SIAM Guidelines for a Professional Master's Degree

Executive Summary

These guidelines are in response to the growing interest in professional master's degree (MS) programs in applied and industrial mathematics. This interest arises from the demand for more broadly-trained, quantitatively competent professionals and from the success of in-place and emerging programs. The influential 1995 SIAM report, "Mathematics in Industry," (MII) documents a need for graduates who are trained in a combination of mathematics, applications, and computation.

A core curriculum is suggested of scientific computing, numerical linear algebra, statistics, applied analysis, as well as a course from among discrete mathematics, differential equations, or optimization. The core includes a concentration in a field of application and an internship in business, government or industry. A complete master's degree curriculum will include course work or training in presentation and writing skills.

The core curriculum is not intended to be a monolith. Examples of master's programs may be found on the web; a search of http://www.sciencemasters.com/students/ will produce several such lists. Successful master's programs demonstrate the wide spectrum of curricula possible. What they have in common is a suite of courses that includes the essentials of analytical, computational, and stochastic modeling; an intern experience; and provision for the learning of presentation and writing skills. These are the sine qua non of a master's program that offers a realistic preparation for working with engineers, scientists, or managers.

1. Introduction
   
   Two articles on the applied master's degree appeared in the September 1985 issue of SIAM News. One article noted that the MS degree is often treated by PhD-granting institutions as a mere way station on the path to a PhD. Unlike such fields as biology, engineering, and psychology, mathematics has not recognized the Master's as a terminal degree [3]. In the second article, it is noted that industry is not hypnotized by the mere presence of a PhD. Thomas Morrisette, with 15 years of industrial experience suggests that, "Most of my employers would regard a PhD as proof that its holder couldn't find, let alone solve, a problem in the 'real world'.[5]" Similar, though more diplomatic, statements can be found in the ground-breaking 1995 SIAM report, "Mathematics in Industry.[2]" This report documents a need for graduates who are trained in a combination of mathematics, applications, and computation, whatever the degree is. The MII report also makes it clear that an applied master's degree is a good match for many positions in business, government, or industry.
   
   Even prior to 1985, there were programs that produced employable MS-holders in applied mathematics. Clemson, the Claremont Colleges, Rensselaer Polytechnic Institute, and Washington State University all had successful programs, yet they were emulated by few other schools. Instead, universities extended their limited resources by going into expensive PhD programs, although there was no indication that their candidates—most of whom were innocent of modeling in any form—would find positions.
   
   The first panel discussion on the applied MS degree was held in 1987 [4]. The panel was held under the auspices of the Mathematical Association of America (MAA), but the organizer, moderator, and panelists were all SIAM members. The ring was closed ten years later with a panel discussion at the 1997 SIAM Annual Meeting at Stanford University, A Proposed Curriculum for the Professional MS Degree [6]. This panel marked the first time that the SIAM Education Committee aired a curriculum.
   
   Much of the credit for generating recent interest in this degree is due to the work of Avner Friedman when he was at the University of Minnesota. Although he prefers the adjective "industrial," this term should be understood as shorthand for "business, government, or industry." (Even that is too restrictive, since it excludes many non-profit organizations.) The term, "Professional Master's Degree" was suggested by Friedman in 1991 at a meeting of the Board on Mathematical Sciences [1].
   
   
2. Rationale
   
   The "Mathematics in Industry" report is perhaps the first official document by any mathematics organization dominated by PhDs that recognizes MS-holders and their contributions [2]. This report is both a symptom and a contributing cause of the increased attention that is being paid to the MS degree by business, government, and industry. It is worth noting that the approximate MS:PhD ratio of employed mathematicians is 9:1.
   
   The goals of the MII report are: 1) to examine the roles of non-academic mathematicians and characterize their working environment; and 2) to use the views of non-academic mathematicians and their managers to suggest changes in the conventional graduate curriculum that will better prepare students for available employment opportunities.
   
