The following is a slightly shortened version of my article in The Mathematical Intelligencer, Vol. 18, No. 1 (1996).

## The Case Against Computers in K-13 Math Education (Kindergarten through Calculus)

In Peru, as in many Third World countries, the system of public education is in crisis. Teachers' pay -- traditionally low -- is falling rapidly because of inflation. The schools are dilapidated, and there is no money for basic supplies. The government has not been responsive to the teachers' protests. Under pressure from the I.M.F. (the International Misery Fund,'' as Egyptian President Mubarak has said), it insists that the state sector -- and that includes public schools -- must be cut back.

Yet President Fujimori has said that he wants to get computers into the schools as soon as possible. The government's priority is to modernize'' the economy and the educational system, and computerized learning is supposedly one way to do this. For teachers who are trying to cope with financial hardship and abysmal working conditions, what could be more demoralizing than the message that machines come before people? To the Peruvian teachers, Fujimori's advocacy of computers adds insult to injury.

On the other hand, not everyone loses. The U.S. computer industry has an interest in creating new markets in the Third World. Thanks to such strategies as interconnected products and planned obsolescence, greater and greater payments will flow to the North as countries like Peru become dependent on U.S. technology in more areas of national life.

Throughout the Third World, for about a decade pressure has been mounting to import computer learning from the wealthy countries. In February 1986, a major conference called Informatics and the Teaching of Mathematics in Developing Countries'' was held in Tunisia. Participants were predominantly from Northern Africa, but many came from the universities and educational establishments of other regions of the Third World. All of the mathematicians and math educators sang the praises of computers and pled for the rapid introduction of computers into their school systems. Not a single note of skepticism was raised, not a single question was explored in depth. One could have asked, for example: Are computers truly what schools in Africa need? In practice, would computers be introduced on a general level, or only in a few of the elite schools for the wealthy? Could resources be better spent in other ways -- to raise teachers' salaries, purchase classroom supplies, expand libraries? It seems bizarre that the Tunisia conference adopted as an axiom the notion that the introduction of computers should be a priority for elementary education in Africa.

It should be noted that much of the support for this conference came from the big French computer companies. Thus, even though the meeting was of little value for people seriously interested in educational issues, the funding companies must have viewed it as a great success, holding open the possibility of lucrative new markets, particularly in the former French colonies of Northern Africa.

### Some Caveats

Despite my skepticism about computers in education, I do not advocate an extreme position -- I do not want to throw out the baby with the bath water. There are some appropriate uses for computers in math education. Thus, I will not argue
• that it is unwise to use calculators in class. (I have my calculus students use them.)
• that computer-based math courses are always a bad idea. (My university has an optional computer lab course, taught in conjunction with third quarter calculus, that seems to be functioning well.)
• that other technology in the classroom should be avoided. (Nice films on area, volume, etc. have been around for decades.)
• that special learning programs based on calculators or computers are never successful. (Almost any teaching method can work well under suitable conditions and with a dedicated and enthusiastic teacher.)

I will argue, however, that there has been too much hype about technology in math education, and it is time to consider the downside. In my opinion, computers should not be a major component in math education reform.

The downside can be divided into several broad areas:

• drain on resources (money, time, energy);
• anti-intellectual appeal;
• corruption of educators.

I shall discuss each of these objections in turn, after a few preliminary remarks.

### Popular Culture

Any discussion of math education reform should take into account the cultural environment in which we live. American popular culture, which has come to dominate the entertainment industry and the mass media in many parts of the world, can easily cause distortions of the movement for reform.

Youngsters who are immersed in this popular culture are accustomed to large doses of passive, visual entertainment. They tend to develop a short attention span, and expect immediate gratification. They are usually ill equipped to study mathematics, because they lack patience, self-discipline, the ability to concentrate for long periods, and reading comprehension and writing skills.

In the U.S., attempts to popularize mathematics tend to be influenced by this ambiance. The public television program Square One'' is an example of a well-intentioned effort that went astray because of too much gimmickry and too little attention to content. The politics of funding adds to the pressure on reformers to water down the content. Funding agencies are impressed if grantees can demonstrate short-term success. This may explain why one often comes across pilot projects in American schools where the math content is not appropriate for the target age group -- it is too trivial. Of course, the students find it easy, and the project organizers can report great success, thereby satisfying the funding agencies. Such an approach to curricular development contributes to the dumbing down'' of the curriculum.

