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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

  • MATH E-3 Quantitative Reasoning: Practical Math (Fall, Spring)
  • MATH E-6 Mathematics and the Greeks (Fall, Spring)
  • MATH E-8 College Algebra (Fall, Spring)
  • MATH E-10 Precalculus (Fall, Spring)
  • MATH E-15 Introduction to the Calculus A (Fall, Spring)
  • MATH E-16 Introduction to the Calculus B and Differential Equations (Fall, Spring)
  • MATH E-21a Multivariable Calculus (Fall)
  • MATH E-21b Linear Algebra (Spring)
  • MATH E-23a Linear Algebra and Real Analysis I (Fall)
  • MATH E-23b Linear Algebra and Real Analysis II (Spring)
  • MATH E-100 Elementary Number Theory with Applications (Fall)
  • MATH E-210 Vector Space Methods for Differential Equations (Spring)
  • MATH E-216 Convexity and Optimization with Applications (Fall)
  • MATH E-217 Probability and Random Processes with Economic Applications (Spring)

Mathematics for teaching

  • MATH E-300 Math for Teaching Arithmetic (Fall)
  • MATH E-302 Math for Teaching Geometry (Spring)
  • MATH E-303 Math for Teaching Algebra (Spring)
  • MATH E-311 Investigating the Mathematical Process: An Introduction to Proofs (Fall)
  • MATH E-314 Classical Mathematics: Linear Algebra and Group Theory (Fall)
  • MATH E-319 Counting Everything: Investigations with Combinatorics (Fall)
  • MATH E-320 Teaching Mathematics with a Historical Perspective (Spring)
  • MATH E-599 Teaching Projects: Math for Teaching Capstone Course (Spring)
MATH E-3 Quantitative Reasoning: Practical Math
Fall term (12500)
Graeme D. Bird, PhD, Lecturer in Extension, Harvard University.
Mondays beginning Aug. 29, 7:35-9:35 pm. Optional sections to be arranged.
Course tuition: undergraduate credit $975.
Spring term (20389)
Graeme D. Bird, PhD, Lecturer in Extension, Harvard University.
Mondays beginning Jan. 23, 5:30-7:30 pm. Optional sections to be arranged.
Course tuition: undergraduate credit $1,025.
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
Fall term (13408)
*** MATH E-6 Fall term (13408) has been CANCELED. ***
Spring term (23689)
Graeme D. Bird, PhD, Lecturer in Extension, Harvard University.
Tuesdays beginning Jan. 24, 5:30-7:30 pm. Optional sections to be arranged.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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.
Thursdays beginning Sept. 1, 6:30-9:30 pm.
Course tuition: noncredit $975, undergraduate credit $975.
Spring term (20393)
Wednesdays beginning Jan. 25, 5:30-7:30 pm. Optional sections Mondays, 6:30-7:30 pm or Thursdays, 5:30-6:30 pm.
Course tuition: noncredit $975, undergraduate credit $975.
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. Prerequisite: satisfactory placement test score. (4 credits)
MATH E-10 Precalculus
Fall term (12572)
David Arias, EdD, Mathematics Instructor, Andover High School.
Mondays beginning Aug. 29, 5:30-7:30 pm. Optional sections for undergraduate-credit students Tuesdays, 6:30-7:30 pm; required seminars for graduate-credit students Mondays, 4:30 to 5:30 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
Spring term (22379)
David Arias, EdD, Mathematics Instructor, Andover High School.
Wednesdays beginning Jan. 25, 5:30-7:30 pm. Optional sections for undergraduate-credit students Tuesdays, 6:30-7:30 pm; required seminars for graduate-credit students Wednesdays, 4:30-5:30 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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 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.
Thursdays beginning Sept. 1, 6-9 pm. Optional sections for all students Mondays, 7:35-8:35 pm; required seminars for graduate-credit students Thursdays, 4:30-5:30 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
Spring term (20399)
Eric C. Towne, AB, Assistant in Instruction, Mathematics, Bates College.
Thursdays beginning Jan. 26, 6-9 pm. Optional sections for all students Mondays, 7:35-8:35 pm; required seminars for graduate-credit students Thursdays, 4:30-5:30 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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)
Wednesdays beginning Aug. 31, 5:30-7:30 pm. Optional sections for undergraduate-credit students Mondays, 7:35-8:35 pm; required seminars for graduate-credit students Wednesdays, 7:35-8:35 pm
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
Spring term (20395)
Eric Connally, BA, Director of Engineering, Mathematics Consortium Working Group.
Thursdays beginning Jan. 26, 6:30-9:30 pm. Optional sections for undergraduate-credit students Tuesdays, 5:30-6:30 pm; required seminars for graduate-credit students Thursdays, 5:30-6:30 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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
Fall term (11648)
Robert Winters, PhD, Instructor in Mathematics, Massachusetts Institute of Technology.
