Return to Department List

## Department - MATH

## MATH E-3 Section 1 (20389)

Spring 2023

**Quantitative Reasoning: Practical Math**

Graeme D. Bird PhD, Lecturer in Extension, Harvard University

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. No previous knowledge of these tools is assumed. Recommendations for calculators are made during the first class.

Prerequisites: A willingness to (re)discover math, appreciate its practical uses, and enjoy its patterns and beauty.

## MATH E-3 Section 1 (12500)

Fall 2022

**Quantitative Reasoning: Practical Math**

Graeme D. Bird PhD, Lecturer in Extension, Harvard University

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. No previous knowledge of these tools is assumed. Recommendations for calculators are made during the first class.

Prerequisites: A willingness to (re)discover math, appreciate its practical uses, and enjoy its patterns and beauty.

## MATH E-4 Section 1 (16456)

Fall 2022

**Mathematical Modeling with Spreadsheets I**

Eric Connally BA, Principal Engineer, Illustrative Mathematics

Mathematical modeling uses a variety of mathematical tools to describe a wide range of real-world situations. Such tools include algebraic expressions, graphs, and statistics to represent critical elements of complicated systems. When done well, it can help us to understand such systems as well as to predict their future behavior. In this course, we employ the highly versatile structure of spreadsheets to explore a wide range of mathematical models involving topics from finance, biology, and physics, among others. This course builds on the basic arithmetical, algebraic, and graphing skills developed in MATH E-3 and is intended for students who want develop more applied mathematical skills, expand their expertise in the use of spreadsheets, or enroll in MATH E-8 in the future.

Prerequisites: MATH E-3 or the equivalent.

## MATH E-5 Section 1 (26316)

Spring 2023

**Mathematical Modeling with Spreadsheets II**

Eric Connally BA, Principal Engineer, Illustrative Mathematics

This course is a continuation of MATH E-4. The course introduces more sophisticated spreadsheet tools for building mathematical models and analyzing data. Whereas MATH E-4 focuses on the basics of spreadsheets, this course covers more advanced topics including applications in statistics, finance, and physics; ranges and array-valued functions; queries, lookups, offsets, matches, indexing, and filters; user interface elements like dropdowns; dashboards for large data sets; specialized built-in functions; and, time allowing, custom built-in functions coded using the script editor.

Prerequisites: MATH E-4 or the equivalent.

## MATH E-6 Section 1 (23689)

Spring 2023

**Mathematics and the Greeks**

Graeme D. Bird PhD, Lecturer in Extension, Harvard University

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 either write a research paper on some aspect of Greek mathematics or 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.

## MATH E-8 Section 1 (20393)

Spring 2023

**College Algebra**

David Abbruzzese, Jr. BSEE

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. This course features some of the same topics as MATH E-10, but at a slower pace and more introductory level. In addition, it does not cover trigonometry and sinusoidal functions, which are discussed in depth in MATH E-10.

Prerequisites: A satisfactory placement test score.

## MATH E-10 Section 1 (22379)

Spring 2023

**Precalculus**

David Arias EdD

An intensive course for students with superior algebra skills who want to enroll in MATH E-15 the following term. During the semester, linear, quadratic, exponential, logarithmic, trigonometric, polynomial, and rational functions are discussed. 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.

Prerequisites: A satisfactory placement test score.

## MATH E-10 Section 1 (12572)

Fall 2022

**Precalculus**

David Arias EdD

An intensive course for students with superior algebra skills who want to enroll in MATH E-15 the following term. During the semester, linear, quadratic, exponential, logarithmic, trigonometric, polynomial, and rational functions are discussed. 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.

Prerequisites: A satisfactory placement test score.

## MATH E-15 Section 1 (20399)

Spring 2023

**Calculus 1**

Eric C. Towne AB, Curriculum Advisor, Advanced Placement Calculus, The College Board

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.

Prerequisites: MATH E-10, or the equivalent, or satisfactory placement test score. The graduate-credit option is available only to students participating in the Extension School's mathematics for teaching program.

## MATH E-15 Section 1 (10436)

Fall 2022

**Calculus 1**

Eric C. Towne AB, Curriculum Advisor, Advanced Placement Calculus, The College Board

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.

Prerequisites: MATH E-10, or the equivalent, or satisfactory placement test score. The graduate-credit option is available only to students participating in the Extension School's mathematics for teaching program.

