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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States 6 POSITIVE LEARNING ENVIRONMENT Doctoral students need feedback and encouragement provided in a cordial atmosphere. Many talented students, uncertain of their talents, leave doctoral programs for want of reassurance or fail because feedback is too intimidating—outcomes that a positive learning environment can help prevent. In a positive learning environment, all students are viewed as potentially successful. How does a program create a positive learning environment, that is, an environment that provides the assistance, encouragement, nurturing, and feedback necessary to attract and retain students and to give them an education appropriate for their future careers? Perhaps the most important step that a department can take is to cluster faculty, postdoctoral associates, and students in specialized areas, as indicated in Chapter 5. Clustering has been common in other sciences and engineering for a long time but has only recently been advocated in the mathematical sciences. Although mathematics is an individual activity and scholarship requires time spent alone, doctoral students and postdoctoral associates benefit from opportunities to discuss, to ask questions, and to try out ideas on interested and knowledgeable colleagues. They must learn to communicate mathematical ideas as well as to discover them, both of which require regular social and intellectual interaction. Clustering of students becomes increasingly important as the courses and research experience of the students become more specialized. Even when the student has a thesis advisor as his/her principal research mentor, the student still needs to be part of a group of students, postdoctoral associates, and faculty working in the area. The department, while large as math departments go, nonetheless concentrates its strength in certain distinct areas of research without trying to cover the waterfront. Consequently a dissertation student is teamed not simply with a thesis advisor, but also with an active cluster of faculty and students with related interests, involvement in ongoing seminars, and strong interactions. Committee Site Visit Report COMMUNICATION AND COOPERATION Effective communication is vital to a positive learning environment. Students need to know how the system works, how and when they will be evaluated, and how decisions are made regarding academic progress and financial support. An orientation meeting for new
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States students and a booklet summarizing the program are useful. Also important is a sense of the faculty's being accessible. The graduate students showed extraordinarily high morale and cohesiveness, and a shared ethic of hard work and high performance. Students repeatedly testified to the concern and accessibility of the faculty and to the mutual support and concern of the students for each other. There was remarkably little evidence of the ill effects of student competitiveness that one finds in many other programs. Committee Site Visit Report The principal attraction for the great majority of students, beyond the reputation of the program, is the friendly, open, cooperative, non-competitive atmosphere. The instructors are approachable, and the students form study groups that are gender-integrated. Committee Site Visit Report The factors that lead to success in attracting and retaining women and underrepresented minorities are the same as those for domestic students in general: a strong and focused mission, a positive learning environment, and relevant professional development. The positive learning environment seems to be the most important of these three factors. The atmosphere is quite inspiring, because the entire department seems to share a common philosophy. There is little talk about standards, but rather a positive, upbeat atmosphere that seems to foster confidence, professionalism, and mutual support among the students, particularly the women and minorities. Committee Site Visit Report Having the offices of faculty and students in one location and providing a “commons” area both encourage an open atmosphere. Their major problem is that they are not provided with office space and thus have little opportunity to interact with other graduate students, especially in the evening. Committee Site Visit Report Communication with colleagues as well as about a program's requirements and procedures is also important for postdoctoral associates and junior faculty. They need to understand clearly the terms of employment, including what is expected of them and when and how decisions will be made on reappointment, promotion, and tenure. Junior faculty need feedback as well as encouragement from senior faculty. It is important for the chair of the department to meet regularly with them.