   The MII Executive Summary lists the first most important backgrounds or skills for a non-academic mathematician. Following is a list of one-word renditions:
   
   Modeling
   Teamwork
   Computation
   Interdisciplinary
   Communications
   
   The MII report urges graduate schools to develop curricula that will impart to their students the five backgrounds and skills listed above. They are features of practically every attempt at curriculum change. It should be noted that the last four can be considered consequences of the first.
   
   Hitherto, no organization has proposed curriculum guidelines for an applied master's degree. The MII report provides the impetus, basic rationale, and many of the concepts for constructing and publishing a set of guidelines for a professional master's degree in the mathematical sciences. In the publication of such guidelines, SIAM exercised its leadership responsibilities in the area of applied mathematical curricula.
   
   
3. Program Guidelines for a Professional MS Degree
   
   The foundation of a professional MS degree includes depth in some field of application and breadth of training in applications. Modeling is the common thread that runs throughout the curriculum. An extended experience in an actual problem-solving situation, preferably in a team setting, is essential. Finally, the student must be provided with opportunities to enhance presentation and communication skills.
   
   These are guides (not a straitjacket) for courses, credit hours, and other experiences. The implementation of a program might require modification as determined by the location of the school and the nature of its mathematics department. Curricula can be quite different and yet capture the spirit and essence of these guidelines.
   
   The program has three distinct aspects: course curriculum, internship or work experience, and communication skills.
   
   Course Curriculum

   CORE COURSES

   Scientific Computing 6 credit hours
   Introduction to numerical methods for solving scientific problems, using a modern program language (such as C/C++) and/or using modern tools like MatLab. Includes an introduction to elements of high performance computing and scientific visualization.

   Numerical Linear Algebra 3 credit hours
   Mathematical and numerical study of direct and iterative methods for solutions of linear systems. Topics include SVD, least squares computations and methods for computations of eigenvalues and eigenvectors.

   Statistics 3 credit hours
   Applied mathematical statistics and data analysis.

   Methods of Applied Analysis 3 credit hours
   Functional analysis with applications to applied mathematics. Topics include metric and normed linear spaces, bounded and compact operators, inner product and Hilbert spaces, self-adjoint operators, orthogonal expansions and Fourier analysis.

   Optimization 3 credit hours
   Linear, unconstrained and constrained optimizations.

   One of the following two courses:
   
   Discrete Mathematics 3 credit hours
   Graphs, combinatorial optimization, integer programming, discrete algorithms.
   
   Differential Equations 3 credit hours
   Introduction to ordinary differential equations from a modern dynamical systems perspective or an introduction to partial differential equations. Focus on modeling and computational method.

   Field of Application 9 credit hours
   Three courses in which the student gains familiarity with the key concepts of the field and is able to contribute to the solution of problems faced by practitioners in the field. Some examples of suitable fields are biology, bioinformatics, engineering, environmental science, finance, genetics, and management science. An alternative arrangement might consist of two courses combined with three credit hours in an internship in the field of application.

   TOTAL: 30 credit hours

   Internship or Work Experience

   Internships at a business, government, or industrial site involving experience on a project. The project should involve cooperation between the site mentor and the student's faculty advisor. The content, scope and objective(s) should be spelled out in advance. This can be done in the three ways listed below, starting with the most desirable:

   1. Full-time experience on a site for at least one month.
   2. One day a week on a site for a period of one semester.
   3. Summer workshops such as those at the Institute for Mathematics and its Applications (University of Minnesota) and North Carolina State University.
   Should none of these three options be available, an extended experience on a campus computer, statistics, or applied mathematics lab might be used as a substitute. One danger is that this could result in a year of similar experiences rather than the progressive build-up expected from a structured, guided project.

   Communication Skills

   Skills in communication are essential when working with a team, the usual mode for solving complex problems. Thus, writing and presenting are an important part of any master's program in applied mathematics. On-the-job experience is an excellent way to learn and to test these skills. Following are a few examples of some ways to bolster a student's confidence in writing and in making presentations.