## MATH E-16 Section 1 (10437)

Fall 2022

**Calculus 2 with Series and Differential Equations**

Srdjan Divac MA, Lecturer on Mathematics and Statistics, Boston University

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.

## MATH E-16 Section 1 (20395)

Spring 2023

**Calculus 2 with Series and Differential Equations**

Srdjan Divac MA, Lecturer on Mathematics and Statistics, Boston University

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.

## MATH E-21b Section 1 (21474)

Spring 2023

**Linear Algebra**

Robert Winters PhD

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, or permission of the instructor; some familiarity with vectors; general familiarity with matrix-capable calculators or mathematical software; the placement test is recommended but not required.

## MATH E-21a Section 1 (11648)

Fall 2022

**Multivariable Calculus**

Robert Winters PhD

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.

Prerequisites: MATH E-16, or the equivalent; placement test is recommended.

## MATH E-21c Section 1 (16432)

Fall 2022

**Ordinary Differential Equations**

Robert Winters PhD

This course covers ordinary differential equations (ODEs); continuous models; analytic, graphical, and numerical solutions; input-response formulation of linear ODEs; systems of first-order ODEs and matrix exponentials; and nonlinear systems and phase-plane analysis.

Prerequisites: One variable calculus; some familiarity with multivariable calculus, linear algebra, and complex numbers.

## MATH E-23a Section 1 (15176)

Fall 2022

**Linear Algebra and Real Analysis I**

Grant Andrew Murray MA - Kris Lokere ALM

This is the first half of an integrated treatment of linear algebra, real analysis, and multivariable calculus. By combining these disciplines into one course, we show important relations between each, which allows us to use results from one topic to gain deeper understanding of other topics. We cover matrices, eigenvectors, dot and cross products, limits, continuity, and differentiability, all in multiple dimensions, with an introduction to manifolds. This course covers both mathematical proofs as well as applications. Students learn to write more than twenty important proofs and see how proof-based mathematics prepares them for applications in engineering, economics, data science, and artificial intelligence.

Prerequisites: A grade of A in MATH E-16 or the equivalent. Some experience with multivariable calculus and linear algebra is not necessary but preferred.

## MATH E-23c Section 1 (25129)

Spring 2023

**Mathematics for Computation and Data Science**

Kris Lokere ALM - Grant Andrew Murray MA

Topics in discrete mathematics, real analysis, linear algebra, and integral calculus, chosen for their relevance to computer science, probability, statistics, and data science. Foundations of probability, logic, vector spaces, infinite series, and integration. Application to statistical problems. Includes an introduction to statistical and graphical tools in the R programming language. Students may not take both MATH E-23c and MATH E-23b for degree or certificate credit.

Prerequisites: Linear algebra, solid single-variable calculus, and introductory multivariable differential calculus. MATH E-23a would be more than sufficient.

## MATH E-115 Section 1 (26355)

Spring 2023

**Introduction to Complex Analysis**

David Arias EdD

Complex analysis is the study of functions of a complex variable. A complex variable (z) can take on the value of a complex number (x + iy), where i is the unit imaginary number and x and y represent real numbers. Differentiation and integration of complex functions involve procedures used to differentiate and integrate functions of real numbers. Thus, if you enjoyed calculus of real variables, you would enjoy complex analysis. During the semester, we discuss limits, continuity, differentiation, and integration involving exponential, logarithmic, power, trigonometric, hyperbolic, inverse trigonometric, and inverse hyperbolic complex functions. Cauchy-Riemann equations, analytic functions, harmonic functions, Cauchy-Goursat theorem, Taylor series, Laurent series, and Cauchy's residue theorem are also discussed.

Prerequisites: Math E-21a or equivalent.

## MATH E-125 Section 1 (26428)

Spring 2023

**Introduction to Vector Calculus and Matrix Algebra Using Python**

Kris Lokere ALM

This is an introductory course in vector calculus and linear algebra, with an emphasis on using Python to build intuition and to do computations. Key theorems are stated, but not proved. Rather, we aim to understand why they work by showing examples in software.

Prerequisites: MATH E-16 or similar background in calculus is required. Either some background with Python or some prior exposure to linear algebra and/or multivariable calculus is preferred.

## MATH E-151 Section 1 (16435)

Fall 2022

**Classic Mathematics with a Modern User Interface**

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

The course presents a variety of topics from linear algebra, abstract algebra, and geometry that lend themselves to an interesting visual display: for example, groups, finite fields, graph theory, finite, Euclidean, and spherical geometry. Students implement key mathematical ideas in R, build an application in R Shiny Dashboard that can be deployed to the internet, and use their software to devise conjectures that they can then prove as theorems.

