November 2000
II. Introduction: A letter from the faculty
B. Special courses in the freshman year.
D-2 Requirements of the traditional mathematics concentration
D-3 Options within the traditional concentration
D-4 The applied mathematics track
D-5 Kinds of upper division mathematics courses
D-6 Undergraduate research opportunities
D-7 The special status of computing
D-8 Teacher preparation program
D-9 Courses in other departments
IV. Mathematics Concentrators After College-What
they do and where they go.
B. The local picture: Our mathematics alumni
Appendix 1: Applied Mathematics in Business, Industry,
and
Government
I. Recognizing mathematicians: jobs titles in business, industry, and government. < /A>
II. Applied mathematicians-problems they solve and subjects they studied.
Appendix 2: What do they earn?
Appendix 3: Publications by William and Mary undergraduates
Appendix 5: Projected course rotation
Appendix 6: Five year BA/MA programs at William and Mary
Appendix 7: Teaching and research careers in mathematics
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This is a revised and updated version of the advising handbook originally prepared in September 1996. Preparation of the original version was a joint project between Mathematics Department faculty and MOST, the Mathematics Organization for Students, with financial support provided by the William and Mary Parents' Association. The Handbook is designed for all current and potential mathematics students - whether or not they might concentrate in mathematics - and we hope it will be a source of answers to many of the standard questions that students ask.
In preparing of the original version of this Handbook, we reviewed the advising handbooks of many other mathematics departments, and borrowed extensively from them and from publications of the National Research Council's Board on Mathematical Sciences and of the Mathematical Association of America. We thank those other mathematics departments, the NRC, and the MAA for their assistance.
II. INTRODUCTION: A LETTER FROM THE FACULTY
William and Mary, like many universities, groups mathematics with the sciences. But is mathematics really a science? Probably not. In terms of its intellectual role, undergraduate mathematics is more like a foreign language: few of us learned to speak it as children, most of us must learn it as a tool for communication, and only some of us study it as a subject in its own right.
Since the beginning, mathematics and its applications were intertwined. Geometry developed in support of the great civil engineering projects of Egypt and Greece. At least since the time of Newton, physical science and mathematics developed in tandem, as can be seen from the fact that many of the major ideas in our discipline have been re-discovered by researchers in the physical sciences and engineering. In recent years, other disciplines have become increasingly mathematical. To a large degree, economics and finance are now the study of specialized mathematical models, and the social sciences use game theory, probability, and statistics as the organizing tools for much of their research. The same is true of industrial applications. Without the insights of operations research, modern industry would not be able to achieve the levels of efficiency required to prosper. These new applications, coupled with remarkable advances in computing, have created whole new fields of mathematics within the last twenty-five years.
In 1960, Eugene Wigner, a renowned physicist of mid-century, wrote an article entitled ``The unreasonable effectiveness of mathematics.'' Wigner focussed on the remarkable interplay between mathematics and the physical sciences, something that was well understood in 1960. His title is even more relevant today because mathematics now contributes to almost every discipline. Mathematics is effective because it finds hidden patterns that unite apparently unrelated phenomena, thereby allowing our intuitions about one thing to illuminate our understanding of others. Mathematics focuses on quantitative relationships, forcing us to isolate the pivotal components of a thing or process, and enables us to explore the ``What if?'' questions that are the basis of thoughtful planning. And mathematics is a vehicle for the most precise communication.
Who studies mathematics? Most of our students come from disciplines that apply mathematics for their own purposes. Many begin by thinking of mathematics only as a tool required for other work, and then find the tool so interesting that they want to study it more. For such students, mathematics is about solving problems. Such mathematical applications often lead to a secondary concentration in mathematics or a mathematics minor, something that employers like to see because it assures them of the student's quantitative skills. Other students are drawn to mathematics for aesthetic reasons, perhaps because they glimpsed the precision and clarity of mathematical thought in their earlier mathematics courses. It is rare in undergraduate study that one can hold a truly important subject in one's mind, feel its theoretical cohesiveness, and see the beauty of it from top to bottom. While such understanding does not come easily, it is possible for an undergraduate in mathematics. And when one grasps the theoretical unity of parts of our subject in that way, mathematics is more like a poem than a tool. But even the most theoretical parts of mathematics - number theory, for example - have surprisingly concrete applications such as protection of financial records and security of electronic data transmissions.
What can mathematics concentrators do after graduation, other than teaching and graduate study? The national picture is outlined in Section IV-A and in Appendix 1 ``Applied mathematics in business, industry, and government'' where we describe some of the career opportunities open to mathematics undergraduates. We also give examples of the kinds of mathematical problems encountered by various applied mathematicians, and suggest undergraduate courses that would be particularly good preparation for various careers in applied mathematics. Appendix 1 would be a good place to start if you want to know ``What can one do with mathematics, other than teach?''
What do William and Mary mathematics concentrators actually do after they leave the College? Most of our concentrators are employed in industry or government. Some pursue further studies in mathematical sciences at the masters or doctoral level. A few enter graduate school in other disciplines, many of which welcome mathematics majors because of their strong quantitative backgrounds. Some mathematics concentrators enter law or medical school, and a small minority go into teaching. Details about the post-graduation lives of our concentrators appear in Section IV , below.
We hope that this advising handbook will serve as an introduction to our department and its faculty, and will answer many of the standard questions that students and parents ask about mathematical study. But we hope that you will also ask us about mathematics after class, or in our offices, or when you meet us in the hallways. Professors Roy Mathias (Jones 133) and David Stanford (Jones 121) are our chief concentration advisors, and if there are questions that you cannot get answered informally, they are available to help you. And so are the rest of us.
Typically, about 70% of every freshman class at William and Mary earns credit for at least one calculus course, either through the AP program or by taking calculus at the College. While the department offers certain non-calculus courses during the freshman year, including freshman seminars, the vast majority of students begin with calculus at some level.
B. Special Courses in the Freshman Year
The department offers five non-calculus courses for freshmen, each being independent of the others. The first is Math 103, Precalculus Mathematics. This course is designed for students who plan to take calculus (Math 108 or Math 111) but whose algebra and trigonometry backgrounds need improvement. The course is supposed to develop skills in operating with functions (including the trigonometric and inverse trigonometric functions), graphs, equations, inequalities, systems of equations and ineaqualities. This course does not satisfy the College's first general education requirement (GER-1).
The second course is Math 104, Mathematics of Powered Flight. Its goal is to study mathematics in an immediately applied context. This course is primarily aimed at non-science students and will fulfill the GER-1 requirement. The course is not open to students who have successfully completed a Mathematics course numbered higher than 210.
The third course is Math 106, Elementary Probability and Statistics. This course emphasizes probability and its application in statistics, with emphasis on underlying principles rather than special techniques. The course is not open to any student who has taken a Mathematics course numbered higher than 210, and it meets the College's GER-1 requirement.
The fourth course, Math 110, Topics in Mathematics, gives an introduction to mathematical thought with topics not routinely covered in other courses. The material is chosen from various areas of pure and applied mathematics, including probability and statistics. This course does not satisfy the College's GER-1 requirement.
The fifth course is Math 150, one of the College's freshman seminars. In recent years, the course has been devoted to such topics as graph theory and cryptography, and the mathematics of voting. In any given semester, several different freshman seminars might be offered, and students should consult the Fall and Spring Registration Bulletins for a description of the seminars' topics. Like all freshman seminars, its enrollment is limited to about 15 students, and the course is reading-, writing-, and discussion-intensive.
There is also a special four credit hour calculus course Math 108, Brief Calculus with Applications, which is designed for non-science students who do not plan to continue further to study calculus. This course fulfills the GER-1 requirement and is not open to any student who has taken a Mathematics course numbered higher than 108 (with the exception of Math 150). To use this course as a prerequisite for Math 112, students need a special approval of the department chair. This course gives an introduction to the calculus of elementary functions, including some elements of multivariable calculus, with numerous applications in buisness, biology, chemistry, physics and other social and life sciences. Maple will be used in this course.
Most students start (or continue) their study of calculus during their first semester at the College. For most, Math 111 (Calculus I) is the right choice, but for others, Math 112 (Calculus II) or perhaps a sophomore level course is the best choice. For students with AP credit from high school, the choice is explained below. Others, including transfer students, should consult the department (Professor George T. Rublein, tel. 757-221-1873) about proper placement.
Many students today enter William and Mary with substantial advanced placement credit. AP credit for Math 111 (Calculus I) is awarded when a student achieves a score of 4 or 5 on the AB test, or a score of 3 on the BC test. AP credit for both Math 111 and 112 (Calculus I and Calculus II) is awarded when a student scores a 4 or 5 on the BC test. In addition, the department offers a program of Credit by Examination for students with strong calculus backgrounds who did not take the AP exam. (For information, call the office of the department 757-221-1873.) In addition, some students simply skip Math 111 based on their high school background and register for Math 112, and these students should consult the department chair during orientation period to have their decision approved by the department.
Students who have AP credit for Calculus I and II should take either Math 212 (Calculus III) or Math 211 (Linear Algebra) in their first semester.
The first college mathematics course taken by most of our students is Math 111 (Calculus I).
Math 111 is a four credit hour course, meeting three hours per week with a faculty member in classes limited to about 35 students. The fourth hour is a required calculus lab, in which students meet with lab assistants and work of additional workbook style material. Both in class and in the laboratory, the course uses graphing calculators as tools for experimentation and visualization. The laboratory assignments complement the lectures and examine themes that run throughout the course. Key ideas in the course include rates of change, continuity, derivatives, approximation, numerical integration via Riemann sums and associated error analysis, techniques of differentiation, and applications of derivatives to graphing and mathematical modelling problems.
