A. Vision, Objectives and Impact
B. Description of Core Components
1. The Tetrahedral Model
2. Postdoctoral Fellows
3. Graduate Traineeships
4. Undergraduate Research Experiences
5. Curriculum Review
a. Case Study Modules
b. Multi-layered project courses
c. Undergraduate Options
C. References
Graduate education in the mathematical sciences has proved to be highly effective in training students for a career in an academic setting. However, more than half of the Ph.D. recipients now find careers in nonacademic settings (Griffiths 1995). The workplace has become more interdisciplinary, collaborative and global, and requires that education be broad (Armstrong 1996), emphasize teamwork and train for communication skills. Below it is shown how the educational and research program in Applied Mathematics at the University of Colorado is already implementing many of the recommendations of the Griffiths report on "Reshaping Graduate Education" (Griffiths 1995); however, the Department certainly has room to improve.
The VIGRE grant will permit us to reshape the training of Applied Mathematical scientists at the University of Colorado, by integrating the training program among faculty, postdoctoral fellows, graduate students and undergraduate students, and focusing on key computational, communication and analytical skills.
The sections below describe the structure of the training program here, beginning with the undergraduate engineering major in applied mathematics through our training of postdoctoral mathematicians. The proposed program is oriented around Tetrahedral Research Groups, consisting of a faculty mentor, a postdoctoral fellow, several graduate trainees and undergraduate majors. The facets of interaction in this group will include
Computation is already being integrated into several of our courses, through the development of Case Study Modules. These will be extended under the VIGRE program. Moreover, this proposal introduces an outreach program, Hands-on Applied Mathematics to take the case study modules to middle and high school teachers in the NSF funded CONNECT systemic reform network of Colorado school districts.
In addition, this proposal enlarges the successful Affiliated Faculty program of the Department. Currently this consists of 35 faculty members at the University and nearby universities who are qualified to lead a Ph.D. thesis for graduate students in Applied Mathematics. This program will be extended to government laboratories and technology companies, such as LANL, LLNL, NCAR, NIST, Ball Aerospace, etc. Many of these are part of the Boulder environment that makes our University a unique educational and research institution.
The VIGRE grant would support four Postdoctoral Fellows. They will receive mentoring from a faculty advisor and teach one course each semester for the Department. Twelve Graduate Trainees will be funded by the proposal. These students will participate in the research tetrahedra, be involved in research proposal development, participate in the development and implementation of case study modules, and receive teacher training through the Department's Teaching and Learning Seminar and the University's Graduate Teacher Program. The proposal will fund four undergraduates each year for Research Experiences. This includes participation in one of the tetrahedra part-time during the academic year, and full-time during the summer.
The proposed initiatives will be sustainable and lead to a number of
permanent structural changes in the Department: successful tetrahedral research
groups will endure and be emulated, case study modules will form an essential
part of our curriculum, and the extended affiliated faculty will result in
a continuing option for training of our students in application areas.
1. The Tetrahedral Model
Vertical integration of the education and research program of the Department of Applied Mathematics will be facilitated through the creation of Tetrahedral Research Groups, each consisting of a faculty mentor, a postdoctoral fellow, several graduate trainees and an undergraduate applied mathematics major. In the subsections below, the impact of this program on each of these individuals is described; in this section the collective activities of the groups are discussed.
Though it is traditional for a faculty member in the mathematical sciences to work individually with graduate students and postdoctoral fellows, research groups are not a standard feature of academic training in this field. Nevertheless many of our current research projects require an interdisciplinary effort among a number of individuals, especially when such projects involve the development of computational tools or the study complex mathematical models of physical systems. A successful research project then requires collaboration with scientists in the modeling process, computational mathematicians in the algorithm development, and analysts in the interpretation and generalization of the results.
The tetrahedral groups will facilitate the acquisition of computational, communication and analytical skills to prepare students for the pivotal fact that "continuous change" is a "central feature of contemporary life" (Griffiths 1995). These skills are already a core focus in our undergraduate major and our graduate training program. For example, each undergraduate and graduate student takes a year long course sequence in computational mathematics. Moreover, computation is broadly used in other courses, such as those in differential equations, linear algebra, dynamical systems and modeling. The Case Study Modules program (see Section 4B.e) is part of our effort to emphasize communication skills in our lower division courses from calculus to differential equations; here students are presented with several projects during the course, and are required to produce a cogent and coherent report. One of the missions of each of the tetrahedra will be the development of case study modules based on research level problems of pedagogical value. In addition, the tetrahedra will develop communication skills by writing reports and proposals. The acquisition of analytical skills is the central result of a mathematical training which "provides methods for organizing and structuring knowledge so that, when applied to technology, it allows scientists and engineers to produce systematic, reproducible, and transmittable knowledge."(Glimm 1991). Our VIGRE program develops these through the tetrahedral interaction among the members of training groups.
The tetrahedra in the VIGRE program will be organized around research in four areas that comprise major focus areas for the Department (here are listed Departmental faculty who could be mentors for the tetrahedra):
Each tetrahedral group will hold a weekly research seminar for members of the group, which is also open to others interested. In the seminar, participants present the status of their work, study journal articles in their area, plan the direction of their research efforts, and collaborate on the development of research proposals and papers.
As part of their educational mission, each tetrahedral group will collaborate in the development of a case study module for one of the lower division calculus or differential equations courses. Modules can involve either a computational or experimental component. They are given to the students as special projects that extend or enhance the regular course content. Students work either in small groups or alone, and produce a word processed report with their results clearly tabulated, explained and placed in the context of the course material. The contribution of the tetrahedral group could include, for example, Java Applets that demonstrate a concept, Mathematica notebooks that lead the students through a complex calculation, or Matlab scripts for data analysis.
In the sections below, we give a detailed discussion of how the tetrahedral
groups will facilitate the development of research and educational skills
for the postdoctoral fellows, the graduate trainees and the undergraduate
research assistants.
Funds are requested for 4 Postdoctoral Fellows (PF's). Our main goal for these is to help young, applied mathematical scientists to further develop their potential for becoming outstanding faculty members in academia or research scientists in industry or government. The Department has previously had Postdoctoral Researchers (focusing on research and often without regular teaching responsibilities) and Postdoctoral Instructors (who have a three year appointment with regular teaching duties of two courses per semester) The training of the proposed PF's would be more beneficial to young scientists than the other forms the Department has had the past through its novel balance between moderate teaching load (one course per semester), emphasis on research programs and enhancements of communication skills and professional development including the writing of proposals.
Our current Instructor program has by all measures been successful. In this ongoing program supported by teaching funds from the University, two outstanding young mathematicians become members of the Department for up to three years. They are selected for both their teaching and research potential, with the aim of enhancing their teaching skills while exposing them to an active research environment. In this program, these young scientists have become effective teachers, have published in refereed journals and have interacted well with faculty and graduate students (possibly less systematically with undergraduate students--an aspect to be remedied through the proposed tetrahedral model). All of our instructors have obtained tenure track academic positions (see Appendix 1). The teaching support and mentoring offered by the Department have been successful in developing the capabilities of these instructors. One of the main objectives of the proposed plan for PF's is to offer improved opportunities in a range of areas while reducing the teaching load to one course per semester. This will allow the newly hired PF's to develop a more individually focused research program, while maintaining--and enhancing--the invaluable teaching and mentorship aspects.
The primary selection criteria for the PF's will be demonstrated research and teaching potential, research and dissertation work that indicates an emphasis and interest in our multidisciplinary objectives, and strong indications of their ability to perform research in a broad scientific environment.
Besides teaching one course each semester, the PF's will be active participants in the Teaching Excellence Program at the University. As such the PF's will have educational experts evaluate their teaching, have videotapes made of some of their lectures, and learn how to construct a teaching portfolio. All of this goes far beyond the standard Postdoctoral Instructor evaluation process. Funds from the grant will be used to significantly enhance the current Teaching and Learning Seminar, which is already strongly committed to developing excellent teachers in the mathematical sciences. The seminar focuses on classroom experiences, professional development and other topics considered key for the successful mathematical sciences professional of the future.
Postdoctoral Fellows would also be exposed to and involved with proposal writing and the associated subprocesses--gradually taking more individual responsibility in this area. This component of professional development has often been overlooked in the past--we will address this shortcoming. The PI's will also be charged with assisting the PF's (and graduate trainees) in obtaining suitable positions when they leave the university.
By having a one course per semester teaching load, the PF's will be in
an excellent position to develop a strong multidisciplinary research program.
At the same time, guidance will be provided so that the PF's will be able
to develop outstanding research and teaching skills in a nurturing setting
(which includes unusual opportunities for external links through the multitude
of organizations such as NCAR, NOAA, NIST, etc., located in Boulder or nearby--see
appendix 2). Existing contacts with these--and with local industry--will
be expanded under this proposal. When the PF's are ready to advance to regular
faculty, research scientist or industrial positions, the goal is that it
will be with an established research record, established course development
skills and a realistic sense of what it means to be an active researcher in
a multidisciplinary scientific environment.
Funds are requested for 12 Graduate Trainees (GT's). The graduate program of the Department has grown from zero when it was founded in 1989 to 51 currently. We expect, even without external funding, that the program will grow to a maximum of 65 students, primarily because research funding for graduate students through our affiliated faculty program has been so successful. The requested funds will not replace current university funding for Teaching Assistant positions in the Department (there are 17 regularly funded TA positions, supporting our teaching of four semesters of calculus and differential equations for engineers)--indeed, the teaching activities supported by the VIGRE proposal will allow us to significantly enhance the educational program of the Department, through the development of case studies and other multi-layered project courses as we discuss below. The VIGRE funds will replace and enhance those of the NSF graduate traineeship program ending in 1998, which currently funds 7 students.
Our current graduate program has been by many measures successful. The eleven students who have earned Ph.D.'s in Applied Mathematics have advanced to careers in fields that use their mathematical training in an essential way (possibly with the exception of one student, who is now in Law school). Though the graduate program received few applicants in the first years after it began in 1989, it now typically receives more than 70 applications, and competes successfully for students from well-established programs in Applied Mathematics.
The average time-to-degree for our Ph.D. students has been 5.1 years. That the Department has had relatively few graduates reflects only that it has been in existence for 8 years, and had a small number of students during the early years.
The educational program at the graduate level is focused towards the acquisition of computational, analytical and communication skills that give the students a broad background for a professional career in the mathematical sciences, whether it be in an academic or an industrial setting. These courses include year long sequences in Numerical Analysis and Applied Analysis. Students can also choose a third sequence in either differential equations and approximation methods or statistics and probability. Advanced courses in our department include dynamical systems, nonlinear waves, stochastic processes, advanced computational algorithms, computational fluid dynamics, etc. Here is a full list of courses and many of the courses have individual web pages.
Along with core applied mathematics courses, each student is required to choose an application area external to the Department in which they take a high level sequence of courses; for example, students can take mathematical courses in engineering, the physical and biological sciences, or business. Typical areas that some of our students have chosen to study include parallel computer architecture, geophysical fluid dynamics, analysis of gene sequences, optimization in finance, topology, signal analysis and combustion dynamics.
In some cases, this study leads to the choice of a Ph.D. thesis topic, often supervised by one of the affiliated faculty members: thirty-five faculty members of the University or nearby universities, whose research programs actively involve applied mathematics. Each of these faculty is qualified serve as co-advisor (to assure that the thesis project satisfies the necessary requirements) of a Ph.D. thesis for our students. Recent theses based on this program include areas such as surface chemistry, solar modeling, and gas dynamics. The affiliated faculty, who support about one third of our students, are uniformly excited to work with students who have the mathematical and computational sophistication to contribute to their research programs, unlike many of the students in their own departments.
The success of this program is exemplified by David Sholl who studied under Professor Skodje of the Chemistry department and obtained a Ph.D. in Applied Mathematics in 1996. His thesis was based on mathematical modeling of surface chemistry; the approaches combined statistical and dynamical methods with a strong computational component. His excellent research record lead to a postdoctoral position in chemistry at Yale University; David is currently an assistant professor of Chemistry at Carnegie-Mellon University.
The VIGRE program will permit the Department to expand the affiliated faculty to include local research laboratories and technology companies (see appendix 2)--as recommended by the NSF Report on "Graduate Education and Postdoctoral Training"(Armstrong 1996). Boulder is truly distinguished in being the location for many research laboratories, such as NCAR, NIST and NOAA, many high technology companies, such as IBM, Storage Technologies, Ball Aerospace, Sun Microsystems and numerous small research and development companies. Moreover, we have already developed research relationships with scientists at local and regional laboratories such as LANL and LLNL to support graduate research projects.
