From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.
Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.
Changes on the Horizon
Donald J. Lewis (UM-Ann Arbor)
For the January 2000 annual MAA meeting, I was asked to chair a panel that was to discuss the mathematics curriculum for 2010. The panel consisted of faculty from a major research department, an engineering university, a small private university, and a regional university. Interestingly, the visions presented had a great deal in common. Rather than present a curriculum for 2010, I chose to talk to some of the parameters that will play a role in forming such a curriculum. Below, I will present a few of these parameters.
Computers will be ubiquitous. Very soon universities will have wireless systems, allowing access wherever the student may be — even the pub. Portables are becoming extremely powerful and have graphic capabilities, and prices are constantly falling. All students will have computers and will expect to use them. Already MSU is requiring all new freshmen to have their own computer. With the availability of the Internet we should see dramatic changes in how we assist students in learning. Of all vocations and professions, collegiate instruction has changed the least. We still rely on the lecture method, which originated in the Middle Ages when universities were first started; books were scarce, and word of mouth was the principal means of communication. While the method of instruction did not change significantly with the printing press, I don't see that being the case with computers, especially with their being an integral part of most people's lives.
There is considerable evidence that the algorithmic (drill) aspects of calculus, linear algebra, and differential equations can be well taught via the computer. Drill must be practiced alone, with feedback as to correctness of the result. There is no need for human intervention; the computer programs are more efficient.
Using the computer to develop facility in algorithmic aspects of elementary mathematics should free the instructor to concentrate on the underlying theory and its use in modeling various aspects of science. If such occurs, the debate about reform calculus should vanish — students should excel in the algorithmic aspects and have an understanding of theory. Ideally, by the end of the second year, the student will have covered a very substantial part of the introductory rigor courses now usually found in the third year. This will be desirable, for as we shall see, there is more that needs inclusion in the undergraduate curriculum than is currently present.
No doubt, in a few years most modern books and journals will be accessible via the Internet, as well as films and lectures by world experts. (Consider what is currently available from MSRI.) These will be the basis for learning for many students and will be much better than sitting in a lecture, unable to go back if a point is missed. Indeed, the distance learning programs now springing up are based on the accessibility of the Internet. The University of Phoenix (exclusively a distance learning institution), which started five years ago, just had its first graduation ceremony. The number of graduates was modest — less than 1000 as I recall — but few physical colleges could start from scratch and graduate that many in five years, especially when most students were working full time. The School of Business Administration at the University of Michigan is now offering courses worldwide, via the Internet. James Duderstadt, former UM President, is hard at work constructing the Virtual University. These endeavors are not unique to Ann Arbor — many universities and colleges are actively seeking to develop distance learning programs. Economic considerations will probably force most post-secondary educational institutions to do so, or to join in consortia doing so. Once that happens, do you think they won't ask about instructional delivery on campus?
All the distance learning programs with which I am acquainted do have a student/faculty relationship. That relationship has the faculty member serving as a critic of the student's problems sets, essays, etc. The faculty member becomes a coach rather than the purveyor of knowledge. I always felt I did my most significant instruction when I critiqued student papers, letting them know when their solution was particularly clever or elegant, letting them know where they had gone wrong and, if possible, trying to save their argument. While Oxford and Cambridge have lectures, all accept that the real instruction occurs in the student tutorials, where student efforts are critiqued. I see the American university of the future relying on the Internet to convey information, with the resident faculty concentrating on critiquing student efforts.
The advent of the computer has greatly changed science: computational modeling now plays an equal role with experimentation/observation and with theory. The best science arises when these approaches are intertwined, interactively challenging and supporting each other. Experimental/observational data drives theoretical models, and computational models based on the theoretical models predict behavior of the systems studied. The predictions often challenge the experimental data, leading to more delicate and more precise experiments to validate the predictions and to shed further light on the theoretical models.
Each of these approaches depends on and makes demands of the mathematical sciences. Attempts to develop good theoretical models have long driven the search for new mathematics; for example, calculus and much geometry was developed to provide a good theory for physics. Computational modeling drives the development of new and faster algorithms. Because of the massive amounts of data now collectable because of new technology, there is a great need for new methods of data collection, pattern recognition, and feature recognition. Further computer modeling is an important substitute for experimentation, when such is dangerous, costly, or time-consuming.
Science cannot move forward without partnering with the mathematical and informational sciences, and with such partnering will come unexpected challenges to the mathematical sciences. Freeman J. Dyson (Institute for Advanced Study), in his article "The Ascent of Science", Civilization (Feb./March 2000), noted that while in the first two thirds of the twentieth century, physics was a king pin, in the later part of the century "the greatest service of physics to science has been to lend its tools to start revolutions in other specialties." I dare say the same could be said of the mathematical sciences. The growing dependency of science on mathematics should be a symbiotic development that enables both fields to develop faster and present great opportunities to our students. Dyson goes on to say, "Sharing tools also leads to increased mobility, a kind of vocational symbiosis, as young people skilled in their use move from one science to another."
Given this growing dependency, how do we prepare our students for the future? Clearly the science and engineering students need to understand what mathematics can contribute and to have a foundation in mathematics sufficient for further learning, and the mathematics students will need to know sufficient science so that they can understand the scientist. Regrettably, in recent times various fields have grown further and further apart and students study a very narrow curriculum. The future demands a much broader experience. Solutions of the complex problems that now face science will require substantial expertise in the mathematical sciences and in one or more other sciences. This is probably beyond what one human can manage, and hence research will be done in teams, as is now usual in industry. I see little being done in curriculum development that recognizes this scenario. How do we provide a broad background and some group experience without extending the time to degree?
A growing phenomenon is that of a research experience for undergraduates (REU). Today's students seek immediate gratification and appear to get great motivation and satisfaction from such experience. More and more, graduate schools look for such experience, but then continue as in the past to require 2-3 years of course work before beginning research. How long before students rebel? Since the student/faculty ratio is large, if all students are to have such experience it will be impossible to have them work alone on research projects. Many NSF-sponsored REU sites have students work in teams. How about cross disciplinary teams working on a project?
The last point I made at the panel was the very high probability of a shortage of instructors at all levels from kindergarten to graduate school. By 2010 over half of the present K-12 teachers, over 60% of the community college teachers, and 40% of the four-year college and university instructors will retire. We are now experiencing the lowest enrollments by math majors in decades, but all the replacements for college and university positions are already enrolled in our colleges. Finally there are growing opportunities in industry and business. The starting salaries in industry are at least 50% higher than what academia pays, whatever the level. Students have become sufficiently aware of these opportunities that at many graduate schools half the new PhD's are choosing industry or business over academia, and many are leaving without the degree. An even larger percentage of BA's and MA's are doing so. Maybe as we incorporate distance learning methods into our programs, we will need fewer faculty. But as indicated above, quality learning still requires the critic/coach.
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