See Changes Affecting Math 320 (fall 2007) General Information The College's general requirements for all majors are described in the Washington University Bulletin.  There are five possible tracks for completing a math major.  These are outlined below.  In the requirements, "upper-level course" always refers to a Math Department course numbered 300 or higher.  Courses numbered in the 500's are graduate-level courses, but they are open to qualified undergraduates as well. In meeting major requirements, note that: There must be at least 18 "stand-alone" upper level math credits, that is, credits not counted toward meeting the requirements (including elective requirements) of any other major, minor, or program.  Students should check with their Undergraduate School with any additional questions about "double-counting" courses toward some other set of requirements. . Both College and department rules require at least half the upper level units required in the mathematics major be earned at Washington University (or Washington University approved overseas programs). Courses transferred from a 2-yr college (such as a community college) cannot count as upper level courses toward the major. The department requires that each graduating senior participate in an exit interview during the senior year. Some seniors are requested to participate instead in a more detailed "extended exit interview." All majors are required to complete the introductory calculus sequence (Math 131-132-233, or its equivalent) and usually also Math 309, 310, and either 318 or 308. We recommend that majors consider taking Math 310 as early as possible since it is a foundation for all upper-level "proof-oriented" courses.  A writing intensive (WI) version of 310, numbered 310W, is offered in fall semesters. The College allows a course to fulfill the WI requirement in the Discovery Curriculum only if it is taken in the junior or senior year.  We strongly recommend that students take Math 310 when they are ready for it and not wait for the junior year just so that 310W can be used to fulfill the WI requirement.  A student may fulfill the WI requirement with a WI course outside the major. Stronger students who are comfortable with proof-oriented courses may be able to bypass such intermediate courses as Math 309, 310, 308 or 318 with permission of an advisor and instead take the sequences 4111-4121 or 429-430 directly.  In all such cases, an upper-level math elective must be substituted in the major requirements for each bypassed course. Freshmen and Sophomores  Usually a major should try to complete the calculus sequence and 309, 310 and 318 (or 308) by the end of the sophomore year.  The following charts indicate some sample course schedules for freshmen and sophomores, depending on where the student enters the calculus sequence.  As long as the prerequisites (described in the Course Listings) are fulfilled, the exact timing of some of the courses may depend on a student's interests.  Asterisks " *** " represent possible elective math courses..

 

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The Five Major Tracks

In addition to the general requirements of the College of Arts and Sciences and the courses in the calculus sequence, each math major must fulfill the requirements for one of the five major tracks described below.  No matter which track is completed, the student's permanent record will simply indicate "Major in Mathematics." Since most majors take 309, 310, and 318 (308), the early requirements for the tracks overlap. Therefore the choice of a track doesn't need to be made immediately. For most students, a decision by the end of the sophomore year is sufficient. Of course, making the decision as soon as it's practical will let a major choose courses in a specific direction and develop the strongest possible program. A grade of C- or better (CR/NCR = "pass/fail" not allowed) is required for each course fulfilling a major requirement.  The only exception is for courses where the credit was earned through AP, IB, or a similar program. No more than one 3-unit research/independent study course can count toward fulfilling major requirements.  Of course, additional independent studies going beyond the major requirements are permitted. Students cannot receive credit toward the major for BOTH 308 and 318,  411 and 4111,  412 and 4121,  449 and (404 or 405) toward the major. No University College (evening) mathematics course can be applied toward the major, even if the course is cross-listed "same as,"  except with prior approval by the Department's Undergraduate Committee.  Approval would involve consideration of a special need for taking the course through University College and the Committee's judgment of the appropriateness of the course in a particular semester.  University College math courses have department number U20 rather than L24. Math 320 no longer exists; for the purposes of major/minor requirements, Math 320 has been replaced by Math 3200.  Students who earlier took Math 320 may still use it to fulfill major requirements.  However, Math 2200 does not count toward upper level requirements for a major or minor.  A course cross-listed as "same as Math 2200" does not count, even  if another department or program cross-lists the course with a 300-level number. See also the additional comments at the bottom of this page.

 

Track A:  The Traditional Major
Math 310 (or 310W)	Foundations for Higher Mathematics
Math 4111 - 4121	Intro. to Analysis / Intro. to Lebesgue Integration
Math 429 - 430	Linear Algebra, Modern Algebra
Math 3200 or Math 493	Elementary to Intermediate statistics with Data Analysis,  or  Probability
Two other upper level math courses	(Math 417-418 is recommended for students considering graduate study in mathematics.)

