Change, growth, and accountability dominate higher education at the dawn of the twenty-first century. According to delegates at UNESCO's recent World Conference on Higher Education, change is unrelenting, "civilizational in scope," affecting everything from the nature of work to the customs of society, from the role of government to the functioning of the economy. Growth in higher education has been equally dramatic, with worldwide enrollment rising from 13 million in 1960 to near 90 million today [2].

Concomitant with change and growth is pressure from governments everywhere for greater accountability from professors and leaders of higher education–for evidence that, in a world of rapid change, universities are working effectively to address pressing needs of society. Autonomy, the prized possession of universities, presupposes accountability. The escalating pressures of global change, growth, and accountability will create, according to UNESCO Director Federico Mayor, "a radical transformation of the higher education landscape–not only more but different learning opportunities" [7].

Much of the upheaval in society and employment is a consequence of the truly revolutionary expansion of worldwide telecommunication, as is the stunning increase in demand for higher education. But this demand is predicated on the belief that universities properly anticipate signals from the changing world of work and create optimal linkages between students' studies and expectations of employers. Unfortunately, few universities have taken up this challenge–at least not unless pushed by external forces.

As the world changes rapidly and higher education grows explosively, universities evolve leisurely. Courses, curricula, and exams remain steeped in tradition, some centuries old, while autonomy and academic freedom rule in the classroom. Few institutions of higher education readily embrace a culture of assessment that is required to ensure relevance and effectiveness of their curriculum. Under these circumstances it is only natural for political leaders to demand stronger connections between the classroom and the community, between the ivory tower and the industrial park.

Undergraduate Mathematics

Demands for relevance and accountability are no strangers to undergraduate mathematics. Indeed, postsecondary mathematics can be viewed as higher education in microcosm. Growth in course enrollments has been enormous, paralleling the unprecedented penetration of mathematical methods into new areas of application. These new areas–ranging from biology to finance, from agriculture to neuroscience–have changed profoundly the profile of mathematical practice. Yet for the most part these changes are invisible in the undergraduate mathematics curriculum, which still marches to the drumbeat of topics first developed in the eighteenth and nineteenth centuries.

It is, therefore, not at all surprising that the three themes identified at the UNESCO conference are presaged in the Discussion Document for this ICMI Study: the rapid growth in the number of students at the tertiary level; unprecedented changes in secondary school curricula, in teaching methods, and in technology; and increasing demand for public accountability [3]. Worldwide demands for radical transformation of higher education bear on mathematics as much as on any other discipline.

Postsecondary students study mathematics for many different reasons. Some pursue clear professional goals in careers such as engineering or business where advanced mathematical thinking is directly useful. Some enroll in specialized mathematics courses that are required in programs that prepare skilled workers such as nurses, automobile mechanics, or electronics technicians. Some study mathematics in order to teach mathematics to children, while others, far more numerous, study mathematics for much the same reason that students study literature or history–for critical thinking, for culture, and for intellectual breadth. Still others enroll in postsecondary courses designed to help older students master parts of secondary mathematics (especially algebra) that they never studied, never learned, or just forgot. (This latter group is especially numerous in countries such as the United States that provide relatively open access to tertiary education [8].)

In today's world, the majority of students who enroll in postsecondary education study some type of mathematics. Tomorrow, virtually all will. In the information age, mathematical competence is as essential for self-fulfillment as literacy has been in earlier eras. Both employment and citizenship now require that adults be comfortable with central mathematical notions such as numbers and symbols, graphs and geometry, formulas and equations, measurement and estimation, risks and data. More important, literate adults must be prepared to recognize and interpret mathematics embedded in different contexts, to think mathematically as naturally as they think in their native language [11].

Since not all of this learning can possibly be accomplished in secondary education, much of it will take place in postsecondary contexts–either in traditional institutions of higher education (such as universities, four- and two-year colleges, polytechnics, or technical institutes) or, increasingly, in non-traditional settings such as the internet, corporate training centers, weekend short-courses, and for-profit universities. This profusion of postsecondary mathematics programs at the end of the twentieth century contrasts sharply with the very limited forms of university mathematics education at the beginning of this century. The variety of forms, purposes, durations, degrees, and delivery systems of postsecondary mathematics reflects the changing character of society, of careers and of student needs. Proliferation of choices is without doubt the most significant change that has taken place in tertiary mathematics education in the last one hundred years.

