One of the most fascinating ideas in your book What Counts: How Every Brain Is Hardwired for Math (Free Press, 1999) is that we are born with a sense of numbers. What exactly is this number sense?
The number sense is having a sense of the manyness, or numerosity, of a collection of things. We believe that babies are born with a kind of start-up kit for learning about numbers that is coded in the genome. Even in the first week of life, babies are sensitive to changes in the number of things that they're looking at, and at six months they can do very simple addition and subtraction. Then, with this start-up kit, they build all the cultural tools—the number words, the counting practices, and the arithmetical procedures and facts that they learn from parents and from school.
How do we know that babies can add and subtract?
If you show a baby a doll, cover it with a screen, and show a second doll being placed behind the screen, the baby will expect there to be two dolls when the screen is removed. If there is a different number of dolls—more or fewer—then the baby will look longer than if there are two. This "violation of expectation" experiment was carried out by Karen Wynn in 1992 at the University of Arizona.
Your book suggests that even prehistoric man had a mathematical brain, as you call it.
Yes, and other species may have different versions of mathematical brains as well. Chimps, for example, can learn to do sophisticated numerical tasks. Ten years ago, David Washburn showed that chimpanzees trained to understand numerals—1, 2, 3—can also be trained to select the larger of two numerals presented to them. More recently, researchers trained monkeys to select the larger of two arrays of objects. Perhaps the best example of wild animals using numbers is the Serengeti lions. Lions defend their territories against intruders, but they will attack only if they outnumber the intruders. They have to figure out how many lions versus intruders there are.
Are you saying that the number sense may be a natural skill linked to survival for animals and related to survival for humans—even in today's society?
In many ways, yes. For example, a study by the Basic Skills Agency here in Great Britain looked at the effects of poor numeracy and poor literacy on getting, keeping, and being promoted in a job. The research showed that poor literacy is something of a handicap—but poor numeracy is even more of a handicap. Although the "why" wasn't investigated, one can imagine that mistakes with numbers lead to financial errors. In that sense, numerical skills are certainly an important survival skill in a numerate society.
Even so, people aren't afraid to say that they're not good at math, yet they wouldn't think of admitting that they couldn't read.
Well, it may not be true in the United States, but in England, literacy and orality are class markers. For example, children at school are taught to say "we were" rather than "we was" because "we was" is the dialect of working-class people, and "we were" is the dialect of the rulers of the country. It's important to try and top the linguistic manners of your so-called betters. This class distinction, however, doesn't apply quite so much to arithmetic. People are more prepared to say "my arithmetic isn't good" rather than say "my grammar isn't good" or "my spelling isn't good."
Does your research reveal any gender differences with respect to mathematical ability?
We find that women are far more willing to say that they are not very good at math than men are. Our research shows, however, that there is no difference on average in the public performance and examination of women and men in this country. The Third International Mathematics and Science Survey showed very similar mean performances by boys and girls on mathematics tests by 8–9 and 13–14-year-old schoolchildren. Actually, in the last set of results, girls scored slightly better than boys did.
Perhaps many people are better at math than they think they are. But if our brains are hardwired for math, why do so many students have such difficulty with it?
Not being good at mathematics can have two main causes. The first is genetic. A minority of people may be born with a condition that makes it difficult for them to learn mathematics; that is, they are born with dyscalculia or are born with dyslexia, which also can have a consequence for mathematics learning. A far more likely cause is that they were taught badly. That means taught in a way that left them failing to understand what they were doing. Thus, everything else that they learned that was based upon what they didn't understand was going to be very fragile. So, they avoided mathematics.
You mentioned dyslexia. Do we use the same parts of the brain that we use for learning to read to learn mathematics?
The parts of the brain that process words are different from the parts of the brain that process numbers. We store words in two areas, Wernicke's area in the left temporal lobe, at least in most right-handers; and Broca's area, in the left frontal lobe. Numbers are stored in the parietal lobe—not that far away, but far enough to be a separate system. No part of the brain is specialized at birth for reading because reading is a very recent skill for which the brain adapts the language areas. The brain, however, does seem to have evolved special circuits for numbers. There's an important difference between those two types of learning. Mathematics is built on a specific innate basis, and reading is not. It's quite important for teachers to remember that when children are learning mathematics, they are using distinctly different brain areas than they use when learning to read.
What is dyscalculia?
Dyscalculia is a condition a child is born with that affects the ability to acquire the usual arithmetical skills. Dyscalculic students may show difficulty understanding even simple number concepts and, as a consequence, will have problems learning the standard number facts and procedures. Even when dyscalculic students can produce the correct answer or the correct method, they may do so mechanically and without confidence because they lack an intuitive grasp of numbers that the rest of us possess. Dyscalculia is rather like a dyslexia for numbers—but unlike dyslexia, little is currently known about its prevalence, causes, or treatment. Dyscalculia often appears in conjunction with other learning difficulties—including dyslexia, dyspraxia, and attention deficit disorders—but most dys-calculic students will have cognitive and language abilities in the normal range and may indeed excel in nonmathematical subjects.
