February 17, 2006
Algebra is a Start but Only a Start
The science wing of the blogosphere has erupted with shock and indignation over a disingenuous piece by Richard Cohen in yesterdays Washington Post. Cohen tells a high school dropout who had trouble with algebra,
Here's the thing, Gabriela: You will never need to know algebra. I have never once used it and never once even rued that I could not use it. You will never need to know -- never mind want to know -- how many boys it will take to mow a lawn if one of them quits halfway and two more show up later -- or something like that. Most of math can now be done by a computer or a calculator. On the other hand, no computer can write a column or even a thank-you note -- or reason even a little bit. If, say, the school asked you for another year of English or, God forbid, history, so that you actually had to know something about your world, I would be on its side. But algebra? Please.
This opinion, Kevin Drum's effort to see a serious point not-with-standing, is not only wrong, it is dangerous. The truth is that however much or little one uses algebra in one's life, one is exposed to probability and statistics every day. And one has no hope of understanding probability if one cannot understand algebra. This is not to down play the importance of algebra. Every science, every field of engineering, and many trades require the frequent use of algebraic manipulation of abstract operators. A simple example, my back yard is an odd shape, vaguely trapezoidal; how much fertilizer do I need to buy to get the recommended coverage?
But what really shocked me was that one could graduate from high school with only algebra. Every time we hear about the results of a poll, probability is involved. Every time we read or hear a politician, a preacher, a scientist or for that matter, read a blog, probability is involved. Advertising is full of probabilistic information and miss-information. Every statement that has any truth claim carries with it a probability that it is or is not true. Of course, formal probability will not always help us in assessing many claims, but it sure can help. For this reason, I am shocked that some probability and statistics are not required for graduation. And to do simple probability one must be able do simple algebra.
I naively thought probability and statistics were required and perhaps they are part of the algebra curriculum today. There was a lot of probability in my high school math curriculum and I know it was a part of our children's education also. But Shirley doesn't remember being exposed to it until college. And in retrospect every member of my immediate family including myself, tended to take the most advanced courses available and so did most of our friends.
Those of you who are regular readers of Abnormal Interests know that I take an overtly probabilistic approach to archaeology and Near Eastern studies in general, for that matter history in general. I am a Bayesian. And I think most people who study the things that interest me are also, whether they know it or not.
When, for example, a politician says, "X is the case." He or she means one of two things; "There is a high probability that X is the case" or "I want you to think that there is a high probability that X is the case." Only by accessing the evidence can one decide which of these two options the case is indeed. Most of this can proceed in a very informal way. But having been through the math, on related issues, helps in developing intuitions that give one the ability to do such an assessment.
I decided to focus this rant on probability but it could have just as well been focused on solid geometry or math-analyses (I think its called pre-calculus today). Everywhere I turn the thought processes that I learned in my high school and college math classes have relevance. And perhaps the most important thing is that people who may never write or solve an equation will still apply their mathematical knowledge or ignorance in a great many real life situations.
In this day and age, algebra is essential but it is just the beginning.
Just a small footnote: our English word "algebra" comes from Arabic al-ğabr as seen in Abū ‘Abd Allāh Muhammad ibn Mūsā al-Khwārizmī's al-Kitāb al-mukhtaşar fī hīsāb al-ğabr wa’l-muqābala ("A compact introduction to calculation using rules of completion and reduction") written by Abū ‘Abd Allāh Muhammad ibn Mūsā al-Khwārizmī. Our word "algorithm" may come from the last word in his name. The word may be related to Babylonian gabru, "equivalence." al-Khwārizmī was born in Baghdad and lived from 790 to 840. While Europe was in a period of intellectual stagnation or at least religious dominance of intellectual life, the Arab and Muslim world was a hot bed of intellectual activity.
The probability of this post being correct is over 90% or at least I want you to think that it is.
Update, February 17, 2006:
Asheesh Siddique of Campus Progress also misses the point when he says,
The point that is being missed is that there are plenty of ways in which modern society makes mathematical illiteracy perfectly acceptable. Be it the tip-table on the bill in the restaurant, tax preparation services, or credit cards that allow you to simply 'swipe and go,' instead of count change, it is more and more acceptable not to know how to do basic calculations and rudimentary applications of math skills. We disparage at this, but at the same time, we like the conveniences of not having to use math skills, clearly because we use credit cards, tax preparation services, etc. I suspect that nobody is going to want to relinquish these conveniences, which simply means that math skills will become more and more irrelevant and unnecessary as technology advances. Such is the paradox of modern society.
Yes, we do enjoy the conveniences of systems that take away out need to do "basic calculations and rudimentary" math. The real issue is that the society that has brought us the wonders that allow us to get by with little or no arithmetic has also brought us increased need to develop ever better math intuitions. Our problem is that while arithmetic may be out, mathematics are definitely in.
Posted by Duane Smith at February 17, 2006 3:36 PM | Read more on Odds and Ends |
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I would like to know how Mr. Cohen proposes that one might program a computer (or for that matter a calculator) without knowledge of Algebra and the science on which it is based. Not only that, but even the ability to post his ranting on a website requires an understanding of abstract variables (read “Algebra”).
The real problem with secondary education is not that requirements are too high, but that they are much to low. Keep up the good work, Duane, and thank you for excusing the frustration of a mathematician.
By the way your discussion of statistics is quite poignant; however, I do not think that most people understand statistics/probability to any competent level. Let me give an example which may be apocryphal, but demonstrates the point regardless:
During the U.S. government's "Strategic Defense Initiative" program, better known as "Star Wars", leading scientists on the project were asked to report their progress to the Minister of Defense. So they gathered their data and brought it together in a presentation. In short, they had discovered that the whole problem of shooting down a nuclear missile at such a great distance was a very tricky problem indeed. They intended to explain to the minister what an impossible problem it was -- well beyond the capabilities of current technology. During the course of their presentation, the following exchange took place.
SCIENTIST: ...and so you can see, Mr. Minister, that in order to achieve an acceptable hit-rate against the missiles, our instruments need to be accurate to one part in ten to the ninth. So far, the best we have been able to achieve is one part in ten to the fifth.
DEFENSE MINISTER: That's tremendous! We're over half way there!
Posted by: HH Hardy at February 17, 2006 9:00 PM
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