What Counts: How Every Brain is Hardwired for Math. Brian Butterworth. 320 pp. Free Press, 1999. $26.
Brian Butterworth, a British cognitive neuropsychologist and founding editor of the journal Mathematical Cognition, has summarized several lines of evidence pointing to the conclusion that the normal human brain contains a "number module"—a highly specialized set of neural circuits that enable us to categorize small collections of objects in terms of their so-called numerosities. When we see three brown cows our brains immediately tell us both that there are three of them and that they are brown. Just as we see colors automatically and involuntarily and without being taught the concept of color, so we immediately recognize and distinguish small numerosities without being taught the meaning of number. In order to communicate, we need to learn the words "brown" and "three," but our perception of small numerosities is as innate and as automatic as is our perception of color.
Of course, through instruction and practice we can greatly extend the capacities of our number module, just as we can similarly improve our ability to read, type or play the piano. But reading, typing and piano playing are not based on hard-wired, specialized, genetically created neural circuits ("cognitive modules"); they depend instead on the slow, purposeful development of general-purpose brain circuits ("central processes").
Butterworth disagrees with the influential Swiss psychologist Jean Piaget, who denied that "any a priori or innate cognitive structures exist in man." For Piaget, a child's understanding of number was founded on years of sensorimotor interactions with physical and temporal realities. Before children can understand number, they must master, for example, transitive inference (if a is less than b and b is less than c, then a is less than c) and must be capable of separating number from the sensory properties of objects. Piaget concluded that the concept of number cannot be understood by children below the age of four or five.
Some of the disagreement between Piaget and Butterworth stems from Piaget's more stringent criterion for what it means to understand number. Piaget did not deny that toddlers recognize the difference between two and three. But Piaget did not regard the ability to make this distinction as proof of an understanding of number. In Piaget's famous experiments, children were shown two identical collections, and then the objects in one collection were moved farther apart. The children were then prone to say that the more-spread-out collection had more objects. Piaget regarded the failure to realize that number is conserved when objects are moved as a failure to understand number.
More recent experiments, described in Stanislas Dehaene's 1997 book The Number Sense (cited by Butterworth), seem to show that Piaget's subjects knew perfectly well that number is conserved when objects are moved; the problem was that they did not understand the questions they were being asked. According to Dehaene, if children are asked to choose between four pieces of candy spread apart and five pieces close together, they are unlikely to be fooled. Piaget may have underestimated children's early understanding of number, but he is probably still correct in claiming, for example, that children below the age of five or six cannot count two sets of objects and compare them unless the collections are simultaneously present. Such capacities for abstract or symbolic representation may plausibly depend on more than the number module. Butterworth's view of the origins of our mathematical abilities is analogous to linguist Noam Chomsky's thesis that the logic of grammar is built into our brain—that spoken language depends on an innate cognitive structure. (Piaget, of course, denies that such structures exist.) Books and conferences have been devoted to attempts to reconcile Piaget's and Chomsky's views of the foundations of cognition.
Although Chomsky and Butterworth have similar theoretical perspectives about knowledge, they disagree about math. Chomsky sees the number concept as a special aspect of language, whereas Butterworth argues (citing, for example, studies of the cognitive consequences of injuries to various parts of the brain) that math and language use different regions of the brain. Butterworth also disagrees with those who explain math as a combination of language, general intelligence and spatial ability.
Modern cognitive science and physical investigations of brain structure may someday resolve or clarify an ancient philosophical issue: Does knowledge have a large innate component (as Kant, Chomsky and various religious philosophers would argue), or is the mind a tabula rasa whose contents are determined by the social and physical environment (Locke and Piaget)?
What are the implications for math education of various cognitive perspectives? Because Piaget believed that number itself was dependent on abstract and logical thought, Piagetians are prone to deduce that premature exposure to mathematics will lead to rote learning without understanding and to disabling confusions and anxieties. "Developmentally appropriate practice" has become a shibboleth in U.S. schools of education—largely reflecting the Piagetian belief in fixed stages of cognitive development. France severely de-emphasized the early teaching of numerals and counting words, believing that such instruction was useless or harmful. Even if Piaget's theories are right, it is an empirical issue whether it is helpful or harmful to teach children to memorize counting words before they can abstractly link these words to collections of objects.
