College Bulletin 2015-2016

Nov 21, 2024

College Bulletin 2015-2016 [ARCHIVED CATALOG]

Mathematics and Statistics

Faculty

PHILIP J. EVERSON, Professor and Chair
VICTOR BARRANCA, Assistant Professor
DEB BERGSTRAND, Professor (part time)
LINDA CHEN, Associate Professor
RALPH R. GOMEZ, Assistant Professor
CHERYL P. GROOD, Professor
THOMAS J. HUNTER, Professor
AIMEE S.A. JOHNSON, Professor
STEPHEN B. MAURER, Professor (part time)
NSOKI MAVINGA, Assistant Professor¹
KELLY MCCONVILLE, Assistant Professor
DON H. SHIMAMOTO, Professor
LYNNE STEUERLE SCHOFIELD, Associate Professor
JANET C. TALVACCHIA, Professor
STEVE C. WANG, Associate Professor
MICHAEL BIRO, Visiting Assistant Professor
RACHEL EPSTEIN, Visiting Assistant Professor
KARAMATOU YACOUBOU DJIMA, Visiting Assistant Professor
AUSTIN LAMPROS, Academic Support Coordinator
STEPHANIE J. SPECHT, Administrative Assistant

¹ Absent on leave, 2015-2016.

Overview of Curriculum

Mathematics and statistics are among the great achievements of human intellect and at the same time powerful tools. As Galileo said, the book of the universe “is written in the language of mathematics.” The goal of the department is to enable students to appreciate these achievements and use their power. To that end, majors and minors in the department receive a firm foundation in pure mathematics and the opportunity to apply it to a variety of disciplines, including statistics, physical science, biological science, computer science, social science, operations research, education, and finance.

Students typically enter our department with strong skills, but there is always room for improvement and new knowledge. Majors and minors grow in:

Reasoning skills: logical argument and abstraction;
Formulation skills: developing mathematical models;
Communication skills: expressing mathematical ideas and information clearly and precisely on paper, orally, and electronically;
Comprehension skills: absorbing mathematical ideas and information presented on paper, orally, and electronically;
Computation skills: mental, by hand, and by machine, as appropriate.

Through core courses, students learn fundamental concepts, results, and methods. Through elective courses, they pursue special interests. In the process, students develop a further appreciation for the scope and beauty of our discipline.

Graduates of the department follow many career paths. These paths lead to graduate school in mathematics, statistics, and other fields; to professional schools; and to the workplace.

Introductory Courses

Most first-year students entering Swarthmore have had calculus while in high school and place out of at least one semester of Swarthmore’s calculus courses, whether they continue with calculus or decide, as is often best, to try other sorts of mathematics. See the discussion of placement later. However, some entering students have not had the opportunity to take calculus or need to begin again. Therefore, Swarthmore offers a beginning calculus course (MATH 015) and several courses that do not require calculus or other sophisticated mathematics experiences. These courses are STAT 001 (Statistical Thinking, both semesters), MATH 003 (Introduction to Mathematical Thinking, spring semester), and STAT 011 (Statistical Methods, both semesters). MATH 003 is a writing course. MATH 029 (Discrete Mathematics, both semesters) also does not require any calculus but is a more sophisticated course; thus, some calculus is a useful background for it in an indirect way. Once one has had or placed out of two semesters of calculus, many other courses are available, especially in linear algebra and several-variable calculus.

Placement and Credit on Entrance to Swarthmore

Placement Procedure

To gain entrance to mathematics or statistics courses at any time during one’s Swarthmore years, students are expected to take at least one of the following exams: the Advanced Placement (AP) or International Baccalaureate (IB) exams, Swarthmore’s Calculus Placement Exam, or Swarthmore’s Math/Stat Readiness Exam. Students who do take AP or IB exams may be required to take the departmental exams as well, or parts thereof. In particular, students intending to take either MATH 15 or MATH 28 must take Swarthmore’s Calculus Placement Exam. Versions of the Calculus Placement Exam and the Readiness Exam are sent to entering first-year students over the summer, along with detailed information about the rules for placement and credit.

Advanced Placement/International Baccalaureate Credit

Placement and credit mean different things. Placement allows students to skip material they have learned well already by starting at Swarthmore in more advanced courses. Credit confers placement as well but also is recorded on the student’s Swarthmore transcript and counts toward the 32 credits needed for graduation.