The most important example of gimmickry in math education reform is computermania.

### A Drain on Resources

We already saw an extreme example of misallocated resources in Peru. But even in a wealthy'' country one has to be concerned about this problem. Many educators in North America share the feeling of betrayal of the teacher who said,
They can give us the axe, but they can spend thousands on computers. We have to fire our music coordinator, we have to fire our music teachers, we have shitty libraries. (Lynn, a Canadian schoolteacher, quoted in [14, p. 41])

At the University of Washington, we also have resource limitations. After considering various ways to reform the calculus course, we selected a low-tech approach using some applications-oriented lecture notes that I had written. Our calculus reform was implemented relatively quickly, painlessly, and inexpensively, largely because it was not based on computers or graphics calculators.

Cost is an issue not only for schools, but also for individual students. Ironically, it is sometimes the colleges with the highest proportion of working-class students that have become most enamored of expensive new gadgetry for teaching mathematics. At a reception for students planning to transfer to my university from community colleges, the students I talked with were complaining about having to spend more than $80 on a graphics calculator, in addition to$60 for a textbook. (By contrast, the required material for my 20-week calculus sequence costs a total of about \$14.)

Another resource issue is that techniques that work in an experimental program with an enthusiastic instructor will not necessarily work under less ideal conditions. One of the lessons of the 1960s new math'' fiasco is that we must look beyond the pilot programs and demonstration classes and selected testimonials, and ask ourselves what is going to happen in a typical classroom with a typical teacher.

Every week, in conjunction with a course I teach for math education majors, I spend a morning visiting just such a school. My students and I work with regular sixth grade classes at Washington Middle School (WMS), an inner-city school in Seattle. The teacher has five math classes every day, with a total of over a hundred students, many of whom have severe personal as well as learning problems.

When I started visiting her classes in 1992, she had several computers in the back of the room. But they were just gathering dust, and so have been removed. What, if not computers, does an overworked teacher like her need in order to be more effective? She herself told me that she very much regrets not having any time in the day to talk as colleagues with the other teachers -- about pupils they have in common, and about teaching methods and materials.

A fundamental problem in education is the failure to treat teachers like professionals. Teachers at schools like WMS need opportunities to upgrade their qualifications, to learn about different teaching materials, and to interact with other teachers as colleagues. This requires release time, light teaching years, or sabbaticals. If money currently used to buy computers and software were instead devoted to improving conditions for teachers, it would be money well spent. If pay and working conditions improved for teachers, then maybe more of our best students would go into teaching.

All of the fuss about computers serves to divert attention away from the central human needs of the school system -- better conditions for teachers and better teacher training.

In response to the grandiose claims of such computers-in-the-schools gurus as Seymour Papert, many experts in child development have pointed out that those claims rest on a flimsy pedagogical foundation, especially where young children (K - 5) are concerned.

The point is that children benefit most from material that stimulates them to exercise their imagination. For example, simple, unstructured play material like clay, sand, blocks, rag dolls, and finger-painting sets are more wholesome entertainment than TV (even educational TV'') and Power Ranger toys.

Harriet K. Cuffaro of the Bank Street College of Education comments on computer painting versus ordinary painting:

...in painting' via the computer, the experience is reduced and limited by eliminating the fluid, liquid nature of paint. In this painting there are no...opportunities to become involved in the process of learning how to create shades of colors; gauging the amount of paint to be mixed; experimenting with and discovering the effects of overlaying colors; understanding the relationship of brush, paint, and paper, the effects achieved by rotating the brush and varying pressure, or how one gains control of or incorporates those unexpected, unintended drips.... There is an absence of texture, of smell, a lessening of qualitative associations with the experience of painting... computer graphics have a stamped-out,' standardized, painting-by-numbers' quality. Though the design or arrangement of colors, lines, and forms will vary with each child, there is a quality of sameness in appearance... individuality is flattened by the parameters of the program. [15, p. 25]