How is dyscalculia diagnosed? Are there tests for it?
Generally, discrepancies between mathematics learning and other cognitive functions, such as reading or I.Q., are taken as diagnostic. We are currently looking at qualitative differences between dyscalculics and other children. By March 2002, we plan to have a fully standardized test battery for this.
How might a classroom teacher recognize when a student might have this difficulty?
Dyscalculic students seem to have an impaired sense of number size. This may affect tasks involving estimating numbers in a collection and comparing numbers. Dyscalculic students can usually learn the sequence of counting words but may have difficulty navigating back and forth, especially in 2s, 3s, or more. They may also find it especially difficult to translate between number words whose powers of 10 are expressed by new names, such as "ten," "hundred," or "thousand" and numerals whose powers of 10 are expressed by the same numerals but in terms of place value, such as 10, 100, and 1,000. These students may be competent at reading and writing numbers, though some dyscalculic students have problems with numbers over 1,000, even in 6th grade.
How common among the student population is this condition?
We are not really sure. Using discrepancy criteria, estimates vary between 3 percent and 6 percent.
Are we as far along in understanding, diagnosing, and treating dyscalculia as we are with dyslexia?
No. We are about 20 years behind in terms of research and, more particularly, in terms of recognition by parents, teachers, education authorities, and the learners themselves.
If the brain uses different brain areas for reading and math, then why might dyslexics have difficulty with math as well as with reading?
There are many different language systems and visual systems that need to be coordinated to become a skilled reader. We don't really know why dyslexics have a much higher rate of problems with arithmetic than do children in general. One possibility is that they have trouble mapping from symbols onto their meanings. On the other hand, you find many extremely severe dyslexics who have no trouble with mathematics at all.
What do we now know about how the brain normally processes numbers?
Research shows that we think about numbers as displayed in a line in our head, a kind of mental representation of numbers. Now, when you ask people if they have a number line, most are not conscious of it. Perhaps only 15 percent of people are conscious of having a number line. In most people, this number line seems to go from left to right.
We suspect that when we can map the parietal lobes with great precision, we will see that separate areas do the separate arithmetical operations—addition, subtraction, multiplication, and division. Each of these operations can be selectively affected by brain damage without the others being affected.
We seem to have separate circuits in the brain, probably all in the left parietal lobe, for facts of these separate operations. There are, of course, procedures that we use when we don't have these facts. For example, we don't learn subtraction facts in school. We typically have to work these out on our own—even quite simple ones like 9–3.
Some would argue that we do learn subtraction facts.
French children certainly are taught to recite subtraction facts, as I learned multiplication facts. Of course, some facts are learned, but, in general, we have a different approach to subtraction and to division, by which a procedure turns these problems into additions and multiplications, respectively. These procedures depend on the facts of addition and multiplication. To solve the problem 9 – 3, you might say, "What do I have to add to 3 in order to get 9?" We suspect that the prefrontal cortex tells us what procedure to apply, and the parietal lobe, which stores the procedure, carries it out.
Do math facts and procedures represent different kinds of memories?
In general, yes. The brain seems to organize its memories in at least two distinct systems. One is called declarative, and the other is procedural. When we learn the capital of Texas, we store that fact in our declarative memory. When we learn how to ride a bike, that's pro-cedural. Now, much of what we know in terms of procedures is implicit, non-conscious. Our declarative knowledge is explicit, or conscious. We can't explain how we ride a bike, at least not very satisfactorily. But we can say what the capital of Texas is.
Most of the facts we store in declarative memory are arbitrary. For example, there is nothing inevitable about Austin being the capital of Texas, but there is something rather inevitable about 3 + 4 being equal to 7. The facts that are stored in numerical declarative memory are systematically organized, with a clear rationale, whereas the facts of geography are much less organized.
Although many procedures are unconscious, such as playing a good backhand shot in tennis, many of the procedures we use in arithmetic are learned quite consciously. We've learned explicitly how to borrow or carry in multiplication or division and can often recall and explain these procedures explicitly.
Are you saying that the brain stores numerical declarative and procedural information in different areas from those in which it stores nonnumerical declarative and procedural information?
That seems to be the case. Much of declarative memory is stored in the temporal lobes, quite near where the language area is, and many of the procedures that you find in ordinary procedural memory are in the frontal lobes. Declarative memory for numbers, however, seems to be in the parietal lobe with other number concepts. The procedures seem to be there as well. When the parietal lobe is damaged, you not only potentially lose arithmetical facts, such as the sum of 4 + 3, but you also may lose a knowledge of arithmetical procedures. The history of mathematics has been in part the history of making new tools, such as calculus, to solve problems. There was a time when only Newton and a few others knew how to solve problems of moving bodies, but now most 18-year-olds can do it because they have the tools, the procedures.