Prerequisites: Solid command of precalculus mathematics. A course in linear algebra is useful but not required. Programming ability in some language (Python, Java, C/C++) is essential, but no prior knowledge of R, HTML, or client-server programming is assumed.

## MATH E-156 Section 1 (26408)

Spring 2023

**Mathematical Statistics**

Dmitry V. Kurochkin PhD, Senior Research Analyst, Faculty of Arts and Sciences Office for Faculty Affairs, Harvard University

This course is an introduction to mathematical statistics and data analysis. It starts by introducing central concepts of probability theory (events, probability measure, random variables, distributions, joint distributions, and conditional distributions) and then moves on to the development of mathematical foundations of statistical inference. Topics covered in the course include random variables, expectations, parameter estimation (method of moments, method of maximum likelihood, and Bayesian approach), properties of point estimators (bias, variance, consistency, and efficiency), confidence intervals, hypotheses testing, likelihood ratio test, data summary methods, and introduction to linear regression. A class of distributions, including chi-squared, t, and F distributions, the distributions derived from normal that occur in many applications of hypothesis testing and statistical inference, are introduced.

Prerequisites: MATH E-15 or equivalent. No prior knowledge of probability is assumed. Students are required to take a short pretest at the beginning of the course. The pretest score does not count toward the final grade but helps students understand whether their background in calculus positions them for success in this course.

## MATH E-156 Section 1 (16470)

Fall 2022

**Mathematical Statistics**

Dmitry V. Kurochkin PhD, Senior Research Analyst, Faculty of Arts and Sciences Office for Faculty Affairs, Harvard University

This course is an introduction to mathematical statistics and data analysis. It starts by introducing central concepts of probability theory (events, probability measure, random variables, distributions, joint distributions, and conditional distributions) and then moves on to the development of mathematical foundations of statistical inference. Topics covered in the course include random variables, expectations, parameter estimation (method of moments, method of maximum likelihood, and Bayesian approach), properties of point estimators (bias, variance, consistency, and efficiency), confidence intervals, hypotheses testing, likelihood ratio test, data summary methods, and introduction to linear regression. A class of distributions, including chi-squared, t, and F distributions, the distributions derived from normal that occur in many applications of hypothesis testing and statistical inference, are introduced.

Prerequisites: MATH E-15 or equivalent. No prior knowledge of probability is assumed. Students are required to take a short pretest at the beginning of the course. The pretest score does not count toward the final grade but helps students understand whether their background in calculus positions them for success in this course.

## MATH E-216 Section 1 (26065)

Spring 2023

**Real Analysis, Convexity, and Optimization**

Grant Andrew Murray MA

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, MATH E-23a, or the equivalent, plus at least one other more advanced course in mathematics. Students need to know linear algebra and multivariable calculus and be comfortable with proofs.

## MATH E-300 Section 1 (25477)

Spring 2023

**Mathematical Foundations for Teaching Secondary School Math**

Carolyn Gardner-Thomas PhD, Director, Mathematics for Teaching Program, Harvard Extension School

Why do students have such a difficult time with basic math concepts such as working with fractions and negative numbers? It could be because arithmetic is significantly more complex than we initially suspect. For instance, the symbol ½ has at least four different interpretations, and students need to be able to quickly figure out which interpretation will be of most use for solving a particular problem. This course was created for middle and high school mathematics teachers to give them a chance to explore the inner workings of fundamental mathematical concepts involved in arithmetic as well as the basis for working with a variety of number systems. The course deconstructs basic math concepts that many people often take for granted, but yet which can continue to give students difficulties throughout their school years. The course emphasizes mathematical reasoning rather than memorizing facts and formulas. In addition to the mathematical content, we also discuss how different methods of teaching affect students differently and we explore a variety of activities and games that teachers can bring to their own classrooms to enhance their students' understanding and enjoyment of mathematics.

Prerequisites: Familiarity with K-12 mathematics.

