Whatever you say it is, it isn't.
A. Bogomolny
Consider the following speculation: it is probably true that more people nowadays heard the word Algebra than the name Alfred Korzybski (1879-1950). "Of course," one could say, "whoever was this Korzybski guy, a solid majority of the population had at one time or another to make a painful and a fateful choice between taking or skipping an algebra course." Well, this is certainly true. But did you know that the quintessentially American gesture of wagging pairs of fingers in the air as if putting quotes around spoken words derives from general semantics - the term and the discipline that originated with A. Korzybski? Korzybski's fundamental idea was that the structure of the language molds human thought process, often in a detrimental manner.
I first heard of Alfred Korzybski a couple of years ago when a nice little book Mathsemantics by Edward MacNeal fell into my lap. (I have mentioned the book on another occasion.) Although his theories were popular in the 1930s and 1940s, Korzybski had been regarded by many as a delusional and idealistic crackpot (e.g., M. Gardner, Fads & Fallacies, New American Library, 1986, Ch. 23). However that may be, I may attest from personal experience that his magnum opus -- Science and Sanity published in 1933 -- does not make an easy reading. And this is so not because of the depth of his arguments, but because of the repulsively repetitive and verbose style of the book. But, as to Korzybski's theories, I am content to let history be the judge. As to his underlying principles, I find some of them tremendously useful. Here are the most basic ones:
The word is not the thing. | |
and | |
Whatever you say it is, it isn't. |
Don't they engender a critical mind frame?
So, let's have a critical look at a five article series from The Sacramento Bee newspaper that started with an editorial in the December 10, 2000 issue. This Fall Go. Gray Davis has signed a law known as the Algebra Bill that makes Algebra 1 a graduation requirement in all California public schools. According to the editorial, the series "attempts to unravel some of the discipline's mystery. The series makes a powerful case for California's new insistence that all students master the subject before graduation." As we shall see, much of the reading should be accompanied by finger wagging.
Day One: The sore subject, December 10, 2000
It is a huge shift for kids and for schools but one that offers lofty rewards. It will open the doors of higher mathematics and college to more students. It will give young people more options in their lives. And beyond all that, the new blanket of algebra will endow students with something invaluable: the ability to ponder, to solve problems and to sort things out no matter how tough.
Algebra, in simple terms, will teach people how to think.
It will open the doors of higher mathematics and college... According to Professor Stein whose statistics is quoted in the Day Three installment, less than 3% of the work force will ever need higher mathematics. I believe the percent to be much lower.
Forced through the open doors, a typically reluctant Liberal Arts student will finalize his or her mathematics education with a Calculus course, the contents of which will be forgotten long before the graduation ceremony. Then what is the argument? Why not keep the doors open?
It will give young people more options in their lives. What about those who would not be able to make it? The second installment -- Day Two: True tales of algebra -- cites several real life stories that would rather suggest that some students are getting fewer options, not more.
The subject now stands between Karon and a community college degree. She is a clerk in the West Sacramento Police Department, where the degree would bring her more pay and a shot at moving up. She has completed 70 credits, more than enough for an associate's degree. But she needs to pass what she calls "dummy math" and an algebra course to complete the degree. "I dread the thought of taking those classes," Karon said. "I'm 38 years old, and I still don't want to do it." |
May we conclude from the above story that Karon has absolutely no need for algebra to carry out her job successfully? Probably so. However, her community college (likely with a view to upholding high educational standards) requires an algebra course to obtain a degree. Karon's employer is probably interested in promoting intelligent employees as certified by an associate degree. Both requirements are quite arbitrary and together block Karon's advancement. It's probably also a reasonable conclusion that were Karon to take an algebra class in high school, she would not have gotten her high school diploma, not to mention an associative degree. Would she be given a job in the West Sacramento Police Department?
It will give young people more options in their lives. Even if we adopt a positive attitude and treat the claim formally, just for what is implied -- a certificate holder has more job choices -- the claim is misleading. Taking and fighting (or enjoying) one's way through an algebra course is a matter of attitude. If it's only done for the sake of high school graduation the wrong kind of attitude may be inculcated in students' minds. People with a wrong kind of attitude are bound to overlook many life's options.
