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Mathematics

See courselist below.

For information about related programs and scholarships see:

Math placement test. Students who want to enroll in mathematics courses are expected to take the online placement test unless in the preceding term they satisfactorily completed at Harvard Extension School or Summer School the prerequisite qualifying course. Students are not expected to study for the test; placement is made assuming students have not taken a mathematics course recently. Every student who takes the test is eligible to register for a course: there is no minimum background required. Students receive placement results as soon as they have finished the test. Advisors are available at the Math Question Center during the first two weeks of class each term.

Students should log in to online services using their ID numbers and registration PINs and select the Math Placement Test link. Students who wish to take the placement test online may need to download and install some software prior to taking the test. It is therefore extremely important that they consult the technical requirements page (follow the link from the Math Placement Test page) before attempting the test. The test is available from August 22 to October 2 and December 5 to February 12.

Students who do not have access to computers or the necessary browser applications may take the test at the Division of Continuing Education's computer facility, 53 Church Street. Beginning September 19, the facility is open Sunday through Thursday, 9 am-12 midnight, and Friday and Saturday, 9 am-10 pm. Students with questions should send e-mail to support@math-placement.dce.harvard.edu.

Math Question Center. For students in MATH E-3, E-8, E-9, E-10, E-15, and E-16, individual help is available at the Math Question Center. The center is open Mondays, 5:30-7:30 pm, and Tuesdays, 7:35-9:35 pm, in Sever Hall 215 beginning the first week of classes each term.

Pedagogical seminars. Graduate-credit students enrolled in courses with weekly lesson study sections must participate in a graduate section for each course in which they are enrolled.

Courselist

Mathematics for Teaching

MATH E-3 Quantitative Reasoning: Practical Math
4 units. Undergraduate credit $625.

Fall term (12500) (Website) (Print View): Graeme D. Bird, PhD, Assistant Professor of English, Gordon College. Monday, Sept. 19 and Thursday, 5:30-7:30 pm, Emerson Hall, Room 108.

Spring term (20389) (Website) (Print View): Charlotte Bonner, EdM. Tuesday, Jan. 31 and Thursday, 5:30-7:30 pm, Science Center, Hall E.

This course reviews basic arithmetical procedures and their use in everyday mathematics. It also includes an introduction to basic statistics covering such topics as the interpretation of numerical data, graph reading, hypothesis testing, and simple linear regression. The personal computer is utilized for the building of spreadsheets, graphs, and some statistical data analysis. No previous knowledge of these tools is assumed. New material is presented during Monday lectures in the fall and Tuesday lectures in the spring; the Thursday class is reserved for all exams, an additional lecture review, and hands-on experience with a calculator. Recommendations for calculators are made during the first class. Prerequisite: placement test is recommended.
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MATH E-8 College Algebra
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.

Fall term (10431) (Website) (Print View): Eric Connally, BA, Analyst, Elytics, Inc. Thursday, Sept. 22, 6-9 pm, Sever Hall, Room 202. Optional undergraduate-credit sections Tuesday, 5:30-6:30 pm; required graduate-credit sections Thursday, 4:45-5:45 pm beginning the second week of classes.

Spring term (20393) (Syllabus) (Print View): David Abbruzzese, Jr., BSEE. Wednesday, Feb. 1, 5:30-7:30 pm, Sever Hall, Room 102. Required undergraduate-credit sections Wednesday, 7:35-8:35 pm; required graduate-credit sections Wednesday, 7:35-8:35 pm beginning the second week of classes.

