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CHAPTER 10: Follies of the Marketplace: A Tirade on Tents

Four species of idols beset the human mind: to which (for distinction's sake) we have asssigned names.. . the third, idols of the market...

Francis Bacon

[208] Curriculum and teachers are the most important factors in education. But there are also texts from which students might learn and which, at the very least, can reinforce the teachers' contribution. Unfortunately, concern for exposition is not one of the hallowed traditions of the mathematical world and the quality of texts at all levels is very low.

The blame for this state of affairs must be laid on the professors. College texts are, of course, written solely by professors. The secondary and elementary school texts are often supposedly cooperative efforts between knowledgeable professors and experienced teachers, but the professors, "obviously" the authorities, dominate the projects. What should we expect of professors insofar as texts are concerned? Many professors are indifferent to pedagogy and others are totally ignorant of it. They receive no training in writing - even of research papers, let alone texts. On the [209] basis of their backgrounds and major concerns one should no more expect effective writing from mathematics professors than good mathematics in mathematical research papers were they written by English professors.

Our expectations are more than fulfilled. Explanations of mathematical steps are usually inadequate - in fact, enigmatic. Because mathematicians do not take the trouble to find out what students should know at any particular level, they do not know how much explanation is called for. But the decision is readily made. It is easier to say less. This decision is reinforced by the mathematician's preference for sparse writing. If challenged, he replies, "Are the facts there?" This is all one should ask. Correctness is the only criterion and any request for more explanation is met by a supercilious stare. Surely one must be stupid to require more explanation. Though brevity proves to be the soul of obscurity, it seems that the one precept about writing that mathematicians take seriously is that brevity is preferable above everything, even comprehensibility. The professor may understand what he writes but to the student he seems to be saying, "I have learned this material and now I defy you to learn it."

Some of the great masters of mathematics did write enigmatically. The most notorious in this respect was Pierre-Simon Laplace. His assistant Jean-Baptiste Biot, who helped Laplace to prepare for the press the latter's masterpiece, the Mécanique céleste, reported that Laplace was frequently unable to reproduce the steps by which he had reached a conclusion and so inserted in the manuscript, "It is easy to see that.. ." Evidently modern textbook writers have taken seriously the precept that one should emulate the masters. Even if one does not become a master thereby, one can at least appear to be one.

There are textbook writers who believe that a mathematical [209]presentation that is logically sound explains itself to the reader who faithfully follows the author step by step. Presumably the meaning need not be stated by the author explicitly but can be grasped by the reader from the details he ploughs through. The authors do not see the need to take the readers into their confidence, to explain where the road is going, why this one is better than another, and what is really achieved. They give no inkling of how a proof was arrived at, why anyone sought the result to begin with, or why anyone should want it now. In effect, the texts are challenges to clairvoyance.

Some textbook writers, unwilling or unprepared to do research, display their "talent" in their texts. They deliberately omit steps that they could not have supplied as students and that they know belong. By pretending that the omitted steps are readily supplied, they seek to put themselves in the position of great masters who have omitted only the trivial. If this condemnation appears too strong, let us remember that with respect to character, mathematicians, whether researchers or teachers, are just a cross section of humanity and, with respect to egotism, a rather disagreeable portion of humanity. In any case, their texts are too often unintelligible.

Surprisingly, many professors object to the few texts that give full explanations and discuss the significance of the ideas being presented. They often complain that such texts are too wordy. Too wordy for whom? These professors prefer to see an enigmatic presentation that leaves the student baffled. Then they, the teachers, can display their brilliance by explaining the text. This preference is well known, but one can also find evidence for it in print. A professor at a good college had the following to say in his review of a text: "So much is here written that is normally spoken by the teacher that the teacher in using the book as a text may find it hard to break away from the book not only in [211] his formal presentation but also in his asides." But if the professor were really teaching ideas and creative thinking he would have so much to do in raising questions, guiding the students' thinking, and improving their suggestions that no book, however helpful, could replace him.

Much poor mathematical writing is due to sheer laziness. There are mathematicians who fail to clarify their own thinking and attempt to conceal their vagueness by such remarks as, "It is obvious that ...," "Clearly it follows that.. . ," and the like. If a conclusion is really evident, it is rarely necessary to say so; and when most authors do say so, it is surely not evident. Often what is asserted as obvious is not quite correct, and the unfortunate reader is obliged to spend endless time trying to establish what does trujy follow.

