IRA FLATOW, host:
Next up, mind your mind over math. A couple of interesting mathematics stories this week. First, it's obvious that a lot of mathematics is just learning rules, right? How do you find the least common denominator? Why you can't divide by zero.
But there are some parts of math that just feel intuitive, like geometry, at least sometimes it feels that way to me. And now a new study suggests that even without mathematical training, the human brain may have certain intuitions about geometry, concepts that we don't learn but may be born primed to understand. We maybe have this hard-wired, right, into our brains.
The study showed that Amazonian villagers with no formal math training seemed to understand these geometric concepts as well as we do, as well as we folks who go to school and study mathematics.
Veronique Izard is a research scientist at the French National Center for Scientific Research at Paris Descartes University in France, and she's the author of that geometry study in the journal Proceedings of the National Academy of Sciences. And she joins us by phone. Welcome to SCIENCE FRIDAY, Dr. Izard.
Dr. VERONIQUE IZARD (Paris Descartes University): Hi, it's good to be here.
FLATOW: Welcome. Tell us about this study. What sorts of questions were you asking the indigenous people in the Amazon?
Dr. IZARD: Oh, yes, so it's a study - I have to say that I didn't do a low enough course. I'm working with a linguist here in (unintelligible), and he goes to the Amazon every year to study these people, the Munduruku.
And so this year, we were interested in how they understand geometry, and particularly we were interested in how they understand concepts of geometry that (unintelligible) that we can perceive, that go beyond what we can perceive, such as the idea of an infinite line or idea of lines that would never cross, lines that are parallels.
And so the way we asked them questions about these kinds of concepts is we asked them to imagine some ideal worlds. We asked them to imagine a world where people lived, which was just, for example, very flat and going on forever and ever.
And then we would say that on these worlds, there would be villages, and village is (unintelligible) to what we geometry-educated people think of as points. And then we would say that on this world, there will also be paths with the constraint that all the paths are just going very straight and always in front of them and that they would never end.
And of course for us, this path corresponds to what we think of as straight lines.
FLATOW: We're going to - stick with us, Dr. Izard, because we're going to take a short break and talk about the results of your studies and how these folks who lived in the Amazon were able to have a concept of geometry as much as we folks who study the math in schools.
Stay with us, our number, 1-800-989-8255. You can tweet us, @scifri, @-S-C-I-F-R-I. Don't go away. We'll be right back.
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FLATOW: I'm Ira Flatow. This is SCIENCE FRIDAY from NPR.
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FLATOW: You're listening to SCIENCE FRIDAY. I'm Ira Flatow. We're talking about math this segment with Veronique Izard, research scientist at the French National Center for Scientific Research, 1-800-989-8255.
Dr. Izard was telling us about this group of folks in the Amazon who had no experience in mathematics, right? They had never seen any math. Yet they were able to have - they sort of have an innate ability to understand geometry.
Dr. IZARD: Well, I was - I'm not sure we would call it innate, but I would say it is a universal ability to understand geometry because we don't know how people come to understand geometry. Is it innate, or is it something they learn early in life? And that's something that is still up for research.
But yes, so we got them to imagine these imaginary worlds and ask them questions where we found that they are able to conceive the fact that there would be straight lines that are infinite, that never end, that two lines may sometimes never cross.
So they have, for example, an understanding of lines can be parallel.
FLATOW: And did they get this ability at a certain point in their lives, at a certain age? Did they need some sort of maturity or experience to understand the concepts of geometry?
Dr. IZARD: So in the Amazon, we only tested adults and children of age seven and more than that. And then we also tested groups of adults and children in the U.S. and in France to compare them with.
But we also tested a fifth group, which were young children in the U.S., children aged five and six in the U.S., and we were interested in them, of course, because at that age, they haven't had education in geometry yet.
And we actually found that these young children in the U.S. weren't doing quite as well as the Amazonians or as the older children in the U.S. and in France. So that actually tells us that if this ability is innate, actually it would be something that comes later in life, that if you can think, for example, of the beard is something that is innate but appears only in puberty, maybe geometry is just like that, that it appears only by age seven.
Dr. IZARD: Yeah, and also - sorry.
FLATOW: Go ahead, go ahead.
Dr. IZARD: Of course, there is another possibility, which would be that geometry is something that we learn before the age of seven. So maybe around five and six, children are actually on the way to learning geometry and grasping these concepts.
Dr. IZARD: That's another possibility.
FLATOW: Interesting, a very fertile area of research. For those of us still struggling with simple numbers, my next guest has been investigating a condition known as dyscalculia. It's a difficulty doing arithmetic where some people have a tough time grasping the meaning of really simple numbers and how the numbers interact with each other.
Brian Butterworth is an emeritus professor of cognitive neuropsychology at the University College London. He's also former chair of the Center for Educational Neuroscience, and he has an article in Science this week on dyscalculia. He joins us by phone. Welcome to SCIENCE FRIDAY, Dr. Butterworth.
Dr. BRIAN BUTTERWORTH (Cognitive Neuropsychology, University College London): Yes, good evening.