This course reviews arithmetic and covers algebraic expressions and equations; their manipulation and use in problem solving; word problems; and an introduction to inequalities, absolute values, and graphing. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: satisfactory placement test score.
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MATH E-9 Functions and Graphs: Introduction to Precalculus (22429)
(Website) (Print View)
Andrew Engelward, PhD, Preceptor in the Mathematics for Teaching Program, Harvard University.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.
Wednesday, Feb. 1, 5:30-7:30 pm, Emerson Hall, Room 104. Optional undergraduate-credit sections Monday, 7:35-8:35 pm; required graduate-credit sections Wednesday, 7:35-8:35 pm beginning the second week of classes. Spring term

This course introduces students to the language of functions and graphing. We work with polynomial, exponential, and trigonometric functions, all at a fundamental level. The course emphasizes problem solving and applications of math to real-world situations. MATH E-9 is intended for students who have successfully completed a college algebra course such as MATH E-8 but who would like to spend more time reviewing basic precalculus topics involving functions than is possible in MATH E-10. Students interested in taking MATH E-15 should take MATH E-10 after completing MATH E-9. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: MATH E-8 or satisfactory placement test score.
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MATH E-10 Precalculus
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.

Fall term (12572) (Website) (Print View): David Arias, EdD, Mathematics Instructor, Andover High School. Thursday, Sept. 22, 6-9 pm, Sever Hall, Room 310. Required undergraduate-credit sections Tuesday, 5:30-6:30 pm; required graduate-credit sections Thursday, 4:45-5:45 pm beginning the second week of classes.

Spring term (22379) (Website) (Print View): Eric Connally, BA, Analyst, Elytics, Inc. Thursday, Feb. 2, 6-9 pm, Sever Hall, Room 202. Optional undergraduate-credit sections Tuesday, 5:30-6:30 pm; required graduate-credit sections Thursday, 4:45-5:45 pm beginning the second week of classes.

An intensive course for students with superior algebra skills who want to enroll in MATH E-15 the following term. Requires the use of a graphing calculator. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: A- grade in MATH E-8, B- in MATH E-9, or satisfactory placement test score.
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MATH E-15 Introduction to the Calculus A
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.

Fall term (10436) (Website) (Print View): Eric C. Towne, AB. Thursday, Sept. 22, 6-9 pm, Emerson Hall, Room 101. Optional undergraduate-credit sections Monday, 7:35-8:35 pm; required graduate-credit sections Thursday, 4:45-5:45 pm beginning the second week of classes.

Spring term (20399) (Website) (Print View): Eric C. Towne, AB. Thursday, Feb. 2, 6-9 pm, Emerson Hall, Room 108. Optional undergraduate-credit sections Monday, 7:35-8:35 pm; required graduate-credit sections Thursday, 4:45-5:45 pm beginning the second week of classes.

A complete course in first-semester calculus. The meaning, use, and interpretation of the derivative; techniques of differentiation; applications to curve sketching and optimization in a variety of disciplines; the definite integral and some applications; and the fundamental theorem of calculus. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: MATH E-10 or satisfactory placement test score.
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MATH E-16 Introduction to the Calculus B and Differential Equations
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.

Fall term (10437) (Syllabus) (Print View): Srdjan Divac, MA. Wednesday, Sept. 21, 5:30-7:30 pm, Sever Hall, Room 103. Optional undergraduate-credit sections Monday, 7:35-8:35 pm; required graduate-credit sections Wednesday, 7:35-8:35 pm beginning the second week of classes.

Spring term (20395) (Syllabus) (Print View): Srdjan Divac, MA. Wednesday, Feb. 1, 5:30-7:30 pm, Sever Hall, Room 214. Optional undergraduate-credit sections Monday, 7:35-8:35 pm; required graduate-credit sections Wednesday, 7:35-8:35 pm beginning the second week of classes.