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Track B:  Probability and Statistics
One of CSE 126, 131,  200	Computer Science (The University College C-programming course, U20-123 may NOT be automatically substituted: that requires written permission from your advisor or Prof. Freiwald.) Additional comments on options in Computer Science (CSE)
Math 309	Matrix Algebra
Math 310 (or 310W)	Foundations for Higher Mathematics
Math 318 or Math 308	Calculus of Several Variables / Math for the Physical Sciences
Math 493-494	Probability/Mathematical Statistics (University College U20 493-494 may NOT be automatically substituted)
Two other upper level probability and statistics courses from the list ** below	Students who plan to use statistics in practical settings are strongly encouraged to learn to use SAS, e.g., via Math 475

* *    For purposes of this requirement, the upper level probability and statistics courses are 3200, 322, 420, 408, 434, 439, 459, 475, 493, 494, 495 (5061, 5062) Business students may substitute the pair of courses QBA 120 and 121 (Managerial Statistics I, II) for Math 3200.  In that case, MAth 3200 cannot also count toward the major. Students with a sufficient background in computing may use Math 475 to fulfill the computing/computer science requirement in this track.  However in that case Math 475  cannot also count as one of the required upper level electives.  Strong incoming freshmen should look at the Honors Program in Statistics

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Track C:  Applied Although the applied track can be done as a "stand-alone" major, most students following this track combine it with substantial work in a math-related area such as physics, computer science, or engineering.	The RED requirements in Track C apply to majors declared after June 30, 2005.   For majors in mathematics declared on or before June 30, 2005 the new requirements are also recommended.  However those majors can fulfill the track under the older requirements displayed in BLUE ITALICS.
Physics 117-118 or 197-198;  or  CSE 131-132   or CSE 126 + a suitable second course in computer science approved by Professor Ron Freiwald.  For more details and other options, see the additional comments on options in Computer Science (CSE) Note:  According to CSE Department, students should NOT take both CSE 126 and CSE 131.  Students who do well in CSE 126 can usually take CSE 132 if they want to, but should consult first with the CSE Department.	Two substantially mathematical courses in a different department can sometimes be substituted, with departmental approval, if the substitution forms part of a coherent "applications" program.
Math 217	Differential Equations
Math 309	Matrix Algebra
Math 310*  (or 310W)	Foundations for Higher Mathematics
Math 318 or Math 308	Calculus of Several Variables / Mathematics for the Physical Sciences
Math 449-450	Numerical Applied Mathematics / Topics in Applied Mathematics
One additional upper level mathematics course useful in applications chosen from the list (**), below
One other upper level math course	This course may or may not be "applications oriented"
For majors declared on or before June 30, 2005, Math 449-450 is recommended but not required.  For those majors, the requirements in red, above, may be replaced by the following older requirements:      i) THREE upper level mathematics courses useful in applications (displayed in the list ** below).         At least TWO of these applied upper level electives courses beyond 309-310-(308/318) must be courses taught by faculty in the Department of Mathematics (not courses taught in other departments, and not courses cross-listed with math but taught by a faculty member of another department). and     ii) ONE additional upper level math elective (which may or may not be applications oriented)

*       Computer Science Majors doing a second major in math and who have received a B or better in CSE 240 (Logic and Discrete Mathematics) may replace Math 310 with an upper level math elective of their choosing. **     Courses counting for the "applications" requirement include: any upper level course in probability and statistics (see list under Track B);  Math 312, 350, 370, 404, 405, 410, 415, 416, 4361, 449, 450, 481, 495, 501-502 (= Physics 501-502);  and from engineering: ESE 317 (Engineering Mathematics), ESE 403 (Operations Research), ESE 411  (Numerical Methods). Note: Only one of ESE 411, Math 404, and Math 449  may be counted toward the major requirements since all of these courses involve numerical methods. If a student chooses to apply ESE 411 toward the major, s/he must choose an upper level applied mathematics elective (L24, and not a cross-listed course home-based in another department) elective to replace Math 449.  It would also be advisable to check with current 449-450 instructor that the material from ESE 411 will provide adequate background for whatever "topic"  is planned for the Math 450 course that year.

      

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Track D:  Secondary Education	This track can be done only as a double major with secondary education.  The courses required in this major track are typical of those required for secondary certification in mathematics.  However, students seeking secondary certification should rely on information provided by the Education Department.
One of CSE 131, CSE 126 or CSE 200	Computer Science (students should check with the Computer Science Department about the semester in which each of these courses is offered) The University College C-programming course, U20-123 may NOT be automatically substituted: that requires written permission from your advisor or Prof. Freiwald. Additional comments on options in Computer Science (CSE)
Math 302	Elementary Geometry from an Advanced Point of VIew
Math 309	Matrix Algebra
Math 310 (or 310W)	Foundations for Higher Mathematics
Math 318 or Math 308	Calculus of Several Variables (or, Mathematics for the Physical Sciences)
Math 3200	Elementary to Intermediate Statistics with Data Analysis
Math 331	Algebraic Systems
One other upper level math course	***

Students in the Secondary Education track usually need to finish the math major requirements by the end of the first semester of their senior year because of student teaching in the spring semester. Therefore, for planning purposes, it is important to note that

• Math 302 and 331 are offered in alternate fall semesters  (Math 302 in fall 2006, Math 331 in fall 2007, etc.)

