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

Courses

This page contains content from the 2008–09 academic year. For current information, visit the Harvard Extension School website at www.extension.harvard.edu.

Mathematics

See courselist below.

For information about related programs and scholarships see:

Math Question Center. For students in MATH E-3, E-8, 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.

Courselist

Mathematics for Teaching

MATH E-3 Quantitative Reasoning: Practical Math
Course tuition: undergraduate credit $800.

Fall term (12500) (Website) (Printable version): Charlotte Bonner, EdM.
Tuesdays beginning Sept. 16, 5:30-7:30 pm, Sever Hall, Room 213. Optional sections to be arranged.

Spring term (20389) (Syllabus) (Printable version): Graeme D. Bird, PhD, Assistant Professor of Linguistics and Classics, Gordon College.
Mondays beginning Jan. 26, 7:35-9:35 pm, Emerson Hall, Room 101. Optional sections Thursdays, 7:35-9:35 pm

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. Recommendations for calculators are made during the first class. (4 credits)

MATH E-8 College Algebra
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.

Fall term (10431) (Website) (Printable version): Eric Connally, BA.
Thursdays beginning Sept. 18, 6-9 pm, Sever Hall, Room 103. Optional undergraduate-credit sections Tuesdays, 5:30-6:30 pm; required graduate-credit sections Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Spring term (20393) (Syllabus) (Printable version): David Abbruzzese, Jr., BSEE.
Wednesdays beginning Jan. 28, 5:30-7:30 pm, Emerson Hall, Room 210. Optional undergraduate-credit sections, required graduate-credit sections Thursdays, 4:30-5:30, 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. (4 credits)

MATH E-10 Precalculus
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.

Fall term (12572) (Website) (Printable version): David Arias, EdD, Mathematics Instructor, Andover High School.
Tuesdays beginning Sept. 16, 5:30-7:30 pm, Sever Hall, Room 106. Optional undergraduate-credit sections Mondays, 6:30-7:30 pm; required graduate-credit sections Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Spring term (22379) (Website) (Printable version): Charlotte Bonner, EdM.
Tuesdays beginning Jan. 27, 5:30-7:30 pm, Sever Hall, Room 102. Optional undergraduate-credit sections, required graduate-credit sections Tuesdays, 7:35-8:35 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. (4 credits)

MATH E-15 Introduction to the Calculus A
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.

Fall term (10436) (Website) (Printable version): Eric C. Towne, AB.
Thursdays beginning Sept. 18, 6-9 pm, Emerson Hall, Room 210. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Thursdays, 4:30-5:30 pm, beginning the second week of classes.

Spring term (20399) (Website) (Printable version): Eric C. Towne, AB.
Thursdays beginning Jan. 29, 6-9 pm, Emerson Hall, Room 210. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Thursdays, 4:30-5:30 pm, beginning the second week of classes.

This is a complete course in first-semester calculus. Topics include 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 the equivalent. (4 credits)

MATH E-16 Introduction to the Calculus B and Differential Equations
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.

Fall term (10437) (Syllabus) (Printable version): Srdjan Divac, MA.
Wednesdays beginning Sept. 17, 7:35-9:35 pm, Sever Hall, Room 214. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Wednesdays, 6:30-7:30 pm, beginning the second week of classes.

Spring term (20395) (Website) (Printable version): Eric Connally, BA.
Thursdays beginning Jan. 29, 6-9 pm, Sever Hall, Room 214. Optional undergraduate-credit sections Tuesdays, 5:30-6:30 pm; required graduate-credit sections Thursdays, 4:30-5:30 pm, beginning the second week of classes.

This course explores integration and differential equations with applications. It covers most of the topics in a second-semester calculus course but with an emphasis on graphical and numerical work. The use of a graphing calculator with the capability of computing (approximating) definite integrals is required. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisites: MATH E-15, or the equivalent—in other words, an excellent working knowledge of first-semester calculus, including the trigonometric and logarithmic functions. (4 credits)

MATH E-21a Multivariable Calculus (11648) (Website) (Printable version)
Robert Winters, PhD, Lecturer in Mathematics, Brandeis University.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Fall term: Thursdays beginning Sept. 18, 7:35-9:35 pm, Harvard Hall, Room 202. Optional sections to be arranged.