Integration and differential equations with applications. Covers most of the topics in a second-semester calculus course but with an emphasis on graphical and numerical work. Requires the use of a graphing calculator with the capability of computing (approximating) definite integrals. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisites: MATH E-15 or equivalent--in other words, an excellent working knowledge of first-semester calculus, including the trigonometric and logarithmic functions--or satisfactory placement test score.
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MATH E-21a Multivariable Calculus (11648)
(Website) (Print View)
Robert Winters, PhD, Visiting Assistant Professor of Mathematics, Wellesley College.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.
Wednesday, Sept. 21, 7:35-9:35 pm, Harvard Hall, Room 201. Optional sections to be arranged. Fall term

Calculus of functions of several variables; vectors and vector-valued functions; parameterized curves and surfaces; vector fields; partial derivatives and gradients; optimization; method of Lagrange multipliers; integration over regions in R2 and R3; integration over curves and surfaces; Green's theorem, Stokes's theorem, divergence theorem. Graduate-credit students complete additional computer-related or other projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16 or equivalent and general familiarity with computer use; placement test is recommended.
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MATH E-21b Linear Algebra (21474)
(Website) (Print View)
Robert Winters, PhD, Visiting Assistant Professor of Mathematics, Wellesley College.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,525.
Wednesday, Feb. 1, 7:35-9:35 pm, Emerson Hall, Room 210. Optional sections to be arranged. Spring term

Solving systems of linear equations; matrices and linear transformations; image and kernel of a linear transformation; matrices and coordinates relative to different bases; determinants; eigenvalues and eigenvectors; discrete and continuous dynamical systems; least-squares approximation; applications in function spaces. Graduate-credit students complete additional computer-related or other projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16 or equivalent and general familiarity with matrix-capable calculators or mathematical software; placement test is recommended.
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MATH E-102 Sets, Counting, and Probability (12591)
(Website) (Print View)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,600.
Tuesday, Sept. 20, 7:35-9:35 pm, 53 Church Street, Room L01. Optional sections to be arranged. Fall term
Online and on-campus options. See the Distance Education website. Lecture 1 video. Lecture 2 video.

This course develops the mathematics needed to formulate and analyze probability models for idealized situations drawn from everyday life. Topics include elementary set theory, techniques for systematic counting, axioms for probability, conditional probability, discrete random variables, infinite geometric series, and random walks. Applications to card games like bridge and poker, to gambling, to sports, to election results, and to inference in fields like history and genealogy, national security, and theology. The emphasis is on careful application of basic principles rather than on memorizing and using formulas. Graduate-credit students learn a little of the programming language PHP and use it to create interactive webpages that illustrate probability models. Prerequisite: excellent command of high school algebra, as indicated by the placement test. Students enrolled for graduate credit should have an elementary working knowledge of a programming language, but no experience with PHP is assumed.
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MATH E-104 Discrete Mathematics with Computer Science Applications (11478)
(Syllabus) (Print View)
Thomas Koshy, PhD, Professor of Mathematics, Framingham State College.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,600.
Tuesday, Sept. 20, 5:30-7:30 pm, Science Center, Room B-10. Optional sections to be arranged. Fall term

An introduction to topics in discrete mathematics that are useful in computer science: mathematical logic, set theory, induction, recursion, combinatorics, relations and functions, and graphs. Includes an introduction to algorithm analysis and techniques for proving correctness of programs. Practical programming examples, as well as proofs of relevant theorems, are discussed. The course is intended both for beginning students of computer science and for computer professionals. Prerequisites: a good working knowledge of Java or other structured language and of precalculus mathematics as evidenced by an appropriate score on the placement test; additional background in computing or in college-level mathematics is desirable but not essential.
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MATH E-105 Groups, Graphs, and Algebraic Structures for Computing (22399)
(Website) (Print View)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,600.
Tuesday, Jan. 31, 7:35-9:35 pm, Science Center, Room 110. Optional sections to be arranged. Spring term