In some cases the difficulty is the writers' sheer ignorance. Even matters that are well understood by reasonably good mathematicians are not understood by numerous authors of high school and college texts. They put out books mainly by assembling passages and chapters from other books, and where the sources are inadequate so is the pirated material. In their books one finds inaccurate statements of theorems, assertions that are not at all true, incomplete proofs, failure to consider all cases of a proof, the use of concepts that are not defined, reliance upon prior results that were not proved or are proved only subsequently, the use of hypotheses that are not stated, two nonequivalent definitions of the same concept, extraneous definitions, assertions of a theorem and its converse with proof of only one part, and actual errors of logical reasoning.

A glaring deficiency of mathematics texts is the absence of motivation. The authors plunge into their subjects as though pursued by hungry lions. A typical introduction to a book or a chapter might read, "We shall now study linear vector spaces. A linear vector space is one which satisfies the[212] following conditions. . ." The conditions are then stated and are followed almost immediately by theorems. Why anyone should study linear vector spaces and where the conditions come from are not discussed. The student, hurled into this strange space, is lost and cannot find his way.

Some introductions are not quite so abrupt. One finds the enlightening statement, "It might be well at this point to discuss. . ." Perhaps it is well enough for the author, but the student doesn't usually feel well about the ensuing discussion. A common variation of this opening states, "It is natural to ask. . . ," and this is followed by a question that even the most curious person would not think to ask.

One need not always precede the treatment of a mathematical theme with the major reason for studying it. The introduction could be the historical reason the topic was studied. But if this is not the reason for its importance today, applications of current importance should immediately follow the treatment. Unfortunately for the authors, most applications involve physical science, with which few mathematicians and high school teachers are familiar. The best that many authors can do is simply to state that there are important practical applications of the subject treated. Some actually promise they will discuss applications but fail to do so.

Problems of science need not be the sole motivation. The mathematics to be taught might be related to the students' world. Perseverance would reveal what excites student interest, but this effort is far more than authors are willing to undertake. The consequence is bare bones and no meat. Even the prettiest woman seeks to enhance her appearance with dress and cosmetics. Similarly, mathematics should be made more attractive by relating it to the interests of neophytes.

The lack of motivation has been criticized by Richard Courant, [213] whom we have cited in other connections:

It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon to the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives....

To begin a text with a statement of the axioms is to write a work that omits the first chapter and that demands of the reader an understanding without which he cannot comprehend the text before his eyes.

Mathematicians wreak additional hardships on the students by indulging their own tastes. As mathematicians they recognize the advantages of generality and abstraction. Surely a general result covers many special cases. Hence, they conclude, it is more efficient to present the general result at once. Logically this is correct; pedagogically it is false. The recklessness with which authors of texts plunge into generalities indicts their judgment.

For example, before students have worked with concrete functions such as y = 2x, y = 2x+3, y = x2 and the like, they are asked to learn a general definition of function in terms of mappings. A mapping from a set A to a set B is a set of ordered pairs with each first component from set A and each second component from set B. Mappings are "wonderfully" broad. The relationship of a set of fathers to a set of sons is a mapping, and knowledge of this fact "clearly" improves the relationship between parents and children. To be sure, the definition of a mapping includes the concrete functions just mentioned; but it also includes relationships that students will never encounter or that are certainly not illuminated by the mathematical definition. Moreover, the vagueness of a [214] general definition leaves students uneasy.

Whereas a generalization extends to a wider class of objects a result known only for a special class - for example, one may prove for all triangles a theorem known previously only for isosceles triangles - abstraction selects from different classes of objects properties common to the classes and studies the implication of these properties. No one would question the value of abstractions for mathematics, but one must question the abstract approach to the concrete. Children do not learn about dogs by starting with a study of quadrupeds. This elementary principle does not seem to have been learned by mathematicians. They love abstractions and indulge in them freely, of course at the expense of the student.

Many other faults already cited apropos of teaching methods are repeated in the texts. Rigorous presentations addressed to beginners in a particular subject are common. From a conceptual standpoint the most difficult mathematical subject is calculus. The concepts can be far more readily understood intuitively, and this is how mathematicians grasped the subject until, after two hundred years of effort, they managed to erect the proper logical foundations (Chapter 7). But many modern authors courageously risk the students' necks. They start their calculus texts with the rigorous formulations of the concepts and at the outset succeed in destroying the students' confidence in their ability to master the subject.