Math 112 completes the study of single variable calculus. Like Math 111, it is a four credit hour course with an attached calculus laboratory. Starting in Fall 2001, Math 212 calculus labs will be based on Maple software. (Maple is one of the powerful symbolic manipulation and visualization software packages that are now available to assist in mathematical and scientific work.) The course focuses primarily on definite integration, using both numerical techniques and integration via anti-derivatives. The course studies applications of the integral to problems in economics and finance, as well as to science and engineering. A special chapter is devoted to a simple class of ordinary differential equations and explains how useful they are for various applied problems. The course closes by returning to the theme of approximation, studying series approximations and their applications.
Math 212 (Multivariable Calculus) is a three credit hour course on the calculus of several variables. It studies surfaces in three-space, vectors, partial differentiation, multiple integration, and line integrals. This course uses Maple as its primary software tool.
Math 211 (Linear Algebra) is a three credit course taken by students interested in mathematics, chemistry, physics, and economics, among others. Linear algebra begins with the study of systems of linear, or first degree, equations. The course then introduces matrices, i.e., rectangular arrays of numbers that provide a compressed way to write down and manipulate systems of linear equations. Matrices can also represent linear transformations, functions of a special type that act on the set of all n-tuples, the standard Euclidean space. In this context, the concepts of eigenvalues, eigenvectors, and canonical forms are introduced and used in the application of linear algebra to topics such as differential equations, geometry, etc. Depending upon the instructor, the course may focus on computational techniques, theoretical aspects, or applications of linear algebra, and the use of computer software such as Matlab or Maple may be integrated into the course. At William and Mary, linear algebra (also called ``matrix analysis'') and the closely related subject ``operator theory'' are the department's strongest research fields, and there is a follow-up course (Math 408, Advanced Linear Algebra) offered each year.
C-4 Foundations of Mathematics.
Math 214 (Foundations of Mathematics) was created a few years ago after students requested a course to prepare them for the more abstract flavor of upper division courses, whether pure or applied. Students report finding Math 214 to be quite challanging, in part because the things that it emphasizes are quite different from what students see in earlier mathematics courses. Topics in this ``bridge course'' can vary from term to term, but whatever topic is the course's organizing theme, the real purpose of the course is to introduce students to the reading and writing of proofs. Somewhere in the course, students will encounter elementary logic (de Morgan's laws, quantifiers, negations, implication, etc.), direct and indirect proofs, proofs by induction, introduction to the various number systems and their inter-relations, equivalence relations, functions of various kinds and their application to cardinality. Students find Math 214 to be very different from mathematics courses that they have taken before and the course should be taken by the end of the second year. Math 214 is a prerequisite for Math 307 and Math 311, two required courses in the mathematics concentration. It can also give students a glimpse of the spirit of modern mathematics.
Every mathematics concentrator needs to be proficient in computer programming, at least at the level of CS 141. Any programming experience that you already have will be helpful (but not required) in this course. The language du jour in CS 141 is C++. If you already have significant experience with Pascal, the Computer Science department offers a special 1-credit course (CS 142) that will enable you to program in C++. Mathematics concentrators report that CS 141 is much different from lower numbered CS courses and can be quite time-consuming for students without any previous programming experience. CS 141 provides crucial tools for students who plan to study or apply mathematics. Other recommended courses include CS 241 (Data Structures), which is an integral part of the applied mathematics track within the mathematics concentration. CS 303 (Algorithms) is alos highly recommended for mathematics students. See also Section D-7 below for more on the special importance of computing in mathematics.
Courses numbered 300 and above are usually courses for juniors and seniors, but you might be ready to begin these courses during your first two years. Indeed, for some students, it would be a mistake to wait. This category includes students who are contemplating the teacher certification program in secondary mathematics. If such students take no 300-400 courses during their first two years, then they will need to take at least three mathematics courses during one semester of their last two years, and for many students, that would be a lot.
D-1 Concentration Declaration and Advising.
Once you have decided to concentrate in mathematics, the next steps are to see one of the Mathematics department's chief concentration advisors, Professors Roy Mathias and David Stanford, and to complete the concentration declaration forms that are available in the Arts and Sciences advising office. The department has a list of faculty members who serve as undergraduate advisors, and normally students choose one of them. Your initial choice of an advisor is easy to change at a later date. Simply contact the chief concentration advisors.
Together with your advisor, you will review the departmental requirements for the concentration, review your progress toward meeting the College's general education requirements, and work out a plan of studies for the next two years. That two year plan is only provisional-we know that your interests and plans might shift as you get deeper into your program of studies. But it is a reasonable first step in designing your concentration, tailoring it to fit your interests and long range plans, and finding ways to build intellectual coherence into it.
D-2 Requirements of the traditional mathematics
concentration.
There are two tracks within the mathematics concentration.The first is described in this paragraph, and is the traditional route to a mathematics concentration. The second is the applied track and is described in Section D-4 , below. The mathematics traditional concentration requires a minimum of 38 hours of mathematical sciences courses, as follows. In addition to a core of freshman and sophomore courses (Math 111, 112, 211, 212, 214), mathematics concentrators must complete Math 307 (Abstract Algebra), Math 311 (Elementary Analysis) and either Math 490 (Seminar) or Math 495-96 (Mathematics Honors). In addition, students must complete four other mathematics courses at the 300-400 level, at least three being numbered 400 or above. Two courses in Applied Science (cross listed with mathematics as Math 441 and 442) may be counted toward this requirement. Experience shows that some students skip part of the freshman/sophomore core described above, without receiving AP credit or credit by examination for the skipped courses. In such cases, each skipped course must be replaced by a mathematics course numbered 300 or higher, or by 441 or 442. Mathematics concentrators satisfy the College's Concentration Writing Requirement by completing either Math 490 or Math 495-96 with a grade of C- or better. Currently, mathematics concentrators satisfy the Concentration Computing Proficiency Requirement by completing CS 141.
The program described above is an adequate undergraduate mathematics program by national standards. However, the Committee on the Undergraduate Program of the Mathematical Association of America has recommended that mathematics concentrators pursue a kind of in-depth study of mathematics that goes beyond the requirements outlined above. Consistent with that recommendation, the department encourages its students to take at least two additional mathematical science courses at the 400-level, in addition to the 38 hour program described above. Ideally these courses should be part of a sequence that builds upon and shows the inter-relations between other courses in the concentration. Whether the courses are applications-oriented or theoretical is not important. This recommendation is particularly important for students contemplating graduate study in mathematics.
By careful planning, students can build a unifying theme into their study-in-depth option. Four relatively common examples are:
Other coherent study-in-depth programs can be designed by students and their advisors.
D-3 Options within the traditional
concentration.
The traditional mathematics concentration at William and Mary is designed to provide a broad mathematical background that will keep open a wide variety of career and graduate study options. Students often plan their upper division mathematics courses with post-college goals in mind and we make specific recommendations about that in Appendix 1. However, experience shows that such goals change over time, and we have designed our program in such a way that students can keep career and graduate study options open throughout their undergraduate years. In concrete terms, this means that we encourage students to pursue a mixture of pure and applied mathematics courses throughout their junior and senior years, and to sample all of the major components of our discipline, as described in Section D-5 below.
Nevertheless, students can focus their studies to some degree. Working with their concentration advisors, they can construct programs of study to prepare them for careers as industrial or government mathematicians or actuaries, or as primary or secondary teachers. Other programs of study prepare students for graduate work in mathematical sciences-mathematics, statistics, operations research-or for interdisciplinary work in such fields as economics, finance, and the social sciences. To help students make decisions about which parts of mathematics to pursue, we encourage our concentrators to sample each of the broad sub-fields of mathematics: algebra, analysis, geometry, and application-oriented courses. In addition, Appendix 1 describes the most popular applied mathematics career options for mathematics concentrators and suggests undergraduate courses that would be important for someone choosing one of these careers.
The new applied track within the concentration provides yet another way for students to focus their mathematics studies. See Section D-4 below.
D-4. The applied mathematics track.
This track is designed for students who want to pursue applications of mathematics or a double concentration in mathematics and another discipline. Students who plan to seek employment immediately after graduating and who want to develop specific skills may be well served by the applied track. Students who plan to pursue graduate study in applied mathematics may be better served by acquiring a strong background in the fundamental branches of modern mathematics via the standard track. You should develop your own program of study in consultation with your professors. Students interested in applications should also consult with the Applied Mathematics Track advisor, Professor Trosset.
The Applied Mathematics Track has the following requirements:
Breadth requirement: three distinct courses, one in each of three of the four applied areas.
Depth requirement: at least three courses in one of the four applied areas.
The four applied areas within the applied track are as follows:
In addition to the above, students in the Applied Mathematics Track should take appropriate courses in computer science, including at least Computer Science 141 (Introduction) and 241 (Data Structures).
Here are some examples of programs of study within the Applied Mathematics Track:
Student A did not take many math courses in high school. She took the standard calculus sequence at William & Mary and opted for the applied mathematics track, emphasizing operations research. After graduating, she took a job with an industrial consulting firm. Here are her mathematics courses:
Fall | Spring | |
Year 1 | 111 | 112 |
Year 2 | 211 | 212,308 |
Year 3 | 214,323 | 302,311 |
Year 4 | 401,CS 628 | 424,490 |
Student B received AP credit for Math 111-112. He majored in physics at William & Mary with a second concentration in mathematics. He opted for the applied mathematics track, emphasizing scientific applications. Here are his mathematics courses:
Fall | Spring | |
Year 1 | 212 | 211 |
Year 2 | 214 | 302,308 |
Year 3 | 311,413 | 417 |
Year 4 | 405 | 490 |
Student C received AP credit for Math 111. She pursued an intensive program of study in mathematics and opted for the applied mathematics track, emphasizing probability and statistics. Her plan was to prepare for graduate study in statistics. Here are her mathematics courses:
Fall | Spring | |
Year 1 | 112 | 211,212 |
Year 2 | 214,323 | 307,308 |
Year 3 | 311,401 | 402,403 |
Year 4 | 413,495 | 414,496 |
D-5 Kinds of upper division
mathematics courses.
