A World of Mathematics How long will oil last? What is the fairest voting system? How can we harvest food and other resources sustainably? To explore such real-world questions we will study a variety of mathematical ideas and methods, including modeling, logical analysis, discrete dynamical systems, and elementary statistics. This is an alternative first mathematics course for students not pursuing the calculus sequence in their first semester. The only prerequisite is an interest in exploring contemporary issues using the mathematics that lies within those issues. (Approval required; This course is not open to students who have had a prior course in calculus or statistics.) 3 hrs lect./disc.

Mathematical Problem-Solving This course is designed primarily for students concurrently enrolled in MATH 121 or MATH 122 who would benefit from structured support to reinforce their mathematical backgrounds, and these students will be given priority at registration. We will emphasize problem-solving rather than a collection of procedures, using problems selected to strengthen students’ conceptual understanding of the material and their strategic competence. In an inquiry-based setting, students will practice and improve their algebra and trigonometry skills, with an emphasis on effective exposition of mathematical arguments.(This is a half credit course.)(Approval required.) 1.5 hrs. disc.

Math and Board Games Have you ever spent minutes agonizing over which move to make in a board game? Out of all the possible options, how could you possibly determine which move is best? Was there even an objectively best decision? In this course, we will explore the mathematics and underlying gameplay structures of several modern board games. In addition to playing these games during class, we’ll use math and logic to assess and quantify the value of a range of possible in-game decisions. Using formal mathematical proofs, papers, and in-class discussions, we’ll analyze the fairness and equity of strategies across a wide variety of games. We’ll finish the course by designing our own board game based on what we’ve learned!

Mathematics for Teachers What mathematical knowledge should elementary and secondary teachers have in the 21st century? Participants in this course will strengthen and deepen their own mathematical understanding in a student-centered workshop setting. We will investigate the number system, operations, algebraic thinking, measurement, data, and functions, and consider the attributes of quantitative literacy. We will also study recent research that describes specialized mathematical content knowledge for teaching.

Making Data Visual Information can be used to inform, persuade, excite, and build community identity. Being able to move between data in a spreadsheet to a story that uses data that highlights information responsibly is a critical skill. In this course we will learn about communication standards for sharing data with experts and non-experts alike. Gaining skills in programs such as Canva, Photoshop and R, we will work to build data visualizations that are accurate, interesting, and responsibly represented. In the final project we will turn to our own community using data to tell a story about experiences at Middlebury.

Introduction to Statistical Science A practical introduction to statistical methods and the examination of data sets. Computer software will play a central role in analyzing a variety of real data sets from the natural and social sciences. Topics include descriptive statistics, elementary distributions for data, hypothesis tests, confidence intervals, correlation, regression, contingency tables, and analysis of variance. The course has no formal mathematics prerequisite, and is especially suited to students in the physical, social, environmental, and life sciences who seek an applied orientation to data analysis. (Credit is not given for MATH 0116 if the student has taken ECON 0111 (formerly ECON 0210) or PSYC 0201 previously or concurrently.)

Introduction to Data Science In this course students will gain exposure to the entire data science pipeline: forming a statistical question, collecting and cleaning data sets, performing exploratory data analyses, identifying appropriate statistical techniques, and communicating the results, all the while leaning heavily on open source computational tools, in particular the R statistical software language. We will focus on analyzing real, messy, and large data sets, requiring the use of advanced data manipulation/wrangling and data visualization packages. Students will be required to bring alaptop (owned or college-loaned) to class as many lectures will involve in-class computational activities. (formerly MATH216) 3 hrs lect./disc.

Calculus I Introductory analytic geometry and calculus. Topics include limits, continuity, differential calculus of algebraic and trigonometric functions with applications to curve sketching, optimization problems and related rates, the indefinite and definite integral, area under a curve, and the fundamental theorem of calculus. Inverse functions and the logarithmic and exponential functions are also introduced along with applications to exponential growth and decay.

Calculus II A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations.

Linear Algebra Matrices and systems of linear equations, the Euclidean space of three dimensions and other real vector spaces, independence and dimensions, scalar products and orthogonality, linear transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule.