Some authors choose the rigorous approach because they are insecure. They are fearful that if they compromise in order to help the student they will appear ignorant to their colleagues. To justify their stand they argue that texts at least must be precise and complete, and often they insist that understanding can best be obtained through the rigorous formulation. The result is elementary mathematics from a complicated standpoint. [215]

Most authors profited as little from their study of English as their readers then profit from the study of mathematics. The writing in mathematics texts is not only laconic to a fault; it is cold, monotonous, dry, dull, and even ungrammatical. The author seeks to remain impersonal and objective. As one reviewer said of the writing in a particular text. "The book is mathematically masterful, grammatically grim, literarily limp, and pedantically pompous. It tells the undergraduate more than he wants to know, presuming, at the same time, that he knows more than he does." Of course good texts should have a lively style, arouse interest, and keep the readers? background in mind. But few do. The books are not only printed by machines; they are written by machines.

One ingredient of style is humor. A relevant story or joke does revive a sagging spirit. But the professors object. They use the same text a number of times and to them the humor becomes stale or the joke palls. But for whom is the book written? What would these professors say about an actor who must repeat for the thousandth time a most dramatic line or a joke as though it were the first time he ever said it?

Beyond their sheer incapacity or unwillingness to write interesting mathematics, authors splurge in terminology that baffles the reader. In addition to using many technical words unnecessarily, mathematicians love to introduce new vocabulary. Thus they have long used the words homomorphism and isomorphism, which at least preserve the etymological meanings of similar structure and same structure, respectively. Instead of saying that a certain homomorphism is an isomorphism, the practice is to say that the homomorphism is faithful - a statement that does nothing to convey its meaning in mathematics, though it may have the merit of suggesting a steadfast, if illicit, romance. Of course, the appearance of a new term gives the impression that a new concept has been introduced. The terms greatest lower bound and least upper bound of, say, a [216] set of numbers were used for years and do describe what they stand for; now, presumably in the interest of brevity and certainly in the direction of making comprehension more difficult, the terms infimum and supremum are used. Single-valued functions and multiple-valued functions are now functions and relations. Further, one no longer speaks of the values of x that satisfy x2 = x + 7 but of "truth values." Apparently, truths can now be obtained readily and we need no longer ponder the mysteries of the universe. New terms to replace old ones appear constantly. This practice disturbed even Cauchy, who shared with Gauss leadership in mathematics in the first half of the nineteenth century, and he felt obliged to complain of the strange terminology introduced in his day:

"One should enable science to make a great advance if one is to burden it with many new terms and to require that readers follow you in research that offers so much that is strange."

Instead of introducing new, meaningless terminology in place of suitable words, mathematicians would do well to replace the old terminology that has misled students. Terms such as irrational, negative, imaginary, and complex, which were historically terms of rejection, remain in the lexicon of mathematics to disturb students. Even the great mathematicians of the past were frustrated by such terms. The resistance to imaginary numbers persisted for three hundred years after their introduction, partly because the word imaginary suggested something unacceptable. As Gauss remarked, if the units 1, -1, and had not been given the names positive, negative, and imaginary units but were called direct, inverse, and lateral units, people would not have gotten the impression that there was some dark mystery in them.

Meaningless terminology is only one evil of the language used. Mathematicians believe in brevity so much that they [217] invent shortened terms. A partially ordered set is now a poset. Still more brevity is achieved by using acronyms a.e. (almost everywhere). This atrocity, added to already barbarically poor writing, makes it almost impossible to read the text, let alone understand it. The l.q.m.w. (low quality of mathematical writing) has no bottom.

The evils of terminology are compounded by the excessive use of symbols. No one would deny that mathematical thinking and processes are expedited by the use of symbols, and one of the great features in the progress of mathematics was the introduction of better and better symbolism. But mathematicians have turned a virtue into a vice. They sprawl symbols over all the pages of their texts just as some modern painters splash paint on canvas. Page after page, almost devoid of exposition, is filled with Greek, German, and English letters, and various other symbols. Some books use as many as a few hundred symbols, presumably in the interest of brevity but more likely to conceal shallowness. Even a gain in brevity hardly compensates for the burden on the reader's memory. What is worse is that a symbol introduced on page 50 is not used again until page 350, with no reminder to the reader of what the symbol stands for. A few authors, somewhat conscious of this problem, include in their texts a glossary of the symbols employed. However, the reader stuck on page 350 must interrupt his reading to find the meaning of the symbol among the several hundred in the list. The natural reaction is exasperation. In modern texts symbols do not facilitate communication; they hinder it.

