This is an archive. See the current website at www.summer.harvard.edu.
This page contains content from the Summer School 2009. For current information, visit the Harvard Summer School website at www.summer.harvard.edu.
For courses marked with a star (*) below, the first class meeting is Tuesday, June 23.
Prerequisites are important in mathematics, especially in the calculus sequence that includes courses through MATH S-21b. Courses numbered MATH S-101 and above do not require calculus.
The math placement test is required for courses that are starred (*) below; eligibility for these courses is determined by test results. The test is recommended for most other courses. The placement test is offered online beginning Sunday, March 1, and on campus at the times listed below. If the test is only recommended for a course, it must be taken online.
If you wish to discuss placement test results or need advice regarding mathematics placement, you may visit Science Center 235, June 22 to 25, 11:30 am to 1:30 pm. You may also send questions to e-mail to support@math-placement.dce.harvard.edu.
Online placement test. The placement test is available online from Sunday, March 1, to Sunday, June 21. You may take the test first online and later on campus if you are interested in taking one of the starred (*) courses.
On-campus placement test. The on-campus placement test is offered Monday, June 22, at 4 pm, in Science Center Hall E. A late placement test is offered on Tuesday, June 23, at 4 pm, in Science Center Hall E. If you take the late placement test, you must make up all coursework that you miss.
If you have questions about placement, send an e-mail to support@math-placement.dce.harvard.edu.
Calculators. MATH S-Y, S-Ar, S-1a, S-1b, and S-1ab use graphing calculators. They are not required for the placement test. Ask your instructor on the first day of class about which models are most conveniently supported.
*MATH S-Y
Mathematical Models and Expressions (31920)
(Printable version)
Rehana Patel
(4 credits: UN, NC) Tuesdays, Wednesdays, Thursdays, noon-2 pm. Optional sections Mondays, Wednesdays, Thursdays, 3-4 pm. Tuition $2,475.
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. The course encourages independent thinking while rigorously reviewing basic algebraic and statistical techniques and notation as needed. This course provides 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) Tuition $2,475.
Section 1 (30388) (Printable version)
Srdjan Divac, Mondays-Thursdays, 1:30-3 pm.
Section 2 (30389) (Printable version)
Srdjan Divac, Mondays-Thursdays, 6:30-8 pm.
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) Tuition $2,475.
Section 1 (30391) (Printable version)
Otto K. Bretscher, Mondays-Fridays, 9-10:15 am.
Section 2 (30392) (Printable version)
Otto K. Bretscher, Mondays-Fridays, 10:30-11:45 am.
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 advanced placement 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. The graduate-credit option is intended for students in the Extension School master's program in mathematics for teaching. Please contact the Mathematics for Teaching Office for details.
*MATH S-1b
Calculus II (30393)
(Website) (Printable version)
John Thomas Hall
(4 credits: UN, GR, NC) Mondays, Wednesdays, Fridays, 9:30-11:30 am. Required sections to be arranged. Tuition $2,475.
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. The course aims 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 advanced placement 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. The graduate-credit option is intended for students in the Extension School master's program in mathematics for teaching. Please contact the Mathematics for Teaching Office for details.
*MATH S-1ab
Calculus I and II (30390)
(Website) (Printable version)
Peter M. Garfield
(8 credits: UN, NC) Mondays-Fridays, 8:45-11:45 am. Required sections Mondays-Fridays, 2-3 pm.
Tuition $4,950.
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) (Printable version)
Oliver Knill
(4 credits: UN, GR, NC) Tuesdays, Thursdays, 8:30-11:30 am. Schedule to be announced. Required sections to be arranged. Tuition $2,475.
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)
(Printable version)
Robert Winters
(4 credits: UN, GR, NC) Mondays, Wednesdays, Fridays, 9:30-11:30 am. Schedule to be announced. Optional sections to be arranged. Tuition $2,475.
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) (Printable version)
Jameel Habeeb Al-Aidroos
(4 credits: UN, GR, NC) Tuesdays, Wednesdays, Thursdays, noon-2 pm. Optional sections to be arranged. Tuition $2,475.
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.
MATH S-104
Discrete Mathematics with Computer Science Applications (32408)
(Website) (Printable version)
Sarah J. Finney
(4 credits: UN, GR, NC) Tuesdays, Thursdays, 6:30-9:30 pm. Optional sections to be arranged. Tuition $2,475.
This course is an introduction to discrete mathematics for computer science: mathematical logic, sets, relations, graphs; proof techniques, induction, recursion; growth of functions; combinatorics; and discrete probability. Computer science applications, such as algorithm analysis, cryptography, search, and state estimation are discussed. This course is suitable for beginning computer science students. Prerequisite: CSCI S-111 or equivalent experience with data structures.
MATH S-152
The Mathematics of Symmetry (32252)
(Printable version)
(4 credits: UN, GR, NC) Required sections to be arranged. Tuition $2,475.
*** MATH S-152 has been CANCELED.***
This course is an introduction to finite groups, finite fields, finite geometry, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. This course is taught in a seminar format, and students gain experience in presenting proofs at the blackboard. Prerequisites: elementary knowledge of vectors and determinants. MATH S-21b would be helpful. Calculus is not required.
MATH S-300
Math for Teaching Arithmetic (32409)
(Website) (Printable version)
Andrew Engelward
(4 credits: GR, NC) Mondays-Fridays, 9:30 am-noon. Tuition $1,240.
June 29-July 17. 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 minus sign (–) has three distinct mathematical meanings, and the symbol ½ has at least four different interpretations. No wonder this can lead to confusion for students. When a student encounters a problem using either of these two symbols he or she must decide which interpretation will best help solve the problem. In this course, we investigate basic operations from a variety of perspectives, and discuss why it makes sense to do arithmetic the way we do. Moreover, everyone taking the course should end up with a newfound respect for how amazing our number system actually is. (Students should try multiplying 97 by 56 using only 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. Although the material covered in this course is predominantly connected with middle school, because of the centrality of arithmetical topics throughout mathematics, it is recommended for anyone teaching math at either the middle or high school level.
MATH S-307
A Journey Into Calculus: Building on the Foundations of Middle School Mathematics (32249)
(Printable version)
Robin Gottlieb
(4 credits: GR, NC) Mondays-Fridays, 12:30-3 pm. Tuition $1,240. Limited enrollment.
July 20-August 7. Harvard College students see additional information.
What are the questions at the heart of calculus? What tools can be used to tackle these questions? How is calculus related to the mathematics of middle school? This course is an active exploration of calculus and its underpinnings, a journey into calculus for middle school mathematics teachers. In addition to the mathematical content we look at the pedagogical advantages of addressing the material from different vantage points: graphical, physical, and symbolic. This course is not meant to prepare participants to teach a calculus course nor is it meant to substitute for a calculus course. Our focus is on mathematical reasoning with the goal of gaining a deeper understanding of the implications and applications of foundational material. Prerequisite: Placement test recommended.
MATH S-310
Graph Theory: Investigating the Mathematical Process (32410)
(Printable version)
John D. Boller
(4 credits: GR, NC) Mondays-Fridays, 12:30-3 pm. Tuition $1,240. Limited enrollment.
June 29-July 17. Harvard College students see additional information.
Is mathematics invented or discovered? Is its language that of the mind or of 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 the way it is taught now. Prerequisite: Placement test recommended.