Thursdays beginning Sept. 1, 7:35-9:35 pm. Optional sections to be arranged.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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
Spring term (21474)
Robert Winters, PhD, Instructor in Mathematics, Massachusetts Institute of Technology.
Thursdays beginning Jan. 26, 7:35-9:35 pm. Optional sections to be arranged.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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, differential equations, and function spaces. 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
Fall term (13313)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
Class times: Tuesdays, Thursdays beginning Sept. 1, 2:30 to 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. Prerequisite: a grade of A in MATH E-16, or the equivalent. (4 credits)
MATH E-23b Linear Algebra and Real Analysis II
Spring term (23175)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Tuesdays, Thursdays beginning Jan. 24, 2:30-4 pm. Required sections to be arranged.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
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-100 Elementary Number Theory with Applications
Fall term (13753)
*** MATH E-100 Fall term (13753) has been CANCELED. ***
This course is an introduction to the beauty and power of number theory. Ideal for inservice teachers and those aspiring to take discrete mathematics or abstract algebra, it emphasizes the various proof techniques and counterexamples, and presents many unsolved problems that have been studied for centuries. Once thought to be the purest of pure mathematics, number theory has a new dimenson since the advent of modern technology: constant practical applications, including computer security checks, such as check digits in UPCs, zip codes, ISBNs, ISSNs, EANs, VINs, credit card numbers, and library book numbers, to name a few. Topics include recursion, Pascal's triangle, binomial theorem, figurate numbers, Fibonacci and Lucas numbers, Fermat numbers, division algorithm, prime and composite numbers, Euclidean algorithm, gcd, lcm, linear diophantine equations, congruences, Pollard Rho factoring method, divisibility tests, perpetual calendar, Chinese remainder theorem, Wilson's theorem, Fermat's little theorem, Euler's theorem, perfect numbers, and Mersenne numbers. Graduate-credit students are expected to write programs in a structured language such as Java. Prerequisites: precalculus and mathematical maturity; knowledge of a structured language such as Java for graduate-credit students; placement test is recommended. (4 credits)
MATH E-210 Vector Space Methods for Differential Equations
Spring term (23498)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Tuesdays, Thursdays beginning Jan. 24, 11:30 am-1 pm. Required sections to be arranged.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
This course develops the theory of inner product spaces, both finite-dimensional and infinite-dimensional, and applies it to a variety of ordinary and partial differential equations. Topics include existence and uniqueness theorems, Sturm-Liouville systems, orthogonal polynomials, Fourier series, Fourier and Laplace transforms, eigenvalue problems, and solutions of Laplace's equation and the wave equation in various coordinate systems. Prerequisites: MATH E-21a and MATH E-21b or MATH E-23a. Some more advanced work in mathematics is strongly recommended. (4 credits)
MATH E-216 Convexity and Optimization with Applications
Fall term (13062)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
Class times: Mondays, Wednesdays beginning Aug. 31, 1-2:30 pm. Required sections Mondays, 3-4 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 MATH E-21b, or the equivalent, plus at least one other more advanced course in mathematics. (4 credits)
MATH E-217 Probability and Random Processes with Economic Applications
Spring term (23624)
Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University.
Mondays, Wednesdays beginning Jan. 23, 1-2:30 pm. Required sections Mondays, 3-4 pm.
Course tuition: noncredit $975, undergraduate credit $975, graduate credit $1,900.
This course is a self-contained treatment of the theory of probability and random processes. Topics include axioms for probability, conditional probability, Poisson and normal distributions, random walks, laws of large numbers and the central limit theorem, Markov processes, martingales, and Poisson and diffusion processes. It also covers applications to games of chance, the design of minimum-variance portfolios, and the Black-Scholes theory of option pricing. While emphasis is on careful analysis of models, occasional guest lectures explore applicability to the real world. Prerequisites: MATH E-21a and MATH E-21b, or MATH E-23a, or the equivalent. Acquaintance with elementary probability is desirable. (4 credits)

