This is an archive. See the current website at www.ssp.harvard.edu.

Mathematics

Not all courses are available to SSP students. For example, some courses are offered only for graduate credit. Note especially any listed prerequisites.

Related Courses

  • ECON S-110 Quantitative Methods in Economics and Business
  • STAT S-100 Introduction to Quantitative Methods

Mathematics courses are listed below.

Mathematics classes begin June 26, except MATH S-21b and GMAT S-1, which start June 25, and MATH S-300 and S-310, which start July 2.

Placement Test

Prerequisites are important in mathematics, especially in the calculus sequence that includes courses through MATH S-21b. Note that courses numbered MATH S-101 and above do not require calculus. Students who wish to enroll in starred (*) mathematics courses (MATH S-Y, S-Ar, S-1a, S-1b, or S-1ab) are required to take the mathematics placement test. The placement test is offered online during registration and on campus at the times listed below. Eligibility for courses at the level of MATH S-Y through S-1ab is determined by the results of the placement test. The placement test is recommended for students who wish to enroll in MATH S-21a, S-21b, S-101, S-300, and S-310. The placement test for these courses is available only online.

Students are responsible for submitting written requests to the Registrar's Office by Friday, June 29, for any course changes that are necessary as a result of the test. If students wish to discuss placement test results or need advice regarding mathematics placement, they may visit Science Center 231, June 26 to 29, 1:30 to 3:30 pm. Students with questions may also send e-mail to support@math-placement.dce.harvard.edu.

Online placement test. The placement test is available online from the beginning of registration through June 24. If students wish, they may take the test first online and later on campus, provided they are interested in taking one of the starred courses.

On-campus placement test. The on-campus placement test is offered Monday, June 25, at 4 pm in Science Center E. A late placement test is offered on Tuesday, June 26, at 4 pm in Science Center E. Students who take the late placement test are responsible for making up all work that is covered in any classes that they miss.

Calculators. MATH S-Y, S-Ar, S-1a, S-1b, and S-1ab use graphing calculators. They are not required for the placement test. Students should ask their instructors on the first day of class which models are most conveniently supported.

Courselist

Mathematics for Teaching

*MATH S-Y Mathematical Models and Expressions (31920)
(Website) (Print version)
Andrew Engelward
(4 credits: UN, NC) T,W,Th noon-2 pm, Science Center, Room 309A. Eight-week session. Optional sections to be arranged. Tuition $2,275.

The course explores basic mathematical models as they arise in real-world situations with the goal of understanding the meaning behind mathematical expressions and functional relationships. Computer spreadsheets are used to analyze actual data sets. The course encourages independent thinking while rigorously reviewing basic algebraic and statistical techniques and notation, as needed. This course provides excellent preparation for statistics courses as well as for quantitative reasoning components of standardized tests at the secondary school level. Prerequisites: Arithmetic and some algebra. Placement test required.

*MATH S-Ar Precalculus Mathematics
(4 credits: UN, NC) Eight-week session. Tuition $2,275.

Section 1 (30388) (Print version)
Srdjan Divac, M-F noon-1 pm, Science Center, Room B-10.

Section 2 (30389) (Print version)
Srdjan Divac, M-Th 6-7:15 pm, Science Center, Room B-10.

A review of algebra is integrated into the study of rational, exponential, logarithmic, and trigonometric functions. Taught in small sections, the course emphasizes applications and problem solving and provides preparation for calculus and basic science. Graphing calculators are used, though no previous calculator experience is required. Prerequisite: A good working knowledge of algebra, as demonstrated by a satisfactory score on the placement test. Students without the prerequisite placement test score are withdrawn from the course.

*MATH S-1a Calculus I
(4 credits: UN, GR, NC) Eight-week session. Tuition $2,275.

Section 1 (30391) (Syllabus) (Print version)
Otto K. Bretscher, M-F 10-11 am, Science Center, Room 507.

Section 2 (30392) (Print version)
Otto K. Bretscher, M-F 11 am-noon, Science Center, Room 507.

This course covers differential and integral calculus in one variable, with applications. We aim to develop conceptual understanding, computational skills, and the students' ability to apply the material to science. The topics covered overlap with the AP calculus curriculum to a large extent. A graphing calculator can occasionally be useful. Students enrolling for graduate credit participate in weekly pedagogical seminars investigating current research in mathematics education. Prerequisites: A good working knowledge of algebra, functions, logarithms, trigonometry, and analytic geometry. Placement test required. Graduate option intended for students in the Extension School master's program in mathematics for teaching. Please contact the Math for Teaching Office for details.

*MATH S-1b Calculus II (30393)
(Website) (Print version)
Thomas W. Judson
(4 credits: UN, GR, NC) M-F 11 am-noon, Science Center, Room 221. Eight-week session. Required sections for graduate students Th 3-4:30 pm; required sections for undergraduate students to be arranged. Tuition $2,275.