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States EFFECTIVE ADVISING On the basis of previous academic record and placement examinations, all incoming students should be placed in courses of an appropriate level. Remedial courses should be prescribed where necessary. “Blanket” solutions to the problem of underpreparation of domestic students that are perceived as not being based on the performance of the students may not succeed. For example, one program that the committee visited set up a “zeroth year” for all students without previous graduate study, a category that includes most domestic students, but it has had little success in recruiting students into it. The committee observed that, in some larger standard programs, a “Darwinian struggle” is characteristic of the first two years, with the faculty accepting only the survivors and the graduate students being guided only minimally in their course work and in their learning how to perform research. In such programs, many faculty members do not exhibit concern about whether graduate students actually finish their degree programs. The seriousness of the problem is underscored by the fact that, in the mathematical sciences, only 1 out of 20 graduate students achieves a PhD, whereas in the physical sciences, 1 out of 10 graduate students receives a doctorate (NRC, 1990b). All students need to be advised about courses and research opportunities available in a department and about the expectations of the department. Advising on entrance to the program helps students to select courses of an appropriate level and to understand the program. During subsequent years, advising should point students toward the required exams. When the interests or abilities of a student do not match the mission or expectations of a program, other possibilities such as transferring to another program or career changes should be suggested. Many students can and do succeed when placed in a more suitable program. Attrition, if it occurs, should occur early. It is during the first two years that an appropriate framework is most important, for it is here that a decision will usually be made as to whether the student is to continue for the PhD or not. An experienced supervisor will probably have little difficulty in deciding by the end of this time, but the student must be able to see the decision as just and fair, as indeed so must others working in the same department. (CGS, 1990, p. 3) Because of the difficulties of selection, it is imperative that doctoral programs evaluate students' progress, identify those students who should make other career choices, and help students who are experiencing correctable problems. Where attrition occurs unnecessarily, students have been mistreated. When attrition occurs unnecessarily deep into the program, students, faculty, and institutions are expending resources to little or no good purpose. (AAU, 1990, pp. 9–10)
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States For advising to be successful, its importance must be recognized. Programs must make certain that enough faculty act as advisors, especially for entering students. Additional effort to ensure thoughtful advising can lead to higher morale and more efficient use of time and resources. The tendency in some programs to saddle the same small group of faculty with responsibility for advising beginning graduate students is shortsighted and can be very costly to the program down the line. Students get lost in the shuffle, and supervision and counseling are often deficient. This affects both the first-year students, many of whom are vulnerable and in need of some confidence building, and later also some of the dissertation students. Committee Site Visit Report COURSE WORK AND SPECIALIZED STUDY Course work is a necessary part of an American doctoral program because most students are not adequately prepared in basic mathematical sciences to proceed directly to research when they start graduate school. Although the horizons of the mathematical sciences have been expanding over the past 50 years, many standard doctoral and postdoctoral programs have continued to focus only on traditional core subjects. In these programs, students are often fit into rigid classical programs based on the specialized research interests of the faculty. As a result, the doctoral curriculum at many institutions lacks sufficient breadth to enable the graduates to teach basic undergraduate mathematics in an effective and innovative manner. The number of mathematics courses taught outside of mathematics departments at the advanced undergraduate level exceeds those offered within mathematics departments. According to one recent report, more (173,237 as compared to 147,000) students are taking advanced work in mathematics outside of mathematics departments than within (Garfunkel and Young, 1990). Graduates are also often unable to engage in interdisciplinary work as a member of a team. Students who limit their studies to mathematics departments are often ill prepared to take positions in industry. Such jobs are often held by people in science or engineering with liberal amounts of mathematics training. To achieve broader experience, students may need to look outside of mathematical sciences departments. Students can be encouraged to take courses, even at the undergraduate level, in other departments to become familiar with applications. Equally important, beginning course work in the mathematical sciences must be at a level appropriate for the students in the program, and remedial instruction for underprepared students should be provided to fill gaps. Advanced courses and research-level courses are critical for students not merely in reaching the frontier of their specialty but also in choosing a thesis advisor. Student seminars with active faculty participation can be a very effective addition to advanced