The upper division mathematics curriculum at William & Mary attempts to strike a balance between courses that provide a rigorous introduction to the fundamental concepts of modern mathematics and courses that study how mathematics is used to solve interesting problems in other disciplines. Both are important! Most mathematical theory was originally motivated by the desire to solve various applied problems, and solving new applied problems often requires developing new theory. Pure mathematicians are generally pleased when their work is found useful in other disciplines, and applied mathematicians need to know a great deal of mathematical theory.
Traditional mathematics concentrators at William & Mary are required to take Math 307 (Abstract Algebra) and Math 311 (Elementary Analysis), two courses that introduce concepts that are central to the study of modern mathematics. Concentrators in the applied track must complete at least one of these two courses. The distinction between algebra and analysis is instructive when trying to understand the interrelationships between various upper division courses.
``Algebra'' is a term for mathematics that studies how mathematical objects can be combined, subject to precise operational rules. The objects of study might be real numbers, or complex numbers, or vectors, or matrices, or functions, and the operations might be adding and multiplying those numbers, vectors or matrices, or combining functions via composition. Math 211 (Linear Algebra) will be the student's first collegiate exposure to a branch of algebra. After completing Math 307 (Abstract Algebra), students will have a better idea of what modern algebra studies and will be able to see how high school algebra is a precursor of modern algebra. Other algebra courses offered by the department include Math 408 (Linear Algebra II), Math 412 (Number Theory), and Math 430 (Abstract Algebra, II). Although algebra is usually regarded as a branch of pure mathematics, it has many important applications. Linear algebra is the basic tool for solving systems of equations, algebraic or differential. Group theory (studied in Math 307 and Math 430) lies at the foundations of cosmology, coding theory, and crystallography. Combinatorics and graph theory (studied in Math 432) are critical tools in operations research and computer science. Number theory is used to secure financial records across the internet.
``Analysis'' is the term used to describe those parts of mathematics that study continuously changing objects, often involving limits, functions, and derivatives of various kinds. Calculus is probably a student's first encounter with analysis. Math 311 (Elementary Analysis) studies the theory of calculus. Math 403 (Intermediate Analysis) looks at related concepts, in more general contexts. Other analysis courses develop aspects of calculus in the context of complex numbers (Math 405: Complex Analysis) and in more general situations such as Hilbert spaces (Math 428: Functional Analysis). Analysis is of fundamental importance in both pure and applied mathematics.
Math 307 and Math 311 were designed to prepare students to take 400-level courses. As a general rule, students should take Math 307 and Math 311 before progressing to 400-level courses. Notice, however, that students who opt for the applied mathematics track are only required to take one of these courses. Although we hope that all students of applied mathematics will elect to take both Math 307 and Math 311, we recognize that not all will. Which of these courses is more important for you will depend on what you choose to study. For example, a student who plans to take Math 401 (Probability) should definitely take Math 311, but might not take Math 307. Please consult with the math faculty to decide which courses will best suit your needs.
Other 300-level courses include Math 302, 308, and 323. Math 302 (Differential Equations) introduces a branch of analysis that is of enormous importance in applied mathematics. Math 308 (Applied Statistics) introduces the basic concepts and techniques used in analyzing data collected from scientific experiments. It is intended to precede Math 401 (Probability) and Math 402 (Mathematical Statistics). Math 323 - 424 (Operations Research) introduces mathematical techniques for solving a variety of applied problems, e.g. the optimal allocation of scarce resources.
Other 400-level courses not mentioned above include Math 413-414, 416, 417, and 426. Math 413-414 (Numerical Analysis) is concerned with the mathematics of computation and the numerical algorithms that computers use to calculate various quantities of interest. Math 416 (Topics in Geometry) is a modern geometry course whose content changes from one year to the next. Classical geometry studied shapes in the plane and in three dimensional space. Today the term ``geometry'' describes the study of more general shapes and spaces, ranging from geometries in finite sets, graphs, properties of convex sets in n-dimensional spaces, and the topology of abstract spaces. Math 417 (Vector Calculus) is a more through study of various topics introduced in Math 212 (Multivariable Calculus). Math 426 (Topology) studies the kinds of spaces in which most of analysis occurs and gives a far more general treatment of some of the topics found in Math 311 (Elementary Analysis). It may also include more geometric ideas such as the use of group theory in classifying surfaces in Euclidean spaces.
D-6 Undergraduate research opportunities.
If you are a mathematics concentrator, or are thinking about becoming one, you already know that learning mathematics is exciting. Your professors can assure you that discovering mathematics is an even more exciting activity, and William and Mary undergraduates can participate in that process through undergraduate research programs.
What is research in mathematics? Mathematical research goes beyond reading someone else's discoveries in a textbook or journal, and beyond solving problems that you have not solved before. Mathematical research involves first determining what are the right questions to ask, and then attempting to answer them. To varying degrees, this involves solving problems that nobody has solved before, often with little idea of what approaches and which techniques might work, and sometimes without a clear initial idea of what the question really is. It involves intuition as much as knowledge, and its unstructured nature is often frightening to beginners, whether they are undergraduates, mathematics doctoral students, or professional mathematicians presented with a challenge they have never seen before. Students are usually introduced to research through a carefully supervised apprenticeship process in which they are treated as junior colleagues whose research skills and insight develop through close interaction with a faculty supervisor.
There are many opportunities for undergraduate mathematics concentrators at William and Mary to participate in mathematical research. These include independent study courses, the Honors Program, summer internships and participation in summer Research Experiences for Undergraduates (REU) programs around the country. Undergraduate mathematical research at the College often results in professional publications. Since 1990, over 43 research articles have been published by mathematics undergraduates who worked with faculty members at the College, and more are being considered by professional journals. See Appendix 3 for a list.
Internships that draw upon one's mathematical skills, can show students what industrial mathematics is really about, and can give students an early start in their post-college job searches. They can also give students valuable experience in working on teams, something that is often missing from a traditional undergraduate mathematics program. Information about internship opportunities is available through the college's Career Services Office in Blow Hall. In addition, messages about jobs and internships from the Career Services Office appear periodically on the WWW home page of the department's undergraduate student organization, MOSt, described in Section V below. Furthermore, the department circulates such information by e-mail to its concentrators. Finally, nearby federal laboratories-NASA's Langley Research Center and the Department of Energy's Jefferson Laboratory-also provide some internship opportunities.
Writing an Honors Thesis, or participating in summer REU programs, can give students a preview of what would be involved in mathematics doctoral study. Students with strong academic records, whose overall performance in mathematics courses (including at least one of Math 307 and Math 311) through the junior year is very strong, should consider the possibility of writing an Honors Thesis. They should also investigate REU programs for the summer between their junior and senior years (and possibly a year earlier), particularly if they are considering graduate school in mathematics.
The National Science Foundation and other federal granting agencies support about two dozen REU programs around the U.S., including one at William and Mary. Most of these programs are small, involving about ten students each. REU participants typically receive modest stipends and free on-campus housing for eight to ten weeks in a summer, and work closely with a faculty member on a research problem in mathematical sciences. Different REU sites typically focus on different research areas, and a list of REU sites and their research foci is published in Notices of the American Mathematical Society and SIAM News. Copies are available in the department. On the Web, the list of REU sites is available from the National Science Foundation. In addition, students may explore REU opportunities via the Web site of the Mathematical Association of America or by sending an e-mail inquiry to REU.dms@nsf.gov. Admission to REU programs is competitive and requires letters of recommendation from faculty members.
D-7 The special status of computing.
All mathematics concentrators are expected to be proficient in computer programming at the level of CS 141, but that course is only an introduction to computing. Today, students who plan to apply mathematics after a bachelors or masters degree, or who plan to teach after completing doctoral study, need more. In addition to programming languages such as C, C++, or Fortran, the ability to use major mathematical software packages such as Maple or Mathematica, familiarity with technical word processors such as Tex or Latex, and experience with different computer operating systems (e.g., Windows and Unix) are important for all mathematics students today. In addition, considerable programming experience developed in connection with formal computer science training is almost sure to be needed by mathematics students who pursue industry or government careers after their bachelors degrees.
With very careful planning of one's undergraduate program, it would be possible to complete a bachelor's degree in mathematics and a masters degree in computer science within five years. This is an extremely challenging program, but the combination of a mathematics undergraduate degree and a computer science masters degree is likely to be a formidable employment credential. For further details, see Appendix 6 , and consult the Computer Science department .
Modern computing is much more than an employment credential for mathematicians. In recent years, significant intellectual interactions have developed between computing and mathematics at the research level. On the one hand, computers and software have now become powerful enough that they are used in almost all applications of mathematics and allow mathematicians to solve problems that were heretofore too complicated to attack. On the other hand, the special needs of computing are helping to shape research developments in mathematics. Examples include numerical linear algebra algorithms and methods for the numerical approximation of solutions of differential equations. Computational mathematics underlies a vast array of the tools used in modern engineering and science.
D-8 Teacher preparation programs.