Advanced Introduction to Statistical and Data Sciences An introduction to statistical methods and the examination of data sets for students with a background in calculus. Topics include descriptive statistics, elementary distributions for data, hypothesis tests, confidence intervals, and regression. Students develop skills in data cleaning, wrangling, visualization, and model fitting using the Statistical Software R.  Emphasis will be placed on reproducibility. (MATH 0121 or APAB 4 or APBC 3, or by waiver) (Not open to students who have taken MATH 0116, ECON 0111 (formerly ECON 0210), PSYC 0201, BIOL 1230, ECON 1230, ENVS 1230, FMMC 1230, HARC 1230, JAPN 1230, LNGT 1230, NSCI 1230, MATH 1230, SOCI 1230, LNGT 1230, PSCI 1230, WRPR 1230, or GEOG 1230.)

Regression Theory and Applications Regression is a popular statistical technique for making predictions and for modeling relationships between variables.  In this course we will discuss the theory and practical applications of linear, log-linear, and logistic regression models.  Topics include least squares estimation, coding for categorical predictors, analysis of variance, and model diagnostics.  We will apply these concepts to real datasets using R, a statistical programming language.  (MATH 0200; and MATH 0116 or MATH 0311) 3 hrs lect./disc.

Introduction to Data Science In this course students will gain exposure to the entire data science pipeline: forming a statistical question, collecting and cleaning data sets, performing exploratory data analyses, identifying appropriate statistical techniques, and communicating the results, all the while leaning heavily on open source computational tools, in particular the R statistical software language. We will focus on analyzing real, messy, and large data sets, requiring the use of advanced data manipulation/wrangling and data visualization packages. Students will be required to bring their own laptops as many lectures will involve in-class computational activities. 3 hrs lect./disc.

Statistical Learning This course is an introduction to modern statistical, machine learning, and computational methods to analyze large and complex data sets that arise in a variety of fields, from biology to economics to astrophysics. The theoretical underpinnings of the most important modeling and predictive methods will be covered, including regression, classification, clustering, resampling, and tree-based methods. Student work will involve implementation of these concepts using open-source computational tools.

Multivariable Calculus The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications.

Topics in Linear Algebra and Differential Equations Topics may include diagonalization of matrices, quadratic forms, inner product spaces, canonical forms, the spectral theorem, positive matrices, the Cayley-Hamilton theorem, ordinary differential equations of arbitrary order, systems of first-order differential equations, power series, and eigenvalue methods of solution, applications.

MATH 0226, Differential Equations This course provides an introduction into ordinary differential equations (ODEs) with an emphasis on linear and nonlinear systems using analytical, qualitative, and numerical techniques.  Topics will include separation of variables, integrating factors, eigenvalue method, linearization, bifurcation theory, and numerous applications.  In this course, we will introduce MATLAB programming skills and develop them through the semester. (MATH 0122 and MATH 0200 or by waiver)

Introduction to Numerical Analysis We will study the development, analysis, and implementation of numerical methods for approximating solutions to mathematical problems. We will begin with applications of Taylor polynomials, computer representation of numbers, and types of errors.  Other topics will include polynomial and spline interpolation, numerical integration and differentiation, rootfinding, and numerical solutions of differential equations.  Accuracy will be quantified by the concept of numerical error.  Additionally, we will study the stability, efficiency, and implementation of algorithms.  We will utilize the software MATLAB throughout to demonstrate concepts, as well as to complete assignments and projects.

Euclidean and Non-Euclidean Geometries In roughly 300 BCE, Euclid set down his axioms of geometry which subsequently became the standard by which people understood the mathematics of the world around them.  In the first half of the 19th century, mathematicians realized, however, that they could remove one of Euclid’s axioms, the one known as the “parallel postulate,” and still produce logically consistent examples of geometries.  These new geometries displayed behaviors that were wildly different from Euclidean geometry.  In this course we will study examples of these revolutionary non-Euclidean geometries, with a focus on Klein's Erlangen Program, which is a modern way of understanding them.

Elementary Number Theory Divisibility and prime factorization. Congruences; the theorems of Lagrange, Fermat, Wilson, and Euler; residue theory; quadratic reciprocity. Diophantine equations. Arithmetic functions and Mobius inversion. Representation as a sum of squares.

Graph Theory       A graph (or network) is a useful mathematical model when studying a set of discrete objects and the relationships among them.  We often represent an object with a vertex (node) and a relation between a pair with an edge (line).  With the graph in hand, we then ask questions, such as: Is it connected? Can one traverse each edge precisely once and return to a starting vertex? For a fixed k/, is it possible to “color” the vertices using /k colors so that no two vertices that share an edge receive the same color?  More formally, we study the following topics: trees, distance, degree sequences, matchings, connectivity, coloring, and planarity.  Proof writing is emphasized.

Abstract Algebra Groups, subgroups, Lagrange's theorem, homomorphisms, normal subgroups and quotient groups, rings and ideals, integral domains and fields, the field of quotients of a domain, the ring of polynomials over a domain, Euclidean domains, principal ideal domains, unique factorization, factorization in a polynomial ring.

Probability An introduction to the concepts of probability and their applications, covering both discrete and continuous random variables. Probability spaces, elementary combinatorial analysis, densities and distributions, conditional probabilities, independence, expectation, variance, weak law of large numbers, central limit theorem, and numerous applications.

Statistical Inference       An introduction to the mathematical methods and applications of statistical inference using both classical methods and modern resampling techniques. Topics will include: permutation tests, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines. This course is taught using R.

Mathematical modeling An introduction into the process of developing and interpreting mathematical models within the framework of numerous applications. In this course, we will utilize discrete, continuous, and probabilistic approaches to explore applications such as population dynamics, epidemiology, and neuron activity.  Time permitting, we may also introduce the derivation of spatiotemporal models.  MATLAB will be used to implement and analyze several of these models.

Operations Research Operations research is the utilization of quantitative methods as an aid to managerial decisions. In the course, several of these methods will be introduced and studied in both a mathematical context and a physical context. Topics included will be selected from the following: classification of problems and the formulation of models, linear programming, network optimization, transportation problems, assignment problems, integer programming, nonlinear programming, inventory theory, and game theory.

Real Analysis An axiomatic treatment of the topology of the real line, real analysis, and calculus. Topics include neighborhoods, compactness, limits, continuity, differentiation, Riemann integration, and uniform convergence.

Complex Analysis An introduction to functions of a complex variable. Mappings of the complex plane, analytic functions, Cauchy Integral Theorem and related topics.

Partial Differential Equations An introduction to partial differential equations (PDEs) with an emphasis on first and second-order  linear equations. Using analytical, qualitative, and numerical techniques, we will study the Laplace,  heat, and wave equations, as well as their applications. MATLAB will be used where applicable.  (MATH  0223 and either of MATH 0225 or MATH 0226) 3 hr lect.

Numerical Linear Algebra Numerical Linear Algebra involves the development, analysis, and implementation of computational algorithms for solving linear algebra problems. These problems frequently arise in applications such as physical simulations, signal processing, neural network design, and many more. This course focuses on numerical methods for linear systems and eigenvalue problems. We will study both direct and iterative approaches, including Gaussian Elimination, LU Factorization, Jacobi and Gauss-Seidel Iterations, Steepest Descent, Conjugate Gradient, the Power Method, and more.  Additional key topics include matrix decompositions, matrix/vector norms, computational efficiency, and stability. MATLAB programming skills will be introduced and developed throughout the semester.

Elementary Topology An introduction to the concepts of topology. Theory of sets, general topological spaces, topology of the real line, continuous functions and homomorphisms, compactness, connectedness, metric spaces, selected topics from the topology of Euclidean spaces including the Jordan curve theorem. (MATH 0122 or MATH 0200 or by waiver)

Differential Geometry This course will be an introduction to the concepts of differential geometry.  For curves in space, we will discuss arclength parameterizations, Frenet formulas, curvature, and torsion.  On surfaces, we will explore the Gauss map, the shape operator, and various types of curvature.  We will apply our knowledge to understand geodesics, metrics, and isometries of general geometric spaces.  If time permits, we will consider topics such as minimal surfaces, constant curvature spaces, and the Gauss-Bonnet theorem. (MATH 0200 and MATH 0223) 3 hr. lect./disc.

Combinatorics       Combinatorics is the “art of counting.”  Given a finite set of objects and a set of rules placed upon these objects, we will ask two questions.  Does there exist an arrangement of the objects satisfying the rules?  If so, how many are there?  These are the questions of existence and enumeration.  As such, we will study the following combinatorial objects and counting techniques: permutations, combinations, the generalized pigeonhole principle, binomial coefficients, the principle of inclusion-exclusion, recurrence relations, and some basic combinatorial designs.

Stochastic Processes Stochastic processes are mathematical models for random phenomena evolving in time or space. This course will introduce important examples of such models, including random walk, branching processes, the Poisson process and Brownian motion. The theory of Markov chains in discrete and continuous time will be developed as a unifying theme. Depending on time available and interests of the class, applications will be selected from the following areas: queuing systems, mathematical finance (Black-Scholes options pricing), probabilistic algorithms, and Monte Carlo simulation.

Bayesian Statistics In this course, we will learn about the Bayesian paradigm of statistics, in which one’s inferences about parameters or hypotheses are updated as evidence accumulates. The goals of the course include understanding basic concepts of Bayesian inference; deriving posterior distributions; assessing the adequacy of Bayesian models; and effectively communicating results. Topics covered include one-parameter models, conjugacy, and Gibbs samplers. Real-world data and applications will feature heavily in this course. (MATH 0311) 2.5 hr. lect.

Galois Theory This course is a tutorial in Galois theory for students who have completed Abstract Algebra.  Starting from the concept of a ring, we will develop the theory of polynomial rings over fields, and use this to carry out an in-depth investigation of field extensions.  Our work together will culminate in proving the fundamental theorem of Galois theory.  Working independently and in small groups, students will explore related areas of algebra and communicate their insights in expository writing and oral presentations.  This course fulfills the capstone senior work requirement for the mathematics major.

Finite Fields Seminar This course is a tutorial in the theory and applications of finite fields, which lie in the intersection of algebra and number theory.  Working in small groups, students will study the fundamental structure and properties of finite fields (also known as Galois fields).  They will then work independently, exploring applications in cryptography, coding theory, or other areas. Students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. This course fulfills the capstone senior work requirement for the mathematics major. (MATH 0241 or MATH 0302; Approval required)  3 hrs. Sem

Quadratic Number Fields In this senior seminar we will explore the algebra and arithmetic of quadratic extensions of the rational numbers. We will study the rings of integers in these extensions, the structure of the unit group in these rings and unique factorization of ideals in Dedekind domains. We will investigate fractional ideals, splitting of primes, the class group and the finiteness of the class number. Some of the ideas and topics introduced are methods, p-adic methods, cyclotomic theory, Dirichlet’s Units Theorem, Quadratic and Biquadratic Reciprocity and quadratic forms. Using these ideas as a springboard students will investigate a topic of their choosing and write their thesis.

Advanced Probability Seminar An introduction to the mathematical foundations of Probability for students who have completed work in Probability and Real Analysis.  The central ideas correspond to the Lebesgue theory of measure and integration.  Probability provides additional perspective and motivates intriguing applications of the theory, which students will explore in their final projects.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights through expository writing and oral presentations.  This course fulfills the capstone senior work requirement for the mathematics major.

Statistics Capstone Seminar In this course we will work with community partners to solve real-world problems using modern statistical and data science techniques. Students will work in small groups to translate research questions into actionable analysis and visualizations. Students will select a project of interest from a subset of community partners, maintain contact and collaboration with the community partner, and present their findings in a final symposium.

Advanced Mathematical Modeling Seminar A tutorial on advanced mathematical model building and analysis for students who have completed work in Differential Equations and Probability. We will study deterministic and stochastic models of interacting populations with a focus on mathematical ecology and epidemiology. Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major. (Approval Only) 3 hrs. Sem.