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States course work. Students in such a seminar can learn the material in an active way and also learn how to communicate it. Specialized study in one or two areas just before research on the dissertation is begun should involve course work, seminars, and individualized study. Clusters of students working in the same area are increasingly important at this stage. Programs in which faculty direct this specialized study find that the often-difficult task of pairing students with thesis advisors is thereby eased. EARLY RESEARCH EXPERIENCE In a standard-model PhD program in the mathematical sciences, research experience is usually postponed until after the second qualifying examination. Most doctoral programs in other sciences and engineering provide exposure to research much earlier. Students work in groups or clusters with academic researchers as a part of a team. 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. (NRC, 1990c, p. 64) Early research experience through problem solving, experimentation, or computation gives students a better idea of how to create and apply mathematics and often provides additional motivation. Many of the specialized programs visited by the committee provide early research experience to their students and, partly because of this, have high completion rates. The committee believes that all doctoral programs, both pure and applied, would benefit from integrating opportunities for early research experience into the program. MASTER'S DEGREE PROGRAMS Viable professional master's degree programs in pure and applied mathematics are not common in the United States. Master's degrees granted in standard doctoral programs are often considered consolation prizes for students unable to achieve a doctorate. This is in contrast to the situation in Europe, where a master's degree (Diplom, diplôme) in mathematics is a valued professional degree. A professional master's degree program would be an excellent complement to a doctoral program. Such a master's degree program could have one of the specialized missions described in Chapter 5. It could be an excellent means for students to gain confidence in their ability without committing themselves to an open-ended doctoral program and
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States would be attractive to them because of the value that government, business, and industry could place on a properly structured and specialized professional master's degree. Some PhD students in engineering and social sciences earn master's degrees in mathematics or statistics to raise their status. A professional master's degree program could be an important source of doctoral students who might otherwise choose to study some other discipline. Finally, a professional master's degree program can provide through its alumni excellent ties to industry and to elementary and secondary education. Appendix C describes how professional master's degree programs could enhance doctoral programs and fulfill needs for mathematical sciences personnel in government, business, and industry. QUALIFYING EXAMINATIONS The first qualifying examination sets the tone for the rest of the predoctoral period. Currently, it often has a negative effect even on those students who readily pass the exam. Sometimes it is viewed by all as the gate to keep out the unworthy, those who should not have been admitted to a program in the first place. In standard-model programs, the examination is often based on a limited number of traditional areas that are not necessarily connected with the student's area of future research. The first exam is seen by first-year students to be a formidable hurdle, with high mortality. It is a source of considerable anxiety, and many students felt that there is insufficient counseling and support given by the department in the first year. Committee Site Visit Report It does not, however, appear possible to receive a broad education in applied mathematics. The first exam is based entirely on core areas, a condition that prompted one applied mathematician to comment, “The program requirements are debilitating to applied mathematicians.” Some of the current fellowship holders commented that there is a “just you wait …” approach to the utility of mathematics, that “there is little or no contact with other departments,” and that “the faculty are disdainful of non-math courses ….” Committee Site Visit Report The mathematical sciences are broad, and the first examination should reflect that breadth in both core and applied areas. Core programs benefit from having applied areas represented in the examination. Programs specializing in applied areas should permit inclusion of some core areas on the first exam. The first qualifying examination should be a diagnostic tool that assists faculty in deciding whether doctoral study is for specific students as well as to assist the students in making
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States that same decision for themselves and in determining whether they are ready to proceed or need additional time and study. Most students who put forth sufficient time and effort should be able to pass the examination, and the faculty should be there to assist. The committee was amazed at how such an approach can transform a doctoral program, an approach that need not be inconsistent with the other purpose of the first examination, namely, to assist students in deciding whether doctoral study is for them. Making assistance and guidance available to students studying for the first qualifying examination can have very positive results. A useful support instrument is a first exam workshop, set up by advanced students, which runs group problem sessions for first-year students. These workshops have been extremely valuable, not only for their instructional content, but also as a setting in which students get to know each other more informally. Committee Site Visit Report The second qualifying examination can be written or oral but should involve more than one faculty member. This examination certifies that a student is ready to start work on a dissertation. It should permit specialization in the direction of the future research of the student. The second exam is an oral exam, based on a topic selected by the student in consultation with a faculty member that may, but need not, turn out to be the student's advisor. This system seems to reduce much of the anxiety usually associated with finding a thesis advisor, as the necessity to form a working relationship with a member of the faculty is built into the program at an early stage. Committee Site Visit Report Research and Thesis Mathematical sciences graduate students, like those in other science and engineering disciplines, are an integral part of the academic research enterprise. However, the faculty in standard mathematical sciences programs are less dependent on graduate students to sustain their research projects than are the faculty in the other sciences and engineering. In the mathematical sciences, research apprenticeships for graduate students before they begin work on a dissertation are not a standard part of the program. When students reach the dissertation stage of their studies, advising is done by their thesis advisors. Currently in many standard programs, selection of a thesis advisor is an idiosyncratic process.
