Andrew D. Todd

Computers and Mathematics Education From First Principles: An Outsider's View.

(July 21, 2001)

There has been a certain amount of controversy about the proper role of computers in mathematics education, of which Steven G. Krantz's "Imminent Danger-- From a Distance" (Notices of the American Mathematical Society, May 2000) is but the latest installment. This controversy has gone on for years without resolution, largely because it has occurred before computers became cheap enough to permit a decisive experiment. At long last, the prices of computers are falling to the point where we will be able to find out precisely what computers are good for.

One truism of computer systems analysis is that before computerizing, one must first understand the system which it is proposed to computerize, in this case, mathematics education.

Mathematics education is a major point of contact between mathematics and the rest of the world (one could say, following Clemenceau, that it is too important to be left to the mathematicians). The mathematician's view of mathematics education is necessarily one-sided. Mathematicians talk about mathematics education as if its purpose were to train mathematicians. This shared assumption unites both those mathematicians who enthuse over computers and those who abhor computers. The pro-computer faction tends to stress the sheer organic biomorphic messiness of mathematical thought, with emphasis on things like empirical disproof (J. J. Uhl, Mar 1996; Epstein & Levy, June 1995). Their claim is in effect that the computer is a means of introducing the student into the glorious messiness of mathematics. They have also chosen to make their stand in the freshman calculus course, the most advanced course with sufficient enrollment to attract the necessary funding. The anti-computer faction tends to stress truth and beauty of final results (Wu, Dec 1996; Kasman, June/July 1997; Kusher and Melman, Mar 1997). Their typical comment on computer-based education is that it reduces the student from an inheritor of an extended cultural tradition to a kind of hobbesian naked savage. Both factions, however, share the assumption that mathematics education exists to produce research mathematicians. This is statistically untrue.

The Uses of Mathematics Teaching

The legitimate purpose of mathematics education is not to train mathematicians. Something like ninety percent of college mathematics courses are given to nonmajors, who represent something like ninety-nine percent of the students passing through the curriculum. There are infiltrators even within the mathematics department in the form of applied mathematicians, whose concerns differ importantly from those of pure mathematicians. Mathematicians are aware of these facts, of course, but when they talk about the whys and wherefores of mathematics education, they practically never do so in terms of the reasonable needs, aspirations, and motives of the neglected ninety-nine percent. However, the privileged position of mathematics as a compulsory school subject can only be justified in terms of this vast majority.

This vast majority of students taking mathematics can further be broken down into two large groups. At the risk of oversimplification, nonmajor students of mathematics can be divided into engineers and bona fide liberal arts students.

Engineers, the first group, are the one of the most numerous groups of scientific and technical workers, the other major group being computer programmers, who typically do not use mathematics as an integral part of their work. Engineering is based on physics, and engineering students are trained as physicists before being trained as engineers. To this end, they are required to take what amounts to a minor in mathematics, and in some cases, what would be a mathematics major if it were not swallowed up in the sheer extent of the engineering curriculum. For the engineer, mathematics serves as a language in which to talk about physical and quantitative reality. Whatever mathematics may be to the mathematician, the engineer's mathematics is a kind of lingua franca, judged by its usefulness for conveying nonmathematical ideas.

Course like Matrix Algebra, Statistics with Calculus, Differential Equations, and Vector Calculus are full of engineers. These courses are all largely rooted in the practices of a time before computers. They focus on analytical solutions which are of severely limited utility for the practical physicist, and by extension, the practical engineer. For example, analytic solutions of ordinary differential equations are only available for linear differential equations, and are therefore singularly useless. Most real-world differential equations are nonlinear. Engineers solve differential equations by numerical methods. In the first place, you don't need to know very much about differential equations to understand how predictor- corrector algorithms work. In the second place, it is possible to package the algorithm up as a "black box," a subroutine whose interior workings are invisible to the user, and of concern only to the specialists who produce mathematical software. Much the same kind of reasoning applies to matrix algebra. A thoughtful engineering school might decide to drop differential equations from the curriculum in favor of key computer science topics. At this point, it is more essential for an engineer to know about stacks and linked lists than for him to know about Bessel equations and Laplace transforms. Come to that, the same principle can be applied to integral calculus itself.