This course explores concepts from discrete mathematics that are useful in many different applications, especially selected and designed for people working in computer programming. We examine proofs and applications concerning topics such as graphs, probability, combinatorics, finite fields, permutations, Markov processes, symmetry groups, and linear algebra. Graduate-credit students have the opportunity to implement many of the mathematical ideas in the course as interactive webpages using the scripting language PHP. Prerequisites: MATH E-10 or satisfactory placement test score required. Some experience with material beyond MATH E-10 is desirable. For graduate-credit students, CSCI E-50a or equivalent. Prior knowledge of PHP is not required.
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MATH E-138 Classical Geometry (21846)
(Website) (Print View)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
4 units. Noncredit and undergraduate credit $625, graduate credit $1,600.
Monday and Wednesday, beginning Wednesday, Feb. 1, 4-5:30 pm, Science Center, Room 110. Optional sections to be arranged. Spring term

An exploration of the many different flavors of plane geometry. The course begins with finite geometry, then surveys the nine possible Cayley-Klein plane geometries, focusing on Euclidean geometry, the Galilean geometry of uniform motion, spherical and elliptic geometry, and geometries related to relativistic physics such as Minkowskian geometry and hyperbolic geometry. An important tool in the study of these geometries is a study of their symmetry groups. Prerequisites: MATH E-21a and E-21b, which may be taken concurrently; placement test is recommended.
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Mathematics for Teaching

MATH E-301 Theory and Practice of Teaching Number Theory (12614)
(Website) (Print View)
Andrew Engelward, PhD, Preceptor in the Mathematics for Teaching Program, Harvard University.
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.
Tuesday, Sept. 20, 4:30-6:30 pm, Science Center, Room 309. Fall term

Topics include rules of arithmetic, the ring of integers, order axioms and well-ordering of the natural numbers, the division algorithm and Euclidean algorithm, primes and divisibility, the fundamental theorem of arithmetic, and congruences and modular arithmetic. Additional topics may include palindromes, Fibonacci numbers, combinatorial formulas, four numbers game, and Egyptian fractions. Prerequisite: familiarity with K-12 mathematics; placement test is recommended.
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MATH E-302 Theory and Practice of Teaching Geometry (12119)
(Website) (Print View)
Andrew Engelward, PhD, Preceptor in the Mathematics for Teaching Program, Harvard University.
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.
Thursday, Sept. 22, 4:30-6:30 pm, Science Center, Room 309. Fall term

Topics from Euclidean geometry include polygons, straightedge and compass construction, Gauss's theorem on constructability, regular polyhedra, symmetry groups and group theory, and area and volume. Additional topics may include golden rectangles, fractals, geometric proof, constructible numbers, dissections, tessellations, packing problems, lattice polygons, and Pick's theorem. Prerequisite: familiarity with K-12 mathematics; placement test is recommended.
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MATH E-303 Theory and Practice of Teaching Algebra
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.

Fall term (12586) (Website) (Print View): Angela Vierling-Claassen, PhD, Preceptor in Mathematics, Harvard University. Wednesday, Sept. 21, 4:30-6:30 pm, Science Center, Room 309.

Spring term (22499) (Syllabus) (Print View): Srdjan Divac, MA. Tuesday, Jan. 31, 4:30-6:30 pm, English High School, Room 407, 144 McBride Street, Jamaica Plain.

Elementary algebraic concepts, including set theory; graphs of linear functions; polynomials, including roots of polynomials, factoring, and graphing; an introduction to algebraic structures, including groups and rings; exponentials and logarithms; and measurement, estimation, and problem solving are discussed. The course also provides a variety of activities that teachers can bring to their classrooms to enhance student understanding of algebra, graphing, and functions. Prerequisites: knowledge of and experience teaching algebra and precalculus; placement test is recommended.
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MATH E-304 Theory and Practice of Teaching Probability (22438)
(Website) (Print View)
Matthew Leingang, PhD, Preceptor in Mathematics, Harvard University.
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.
Monday, Jan. 30, 4:30-6:30 pm, Science Center, Room 309. Optional sections to be arranged. Spring term