FLATOW: How would you define that? How simple, you know, how much of a problem is it with just simple numbers? Can you give us an example?
Dr. BUTTERWORTH: Sure. One of the standard tests for dyscalculia now is just to say how many dots there are in an array, like, you know, three dots or seven dots or nine dots. And dyscalculics are bad on that test. I mean, they're slower, and they're less accurate.
So the problem isn't just being bad at long division or multiplication tables. It seems to be a much more profound difference between dyscalculics and normally developing individuals.
FLATOW: Can people add simple numbers, like five and two and seven and 12 or something like that?
Dr. BUTTERWORTH: Sure, you know, if they get enough practice. But they'll do it in a different way, and they'll do it rather slowly. So they might, for example, use their fingers where typically developing adults would not use their fingers. They'd just know that, you know, five and seven is 12. But the dyscalculics will go, well, let's see, one, two, three, four, five, six, seven, and then another five, eight, nine, 10, 11, 12. So they'll be slow, and they might use a different strategy.
FLATOW: I see a lot of people doing that. Can this be an undiagnosed problem in the population?
Dr. BUTTERWORTH: Well, unfortunately, I think it's very widely diagnosed, and our best estimates, at least five percent of people suffer from this condition.
And there have been studies in the U.S. and Israel and Cuba and many places which suggest that the prevalence rate is at least five percent. So that means that a lot of kids who are bad at maths, and indeed adults who are bad at maths, are bad at maths because they have this condition.
And the trouble is that if you've got this condition and it's not diagnosed, people think that because you can't do arithmetic, because you can't add nine and seven, you must be stupid. And so teachers think that kids who can't do this are stupid. Parents who think their kids can't do this think they're stupid, and the kids themselves think they're stupid.
And then their classmates think they're stupid. So it's - it really is a problem if it's undiagnosed, and unfortunately, not many education authorities actually recognize this as a serious problem.
But actually it is a serious problem. It's a serious problem for individuals because individuals with low numeracy are much more likely to be unemployed. If they are employed, they'll earn less.
At least in studies in the U.K., they're also likely to be more depressed, more likely to be in trouble with the law, and so the outlook for individuals is extremely problematic.
But also the outlook for the non-dyscalculic taxpayer is bad, too, because in the U.K., where they've done the sums on this, they reckon it costs $2.4 billion pounds a year in lost revenues and problems with justice and additional teaching.
And so it's very expensive for the rest of us.
FLATOW: Interesting. Let me go to the phones, to Kent(ph) in Dallas. Hi, Kent.
KENT (Caller): Hello.
FLATOW: Hi there. You sent us a note on our Facebook page that you have dyscalculia.
KENT: Yes, I do. I've lived with it all my life.
FLATOW: Can you describe for us the problems you face? And how do you overcome it?
KENT: Well, a lot of the strategies that he mentioned are what I use. I'll count by fives. I use calculators. I use my kids because both of them are better than I am at it.
When I was younger, it was very frustrating because I would have, you know, high science scores, high language scores, and then math was just like this black hole. And so we would do a lot of individual studies.
My parents worked with me very hard. They were actually key in getting me as far as I got.
FLATOW: Interesting. So you've learned little techniques to get around it and to get help from other people.
KENT: Yes, and then...
FLATOW: And did your teachers understand your problem at all?
KENT: Some did. Some didn't. It was kind of hit or miss. I mean, the ones who knew me, that knew that I was really applying myself, you know, I had a trig test one time. I studied, studied, studied. I got a nine out of 100. I studied, studied, studied to retake it. I doubled my score.
You know, I did end up passing the class, but it was because the teacher knew what I was - she knew the thought process that was going on, what I was trying to do.
FLATOW: Well, thanks for sharing your experience with us, Ken, and good luck to you.
KENT: Thank you.
FLATOW: You're welcome. 1-800-989-8255. Veronique, do the Munduruku people have a number system, counting, stuff like that?
Dr. IZARD: Well, they have a very special number system. They actually have number words only up to five, which is something that is quite common in the Amazon. But so that's how we got interested in them because we also first did studies of their numeracy.
And we looked at how they could perceive numbers, even though they had this limited lexicon, and we actually found that if you show them images with lots of dots, you ask them to compare where there would be more dots or less dots, they are still able to do that.
So even without a linguistic system, you still have an intuition of numerical quantity.
FLATOW: Why don't they have larger numbers? Do they not have to keep track of larger items, numbers of items?
Dr. IZARD: Well, they - it's a hard question to answer because of course if you don't have a large number system, then you are not going to keep track of large numbers of items. But I think it would probably be of some use to them.
But they have also developed alternative systems, like they would use configurations for example, for keeping track of numbers. They would put five on the side, five on another side, and then, they might complete the picture with even more fives or things like that.
So for example, when they basket, at first, they have to make sure that they have enough lines to make the basket. And to make sure that they do, what they do is choose some geometric configuration where they make squares of three by three lines, and then they make big squares of smaller square of three by three. And then, once the configuration is complete, they know they have the exact number they need.
