Learning:

Optimists think they can. Traditional teaching yields in many students only a thin veneer of procedural skills that lack a web of essential connections. Students must argue and think about mathematics to build understanding. Mathematical intuition comes from mathematical thinking, not vice versa. Computers, however, have the potential to change the way you think: they "mess with your mind" and stimulate new patterns of thought. In this view, understanding is a natural consequence of stimulated thinking.

2. Can students develop mathematical intuition without performing extensive mathematical manipulations?

Skeptics worry that intuition requires not only thought, but extensive manual "patterning" in the mechanisms of mathematics. Without sufficient experience or research, we don't really know whether it is doing algorithms or examining results that really matters. Many engineers, scientists, and mathematicians doubt that students who have learned their mathematics using powerful computer packages will remain in personal control of the processes of mathematics. Teachers observe that students who rely on calculators often lack the geometric insight that comes from constructing solutions through hand-based methods. Some wonder how students will learn the value of exact answers; others wonder whether the value of closed-form solutions isn't over-rated. Even advocates of computer-enhanced instruction admit that the evidence on these issues is insufficient to yield definitive conclusions.

3. Do the mechanics of computing obscure mathematical insight?

Early explorers of computer-enhanced instruction often found that student frustration with syntax and software impeded learning of mathematics. Newer systems have reduced up-front special learning, so that the return on investment comes sooner and with greater impact. But the question remains whether students who are having difficulty learning both mathematics and computing will find their problems compounded rather than ameliorated by having to do both at the same time.

4. Will using computers reduce students' facility to compute by hand?

Many well-informed observers fear that students who learn with computers will become excessively reliant on them. This fear is exacerbated by widespread reports of college mathematics students who use a calculator to multiply 4 times 10. The evidence on this question--mostly from calculator-based school mathematics--is sparse and inconclusive. One might better ask: does it matter? Is society worse off today because fewer high school graduates know how to perform long division accurately? Perhaps we have been hurt more by the school-generated perception that mathematics is computation, and that any mathematics problem should be able to be solved in under three minutes.

Curriculum:

Three examples can make clear the importance of this issue: geometry, statistics, and computational science. All three fields have experienced a virtual renaissance in the last 10-15 years as a consequence of computer methods. Yet the traditional mathematics curriculum pays slight attention to these areas, in contrast to fields that rely heavily on formal symbolism such as analysis and algebra. Should computational science be taught alongside laboratory and theoretical methods from the very beginning? Should geometry and statistics be continuous strands in mathematical development, not just special courses largely isolated from the mainstream curriculum? Isn't a large part of the traditional calculus curriculum obsolete, being designed primarily to yield information in the absence of accurate graphical or numerical methods?

6. How does computing change what students can learn about mathematics?

Two examples here will suffice: patterns and parameters. Mathematical insight grows in proportion to the stock of examples that one can draw on for analogy and similarity. Computers enable students to observe and create patterns from which mathematical connections grow. Varieties of patterns (e.g., of various polynomial graphs with corresponding algebraic representations) can make clear the significance of parameters in mathematical formulas; computer images can remove the mystery from variables, parameters, and constants in a way that does not depend on which part of the alphabet the symbol is drawn from. If mathematics is a science of patterns, computers are the laboratory of that science.

7. Where in the curriculum is computing most appropriate?

Perhaps a better questions is the reverse: Is there any place where computers would not be appropriate? Surely on a local scale--from one problem to the next, from one class period to another--there will be situations where computers are either irrelevant or inappropriate. But at the course level, from kindergarten on, calculators and computers have significant, appropriate roles in every course. A tougher question for college teachers is this: Are any undergraduate courses immune to computer-inspired revolution? So far most attention has been focused on calculus, linear algebra, differential equations, statistics, and other courses where computation plays a natural role. But it doesn't take too much imagination to foresee equally intensive roles for computers in courses like modern algebra or topology.

8. Will use of computers reduce the need for remediation?

Some have argued that the availability of computational tools will enable students who do not have strong, formal manipulative skills to continue advancing in mathematical-based subjects. If this conjecture is correct, it could significantly improve the prospects for science education in the United States. Even if this were possible, it surely will not happen unless advanced courses build on appropriate computer skills. Since most remediation now is a failure, we might better ask the question in a different way: Can computer-based instruction provide improved opportunities for at-risk students to attain appropriate standards in mathematics-based courses?

Resources:

The important subtext is a concern about equity: Will unequal access to computers aggravate inequity in opportunity to learn? How much computer power is needed, on average, by mathematics students in different courses? How much will it cost--in equipment, in space, and in added staff? A back-of-the-envelope calculation suggests that true additional costs could double the instructional expense of mathematics instruction. Equally vexing are questions about classroom computers: Is classroom use of computers essential for effective courses? Is it sufficient for the instructor to demonstrate from the front of the room? What kind of displays are effective and readable? How can one equip a classroom so that computers are both secure and available?

10. How much time and distraction is computing worth?

Who will provide technical support to departments with a computer-intensive curriculum? Will universities recognize faculty effort to introduce computers in the curriculum? Faculty time is critical to the success of a computer-enhanced course, and experience shows that such courses take considerably more time than traditional styles of instruction. Will universities provide to mathematicians the same type of special lab credit that science faculty receive? What is the optimal distribution of resources for a department to support computer-intensive instruction?