In Virginia, all students seeking teacher certification must have an Arts and Sciences concentration, plus a block of up to 24 hours in the School of Education. Students seeking high school mathematics certification complete the mathematics concentration's 38 hour requirements (at least), and the School of Education recommends using courses in geometry (Math 416), number theory (Math 412), and probability and statistics (Math 401-402 or Math 308) to fulfill the departmental requirement of ``four other mathematics courses at the 300-400 level, three being numbered 400 or above'' in the traditional mathematics concentration (see paragraph D-2 , above). In addition, it is sometimes possible to meet the department's Math 490 requirement by choosing a seminar that is particularly relevant to high school mathematics teaching. The requirement in the School of Education includes Ed 301, Ed 310, and Ed 423, plus 15 credit hours in the spring of the senior year. For details, see the course requirements and professional semester paragraphs in the School of Education section of the Undergraduate Program Catalog.
If you are interested in secondary mathematics certification, you should contact the School of Education very early in your College career. In addition, you and your mathematics advisor will need to plan your sequence of courses very carefully starting in your sophomore year, to make sure that you will be able to complete the required and recommended courses. This is because the teacher certification program will take all of your time during the spring semester of your senior year, and because some of our courses are offered only in alternate years. See Appendix 5 about projected course rotations.
D-9 Courses in other departments.
Particularly if you plan employment immediately after your bachelor's degree (as opposed to attending a graduate or professional school), it would be important for you to include elective courses that are related to mathematics. As an industrial mathematician, you are most likely to be acting as part of a consulting team whose job is to support various client groups within your business. It is very helpful if you have at least mastered the basic vocabulary of your clients. In the past, such clients were usually scientists or engineers, and mathematics departments typically recommended a science minor, usually physics or chemistry. Today, with the discovery by other disciplines of the unreasonable effectiveness of mathematics (to borrow Wigner's phrase from Section 1, above), other related areas are available. Two of the most exciting are economics and finance, and careful choice of elective courses from these areas can introduce you to significant applications of mathematics and make your studies more relevant to a significant group of potential employers.
In a recent survey of our graduating seniors, we asked for suggestions about other courses around campus that mathematics concentrators had found very interesting. Physical Chemistry, Classical Mechanics, and Electricity and Magnetism were mentioned, as were several of the Linguistics courses offered by the English department. Philosophy offers Symbolic Logic classes that were described as giving a different perspective on Math 214 material. In addition, several of the Social Science departments have statistical and mathematical methods classes that might be of interest. Usually these courses have prerequisites in their own discipline, and students may have taken the pre-requisite courses when fulfilling the College's general education requirement. However, students should be aware that the College limits the number of statistics-related courses that can count toward graduation. See the discussion under Requirements for the Baccalaureate Degree (p.63 of the 2001 College catalog). Mathematics concentrators will find such courses to be quite different in spirit from Math 401-2.
D-10. Requirements for the minor.
A minor in mathematics requires at least four courses above the 110 level, plus another two courses in mathematics at the 300-400 level. Normally minors in mathematics include Math 111, 112, 211, and 212. Many students choose applications oriented courses to complete their minors. For a discussion, see section D-4 below. Students who plan to take more than the minimum requirements for the minor often include Math 214 in their plan of studies. If a student skips freshman or sophomore courses without receiving AP credit or credit by examination for them, such courses cannot count toward the requirements of the minor in mathematics. Each skipped course must be replaced by a college course at the 300-400 level. A student must earn at least a 2.0 grade point average in those courses submitted to meet the minor requirements.
IV. MATHEMATICS CONCENTRATORS AFTER COLLEGE - WHAT DO THEY DO AND WHERE THEY GO
Mathematics concentrations prepare students for jobs. Contrary to a prevailing stereotype, teaching is one of the things that mathematics concentrators do least! And if salary is a measure of employability, Appendix 2 shows that mathematics concentrators do very well indeed. A detailed discussion of opportunities for mathematics concentrators in various areas appears in Appendix 1 , below.
What do mathematics concentrators do after college? The three main mathematics professional societies (the American Mathematical Society (AMS), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM)) use the World Wide Web to publicize what mathematics students do in the outside world. A good place to start your search is to use some network software to review a career-related site of the American Mathematical Society .
The Mathematics department also maintains a library of career publications prepared by our professional societies. Just as important are the resources available through the College's Career Services Office. These include commercially available publications (e.g., Careers for Number Crunchers and Other Quantitative Types) and publications of the U.S. Department of Labor and Bureau of Labor Statistics about career opportunities. Students sometimes wait until their senior years to contact Career Services, and that is probably a mistake.
The outlook for mathematics concentrators in business, industry, and government has been strong for many years, with fluctuations from year to year depending upon the general economic outlook. Mathematics concentrators who plan to enter business or government positions immediately after their bachelors degrees would do well to link their mathematics studies with related course work in science, computer science, economics, or business.
There is a perennial shortage of high school mathematics teachers with strong mathematics backgrounds, and high school teaching is an employment option for mathematics students. State certification normally requires a fixed set of courses taken in the School of Education as well as a mathematics concentration. (See Section III-D-8 , above.) However, there are other routes to teaching for mathematics concentrators, e.g., through the national ``Teach for America'' program [MA], or through the Peace Corps. Furthermore, teaching in private high schools often does not require state certification.
Post-baccalaureate study in professional schools has always been an option for mathematics concentrators. For further information about Medicine students should consult the College's pre-medical advisor Professor Randolph Coleman (phone 221-2476; e-mail: racole@wm.edu]. For information about Law the contact persons are the College's pre-law advisor Professor John McGlennon (phone 221-304; e-mail: jjmcgl@wm.edu) and Faye Shealy, Associate Dean for Admission, Law School (phone 221-3784; e-mail: ffshea@wm.edu). MBA programs welcome students with strong quantitative backgrounds, and some draw a majority of their students from undergraduate programs in science, engineering, and mathematics. Interested students should contact Mary Catherine Bunde (phone 221-2910; e-mail: marycatherine.bunde@business.wm.edu) for information about Schools of Business here and elsewhere.
Recent years have seen a marked increase in the number of mathematics concentrators who pursue graduate study in disciplines such as engineering, economics, or other sciences. In some universities, twenty-five percent of graduating mathematics majors pursue graduate study in other Arts and Sciences disciplines. Recent articles include [CP] and [FI]. Students contemplating these options must carefully plan their undergraduate programs to make sure that they have the required basic courses in the other field to make it clear to graduate admissions officers that they have a reasonable chance of success in that other field. Appendix 1, Section II-F contains a discussion of career opportunities for mathematics concentrators in other fields.
Continued study of the mathematical sciences in graduate school is an excellent option for some students. A masters degree in operations research, statistics, applied mathematics, or computer science is a strong employment credential, as a recent SIAM study shows [SI]. In many cases, students can obtain financial aid during such study through tuition waivers, graduate assistantships, and fellowships. A listing of such fellowship possibilities is published annually by the American Mathematical Society [A1] and is available through the department. In other cases, a student decides on graduate study after a year or two of employment, and in such cases, the student's employer may have programs to offset the cost of graduate study considered relevant to the employer's needs.
Finally, some mathematics concentrators pursue doctoral study in mathematics. It is hard to know how to advise undergraduates about doctoral study in mathematics. A doctoral program usually takes five or six years of additional study. From time to time, the job for mathematics Ph.D.s has been terrible. Right now it is good. Six years from now, who knows? In addition, there are major changes afoot regarding the kinds of employment available to new mathematics doctoral recipients. In the past, the goal of mathematics doctoral study was to produce academic mathematicians, but that is changing as industry and government discover that they have problems that we can help solve. As a result, a number of students decide to pursue doctoral study after a year or two of employment.
One thing that we can assure prospective doctoral students is that very few students pay for their doctoral study in mathematics. Fellowships and assistantships are normally available to support such study. A listing, published annually by the AMS [A1], is available through the department. For more information, see Section C below.
B. The Local Picture: Our Mathematics Alumni
Our most recent large scale survey of mathematics alumni was completed in 1990, giving us a picture of the career paths of our mathematics concentrators over the previous eleven years. The response rate was only 52%, making the resulting data somewhat anecdotal.
At the time of our survey, 8% of our alumni described themselves as full time students, pursuing an advanced degree. The following table shows how the other 92% of our bachelors graduates classify their current positions. (Due to round off, the percentages total more than 92%.)
Description | Percentage | |
1. | systems analyst, network analyst, information analyst, software engineer, network analyst, programmer, operations research analyst | 28% |
2. | actuary/statistical analyst | 10% |
3. | secondary school teacher | 10% |
4. | securities analyst, budget analyst, market analyst | 9% |
5. | applied mathematician | 4% |
6. | lawyer | 4% |
7. | medical/dental/health professions | 3% |
8. | manager | 3% |
9. | college professor | 3% |
10. | nuclear power specialist | 2% |
11. | electrical engineer | 2% |
12. | other | 16% |
Two-thirds of our graduates pursued some
post-baccalaureate education and just over half
had received at least one post-graduate
degree. The most popular graduate degree field was
operations research, followed by mathematics,
computer science, business, physics, economics,
and education. About 4% of our alumni held or
were pursuing degrees in medical, dental, or
veterinary fields, and about the same number held
or were pursuing law degrees. Other graduate
degrees obtained by our mathematics concentrators
include chemistry, electrical engineering,
materials science, biochemistry, biostatistics,
environmental affairs, statistics, biomedical
engineering, history of science, philosophy,
theology, and business.
C. Graduate Study in the Mathematical Sciences
Different people attend graduate school at different points in their lives. Some students apply in the fall of their senior years for admission to graduate school in the next fall term. Other students work in business, industry, or government for a few years, and then apply to graduate school on a part-time basis, often with financial support from their employers. Others choose to return to school full-time, after a few years of work.
Letters of recommendation and scores from the Graduate Record Examination (GRE) are required for admission to graduate programs. Several of the letters should be from faculty members who know the student's work in advanced mathematics courses. You would normally take the GRE General Test and the Advanced Mathematics Test during the fall before they plan to enter a graduate program, usually no later than October and November. Information about the GRE test is available from Professors Margo Schaefer and Roy Mathias, as well as from the Career Services Office.
