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Mathematics

See also Graduate Program in Mathematics for Teaching and Graduate Program in Information Technology.

Math Question Center. For students in MATH E-3, E-6, 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

Fall term (12500)

Charlotte Bonner, EdM.

Class times: Wednesdays beginning Sept. 2, 5:30-7:30 pm. Optional sections to be arranged.

Course tuition: undergraduate credit $900.

Spring term (20389)

Graeme D. Bird, PhD, Associate Professor of Linguistics and Classics, Gordon College.

Class times: Mondays beginning Jan. 25, 7:35-9:35 pm. Optional sections Thursdays, 7:35-9:35 pm

Course tuition: undergraduate credit $900.

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-6 Mathematics and the Greeks (13408)

Fall term

Graeme D. Bird, PhD, Associate Professor of Linguistics and Classics, Gordon College.

Class times: Mondays beginning Aug. 31, 7:35-9:35 pm. Optional sections Thursdays, 6:30-7:30 pm.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

In this course we seek to understand how the ancient Greeks thought about mathematics by focusing on three activities: finding solutions and proofs for simple numerical problems, drawing geometrical constructions using compasses and straightedge, and reading brief historical abstracts by and about early Greek mathematicians. Students also learn the Greek alphabet to enable them to read a few common mathematical terms. Graduate-credit students prepare a series of lesson plans showing how a section of the course material could be taught in high schools. Prerequisites: high school algebra or MATH E-8. (4 credits)

MATH E-8 College Algebra

Fall term (13477)

Eric Connally, BA, Director of Engineering, Mathematics Consortium Working Group.

Class times: Thursdays beginning Sept. 3, 6:30-9:30 pm. Optional sections for undergraduate-credit students Tuesdays, 5:30-6:30 pm; required sections for graduate-credit students Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Spring term (20393)

David Abbruzzese, Jr., BSEE.

Class times: Wednesdays beginning Jan. 27, 5:30-7:30 pm. Optional sections for undergraduate-credit students Mondays, 5:30-6:30 pm; required sections for graduate-credit students Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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. (4 credits)

MATH E-10 Precalculus

Fall term (12572)

David Arias, EdD, Mathematics Instructor, Andover High School.

Class times: Tuesdays beginning Sept. 1, 5:30-7:30 pm. Optional sections for undergraduate-credit students Mondays, 6:30-7:30 pm; required sections for graduate-credit students Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Spring term (22379)

Charlotte Bonner, EdM.

Class times: Tuesdays beginning Jan. 26, 5:30-7:30 pm. Optional sections for undergraduate-credit students to be arranged; required sections for graduate-credit students Tuesdays, 7:35-8:35 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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, or satisfactory placement test score. (4 credits)

MATH E-15 Introduction to the Calculus A

Fall term (10436)

Eric C. Towne, AB, Assistant in Instruction, Mathematics, Bates College.

Class times: Thursdays beginning Sept. 3, 6-9 pm. Optional sections for undergraduate-credit students Mondays, 7:35-8:35 pm; required sections for graduate-credit students Thursdays, 4:30-5:30 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Spring term (20399)

Eric C. Towne, AB, Assistant in Instruction, Mathematics, Bates College.

Class times: Thursdays beginning Jan. 28, 6-9 pm. Optional sections for undergraduate-credit students Mondays, 7:35-8:35 pm; required sections for graduate-credit students Thursdays, 4:30-5:30 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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, or satisfactory placement test score. (4 credits)

MATH E-16 Introduction to the Calculus B and Differential Equations

Fall term (10437)

Srdjan Divac, MA.

Class times: Wednesdays beginning Sept. 2, 7:35-9:35 pm. Optional sections for undergraduate-credit students Mondays, 7:35-8:35 pm; required sections for graduate-credit students Wednesdays, 6:30-7:30 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Spring term (20395)

Eric Connally, BA, Director of Engineering, Mathematics Consortium Working Group.

Class times: Thursdays beginning Jan. 28, 6:30-9:30 pm. Optional sections for undergraduate-credit students Tuesdays, 5:30-6:30 pm; required sections for graduate-credit students Thursdays, 4:30-5:30 pm, beginning the second week of classes.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

This course covers integration, differential equations, and Taylor series with applications. It covers most of the topics in a second-semester calculus course with the emphasis on applications as well as 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, or satisfactory placement test score. (4 credits)

MATH E-21a Multivariable Calculus (11648)

Fall term

Robert Winters, PhD, Lecturer in Mathematics, Wellesley College.

Class times: Thursdays beginning Sept. 3, 7:35-9:35 pm. Optional sections to be arranged.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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 may be asked to complete additional projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16, or the equivalent; placement test is recommended. (4 credits)

MATH E-21b Linear Algebra (21474)

Spring term

Robert Winters, PhD, Lecturer in Mathematics, Wellesley College.

Class times: Thursdays beginning Jan. 28, 7:35-9:35 pm. Optional sections to be arranged.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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 may be asked to complete additional 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; placement test is recommended. (4 credits)

MATH E-23a Linear Algebra and Real Analysis I (13313)

Fall term

Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Tuesdays, Thursdays, beginning Sept. 3, 2:30-4 pm. Required sections to be arranged.

This course is an integrated treatment of linear algebra and multivariable differential calculus, with an introduction to manifolds. Students are required to learn 22 important proofs. Prerequisites: a grade of A in MATH E-16, or the equivalent. (4 credits)

MATH E-23b Linear Algebra and Real Analysis II (23175)

Spring term

Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.