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States Students must work to find a thesis advisor. There are almost no research assistantship inducements, and most faculty do not advertise that they are seeking or have room for a thesis student. Students often feel intimidated about approaching faculty members about their research. This is probably not due to excessive reserve on the faculty's part, but there is also very little outreach. Committee Site Visit Report By designating a faculty member as an advisor for specialized study and the second examination, a program can facilitate the process of matching a student with a thesis advisor. A program-sponsored forum for faculty to discuss their research interests can also help, as can opportunities for students and faculty to meet informally. A program should monitor both the research progress and the relationship between the student and the thesis advisor. The thesis advisor has responsibility for the progress made in the research of the student, including attendance and participation in colloquia and conferences. The thesis advisor also has responsibility for the student's professional development, including his/her learning how to communicate orally, how to publish, and how to teach. There are two aspects to supervision [by the thesis advisor]. The first and more important has to do with creativity and involves the ability to select problems, to stimulate and enthuse students, and to provide a steady stream of ideas. The second aspect is concerned with the mechanics of ensuring that the student makes good progress. (CGS, 1990, p. 1) Although the thesis advisor has primary responsibility for the student completing the dissertation, the program also has a share. In addition, the student still needs to be part of a cluster of students, postdoctoral associates, and faculty working in the same area. The time necessary to move through the various stages of a doctoral program varies from student to student. Flexibility in time requirements is important, often critical, to students with family or other obligations. When more is expected of students and when assistance in reaching the higher standards is provided, students achieve more. At the dissertation stage nothing should be done to diminish the necessary rigor of the research apprenticeship, but much can be done to minimize unnecessary frustration and to improve the process. (CGS, 1991, p. 32) The thesis defense should be used to demonstrate a student's expertise and to show other students the level of expectation and accomplishment in the program. Attendance by faculty in addition to the appointed committee is important.
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States For the student, the defense should be a “crowning experience,” the ultimate opportunity to demonstrate his or her expertise after years of research, reporting, and writing. It is also excellent preparation for future professional presentations where defense of one's work is an accepted part of standard professional meeting structure. For other doctoral students who attend the examination it is a learning experience, conveying guidance on the formulation and completion of a dissertation project. (CGS, 1991, p. 28) POSTDOCTORAL FELLOWSHIPS Education does not end with completion of a PhD. It is very useful to the new PhD and to society in general to extend the learning period by a few more years. However, the opportunities to do so in the mathematical sciences are limited. The number of postdoctoral positions available to mathematical scientists is far below those available in the other sciences. In 1988, there were only 188 federally funded postdoctoral positions in the mathematical sciences vs. 1280 in physics and 2587 in chemistry (NRC, 1990c). In the physical and biological sciences many new doctorate degree holders seeking an academic career start with a three-year postdoctoral position, most often federally supported. This position is viewed as the logical next step after completion of the doctorate for the good student, not as a highly competitive prize for a select few. In the mathematical sciences, postdoctoral fellowships are viewed as prizes for the very best rather than as a logical next step in research for many new doctorates, so that mathematics PhDs from American graduate schools that are not among the elite included in a postdoctoral-fellowship or research/instructorship program are often left woefully unprepared to compete with new PhDs from abroad. The restricted opportunity for postdoctoral education in the United States contrasts sharply with the practice in the European mathematical community. Although European undergraduates specialize earlier than their American counterparts and hence know more mathematics when they begin graduate study, many European countries still provide an extended research apprenticeship program for new PhDs to deepen their research experience. The problem of limited opportunity for postdoctoral education is compounded by the fact that departmental committees selecting postdoctoral associates often focus on attracting the best new doctorates, independent of a candidate's field. The “postdocs” are appointed predominantly on the grounds of scientific achievement, independent of field. As a result many of them arrive with no natural scientific bonds within the department. Committee Site Visit Report
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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States Such an approach often results in a lonely, isolated tenure with limited departmental interaction and little if any “postdoc”-mentor relationship with the senior faculty. Postdoctoral associates should be hired in the areas of expertise of the program, not in other areas. Each should have a mentor from the senior faculty and be a member of a group of faculty, postdoctoral associates, and doctoral students working in the same area. The postdoctoral positions that are currently available are usually designed to deepen knowledge in a specific area of research. One could design postdoctoral fellowships that could be used to advantage to broaden the knowledge base of the fellow or to expand the fellow's expertise in teaching or applications. The shortage of opportunities for postdoctoral positions places added responsibilities on departments to help new faculty who have not had a postdoctoral fellowship to continue growing toward their full potential. This is especially true for new faculty with temporary appointments. A POSITIVE LEARNING ENVIRONMENT FOR ALL PROGRAMS The clustering of faculty, postdoctoral associates, and doctoral students that typically takes place in a specialized program is highly beneficial for the learning environment. Nevertheless, all doctoral/postdoctoral programs, including standard programs in core areas, can achieve a positive learning environment and can benefit from it.
Representative terms from entire chapter: