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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 for?” but the shorter to prove . 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 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 as much as it When was solving Pell’s 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 algebra, , Of course, you may think, those were the an- , , and so on: all those subjects cients; in modern we have learned better, beyond . 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. , , surprise to you, as it did to me, that arithmetic , , : 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 . Almost 1970) we read all citizens can and do get through 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 and received without a population competent in arith- training in , but there were not many. Of metic? Well, it did, and it even thrived. Arithmetic course, in ancient 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 , 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 , 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 57, 5 Prior to this [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 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 . 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 . But mathematics civil/chemical ; 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 - its program a talk on the mathematics used by ship of chapter 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 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 . 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 , 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 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 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. , 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 for solving them, though, is not list in 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 ’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 . There will be a , or rule, that gives the , 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. But before, so new solutions by workers will not be two members left the club, and the needed. remaining members had to pay $50 I am glad that we do not have to depend on more apiece. How many members workers’ ability to solve algebra problems to get are in the club? through the day because, as every teacher of Do you detect the similarity to the career application mathematics knows, students don’t always get in the first text? The two problems are the same, problems right. The chair of the department of a with different numbers. The second is not practical, Big Ten university once observed, probably after any more than the first comes up as part of a job. a bad day, that it was possible for a student to The reason that this problem—well worth doing graduate with a mathematics major without ever by students—appears in more than one text is that having solved a single problem correctly. Partial it is a superb problem, so superb that has been credit can go a long way. This was in the 1950s, appearing in texts for hundreds of years, copied looked on by many as a golden age of mathematics from one author to another. If you want a problem education. that makes students solve a quadratic equation, In one of those international tests of mathemat- here it is. ical achievement appeared the problem of finding I keep looking for uses of algebra in jobs, but which of two magazine subscriptions was cheaper: I keep being disappointed. To be more accurate, 24 issues with (a) the first four issues free and $3 I used to keep looking until I became convinced each for the or (b) the first six issues that there were essentially none. For this article I free and $3.50 each for the remainder. This is not searched again and found a website that promised a tough problem, so I leave its solution to you. As applications of “college algebra” to the workplace. easy as it is, only 26% of United eighth-graders The first was could do it correctly. That was above You are a facilities manager for the international average of 24%. Even the Japanese a small town. The town contains eighth-graders could manage only 39%. No doubt approximately 400 miles of road when the eighth-graders become adults they will that must be plowed following be better at solving such problems, but even so I a significant snowfall. How many do not want them having to solve problems that plows must be used in order to when solved incorrectly can do me harm. complete the job in one day if the Though people know that they do not use plows can travel at approximately algebra every day, or even every month, many seem 7 miles per hour when engaged? to think that there are hosts of others who do. Perhaps they have absorbed the textbook writers’ This is another textbook “application” made up, I insistence on the “real world” uses of algebra, even think, by its writer with no reference to external though the texts actually demonstrate that there reality. (It’s a big small town that has 400 miles of are none. Were uses of algebra widespread in the streets.) The facilities manager knows how many world of work, all textbook writers would have to do plows there are and can estimate how many more, is to ask a few people about their last applications if any, are needed. The next problem, I think, did of algebra, turn them into problems, and put them arise outside of the head of a textbook writer: in their texts. If 75% of all jobs require algebra, How much ice cream mix and vanilla they could get a problem from three of every four flavor will it take to make 1000 gal- people they ask. However, such problems do not lons of vanilla ice cream at 90% appear in the texts. We get instead the endlessly overrun with the vanilla flavor us- repeated problems about investment clubs losing age rate at 1 oz. per 10 gallon mix? two members and all of the other chestnuts, about (90% overrun that enough cars going from A to B and farmers fencing fields

