*What we urgently need today is a more inclusive view of what it means to be a scholar—a recognition that knowledge is acquired through research, through synthesis, through practice, and through teaching.*

*—Scholarship Reconsidered, 1990*

**A**merica's system of higher education is the envy of the world. Institutional variety matches the variety of our population: two-year colleges, liberal arts colleges, women's colleges, historically Black colleges and universities, comprehensive universities, research universities—not to mention a variety of specialized institutions. Diverse, robust, and infinitely varied, U.S. higher education offers students from around the world courses to suit every whim and programs to meet every need. Nevertheless, the overall balance of the system seems no longer to serve well the needs of U.S. society. Industry, education, military, and government personnel offices all report difficulties in finding employees with the skills required for productive work. The American pub-

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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
FACING THE MYTHS
What we urgently need today is a more inclusive view of what it means to be a scholar—a recognition that knowledge is acquired through research, through synthesis, through practice, and through teaching.
—Scholarship Reconsidered, 1990
America's system of higher education is the envy of the world. Institutional variety matches the variety of our population: two-year colleges, liberal arts colleges, women's colleges, historically Black colleges and universities, comprehensive universities, research universities—not to mention a variety of specialized institutions. Diverse, robust, and infinitely varied, U.S. higher education offers students from around the world courses to suit every whim and programs to meet every need. Nevertheless, the overall balance of the system seems no longer to serve well the needs of U.S. society. Industry, education, military, and government personnel offices all report difficulties in finding employees with the skills required for productive work. The American pub-

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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
lic is now demanding accountability—both from schools and from higher education.
Calculus and Computers
An innovative calculus course at the University of Illinois uses the full symbolic, numeric, graphic, and text capabilities of a powerful computer algebra system. Significantly, there is no textbook for this course—only a sequence of electronic notebooks.
Each notebook begins with basic problems introducing new ideas, followed by tutorial problems on techniques and applications. Both problem sets have "electronically active" solutions to support student learning. The notebook closes with a section called "Give-it-a-try," where no solutions are given. Students use both the built-in word processor and the graphic and calculating software to build their own notebooks to solve these problems, which are submitted electronically for comments and grading.
Notebooks have the versatility to allow re-working of examples with different numbers and functions, to provide for the insertion of commentary to explain concepts, to incorporate graphs and plots as desired by students, and to launch routines that extend the complexity of the problem. The instructional focus is on the computer laboratory and the electronic notebook, with less than one hour per week spent in the classroom. Students spend more time than in a traditional course and arrive at a better understanding, since they have freedom to investigate, rethink, redo, and adapt. Moreover, creating course notebooks strengthens students' sense of accomplishment.
America's record in mathematical research is, likewise, the envy of the mathematical world. This preeminence is documented not only in objective measures (e.g., research productivity, international awards) but also in the steady stream of international students and researchers who visit our premier universities. The challenge we now face is to elevate U.S. mathematics education to the same high level of achievement and reputation as U.S. mathematical research.
To attack this problem successfully, mathematicians and university administrators need first to examine carefully the circumstances that give rise to inadequacies and deficiencies. Surely the twin burdens of increased enrollments and erosion of support have left many departments without the resources needed to fulfill their educational and intellectual responsibilities. But there is more to the problem than inadequate resources. A plan to revitalize mathematics education must be built on a careful analysis of current deficiencies.
Wasted Breath
Since virtually all mathematics teachers received the majority of their mathematics instruction in traditional lecture courses, it is not surprising that lecturing has continued to be the most common way that mathematics is taught, both in high school and in college. To believe that one can teach mathematics successfully by lectures, one must believe what most mathematicians know to be untrue—that mathematics can be learned by watching someone else do it correctly. Research shows clearly that this method of teaching does little to help beginning students learn mathematics, a fact underscored by the staggering rates of withdrawal or failure among students who take introductory college mathematics courses.
It is widely recognized that lectures place students in a passive role, failing to engage them in their own learning. Even students who survive such courses often absorb a very misleading impression of mathematics—as a collection of skills with no connection to critical reasoning. Yet despite