So I think they developed this on-the-spot strategy to keep - to track these precise numbers that they need in precise situations, but they don't have this abstract system that we have to track any number of anything.
FLATOW: Very interesting. And, Brian, is there any crossover between...
Dr. BUTTERWORTH: I think there is, actually...
Dr. BUTTERWORTH: ...because in previous work, Veronique and her team have found that the Munduruku are not very good at doing exact calculation. I mean, they have to go to these elaborate strategies, but they are very good at geometry, which they've never been taught.
So that suggests that in evolutionary times, geometry and numbers are separate, and also, we know from studies of brain function and structure that the part of the brain that deals with space and geometry is not exactly the same as the part that deals with numbers.
And it turns out, actually, that quite a lot of people who have dyscalculia, that is they're really bad at arithmetic and numbers, can be actually quite good at geometry. So it seems to me that there's a convergence between the work that Veronique is doing in the Amazon and the work we're doing with normal children in a numerate society and actually converges to a certain extent.
FLATOW: 1-800-989-8255. What about, you know, we call it dyscalculia. There's a dys like in a dyslexia. Is there any connection between the two?
Dr. BUTTERWORTH: Well, that's an interesting question. They're both developmental disorders. They both seem to run in families. They both have particular neural signatures.
But as far as we can tell, the neural basis of the two conditions is really quite different. So there aren't overlaps in the brain between the reading brain and the calculating brain, at least broadly speaking.
Also, as far as we know, the genetics of dyscalculia and the genetics of dyslexia are different. So they seem to be independent.
On the other hand, what happens in, as a matter of fact, is that they're much likely to be dyscalculic if you're dyslexic and much more likely to be dyslexic if you're dyscalculic than you might expect by chance.
So there's a kind of mysterious co-occurrence of these two developmental disorders. We don't know why that should be, but it does seem to be the case.
FLATOW: This is SCIENCE FRIDAY from NPR. I'm Ira Flatow talking about mathematics and the mind this hour. 1-800-989-8255 is our number if you'd like to talk about it.
Let's see if we can get one more phone call in. Paula(ph) in Wisconsin.
PAULA (Caller): Hello.
FLATOW: Hi there.
PAULA: Thanks for taking my call. I am almost 60 years old, and listening to your program is the first time I've known there was a name for what's been wrong with me.
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PAULA: I have - oh, I struggled as a child. I mean, my - I'm adopted, and my father was an accountant, and it drove him crazy that I could not do numbers. And to this day, of course, I still can't, and my friends know that when we were like going to play a game of cribbage, I - they have to add it for me.
I can't keep more than like two numbers together in my head at one time. They just don't go there. They just - and I do have to use tools to count, either my hands or - and I have to add weird like if somebody wants me to get, you know, subtract seven from 23. First, I bring the 23 down to a 20, and then, I could take seven from that. And then, I get three more to rid off, and, you know, I've just over these years learned.
FLATOW: Wow. What do you say that, Dr. Butterworth?
Dr. BUTTERWORTH: Well, it's not time - it's not the first time I've heard a story like this. And we tested a series of what we call high-functioning dyscalculics. These are people who - of high intelligence who've been very successful in life in all sorts of different areas, including science, by the way, but who nevertheless have this profound problem with numbers. So whatever they've tried to do, they haven't managed to succeed in doing simple arithmetic.
Now, one of the problems here is that we think we have a way of helping them, but these adults didn't have the opportunity to use that method when they were growing up, and we still don't really know whether our methods are going to work in the long term.
FLATOW: What - can you, briefly, tell us what that is.
Dr. BUTTERWORTH: Well, you have to work with concrete materials for much longer with dyscalculics than you would with normal people. That is you have to get them to use beads and blocks and so on, until they really understand what (unintelligible) and what happens when you add a set of five objects to a set of three objects to get eight objects.
So you really got to make sure that they understand those processes before you start giving them five plus three or the number bonds to 10 or the multiplication table. You've really got to make sure that they understand these concepts in either in real concrete materials or what we've been developing recently are virtual concrete materials where you have software, computer games which simulate or emulate what a good teacher would do with concrete materials.
This, we think, really does help these kids. Whether they'll ever get to be normally good calculators, we don't know yet because we haven't been running the study for long enough, so we need to be running it for some years now.
Dr. BUTTERWORTH: So I'm sorry your caller didn't have the benefit of these methods by...
FLATOW: Well, I think my caller is going to alert a lot of other people listening, and other people around, you know, the world who never knew what they had, you know? And thank you very much, Dr. Butterworth, for letting us know. And good luck to you.
Prof. BUTTERWORTH: You're welcome.
FLATOW: Brian Butterworth is former chair of the Center for Educational Neuroscience, and also with us was Veronique Izard, a research scientist at the French National Center for Scientific Research in Paris Descartes University in France.
We're going to take a short break. And when we come back, we're going to talk about the history of syphilis. Stay with us.
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FLATOW: I'm Ira Flatow. This is SCIENCE FRIDAY from NPR.