This course covers the following topics: 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 the equivalent and general familiarity with computer use. (4 credits)

MATH E-21b Linear Algebra (21474) (Website) (Printable version)
Robert Winters, PhD, Lecturer in Mathematics, Brandeis University.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Spring term: Thursdays beginning Jan. 29, 7:35-9:35 pm, Harvard Hall, Room 201. Optional sections to be arranged.

This course covers the following topics: 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 the equivalent, and general familiarity with matrix-capable calculators or mathematical software. (4 credits)

MATH E-104 Discrete Mathematics with Computer Science Applications (11478) (Website) (Printable version)
Thomas Koshy, PhD, Professor of Mathematics, Framingham State College.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Fall term: Tuesdays beginning Sept. 16, 5:30-7:30 pm, Science Center, Room B-10. Optional sections to be arranged.

This course is 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. It 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 another structured language and of precalculus mathematics; additional background in computing or in college-level mathematics is desirable but not essential. (4 credits)

MATH E-138 Classical Geometry (22919) (Website) (Printable version)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Spring term
Mondays, Wednesdays beginning Wednesday, Jan. 28, 4-5:30 pm.

This course is an exploration of the many different flavors of plane geometry. It 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. (4 credits)

MATH E-140 Scientific Computing with Matlab (13085) (Website) (Printable version)
Marie Dahleh, PhD, Lecturer on Engineering Sciences, Harvard University.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Fall term: Mondays beginning Sept. 15, 5:30-7:30 pm, 53 Church Street, Room 201.

This course uses Matlab to introduce students to the fundamentals of numerical methods. The emphasis is on understanding and using numerical methods for the following types of problems: nonlinear root finding, solutions of linear equations, curve fitting, integration, differentiation, eigenvalues, and the solution of ordinary differential equations. Prerequisites: MATH E-21a, or the equivalent; the course requires background in calculus, linear algebra, and ordinary differential equations. (4 credits)

MATH E-216 Convexity and Optimization with Applications (13062) (Website) (Printable version)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Course tuition: noncredit and undergraduate credit $800, graduate credit $1,725.
Fall term: Tuesdays, Thursdays beginning Sept. 16, 4-5:30 pm, Science Center, Room 110.

This course develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students are expected to understand and invent proofs of theorems in real and functional analysis. Prerequisites: MATH E-21a and E-21b, or the equivalent, plus at least one other more advanced course in mathematics. (4 credits)

Mathematics for Teaching

MATH E-300 Math for Teaching Arithmetic (13089) (Website) (Printable version)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Course tuition: noncredit $450, graduate credit $1,725.
Fall term: Wednesdays beginning Sept. 17, 4:30-6:30 pm, English High School, Room 407, 144 McBride Street, Jamaica Plain.

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 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. In addition to the mathematical content, we also discuss how different methods of teaching affect students differently. Prerequisite: familiarity with K-12 mathematics. (4 credits)

MATH E-301 Math for Teaching Number Theory (12614) (Website) (Printable version)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Course tuition: noncredit and graduate credit $1,725.
Fall term: Thursdays beginning Sept. 18, 4:30-6:30 pm, 53 Church Street, Room L01. Optional sections to be arranged.
Online and on-campus options. See Distance Education. Lecture 1 video.

There are still many mysteries surrounding number theory and prime numbers as we see in this course. Once considered purely abstract, now number theory is critical for almost all of the security codes used on the Internet. In this course we investigate the makings of basic number systems such as the rational and real numbers, study their properties, and work through the division algorithm, Euclidean algorithm, and modular arithmetic to prove the fundamental theorem of arithmetic. Additional topics may include palindromic numbers, Fibonacci numbers, perfect numbers, numbers in different bases, repeating decimals, and Egyptian fractions. Prerequisite: familiarity with K-12 mathematics. (4 credits)

MATH E-302 Math for Teaching Geometry (22945) (Website) (Printable version)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Course tuition: noncredit $450, graduate credit $1,725.
Spring term: Wednesdays beginning Jan. 28, 4:30-6:30 pm, Science Center, Room 216. Optional sections to be arranged.