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its recognized ineffectiveness for most mathematics students, lecturing continues as the dominant form of instruction in mathematics classrooms because it is inexpensive: neither faculty members nor administrators have to invest heavily in mathematics courses taught in the traditional lecture style.
NSF Calculus Initiative
As the pivotal course in the scientific curriculum, calculus has long been a natural focus for debate when discussions turn to educational reform. In the mid-1980s, talk turned to action when the National Science Foundation launched a major initiative to foster improvement in the quality of calculus instruction on a national scale.
Begun in 1988, the program made over 50 awards totaling nearly $7 million during its first three years of operation. The results of the individual calculus projects are still to be evaluated, but one great benefit is already evident—the enthusiasm generated in hundreds of mathematical sciences departments for improvement in the teaching of calculus.
NSF expects to expand the scope of this initiative to eventually encompass the entire undergraduate curriculum.
Perhaps even more mysterious is the virtual absence of computers from undergraduate mathematics. Computers have changed significantly the practice of mathematics at all levels, from routine application to advanced research. Computers also enhance student motivation, provide natural catalysts for teamwork, and focus faculty attention on the process of learning. Unless undergraduate mathematics courses are revised to reflect the impact of computers on the practice of mathematics, students will continue to perceive mathematics as a discipline disconnected from reality.
Missing Context
The unease with which many students regard mathematics begins early in their education and is all too regularly reinforced through their school years. The situation seldom improves, not even in college. Students often find themselves in classes in which little effort is made to place the subject matter in a context that is meaningful to the student. Mathematics becomes not a powerful tool but an overwhelming barrier that students must surmount to enter their chosen disciplines. For such students, undoubtedly the majority, mathematics continues to be a mystery unrelated to other subjects or problems in the real world.
"The real impact of technology is the opportunity it provides for students to explore, to work in groups, to write laboratory reports, and undertake projects."
—Priming the Calculus Pump, 1990
Absent a conscious effort to set mathematics in the context of learners' experiences, mastery of skills (including vocabulary, notation, and procedures) serves no legitimate educational purpose. Students who endure such instruction come to regard mathematics as a ritual irrelevant to their own lives. Upon becoming parents, such students impart the same attitude about mathematics to their children.
It is virtually impossible to cover well in a single course both the many ways in which mathematics is applied and the range of topics needed to support authentic applications. As

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a result, many introductory college courses offer students—frequently with disastrous results—a distilled mathematical essence that is, from a student's perspective, devoid of meaning. Particularly at beginning levels, where students from various disciplines take the same courses, this common approach isolates mathematics from the very subjects for which mathematics students are preparing themselves.
Priming the Pump
In response to the growing concern in recent years about undergraduate mathematics, many projects have emerged to explore new approaches to instruction or to study particular issues in depth. Examples:
Preparing Teachers. A guide for mathematics departments to curricula that will educate intending teachers in a way to meet contemporary standards is contained in the 1991 MAA report A Call for Change: Recommendations for the Mathematical Preparation of Teachers of Mathematics and the 1991 NCTM document Professional Standards for Teaching Mathematics.
Undergraduate Research. Reports on a wide variety of effective programs that provide research experiences to undergraduate students of mathematics appear in the 1990 MAA volume Models for Undergraduate Research in Mathematics.
Calculus Reform. Nearly one hundred projects, the tip of the iceberg, are presented in the 1990 MAA publication Priming the Calculus Pump: Innovations and Resources.
Doctoral Study. The Board on Mathematical Sciences is conducting a study to identify features of successful doctoral and postdoctoral programs which contribute to the preparation of productive members of the mathematical community.
Upper-division courses often exhibit a similar profile, albeit for different reasons. Many advanced mathematics courses are taught primarily in a style that will prepare students for graduate study in mathematics. Consequently, they address insufficiently the needs of the majority of students who intend either to enter the job market with a bachelor's degree or to pursue graduate work in another discipline. From beginning to end, undergraduate mathematics has been relatively unresponsive to the changing role of mathematics in society and to the changing needs of students in our colleges.
Casual Teaching
In many university mathematics departments, mathematical stature is defined by research. Faculty play an effective role in mentoring graduate students from their role as student to the position of junior colleague. But undergraduate teaching, especially elementary teaching, can under these circumstances be too easily viewed as an ancillary responsibility. Because professional attitudes are shaped primarily while students are in graduate school, the culture of the leading research departments has a subtle but nonetheless very real influence even in the majority of institutions where teaching is unquestionably the primary institutional objective. Although some institutions have taken steps to increase the priority of undergraduate teaching, the overall effect of graduate education is to perpetuate a system of rewards that undervalues teaching.
Faculty attitudes towards teaching are both reflected and perpetuated by the role assigned to graduate teaching assistants. At most universities that have graduate pro-