This affiliation with laboratories and industry will allow us to implement the recommendation of the Griffiths report for a "third option" beyond the terminal masters and traditional academic Ph.D. thesis (Griffiths 1995). In this program a student will develop a research project based on a problem proposed by a mentor at one of the local laboratories or companies. This project will become a portion of the Ph.D. thesis for the student, which will be supervised jointly by the outside mentor and an applied mathematics faculty member. Apart from producing a thesis and resultant scientific journal articles, the student will also spend time with the mentor to develop strategies for effective communication of the results in the laboratory or industrial environment.
Even GT's that do not choose to work directly with a laboratory or industrial affiliated faculty member, will participate in this program by "shadowing" one of our external affiliated faculty members for several periods during their regular work schedule to observe and interact in a job environment similar to one they could pursue after their degree.
Each GT will also develop teaching skills though participating in the University Teaching Excellence program where they can receive expert instruction about teaching strategies and methodologies through videotaping, individual instruction and seminars. Each of the graduate trainees will also participate in the Department's teaching and learning seminar. The main goals of this course are to help teaching assistants in their first year, to prepare students for jobs in a university setting, and professional development. These are attained by reading mathematics education articles, inviting guest speakers, and by discussing teaching related issues, course development, and problems that arise in the recitations. In addition, through the Preparing Future Faculty (PFF) network our graduate students gain first hand knowledge about the actual duties of a faculty member at a suite of institutions including community or small colleges, nonresidential institutions and engineering schools.
The GT's will initially teach a recitation section for one of the engineering calculus or differential equations courses. The VIGRE funding will permit the GT's to teach a single recitation of 25 students, reduced from the normal load of 3 recitations for a TA. The GT's will be supervised by the coordinating professor for the course, as well as the lead TA and professor who run the Teaching and Learning seminar. During their second year, the graduate trainee will facilitate the development of a case study module by their tetrahedral group for the course that they taught in the first year. They will serve as a laboratory assistant for this course, and interact with the undergraduate students in a computational lab setting, as well as individually for the duration of the projects. After passing the comprehensive examination, and being advanced to candidacy for the Ph.D. degree, the University permits graduate students to become Graduate Part-Time Instructors (GPTI's). At this stage, the GT's will have full teaching responsibility for a small section of their course, again under supervision of the coordinating professor. This program will permit the Department to institute smaller sections of its calculus courses, as well as give valuable teaching experience to the GT's.
The GT's will be selected by the Graduate Committee for their academic skills and for their interest in and suitability for research in one of the focus areas of the VIGRE program. However, it is rare that a student will have decided upon a specific research area when they apply to graduate school, so they will not be initially assigned to a specific tetrahedral group. Each GT will be initially advised by the Chair of the graduate committee. During their first year, they will attend research group meetings of prototype or existing tetrahedra, and will be recruited for a tetrahedral group based on their indication of preference to begin in the first summer.
As part of the VIGRE program, the Department will continue its efforts to track the career path of its graduates. Moreover, we plan to devise an outcomes questionnaire that will determine which portions of the course work and educational experience of each student were most and least useful to the attainment of career goals. This will be used to facilitate the production of career advice for our students, as well as to tune the graduate program to better serve our students.
The graduate program has been successful in recruiting woman graduate
students (15 of the 51 current students are woman) though somewhat less successful
in recruiting minority students (3 of the current students are from minority
groups). Every effort will be made to improve these rates, and to develop
more extensive mentoring programs to improve the retention rates for protected
class students.
Undergraduate students who major in applied mathematics are students in the College of Engineering at the University of Colorado. The applied mathematics major is a rigorous, but flexible major) and requires the students to select an area of application external to the Department where they take a sequence of 24 credits (8 semester courses). The application area can be chosen from many options that include, for example, electrical, computer, civil and mechanical engineering, engineering physics, actuarial science, and biomedical engineering. Our students are of high quality: 50% are on the dean's list (GPA higher than 3.5), and many continue their studies at outstanding graduate schools.