610 Notices of the AMS Volume 57, Number 5 and so on, that I lack the to display. The Perhaps you’ve had only one op- reason that problems drawn from everyday life do portunity to use geometry in your not appear in the texts is not that textbook authors life, but there are a number of lack and initiative; it is that they do not occupations in which it’s a must. exist. Myself, I’m pleased that my house Though they may not use algebra themselves, was designed and built by people people are solidly behind having everyone learn who were capable of calculating the algebra. Tom and Ray Magliozzi, the brothers who correct rise of a roof for proper are hosts of National Public Radio’s popular “Car drainage or the number of cubic Talk” program, like to pose as vulgarians when they feet of concrete needed for a strong are actually nothing of the kind. On one program, foundation. brother Tom made some remarks against teaching Here is the common error of supposing that geometry and trigonometry in high school. I doubt problems once solved must be solved anew every very much that he was serious. Whether he was they are encountered. House builders have serious or not does not affect the content of his handbooks and tables, and use them. Indeed, remarks or the reaction of listeners. The reaction houses, as well as pyramids and cathedrals, were was unanimous endorsement of mathematics. being built long before algebra was taught in the When mathematics is attacked, people leap to its schools and, in fact, before algebra. defense. Another common misconception occurs in In his piece Tom alleged that he had an octagonal another response: fountain in his back yard that he wanted to surround You sure laid a big oblate spheroid with a border and that he needed to calculate the shaped one when you went on length of the side of the concentric octagon. your tirade against having to learn After succeeding, using, he said, the Pythagorean geometry, trigonometry and other , he reflected things mathematical. Who uses this stuff? Geologists, That this was maybe the second aircraft designers, road builders, time in my life—maybe the first— contractors, surgeons and, that I had occasion to use the yes, even radio broadcast techni- geometry and trigonometry that I cians (amplitude modulation and had learned in high school. Further- frequency modulation are both more, I had never had occasion to based on manipulating forms use the higher mathematics that described by trig functions—don’t the high school math had prepared get me started on alternating cur- me for. rent). Never! So, Tommy, get a life. The only Why did I—and millions of other people who don’t use these princi- students—spend valuable educa- ples every day are those who can’t tional hours learning something do and can’t teach, and thus are that we would never use? suited only for as politicians Is this education? Learning skills or talk show hosts. that we will never need? People seem to think that because something After some real or pretended populism (“The involves mathematics it is necessary to know people who run the education business are money- mathematics to use it. Radio does indeed involve grubbing self-serving morons”), he concluded and cosines, but the person adjusting the dials that needs no trigonometry. Geologists searching for oil do not have to solve differential , though The purpose of learning math, differential equations may have been involved in which most of us will never use, is the creation of the tools that geologists use. only to prepare us for further math I am not saying that mathematics is never courses—which we will use even required in the workplace. Of course it is, and less frequently than never. it has helped to make our what it is. There were answers, quite a few of them, posted However, it is needed very, very seldom, and we do at the “Car Talk” website. All disagreed with Tom’s not need to train millions of students in it to keep conclusion, which actually has elements of truth. businesses going. Once, when I was an employee (A reply that started with “I agree” might be of the Metropolitan Life Insurance Company, I thought to be a , but the irony was given an annuity rate to calculate. Back then, that followed was at least as heavy as lead.) One insurance companies had rate books, but now and response included then there was need for a rate not in the book.