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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
grams, teaching assistants staff the majority of lower-division courses, often assuming virtually complete responsibility for instruction and evaluation. Indeed, the practice of using teaching assistants is a structural feature of graduate support so ingrained that without it most departments could not survive on their present budgets. Heavy reliance on the use of graduate teaching assistants, many of whom have limited experience or training for the responsibilities placed on them, has far-reaching consequences.
Study in Depth. Advice for effective approaches to undergraduate majors is the focus of Challenges for College Mathematics: An Agenda for the Next Decade, a report prepared by a joint task force of the MAA and the Association of American Colleges as part of a multidiscipline study of undergraduate majors.
Approximately half of the graduate mathematics students in the United States are nonresident foreigners. These students both need and merit support for their graduate studies: they contribute much to the strength and sophistication of graduate programs, and many will contribute significantly to mathematics and to society after they finish their degrees.
''Beginning graduate students in the laboratory sciences may learn as much from advanced graduate students and postdoctorals as they do from the principal investigator—the group provides a mutually supportive and nurturing learning environment for all. The challenge for the mathematical sciences is to create an analogous environment for their own graduate students and postdoctorals."
—Renewing U.S. Mathematics, 1990
Few graduate students, however, are ready to serve well the educational needs of first-year college students; teaching in a foreign culture is particularly difficult for those who were not themselves educated in U.S. schools. Experience in teaching, mathematical expectations, communication skills, cross-cultural sensitivity, and familiarity with technology are vitally important issues that frequently take considerable time to develop—especially among those who are new to this country. The special need for sensitivity toward all students that is required to attract more women and underrepresented minorities into mathematics underscores the important role that graduate teaching assistants play in nurturing students in the mathematical sciences.
Few universities recognize explicitly in the design of their graduate mathematics programs that the future careers of most of their doctoral students will be devoted primarily to undergraduate teaching. Few if any mathematics graduate programs attempt to familiarize graduate students with important curricular and policy issues of undergraduate education. Few graduate assistants undergo systematic training to prepare them for their lifelong role as teachers.
Our system of graduate education produces a cadre of college and university teachers who have little concept of teaching as a profession. Not only are mathematics graduate students denied significant opportunities to develop this

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aspect of professional competence, but their graduate training frequently conveys the subtle message that undergraduate teaching is a second-class activity rather than a critical aspect of the profession.
Linking with Schools
Many colleges and universities are developing outreach programs to help increase the professional competence of teachers of mathematics. Two examples:
The Special Projects Office in the School of Mathematics at the University of Minnesota coordinates a variety of outreach projects that bring together business leaders, government officials, and educators from the elementary school level through the university level to discuss policy issues and to advocate change. The activities include direct service projects such as the University of Minnesota Talented Youth Mathematics Program (UMTYMP) and an NSF Teacher Renewal Project, as well as networking projects such as the Minnesota Mathematics Mobilization, the Twin Cities Urban Mathematics Collaborative, and the Mathematicians and Education Reform network. Each of these activities builds on the expertise of college and university mathematicians.
The Bay Area Mathematics Project, now in its eighth year, brings school mathematics teachers to Berkeley for a four-week summer program designed to improve their effectiveness as teachers. To increase the scope of the program, participants serve as teachers of other teachers and administrators when they return to their schools. The program has grown each year with funding provided by local community and civic organizations. Teachers from different grade levels are selected as participants so
Flawed Models
From grade school to graduate school, mathematics education revolves around the hub of undergraduate mathematics. Unfortunately, few university mathematics departments maintain meaningful links with mathematics in school or with the mathematical preparation of school teachers. Although one in four mathematics majors eventually teaches in school, instructional methods that are widely used in undergraduate programs foster a model of teaching—blackboard lectures, template exercises, isolated study, narrow tests—that is inappropriate for elementary and secondary school teachers. Similarly, most graduate doctoral programs place scant emphasis on preparing students to be effective at what most of them will do for their entire career—undergraduate teaching. All too often, new teachers embark on their careers with serious deficits of preparation in broad areas such as curriculum development, problem solving, and connections between mathematics and other disciplines.
Despite widespread efforts to establish effective standards for curriculum and instruction in school mathematics, undergraduate mathematics programs frequently perpetuate modes of delivery that are ineffective for most students and choices of content that are inappropriate for most prospective teachers. Only when college faculty begin to recognize by deed as well as word that preparing school teachers is of vital national importance can we expect to see significant improvement in the continuity of learning between school and college.
Few college mathematicians pay as much attention to advances in the study of teaching and learning as to advances in mathematical research. It is rare to find mathematics courses taken by prospective teachers that pay equal attention to strong mathematical content, innovative curricular