The Swarthmore Calculus Placement Exam is used for placement only, not credit. Credit is awarded on the basis of the AP and the IB exams, as follows:

1 credit (for STAT 011) for a score of 4 or 5 on the Statistics AP Test of the College Board.
1 credit (for MATH 015) for a score of 4 on the AB or BC Calculus AP Test of the College Board (or for an AB subscore of 4 on the BC Test) or for a score of 5 on the Higher Level Mathematics Test of the IB.
1.5 credits (for MATH 015 and the first half of MATH 025) for a score of 5 on the AB Calculus AP Test (or for an AB subscore of 5 on the BC Test) or a score of 6 or 7 on the higher-level IB. Students who receive this credit and want to continue calculus take MATH 026.
2 credits (for MATH 015 and 025) for a main score of 5 on the BC Calculus AP Test.

Alternatively, any entering student who places out of MATH 015 or 025 may receive credit for those courses by passing the final exams in these courses with a grade of straight C or better. These exams must normally be taken during the student’s first semester at Swarthmore, at the time when the final exam is given for the course. Students who wish to take these exams must arrange to do so with the departmental placement coordinator and should do so during their first semester at Swarthmore. Students who are eligible on entrance for credit for a course, but who take the course anyway, will lose the entrance credit.

First-year students seeking advanced placement and/or credit for calculus taken at another college or university must normally validate their work by taking the appropriate external or Swarthmore placement examination, as described earlier. The department does not grant credit directly for college courses taken while a student is in high school. For work beyond calculus completed before entering Swarthmore, students should consult the departmental placement coordinator to determine the Swarthmore courses into which they may be placed and additional materials they may need to present for this placement. The department will not normally award credit for work above the first-year calculus level completed before entering Swarthmore.

The Academic Program

Major and Minor Application Process

Students apply for a major in the middle of the second semester of the sophomore year. Students should consult the department webpage during the College’s Sophomore Plan process for more details on how to apply for the major. After the Sophomore Plan process is over, students may apply to add or change a major or minor at any time, but applications will normally be held until the next time that sophomore applications are considered (around March 1).

Course Major

Acceptance into the Major

The normal preparation for a major in mathematics is to have obtained credit for, or placement out of, at least four of the following five course groups by the end of the sophomore year: Calculus I (MATH 015), Calculus II (MATH 025 or 026), Discrete Mathematics (MATH 029), Linear Algebra (MATH 027 or any flavor of 028), and Several Variable Calculus (MATH 033, 034, or 035). In any event, all majors must complete the Linear Algebra and Several Variable Calculus requirement by the end of the first semester of the junior year.

To be accepted as a major or a minor, a candidate normally should have a grade point average of at least C+ in courses taken in the department to date, including courses in the fall term of the first year, for which we have shadow grades. A candidate should have at least one grade at the B level. Students should be aware that upper-level courses in mathematics are typically more demanding and more theoretical than the first-and second-year courses. This is an important factor in considering borderline cases. In some cases, applicants may be deferred pending successful work in courses to be designated by the department.

Basic Requirements

By graduation, a mathematics major must have at least 10 credits in mathematics and statistics courses. At least 5 of the credits counted in the 10 must be for courses numbered over 040. (Courses numbered under 10 do not count toward the major in any event.) Furthermore, every major is required to obtain credit for, or place out of, each of the following course groups: MATH 015; MATH 025, or 026; MATH 027, 028, or 028S; MATH 033, 034, or 035; MATH 063; and MATH 067. The two upper-level core courses, MATH 063 (Introduction to Real Analysis) and MATH 067 (Introduction to Modern Algebra), will be offered at least every fall semester. At least one of these two should be taken no later than the fall semester of the junior year. Majors are expected to complete both MATH 063 and 067 before the spring semester of the senior year; permission to delay taking either course until the senior spring must be requested in writing as early as possible but in any event no later than the beginning of the fall semester of the senior year. Finally, course majors must satisfy the departmental comprehensive requirement by passing MATH 097, Senior Conference. Normally, at least 3 of the 5 credits for courses numbered over 040 must be taken at Swarthmore, including MATH 097 and at least one of the core courses MATH 063 and 067. Note that MATH 097 is given in the fall only.

