Janet Stemwedel has a long post which elucidates various angles of the Cohen & algebra story. I agree with many of Janet’s points, and I tend to believe that knowing algebra is an important necessary precondition for being a well rounded modern intellect. But I want to emphasize modern, I’ve mentioned before that John Derbyshire is writing a history of algebra, Unknown Quantity. Derb mentioned to me that though the Greek mathematician Diophantus lurched toward symbolic algebra 2,000 years ago, his work did not lay the seeds for any further developments because a scientific culture did not exist which could make the next leap. The ancient Greeks were not unintelligent, so the fact that many of us (rightly I believe) take symbolic algebra for granted as a necessary feature of our cognitive landscape is something to reflect upon. Maths that we assume to be fundamental elements of our mental toolkits would have been beyond the very conception of the most brilliant minds of our species over one thousand years ago. I am somewhat skeptical that the solid majority of American students could not pass algebra I, the very basics, with proper instruction. Nevertheless I do suspect a minority of humans (not subject to pathological cognitive impairment) might never be able to grasp algebra. I do not know if “Gabriella” or Richard Cohen is in that minority, and they would be no less worthy of respect if it was truely so that they lacked an aptitude for abstract mathematics. 18th century mathematics is necessary for the modern life of the mind, but we should recall that 8th century genius (eg., the Venerable Bede) did well without it, and moderns whose focus in life is less reflective and cognitive will do fine as well.
Addendum: In The Number Sense cognitive neuroscientist Stanislas Dehaene reviews literature which points to the existence of mathematical aphasiacs. For example, individuals who suffer brain damage and lose the ability to comphrehend algebra, or do basic addition, subtraction and multiplication, but remain high functioning in other ways. Dehaene’s book points to two important points that are salient in relation to mathematical abilities. First, math is an extension of our innate analog numeracy, our gestalt sense of proportions and ranks which we inherited from our ancestors and share with other animals. For example, there are forms of brain damage which simultaneously render individuals unable to move their fingers and count, strongly suggesting that counting is an abstraction of tacking off numbers with one’s fingers. Additionally, mathematical abilities draw from various cognitive subfunctions, both in concert and separately. This might explain why some people can lose the ability to do arithmetic, but retain a capacity to understand basic algebra, or vice versa.
There are likely a non-trivial number of “normal” people walking around who might have difficulty with mathematics because of their cognitive architecture. Our modern world has been characterized by a progressive ratcheting up of the minimum mathematical competencies demanded of individuals, and as this occurs it is expected that eventually one will encounter a piont at where a large portion of humanity simply can not be reached by practically implementable didactic methods. In mathematics I suspect that that point is somewhere around algebra and geometry, so demanding that all high school graduates complete a course in these subjects might be setting people up for failure. The American educational system is different than many European models because we do not emphasize tracking and segregation of the college bound and non-college bound, our schools produce well rounded citizens. Operationally tracking does exist, as does extreme segregation socioeconomic status, but the ideal remains egalitarian. There needs to be a place for the Richard Cohens of the world, just not to as columnists.