Many authors seem to believe that symbols express ideas that words cannot. But the symbolism is invented by human beings to express their thoughts. The symbols cannot transcend the thoughts. Hence, the thoughts should first be stated and then the symbolic version might be introduced [218] where symbols are really expeditious. Instead, one finds masses of symbols and little verbal expression of the underlying thought.

As in the case of terminology, much could be done to improve older symbolism. Perhaps the most imperative need is to replace the symbol dy/dx of calculus. The most important idea to be transmitted in calculus is that the derivative is not a quotient but the limit of a quotient. However, the symbol dy/dx, though intended to be taken as a whole, looks like a quotient of dy by dx. (In fact, this is what it was for Leibniz, who failed to formulate the precise concept.) Hence, the notation seems to refute what the teacher must attempt to convey. Superior notations have been proposed, but professors resist change in this area as zealously and perversely as they promote it where the traditional symbolism is altogether adequate.

A mathematician of the sixteenth or seventeenth century often presented his discoveries in the form of an anagram that was intended as evidence to his rivals that he had solved a problem but that was also intended to be undecipherable to the rivals so that they could not claim they, too, had solved the problem. When challenged, the composer of the anagram could then reveal what the anagram stood for and establish his claim. This practice continues today, except that the anagram is called a textbook.

Mathematicians claim to teach thinking, and this can be promoted by getting students to help discover theorems and proofs. But the texts do no such thing. Definitions, axioms, theorems, proofs, and obscurity are the style and content, the sum and substance. This type of presentation has the advantage - for the author - of facilitating the writing. One does not have to think about what to say because the theorems and proofs of the usual undergraduate textbook [219] are well known. However, as we have previously noted (Chapter 6), most theorems of any consequence have been reproven many times; each time some refinement or modification is achieved that makes the theorem more general or the proof shorter. Often an ingenious trick will do the latter. Since even the original proof may have been the product of weeks, months, and perhaps years of thought, to which a succession of mathematicians may have contributed, a modern proof is almost sure to be sophisticated and highly artificial, though mathematicians would describe it as elegant. These refined proofs, presented in a page or so, stun and humble the students. They cannot help imagining themselves being called upon to make such proofs and readily realize that the task would be inordinate for them. The inevitable consequence is that they lose confidence in their ability. To pass examinations they memorize the proofs.

Good writing, like good teaching, calls for letting the students in on the struggles mathematicians have undergone to arrive at the proofs. Students should be told how long and hard the best mathematicians worked to obtain the proofs, and how many false proofs were often published in the belief that they were correct. This history not only would avert discouragement and loss of confidence but also would dispose students to the kind of effort they must be prepared to make when attempting a proof on their own.* But authors are reluctant to level with the students. By presenting proof after proof with no mention of how these were obtained, the authors seem to suggest that the clever proofs are due to them and, very likely, this is the impression some wish to give. Authors do not recognize the psychological damage of a bare logical presentation.

*Texts do present historical material, usually to the effect that Descartes was born in 1596, died in 1650, and had one illegitimate child. [220]

Why can't texts be more informal, almost conversational? Suppose an author is about to present the theorem that the three altitudes of a triangle go through the same point. This fact should surprise anyone. Should not the author remark on this, perhaps state that it is not an expected fact, and first give some intuitive reason that it should be so before proving it? Calculus texts treat the derivative of the product of two functions. Students expect that the result should be the product of the derivatives and are surprised to find that it is not so. Even Leibniz stumbled on this point and spent a whole month getting the correct result. The authors could state what superficial argument suggests and then point out why it is not correct. The fact that Leibniz struggled to understand this matter might also be mentioned and would reassure students that they are not so far below the Leibnizes in intellectual capacity. Such discussions should precede the formal presentation.

To rebut the charge that the texts proper do not call for student participation and thinking, the authors point to the exercises. Good exercises could be some redress for the dogmatic text. However, the texts usually work out half a dozen typical problems in each section and then assign exercises of the same type. The students, called upon to do an exercise, look among the illustrative examples to find one that fits the exercise. They then repeat the steps made in the illustrative example without necessarily understanding them and certainly without having to do any thinking for themselves. Thus, the students do the homework successfully and feel satisfied. The professors, in turn, congratulate themselves on their successful pedagogy.

Of course, some illustrative examples are needed. Students cannot be expected to acquire techniques without guidance. But the examples should be accompanied by a discussion of how the theory is involved, why the solution [221] should take one course rather than another, and any other pertinent comments. In fact, rather than being set out as examples, these illustrations are best incorporated in the text proper to oblige the students to read the text - something that students often shirk if not compelled to do it. Moreover, some of the exercises could raise questions about the examples. Alternative methods might be proposed that may or may not work, and the student might be asked to evaluate them. Mere repetition of a process that is illustrated will teach technique, but it will not inculcate understanding or foster thinking. The usual exercises are intellectual slavery rather than intellectual challenges.

