene", has argued that maths is nothing more than an extension of language. Now modern scientific approaches to how the brain deals with numbers and other aspects of mathematics show that really there are separate parts of the brain that deal with maths on the one hand, deal with reasoning on the other deal and deal with language on the third hand.
So there is evidence that there is independence in the brain. Of course it doesn't mean that there's functional independence. Clearly you learn most maths through language. But once you've learnt it, does it get stored with other things that you've learnt through language or does it get stored somewhere else? One of the things that we've been working on - and we work on these things entirely opportunistically, it depends who comes into the clinic, is about whether reading words, reading numbers and recently, reading music all use the same brain circuits or whether they use separate ones.
There's a very interesting novel by Ken Follett called the "Hammer of Eden" in which one of the central characters, I'd better not say whether he's a hero or a villain, is unable to read either letters or numbers but actually truth, as is so often the case, is stranger than fiction. We've found that there are people who can read numbers but can't read words, and my colleague Lisa Cipolotti here at the National Hospital for Neurology, has found that there are people who can read words much better than they can read numbers.
I mean, they've had brain damage which has affected their number reading but left their word reading intact. Now recently with some colleagues at St. Thomas's Hospital, we examined a patient called Mrs C. Now Mrs C was a professional musician and a professional musical composer, and she had some brain disease which affected not the language part of the brain and not what we believe to be the number part of the brain, but a part of the brain that we haven't really paid much attention to before, in the right hemisphere.
As a consequence of this she was left with a very, very specific deficit. Although she could still play music, although she could still sing, she could no longer sight-read music; something that she was able to do fluently for over 30 years, nearly 40 years in fact.
Now she can't sight-read music at all. In fact she can't read musical notes at all. Nevertheless she can still read numerals, she can still read words perfectly well, and this shows that very soon after the visual information enters the brain it gets split into separate channels. There's a channel for words, there's a channel for numerals and there's a channel for musical notation as well. Robyn Williams: But is it just a matter of the numbers, recognising the numbers or is it this ability for computation?
In other words, putting them together Is it all in one bag? Brian Butterworth: Well we actually have found a disassociation here.
In a subject I called in my book Charles. He was able perfectly well to recognise 17 and he could read aloud the number 17 but he had no idea what it meant; he couldn't do computation with it.
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So deeper into the brain than, if you like, the recognition centre, there's comprehension and calculation and that's clearly something that goes beyond the reading and comprehension. Actually, we found another patient who was rather poor at reading numerals but was very good at calculating with them.
So if you asked what is 17 plus 35 he would give you the correct answer, which is 52, I think. But if you asked him to read aloud the numerals 17 plus 35 he would often make mistakes in doing that.http://sachenbesorgen.de/error/verizon/vem-reicher-mann-sucht.php
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So there seems to be something rather separate about the calculation. Robyn Williams: Yes. What are the implications of this independence of mathematical ability? Brian Butterworth: Well, one of the things that's coming up now as an important social issue is people's numeracy.
Certainly our government and the government of the United States have been worried that the numeracy of our citizens is really much lower than our economic competitors and we wonder what we should do about that.
Now one of the questions that comes up is can we treat mathematics and arithmetic as being separate from other school subjects. If it's separate in the brain what are the education implications of this? Now that's not completely clear yet, but one thing is becoming clear - and this is something that I have to say neither government recognises - and that is that there are people who are born without a properly developed number centre in the brain and these people are always going to be very bad at calculation.
They might be quite good at other aspects of maths but at calculation and understanding numbers they'll always be awful. They are in a similar position to dyslexics twenty or thirty years ago. It's not a condition that's recognised, no one really knows how to treat it. We're beginning to assemble guide lines for best practice here and it's not absolutely clear how you diagnose it, but I get frantic parents ringing me up and saying, "Look, my kid does very well at school in geography, history, English.
He's not dyslexic but he's just hopeless at maths.
Is it bad teaching? Can I get him extra tuition, will that help? And often we find that these kids are very poor at stuff to do with numbers that really doesn't depend upon education at all. Like, for example, counting dots or saying which of two numbers is larger; is 6 larger than 2? These kids have trouble with that in a way that's very striking.
As I say, it's not absolutely clear how to help these children but it is becoming clear how to diagnose them and that's by using these very simple, non-education-dependant tests like dot counting or number comparison. Robyn Williams: I don't suppose the same thing would happen with music comprehension because music is often considered to be an add-on. You won't have someone brought to you - "Look I'm afraid my child can't play the violin to concert performance or sight-read Bach?