More generally, according to Douglas Sloan of Columbia Teachers College:

For the healthy development of growing children especially, the importance of an environment rich in sensory experience -- color, sound, smell, movement, texture, a direct acquantance with nature, and so forth -- cannot be too strongly emphasized.... At what points and in what ways will the computer in education only further impoverish and stunt the sensory experience so necessary to the health and full rationality of the human individual and society?... What is the effect of the flat, two-dimensional, visual, and externally supplied image, and of the lifeless though florid colors of the viewing screen, on the development of the young child's own inner capacity to bring to birth living, mobile, creative images of his[/her] own? [15, pp. 5,8]

Some have also questioned the effect of computers on teacher-student interaction. Larry Cuban, who has made a detailed study of the history of attempts since 1920 to introduce technology into American schools, writes:

In a culture in love with swift change and big profit margins, yet reluctant to contain powerful social mechanisms that strongly influence children (e.g., television), no other public institution [besides schools] offers these basic but taken-for-granted occasions for continuous, measured intellectual and emotional growth of children.... The complex relationships between teachers and students become uncertain in the face of microcomputers... introducing to each classroom enough computers to tutor and drill children can dry up that emotional life, resulting in withered and uncertain relationships. [2, pp. 89, 91]

Several educators have criticized the public's unquestioning acceptance of so-called computer literacy'' as a goal for education. Computer scientist Joseph Weizenbaum has said, There is, as far as I know, no more evidence [that] programming is good for the mind than [that] Latin is, as is sometimes claimed'' (quoted in [2, p. 94]). According to John M. Broughton of Columbia Teachers College:

Inherent in that term [computer literacy'] is the promise of generalizability comparable with [that] of reading and writing. However, there is no evidence that programming skills transfer to other areas of psychological development, even cognitive ones. In fact, a recent comprehensive review of the literature by Pea and Kurland suggests that virtually all the claims made about the beneficial educational effects of learning to program are not only inflated, but probably incorrect. Moreover, Pea and Kurland reveal that there is not even support for the...notion that learning to program aids children's mathematical thinking. Their own research study on transfer revealed that Logo programming experiences had no effect on the planning skills that are deemed central to problem-solving skills. The tradition of grossly inflated claims identified in the artificial intelligence literature...appears to have carried over into the...area of electronic learning. [15, p. 109]

The inability to develop good translation software has been one of the most embarrassing failures of Artificial Intelligence. If the best computers in the world are unable to translate from French into English, then they certainly cannot help my calculus students do what is the main point of the course: translating word problems into mathematics. Computers in this course would only be a distraction and a waste of time and resources. If the focus in beginning calculus is put where I believe it belongs -- on analyzing real world problems and choosing the appropriate mathematical techniques -- then the course cannot be centered around computers or graphics calculators.

At my university we have an optional one-credit computer lab for students in their third quarter of calculus. At this point it makes sense to offer the technology, because

• the emphasis shifts from word problem applications to geometric applications of calculus in three dimensions, where computer graphics serve a useful purpose, and
• students at that level are sophisticated enough to benefit from computers.

At all levels of schooling we have to ask: Do the students learn to punch buttons, or do they learn mathematics? One day at Washington Middle School, we had the children play a math game that involves dividing by 7 and rounding off to the nearest integer. When the sixth graders had to find 60/7, they punched it correctly into their calculators, which displayed 8.5714... But most of them could not read or interpret the answer: they did not understand the significance of the decimal point.

Similar dangers exist among older students. For this reason, in the calculus final exams at my university we usually ask for exact (not decimal) answers. For example, sin(60°)=\sqrt{3}/2, not 0.866; the circumference of a circle is 2 pi r, not 6.283r.