Many textbook authors boast of the number of illustrative examples their texts contain. What they are really saying is that the students do not have to read the texts or do any thinking. These texts are rightly called cookbooks. Actual cookbooks usually offer recipes: pour a half cup of flour into a bowl, add one quarter cup yeast, sprinkle with vinegar and bake for one hour. Lo and behold, a cake appears. But the cookbook gives the cook no idea why such a mixture produces a cake. The illustrative examples are likewise recipes for getting answers. If the recipes were changed and produced absurd answers, the students would not have the insight to recognize this fact and would be content as long as their answers agreed with those given by the text. Clearly, the valuable role that texts could play in the educational world is nullified by the various defects we have cited. The students certainly do not read the texts, because the texts are unreadable.

Surprisingly, when choosing texts for their classes many professors have said openly that they do not care about text exposition. They look only at the illustrative examples and the exercises. But in our civilization, learning to think and learning to use books are surely some of the objectives of [222] higher education. Apparently these objectives have been abandoned.

Since the texts are so bad, one is impelled to ask, why are such texts chosen? The reasons are numerous. The poor exposition is not recognized by most professors because they themselves are not trained in writing. Lack of motivation and application in textbooks is even welcomed by professors. Such material must usually draw on subject matter that lies outside mathematics proper, and to teach it would require that the professors know and feel secure about, for example, a bit of science. But the professors do not know science, and they are not willing to learn it just to do a better teaching job. In fact, many professors fear any book that would make them deal with the history of mathematics, science, or cultural influences. Hence, they choose one that takes the straight and narrow path of mechanical, technical mathematics and routine exercises. Terminology, symbolism, rigor -these are dear to the hearts of mathematicians. From their point of view one could not possibly overdo such features.

A major reason for the choice of poor college texts is that the bulk of the undergraduate courses is taught by graduate students. For such teachers, stock material and routine presentations are musts. Any felicitous or unusual approach, especially if it calls for pedagogical skill, would be ignored or bungled.

Many professors choose a text because the topics treated are what they want to teach. But they do not care about the text's presentation. They give their own. The student is then faced with the task of reconciling what the teacher says and what the book says. This is difficult to do in mathematics. And since the book is most likely to be poor, the difficulty is compounded. If the professor really has a better presentation than what is available in an existing text, he should write [223] up his material and distribute it so that students will not have to spend the class time in copying. In many cases the professor's presentation is not better, but he considers it demeaning to follow someone else's.

There are even university professors who deliberately adopt a difficult book because it bolsters their ego to be able to say that they are using it. They hope that others will judge them and their students favorably, because presumably both can master such a book. Actually, many of the professors who choose such books are hard put to understand them, but the students are so much more bewildered that the professors can get away with almost any kind of explanation.

Professors at four-year colleges often feel inferior to those at the prestigious universities. To overcome the feeling of inferiority many four-year college professors try to "outdo" the university professors by adopting texts that are far too difficult for the students. When asked at a professional society meeting what texts they are using, they can name them and imply that they are really doing wonders with their students and, of course, have no trouble themselves in teaching on the advanced and sophisticated levels that these texts bespeak.

Teachers at the two-year community and junior colleges, which, on the whole, have the weakest students, also use difficult texts just to be able to boast that they are teaching on a high level. They claim they must use these books to prepare students who will transfer to a four-year college. But they kill off the students and so make transfer impossible. Only about 25 percent go on to a four-year college, and most of these students do not take any more mathematics. The phenomenon of low-level institutions using high-level texts is especially prevalent in areas dominated by a major university.

Many texts are chosen by a committee. If a book contains [222] any applications, some professors who are unfamiliar with them will object. Other professors may rightly or wrongly object to the level of presentation. Still others may object to the "wordiness." The consequence is a compromise that almost necessarily is a dull, meaningless, inept book. Even where a text is chosen for departmental use by a single professor, he may, like so many others, lack insight, conviction, and determination, and pick a "safe" book - which usually means a mediocre one that will satisfy most of the professors.

Quite often a department decides to change the text in use because some members complain that it is not satisfactory; or it may be going out of print. One would think it might be replaced by a good text, but that rarely happens. Most of the staff prefer a text that is old hat, so they do not have to read it and do new exercises. Hence, the replacement is usually a "copy" of the previous text. After all, change is sufficient evidence of progress in our society.