Topics include elementary ideas of probability theory, set theory and its relation to probability theory, introduction to sample spaces and events, basic facts about binomial coefficients and counting techniques, independent and dependent events, advanced combinatorial formulas, random variables, probability distributions, statistical methods, and testing statistical hypotheses. Prerequisite: familiarity with K-12 mathematics; placement test is recommended.
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MATH E-305 Theory and Practice of Teaching Advanced Algebra (22440)
(Website) (Print View)
Angela Vierling-Claassen, PhD, Preceptor in Mathematics, Harvard University.
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.
Wednesday, Feb. 1, 4:30-6:30 pm, Science Center, Room 309. Spring term

Topics include rings and an introduction to algebraic structures; polynomials and the algebra of polynomials; polynomial arithmetic and the division algorithm for polynomials; order relations; linear and quadratic inequalities; progressions, sequences, and series--arithmetic, geometric, and otherwise; complex numbers and their geometry; and the fundamental theorem of algebra. Prerequisites: MATH E-303 or equivalent and familiarity with K-12 mathematics; placement test is recommended.
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MATH E-311 Analysis: Investigating the Mathematical Process (22428)
(Website) (Print View)
Andrew Engelward, PhD, Preceptor in the Mathematics for Teaching Program, Harvard University.
4 units. Noncredit $325, graduate credit $1,525. Limited enrollment.
Thursday, Feb. 2, 4:30-6:30 pm, Science Center, Room 309. Spring term

There is more to mathematics than formulas and procedures. Ever wonder where a theorem comes from or why you should believe it? What does it mean to prove a theorem and how do mathematicians discover or create proofs? In this course, we begin with basic assumptions (no calculus necessary) and reason our way together until we reach some interesting and sophisticated conclusions, including beautiful results from basic analysis, set theory, and topology. Prerequisites: an interest in puzzles, a solid foundation in precalculus, and the patience to discuss theoretical mathematics as well as an interest in making and critiquing arguments. Placement test is recommended.
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MATH E-312 Classical Mathematics I: Geometry and Algebra (12266)
(Syllabus) (Print View)
Roberto E. Martínez II, SM, Research Assistant, Division of Engineering and Applied Sciences, Harvard University.
4 units. Noncredit $325, graduate credit $1,525.
Saturday, Sept. 24, 10 am-12 noon, Science Center, Room 110. Required sections to be arranged. Fall term

This is a course in geometry in which we endeavor to study its interweaving relationship with algebra spanning roughly two and a half millennia. Equipped primarily with trigonometric and algebraic tools, we follow the masters through timeless vignettes of mathematical triumphs and genius. From the unknown Egyptian author of the Rhind papyrus to Euclid, Gauss, and recently Einstein and Minkowski, geometry has proven its worth in the measurement of land partitions and straightedge and compass constructions to solutions of cubic equations and even to its manifestations in curvature and fundamental processes of the physical universe. Prerequisites: familiarity with K-12 mathematics and proof methodology; placement test is recommended.
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MATH E-313 Classical Mathematics II: Number Theory and Analysis (22304)
(Syllabus) (Print View)
Roberto E. Martínez II, SM, Research Assistant, Division of Engineering and Applied Sciences, Harvard University.
4 units. Noncredit $325, graduate credit $1,525.
Saturday, Feb. 4, 10 am-12 noon, Science Center, Room 110. Required sections to be arranged. Spring term

This is a course in number theory and real analysis in which the historical development of geometry takes a central role. We investigate the real numbers, sequences, convergence, limits, series and products, continuity, differentiation and integration, elucidating the connection with geometry along the way. In addition, we study irrationality and transcendence to understand the impossibility of circle squaring, the greatest problem of antiquity. Finally, we sample a variety of computational methods for determining digits of pi. In this way, students gain experience with Mathematica, a powerful, symbolic program used by scientists and mathematicians worldwide. Students may not count both MATH E-130 and MATH E-313 toward a degree. Prerequisites: familiarity with K-12 mathematics (including calculus) and proof methodology; placement test is recommended. MATH E-312 is not a formal prerequisite, but the material complements this course.
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