### Anti-Intellectual Appeal

Computers reinforce the fascination with gadgetry, as opposed to intellect, that is endemic in American popular culture. As pointed out by Brian Simpson, former education advisor to IBM in the U.K.,
Technological solutions to educational problems often have a seductive appeal, promising to make education easier and more enjoyable than ever before. In the past, extravagant claims have been made for teaching machines, educational television, language laboratories, and even such improbable, esoteric phenomena as sleep learning and learning under music-induced hypnosis. [15, p. 84]

There is a long history of people wanting and expecting technology to transform education. It has been over seventy years since the following prediction was made by a famous American:

I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks. (Thomas Edison, 1922; quoted in [2, p. 9])

Perhaps the most frequent argument for computers in the schools -- and also the most illogical -- is the inevitability argument: Calculators/computers are going to be everywhere, so we might as well incorporate them into math class. One can't fight against the tide.'' Using the same reasoning, one could say that, since automobiles are everywhere in our society and play a crucial role in all of our lives, therefore cars should be used as much as possible in education, and driver education should be regarded as a centrally important subject in school, much more so than such relatively useless subjects as music and art. The inevitability argument for computers in the schools is exactly the same sort of anti-intellectual non sequitur.

Computers are usually used in the classroom in a way that fosters a Golly-Gee-Whiz attitude that sees science as a magical black box, rather than as an area of critical thinking. Much computer use is teaching by demo'' with the student as spectator. There is then little difference between so-called electronic learning and simply watching television.

Most software is based on immediate gratification, and does not encourage sustained mental effort. While physically playing an active role, the pupil is intellectually passive, and has little opportunity to be creative. That is, the pupil is programmed to follow a path already laid out in detail by others.

Even when software designers try to get the children into a creative mode, in many cases the same could be better accomplished without the technology. Educators tend to put the cart before the horse: instead of asking whether or not technology can support the curriculum, they try to find ways to squeeze the curriculum into a mold so that computers and calculators can be used.

Like a quack cure in medicine, perhaps the most harmful effect of the computer craze is that it diverts people from other, more solidly grounded approaches to treating the ailments of mathematics education. Might not the Golly-Gee-Whiz-Look-What-Computers-Can-Do school of mathematical pedagogy eventually come to be regarded as a disaster of the same magnitude as the new math'' rage of the 1960s?

### Computer Science is Not the Same As Computers

One can strongly advocate increased teaching of computer science (and related areas, such as discrete math), while opposing the use of computers. Computer scientist Michael Fellows, who is an active campaigner for computer science in grade school (see [3]), has said: Most schools would probably be better off if they threw their computers into the dumpster.'' Fellows uses the term Cargo Cult'' to refer to the fetishization of computers by the media and educational establishment.

Definition of Classical Cargo Cult: An isolated civilization comes into initial contact with European technology. Ignorant of modern science, they interpret the benefits of technology in terms of their familiar world and their familiar mode of operation. They pray and perform sacrifices, or do whatever seems to be necessary to induce the deities to bring them the Cargo.

Modern Cargo Cult: In the U.S. and many other countries, most of the general public is prescientific, in the sense of having no rational understanding of the intellectual processes that go into scientific advances or their application to the real world. On the other hand, like the classical Cargo Cultists, they realize that technology is associated with economic well-being, and that something must be done so that youngsters will later be able to reap the benefits of the computer age.'' The natural response, then, is to fetishize computers and fit them into the familiar world of traditional mindless school math.

The public needs to understand that math and computer science are not about computers, in much the same way that cooking is not about stoves, and chemistry is not about glassware. That is,

### COMPUTER SCIENCE not= COMPUTERS

The meaning of this inequality is: What children need in order to become mathematically literate citizens in the computer age is not early exposure to manipulating a keyboard, but rather wide-ranging experience working in a creative and exciting way with algorithms, problem-solving techniques and logical modes of thought.

### Money Corrupts

Much of the effort to introduce technology in the classroom is profit-driven. As Douglas Sloan says,
It does not take a flaming Bolshevik, nor even a benighted neo-Luddite, to wonder whether all those computer companies, and their related textbook publishers, that are mounting media campaigns for computer literacy and supplying hundreds of thousands of computers to schools and colleges really have the interests of children and young people as their primary concern. [15, p. 3]

In the words of Joseph Menosky, an American writer and former science editor at National Public Radio,

Certainly those who have a great deal to gain from a universal acceptance of computer literacy -- microcomputer firms selling hardware, textbook companies selling educational software, organizations selling worker and teacher retraining courses, and writers and publishers selling books and instructional guides -- have done a brilliant, if morally indefensible, job of commercial promotion. [15, p. 77]

Corporate domination of math education reform has corrosive effects. The profit motive has an excessive influence; the intrinsic value of a pedagogical idea is not considered to be as important as its saleability. Educational ideas that are not based on expensive gadgetry or new textbooks are not likely to be supported strongly. There is an excess of faddism and hype.