Class times: Tuesdays, Thursdays beginning Jan. 26, 2:30-4 pm. Required sections to be arranged.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

This course covers Riemann and Lebesgue integration in n dimensions, differential forms, and Stokes's theorem. Students are required to learn 20 important proofs. Prerequisite: MATH E-23a, or the equivalent. (4 credits)

MATH E-104 Discrete Mathematics with Computer Science Applications (11478)

Fall term

Thomas Koshy, PhD, Professor of Mathematics, Framingham State College.

Class times: Tuesdays beginning Sept. 1, 5:30-7:30 pm. Optional sections to be arranged.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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)

Spring term

Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.

Class times: Mondays, Wednesdays beginning Jan. 25, 4-5:30 pm. Required sections to be arranged.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

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-216 Convexity and Optimization with Applications (13062)

Fall term

Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.

Course tuition: noncredit and undergraduate credit $900, graduate credit $1,800.

Tuesdays, Thursdays, beginning Sept. 3, 4:30-6 pm. Optional sections Tuesdays, 7-8 pm.

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 (23236)

Spring term

Andrew Engelward, PhD, Lecturer in Extension, Harvard University.

Class times: Wednesdays beginning Jan. 27, 4:30-6:30 pm.

Course tuition: graduate credit $1,800.

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)

Fall term

Juliana Belding, PhD, Preceptor in Mathematics, Harvard University.

Class times: Mondays beginning Aug. 31, 4:30-6:30 pm.

Course tuition: graduate credit $1,800.

There are still many mysteries surrounding number theory and prime numbers as we explore 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, divisibility tests, and application to security codes and cryptography. Prerequisite: familiarity with K-12 mathematics. (4 credits)

MATH E-302 Math for Teaching Geometry (13393)

Fall term

Andrew Engelward, PhD, Lecturer in Extension, Harvard University.

Class times: Tuesdays beginning Sept. 1, 5:30-7:30 pm.

Course tuition: graduate credit $1,800.

Online option available. Lecture 1 video.

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 and Pick's theorem. Along the way we also investigate golden rectangles, 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)

Spring term

Srdjan Divac, MA.

Class times: Tuesdays beginning Jan. 26, 4:30-6:30 pm. Optional sections Tuesdays, 3:30-4:30 pm, beginning the second week of classes.

Course tuition: graduate credit $1,800.

This course examines the mathematical underpinnings behind what is taught in secondary level algebra courses. It considers what, why, and how we teach what we teach, and investigates different strands of algebraic competence with particular emphasis on how we assess the students' proficiency in these various strands. Prerequisite: familiarity with K-12 mathematics. (4 credits)

MATH E-304 Inquiries into Probability and Combinatorics (13392)

Fall term

Andrew Engelward, PhD, Lecturer in Extension, Harvard University.

Class times: Wednesdays beginning September 9, 4:30-6:30 pm. Please note that this class begins a week later than other classes.

Course tuition: graduate credit $1,800.

Probability is the study of uncertainty, something that surrounds us every day. Will the stock market rise? Will it rain tomorrow? Beginning with the intuitive frequency interpretation of probability we develop language necessary to answer interesting questions and resolve puzzling paradoxes (consider the Monty Hall problem and birthday matches). In this course we work our way through such topics as sample spaces and events, counting principles, basic combinatorics, and independent events and conditional probability, along with probability distributions and various statistical tests. Prerequisites: familiarity with K-12 mathematics. (4 credits)

MATH E-311 Analysis: Investigating the Mathematical Process (22428)

Spring term

John Thomas Hall, PhD, Preceptor in Mathematics, Harvard University.

Class times: Thursdays beginning Jan. 28, 4:30-6:30 pm. Optional sections Thursdays, 3:30-4:30 pm.

Course tuition: graduate credit $1,800.

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 and set theory. 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)

Fall term

Roberto E. Martínez II, SM.

Class times: Saturdays beginning Sept. 12, 10 am-1 pm.

Course tuition: graduate credit $1,800.

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-320 Teaching Math with a Historical Perspective (23238)

Spring term

Oliver Knill, PhD, Preceptor in Mathematics, Harvard University.

Class times: Mondays beginning Jan. 25, 4:30-6:30 pm.

Course tuition: graduate credit $1,800.

The difficulty of both learning and teaching math is evident in its history. The struggle of early research mathematicians who developed and formalized a topic parallels the struggle of students and teachers in the modern classroom. Students learning about the concept of limits and series undergo a similar process as the pioneers of calculus did when they developed the subject. Archimedes, Zeno, Cavalieri, Newton, Leibniz, and Cauchy had to find or invent structure. This struggle goes on today, as new flavors of calculus are developed and studied. Each week, this course considers a different math subject and pinpoints a moment when something interesting happened. This moment is condensed into a specific and concrete mathematical problem. Each story is tied to a particular mathematician. (4 credits)

MATH E-599 Teaching Projects: Math for Teaching Capstone Course (22946)

Spring term

Andrew Engelward, PhD, Lecturer in Extension, Harvard University.

Class times: Tuesdays beginning Jan. 26, 4:30-6:30 pm.

Course tuition: graduate credit $1,800.

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, participants 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, participants are asked to investigate how use of a particular technology can be used to enhance classroom math lessons. In addition, everyone gets a chance to participate in an alternative math teaching experiment. 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, or with prior approval of the instructor. Students who do not meet these requirements are dropped from the course. (4 credits)



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