May 2010 Notices of the AMS 611 Using my of the mathematics of life Our forebears were not so diffident. In 1906 J. D. contingencies, I calculated the rate. When I gave Fitch wrote it to my supervisor he said, “No, no, that’s not Our future lawyers, clergy, and right. You have to do it this way.” “But,” I said, statesmen are expected at the Uni- “that’s three times as much work.” Yes, I was told, versity to learn a good deal about but that’s the way that we calculate rates. My , and angles, and number knowledge of life contingencies got in the way and proportions; not because these of the proper calculation, done the way it had subjects have the smallest relation been done before, which any minimally competent to the needs of their lives, but employee could have carried out. because in the very act of learn- It may be that there could arise, say, a partial ing them they are likely to acquire differential equation that some company needed that habit of steadfast and accurate to solve, the likes of which it had never seen before. thinking, which is indispensible in If so, there are plenty of available all the pursuits of life. to do the job. They’d work cheap, too. I do not know who J. D. Fitch was, but he was Jobs do not require algebra. I have expressed correct. Thomas Jefferson said this truth many times in talks to any group who would listen, and it was not uncommon for a Mathematics and natural philoso- member of the audience to tell me, after the talk phy are so peculiarly engaging and or during it, that I was wrong and that he used delightful as would induce every- algebra or calculus in his job all the time. It always one to wish an acquaintance with turned out that he used the mathematics because them. Besides this, the faculties he wanted to, not because he had to. of the mind, like the members Even those who are not burdened with the of a body, are strengthened and error that algebra is necessary to hold many improved by exercise. Mathemati- jobs the teaching of algebra. Everyone cal reasoning and deductions are, supports the teaching of algebra. The public wants therefore, a fine preparation for more mathematics taught, to more students. The investigating the abstruse specula- requirements keep going up, never down. tions of the law. The reason for this, I am convinced, is that In 1834, the Congressional Committee on the public knows, or senses, that mathematics Military Affairs reported develops the power to reason. It shows, better Mathematics is the study which than any other subject, how reason can lead to forms the foundation of the course truth. Of course, other sciences exhibit the power [at West ]. This is necessary, of reason, but there’s all that overhead—ferrous both to impart to the mind that and ferric, dynes and ergs—that has to be dealt combined strength and versatility, with. In mathematics, there is nothing standing the peculiar vigor and rapidity of between the problem and the reasoning. comparison necessary for military Economists reason as well, but sometimes two action, and to pave the way for economists reason to two different conclusions. progress in the higher military Philosophers reason, but never come to any sciences. conclusion. In mathematics problems can be solved, Here is testimony from a contemporary student: using reason, and the solutions can be checked and shown to be correct. Reasoning needs to be The summer after my freshman learned, and mathematics is the best way to learn year I decided to teach myself it. algebra. At school next year my People grasp this, perhaps not consciously, and grades improved from a 2.6 gpa to a hence want their children to undergo mathematics. 3.5 gpa. Tests were easier and I was Many times people have told me that they liked much more efficient when taking mathematics (though they call it “math”) because them and this held true in all other it was so definite and it was satisfying to get the facets of my life. To sum this up: right answer. Have you not heard the same thing? algebra is not only mathematical They liked being able to reason correctly. They principles, it is a or way knew that the practice was good for them. No one of thinking, it trains your mind has ever said to me, “I liked math because it got and makes otherwise complex and me a good job.” overwhelming tests seem much We no longer have the confidence in our subject easier both in school and in life. that allows us to say that. We justify mathematics Anecdotal evidence to be sure, but then all history on its utility in the world of getting and spending. is a succession of anecdotes.

612 Notices of the AMS Volume 57, Number 5 That is what mathematics education is for and what it has always been for: to teach reasoning, usually through the medium of silly problems. In the Rhind Papyrus, that Egyptian textbook of mathematics c. 1650 BC, we find Give 100 loaves to five men so that the shares are in arithmetic progression and the sum of the two smallest is 1/7 of the three greatest. The ancient Egyptians were a practical people, but even so this eminently unpractical problem was thought to be worth solving. (The shares are 1 2/3, 10 5/6, 20, 29 1/6, and 38 1/3.) George Chrystal’s Algebra (1886) has on page 154 more than fifty problems, all with the instruction “Simplify”, including

1 1 1 1 + − x2 y 2 x2 y 2 − 1 1 1 1 − + x2 y 2 x2 y 2  8  ! !  x + y x − y x2 y 2   + + − 2  x − y x + y y 2 x2

There is no reason given, anywhere in his text, why anyone would want to simplify such things. It was obvious. That is how algebra is learned. As for the reason for learning algebra, that was obvious as well, and it was not for jobs. (The answer to the problem—what fun Chrystal must have had in making it up—is −1.) I am not so unrealistic as to advocate that textbook writers start to produce texts with titles like Algebra, a Prelude to Reason. That would not fly. We do not want to make unwilling students even more unwilling. We cannot go back to texts like Chrystal’s. But could we perhaps tone it down a little? Can we be a little less insistent that mathematics is essential for earning a living? What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better. Furthermore, it is worth teaching. Were I given to hyperbole I would say that mathematics is the most glorious creation of the intellect, but I am not given to hyperbole so I will not say that. However, when I am before a bar of judgment, heavenly or otherwise, and asked to justify my life, I will draw myself up proudly and say, “I was one of the stewards of mathematics, and it came to no harm in my care.” I will not say, “I helped people get jobs.”

May 2010 Notices of the AMS 613