Note that placement counts for satisfying the requirements but not for the 10-credit rule. Those students who are placed out of courses without credit must take other courses to obtain 10 credits. If you believe you are eligible for credit for courses taken before Swarthmore (because of AP or IB scores) but these credits are not showing on your transcript, please see the registrar.

The two required core courses, Introduction to Real Analysis (MATH 063) and Introduction to Modern Algebra (MATH 067), are offered every fall semester, and we try to create enough sections to keep them relatively small and seminar-like. We hope, but cannot promise, to offer one or the other of 063 and 067 each spring as well.

Mathematics majors are encouraged to study in some depth an additional discipline that makes use of mathematics. We also recommend that they acquire some facility with computers. Students bound for graduate work should obtain a reading knowledge of French, German, or Russian.

Special Emphases

The preceding requirements allow room to choose an optional special emphasis within the mathematics major. For instance:

A student may major in mathematics with an emphasis on statistics by taking the following courses at the advanced level: (1) the core analysis course (MATH 063); (2) Probability (STAT 051); (3) Mathematical Statistics I (STAT 061); (4) Mathematical Statistics II (STAT 111); (5) Data Analysis and Visualization (STAT 031); (6) the Senior Conference (MATH 097); and (7) credit for or placement out of Introduction to Computer Science (CPSC 021). When a student does an emphasis in statistics, STAT 031 counts as if it were numbered over 040. Students in the class of 2016 may complete the major without STAT 051.

Students interested in mathematics and computer science should consider a mathematics major with a minor in computer science or an Honors Program with a mathematics major and a computer science minor. Details on these options are in the catalog under computer science.

Students thinking of graduate work in social or management science, or a master’s in business administration, should consider the following options.

Basic courses: single-variable calculus (two semesters), one or more practical statistics courses (STAT 011 and 031), linear algebra, discrete math, several-variable calculus, and introductory computer science; advanced courses: (1) Modeling (MATH 056); (2) at least one of Probability (STAT 051), Mathematical Statistics I (STAT 061), and possibly Mathematical Statistics II (STAT 111); (3) at least one of Combinatorics (MATH 069) or Operations Research (ENGR 057); (4) the three required core courses (MATH 063, MATH 067 and MATH 097); and (5) Differential Equations (MATH 043 or 044). Because this program is heavy (one who hopes to use mathematics in another field must have a good grasp both of the relevant mathematics and of the intended applications), one of the core course requirements may be waived with permission of the department.

Students thinking of graduate work in operations research should consider the following options. Basic courses: same as previous paragraph. Advanced courses: (1) the three required core courses (MATH 063, MATH 067 and MATH 097); (2) Combinatorics (MATH 069) and Topics in Discrete Mathematics (MATH 059 or 079); (3) Probability (STAT 051); (4) Mathematical Statistics (STAT 061); and (5) at least one of Number Theory (MATH 058), or Modeling (MATH 056).

Students interested in quantitative economics, mathematical finance, or similar fields should consider a double major in mathematics and economics, or a major in mathematics with a minor in economics. Students thinking of graduate work in quantitative economics or mathematical finance should consider a math major with a program including at least MATH 43, MATH 54, MATH 63, STAT 51 and STAT 61 together with appropriate additional coursework to round out a mathematics major or a mathematics major with emphasis in statistics.

Course Minor

Acceptance into the minors

The requirements for acceptance into either course minor, such as prerequisite courses and grade average, are the same as for acceptance into the major. Students may not minor in both mathematics and statistics.

Basic requirements to complete the mathematics course minor

By graduation, a mathematics course minor must have 6 credits in mathematics or statistics, at least 3 of which must be for courses numbered 045 or higher. Also, at least 1 of these 3 credits must be for MATH 063 or 067. Also, at least 2 of these 3 credits must be taken at Swarthmore.

Basic requirements of the statistics course minor (for Class of ‘17 and later)

By graduation, a statistics course minor must have 6 credits in mathematics or statistics. Every statistics course minor must obtain credit for, or place out of, CPSC 021, STAT 031, STAT 051 and STAT 061. At least one of STAT 031 and STAT 061 must be taken at Swarthmore. Students are advised to take CPSC 21 as early as possible, as it can be difficult to add the course in junior and senior years.