## MATH E-300 Section 1 (13787)

Fall 2022

**Mathematical Foundations for Teaching Secondary School Math**

Andrew Engelward PhD, Lecturer in Extension, Harvard University

Why do students have such a difficult time with basic math concepts such as working with fractions and negative numbers? It could be because arithmetic is significantly more complex than we initially suspect. For instance, the symbol ½ has at least four different interpretations, and students need to be able to quickly figure out which interpretation will be of most use for solving a particular problem. This course was created for middle and high school mathematics teachers to give them a chance to explore the inner workings of fundamental mathematical concepts involved in arithmetic as well as the basis for working with a variety of number systems. The course deconstructs basic math concepts that many people often take for granted, but yet which can continue to give students difficulties throughout their school years. The course emphasizes mathematical reasoning rather than memorizing facts and formulas. In addition to the mathematical content, we also discuss how different methods of teaching affect students differently and we explore a variety of activities and games that teachers can bring to their own classrooms to enhance their students' understanding and enjoyment of mathematics.

Prerequisites: Familiarity with K-12 mathematics.

## MATH E-301 Section 1 (16762)

Fall 2022

**Elementary Number Theory**

David Arias EdD

Number theory can be used to find the greatest common divisor, determine whether a number is prime, and solve Diophantine equations. With the improvement of computer technology, number theory also helps us to protect private information by encrypting it as it travels through the internet. During the course, we discuss mathematical induction, division and Euclidean algorithms, the Diophantine equation ax + by = c, the fundamental theorem of arithmetic, prime numbers and their distribution, the Goldbach conjecture, congruences, the Chinese remainder theorem, Fermat's theorem, Wilson's theorem, Euler's theorem, and cryptography. Additional topics may include number-theoretic functions, primitive roots, and the quadratic reciprocity law.

Prerequisites: MATH E-8.

## MATH E-302 Section 1 (26524)

Spring 2023

**Math for Teaching Geometry**

Carolyn Gardner-Thomas PhD, Director, Mathematics for Teaching Program, Harvard Extension School

Geometry is about symmetry, shape, and space. This course emphasizes mathematical reasoning and the role of mathematical discourse in geometry classrooms. Our explorations begin with the classic work on geometry, Euclid's The Elements. We study straightedge and compass constructions; investigate golden rectangles, constructible numbers, and geometry in higher dimensions; and work to more modern topics such as tessellations and Pick's Theorem.

Prerequisites: Knowledge of number systems, algebra, and other standard precalculus mathematics. Experience teaching geometry would be useful, but not essential.

## MATH E-320 Section 1 (25678)

Spring 2023

**Teaching Mathematics with a Historical Perspective**

Oliver Knill PhD, Preceptor in Mathematics, Harvard University

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 gives an overview as well as discusses some core results in that area.

Prerequisites: Single variable calculus is helpful.

## MATH E-345 Section 1 (16409)

Fall 2022

**Math Teacher Leadership**

Carolyn Gardner-Thomas PhD, Director, Mathematics for Teaching Program, Harvard Extension School

This course supports math teacher leadership knowledge and skill development through investigations of practice, reflections, design, and implementation of programs oriented to address dynamic contextual school situations. Using research-based frameworks for teacher leadership development, we explore strategies to drive school improvement efforts in mathematics teaching, learning, and school culture. Students engage with collaborative and system-based approaches for timely, relevant, data-informed, and sustainable mathematics education improvement. The course is designed for math teachers and administrators interested in formal and informal coaching and mentoring of math teachers, the design and facilitation of professional development experiences for math teachers, and transformational leadership in mathematics education.

Prerequisites: Knowledge of number systems, algebra, and other standard precalculus mathematics. A minimum of three years of teaching mathematics is assumed.

## MATH E-599 Section 1 (16408)

Fall 2022

**Teaching Projects: Math for Teaching Capstone**

Carolyn Gardner-Thomas PhD, Director, Mathematics for Teaching Program, Harvard Extension School

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: Registration is limited to officially admitted degree candidates for the Master of Liberal Arts, mathematics for teaching, capstone track. Prospective degree candidates and students with pending admission applications are not eligible. Candidates must be in good academic standing, ready to graduate in March, with only the capstone left to complete (no other course registration is allowed simultaneously with the capstone). Candidates who do not meet these requirements are dropped from the course.

## MATH E-599 Section 1 (22946)

Spring 2023

**Teaching Projects: Math for Teaching Capstone**

Andrew Engelward PhD, Lecturer in Extension, Harvard University

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: Registration is limited to officially admitted degree candidates for the Master of Liberal Arts, mathematics for teaching, capstone track. Prospective degree candidates and students with pending admission applications are not eligible. Candidates must be in good academic standing, ready to graduate in May, with only the capstone left to complete (no other course registration is allowed simultaneously with the capstone). Candidates who do not meet these requirements are dropped from the course.