Brian Butterworth: That's absolutely true. It's not regarded as a great national problem that we're not very musical. But actually not being able to multiply 8 x 7, which our current First Secretary of the Treasury was unable to do in public, isn't regarded as a great social problem, whereas saying "we was" instead of saying "we were" still is regarded as a great social problem.
Robyn Williams: Going to the brain itself, I can actually work out how words might be represented in the brain but how on earth are the numbers? Brian Butterworth: Well, there's a big debate going on at the moment. On the one hand there are people who think that the way in which we represent the meaning of numbers and what 5 stands for really isn't specifically numerical. Their idea is that we have a kind of quantity sense, so 5 is kind of approximately this much; it's a bit more than 4 and a bit less than 6 but it's all rather approximate.
It's as though you have a line in your head and 5 is approximately here and 4 is approximately there on this line. Now I take a different view. I think that we're born with a sense of a collection of things and a number for that collection so that a collection with an extra member has a very definite precise extra representation. At the moment we are going through some experiments trying to see whether I'm right or whether most of the rest of the scientists in this area are right. They've actually got some rather good evidence, but we've got some experiments that we're doing now which I think may well tip the balance in our favour, but I can't tell you about that at the moment because we haven't finished the analysis.
Robyn Williams: Well I won't ask you about it if you haven't finished the analysis, but what I will ask you about are those animals which, I think, dogs can count fairly comfortably up to 4, specially mine which are border collies which are the most intelligent and can go even further. But parrots can do calculation and not simply say, well there are two things over there but two entities we've brought together, you know two apples and one pear, they can say, the parrot or one or two parrots can say, This is 3, as if animals indeed have a lower level of cognition evolution have got this wired in as well.
Brian Butterworth: Well, that's certainly my belief which is that our specific number sense is something that we've inherited from animals. We know that animals can do quite a lot of number tasks, some of which they have to be taught and some of which seem to come rather naturally. They can be taught for example to pick the larger of two numerical arrays, so they can pick, if you want three things and four things they can pick four. If you're a very smart animal like a chimp you can be trained to understand what the numeral 3 means and what the numeral 4 means and then pick the numeral 4 over the numeral 3.
They can do some extremely primitive calculation - extremely primitive. The problem is, if you want to say that these are ancestral abilities to our own, is to show that animals use homologous brain areas to our numerical brain and so far nobody knows which bit of the brain animals use to do these tasks. So until we know that, we won't know whether it's just in general, kind of handy to make numerical distinctions or whether actually some ancestral creature got this idea and then the rest of us inherited it.
Robyn Williams: Why is it important to know? Brian Butterworth: Well, we want to know whether our genome contains information to enables it to build brain areas that are specific to mathematics.
If it does then that will explain why some people are born without this ability. Also it will help us actually to understand some archaeology. There are lots of interesting arrays of dots in Palaeolithic caves in Europe particularly and no one knows what they mean. My guess is that our Palaeolithic ancestors 20, years ago in the depths of the Ice Age were counting things. We don't know what they were counting, they might have been counting bison they might have been just counting the number of people who turned up for a party, but I leave that questions to the archaeologists.
His book is called "The Mathematical Brain" and it's published by Macmillan. Robyn Williams. Brian Butterworth: What's fascinating about him is that he can only read one sort of word. He can only read number words and it's never been reported before and that's why we're so interested in it.
But it's theoretically very important because the standard theory of reading goes something like this: that when you see a string of letters you do two things simultaneously; you try and find the sound of that string of letters.
You might do that by taking each letter and finding the sound of each letter and putting them together. You might try and recognise it as a whole word and try and find the sound of the whole word but, at the same time, you're trying to interpret the meaning of that string of letters, you're trying to find the meaning of that word and these two processes converge to give you the pronunciation of the word. Now the problem with the theory up to now is that it's never really been tested properly.
We don't know whether you can pronounce the word accurately just going via the meaning. So for example, if you see a letter string like t-r-e-e- you get the meaning of a tree and then you have to say what that meaning is. Now what we want to know is: is that route via meaning sufficient to give you just "tree", and we haven't been able to do that up to now. This case is very interesting because this man can't use the route via sound alone. We know this because if you give him a very simple string of letters he hasn't seen before, like y-i-t- he can't say it.
He used to be able to say it but he can't say it anymore. Even if the letter string sounds like a real word, like y-o-t- he still can't say it. Now the other really interesting thing is that he can't say words that he doesn't understand. So he can't say y-a-c-h-t says yacht either, he can only say words that he does understand and if he does understand them then he says them accurately.
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