It is regrettable that computers have been so aggressively marketed to teachers and school systems. In speaking to parents, teachers and school boards, many company sales representatives have taken the hard-sell approach: If you don't buy our latest products, you will be neglecting to prepare your children for the 21st century.''

Foundations and government agencies, such as the N.S.F. in the United States, compound the problem, because they share the biases of the commercial interests. Money from corporate and foundation sources seduces educators, whose objectivity and independent judgment become compromised. Grants can corrupt.

The technology-in-education movement has some of the characteristics of a religious evangelical campaign, fueled by corporate and foundation money. One sees the same reliance on testimonials. A technology enthusiast might proclaim How graphics calculators have changed my life!'' -- just as a born-again Christian talks about rediscovering Jesus. Like their religious counterparts, technology boosters tend to adopt a Manichaean view, dividing educators into two camps: those who have seen the light, and the fractious infidels.

I recently had a personal encounter with the closed-mindedness of technology advocates. About two years ago, N.S.F. asked me to help evaluate calculus reform proposals. But when they learned that I am skeptical about computers and graphics calculators in calculus, they changed their minds, and decided not to send me any proposals to review. They did not want any input from a nonbeliever.

### Low-Tech Is Better

To end on a positive note, the good news is that the vast majority of enrichment topics do not require high technology, but only pencil and paper and inexpensive manipulatives, such as geoboards and abacuses. (According to a Nov. 22, 1994 article in the Wall Street Journal, the abacus is making a comeback in Asian schools, replacing calculators. The sound and tactile sense give children a feel for the algorithmic dynamics -- kids say things like to subtract 7 you add 3 and then subtract 10'' -- whereas the use of calculators made the children feel alienated from the arithmetic.) All of the material that my math education students and I present to the Washington Middle School youngsters is low-tech.

### Epilogue: First Time It's Tragedy, Second Time It's Farce

At the beginning of the article I mentioned how President Fujimori of Peru has become a fervent advocate of computers in the schools. It is interesting to note that in the late 1960s Peruvian education officials came under influences that were in some ways a foretaste of the computer rage. The new math'' was then the reigning paradigm in math education in the wealthy countries. The Peruvian education ministry, which at that time was heavily French-influenced, imported the new pedagogy despite the strenuous opposition of the leading Peruvian mathematicians (particularly, Professors C. Carranza and M. Helfgott). Of course, the introduction of Bourbaki-style mathematics in the schools was a disaster in Peru, as elsewhere. The Peruvian mathematicians had been right, and the French had been wrong.

Will history repeat itself? Will countries around the world again import a poorly thought out and unworkable pedagogy from the U.S. and France? This time the cost will be higher. The new math,'' for all its foolishness, at least was relatively inexpensive.

### References

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2. L. Cuban, Teachers and Machines: The Classroom Use of Technology Since 1920, New York: Teachers College Press, 1986.
3. M. R. Fellows, Computer science and mathematics in the elementary schools, in N. D. Fisher, H. B. Keynes, and P. D. Wagreich, eds., Mathematicians and Education Reform 1990-1991, Providence: Amer. Math. Society, 1993, pp. 143-163.
4. M. R. Fellows, A. H. Koblitz, and N. Koblitz, Cultural aspects of mathematics education reform, Notices of the Amer. Math. Society, 41 (1994), pp. 5-9.
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6. N. Koblitz, Math 124/125 (Calculus Lecture Notes), University of Washington Mathematics Department, 1994.
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13. R. D. Pea and D. M. Kurland, Logo Programming and the Development of Planning Skills, Technical Report No. 16, New York: Bank Street College of Education, 1983.
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15. D. Sloan, ed., The Computer in Education: A Critical Perspective, New York: Teachers College Press, 1985.
16. C. Stoll, Silicon Snake Oil: Second Thoughts on the Information Highway, New York: Doubleday, 1995.