Mathematics for teaching

MATH E-300 Math for Teaching Arithmetic
Fall term (13787)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Tuesdays beginning Aug. 30, 4:30-6:30 pm.
Course tuition: graduate credit $1,950.
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-302 Math for Teaching Geometry
Spring term (23646)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Wednesdays beginning Jan. 25, 4:30-6:30 pm.
Course tuition: graduate credit $1,900.
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
Spring term (22499)
Tuesdays beginning Jan. 24, 4:30-6:30 pm.
Course tuition: graduate credit $1,900.
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-311 Investigating the Mathematical Process: An Introduction to Proofs
Fall term (13592)
Jameel Habeeb Al-Aidroos, PhD, Senior Preceptor in Mathematics, Harvard University.
Wednesdays beginning Aug. 31, 5:30-7:30 pm. Optional sections to be arranged.
Course tuition: graduate credit $1,900.
Ever wonder where a mathematical fact comes from or why you should believe it? What does it mean to prove a theorem and how do people discover or create these proofs? In this course, we begin with basic tools of deduction and reason our way together until we reach new and interesting conclusions. Along the way we see some beautiful results about numbers, sets, and shapes. The class is interactive. Prerequisites: a solid foundation in precalculus, curiosity about the theory behind the math that you've been learning, and an adventurous spirit of inquiry. (4 credits)
MATH E-314 Classical Mathematics: Linear Algebra and Group Theory
Fall term (13791)
Saturdays beginning Sept. 10, 10 am-1 pm. Required sections to be arranged.
Course tuition: graduate credit $1,900.
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-319 Counting Everything: Investigations with Combinatorics
Fall term (13788)
John Thomas Hall, PhD, Preceptor in Mathematics, Harvard University.
Thursdays beginning Sept. 1, 4:30-6:30 pm. Optional sections Thursdays, 6:30-7:30 pm.
Course tuition: graduate credit $1,900.
Combinatorics is the study of counting things. In this course, we explore a wide range of intriguing topics in which combinatorial problems arise, with a particular emphasis on discrete probability (every discrete probability problem really just comes down to counting all possible outcomes). Having a solid background in combinatorics is thus particularly useful for teaching probability. The course is primarily concerned with inquiry: the majority of class time is spent exploring problems, making and testing conjectures, and generally playing around with math. Topics include the basic principles of counting (the sum and product rules, injections, surjections, and bijections, the Pigeonhole Principle, and inclusion/exclusion), recursions and mathematical induction, permutations and permutation statistics, integer partitions and compositions, Catalan numbers, and generating functions. Prerequisite: comfort with polynomials is helpful. (4 credits)
MATH E-320 Teaching Mathematics with a Historical Perspective
Spring term (23238)
Oliver Knill, PhD, Preceptor in Mathematics, Harvard University.
Mondays beginning Jan. 23, 4:30-6:30 pm.
Course tuition: graduate credit $1,950.
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. Prerequisite: single variable calculus is helpful. (4 credits)
MATH E-599 Teaching Projects: Math for Teaching Capstone Course
Spring term (22946)
Andrew Engelward, PhD, Lecturer in Extension, Harvard University.
Tuesdays beginning Jan. 24, 4:30-6:30 pm.
Course tuition: graduate credit $1,900.
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 prior approval of the instructor. Students who do not meet these requirements are dropped from the course. (4 credits)

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