Much the same kind of analysis goes for statistics. Statistics is based around trying to fit very simple models (eg. polynomials) to complicated data. It is often possible to construct programs, full of conditional branching, which present much more credible models, explaining much more of the variation. With such a model, one can get by with rudimentary statistical techniques. Again, these techniques are likely to be "black- boxed" and built into a simulation language such as SIMSCRIPT.

One must also remember that for much of physics, and by extension, much of engineering, mathematics is not the primary language of description and problem formulation. For that purpose, people use conventional diagrams-- for example an electronic circuit diagram-- which they afterwards translate into mathematical equations. New "problem-oriented languages" such as SPICE receive their input in a form closer to the diagram, so that the user never gets around to formulating equations at all. He merely names the key landmarks of the circuit diagram, and makes a series of statements, eg. "there is a resistor of resistance r between junctions ABLE and BAKER." In effect, SPICE has black-boxed mathematics out of electronic circuit theory. The next stage, of course, is to integrate SPICE into a CAD/CAM system, permitting people to draw their diagrams directly into the computer. Problem-oriented languages are being formulated to cover all the conventional diagram-types of physics.

To make matters even worse, much of engineering is concerned not with designing machines as such, but rather with designing controls for the machines. There has been a tendency to substitute small computers for other control mechanisms (see Ceruzzi, 1989). Instead of using mathematics to design a system of resistors, coils, and capacitors which behave in a specified way, the designer simply writes a program describing the desired behavior. In an extreme case, there is an ongoing tendency for engineers, mathematicians, and various scientists to become computer programmers, outright. They simply choose not to use their special training on the grounds that it is not sufficiently useful compared to programming.

Mathematics is on probation. There is a practical alternative in the form of computer programming. At every stage of the proceedings, mathematicians will have to prove that they are making better use of the student's time and energy than the Computer Science department would. Computers can be, and in fact are, used to teach computer programming. The normal mode of instruction is interactive debugging, either with compiler error messages, or with a symbolic debugger. Computer Science is gradually becoming an alternative to Mathematics in college distribution requirements. There are few if any mathematical prerequisites for learning to program. An intelligent course of instruction is text-oriented, with the first example encountered being the classic "hello, world" program. Certainly, in fields such as engineering, computationally intensive techniques (numerical analysis, discrete event simulation, finite elements methods, etc.) are superseding the analytic solution.

If engineering is increasingly hostile to mathematics, the situation in the liberal arts, the source of the second great body of students, is even worse. The liberal arts are not merely hostile to mathematics: they are hostile to quantitative analysis in general. Mathematics is of little value in the liberal arts, which leads to the enrollment drops noted by Richard Maher (Dec 1999). Liberal arts are about man, and precise quantitative measurements of man rarely stand up to close examination. Even biometric measurements, such as the dimensions of bones, are surprisingly elusive. The use of elaborate statistical techniques is usually a case of comparing apples and oranges, and usually generates a more or less refined form of garbage. If numerical relationships are not self-evident from raw data, use of the techniques of the mathematician will only muddle the issue. The use of statistics in the social sciences was founded on a false promise that it would somehow make the social sciences into "hard science." One of the fullest expressions of this promise was George Peter Murdock's _Social Structure_(1949). However, the promise of statistical precision was doomed to failure, and quantitative methods in the social sciences have never delivered on their promise of absolute truth. What we got was large numbers of people substituting the use of packaged statistics programs for systematic thought of either the social or philosophical variety. The recent major trends in the social sciences, such as hermaneutics and deconstruction owe very little to mathematics.

Furthermore, small quantities of mathematics are of only limited value for understanding modern science and technology. One either learns enough to use mathematics as a language, or one is wasting one's time. Mathematics courses for liberal arts students reflect this fact: they are generally at a very elementary level, and they tend to be conspicuously abortive, in the sense that the teacher is trying to teach students material which they have already failed to learn, and are failing to learn yet again. If there is one course which forms an archetype of the relationship between mathematics and the liberal arts, it is "Mathematics for Elementary School Teachers."