Although Butterworth sensibly rejects Piagetian-based pessimism about what children are capable of learning at various ages, he is implausibly optimistic about our mathematical potential. Although his central thesis, the number module, is genetic, he argues that the main sources of individual differences in developed math ability are environmental: "provided [that] the basic Number Module has developed normally . . . differences in mathematical ability ? are due solely to acquiring the conceptual tools provided by our culture. Nature, courtesy of our genes provides the piece of specialist equipment, the Number Module. All else is training. To become good at numbers, you must become steeped in them." Butterworth denies that there is any "essential and innate difference between children . . . who find maths [the British usage for math] really easy and those who find it a struggle. There may have been differences in their capacity for concentrated work or in what they found interesting ? but there was no difference in their innate capacity specifically for maths."
Butterworth cites international comparisons that show large differences in performance (for instance, a test on which the average score of Iranian children is equal to that of the lowest 5 percent of children in Singapore). Cultural resources and pedagogy clearly matter. But it does not follow that all individual differences in developed math ability are due to temperamental and environmental factors (ability to concentrate, ambition, interest and time devoted to math). Intelligence, verbal aptitude and spatial ability are also likely to be important for math.
According to Piaget, children must discover or construct for themselves certain regularities about the world (that objects continue to exist even when we can't see them, for example). Some constructivists go beyond Piaget, claiming that all genuine knowledge must be gained through a process of discovery. Although infants and toddlers do need to learn basic facts and distinctions (hard versus soft, solid versus liquid) through their experiences with external objects, it does not follow that more advanced material must be learned by recapitulating the original process of discovery. If so, the potential for human progress in science and other areas would be severely limited.
Although Butterworth rejects Piaget's theoretical framework, he agrees with most Piagetians in advocating discovery learning. Butterworth argues that schools limit children's potential for growth when they insist that there is a preferred way to do math problems, then drill students in approved methods. He reasons that since we all have a number module, we all have the capacity to work out our own approach. Butterworth approvingly quotes educational researcher Lauren Resnick:
The failure of much of our present teaching to make a cognitive connection between children's own math-related knowledge and the school's version of math feeds a view held by many children that what they know does not count as mathematics. This devaluing of their own knowledge is especially exaggerated among children from families that are traditionally alienated from schools, ones in which parents did not fare well in school and do not expect--however much they desire--their children to do well, either. In the eyes of these children, math is what is taught in school.
But a large body of empirical evidence (not cited by Butterworth) shows that discovery learning is ineffective with all but the most basic material. Few children will discover for themselves efficient ways of multiplying three-digit numbers, and virtually none will discover Archimedes' law by experimenting with floating bodies. To be sure, children may work out ways of doing simple arithmetic problems. Butterworth cites as an example an untutored Brazilian coconut seller, who calculated the price of 10 coconuts without understanding decimal place notation. But the ad hoc methods children discover for themselves are most unlikely to be suitable building blocks for more advanced knowledge. Even when successful, discovery learning is inefficient, taking time that could be better devoted to practice. Butterworth correctly relates that great mathematicians all steeped themselves in mathematics. Yet he disparages school practice and somehow regards it as antithetical to understanding.
Butterworth sees the international comparisons he cites as proof that children can learn more math than they typically do. But the best countries (such as Singapore) are the ones that emphasize direct instruction and drill, not the student-centered discovery methods he advocates. Butterworth's findings and views of mathematical cognition may well be sound. But the existence of a number module does not in and of itself establish the relative soundness of various educational methods. Doing so would require an evaluation of empirical research in educational settings, and this Butterworth has not done. Cognitive theorists are too prone to jump from models of cognition to classroom practice without empirical testing under realistic classroom circumstances.