There are many other reasons that a professor will pick a poor text. Professors are most likely to be narrow specialists. An algebraist who is called upon to teach differential equations is not interested in how to teach that subject. He wants a book that is easy to teach from, and this means one that presents either a series of techniques or a canned sequence of theorems and proofs that need only be repeated.

Even if all or a majority of the texts were good, the students in many institutions would still suffer. Some professors choose texts that interest them and from which they can learn new ideas or new proofs, whether or not these texts are right for the students. Thus, an algebraist might pick an advanced text for an elementary course not because the students can learn from it, but because he can. At one respectable institution the professors used a text that was [225] two or three levels higher than the course, and a large percentage of good students failed. Others, also highly qualified, became discouraged and abandoned mathematics. When the professors were asked why they used such an unsuitable text they replied that they were conducting an experiment. They might just as well have said that they had fired six bullets into a man's heart to see if he would die.

Why are so many poor texts written? The main reason is obvious - greed. Texts bring in royalties, and money does interest some people. To make money, one must write a text that sells well. But the poor texts sell best, and the money-minded author caters to the market. Most professors write with more than one eye on the market. They rivet their attention on it. What happens is well illustrated by the history of calculus texts. For years only mechanical or cookbook treatments of calculus were used. Authors, accordingly, wrote cookbooks. As American professors became better educated, they decided that students should receive the benefit of professorial enlightenment and that calculus should be taught with a full background of theory. A spate of rigorous calculus books soon appeared on the market. When this pedagogical blunder became apparent and the intuitive approach became popular, professors showed their open-mindedness and flexibility by turning to an intuitive approach. It did not take long before the very authors of rigorous texts wrote intuitively oriented texts and even boasted that they offered this approach. Professors do learn remarkably fast - what the market wants.

Because most authors aim for the largest possible market, they repeat endlessly books that sell well. All that is required is a minimum of knowledge, shoddy writing, standard exercises, and reasonable caution against outright plagiarism. One need only vary the order of the topics to make a book seem different, and since there are about twenty-five [226] topics in the usual text, the possible permutations are large enough to allow for many thousands of "different" college algebra, trigonometry, calculus, and other texts to be written. The fact that the sources may be incorrect or poorly written is a minor concern compared to the expected gain. To hide obvious repetition of existing texts, some authors introduce a few variations, such as contrived proofs even though more natural ones are available, sophisticated definitions, new terminology, and their own brand of symbolism. When accused of plagiarism the professors can always retort that the truth never changes.

One must of course have a different title. But then one can use College Algebra, Elementary College Algebra, College Algebra: A Full Course, College Algebra: A Short Course, and College Algebra: A Seven-Eighths Course. The possibilities are clearly infinite. In fact, since there are irrational numbers, one could use Algebra: An Irrational Course.

The outright imitation of successful texts - successful financially though usually not at all pedagogically - is a fact. Many authors do not hesitate to admit this. They speak proudly of their books as being in the mainstream of mathematics education, as though this fact is an assurance of quality. Actually, in view of what books sell best, a book in the mainstream is sure to be dull, unoriginal, and pedagogically disastrous.

Are all texts repetitious of each other? No. Another spate of bad texts comes from professors who have achieved a reputation for research in their specialty or whose name is well known in the mathematical world, perhaps because they have held high office in a professional society. These authors, most of whom have never or only rarely taught the courses for which the books are intended and are unconversant with how college students think and what backgrounds [227] the students have, nevertheless decide to cash in on their names and plunge unhesitatingly into the writing of texts. "Genius" transcends mediocrity; so these texts contain innovations in concepts and proofs that students cannot possibly grasp. The exposition of the topics is shoddy and the writing is shameful. The books are hastily written and often contain numerous errors. Chapters or sections begin with one objective and end up with another. Within the same section authors shift from one topic to a totally unrelated one. They ask the students to do exercises that are not workable on the basis of the material in the text or, if related to the text, require a Newton. To make a token gesture to that sector of the market that wants some applications, these researchers include some brief mention of relativity or quantum mechanics, topics that mean nothing to undergraduates at the levels for which the texts are written. It is clear that these professors dash off the books as fast as they can just to get them out and "earn" royalties. Were these authors judged by their texts they would not be admitted as graduate students to any decent graduate school. Nevertheless, many schools adopt such texts on the basis of name alone. Usually the texts are so bad that they are dropped after one year's use. About all one can say of them is that they are flops d'estime. Ironically, these prestigious professors, who rush to write texts for low-level courses, would disdain teaching them or, if obliged to do so, would be ashamed to admit that they were teaching such lowly work.