The Department has an active undergraduate chapter of SIAM, which sponsors faculty-student lunches, an evening seminar series, and a web page contest. Moreover, the undergraduate computing lab acquired three Silicon Graphics O2 workstations with funds from the University based on a proposal written by the chapter. The chapter also sponsors teams to enter the undergraduate modeling competition sponsored by the Consortium for Mathematics and its Applications (COMAP). A team from our Department won the contest in 1992 (presenting their work at the OR Society international meeting), and last year our two teams both were awarded the "Meritorious" designation (of the 409 entries in all, our teams were ranked among the top fifth). This year the Department has also fielded two teams in the competition.
We propose to select four undergraduate students each year during the spring semester of their junior year for VIGRE Research Experiences. Each selected student will join a research tetrahedron during the summer and for their senior year. This "capstone" research experience will include a weekly research seminar, where the group discusses its problems and progress, and the development of writing skills through the preparation of a research proposal. For the academic year, the students will carry out their proposal, give regularly scheduled talks on the progress of the research, and will be paid on a 50% time basis. The experience will culminate with the student writing a scientific report on their research during the spring semester, and presenting the work in the SIAM undergraduate seminar to their peers. Each of the participants in this program will sign up for course credit in APPM 4840, Reading and Research in Applied Mathematics, supervised primarily by the Postdoctoral Fellows in the tetrahedra.
For example, we mention a recent experience for a research group including two of our faculty, Robert Easton and James Meiss, in dynamical systems, who were studying the problem of the onset of chaotic motion in dynamical systems. During the summer they hired a junior undergraduate (Sean Carver, currently attending graduate school at Cornell), using funds from the NSF REU program. The student was asked to develop a computer algorithm to detect the transport times for simple dynamical systems. Regular group meetings allowed the faculty to realize that the student's expertise far exceeded the mere "computer programmer" level. Together this group developed new diagnostics for transport (exit time distributions, the concept of "width," etc.), which culminated in writing a research paper, which was subsequently published, including the student in the list of authors.
This experience was repeated (though perhaps not quite as successfully) with other recent undergraduates including Laura Mather, Rupa Patel, and Eric Phipps. Moreover, for the past three summers we have participated in the SMART (Special Access for Minorities for Research and Training) program at the University. Together, these show that the Department has an excellent track record of taking our best students and working with them for the summer and for extended lengths of time.
In the VIGRE program, the undergraduate majors will also participate
in our curriculum development program, through the development of case study
modules for the lower division courses. Undergraduates have already been important
contributors to the Department's development, through our capstone Undergraduate
Seminar course, where undergraduate students have helped develop curriculum
and a critique of various efforts in the Department.
The traditional mode for education in the mathematical sciences is through the lecture: "revelations from on high." While there has been much discussion of modifying this at the lower division level, a goal of our VIGRE proposal is to use scientific computation throughout the curriculum to introduce students to a methodology that lends itself to solving problems of considerable interest, problems that often do not have solutions in their original formulation. This will be accomplished through the development of case study modules, the development of multi-layered projects, and through the addition of new options for our undergraduate majors. Each of these initiatives emphasizes the recommendation of the Glimm report "to support the teaching of modeling and of industrial applications of mathematics."(Glimm 1991),
The President of the four campus University of Colorado system has made the development of a "Total Learning Environment" a primary goal. The TLE's focus is to provide a more enriching and relevant learning experience for students by:
Since the Fall of 1996 we have been developing Case Study Modules for the fourth semester course, Differential Equations and Linear Algebra (APPM 2360). The four current modules extend and enhance the regular course content beyond normal homework assignments. Students work either in small groups or alone, and produce a word processed report explaining the mathematics of the project, its significance for applications, and carefully documenting their procedures and results. The modules are presented to the students during class time either by the instructor, by the laboratory TA, or by a professor in engineering or science whose expertise in the area demonstrates conclusively that mathematics is central to applications. Generally, students have one to two weeks to complete the module.