Honors Major

All current sophomores who wish to apply for Honors should indicate this in their Sophomore Plan, should work out a tentative Honors Program with their departmental adviser, and should submit the College’s Honors Program Application along with their Sophomore Plan. (All Sophomore Plan forms and Honors forms are available from the registrar or the registrar’s website.) Honors applications are also accepted at the end of the sophomore year or during the junior year. Students, in consultation with their advisers, often change their Honors Programs anyway as time goes on.

Basic requirements

To be accepted as an Honors major in mathematics, a student should have a grade point average in mathematics and statistics courses to date of at least B+.

An Honors math major program consists of three preparations of two credits each, for a total of six distinct credits. One preparation must be in algebra and one in analysis (real or complex). The student must also satisfy all requirements of the mathematics major with the exception of the comprehensive requirement (MATH 097, Senior Conference).

Preparations

The department offers preparations in the fields listed below. Each preparation is subject to External Examination, including a 3-hour written examination and a 45-minute oral examination. Each preparation consists of a specified pair of credits. The specified credits are listed after each field.

Algebra (067 and 102)
Real Analysis (063 and 101)
Complex Analysis (063 and 103)
Geometry (either 055 or 075, and 106)
Statistics (061 and 111)
Topology (104, a 2-credit seminar)

No course is allowed to count in two honors preparations, so it is not possible for a student to do honors preparations in both Real Analysis and Complex Analysis.

The external examination component of the program is meant to prompt students to learn their core subjects really well and to show the examiners that they have done so-that is, show that they deserve Honors. However, no three fields cover everything a strong student would ideally learn as an undergraduate. Honors majors should consider including in their studies a number of advanced courses and seminars beyond what they present for Honors.

Senior Honors Study/Portfolio

None is required or offered.

Honors Minor

For the honors portion of their program, minors must complete one preparation chosen from those in the previous section.

Transfer Credit

Courses taken elsewhere may count for the major. However, the number of upper-level transfer credits for the major is limited. Normally, at least 3 of the 5 upper-level courses used to fulfill the major must be taken at Swarthmore, including at least one of the core courses MATH 063 and MATH 067. Exceptions should be proposed and approved during the Sophomore Plan process, not after the fact. Also, the usual College rules for transfer credit apply: students must see the professor in charge of transfer twice: in advance to obtain authorization, and afterwards to get final approval and a determination of credit. In particular, for MATH 063 and 067, students are responsible for the syllabus we use. If a course taken elsewhere turns out not to cover it all, the student will not get full credit (even though the transfer course was authorized beforehand) and the student will not complete the major until he or she has demonstrated knowledge of the missing topics.

Similarly, for honors preparations students are responsible for the syllabi we use; we will not offer special honors exams based on work done at other institutions.

Off-Campus Study

Students planning to study abroad should obtain information well in advance about the courses available at the institution they plan to attend and check with the department about selecting appropriate courses. It may be difficult to find courses abroad equivalent to our core upper-level courses, or to our honors preparations, since curricula in other countries are often organized differently.

Teacher Certification

Swarthmore offers teacher certification in mathematics through a program approved by the state of Pennsylvania and administered by the College’s Educational Studies Department. For further information about the relevant set of requirements, please refer to the Educational Studies section of the Bulletin. One can obtain certification either through a mathematics major or through a Special Major in Mathematics and Education, in either case if taken with appropriate electives.

Mathematics and Statistics Courses

Note 1: For courses numbered under 100, the ones digit indicates the subject matter, and the other digit indicates the level. In most cases, a ones digit of 1 or 2 means statistics, 3 to 6 means continuous mathematics, and 7 to 9 means noncontinuous mathematics (algebra, number theory, and discrete math). Courses below 10 do not count for the major, from 10 to 39 are first- and second-year courses, from 40 to 59 are intermediate, in the 60s are core upper-level courses; from 70 to 89 are courses that have one or more core courses as prerequisites, and in the 90s are independent reading courses.

Note 2: There are several sets of courses below where a student may not take more than one of them for credit. For instance, see the descriptions of MATH 033, 034 and 035. In such cases, if a student does take more than one of them, each group is treated for the purpose of college regulations as if they have the same course number. See the Repeated Course Rule in section 8.2.4.