Geometry is all about symmetry, shape, and space. We begin our exploration by going back to the classic work on geometry, Euclid's The Elements; studying straightedge and compass constructions; and then working our way to more modern topics such as tessellations, Pick's theorem, and fractals. Along the way we also investigate golden rectangles, symmetry groups, constructible numbers, and geometry in higher dimensions. We emphasize mathematical reasoning, and communicating mathematics plays an important role in the course. Prerequisite: familiarity with high school geometry. (4 credits)

MATH E-303 Math for Teaching Algebra (22499) (Syllabus) (Printable version)
Srdjan Divac, MA.
Course tuition: noncredit $450, graduate credit $1,725.
Spring term: Tuesdays beginning Jan. 27, 4:30-6:30 pm, English High School, Room 407, 144 McBride Street, Jamaica Plain. Optional sections Tuesdays, 3:30-4:30 pm, beginning the second week of classes.

This course covers the following topics: 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. 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. (4 credits)

MATH E-306 Theory and Practice of Teaching Statistics (22927) (Website) (Printable version)
Joan Weinstein, CAS, MA, Professor of Mathematics and Computers, Pine Manor College.
Course tuition: noncredit $450, graduate credit $1,725.
Spring term: Tuesdays beginning Jan. 27, 4:30-6:30 pm, Science Center, Room 112. Optional sections to be arranged.

This course is an introduction to descriptive and inferential statistics, with attention to current trends and practices in statistics education. The class is activity-based, emphasizing understanding statistical concepts through the use of software for calculations, simulations, and demonstrations. This course would be particularly useful preparation for teaching Advanced Placement Statistics. Prerequisite: working knowledge of algebra. (4 credits)

MATH E-309 Teaching Math with Technology: A Critical Examination of the School Curriculum (12979) (Website) (Printable version)
Benjamin Levy, MAT, Consultant, Mathematics Education and Technology.
Course tuition: noncredit $450, graduate credit $1,725.
Fall term: Mondays beginning Sept. 15, 5:30-7:30 pm, Science Center, Room 111.

This course, for both current and future teachers, examines standard presentations of selected topics from the secondary mathematics curriculum and considers how the presentation of each topic reflects what methods and tools are assumed. Features of graphing calculators, dynamic geometry software, and computer algebra systems are introduced and explored. Teachers write exercises, lessons, and curricula that assume students will make essential use of technology to learn mathematics and to solve problems. Prerequisite: familiarity with secondary school mathematics, from basic algebra and geometry through precalculus. (4 credits)

MATH E-311 Analysis: Investigating the Mathematical Process (22428) (Website) (Printable version)
Roberto E. Martínez II, SM.
Course tuition: noncredit $450, graduate credit $1,725.
Spring term: Saturdays beginning Jan. 31, 10 am-noon, Science Center, Room 112. Optional sections to be arranged.

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. (4 credits)

MATH E-314 Classical Mathematics: Linear Algebra and Group Theory (13088) (Syllabus) (Printable version)
Roberto E. Martínez II, SM.
Course tuition: noncredit $450, graduate credit $1,725.
Fall term: Saturdays beginning Sept. 20, 10 am-noon, Science Center, Room 309.

This is a course in both linear algebra and group theory that develops the general structure and formulation of geometries over sets in terms of their transformation groups (Klein Erlangen Programme). By introducing elementary notions of Galois theory and symmetric groups, we also prove the impossibility of solving by radicals the quintic or higher order polynomial equations (in a single variable). Prerequisites: familiarity with K-12 mathematics, including calculus and proof methodology. (4 credits)

MATH E-599 Teaching Projects: Math for Teaching Capstone Course (22946) (Website) (Printable version)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Course tuition: graduate credit $1,725.
Spring term: Thursdays beginning Jan. 29, 4:30-6:30 pm, Science Center, Room 309.

This course is intended to give current and aspiring secondary math teachers an opportunity to become engaged in a variety of teaching-related projects. In the first part of the course, students are given a chance to research a current topic in mathematics education through use of journal articles, giving a presentation of their findings to the math for teaching community. In the second part, students are asked to investigate how use of a particular technology can be used to enhance classroom math lessons. In the final part, students get to apply the knowledge they have gained in the program to develop a short course on a nonstandard middle or high school topic. Designs are shared with reviewers and students are given an opportunity to videotape themselves teaching their material during a practice teaching session. Prerequisites: students must be candidates for the Master of Liberal Arts in Mathematics for Teaching and in their final semester of the program; successful completion of MATH E-15. (4 credits)