Deciding whether or not to attend graduate school is a major decision and you should take care not to discount the possibility of going on to graduate school too early in your mathematical studies. The more abstract emphasis in certain courses-Math 211, 214, 307, and 311, for example-might be somewhat disconcerting upon first acquaintance and students sometimes rule out graduate study on the basis of their first contact with abstract mathematics. Such a snap judgment is usually a mistake.
At least one of Math 307 and Math 311 should be completed no later than the student's junior year. This allows you to make a more informed judgment about applying to graduate school in the fall of your senior year. It is certainly not the case that graduate mathematical studies all deal with theoretical mathematics. However, even in the most applied parts of the mathematical sciences, students will encounter a progressively more abstract emphasis as they delve deeper, and you need experience with several theoretical courses before making a decision about further study. In addition, graduate admissions officers, particularly in doctoral programs, want to see the results of courses like Math 307 and 311 before making admissions decisions.
You should ask faculty advisors for advice on choosing a graduate school. Depending on your mathematical interests, some programs will be more appropriate than others, and faculty advisers can help you judge the relative strengths of various programs. One place to start comparing graduate programs is the annual AMS publication [A1]. It lists mathematical sciences graduate programs and shows the areas in which each specializes by giving the broad research areas studied by recent doctoral graduates. Just as important, it gives details about financial support available for graduate study at various universities. Another important source of information is the periodic National Research Council report on the quality of research graduate programs. The American Mathematical Society publishes rankings of mathematics doctoral programs based upon these reports. A new report was issued in 1995 [A2,A3]. A web version of the doctoral program ranking can be found at http://www.ibc.wustl.edu/nrc_{-}rankings/view.cgi. The magazine US News & World Report also ranks PhD programs in mathematics at its website.
It is an unusual senior in college who knows for sure what he or she will choose as a dissertation area after two or three years of graduate study. One reasonable approach to address this issue is to choose a large, high quality progam (as judged by the National Research Council). Such a program will offer a wide spectrum of good choices for an eventual research area.
The Mathematics Department has several co-curricular activities designed to enhance the undergraduate experience here. These include the Mathematics Organization for Students (MOSt), periodic colloquia such as the annual Cissy Paterson Lectures that are aimed at undergraduates, and various departmental honors and prizes.
MOSt is a student organization whose goal is to bring together mathematics students, whether or not they are mathematics concentrators, for periodic meetings, talks by students, field trips to nearby federal laboratories that apply mathematics in a significant way, and career-related talks by mathematics alumni. In addition, MOSt assists the department by conducting the annual survey in which graduating seniors are asked to evaluate their experience as mathematics concentrators, and MOSt cooperates with the department on other student-related projects such as review sessions for the mathematics GRE exam and the preparation of this handbook.
Using funds from the Cissy Patterson endowment, the department sponsors annual undergraduate-oriented lectures by distinguished visitors. Recent speakers include:
Mathematics competitions continue to be a national tradition in undergraduate mathematics, and William and Mary students are frequent participants in the Putnam Examination and in the national Mathematical Contest in Modelling. The first of these is a one day examination that focusses on clever use of knowledge from the first year or two of college by individual students. The second is an examination in which teams of students spend a weekend studying, solving, and writing up an applied problem that draws on their entire undergraduate program of studies. A regional mathematics competition is sponsored annually by Virginia Tech, and William and Mary mathematics students usually do well.
Each year at graduation, the department makes two awards to graduating seniors. These are the William and Mary Prize in Mathematics, awarded for superior achievement in, and unusually strong dedication to the study of, mathematics, and the Luther Connor Prize, awarded for contagious enthusiasm for mathematics and an attitude of respect and concern for others. The names of recipients of these two prizes are displayed on plaques in the main lobby of the first floor of Jones Hall.
APPENDIX 1: APPLIED MATHEMATICS IN BUSINESS, INDUSTRY, AND GOVERNMENT
In this appendix we give more details about
professional opportunities for mathematics
concentrators nationally. Most of the following
material consists of updated and abbreviated
excerpts from a booklet Professional
Opportunities in the Mathematical
Sciences published by the Mathematical
Association of America [M1], quoted with
permission. Another publication of the
Mathematical Association of America, 101
Careers in Mathematics [ST]
describes the careers of 101 young mathematical
scientists and contains reprints of many
career-related articles from Math Horizons.
The book [LD] contains useful information
concerning career opportunities for math majors.
Some good web resources are:
http://www.maa.org/careers/index.html
This appendix is organized as follows:
Private industry, business, and government are major employers of bachelors and masters level mathematicians, but it is often hard to recognize mathematicians by their job titles. As a start, we will give a sample of the job titles that one might expect an applied mathematician in business, industry, or government to have, and give brief descriptions of what such a person might do.
A. Job title: Operations Researcher, Operations Analyst, Systems Analyst. This type of applied mathematician constructs mathematical models of complex structures - social, economic and technical, civil and military, government, business, and industrial. The models must be complex enough to approximate the real world with some predictive value, and simple enough to be analyzed. Part II-B of this appendix discusses operations research in more detail. Essentially, it is the ``mathematics of the decision sciences'' and uses mathematics, statistics, and computer science to provide the quantitative foundation for business or governmental decisions.
B. Job title: Programmer. Particularly in companies and government agencies with large computers, there is a demand for people who can communicate with a computer correctly and efficiently. Large computers are expensive to operate and companies place a premium on efficient usage. The same is true for software design companies that create applications programs for millions of small computer users. In such situations, someone who can understand vaguely worded problems and translate them into efficient algorithms becomes very valuable. The advent of parallel computing has increased the need for qualified mathematical programmers. Mathematical problems abound in programming, even though most users of computers do not understand such things as the non-commutativity of finite arithmetic, or the logic of serial and parallel algorithms, or how to estimate errors accurately in approximations. To understand such problems requires a sound knowledge of mathematics, and anyone who expects to work as a programmer in government, business, or industry should have considerable computer experience, the more the better.
C. Job title: Statistician. This job title can easily overlap with the Operations Research category above and includes mathematicians who can utilize data and say what data are appropriate. Data abound in government, business and industry, and those who know how to extract usable, reliable information from data are very useful. Statistically sound design of experiments, together with planned analysis of data in accord with the experimental design, can be used to great advantage in improving products and processes, and so are of great importance to a variety of firms and agencies. More information about statistics may be found in Part II-C.
D. Job title: Applied Mathematician. Traditionally the term ``applied mathematician'' has meant someone with a differential equations, physics, and engineering orientation. This remains a fundamental field in government, industry, and military applications. There is still a tremendous interest in analyzing equations of motion and those of steady state fields, and recent computer advances have made formerly impractical problems routinely solvable. As computers become more powerful, mathematical modeling has emerged as a new way of conducting experiments. For example, today mathematical models are routinely used to simulate fluid flow problems associated with aircraft wing design, thereby saving the immense costs of wind tunnel testing of many prototypes. Part II-A gives more information about Classical Applied Mathematics and Engineering.
E. Job title: Actuary, and Financial Analyst. An actuary is an applied mathematician who specializes in the design, financing, and operation of insurance plans of all kinds, and of annuity and welfare plans [Mi]. In the past, actuaries were employed primarily by life and property insurance companies, but more recently actuaries can be found in other large financial institutions and government agencies that have a need to understand the consequences of proposed changes in pension systems or government regulation. See Section II-D below for more information on actuarial sciences and on the preparation of actuaries.
Recent years have seen the rise of a new mathematical applications area, called ``mathematics of finance,'' and its practitioners in business and industry who are sometimes called Financial Analysts, although that term has many other meanings too. New financial investment instruments called ``derivative securities'' are used by companies to protect themselves against currency fluctuations and shifts in interest rates, for example, and creating mathematical models of such phenomena has become a vigorous part of applied mathematics. See Section II-E for further information about the mathematics of finance.
F. Job title: Information Scientist (and other job titles containing the words ``Control Systems,'' or ``Communications Systems.'') Information scientists are concerned with problems involving the information-bearing characteristics of signals, patterns, and observations; with information conversion from one form to another; and with storage, retrieval, transmission and reception of information. They use many of the same tools as statisticians. Perhaps the primary distinction to be made between statistics and information sciences is that the former tends to be concerned with after-the-fact analysis of relatively slow processes such as interpretation of national economic data, whereas the latter are more concerned with the interpretation as they occur of fast events, such as modulated radio waves.
G. Job title: Consultant. A mathematical consultant usually has an established reputation for solving problems and carrying out research, and frequently works for clients on a short-term (even daily) basis. For example, if the client company or agency employs its own mathematicians, they may have formulated the scope and content of a mathematical investigation and identified the particular kind of mathematical talent and experience required to carry it forward. In such a case, the consultant reviews their work, advises them, and suggests methods of approach to be worked out in detail by the client's resident staff. In other cases, it may be necessary that the consultant master the fundamentals of one or more fields of science, engineering, business or government operations or management in which complex quantitative issues susceptible to mathematical treatment can become urgent and economically important. Most mathematical consultants have full time positions elsewhere and consult ``on the side.'' For example, university professors are often consultants. Some consultants have private consulting practices while others are members of consulting firms, sharing a practice with other mathematicians or with a more diversified professional staff capable of under taking total responsibility for interdisciplinary assignments.
II. Applied Mathematicians - Problems They Solve, and Subjects They Studied.
In this section we will describe the kinds of problems that mathematicians employed in business, industry, and government might encounter, and the kinds of undergraduate course work that would be good preparation for each kind of application. The titles of the sections below are not consistent with the job titles in Section I above, but they are traditional.