Mathematical Practice

The primary purpose of mathematics programs in higher education is to help students learn whatever mathematics they need, both for their immediate career goals and as preparation for life-long learning. Today's students expect institutions of higher education to offer mathematics courses that support a full range of educational and career goals, including:

Even without exploring details of specific curricula or programs, it should be obvious that the multiplicity of student career interests requires, if you will, multiple mathematics. Consider a few examples of how relatively simple mathematics is used in today's world of high performance work:

• Precision farming relies on satellite imaging data supplemented by soil samples to create terrain maps that reflect soil chemistry and moisture levels. These methods depend on geographic information systems that blend spreadsheet organization with a variety of algorithms for geometric projections (e.g., for rendering onto flat maps oblique satellite images of earth's curved surface).
• Technicians in semiconductor manufacturing plants analyze real-time data from production processes in order to detect patterns of change that might signal an impending reduction in quality before it actually happens. These methods involve measurement strategies, graphical analyses, and tools of statistical quality control.
• Teams that design new commercial airplanes now engage designers, manufacturing personnel, maintenance workers, and operation managers in joint planning with the goal of minimizing total costs of construction, maintenance, and operation over the life of the plane. This enterprise involves teamwork among individuals of quite different mathematical training as well as innovative methods of optimization.
• Emergency medical personnel need to interpret quickly and accurately dynamic graphs of heart action that record electrical potential, blood pressure, and other data. With experience, they learn to recognize both regular patterns and common pathologies. With understanding, they can also interpret uncommon signals.

These examples are not primarily about the relation of mathematical theory to applications–the traditional poles of curricular debate–but about something quite different: mathematical practice. Behind each of these situations lurks much good mathematics (e.g., projection operators, optimization algorithms, fluid dynamics, statistical inference) that can be applied in these and many other circumstances. However, most students are not primarily motivated to learn this mathematics, but rather to increase crop yield, minimize manufacturing defects, reduce airplane costs, or stabilize heart patients. Although a mathematician will recognize these as mathematical goals–to increase, minimize, reduce, stabilize–neither students nor their teachers in agriculture, manufacturing, engineering, or medicine would recognize or describe their work in this way. To these individuals, the overwhelming majority of clients of postsecondary mathematics, mathematical methods are merely part of the routine practice of their profession.

In sharp contrast to this profligate flowering of practical mathematics in diverse postsecondary settings, university mathematics–what mathematicians tend to think of as "real mathematics"–matured in this century as a tightly disciplined discipline led by professors of world-wide renown who held major chairs in leading universities and research institutes. However, this university mathematics, "real mathematics" as practiced in real universities, now constitutes only a tiny fraction of postsecondary mathematics. One data point: in the United States, fewer than 15% of traditional undergraduate mathematics enrollments are in courses above the level of calculus [5]. And this does not count non-traditional enrollments, where the variety of offerings is even greater. A realist might well argue that "real mathematics" is found not in the traditional curriculum inherited from the past but in the today's widely dispersed courses where a multitude of students learn a cornucopia of mathematics in diverse situations for a plethora of purposes.

Learning Mathematics

Where do students learn mathematics? Some take traditional mathematics courses such as calculus, geometry, and statistics. Some take courses specifically designed for certain professions–mathematics for nurses, statistics for lawyers, calculus for engineers–that are offered either by mathematics departments or by the professional programs themselves. But many, perhaps even most, pick up mathematics invisibly and indirectly as they take regular courses and internships in their professional fields (e.g., in physiology, geographic information systems, or aircraft design).

Any university dean knows that statistics is more often taught outside of statistics programs than inside them. The same is true of mathematics, but is not as widely recognized. Every professional program, from one- and two-year certificates to four- and five-year engineering degrees, offers courses that provide students with mathematics (or statistics) in the context of specific professional practice. This is entirely natural, since most students find that they learn mathematics more readily, and are more likely to be able to use it when needed, if it is taught in a context that fits their career goals and in which the examples resonate with those that appear in their other professional courses.

The appeal of context-based mathematics is no surprise, nor is its widespread presence in university curricula. But what is somewhat new–and growing rapidly–is the extent to which good mathematics is unobtrusively embedded in routine courses in other subjects. Anywhere spreadsheets are used (which is almost everywhere) mathematics is learned, as it is also in courses that deal with such diverse topics as image processing, environmental policy, and computer-aided manufacturing. From technicians to doctors, from managers to investors, most of the mathematics people use is learned not in a course called mathematics but in the actual practice of their craft. And in today's competitive world where quantitative skills really count, embedded mathematical tools are often as sophisticated as the techniques of more traditional mathematics.