Sometimes these prestigious professors resort to second and third editions and, having learned by this time how to meet the market on its terms, sell more books. The venality of such professors and their crass commercialism are disgraceful. In these later editions they may succeed in selling more books, but they also succeed in sacrificing students and vitiating educational goals. Cheap fiction, [228] potboilers, are far more excusable because the authors make no pretense to ethical principles and are not under any obligation to develop young minds. If these professors are really capable research men or seek to exert beneficial influence through office-holding in professional societies, why do they lower themselves by writing the hundredth facsimile of cheap, commercial texts? Or are the supposedly intelligent professors as badly confused about their role and goals in life as any adolescent?

The problem of writing for financial gain does call for keeping up with the market. As we have already observed (Chapter 7), mathematical teaching as well as mathematical research is swept by fads. Analytic geometry, formerly taught as an independent course preceding calculus, is now submerged in calculus. The successful author must yield to this fad or his book will not sell. If the fad is to incorporate linear algebra in the calculus or the differential equations text, whether or not there is any point to doing so, one must incorporate the linear algebra. To keep up with fads one must put out new editions every few years. But professors do not object because this eliminates the secondhand market for the older edition and students are obliged to buy the new one.

The determination of what the market wants is made rather scientifically. The publishers canvass the colleges for what they would like to see in the texts, and then the authors willingly set about supplying the common denominator of those wants. The author's own convictions, if he has any, as to what a text should contain are irrelevant.

The normal market can be enlarged by special devices. One such device is to offer applications but to crowd them all into the last chapter. There is method in this madness. Applications are desired by some professors but frighten off others. If they are placed at the end, professors who do not [229] want to teach them manage to end the semester before reaching the last chapter, thereby omitting them with least embarrassment. Applications placed at the end of a text serve little purpose in any case, because whatever value these applications might have as motivation and meaning for the mathematics proper comes too late.

To enlarge their market many authors employ ruses that are deliberately fraudulent. When the New Mathematics became popular these authors took traditional books, inserted a few pages of New Mathematics material here and there, changed terminology in spots, and sprinkled words such as sets, commutative law, inverse, and the like throughout; they then proclaimed that they were presenting the New Mathematics. Many teachers aided in~ this fraud because they could convince their superiors that they were teaching the New Mathematics, while actually continuing to teach the material they either preferred or knew better. Calculus texts often contain a facade of rigor to please those professors who wish to include some theory but the rigor, usually in the first chapter, is thereafter never utilized.

There are other types of deception. One would expect that a text entitled Mathematics for Biologists would contain not only the mathematics that biologists use but also some indication of how biologists use it. But the contents are the same as any traditional text that covers the same level of mathematics.

Many authors know that students come to college disliking mathematics. However, some colleges still require a course in mathematics as a degree requirement. Even if they don't, the professors wish to attract students to a mathematics elective so there will be more jobs. Hence, many authors write texts that purportedly offer an appreciation of the role of mathematics in our civilization. The titles are inviting: Mathematics, An Intellectual [230] Endeavor; Mathematics, the Science of Reasoning; An Appreciation of Mathematics; Mathematics, the Creative Art; Mathematics, Art and Science. But the texts teach axiomatics, symbolic logic, set theory, topics of the theory of numbers such as congruences, the binary number system, finite geometries, matrices, groups, and fields and so do not really live up to their promise (Chapter 6). Clearly, one can't judge a book by its title.

Since a course in mathematics proper does not attract liberal arts students and has little value for them, some professors have taken another tack. For their books they gather together curiosities, trivia, puzzles, and bits and pieces of standard topics that never get to any serious level and do not require any thinking on the part of the student. The chapters are deliberately unrelated to each other so that the student will not have to carry an extended train of thought and so that the teacher can pick and choose what pleases him. Since these measures rarely succeed in interesting students, some professors have resorted to the ingenious device of including cartoons. There are even calculus texts "enlivened" by cartoons. Why not? After all, isn't mathematics supposed to be fun? Perhaps pointed, truly humorous cartoons can be admitted as a pedagogical device on the college level, but shallow sequences of drawings that would hardly elicit a smile from six-year-olds make no contribution. Something can be said for cartoons:

They do enlighten us as to the intellectual level of the authors. Mathematical texts do not as yet resort to pornography, though this means of attracting students would be more acceptable because it would not be mistaken for a pretense to education in mathematics.