Mathematics teaching has failed to meet the needs of both of its major student constituencies. Predictably, students are finding ways to evade mathematics requirements.

Deficient Teaching of Beginning Mathematics

Beginning Mathematics is unskillfully and inefficiently taught, by people who cannot use mathematics effectively, and this is a product of mathematics teaching's comparatively low productivity.

Engineers and computer programmers produce designs to be mechanically mass-produced. The extreme case of this is of course the development of shrink-wrap software. An engineer's high pay is not arbitrary, but is the sum of modest "royalties" on the vast number of copies of his work. By contrast, classroom teachers are still at the handicraft stage, teaching small classes, and correcting one paper at a time; and their low pay reflects this. It is all very well to talk about what a good mathematics teacher does or does not do, but the reality is that a good mathematics teacher necessarily understands mathematics well enough to use it, and, more to the point, is demonstrably capable of learning other formal and logical systems, and is therefore liable to go off and become an engineer or (especially) a computer programmer. People have from time to time instituted special courses to train efficient mathematics teachers, but the only result has been that their students have promptly been raided by someone higher on the priority scale (for example, see Stein, 1997). If the engineers have refrained from enticing away a particular high-school mathematics teacher, that is a reflection of their judgment that she is so unsalvageable as not to be worth stealing. Beginning Mathematics teaching is a sort of economic punching bag.

Alternatively, one can employ foreign-born mathematicians, who do not speak English well enough to be employable in industry. The problem is that they do a poor job of communicating with students, and, to add insult to injury, the classroom becomes their language laboratory. Teaching requires better communications skills than purely technical work, and the result is that the foreign mathematician becomes linguistically qualified for industrial employment long before he becomes an efficient teacher. Again, the simple fact is that industry and research have first call on technical skills of all kinds, and low-productivity teaching inevitably has to make do with their leavings. The radical growth of computer programming in the last four decades, from 13,000 programmers in 1960 to two million today, has heightened this problem by increasing the number and attractiveness of the escape hatches from teaching. Womens' Liberation has had the same effect, and, indeed, there have been far more women programmers than there have been women engineers. A whole constellation of technical and social tendencies combine to lure away the best beginning mathematics teachers.

That leaves beginning mathematics teaching with the inept mathematicians by definition. By the time one gets down to the ninth grade, it is a case of the blind leading the blind, as H. Wu (May 1999) and others have documented. As for extending mathematics education still further down, as proposed by Siegel (1998), there is no realistic prospect that funding will become available to pay the salaries required to recruit teachers who are both capable mathematicians and capable teachers of young children. The fact is that there are a significant number of human mathematics teachers who:

a) know no mathematics and do not want to know.

b) are afraid of mathematics, and manage to project their fear and hostility.

c) follow the teachers' handbook so closely as to become inferior sorts of computers, with the handbook as their program.

d) are available for only a severely limited contact time, per student. Nominally, a Carnegie unit would be five hours a week for 36 weeks, or 180 hours, and assuming a class of 30, each student's pro rata share would be three hours per year, but when one deducts time spent taking roll call, proctering exams, dealing with discipline problems, etc., the actual figure is considerably less. This is obviously trivial compared to the time the student spends doing homework. When the teacher has not made herself into a defective computer, she makes herself into a defective video player.

The students who succeed under this regime are those who would have learned from a book, or under some kind of independent study arrangement, such as the Keller Plan. In effect, they ignore the teacher and set about learning mathematics on their own. This point should be stressed. Whatever the merits of personal teaching, it is, at all but the highest levels, extremely rare-- available to most students only in minute doses. For the less inspired students, there is enormous wastage. Some fail outright. Others get low passes (B or C in a typical high school), and go on to college to take remedial mathematics from someone who cannot speak English-- with about the same results.

The situation is likely to grow worse of its own accord. Many of the people in the school system made their career decisions years ago, working in terms of a different set of facts. The distribution of manpower is still in the process of catching up with current realities. Any reasonable plan or prediction for school mathematics has got to be based on the assumption of the terminally unmathematical teacher.