So tertiary mathematics now appears in three forms: as traditional mathematics courses (both pure and applied) taught primarily in departments of mathematics; as context-based mathematics courses taught in other departments; and as courses in other disciplines that employ significant (albeit often hidden) mathematical methods. I have no data to quantify the "biomass" of mathematics taught through these three means, but to a first approximation I would conjecture that they are approximately equal.

If that is true, then I can with some confidence suggest that the biomass of mathematics that is learned, remembered, and still useful five years later is decidedly tilted in favor of the hidden curriculum. Most of what is taught in the traditional mathematics curriculum is forgotten by most students shortly after they finish their mathematical studies, but most of what students learn in the context of use is remembered much longer, especially if students practice in the field for which they prepared.

Thus this paradox: As widespread use makes postsecondary mathematics increasingly essential for all students, mathematics departments find themselves playing a diminishing role in the mathematical education of postsecondary students. The obvious corollary poses another paradox: to exercise their responsibility to mathematics, mathematicians must think not primarily about their own department but about the whole university. Mathematics pervades not only life and work, but also all parts of higher education.

Many people in higher education can think of nothing more dangerous than to have mathematicians takes seriously their leadership responsibility for university-wide mathematics. I daresay that most faculty and students think that mathematics is too important to be left to mathematicians. After all, mathematicians tend to think that they alone know what mathematics is and how it should be taught. Courses that do not fit their standards for real mathematics–for example, mathematics for elementary school teachers, or a reprise of secondary school algebra–are often marginalized, if not ignored. Changes in the secondary school programs are resisted or deflected under the pretense that they should not apply to university-bound students. (In fact these changes have a major impact on university mathematics because so many students–in some countries, the majority–now take some form of postsecondary education.) Also ignored, but for different reasons, is the vast array of mathematics taught in other departments.

Responding to Needs

The growth of higher education, the pervasiveness of mathematics, and the pace of change in society fuel legitimate public demand for tertiary programs in mathematics that are effective, flexible, and responsive to current conditions. Mathematicians know perfectly well how to make systems that are effective, flexible, and responsive: they require feedback loops. But in their own completed work, mathematicians disdain feedback loops. Feedback smacks of approximation, of trial and error, of processes that overshoot and undershoot. Good mathematics hits the nail on the head. It provides a clean argument based on solid principles and strict reasoning. Who ever heard of a completed proof that relied on a feedback loop to correct its errors?

Mathematicians tend to design curricula the same way we design proofs. We select our goal, we think carefully and systematically about all relevant factors, and then we write down all the steps that are logically necessary to reach that goal. What we do not do very often is to listen carefully for feedback–from students, from other faculty, and especially from professionals who employ mathematical methods in their daily work. If we listen carefully, here is some of what we might hear:

• The panic of an architecture student who took several years of secondary mathematics but is still mystified by simple equations and graphs.
• The fear of a minority-culture student who is afraid of mathematics but wants to return to his or her home community to teach young children.
• The frustration of an employer over a mathematics tradition that favors esoteric concepts over mundane examples.
• The anger of college-educated adults who despite their education are unable to understand the economic implications of changes in interest rates.
• The confusion of consumers trying to decide among competing offers for mortgages or cellular telephone service.
• The embarrassment of recent university graduates who find themselves inexperienced with common mathematical, statistical, and graphical software.

Rarely if ever would we hear the lament of a former student who felt disabled for lack of understanding the Lebesgue integral or the Sylow theorem.

It turns out that many business leaders, politicians, and innovative educators are listening more attentively than are mathematicians. Having failed to persuade mathematicians to expand their horizons, these outsiders are busy setting up alternatives to university mathematics –indeed, alternatives to the university itself. The Open University in Great Britain was perhaps the first major alternative of this type. A more recent example is the for-profit University of Phoenix which, in less than a decade and with a faculty that is over ninety percent part-time, has become one of the ten largest universities in the United States. It is fully accredited, intensely evaluated, and traded on a major stock exchange [1, 4, 9, 13].

Recently several internet-based "virtual universities" have begun, one sponsored by the State of California, another by a dozen states in the western half the U.S. where students in need of education are often geographically isolated and employed in jobs that prevent them from studying full time on a traditional campus. As satellites enable wireless worldwide telecommunications, courses and credits from these alternative universities will soon be available almost everywhere [6, 10, 12]. Economic viability will depend on feedback from students, now called "consumers." Convenience learning will transcend both institutional and even national boundaries. In this networked future the choice of what mathematics to study, under whom, and at what time will be made not by professors, but by students.