"I hate algebra. I took it three times in high school, and I never understood it. I don't know why I should bother. It feels totally pointless to me," said Mary Ann Colby, a student at Sacramento City College. Colby, 25, is taking beginning algebra this fall at the community college. She spends several hours a day in math labs and with tutors, trying to pass. Once she does, she must take Algebra 2 to reach her dream of finishing college and becoming a social worker. Then, she plans to ignore algebra for the rest of her life. |
Algebra, in simple terms, will teach people how to think. Of course this only applies to those who can make it. What about the rest? But a more important question is, Does one need algebra to learn thinking? There is any number of books that teach thinking under various guises (e.g., Smart Choices, de Bono's Thinking Course, A Whack on the Side of the Head, etc.), none requiring mathematics beyond common arithmetic.
Imagine a car commercial that trumpets the car's ability to withstand a ride from one point to another. Would one fancy the car for that reason? Do you know of a subject matter that, if taught correctly, would not improve students' mental abilities?
And from another angle, may it be that some other mathematics discipline -- Geometry is one possibility -- may do at least as good a job?
Algebra, in simple terms, will teach people how to think. Algebra of course will do nothing of the sort. Algebra teachers may. However, we learn from the second installment that
In California, 40 percent of math teachers in grades seven through 12 have neither a major nor a minor in math, with the numbers running even higher in poor and urban neighborhoods. |
Day Three: Applied Math, December 12, 2000
Have you crossed a bridge lately? Picked up a cellphone? Paid an insurance premium? Walked into a building? Earned or paid some compound interest? Been in a hospital? Signed on to a computer?
If the answer is yes to any of these, you have encountered algebra -- in all its glorious usefulness.
... you have encountered algebra ... Or have you? The next sentence reads:
Many people never give algebra a thought, save for the year or two they may study it in high school or college. |
Can one encounter anything without giving it a thought? By the same token, while driving a car you encounter chemistry (oil refining processes), politics (OPEC prices), state governance (environmental policy), physics (Newton laws), computers (cruise control) and, which is most likely if you do not pay attention, mysticism (What are these flashes just behind my car?)
There is more in the same spirit, now with a support from a well known mathematician.
"You are surrounded by algebra for heaven's sake," said Sherman K. Stein, a professor emeritus of mathematics at the University of California, Davis.
"Every time you pick up your cellphone, you are identified as an algebraic formula," he said. "You turn on the radio, if you looked at how the antennas are designed, you would find algebra. If you're buying insurance, all those calculations about the life span of people who smoke and so forth, all of it involves algebra."
In his 1996 book, "Strength in Numbers," Stein went to considerable length to document the mathematical skill levels required in several hundred occupations. In the chapter called "What's In It For Me?" Stein found that two-thirds of the American work force -- about 80 million people out of 121 million -- do not use algebra or math skills higher than basic arithmetic.
Sherman Stein gave a very personal estimate with which I disagree and so would many others. For example, according to Stein algebra is used by executives. According to Keith Devlin, "Many -- perhaps most -- of the CEOs of top companies have virtually no mathematical knowledge. If their company needs to have someone with mathematical expertise, such a person will be hired." Stein lumps into a single category professionals that work in computers, mathematical, operation research systems whose work require the core of a college level mathematics program. Similar requirements are imposed on another category of chemists, physicists, meteorologists, etc. As a professional, I may attest that a rare programmer needs any kind of mathematics to perform a job. Among my personal acquaintances are biologists and chemists, none of whom needs any mathematics either. I also disagree with Stein's estimate that elementary school teachers need algebra in their jobs. Summing up, I believe that considerably more than 80 million people -- probably more like 90 million people -- do not need and do not use algebra in their jobs. A tremendous number of those who do, do not use but most rudimentary concepts.