These puerile "liberal arts" texts also sell well. Students, deceived or not as to the worth of the material, can earn credit for the course without really being pressed into [231] thinking. The professors can teach such material without any effort and thereby "solve" the problem of what to teach the liberal arts student.

Many professors express concern that the lowering of standards for admission to college and automatic admission of any high school graduate will result in the lowering of the educational level for all students. But by publishing the liberal arts and cookbook texts we have described, which are used in hundreds of colleges, these professors have reduced standards to about the lowest level possible.

The financial gain to be derived from textbook writing has corrupted many professors. Some authors ask publishers for guarantees up to $100,000. Apparently the authors have no confidence in their works and seek to ensure profit by a guarantee. The argument is sometimes made by authors that a publisher will work harder to push a book on which it has given a guarantee. But this is hardly a justification. The investment of the publisher can range from $50,000 to $100,000. Surely no publisher will invest such a sum and then fail to promote the book. If the author is asking for sales promotion beyond the merits of the book, he is certainly culpable.

There are two possible controls over the quality of texts. The first is reviews. Most texts are reviewed in one or more of the professional journals. However, some reviewers are apparently too polite to write the condemnation that most texts should receive. Instead, they merely describe the contents or compare the book with similar ones and often end with the "compliment" that it is good because it is just like the others on the market. Other reviewers respect the principle of honor among malefactors. What we need is honest, damning reviews of books that impose unnecessary hardships on students and fail to teach the values that the course in question should offer.[232] Though critical reviews of texts are rare, one does find some. One reviewer of a new calculus text said that the "exercises are presented with less imagination than" - and here he mentioned a best-selling calculus book - "if that is possible."

One might look to the publishers to control the quality of texts. But this is not fair. Publishers do have manuscripts reviewed before they accept them; but generally the reviewers are men who teach at the same level as the proposed book, and they are no more critical and no more demanding about pedagogy than the authors. If they see material they are able to teach, they approve the manuscripts.

Moreover, publishers are in business to make money. This is their avowed purpose, and they cannot alter the market. If they do not yield to it, they will fail. They certainly cannot exist on the rewards of virtue. No doubt many exaggerate the qualities of what they publish. Often, too, they seek to anticipate a trend and accelerate it by promoting books that further it and give the impression that such texts are already in wide demand. They did this when the New Mathematics was in the offing and are now hastening to break from the New Math because they foresee its doom. Publishers are often criticized because they publish books just like dozens of others already on the market. But if a publisher is to stay in business he must have saleable books in each of the subjects and perforce must duplicate existing books. The better publishers do compensate somewhat for publishing junk by putting out high-quality monographs and treatises on which they lose money though they may gain prestige.

The responsibility for good texts definitely rests with the professors, who, unfortunately, regard their station as practically a license to publish. In these times the only[233] concession they must make to secure the full imprimatur is to have an opening chapter on set theory, whether or not it is relevant to the body of the book or referred to in later chapters.

Good texts, so sorely needed, would raise the educational level immeasurably. Not only students would benefit. Young teachers, older ones when called upon to teach a course in an area unfamiliar to them, and even knowledgeable and competent teachers can learn much from a good text because the author would have devoted months and years to the selection and presentation of the material, whereas the teacher could not hope to do that in more than one area.

The low quality of the texts is the severest indictment of the professors. Those who deliberately cater to the market even when capable of doing a better job besmirch their character. Those who write texts for courses they have never or only rarely taught impugn their integrity. And experienced teachers who sincerely attempt to write well demonstrate that the arts of pedagogy and writing are rare gifts.

This derogation of the quality of American mathematics texts may seem overdrawn or grossly exaggerated. It is not. The low quality is as much a consequence of the development of the nation?s educational efforts as are the poor content and pedagogy of the courses and curricula. The principle of universal education from the elementary school to the highest levels students can attain certainly was and is desirable. But the constant immigration of mainly poor and uneducated people has placed a burden on the country that would be difficult to carry under any conditions. To make matters worse, the emphasis on research in the last thirty or forty years has diverted manpower from teaching and so has[234] cut off the flow into the fountain of all our educational efforts, the teaching in the colleges. Perhaps rather belatedly we shall develop sincere and capable cultivators of mathematics - a science and an art - who will recognize that exposition is as vital in their medium as it is in painting, music, and literature.

I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.

Copyright © Helen M. Kline & Mark Alder 2000

Version: 22nd March 2001

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