Since this version of the major requires an additional major in secondary education (including student teaching), students pursuing this track who are uncertain about continuing with the education major should plan their math courses to make possible a later switch into one of the other tracks.  Certification by the Department of Education for student teaching requires a higher mathematics GPA (near a B average--check with the Department of Education) than the math major itself.

With special permission from the department's Undergraduate Committee, students who complete a major in elementary or middle school education may also complete a mathematics major using the "Secondary Education" track and subject to the same requirements.  However, such students should be aware that the course scheduling of Math 302 and 331 is set up to accommodate the schedules (including student teaching) of students seeking secondary certification.  There might be irreconcilable conflicts with required education courses or student teaching for those in elementary or middle school education.

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Track E:  Mathematics (Economics Emphasis ) This track provides an opportunity for math majors to acquire a strong mathematics background in a track resembling the "traditional" Track A, but with an emphasis in economics.  (Two algebra requirements in the traditional major are replaced with requirements appropriate to this track--although Math 429, Linear Algebra, might still be a useful elective.) On the mathematics side, this track would be an excellent mathematical  preparation for a graduate program in finance or economics.  But students interested in such graduate programs should also consult with the Business School or Economics Department to see what additional work in finance, business or economics is suggested.	The track could also be a good background, for example, for a career in investment banking where pure math/economics candidates are sometimes attractive. Of course, rather than following this track, a student interested in economics might follow one of the other major tracks combined with a minor or second major in economics or finance.  Students should talk with appropriate advisors in mathematics, economics, or business about the best way to structure their undergraduate program in light of particular personal goals.
Math 309	Matrix Algebra
Math 310 (or 310W)	Foundations for Higher Mathematics
Math 4111-4121	Introduction to Analysis / Introduction to Lebesgue Integration
Math 3200 or Math 493	Elementary to Intermediate Statistics with Data Analysis     (or Probability)
Two other upper level mathematics courses chosen from the list (**) below.	**, **
In addition, the following Economics courses are required: Economics 103 and 104 (Microeconomics and  Macroeconomics) Economics 413  (Econometrics) At least one of Economics 401/402 (Price Theory / Income and Employment Theory)

(**) Math 429 (Linear Algebra), Math 439 (Linear Statistical Models), Math 456 (Topics in Financial Mathematics), Math 493 (Probability), Math 494 (Mathematical Statistics), Math 475 (Statistical Computation with SAS), Math 495 (Stochastic Processes) For students in the Business School only: MECO 290 (Microeconomics) may substitute for Economics 103.  Students who take MECO 290 should probably not also take Economics 401.  For the purposes of this major, they may take 402 or may propose another mathematically oriented upper level economics course for approval instead of 402  (Examples might include Economics 4111  (Optimization and Economic Theory) or Economics 467 (Game Theory)) MECO 292 (Global Economy) may be used in place of Economics 104.  However, such students might want to consider taking Economics 402. The pair of courses QBA 120,121 (Managerial Statistics I, II) may replace Math 3200. In that case, MAth 3200 cannot also count toward the major.

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Additional Comments About Major Requirements There are a number of courses in other departments which, with the prior approval of your advisor, can sometimes be used to satisfy one of the department requirements for "upper level math courses."  These include: ESE 403 (Operations Research),  Philosophy 401 (Set Theory), Philosophy 403-404 (Mathematical Logic I, II) An occasional substitution may be made in requirements if it serves a reasonable mathematical purpose.  Such substitutions must have prior approval of the department's Undergraduate Committee. The introductory graduate level math courses (5021-5022, 5031-5032, 5041-5042-5043, 5051-5052, 5061-5062) are open to qualified undergraduates. Undergraduates who have satisfied the prerequisites and who are willing to put in the extra effort can find these courses very rewarding. Sometimes undergraduates can enroll in supervised independent study.  A student who has a plan for independent work and who has found a faculty member who will supervise the work should discuss the possibility with his/her advisor.  With the instructor's approval, they may enroll for L24-400, Undergraduate Independent Study, using the section number corresponding to the supervising faculty member.  (The section numbers are in the Course Listing Book in the department header information.). Students are encouraged to consider a senior project (whether or not for honors).  Some of our faculty have listed ideas for projects at various levels.  Students working on an honors project can register, with faculty permission, in L24-499, Study for Honors--although formal enrollment in 499 is not required to complete an honors project. Students planning on spending a semester or year abroad should consult with their advisors or Prof. Ron Freiwald as early as possible. (If the plan is to spend only a semester abroad, the spring semester abroad usually coordinates better with the Washington University math program.)  The Budapest Semesters in Mathematics program is an excellent opportunity for serious mathematics students.