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materials, and awareness of what research reveals about how children learn mathematics. Unless college and university mathematicians model through their own teaching effective strategies that engage students in their own learning, school teachers will continue to present mathematics as a dry subject to be learned by imitation and memorization.
that they can appreciate how their students are taught before and after they teach them. These teachers participate in hands-on problem sessions to learn how students learn, and they practice giving presentations that they can use to instruct their fellow teachers at their home schools. The success of the Bay Area Mathematics Project led to the American Mathematics Project, a collection of over 20 similar projects throughout the country jointly sponsored by the MAA and NCTM.
A similar concern must be expressed regarding the experiences of the graduate students who will become the next generation of college teachers. Only infrequently—more by accident than by plan—does the education of our future college teachers provide models of appropriate instructional techniques as well as intellectually challenging opportunities to address issues of how mathematics is taught and learned.
Outmoded Values
Because research institutions tend to establish values for the discipline, any reform of undergraduate mathematics education must address the question of how those values can be reshaped. Although attitudes are gradually changing on some campuses, it is still the case that at many institutions, teaching and research are viewed as competitive activities. In those universities whose basic charge is research and the granting of advanced degrees, academic survival depends on published research; in such institutions, undergraduate teaching offers few rewards. In contrast, at many colleges teaching loads are heavy and time for professional life is limited. The result is a schism in the profession that sunders the mathematical community into separate but decidedly unequal sectors.
"One-third of the first and second year college students in the United States are enrolled in two-year colleges, including over two-thirds of Afro-American, Hispanic, and Native American students. It is clear from these figures that any effort to strengthen the undergraduate mathematics major, especially to recruit more majors among minority students, must be carried out in a manner that includes two-year colleges as a full partner...."
—Challenges for College Mathematics: An Agenda for the Next Decade
Due to constraints on resources in many institutions, there is little mobility for faculty. Funding for research, which could support visits to other institutions, is very scarce. Financial support for activities that would contribute to the improvement of teaching is even more difficult to obtain. Thus many mathematics faculty members—especially those at smaller institutions—are professionally isolated, unable to keep abreast of modern developments in either teaching or research.

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Invisible Instructors
Effective Cooperation
At Austin Community College in Texas, nearly nine out of every ten students who transfer to public colleges and universities are still enrolled one year later. This measure of "one year survivability" of transfer students is nearly 40 percent higher than the state-wide average. Transfer success comes from paying attention to curriculum.
Almost every course at the college has been structured so that it will transfer directly to the two large neighboring state universities. The operative philosophy is that students should never find comparable courses at the community college easier because content is reduced or standards diluted. Classes should be easier only because they are smaller in size, because the faculty is more accessible and effective, or because the atmosphere is more supportive. The faculty is firm in its standards but caring in its approach to students.
College faculty have frequent contact with faculty at nearby universities, sometimes even influencing the university curriculum structure or choice of text. For example, a special program in mathematics, in place for over a decade, brings Austin Community College onto the University of Texas campus two nights a week to teach college algebra to university students. For these students, the university counts hours of enrollment at Austin Community College toward a student's minimum full-time enrollment obligation at the university.
The great majority of mathematics faculty in U.S. colleges view themselves primarily as teachers, not as researchers. Moreover, most of these teachers' time and energy is devoted to developing the mathematical power of students who never will use higher mathematics. What their students need is quantitative literacy sufficient for life—for trades and vocations, for public affairs and private lives. The mathematical education of the great majority of students in American higher education is confined to introductory courses taught most often in two-year colleges or in the margins of university departments of mathematics.
The infrastructure of U.S. business depends on the fruits of these instructors' efforts. Their students will become the technicians and practitioners that support American manufacturing and transportation, farming and commerce. More often than not, minority students get their first chance at higher education in a two-year college, or in a small regional college; many such students are first-generation college students. Their access to advanced degrees depends on the momentum provided by this initial experience with higher education.
Yet in the world of higher education, instructors who undertake these tasks are virtually invisible. With rare exceptions, no graduate programs in mathematics focus on the task of preparing individuals for careers of teaching introductory, vocational, and technical mathematics. Few natural career paths exist for those who teach in two-year colleges, nor is there any visible reward system for those who devote their lives to this most fundamental work. In reality, the "mathematical community" is not really a community at all, since many of those who do the most important instructional work feel like outsiders in their own world.