(As a later addition to the above, a book I recently purchased (What The Numbers Say by D. Niederman and D. Boyum, Broadway Books, 2003), we find the following remark: ... large amounts of quantitative information are a feature of many such jobs, and good quantitative thinking is crucial to doing the jobs well. But matrix algebra is not required. Nor are high school staples such as quadratic equations, analytic geometry, and imaginary numbers.)
Day Five: Shaking up schools, December 14, 2000
Teachers and principals have expressed private doubts, saying middle school is too early for many to learn algebra. Even at the high school level, some teachers believe that not all students can or should be required to complete the subject.
To the naysayers, the algebra allies point across the sea.
"I don't believe it for a second. I see too much evidence in other countries, where algebra is just the norm for everyone," said Sue Stickel, an assistant superintendent for the Elk Grove Unified School District.
Accross the sea students start receiving a fundamentally different education in elementary school. In this respect a recent book Knowing and Teaching Elementary Mathematics by Liping Ma is quite revealing. In a certain sense, American school teachers know more of mathematics than their Chinese counterparts. The knowledge of Chinese teachers, however, is much deeper and more conceptual. Ma's research correlates teaching practices with teacher's understanding of mathematical concepts. As a result, Chinese students, unlike their American peers, reach 7-8 grades fully prepared for a switch from arithmetic to algebraic thinking.
Indicative of the problem is a message at the Algebra Discussion Forum posted by a district-wide coordinator of mathematics from Georgia where Algebra is being offered in grades 6-7.
My fourth middle school is very good-hearted, and feels as if they have students who just can't be a part of this plan--they need more study of basic skills. Can someone provide me with some evidence that these lower kids can learn algebra without proven mastery of all basic skills? Scott Farrand's Response (Scott Farrand is a professor of mathematics at California State University, Sacramento): If mastery means that a very high level of performance, then there is lots of evidence that students actually attain this sort of mastery in taking the subsequent course. This is an interesting phenomenon in the world of testing -- students perform much better in a test on say prealgebra when they are tested a year later, after they have had a year of algebra. But if by mastery you mean a sort of functional ability, then there is lots of evidence that students do need this to be able to succeed in algebra. |
There indeed appears to be an entrenched pattern in the American education system. Ralph J. Boas once wrote that "... the students always have to be taught what they should have learned in the preceding course." However, in the Algebra Discussion Forum we encounter the following remark by Michael Paul Goldenberg:
Last summer at Rutgers University I heard Liping Ma make a comment similar to that reported by Dr Askey. But what I heard her say was that STUDENTS, not teachers, considered algebra easier than arithmetic. When asked why, she stated that the Chinese study arithmetic in such depth that when students encounter algebra, it doesn't seem hard. That would be consonant with my limited experience teaching the subject: the better a student grasps arithmetic, the less trouble s/he seems to have with algebraic manipulations and concepts. |
What do other newspapers write?
Los Angeles Times (December 17, 2000) reports that the state Board of Education in Sacramento proposed to make California's test easier and shorter than originally envisioned. "Gripped by fears that too many students will flunk, California and many other states are pulling back on plans to require passage of a single high-stakes exam before seniors can earn diplomas."
San Francisco Chronicles (December 29, 2000) reports that the odds are that more than a third of new teachers in California will leave their jobs within 3 years.
Howard Gardner in Los Angeles Times (December 31, 2000). "With California leading the way, the nation is going through a frenzy of testing its public school students. Never before have so many students been given so many formal standardized tests." In the quest to improve public schools, we've made test performance more important than education.
More from Los Angeles Times (January 14, 2001). California's standardized testing program is having a variety of unintended consequences. Teachers are dumping whole books in favor of short passages geared toward the test. Science and social studies are being thrown over to make time for test-related drills. And some family law lawyers are using school test scores to argue in favor of a child being placed with a particular parent in custody disputes.
Acknowledgement
I thank Jerry Becker from Southern Illinois University for keeping my name on his list server. As other lucky participants on the list, I receive regular news cuts and various updates of interest to mathematics educators.
- Why Not Geometry?
- A critical look at a newspaper article
- A nice piece of geometry
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