   • Invite engineers, managers, or scientists from local industry to give talks
   • Have students give presentations, with constructive evaluation from peers and faculty
   • Invite campus engineers, scientists, or administrators to make presentations
   • Invite staff from your communication and personnel departments to give seminars
   • Encourage students to give papers at professional meetings.
   
   
   
4. Some Working Programs
   
   A list of university programs that satisfy the essentials of the criteria in each of the three categories (Course Curriculum, Internship or Work Experience, and Communication Skills) will be posted on the SIAM Web site in the near future.
   
   The most challenging aspect of designing this degree is the Internship and Work Experience category. It is very difficult to learn how to approach off-campus firms for internship arrangements from reading a pamphlet. Consultants should have had such experiences, as well as fairly intimate knowledge of how to organize and secure funding for clinics or similar on-campus centers for providing students with the equivalent of on-the-job experience.
   
   
5. References
   1. Board on Mathematical Sciences, National Academy of Science (1991). A Master's Degree in Applied Mathematics, 16 April 1991, Washington, DC.
   2. Davis, P.W., Chair (1995). Report on Mathematics in Industry, SIAM Mathematics in Industry Steering Committee, 34 pages, October 1995.
   3. Fusaro, B. A. (1985). It's Time to Recognize the Value of a Master's Degree in Applied Math, SIAM News, p.6, September 1985.
   4. Mathematical Association of America (1987). What is a Master's Program in the Mathematical Sciences at a Master's Granting University?, Panel Discussion, 22 January 1987, MAA 70th Annual Meeting, San Antonio, TX.
   5. Morrisette. T.M. (1985). Bridging the Gap is Rewarding, SIAM News, September 1985.
   6. Society for Industrial and Applied Mathematics (1997). A Proposed Curriculum for the Professional MS Degree, Panel Discussion, 15 July 1997, SIAM 45th Anniversary Meeting, Stanford University, CA.
   7. IMA Workshop, November 1996, Minneapolis, MN
   
   
   
6. Annotated Source Materials
   • Cochran, J.A. (1987), Master's Degrees: A Current Perspective, SIAM News, May 1987.
   • Friedman, A. and Lavery, J. (1993). How to Start an Industrial Mathematics Program in the University, SIAM, Philadelphia, PA.
   • McCartin, B.J., Salacuse, J.L, and Green, David Jr. (1998). A Well-Kept Secret: The Kettering University Experience, SIAM News, July-August 1998.
     This article describes an extraordinary five-year bachelor's program in applied mathematics. Although far too structured and specialized for most undergraduate schools, it has many excellent ideas for any institution that is fashioning a curriculum for a professional MS degree.
   • Spencer, D.S. (1987). Perspectives on the Master of Applied Math Degrees, SIAM News, July 1987.
   
   
   
7. Acknowledgements
   • Terry L. Herdman, SIAM Vice President for Education, 1996-2002
   • Gilbert Strang, SIAM Vice President for Education, 1993-96
   • Samuel M. Rankin III, Associate Director, AMS
   • Richard Haberman, SIAM Education Committee Chair, 1987-90
   • B. A. Fusaro, SIAM Education Committee Chair, 1984-87
   • William G. Kolata, SIAM Technical Director
   • James M. Crowley, SIAM Executive Director
   • SIAM Education Committee Working Group on Professional Master's Programs: Kathryn Brenan (Aerospace Corporation), Steven Cox (Rice University), B. A. Fusaro (Florida State University), Terry Herdman (Virginia Polytechnic Institute and State University), Donald Miller (St. Mary's College, University of Notre Dame), David S. Ross (Eastman Kodak Company).
   • Editor – B. A. Fusaro (Florida State University)
     Co-Editors – Robert E. Fennell (Clemson University), Donald Miller (St. Mary's College, University of Notre Dame

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