A. Classical applied mathematics
1) What problems does it solve?
The most traditional role of the applied mathematician in a professional setting has been in the solution of problems arising from physical phenomena and engineering. From its very inception, calculus has been applied to laws of motion and to understanding the consequences of interacting forces. While the early applied mathematicians were necessarily physicists and engineers as well, the modern setting calls for the mathematician to serve as a member of a team of specialists, each bringing a particular talent to bear on problems.
In a broad sense, the applied mathematician is instrumental in designing and analyzing models of systems and in testing and evaluating performance. It is a characteristic of this field that the technical questions readily move across once clearly distinguished boundaries. Whether in research and development or in industrial production, the applied mathematician must interact with engineers, physicists, programmers, and other specialists. The common goal is to find ways to improve quality, reduce cost, and increase productivity. The analytical skills of the mathematician are particularly valuable in consulting for technical services or trouble shooting.
Recent mathematical research in combination with increasing computer sophistication has opened fields that saw little development in the past due to their intractability to classical analytic techniques. These include the solution of problems involving enormous numbers of equations, the numerical simulation of complex systems such as power grids, and the application of control theory and other mathematical tools to the management of traffic or industrial processes.
The tasks of the applied mathematician are as diverse as the constituencies served. The broad category of engineering disciplines is a rich source of mathematical problems. In the aeronautical field a mathematician may help to develop models for atmospheric flight including the analysis of performance in search of optimal trajectories. Biomedical engineers may rely on mathematicians when designing and interpreting theoretical models of chemical and biological processes. A mechanical engineer may require a study of heat transfer by conduction, convection, and radiation resulting from a gas turbine.
Many problems involve scientific or engineering data and the use of computer techniques to answer questions arising in research, plant operations, product distribution systems, inventory controls, and business system analyses. Mathematicians seek efficient and reliable computer programs for the numerical solution of initial value problems or special function routines capable of delivering accurate answers over a wide range of parameters. While the methods most frequently applied are based in ordinary and partial differential equations, there is increasing involvement of probability, statistics, and computing.
2) What should one study in college?
For a career in classical applied mathematics, a student should obtain a thorough background in calculus, linear algebra, ordinary and partial differential equations, probability, statistics, numerical analysis, and vector calculus. These courses should include some extensive use of computing, or they should be supplemented by appropriate courses in computer sciences. Supporting work should include physics and basic engineering courses.
3) Additional Resource:
Careers in Applied Mathematics and
Computational Mathematics, by the Society for
Industrial and Applied Mathematics (SIAM),
at
the weblink: http://www.siam.org/careers/
1) What problems does it solve?
Operations Research (OR) has been defined as a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control. A great deal of operations research today deals with determining the optimum way to achieve a goal based on some mathematical or statistical model of a situation. The task of the operations research worker is to present the quantitative aspect in an intelligible form and to point out, if possible, the non-quantitative aspects that may need consideration by the executive before reaching decisions.
Operations Research as we know it today is primarily an out-growth of military research in World War II that sought optimal ways to allocate scarce resources. This included questions such as ``How should patrol aircraft be deployed to maximize the expected number of enemy submarines detected in a limited number of hours of search?'' and ``How should a limited inventory of spare parts be distributed among units in the field, advance depots, and distribution warehouses to minimize equipment down-time due to parts shortages?'' At the end of the war, most operations researchers moved into industry, where similar questions in budgeting, planning, marketing, decision-making, and other aspects of management were in need of answers.
A sample problem might be the optimum operation of toll booths on a bridge or turnpike. The problem is to achieve the best balance between having idle attendants during slack hours and too much delay during rush-hour. One would have to determine the statistics of the traffic flow, construct a mathematical model of the queuing system, determine the expected number of idle attendants and the expected delay as a function of the number of attendants. The resultant expression is then analyzed to determine the optimum performance, given any other restraints imposed.
The solution to the toll-booth problem is well understood, but there are other OR problems that are mathematically challenging and still unsolved. One that is far from solved in general was originally described as the following traveling salesman problem. Sales representatives of a corporation have customers in each of a list of cities. The goal is to find the shortest tour enabling them to visit all their customers exactly once. This can be translated into a problem in graph theory - how to characterize the shortest path joining a certain number of nodes. It can also be translated into a problem of minimizing a certain function subject to constraints. Linear and integer programming are among the techniques utilized in attempting to solve problems of this type.
2) What should one study in college?
The major mathematical tools of OR are vector calculus, linear algebra, differential and difference equations, probability, statistics, and computer programming. Other courses particularly relevant to this field include number theory, abstract algebra, graph theory and combinatorics. Still other relevant courses may be given in or outside of the mathematics department, e.g., linear programming, control theory, integer programming, dynamic programming, game theory, and queuing theory, as well as computer science courses and simulation.
An individual with a bachelor's degree in mathematics and an applied minor can possibly obtain direct employment in operations research, but a masters degree in OR is the credential preferred by most employers. A good working knowledge of economics, finance, and organization theory is also valuable, and that is something that a mathematics undergraduate can pursue, e.g., as a minor.
3) Additional Resource:
Careers in Operations Research , by the
Institute for Operations Research and the
Management Sciences (INFORMS),
weblink:
http://www.informs.org/Edu/Career/booklet.html
1) What problems does it solve?
Statistics has been described as the science concerned with making sense out of numbers, and as the science of dealing with uncertainty. Statisticians give advice on the statistical design of experiments, conduct surveys, analyze data with the help of existing statistical techniques, or devise new methods for analyzing data. Today, computing plays an important role in the work of statisticians. Widely available software packages such as SAS, SPSS, and S-Plus, are standard tools of the trade. In business, industry, and government, statisticians rarely work by themselves. They collaborate with specialists in fields such as agriculture, biological and health sciences, economics, psychology, sociology, as well as business and industry. Other fields for statistical applications are law and public policy. Many statisticians hold positions in government agencies such as the National Institute of Standards and Technology, the Bureau of the Census, the Bureau of Labor Statistics, the Department of Agriculture, the Department of Defense, the National Institutes of Health, and the Environmental Protection Agency.
2) What should one study in college?
In recent years more and more colleges and universities have begun to offer undergraduate majors in statistics, though more often than not students will take statistics as a minor along with programs in mathematics and/or computer science, the biological or social sciences, or business. A student's undergraduate education should include computer science courses as well as calculus, linear algebra, and probability and statistics. Other courses that make use of statistics (e.g., stochastic modeling) can also be useful, and students should be aware that statisticians can sometimes be found in departments such as biology, business, economics, psychology, sociology, and political science. If job opportunities are good for mathematics majors with strong undergraduate training in statistics, they are even better for undergraduate mathematics majors who obtain a masters degree in applied statistics.
3) Additional Resource:
Careers in Statistics, by the American
Statistical Association,
weblink:
http://www.amstat.org/education/careers.html
1) What problems does it solve?
Actuarial Science is a part of applied mathematics that originated at least 200 years ago. Two of the earliest developments were in regard to mortality or life tables and to compound calculations. These were combined in the eighteenth and nineteenth centuries to provide the scientific basis for individual life insurance and life annuity contracts. In this century, applications of actuarial mathematics have become more numerous and more complicated. Individual insurance contracts have been refined greatly, and there has been a tremendous growth in group plans providing life, health, disability and pension benefits, together with the parallel development of Social Security. Another part of actuarial science is concerned with the evaluation of non-life insurance risks such as those covered by automobile or fire insurance. The extent and complexity of these varied insurance plans, and the maturing of pension and Social Security systems, have created strong demand for competent actuaries.
In insurance companies, where most actuaries work, the actuary is responsible for seeing that the risk is properly defined and evaluated, that a fair price is charged for assuming the risk, and proper provision is made to pay all claims and expenses as they occur. Insurance company actuaries engage in a variety of other important activities ranging from research to management functions. For example, an actuary may study the claims experience, in particular, the mortality and survival experience of insured persons. Or they may apply mathematical models or techniques of operations research to insurance company problems and may engage in corporate planning.
Consulting actuaries, who now include more than one third of all active actuaries in America, offer professional advice to corporations, insurance companies, federal, state, and local governments, labor unions, joint labor-management trustees, and attorneys. The need for actuaries in government work is steadily increasing because of governmental involvement in old age, survivors, disability, and medical benefits provided by Social Security, and in the supervision of insurance and pension funds.
2) What should one study in college?
Professional certification as an actuary comes from the Society of Actuaries for those working in life or health insurance and from the Casualty Actuarial Society for those who work in property and liability insurance. Certification requires passing a sequence of examinations. After passing the first few examinations, one becomes an Associate of the Society. After completing all examinations, one is eligible to become a Fellow of the Society. The material for many of these exams is given in courses offered by the Mathematics Department (e.g., calculus, probability, statistics, operations research). Information about the examinations (typically given in November and May) required to become an Associate or Fellow of the Society of Actuaries can be found on their web site at www.soa.org. Study materials for the exams can be purchased from Mad River Books (A Division of ACTEX Publications) at www.actexmadriver.com. For further information, see the actuarial advisor Professor Roy Mathias who administers the exams on campus.
Typically, a student should aim to pass at least two of the Associateship examinations before leaving college. The other actuarial examinations are typically taken after the student is employed in actuarial mathematics. Courses in calculus, linear algebra, probability, statistics, numerical analysis, operations research, and computer science are particularly important in preparing for the first four examinations, and courses in accounting, economics, finance, and marketing can also prepare students for later examinations. [MI]
3) Additional Resource:
Actuaries Make a Difference , by the Casualty
Actuarial Society and Society of Actuaries,
weblink:
http://www.BeAnActuary.org/
1) What problems does it solve?