There seems to be a relationship between expressing what you know verbally and gaining a better understanding of it. Does brain research offer any insights into why this might be so?
Trying to express something can help you understand it better. We know from a number of studies that language and numbers occupy different regions of the brain. It's nevertheless the case that some of what we know about numbers is stored linguistically. For example, if we learned our multiplication tables by rote, they lodged in the language part of our brain as a kind of poem or perhaps even a nonsense verse, depending on how well we understood them.
There is an important transmission component of learning about numbers through language. Some of our knowledge remains linguistic, but most of what we know about numbers goes to the parietal lobes and is stored in a numerical rather than a linguistic way. Trying to explain mathematical ideas does help us understand what we might know implicitly. For example, children of about 5-years-old know that 5 + 3 is the same as 3 + 5. They probably have never expressed this as a formal rule—that addition is commutative, or N + M = M + N. If you ask them to explain why it is that they think that those two problems are the same, however, they might be able to formulate the concept in such a way that it will help them fix in their mind that it doesn't matter in which order they perform additions.
How might knowing about the brain's natural ability to process numbers help teachers teach math facts, procedures, and concepts?
The fundamental principle that must guide the teaching of mathematics is that children have to understand what they're doing. The work has to be meaningful for them. Children come to school from different backgrounds and with different information about numbers. It is important for teachers to adjust the way that they teach to fit the skills that students already have. Otherwise, students are going to start getting left behind, even in the first year of school. Learning mathematics is a cumulative process, and if you fail to understand one stage, then anything that is built upon that stage is going to be rather fragile.
Traditionally, schools have emphasized drill and practice for learning mathematics, assuming that understanding would come as a natural result of learning facts and skills. Is this a good practice?
We don't know whether fluency with number facts actually leads to a better understanding of the number system, but we do know that good understanding makes you much better with number facts. Try to make sure that students understand what they're doing before you start drilling with number facts. Going over the same thing again and again gets information from short-term memory into long-term memory, but you have to rehearse reflectively. Recent studies of musicians suggest that what's really important in becoming a truly excellent musician is reflective practice. This means playing a Bach cantata over and over again, but not just in a rote way. The musician must rehearse in a reflective way, thinking about how the parts of that piece are connected together and what those parts and what the whole means.
If the number sense is natural for us, why are some number concepts so difficult to understand?
My view is that the kinds of numbers that are easy to learn about are those that correspond most closely to innate numerical concepts, the idea of a collection and its numerosity. Whole numbers are quite easy to learn about for just that reason. Now, you can't have negative collections. It doesn't make any sense. So you have to think about negative numbers in a rather different way. You have to think about moving along a number line, and that is complicated. Fractions, too, are hard because they don't fit easily into our natural concept of whole numbers.
You've emphasized the importance of keeping mathematics meaningful. What can teachers do to promote understanding?
The nature of mathematics is that you can come to the right answer in many different ways. Encouraging different approaches to the same problem helps students grasp this idea. Using a range of examples is also important because numbers are abstract. They don't apply to particular things. Anything can be counted. You can count eggs. You can count jumps. You can count things you can't even see, like wishes. Using numbers in a variety of contexts helps students understand the abstract nature of the number concept.
You can make mathematics meaningful by actively engaging students with numbers in different ways. Manipulatives, for example, make use of the innate number sense, the numerosity of collections. Many manipulatives are little collections that can be put together. This process helps students understand that operations on the collections have an effect on the numerosity of the union of those collections. That is the basis of addition.
Because mathematics is a cumulative subject, it is also essential for teachers and students to identify and correct misconceptions as soon as possible. This means spending quite a lot of time with each child. Certainly, in England, where we have class sizes of 20 to 30, this is asking rather a lot of the teacher, but it is absolutely vital to know what each child does and does not understand.
Despite the best efforts of caring teachers, many students seem to develop a fear or dislike of mathematics. Why might this be so?
When understanding breaks down in mathematics, students do feel as though they're swimming in a sea of incomprehension. Because this drowning sensation is so anxiety-provoking, they avoid situations that give rise to it. This leads to more anxiety, and they get worse and worse. It's a vicious circle. To make sure that students don't fall into this sea of incomprehension and drown, you have to make sure that they understand what they're doing at each stage.
Educators are eager to learn about neuroscience in hopes that it will lead to improved ways to help students learn. What do you think will come of educators' interest in neuroscientific developments?
Education and neuroscience are just starting out on a great adventure. The more we talk to each other, the more we will begin to understand the kinds of problems that we're each interested in and seek common solutions. Perhaps most of the interest will be in the area of special education needs—teaching mathematics to dyscalculics, to dyslexics, and to others who have inherited disorders that make it hard for them to learn. In the future, we might be able to move to more general theories of how the brain learns abstract concepts. And then, I think the great adventure will really have taken off.