A Deadly Combination

Taken together, mathematics education is simultaneously becoming less useful to the student, and less efficient. That is not a winning combination. Mathematics education is marching ahead into a lethal "coffin corner." Mathematicians, of course, have an alternative. They do not have to be mathematics educators. They can become engineers or computer programmers who do mathematics in their spare time, either in a university or somewhere else. In practice, most holders of the B.S. in Mathematics will not spend much time agonizing about the choice. The agony will be mostly confined to the Ph. D.'s, particularly, that kind of Ph.D. who uses "engineer" as a swear word. Engineering is no more demeaning than teaching. Engineering does not necessarily mean being a cog in a giant company. If academic scientists (including mathematicians) want to make their living at engineering, a good way to do it is to practice engineering in the public interest. Nowadays, the most distinguished form of engineering in the public interest is producing general-purpose public-domain software, in the tradition of Richard M. Stallman. Government grants are readily available to fund such software development, based on the general usefulness of what the grantees produce. I would not presume to tell mathematicians what kinds of programs they are best qualified to write. But there _must_ be _something_ they could do. One piece of advice is to pick out the most expensive and/or frustrating commercial software one uses regularly-- and then clone it. Another approach is to pick out a regular task which one finds disagreeable or tedious-- and automate it.

The Precedent of Latin

If the mathematicians either cannot or will not do something with computers, they are likely to be superseded by someone else with more of a constructive attitude. A hundred years ago, Latin occupied the place now occupied by mathematics. Its practitioners exhibited the same backwards looking, obscurantist attitude that many mathematicians take to computers, and they had a consuming preoccupation with their own social superiority, which again is not altogether unfamiliar. Arguments about Latin teaching one to think proliferated, even as the number of graduates who could read a Latin book or hold a serious conversation in Latin with a foreigner diminished.

Latin was of course in long-term fundamental decay. Like mathematics, its difficulties began with a machine-- in this case the invention of the printing press in the fifteenth century. Printing presses meant mass-production of books, and mass- production of books meant mass-literacy, so there would be someone to read the books. Mass literacy was impossible on the basis of the Latin language, but feasible in the vernacular. Increasing numbers of books began to be published in the vernacular. National intellectual communities emerged, with the intellectuals increasingly involved in the concerns of the literate mass audience. The writing of new Latin books had practically ceased by about 1700, and the translators were therefore catching up.

The result was that for an increasing number of people, Latin was not a useful skill, and there was no incentive to learn it well. Another factor was that the best latinists were often men who were good at languages in general. A man of talent was apt to speak five or six major European languages, usually acquired during the "grand tour," a multi-year peregrination across the length and breadth of Europe. The Grand Tour also involved gaining a detailed knowledge of the geography, economics, and customs of other nations. It was the graduate school of its time. Such men tended to be snapped up by the diplomatic corps or the higher civil service. They were much too valuable to be wasted in teaching school. So Latin was likely to be taught by lesser talents.

Latin's final collapse practically coincided with the "American Century," following the "English Century." For a period of two hundred years, the english-speaking nations enjoyed an extraordinary ascendancy, and English emerged as the world language. It was no longer necessary for a capable continental European to know many languages-- it sufficed for him to know his own native language and English. For the english-speaking peoples, their own native language sufficed. English was not taught as an arduous discipline. In the nineteenth century. Italian industrialists and Russian princes did not send their sons to demanding, elite schools to learn English-- they hired English nannies and governesses, to gain English as a "cradle tongue." In the twentieth century, Chinese millionaires sent their children to American boarding schools, to pick up English by living with American children, before going on to American universities. The rise of the English language left a gap in the educational system of the english-speaking world. After a period of confusion, mathematics began to occupy the place of Latin in the educational system. Mathematics was seen as a passport to the expanding technology which drove the twentieth century.

Pure mathematics is analogous to classical philology. And where is Latin now? A typical Classics department might have three or four professors, ten undergraduates, and no room for graduate students without private means. Most of the courses taught are "literature in translation," and as such, not suitable for Classics majors. That is a consequence of the classicists having lost the battle to make Latin a mandatory core of general education. That in turn was the inevitable consequence of the failure of Latin as a means of communication. Don't think it can't happen to mathematics. Can one be sure that Mathematics will not follow Latin into decline?

