In one test Charles was shown a pair of digits and asked to name the larger number. The bigger the gap, the faster most people can do this: they say “nine” faster when shown 9 and 2 than when shown 9 and 7. But with Charles, the reverse was the case—and the researchers could see why. Rather than telling the answer directly, he was counting on from one number (on his fingers) until he got to the other, which meant he must have started at the smaller, or he got to ten, in which case he must have started at the bigger. Most strikingly, he lacked the fundamental numerical ability possessed by most newborns: being able to tell the number of objects in a small group simply by looking. When asked how many dots were on a sheet of paper, he counted on his fingers—even when there were only two.
Charles’s deficit, though severe, seemed to affect his numerical abilities alone. Numerical deficits in people of otherwise normal abilities can be even more striking in cases of brain damage. Lisa Cipolotti, a neuropsychologist, studied a Signora Gaddi, who used to run a hotel and keep its accounts. After a stroke she could find the number of things in a small group only by counting—when asked how many arms a crucifix had, she got Dr Cipolotti to hold out her arms so she could count them. Signora Gaddi’s problems seemed to affect only numbers. She could still read, speak and reason, remember historical and geographical facts, and order objects by their physical size.
In fact, Signora Gaddi’s difficulties went even deeper than Charles’s. The stroke which damaged her innate understanding of small numbers also robbed her of the entire numerical edifice built on that foundation. For her, numbers stopped at four. When asked to count up from one, she got to four and no further. If there were more than four dots on a page she could not count them. She could not say how old she was or how many days were in a week, or even tell the time.
The article includes the interesting comment that a fairly eminent geometrician had little number sense of the kind described in the article. One of the wonderful things about mathematics is that it combines reasoning about discrete, countable things, that is numbers, with reasoning about shapes, that is geometry. An especially good book about both aspects of mathematics is John Stillwell's Numbers and Geometry.
I'll quote a bit of Stillwell's preface to give the flavor of the book:
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
A more controversial, and plenty spicy, comment on the algebraic approach to mathematics as contrasted with the geometric approach to mathematics is Paul Halmos's article, "Applied Mathematics Is Bad Mathematics,"
Meta-comment: Could we please fix the quoted-text-not-wrapping 'problem' on this site? I hate having to scroll x,000 pixels to read a comment just because someone is using <pre> or whatever it is.
Although if they're serving a custom .css file to my Firefox 3 that doesn't include that precisely because it doesn't work, I wouldn't know. wget got the same pre stuff in the last two lines, though.
Charles’s deficit, though severe, seemed to affect his numerical abilities alone. Numerical deficits in people of otherwise normal abilities can be even more striking in cases of brain damage. Lisa Cipolotti, a neuropsychologist, studied a Signora Gaddi, who used to run a hotel and keep its accounts. After a stroke she could find the number of things in a small group only by counting—when asked how many arms a crucifix had, she got Dr Cipolotti to hold out her arms so she could count them. Signora Gaddi’s problems seemed to affect only numbers. She could still read, speak and reason, remember historical and geographical facts, and order objects by their physical size.
In fact, Signora Gaddi’s difficulties went even deeper than Charles’s. The stroke which damaged her innate understanding of small numbers also robbed her of the entire numerical edifice built on that foundation. For her, numbers stopped at four. When asked to count up from one, she got to four and no further. If there were more than four dots on a page she could not count them. She could not say how old she was or how many days were in a week, or even tell the time.
Scary.