Over the past two decades, new quantitative techniques have transformed the investment process and the finance industry. Today, banks and other financial institutions gain competitive advantage through technical innovation. Powerful mathematical models are used to measure risk, and value complicated transactions. Computational methods transform these theories into tools that sit at the fingertips of traders, portfolio managers, regulators, and risk managers, bringing greater efficiency and rigor to financial markets. These developments have led to a large and growing demand for talented people trained in the mathematics of finance.
New kinds of financial instruments called ``derivative securities'' have become major tools of financial planning for corporations, banks, mutual funds, and other large financial institutions. Called derivative securities because their value is derived from values of other commodities in the market, these include (mixtures of) currency repurchase agreements, put and call options, and futures contracts of various kinds. Making mathematical models of these financial instruments has become a rapidly growing part of applied mathematics, and the mathematical models are used to understand the value and the hedging structure of many of the derivatives. Typical models are based upon stochastic (i.e., time-dependent) partial differential equations that are usually solved by numerical techniques. See [PR] for a discussion of this relatively new field of mathematics. Several Mathematics in Finance Master Programs have been established in some of nation's most prestigious institutes, and a list can be found at the weblink below in 3).
2) What should one study in college?
Mathematical topics that are of particular use in the mathematics of finance are calculus, differential and finite difference equations, probability and statistics, numerical analysis, and modern algebra. Stochastic modeling courses could also be valuable as might courses in mathematics and other departments that study the diffusion, or heat, equation. Other valuable courses outside of mathematics include courses in finance, mathematical economics, and perhaps other social science courses that use game theory to model human behavior.
3) Additional Resource:
F. Mathematics in interdisciplinary areas.
1) What problems does it solve?
Mathematics has always been used as a tool for organizing and understanding the physical sciences. Today mathematics is also applied to other disciplines such as biology, medicine, management, linguistics, and the social sciences.
Initially the area of mathematics which was of primary importance outside the physical sciences was classical statistics, used in the collection and analysis of data. More recently there has been a growing interest in the exploration of other areas of mathematics for the construction of non-statistical models. These endeavors coexist and sometimes overlap, but are considered to be different aspects of their respective disciplines. The professional nomenclature reflects this difference. Economists who specialize in the application of statistics to their field are called econometricians; biologists, biometricians; psychologists, psychometricians, and so on. Economists who are primarily interested in non-statistical modeling are called mathematical economists. Similarly there are mathematical biologists, mathematical psychologists, and so on. The growing importance of these professions provides an opportunity to combine mathematical training with a serious interest in another discipline.
The role of statistics in applications of mathematics to the social sciences, biology and medicine is extremely influential, especially, as has been noted above, in a major category of interdisciplinary research. The mathematics used in non-statistical modeling varies with the kind of problem under consideration. The construction of a mathematical model entails the formulation of laws or axioms which describe in mathematical terms the (necessarily idealized) underlying structure of a system. Examples of systems range from free competitive economics to neural networks. Since we are discussing such a variety of fields and diversity of approaches within each field it is hardly possible to enumerate all the branches of mathematics used. Furthermore it must be remembered that many of these efforts are still young; the number of mathematical tools drawn upon and their level of sophistication are continually increasing. We offer below a sampling of the kinds of problems treated and the kinds of mathematics used.
Mathematical economics is the oldest, and probably the best developed of these interdisciplinary pursuits. The first Nobel Prize in Economics went to a principal founder of mathematical economics, and a more recent Nobel Prize in Economics was awarded for the mathematical study of derivative securities pricing, the Black-Scholes equation. One topic which has been the subject of research in this area is the existence of equilibrium in a competitive economy. The problem simply stated is this: given a free market in which prices respond to the law of supply and demand and a set of assumptions about the behavior of consumers and producers, will prices eventually regulate themselves to values at which supply and demand exactly balance? Other topics which occur concern individual behavior, stability of equilibria, oligopolic systems and the economics of the welfare state. Sociologists and political scientists have adopted some of the techniques of mathematical economics to study social and political issues. Linear algebra and real analysis are heavily used, as well as differential and difference equations, topology, set theory, logic, combinatorial mathematics, and game theory.
One of the earliest uses of mathematics in biology was in the study of population growth. If we assume that the growth of a population of organisms is not affected by pressure of resources, then we arrive quickly at the conclusion that the number of organisms existing after a given period of time is a constant multiple of an exponential function of the time period elapsed. However, as we take into account additional factors such as availability of resources, the model becomes more complicated and the mathematics more sophisticated. Other areas in biology and medicine which are studied by means of mathematical models are immunology, epidemiology, ion transfer across membranes, and cell differentiation. Neurophysiology is closely associated with psychology in the study of models of perception and learning. Frequently used mathematical tools are ordinary and partial differential equations, difference equations, dynamical systems, control theory, optimization theory, stochastic processes, and computer science, as well as some topology. The last decade has seen a significant growth in applications of mathematics to biological problems, to such an extent that every national mathematics meeting seems to have special sections devoted to mathematical biology.
In psychology, one finds mathematical modeling closely associated with experimentation. For example, consider the ``simple learning'' model. A subject is placed in a repetitive choice situation in which different responses carry different rewards. As the reward pattern reveals itself to the subject, the subject's responses slowly change. The problem is to explain the laws governing the evaluation of the choice pattern within the framework of the experiment. More complex learning situations are studied, as well as problems in stimulus response, reaction time, preference behavior, and social interaction. Computer modeling is used to simulate the organization of the nervous system. Another kind of problem which arises in mathematical psychology occurs in the theory of measurement and scaling. The categories of mathematics which have been heavily used are probability and stochastic processes, ordinary and partial differential equations, computer science, combinatorial mathematics, set theory, and some analysis.
Mathematical linguistics has become a major force in the study of linguistics, the science of languages. It has some relationship with mathematical psychology since it is concerned with the range of humanly possible linguistic structures rather than with the particular qualities of any given language. This area makes use primarily of set theory, logic, algebra, automata theory, and computer mathematics.
2) What should one study in college?
There is no well-defined educational path for students wishing to enter these interdisciplinary areas. A sampling of those now engaged in each of the various fields would show considerable diversity in patterns of formal education, although it can be safely said that there is little opportunity without a doctoral degree. A strong undergraduate mathematical education with a double major would be the ideal start. Short of that ambitious program, a major in mathematics with considerable course work in the other field would be a good beginning. It does appear to be important that the mathematical training be started early, and preferably that it include some work in statistics and computer science. There is no prescription for graduate study. This depends very much on finding an individual or group working in the area one would like to pursue. Mathematical biologists and psychologists may be found in some departments of biology and psychology, respectively, but often are based in departments of mathematics or applied mathematics. A student who is interested in entering one of these interdisciplinary fields, or any of the others involving social sciences would do well to engage in some preliminary research to locate an appropriate graduate department.
3) Additional Resource:
weblinks: The Econometric Society at http://www.econometricsociety.org/ and The Society for Mathematical Biology at http://www.smb.org.
APPENDIX 2: WHAT DO THEY EARN?
In 1993, the College Placement Council published average salary data by major for bachelors degree recipients entering their first jobs. [DA]
College Major | Average Salary |
Engineering | $32,800 |
Computer Science | $30,900 |
Accounting | $27,800 |
Mathematics and Actuarial Science | $27,600 |
Mathematics | $26,100 |
Business | $25,600 |
Science other than mathematics | $25,300 |
More recent (and more detailed) information shows the following data. [NACE], [SA]
College Major | Average Salary | Average Salary | Average Salary |
1996 | 1997 | 1998 | |
Accounting | $29,375 | $ 30,154 | $ 32,825 |
Business | $27,274 | $ 29,346 | $ 31,454 |
Marketing | $26,777 | $ 28,031 | $ 29,231 |
Engineering, Civil | $31,308 | $ 33,031 | $ 35,335 |
Engineering, Chemical | $41,443 | $ 42,802 | $ 45,104 |
Engineering, Computer | $37,529 | $ 40,093 | $ 43,865 |
Engineering, Electrical | $38,025 | $ 39,546 | $ 43,282 |
Engineering, Mechanical | $37,036 | $ 38,287 | $ 41,260 |
Engineering, Nuclear | $37,453 | $ 37,050 | $ 41,517 |
Engineering, Petroleum | $39,770 | $ 43,444 | $ 49,926 |
Engineering, Technology | $33,826 | $ 35,498 | $ 39,390 |
Chemistry | $29,743 | $ 34,135 | $ 33,892 |
Mathematics | $29,745 | $ 32,151 | $ 36,203 |
Physics | $30,484 | $ 35,554 | $ 36,139 |
Humanities | $24,285 | $ 25,078 | $ 28,447 |
Social Sciences | $24,635 | $ 25,103 | $ 27,149 |
Computer Science | $35,222 | $ 37,215 | $ 41,949 |
It is likely that the trend of competitive earning potential for mathematics concentrators continues.
APPENDIX 3: REFEREED PUBLICATIONS BY UNDERGRADUATE RESEARCH STUDENTS AT WILLIAM AND MARY.
For a list of refereed publications by
undergraduates, please click here.
The following list shows the department's
full-time faculty, plus a bit of information on
each. Further data can be found through the
faculty section of the department's home page,
whose address is http://www.math.wm.edu.
Vladimir Bolotnikov (B.S. and M.S., Kharkov
State University; Ph.D., Ben-Gurion University)
Vladimir's mathematical interests include operator and function theory, and problems of interpolation.
John Drew (B.A., Case Western University; Ph.D., University of Minnesota)
John specializes in matrix theory with a current emphasis on completely positive matrices and matrix completion problems. His outside interests include mathematical puzzles, bicycling, woodworking, and photography.