In 1873?, when Latin was already a sort of walking corpse, a pole, Ludwig Zamenhoff, invented a new language, esperanto. Esperanto was essentially a "corrected" Latin. All aspects of grammar and syntax, including the formation of derived words, had been relentlessly regularized, so that esperanto could be learnt in the least possible time.

Esperanto appeared extremely late, at a time when state-sponsored linguistic nationalism was in full flower. Even backwards nations such as Russia were producing their second crop of major writers (eg. Tolstoy, Dosteyevsky). Esperanto did not succeed in resurrecting the universal Latin community. However, something like a million people learned it.

What is to be Done?

The weakness of mathematics is at the bottom, in the ninth grade and in the middle school. There are fundamental socioeconomic reasons why it is not possible to supply these grades with properly trained mathematics teachers. These socioeconomic reasons do not operate with respect to computers and software. On the contrary, industry has long tended to flood the school and college system with hand-me-down computers, as well as driving component prices downwards. The only question is how cheap computers have to become in order to become truly ubiquitous in the schools (as distinct from the present ratio of one computer for every five or seven children) or in the home (i.e. reaching levels of several computers per household).

If it is possible to construct suitable educational toys for ten-year-olds, covering the work of ninth grade algebra, these toys could simply be distributed, in much the same way as books. If playing with the toys is even one-fifth as effective as attending a conventional class, that will still move the student along fast enough to be exempted from ninth-grade algebra, and to take calculus in high school. Put another way, it can be argued that a Carnegie unit represents the equivalent of say, a hundred hours of videotape, fifty hours of study hall, thirty hours of roll call and other delays, and about an hour of actual human feedback. By comparison, a computer can deliver hundreds of hours of inhuman feedback to each student. This inhuman feedback only has to be ten percent as efficient as a human teacher to deliver superior results.

Andrew D. Todd
1249 Pineview Dr., Apt 4
Morgantown, WV 26505 (formerly  U46A8@WVNVM.WVNET.EDU)

Table of Contents  Homepage


Paul Ceruzzi, "Electronics Technology and Computer Science, 1940-1975: A Coevolution," Annals of the History of Computing, 1989, 10[4]:257-275.

David Epstein and Silvio Levy, Experimentation and Proof in Mathematics, Notices of the American Mathematical Society [hereafter, "Notices..."], June 1995, Vol. 42[6], pp. 670-74.

Alex Kasman, Letter to the Editor: "On Two Points in the Article by Wu," Notices..., June/July 1997, Vol. 44[6], p. 653.

Boris A. Kusher and Marc H. Melman, Letter to the Editor: "Mathematics at Second-Tier Institutions," Notices..., Mar 1997, Vol. 44[3], pp.310-11.

Steven G. Krantz, "Imminent Danger-- From a Distance," Commentary: In My Opinion, Notices..., May 2000, Vol. 47[5], p. 533.

Richard J. Maher, Letter to the Editor: "Dealing With Reduced Enrollments," Notices..., December 1999, Vol. 46[11], p. 1350

George Peter Murdock, _Social Structure_, The Free Press, New York, 1965, orig. pub. 1949

Murray H. Siegel, Letter to the Editor: "Look to the Elementary Schools," Notices..., December 1998, Vol. 45[11], p. 1454

Sherman Stein, University of California, Davis, Letter to the Editor: "Preparation of Future Teachers", Notices..., March 1997, Vol 44[3], pp. 311-12.

J. J. Uhl, Letter to the Editor: "Steven Krantz versus Calculus&Mathematica," Notices..., March, 1996, Vol 43[3], pp. 285-86.

H. Wu, Forum: "The Mathematician and the Mathematics Education Reform," Notices..., December 1996, Vol. 43[12], pp. 1531-37.

--------, "Professional Development of Mathematics Teachers," Notices..., May 1999, Vol. 46[5], pp. 535-42