The Next Century

As we approach the twenty-first century we are greeted by an astonishingly rich landscape of undergraduate mathematics that extends vast distances in several different directions:

• In terms of level,it ranges from elementary topics ordinarily taught in the primary and secondary schools to advanced courses at the interface with graduate education.
• In terms of content,it ranges from classical topics of analysis and algebra to modern interdisciplinary subjects such as neural networks, image processing, and the pricing of financial derivatives.
• In terms of context,it ranges from traditional mathematics courses offering basic theory and selected applications to authentic scientific or work-related situations with embedded mathematics that is learned through use but rarely studied separately.
• In terms of setting,it ranges from scheduled classes in traditional universities and corporate short-courses on weekends and after working hours, to on-line courses designed for professional development or pre- professional qualification and for-profit educational corporations that constantly re-invent the means of delivery.

One corner of this landscape shelters a tiny village somewhat isolated from the new superhighways of postsecondary education. This village is world-famous for its continued commitment to mathematics courses and degrees using traditions inherited from the nineteenth century. However, many of today's young people disdain these traditions as largely irrelevant for the twenty-first century. Others learn these traditions and come to respect them, but nonetheless choose to follow a different path. Only a few follow the tradition faithfully, preserving the past and advancing its accomplishments.

Outside this village, across the vast landscape of postsecondary education, a more contemporary and flexible form of university-level mathematics thrives. This mathematics pervades life and work, is required of virtually every profession and career, is taught both explicitly and implicitly in courses throughout higher education, and is available at any time and in any place–in colleges, at work, after work, on-line, and at home. The reality is clear to anyone overlooking this terrain: postsecondary mathematics is no longer a craft practiced only in the village of university mathematicians.

Nor should it be. The critics are right: mathematics is too important to be left to mathematicians. The forces of change, growth, and accountability have spread both innovation and mathematics across the postsecondary landscape, freeing mathematics from the confining traditions of university departments and opening it to innovative content and pedagogy of other fields. Abetted by these external forces, mathematics the discipline–if not mathematics the department–is thriving. As observers of this vast landscape, we become witnesses to a revolution by stealth, a virtual takeover of undergraduate mathematics by its entrepreneurial clients who now set the agenda. As mathematicians, we should welcome our new colleagues and thank them for propelling mathematics into the twenty-first century.

References

1. Fischetti, Mark, et al. "Education? The University of Phoenix is Just Around the Corner." University Business, (March/April 1998) 45- 51.
2. "Higher Education in the Twenty-First Century." World Conference on Higher Education, 5-9 October 1998. Paris: UNESCO, 1998.
3. Holton, Derek, et al. "ICMI Study on the Teaching and Learning of Mathematics at the University Level." Bulletin of the International Commission on Mathematical Instruction 43 (December 1997) 3-13.
4. Leatherman, Courtney. "University of Phoenix's Faculty Members Insist They Offer High-Quality Education." The Chronicle of Higher Education, (October 16, 1998) A14-A16.
5. Loftsgaarden, Donald C. Rung, and Ann E. Watkins. Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States. Washington, DC: Mathematical Association of America, 1997.
6. Marchese, Ted. "Not-so-Distant Competitors: How New Providers are Remaking the Postsecondary Marketplace." Bulletin of the American Association of Higher Education (AAHE) 50:9 (May, 1998) 3-11.
7. Mayor, Frederico. World Conference on Higher Education, 5-9 October 1998. Paris: UNESCO, 1998.
8. Phipps, Ronald, et al. College Remediation: What it Is, What it Costs, What's at Stake. Washington, DC: The Institute for Higher Education Policy, December, 1998.
9. Shea, Christopher. "Visionary or 'Operator'? Jorge Klor de Alva and his Unusual Intellectual Journey." The Chronicle of Higher Education, (July 3, 1998) A8-A10.
10. Slaughter, Sheila and Leslie, Larry L. Academic Capitalism: Politics, Policies, and the Entrepreneurial University. Baltimore, MD: Johns Hopkins Univ. Press, 1997.
11. Steen, Lynn Arthur (Ed.) Why Numbers Count: Quantitative Literacy for Tomorrow's America. New York, NY: The College Board, 1997.
12. Strosnider, Kim. "For-Profit Higher Education Sees Booming Enrollments and Revenues." The Chronicle of Higher Education, (Jan 23, 1998) A36-A38.
13. Traub, James. "Drive-Thru U. Higher Education for People who Mean Business." The New Yorker, (October 20 & 27, 1997) 114- 123.