11. When will there be good software and compatible hardware?

To be honest, one must admit that most of the hardware and software available for college-level mathematics, and even for much of high school mathematics, has not been of sufficient overall quality to make clear any compelling advantages to its use. One had to have a pioneer spirit--and a class of students with similar inclination--to enjoy "roughing it" in the wilds of early computer systems. Today, however, circumstances are notably different. The development of modest workstation-class machines with windowing environments and integrated software makes available for the first time desk-top computers that can really enhance the use and study of undergraduate mathematics. We have crossed a threshold where the cost-benefit analysis went from negative to positive.

12. Can textbooks ever reflect contemporary computer examples?

The publication pipeline of a good textbook is about one computer generation--roughly five years. This guarantees that textbook-based learning will always be out-of-date in a rapidly changing computer environment. The recently-announced McGraw Hill 48-hour customized textbook may change that, but only in the largest markets. Alternatively, the notebooks of Mathematica may render textbooks obsolete. One might note here the recent tendency of computer manufacturers to "publish" their documentation mostly on-line, expecting users to search for and print only the parts that interest them at particular moments. Maybe a better question would be: Should printed textbooks survive?

Teaching:

Broadly speaking, there are three ways in which computing is used in mathematical instruction: programming, computer-assisted instruction (CAI), and computational packages. Each approach has strong adherents, and strong critics. Programming is useful as a device for forcing students to think through algorithms with uncommon care. CAI is useful for creating special micro-world environments, and is most often used to enhance student skills. Computational packages, by which I mean to include computer algebra systems, statistical packages, numerical packages, and graphical packages, are productive tools of the scientific workplace that students learn to use as they learn the subject they are studying. My preference is to emphasize the latter since these workplace packages provide students with real mathematical tools that they can use long after their course is over.

14. Can pure mathematicians convey an appropriate computational perspective?

Statisticians have argued that mathematicians who have not had statistical experience should not teach statistics since mathematicians approach statistics from an axiomatic perspective that is fundamentally at odds with that of the data-driven statistician. Is the same true for computational mathematics or computer-enhanced mathematics courses? Is there a distinctive perspective appropriate to this teaching that is fundamentally different from what a mathematician is educated for in graduate school?

15. How will new faculty fit into computer-enhanced programs?

Many students who serve in traditional university TA roles have had no opportunity to think about or experience a computer-enriched approach to teaching mathematics. International graduate students, especially, are most unlikely to have any background for this style of instruction. Who can we hire to teach such courses? How much training is required to be comfortable teaching a computer-enhanced course? Will instructors with limited computer background be able to stay ahead of their students?

16. Will use of computers improve teaching of mathematics?

Does it really matter what or how we teach? Many mathematicians act as if they believe that good students learn regardless of what happens in class, and that weak students never learn. Those working with computer algebra systems universally report an increased thoughtfulness about pedagogy as an important side effect of teaching in this environment. They discover, for example, that a mathematics class can be a place where mathematics is done, not just where it is presented. They examine more closely their implicit beliefs about teaching and learn anew the value of students observing their teachers learning--as role models. Perhaps renewed emphasis on active, open-ended exploratory styles of instruction will emerge as an important beneficial side effect of computer-enhanced courses. Indeed, computers may be the only force of sufficient power to change attitudes towards teaching: without the challenge posed by computers, mathematics teaching today would be stagnant.

Dilemmas:

Mathematics departments are full of tales of muddy reasoning emerging from students who take social science statistics courses and then run every data set they find through an enormous computer package to see if anything "significant" emerges. What is to prevent analogous excesses in calculus once students have tools as powerful as Mathematica ? Will computer tools become a substitute for hard work and precise thinking? Perhaps the question should be phrased in more useful forms: What style of instruction will minimize inane use of mathematical software? How can computers enhance precise thinking?

18. If computers handle routine calculations, what will students do instead?

Because they involve less-routine calculation, computer-based mathematics courses tend to be more difficult than traditional courses. They take more time and require more thought for both the student and for the instructor. Moreover, in experimental sections where parallel classes are pursuing more standard material, students and faculty are subject to a dual standard--both the old course expectations and the new one.

19. What are appropriate prerequisites for computer-based calculus courses?

Two issues emerge in this question: the signals that are sent to secondary schools by college placements practices, and the effectiveness of the placement procedure in identifying students who are prepared for the course. Traditional placement testing rarely examines students' abilities to employ calculators effectively, and almost never seeks to determine students' experience with using computers to solve mathematics problems. Effective placement procedures would have to look at both the student's experience in a technology intensive course and at indicators of attitude and habits of mind that suggest likelihood of success in such a course. Moreover, as high schools increase their use of calculators and computers, students entering college will expect courses that build on and extend these prior mathematical experiences.

20. Should mathematics be a lab science?

There are enormous pedagogical advantages to well-designed laboratory experiences. Instructors can watch students doing mathematics, and students can see their teacher as a human being who often might say "I don't know; let's find out." Labs help instructors get to know their students well; by listening rather than lecturing, instructors soon discover that students don't know what we think they know, nor think the way we think. Labs promote important social aspects of doing mathematics, ideas that resonate with the principles that underlie much of Uri Treisman's work with minority students. Labs in mathematics can also promote collaboration in research-like experiences with faculty, thereby attracting students to careers in mathematics. However, virtually no mathematics department is presently equipped with facilities, budget, or staff to operate like a laboratory science. What will it take to convince colleges and universities to make the investment to shift mathematics from chalk and blackboards to computers and labs?