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#### Related courses

- APMA S-111 Introduction to Scientific Computing
- STAT S-100 Introduction to Quantitative Methods

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This page contains content from Summer School 2012. For current information, visit the Harvard Summer School website at www.summer.harvard.edu.

# Mathematics Courses

**Math placement test.** To enroll in courses marked with a star (*) below, you must take the math placement test. For test dates, see the math placement test information. Starred courses begin Tuesday, June 26.

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.

**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.

## Courselist

- MATH S-Y Mathematical Models and Expressions
- MATH S-Ar Precalculus Mathematics
- MATH S-1a Calculus I
- MATH S-1b Calculus II
- MATH S-1ab Calculus I and II
- MATH S-21a Multivariable Calculus
- MATH S-21b Linear Algebra and Differential Equations
- MATH S-101 Spaces, Mappings, and Mathematical Structures: An Introduction to Proof

#### Mathematics for Teaching

- MATH S-310 Graph Theory: Investigating the Mathematical Process
- MATH S-317 Math For Teachers: An Algebraic View of Geometry

#### *MATH S-Y Mathematical Models and Expressions (31920)

Class times: Tuesdays, Thursdays, 8:30-11:30 am. Optional sections to be arranged.

Course tuition: noncredit and undergraduate credit $2,700.

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 algebra. Placement test required. (4 credits)

#### *MATH S-Ar Precalculus Mathematics

Section 1 (30388)

Class times: Mondays-Thursdays, 10-11:30 am.

Course tuition: noncredit and undergraduate credit $2,700.

Section 2 (30389)

Class times: Mondays-Thursdays, 4:45-6:15 pm.

Course tuition: noncredit and undergraduate credit $2,700.

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. (4 credits)

#### *MATH S-1a Calculus I

Section 1 (30391)

Class times: Mondays-Fridays, 9-10:15 am.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

Section 2 (30392)

Class times: Mondays-Fridays, 10:30-11:45 am.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

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 graduate program in mathematics for teaching. Please contact the Mathematics for Teaching Office for details. (4 credits)

#### *MATH S-1b Calculus II (30393)

Class times: Mondays, Wednesdays, Fridays, 9:30-11:30 am. Required sections Tuesdays, Thursdays, 10:30-11:30 am.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

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 graduate program in mathematics for teaching. Please contact the Mathematics for Teaching Office for details. (4 credits)

#### *MATH S-1ab Calculus I and II (30390)

Andrew Engelward and Peter M. Garfield.

Class times: Mondays-Fridays, 8:45-11:45 am. Required sections to be arranged.

Course tuition: noncredit and undergraduate credit $5,400.

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. (8 credits)

#### MATH S-21a Multivariable Calculus (30189)

Class times: Tuesdays, Thursdays, 8:30-11:30 am. Optional sections Thursdays, 1-2 pm.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

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. (4 credits)

#### MATH S-21b Linear Algebra and Differential Equations (30190)

Class times: Mondays, Wednesdays, Fridays, 9:30-11:30 am. Optional sections to be arranged.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

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. (4 credits)

#### MATH S-101 Spaces, Mappings, and Mathematical Structures: An Introduction to Proof (31859)

Class times: Tuesdays, Wednesdays, Thursdays, noon-2 pm. Optional sections to be arranged.

Course tuition: noncredit, undergraduate, and graduate credit $2,700.

Ever wonder where a theorem comes from or why you should believe it? There is much more to mathematics than learning formulas and procedures: studying mathematics is about building and describing mathematical structures, discovering their properties (your theorems), and convincing yourself and others of your discoveries (by writing proofs). In this class, we start from some basic assumptions (no calculus necessary) and reason our way together until we have built and described some important concrete and abstract structures. Along the way, students learn the subtle art of mathematical reasoning, and we convince ourselves of some surprising and sophisticated conclusions, including beautiful results from basic number theory, group theory, and topology. Prerequisites: imagination, a solid mastery of precalculus, as well as a serious interest in making and critiquing arguments. Placement test recommended. (4 credits)

## Mathematics for Teaching

#### MATH S-310 Graph Theory: Investigating the Mathematical Process (32838)

Class times: Mondays-Fridays, 12:30-3 pm. Optional sections to be arranged.

Course tuition: graduate credit $1,350.

This course meets Tuesday, June 26 to Friday, July 13; the last date to withdraw for a WD grade is July 9. Harvard College students see additional information.

Limited enrollment.

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. (4 credits)

#### MATH S-317 Math For Teachers: An Algebraic View of Geometry (32707)

Class times: Mondays-Thursdays, 8:30-11:30 am. Required sections to be arranged.

Course tuition: graduate credit $1,350.

This course meets Tuesday, June 26 to Friday, July 13; the last date to withdraw for a WD grade is July 9. Harvard College students see additional information.

This course introduces little-known topics in plane geometry that are accessible to secondary school students. Topics include finite affine and projective geometry, isometries of Euclidean geometry, construction of regular polygons with 5 and 7 sides, origami constructions, and models for non-Euclidean plane geometries. Axioms for groups, fields, and vector spaces are presented as algebraic background. Prerequisite: secondary school mathematics through trigonometry. (4 credits)