Galileo wrote that "the book of the universe is written in the language of mathematics." Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. We aim to balance applications and theoretical understanding. Graphing calculators can help with understanding certain concepts and are recommended, but exams do not require them. The topics covered are not identical to those of a BC advanced placement class, but do overlap with the AP calculus curriculum to a large extent. Students enrolling for graduate credit participate in weekly pedagogical seminars investigating current research in mathematics education. Prerequisite: A good working knowledge of differentiation and an acquaintance with integration, as demonstrated by a satisfactory score on the placement test. Graduate option intended for students in the Extension School master's program in mathematics for teaching. Please contact the Math for Teaching Office for details.

*MATH S-1ab Calculus I and II (30390)
(Print version)
Matthew Leingang
(8 credits: UN, NC) M-F 10 am-noon, Science Center, Room 309. Eight-week session. Required sections M-F 2-3 pm. Tuition $4,550.

This is a very intensive course covering differential and integral calculus in one variable, including series and some differential equations. We aim to develop theoretical understanding and practical skills. Some students leave prepared for multivariable calculus; others leave having previewed one-variable calculus. Graphing calculators are recommended but are not used in exams. The topics covered are not identical to those of a BC advanced placement class but do overlap to a large extent. Prerequisite: A strong interest in mathematics plus an excellent facility with geometry, algebra, and analytic geometry, including functions, graphs, exponentials and logarithms, and trigonometric functions. Placement test required.

MATH S-21a Multivariable Calculus (30189)
(Website) (Print version)
Oliver Knill
(4 credits: UN, GR, NC) T,W,Th 9:30-11 am, Emerson Hall, Room 101. Eight-week session. Optional sections Th 8-9 am or 1-2 pm. Tuition $2,275.

To see how calculus applies in situations described by more than one variable, we study vectors, lines, planes, and parameterization of curves and surfaces; partial derivatives, directional derivatives, and gradients; optimization and critical point analysis, including the method of Lagrange multipliers; integration over curves, surfaces, and solid regions using Cartesian, polar, cylindrical, and spherical coordinates; vector fields, and line and surface integrals for work and flux; and the divergence and curl of vector fields together with applications. Prerequisite: Two semesters of calculus. Placement test recommended.

MATH S-21b Linear Algebra and Differential Equations (30190)
(Website) (Print version)
Robert Winters
(4 credits: UN, GR, NC) M-F 11 am-noon, Science Center, Room 110. Eight-week session. Optional sections to be arranged. Tuition $2,275.

Topics to be covered include Gauss-Jordan reduction and systems of linear equations; matrices and linear transformations; linear independence; subspaces; matrices and coordinates relative to different bases; general linear spaces; orthogonality and least-squares approximation; inner product spaces; determinants; eigenvalues, eigenvectors, and the spectral theorem; discrete and continuous dynamical systems; phase-plane analysis of linear and nonlinear systems of ordinary differential equations; and function spaces and differential operators. Prerequisite: MATH S-21a (taken concurrently if necessary) or the equivalent. Placement test recommended.

MATH S-101 Spaces, Mappings, and Mathematical Reasoning: An Introduction to Proof (31859)
(Website) (Print version)
Thomas W. Judson
(4 credits: UN, GR, NC) T,W,Th 9:30-11 am, Science Center, Room 221. Eight-week session. Required 1-hour sections to be arranged. Tuition $2,275.

There is more to mathematics than formulas and procedures. Ever wonder where a theorem comes from or why you should believe it? In this class, we start from some basic assumptions (no calculus necessary) and reason our way together until we convince ourselves of some surprising and sophisticated conclusions, including beautiful results from basic topology, analysis, and group theory. Prerequisites: Imagination, a solid mastery of precalculus, as well as a serious interest in making and critiquing arguments. Placement test recommended.

Mathematics for Teaching

MATH S-300 Math for Teaching Arithmetic (32122)
(Print version)
Bret Benesh
(4 credits: GR, NC) M-F 9-11:30 am, Science Center, Room 216. Short session I. Tuition $1,100. Limited enrollment.
July 2-July 20. Harvard College students see additional information.

Why do students have such a difficult time with arithmetic? It could be because arithmetic is much more complex than we initially suspect. For instance, the symbol ½ has at least four different interpretations. When a student encounters this fraction he or she must decide which interpretation will best help solve the problem. In this course, we study why it makes sense to do arithmetic the way we do. Moreover, we see how amazing our number system actually is. (Students should try multiplying 97 by 56 by only using Roman numerals if they are skeptical.) This course emphasizes mathematical reasoning rather than mathematical skills and results, and communication is a large part of the course. In addition to the mathematical content, we also discuss how different methods of teaching affect students differently. Placement test recommended.

*MATH S-310 Graph Theory: Investigating the Mathematical Process (32161)
(Print version)
John D. Boller
(4 credits: GR, NC) M-F 12:30-3 pm, 53 Church Street, Room L01. Short session I. Tuition $1,100. Limited enrollment.
July 2-July 20. Harvard College students see additional information.

Is mathematics invented or discovered? Is its language that of the mind or the world? How does one actually go about exploring a mathematical topic? Questions such as these inform this course as we develop a specific branch of mathematics—graph theory—and apply it to a vast spectrum of subjects, from algebra, topology, and computer science to tournaments and transportation. What, for example, is the most efficient street cleaning system for New York City? The course interests those who teach or are thinking of teaching math, or those who want to help improve some of the ways it is taught now. Prerequisite: Placement test recommended.

 



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