What Is Mathematics For? Underwood Dudley
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What Is Mathematics For? Underwood Dudley more accurate title is “What is mathe- do it to fill classrooms with students learning how matics education for?” but the shorter to prove theorems. Compared to them, the ancient one is more attention-getting and al- Romans were a mathematical blank. The Arab lows me more generality. My answer scholars who started to develop algebra after the Awill become apparent soon, as will my fall of Rome were doing it for their own pleasure answer to the subquestion of why the public and not as something intended for the masses. supports mathematics education as much as it When Brahmagupta was solving Pell’s equation a does. millennium before Pell was born, he did not have So that there is no confusion, let me say that students in mind. by “mathematics” I mean algebra, trigonometry, Of course, you may think, those were the an- calculus, linear algebra, and so on: all those subjects cients; in modern times we have learned better, beyond arithmetic. There is no question about what and arithmetic at least has always been part of arithmetic is for or why it is supported. Society everyone’s schooling. Not so. It may come as a cannot proceed without it. Addition, subtraction, surprise to you, as it did to me, that arithmetic multiplication, division, percentages: though not was not part of elementary education in the United all citizens can deal fluently with all of them, we States in the colonial period. In A History of Mathe- make the assumption that they can when necessary. matics Education in the United States and Canada Those who cannot are sometimes at a disadvantage. (National Council of Teachers of Mathematics, Algebra, though, is another matter. Almost 1970) we read all citizens can and do get through life very well Until within a few years no studies without it, after their schooling is over. Nevertheless have been permitted in the day it becomes more and more pervasive, seeping down school but spelling, reading, and into more and more eighth-grade classrooms and writing. Arithmetic was taught by a being required by more and more states for few instructors one or two evenings graduation from high school. There is unspoken a week. But in spite of the most agreement that everyone should be exposed to determined opposition, arithmetic algebra. We live in an era of universal mathematical is now being permitted in the day education. school. This is something new in the world. Mathematics has not always loomed so large in the education of Opposition to arithmetic! Determined opposi- the rising generation. There is no telling how many tion! How could such a thing be? How could society children in ancient Egypt and Babylon received function without a population competent in arith- training in numbers, but there were not many. Of metic? Well, it did, and it even thrived. Arithmetic course, in ancient civilizations education was not was indeed needed in many occupations, but those for everyone, much less mathematical education. who needed it learned it on the job. It was a system Literacy was not universal, and I suspect that many that worked with arithmetic then and that can who could read and write could not subtract or work with algebra today. multiply numbers. The ancient Greeks, to their Arithmetic did make it into the curriculum, but, glory, originated real mathematics, but they did not then as now, employers were not happy with what the schools were turning out. Patricia Cline Cohen, Underwood Dudley is a mathematician, retired from in her estimable A Calculating People: The Spread DePauw University. His email address is ddunx46135@ of Numeracy in Early America (U. of Chicago Press, yahoo.com. 1983; Routledge paperback, 1999) tells us that 608 Notices of the AMS Volume 57, Number 5 Prior to this act [1789] arithmetic Janitor service, Janitors’ equipment had not been required in the Boston and supplies, Jewelers, Karate and schools at all. Within a few years other martial arts, Kennels, Label- a group of Boston businessmen ing, Labor organizations, Lamps protested to the School commit- and lamp shades. tee that the pupils taught by the In which six is algebra required, even for training method of arithmetic instruction or license? I again looked very hard for evidence in then in use were totally unprepared the NRC’s publication but couldn’t find any. for business. Unfortunately, the ed- It may be that no evidence is presented because ucators in this case insisted that none is needed: everybody knows that algebra is they were doing an adequate job needed for all sorts of jobs. For example, there was and refused to make changes in an algebra book whose publisher advertised that it the program. contained Both sides were right. It is impossible to prepare “Career Applications”—Includes ex- everyone for every possible occupation and it is planations, examples, exercises, foolish to try. Hence many school leavers will be and answers for work in electronics; unprepared for many businesses. But mathematics civil/chemical engineering; law en- teachers, then as now, were doing an adequate job. forcement; nursing; teaching; and A few years ago I was at a meeting that had on more. Shows students the relation- its program a talk on the mathematics used by ship of chapter concepts and job the Florida Department of Transportation. There skills—with applications developed is quite a bit. For example, the Florida DoT uses through interviews and market re- Riemann sums to determine the area of irregular search in the workplace that ensure plots of land, though it does not call the sums relevance. that. After the talk I asked the speaker what Of course I requested an examination copy, and mathematical preparation the DoT expects in its the publisher graciously sent me one. To return new hires. The answer was, none at all. The DoT the favor, I will refrain from naming the publisher has determined that it is best for all concerned or the author. The career applications were along to assume that the background of its employees the lines of includes nothing beyond elementary arithmetic. What employees need, they can learn on the job. In preparation for the 2002 Win- There seems to be abroad in the land the delusion ter Olympic Games in Salt Lake that skill in algebra is necessary in the world of City, several people decide to pool work and in everyday life. In Moving Beyond Myths their money and share equally the (National Academy of Sciences, 1991) we see $12000 expense of renting a four- bedroom house in Salt Lake City Myth: Most jobs require little math- for two weeks. The original number ematics. of people who agreed to share the Reality: The truth is just the oppo- house changed after two people site. dropped out of the deal because I looked very hard in the publication for evidence they thought the house was too for that assertion, but found none. Perhaps the small. Those left in the deal must NAS was equating mathematics with arithmetic. now pay an additional $300 each Many people do this, as I have found in asking them for the rental. How many people about how, or if, they use mathematics. Almost were left? always, the “mathematics” they tell me about is Exactly what career this applied to was not specified. material that appears in the first eight grades of Nor was it mentioned that the best way to solve school. this problem is to find a member of the group Algebra, though, is mentioned explicitly in and ask. The answer should be forthcoming. If the Everybody Counts (National Research Council, person’s reply is the conundrum in the text, the 1989): member of the group should be beaten about the Over 75 percent of all jobs require head until he or she promises to behave in a more proficiency in simple algebra and civilized manner. geometry, either as a prerequisite This is not to say that the problem is not a good to a training program or as part of one. It is a good one, a very good one, and one that a licensure examination. students should try to solve. Students should be I find that statement extraordinary. I will take made to solve many word problems, the more the my telephone Yellow Pages, open it at random, and better. The reason for solving them, though, is not list in order the first eight categories that I see: that they will arise in their careers. May 2010 Notices of the AMS 609 Another text, whose author and publisher I air is put into the frozen mix to will not name—alas, still in print in its third increase its volume by 90%.) edition—asserts Though dressed up with x’s and y’s, the solution This text aims to show that mathe- amounts to calculating that you need 1000/(1 + matics is useful to virtually every- .9) = 526.3 gallons of mix to puff up into 1000 one. I hope that users will complete gallons of ice cream, so you will need 526.3/10 = the course with greater confidence 52.6 ounces of flavor. in their ability to solve practical The employee adding the flavor will not need problems. algebra, nor will he or she need to think through Here is one of the practical problems: this calculation. There will be a formula, or rule, that gives the result, and that is what happens An investment club decided to buy on the job. Problems that arise on the job will be $9000 worth of stock with each for the most part problems that have been solved member paying an equal share.