Shandelle Henson (B.S., Southern College of Seventh-day Adventists; M.A., Duke University; Ph.D., University of Tennessee, Knoxville)
Shandelle studies dynamical systems as applied to population biology. She also enjoys logic, the philosophy of science, biking, running and hiking.
Charles Johnson (B.A., Northwestern; Ph.D., California Institute of Technology)
Throughout his career, Charles has studied mathematics and economics. His research focusses on matrix theory and uses tools from combinatorics and optimization. Outside interests include bridge (he is a life master), current events, and Chinese history.
Dana T. Johnson (B.A., Northwestern; M.Ed., University of Maryland)
Dana's interests are in teaching of mathematics. Her outside interests include sewing and baking bread.
Rex Kincaid (B.A., Depauw University; Ph.D., Purdue University)
Rex's mathematical interests lie in the area of operations research, and he studies discrete optimization problems and meta- heuristics. Outside interests include volleyball, frisbee, golf, and reading fiction.
Larry Leemis (B.S. and Ph.D., Purdue University)
Larry works in the general area of operations research, with a focus on reliability theory and simulation. He lists photography and frisbee as outside interests.
Robert Michael Lewis (B.S., Rice University; Ph.D., Rice University)
Michael's research interests lie in the area of optimization, particularly optimization techniques for scientific and engineering applications, and the optimization of systems governed by differential equations. He admits to reading and botany as outside interests.
Chi-Kwong Li (B.A. and Ph.D., University of Hong Kong)
Chi-Kwong's mathematical interests lie in matrix analysis, operator theory, and combinatorics. Specific research topics include problems dealing with numerical ranges, normed spaces, linear preserver problems and matrix inequalities. He includes Chinese literature and music (particularly the Chinese flute) among his outside interests.
David Lutzer (B.S., Creighton University; Ph.D., University of Washington)
David's research interests lie in set-theoretic topology, particularly the study of ordered spaces and function spaces. Outside interests include reading political biographies, listening to classical music, and Chinese cooking.
Roy Mathias (B.A., Cambridge University; Ph.D., Johns Hopkins University)
Roy studies numerical linear algebra, a part of numerical analysis, and has strong interests in matrix theory and probability. His outside interests include hiking, tennis, chess, and the game of Go.
Leiba Rodman (B.A., Latvian State University; Ph.D., Tel-Aviv University)
Leiba's research involves operator theory, linear algebra, control systems, and function theory. His outside interests include reading and classical music.
George Rublein (B.A., Saint Mary's of Texas; Ph.D., University of Illinois)
In recent years, George has shifted his interests to mathematics education, with particular attention to curricular reform of general mathematical education. Outside interests include music and bridge, and not gardening.
Margaret Schaefer (B.A., Smith College; Ph.D., Northwestern University)
Margaret's interests lie in operations research, and particularly in the areas of resource allocation, inventory optimization, and decision analysis. Outside interests include travel, reading, gardening, and researching stocks for an investment club.
Junping Shi (Ph.D., Brigham Young University)
Junping's research interests lie in the theory and application of differential equations and nonlinear analysis, particularly bifurcation theory, nonlinear elliptic and parabolic equations, infinite dimensional dynamical systems, and applications to mathematical biology, material sciences. Outside interests include reading, travel, music and surfing the net.
Ilya Spitkovsky (B.A. and Ph.D., University of Odessa; D.Sc., Georgian Academy of Sciences)
Ilya studies operator theory, complex analysis, integral equations, and matrix theory. In particular, he is interested in Toeplitz operators, Weiner-Hopf factorizations, extension and interpolation problems, and properties of numerical ranges. In his spare time, he enjoys listening to Russian bards' songs, reading science fiction, and travelling.
David Stanford (B.A., Hartwick College; Ph.D., University of North Carolina)
David's mathematical interests lie in matrix theory. His outside interests include music and music-making, and include playing early music on the recorder.
Michael Trosset (B.A., Rice University; Ph.D., University of California - Berkeley)
Mike's general interests are in statistics and computational mathematics. His current research involves distance geometry and stochastic optimization. Other interests include cinema, hiking, tennis, and various board games.
Hugo Woerdeman (B.A. and Ph.D., Vrije Universiteit, Amsterdam)
Hugo studies operator theory, with special emphasis on completion, extension, and interpolation problems. His outside interests include biking, tennis, squash, and racquet ball.
Nahum Zobin (M.S., Kazan State University; Ph.D., Voronezh State University)
Nahum's mathematical interests are mainly in analysis and operator theory, very broadly interpreted, and also in mathematical physics, convex geometry, Coxeter groups, affine algebraic geometry, and applied mathematics, where his main interests are hydrodynamics and medical imaging. Besides mathematics, Nahum is interested in poetry and literature.
In addition, the department has several
part time, or adjunct, faculty who teach in our
program. Adjunct appointments change from one
semester to the next and a list can be obtained
from the department office.
APPENDIX 5: PROJECTED COURSE ROTATION
Normally, the department offers Math 104,
106, 108, 110, 111, 112, 150, 211, 212, and 214
every semester, and Math 103 every Fall
semester. The upper division courses are normally
rotated as follows:
Several of our upper division courses are given in alternate years. In planning their programs, students in the K-12 certification programs will need to plan very carefully, because the Spring semester of their senior years will not be available for taking mathematics courses.
APPENDIX 6: FIVE YEAR BA/MA PROGRAMS AT WILLIAM AND MARY
By very careful planning, mathematics concentrators can design their undergraduate programs in such a way that a fifth year of study can lead to an M.S. in Computer Science with a specialization in Computational Operations Research or an M.S. in traditional Computer Science. Either of these masters degrees would be extremely attractive to potential employers.
Students interested in a one year masters degree in Computer Science must plan their undergraduate years very carefully, and there is no guarantee that such a student will be admitted to the CS graduate degree program. Decisions concerning admission are made by the Department of Computer Science .
Students interested in exploring the possibility of a one-year program to complete the M.S. in Computer Science with a specialization in Computational Operations Research should meet with Rex Kincaid, the chief adviser for graduate studies in the Department of Mathematics. In general, the timetable for this option is similar to that discussed next for a traditional M.S. in computer science; however, the choice of courses differs.
For a traditional M.S. in computer science, the following guidelines show how a William & Mary undergraduate can, with proper preparation, complete a conventional four-year undergraduate B.S. or B.A. degree in an appropriate concentration area (physics, mathematics, chemistry, business, etc) and then, with one additional year of study, earn the M.S. in computer science. The key here is that this one-year M.S. degree would be based on the current computer science 32-hour, no-thesis M.S. degree option (see page 69 in the 2001 graduate catalog).
Two of these 32 hours must be satisfied by passing CSci 710; the other 30 hours correspond to ten conventional three-hour computer science graduate courses. To complete a M.S. degree in just one year of study two of these ten courses must be taken for graduate credit while still an undergraduate (see page 51 of the 2001 undergraduate catalog). The other eight courses would be taken, four per semester, in two intensive semesters (and one summer) of graduate study.
The schedule of study would be as follows:
Note that if a student can't satisfy condition (3), then the one-year M.S. option would become a three-semester option instead, with graduation in December, not August. Note also that the Undergraduate College Catalog states ``¼ undergraduate students of the College who have a grade point average of at least 3.0 may take for graduate credit in their senior year up to six hours of courses normally offered for graduate credit, provided that these hours are in excess of all requirements for the bachelor's degree and that the students obtain the written consent of the instructor, the Chair of the Department ..., the Chair of the Committee on Degrees, and the Graduate Dean of Arts and Sciences, at the time of registration. Such students will be considered the equivalent of unclassified (post baccalaureate) students as far as the application of credit for these courses toward an advanced degree at the College is concerned.''
APPENDIX 7: TEACHING AND RESEARCH CAREERS IN MATHEMATICS
In early school grades, we all get a first impression of mathematics as a random collection of procedures used to solve textbook problems. Some of us later come to realize that mathematics comprises many rich and beautiful structures, designed and built by human beings. And much to the surprise of some, this activity continues today and will surely go on for a long time to come.
Some of the most important players in this enterprise are high-school mathematics teachers. It is certainly the case that many of our mathematics majors have been inspired by one or more of their high-school teachers to study the subject more deeply. Some of those majors want to emulate their teachers by becoming high-school teachers themselves. To teach and learn mathematics in this way can make for a gratifying career.
For those teaching in high-school, there is a vast supply of mathematics that can be exploited for the benefit of their students. For instance, fascinating material may be found in a variety of journals devoted to ``elementary" problems of a strictly mathematical nature. A teacher can return the favor of his or her own teachers by stimulating the more able students with these kinds of resources. Moreover, other courses in the high-school curriculum, especially the science courses, employ mathematics to their own ends. A teacher who sets out to understand how mathematics contributes to these external subjects has established a lifetime agenda of interesting work to do, work that will enhance the scientific and technical skills of students at all levels of mathematical capacity.
Some details about appropriate course selection for those interested in high-school teaching are found in Section III-D-8 of this handbook.
We would like to think that some of our students will also be inspired to study mathematics more deeply by William and Mary faculty. Virtually all of our departmental faculty are engaged in mathematics research of some kind. Research means working on problems whose resolution is unclear. Working at the edge of the known world is the ambition of most academicians, and mathematicians are no exception in this regard. Someone who participates in this lifetime project, and at the same time, can form and re-form the mathematical perceptions of young students, is in an enviable position.
Access to jobs in university mathematics departments ordinarily requires doctoral study. As has already been demonstrated in this handbook, there are many kinds of mathematicians, and mathematics departments are always on the lookout for talented and energetic new faculty of any flavor. Those interested in this professional possibility should look at Section IV-C of this document.