Modules can be either primarily computational or experimental. For example, one module involves an apparatus consisting of fluid being pumped through a series of reservoirs--similar to the classic example given in the word problems section of most differential equations texts. Students take data from the apparatus using digital flow and pressure sensors; they then examine how the data fit a mathematical model consisting of a coupled system of differential equations. Other experimental modules that we hope to develop include a demonstration of stability concepts using model airplanes in a wind tunnel, a demonstration of resonance using a tunable motor on a spring loaded platform, and a demonstration of chaotic behavior using a spherical rigid body floating on an air bearing. Each of these would be facilitated by our colleagues in engineering together with the new Integrated Teaching and Learning Laboratory (ITL) at the College of Engineering. For example, Professor Dale Lawrence in Aerospace engineering has a prototype wind tunnel that will be used for the module development, and Professor Patrick Weidman of Mechanical Engineering has a prototype air bearing. Dr. Trudy Schwartz of the ITL will coordinate the fabrication of the projects and the integration of the projects into the ITL workspace.
Additional modules have been developed for 2360 to introduce the students to numerical solutions of differential equations and the visualization of dynamical phenomena. Currently these use Matlab scripts and rudimentary Java Applets. As part of the curriculum reform aspect of the VIGRE program, and supported by the Engineering Excellence Fund of the University, we propose to develop web based Java Applets for the solution and visualization of dynamics so that the students can enter the equations in natural mathematical notion, and interactively control the parameters (through sliders) and initial conditions (through the mouse) for the system.
As part of the curriculum reform effort for VIGRE, we propose to extend the modules to other lower division applied mathematics courses, beginning with Multivariable Calculus (APPM 2350). In this course the development of visualization skills for surfaces and curves in three dimensional space is essential, and yet, much of the current teaching methodology involves sketching these objects on a two dimensional blackboard. The development of modules to demonstrate key concepts for this course will be done using mathematical software such as Maple or Mathematica.
Additional modules for these courses, and for other undergraduate applied
mathematics courses will be proposed and developed by the tetrahedral groups.
During their second year, each graduate trainee will lead their group in
the development of such a module.
The Department plans to continue the development of projects and modules in other courses, such as a first semester freshman course called Applied Mathematics: The Next Generation (APPM 1400) which will introduce students to the problems and methodology of applied mathematics through the exploration of mathematics that influences our everyday experiences, and Statistical Methods (APPM 4580/5580) which introduces students to statistical tools for data analysis.
In addition, we have been developing a senior level course, Modeling in Applied Mathematics (APPM 4380) with multiple goals:
At present, this course is taught as APPM 4380, i.e., a senior-level undergraduate course. Discussions have been held whether to bring it down to lower level, or to increase the level to make it appropriate for graduate credit. Reasons for lowering the level would be to use it as a motivational tool for students, allowing them to experience the utility of, say, calculus and differential equations before (or in parallel with) encountering these topics in other courses. Also, it would allow students to get their first exposure to many techniques in a less rigorous and more heuristic setting, helping their understanding in later and more formal courses. Reasons to increase the modeling course level would be to allow our M.S. and Ph.D. students to acquire modeling experience, thereby making them better prepared for the work place (the lack of available graduate credit now strongly discourages their participation).
This VIGRE proposal would allow us to entirely avoid the dilemma of whether to raise or lower the course level by doing both--offering simultaneously an undergraduate and a graduate version, based on the similar modeling tasks and the same text (which is structured so that reference areas are separated from the applications). We believe this course could be vertically integrated in this manner and it would then become an excellent vehicle for uniting the four tetrahedral units. A faculty member, and a mentored postdoctoral fellow would teach the duplicate courses (the undergraduate and the graduate versions), bringing the two courses into contact for joint discussions and presentations at strategic intervals. For the undergraduates, this would offer an unusual opportunity of experiencing combined education and research. Both undergraduate and graduate students would gain experiences in a range of applications of mathematical sciences, and how its techniques can be brought to bear in both academic and industrial environments.
Currently, undergraduate majors in applied mathematics select an applied option in which they are required to take eight courses (24 credits). There are nine options currently recommended by the Department,
As part of the Curriculum review under the VIGRE program, we plan to develop additional options that reflect career opportunities for students with training in the applied mathematical sciences. In conjunction with the external affiliated faculty program, we will learn more about the types of careers available to our students. Moreover, we are completing a survey of the graduates with applied mathematics degrees to learn more about the careers that they have chosen, and what aspects of their training that they found valuable and not so valuable.