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.
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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.
Hmm. In my country, ALL high school students have to learn algebra and geometry. The difference between those which will go to university and those which won't is how much algebra they learn, and that the latter certainly won't learn more advanced concepts, like derivatives, calculus or infinite series. But algebra is the basic stuff EVERYONE has to learn in high school.
As I've written to others, one really needs to read the original LA Times article (reg required) that Cohen left uncredited as how he found out about "Gabriella". The problem was not the increase of the standard or that *some* people will never understand algebra no matter what we do. The problem was an arbitrary standard suddenly applied to a generation of high school students who were totally unprepared for it because the lower grade teachers were not teaching them as if they would need to take this higher standard when they got to it.
The article talks about some schools with 50% failure rates in the first class that had that requirement thrust upon them.
There are reams of studies and proven techniques for how a school and school system should increase their standards, and all of them insist that you have to start young in the earliest grades so they're prepared for the higher standard at high school when they get there, and only then do you raise the standard for high school graduation itself. There are no instant techniques to suddenly get "better", and ignorant jerk (who's no longer in the school system to take the heat for the failures he wrought) decided that because a more challenging system worked for him, it must work for everybody. He even admitted that there would be an increase in the drop-out rate and seemed to not care at all.
Virginia, at least in my day, actually had two graduation diplomas. One was for the college-bound and had the requirements of 3 years of math (all algebra 1 and above), 3 years of science (including chemistry with mathematics), 3 years social studies, etc.
The second was for those knowing they're just entering the job market right away, and didn't have the higher math and science requirements, only 2 years of "something", so at least they could be able to say they finished high school. They still had to pass the minimum standards tests to escape 11th grade, but those usually just involved arithmetic and word problems.
While Cohen is certainly out to lunch, the fact that so many students are unable to handle formal reasoning using abstract sysmbols is consistent with Piaget's theory of cognitive development. According to Piaget, formal operational thinking is the last of four stages of cognitive development, and many adults never reach it. Transitions between the earlier stages occur naturally as the individual matures, but it appears that reaching the last stage requires help; few people do it on their own.
So we should probably look to the teaching methods these students are being exposed to, and ask if they are adequate. From what I understand, the answer is no.
The main flaw with Cohen's argument that algebra is not necessary in "real life" anyway is that it would have to be extended to all forms of formal reasoning. This leaves flipping hamburgers.
What I don't understand is the focus on algebra in the particular case Cohen was describing. Gabriela couldn't do basic arithmetic. I don't know enough about cognitive development to say whether the inability to do arithmetic ties into what you discuss in the addendum.
Ha, forget unknown algebra methods -- the staircase-looking algorithm for long division wasn't even published until 1491!
Speaking of inventing algebra, this gets back to a pet interest of yours: Central Asian history. How come almost all the big names in Persian math/sci were from what's now Uzbekistan? Al-Biruni, Abu Nasr Mansur, and big guns Avicenna & Al-Khwarizmi (named after a region in Uzbekistan). That was before the Turks & Mongols arrived, who smashed the capital lest it rival Samarkand. I can hear Genghiz Khan upon arrival: Math schmath, where your women at? I bet he could've played a mean game of Tetris, though...
> Hmm. In my country, ALL high school students have to learn algebra and geometry.
But do they /really/ learn it? In my country, everybody has to pass the test, but few indeed have conceptual understanding of algebra when they're done.
While modern algebraic symbols are a relatively new invention, that doesn't mean that algebraic notation is difficult. In fact, their purpose and effect is to make algebra far easier to do and learn. In medieval and classical times only a few of the brightest could master algebra because it was so difficult to do without a good notation. Today we can teach it to school children because the notation has reduced algebra to a simple task of symbolic manipulations that can even be performed without thought by machines.