Mathematical Methods in Fluid Dynamics This course is an introduction to the mathematical models and methods used in modern fluid dynamics.  Students will derive and analyze fundamental equations of fluid flow, explore their applications, as well as examine theoretical and practical solution techniques.  Equations of study will include the Poisson, diffusion, and Navier-Stokes equations.  We will also introduce basic methods of computational fluid dynamics.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major. 3 hrs. Lect./Lab

Advanced Number Theory A senior tutorial on some topics in advanced elementary number theory and an introduction to analytic number theory.  In this course we will review key areas of elementary number theory and abstract algebra followed by the study of integer partitions, continued fractions, rational approximations of irrationals, primes and primality testing, the average order of magnitude of several number theoretic functions, the Basel problem, Bernoulli numbers, and the Riemann zeta function.

The Polynomial Method A tutorial in the Polynomial Method for students who have completed work in Abstract Algebra and at least one of Combinatorics, Graph Theory, and Number Theory.  We will study Noga Alon’s Combinatorial Nullstellensatz and related theorems, along with their applications to combinatorics, graph theory, number theory, and incidence geometry.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations.  Fulfills the capstone senior work requirement for the mathematics major.  (Approval required; MATH 0302 and one of the following: MATH 0241, MATH 0247, or MATH 0345).

Linear Algebra Methods Seminar A tutorial in linear algebra methods for students who have completed work in Linear Algebra (and possibly Abstract Algebra) and at least one of Combinatorics, Graph Theory and Number Theory. We will study the linear algebra method through applications to combinatorics, graph theory, number theory, and incidence geometry. Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major.

The Game of Go Go is the most ancient of all East Asian board games and the most challenging in play. It combines the logic of mathematics with the aesthetic appeal of music and art. Despite this, it is a simple game to learn and is enjoyed by approximately 40 million enthusiasts the world over including over 1000 dedicated professionals. The course will involve playing, recording, analyzing, and critiquing our games and learning about its history and the cultures in which it flourishes. We will also read and write about various related Japanese arts and traditions including the latest AI developments.

Introduction to Networks In this course we will explore the ubiquity of networks and the beautiful mathematics that helps us understand them. Together we will cover the basics of graph theory, introduce real world social, informational, and biological networks, explore how information (or a virus) can diffuse or cascade through a network, and learn about popular social and graph phenomena like the six degrees of separation and the friendship paradox. We will utilize jupyter notebooks and python libraries to build a toolset for studying networks and you will have the opportunity to analyze an empirical network using the ideas and tools you develop over the course of this class. No previous coding or mathematical experience is necessary: we will cover all concepts together. Izabel Aguiar is a PhD candidate in Computational and Mathematical Engineering at Stanford University where she is lucky to be advised by Johan Ugander./

Math Circus Each class will entail small group work solving various logic and puzzle problems, mathematical conundrums, and devising winning strategies for various games and challenges.  Students will present their solutions to the class and work on their own puzzles and games pamphlet to be completed by the end of the term.  Elementary mathematical, logical, game theoretic, and probability concepts will be introduced as needed with the goal of increasing everyone's problem solving and analytical thinking skills. The class will meet four days a week (2 hours per class).

Data Science Across Disciplines In this course, we will gain exposure to the entire data science pipeline—obtaining and cleaning large and messy data sets, exploring these data and creating engaging visualizations, and communicating insights from the data in a meaningful manner. During morning sessions, we will learn the tools and techniques required to explore new and exciting data sets. During afternoon sessions, students will work in small groups with one of several faculty members on domain-specific research projects in Geography, Political Science, Restorative Justice, or Healthcare. This course will use the R programming language. No prior experience with R is necessary.  MATH/STAT 1230: Students will explore pediatric healthcare data to better understand the risks correlated with various childhood illnesses through an emphasis on the intuition behind statistical and machine learning techniques. We will practice making informed decisions from noisy data and the steps to go from messy data to a final report. Students will become proficient in R and gain an understanding of various statistical techniques.  GEOG 1230: In this section, students will use data science tools to explore the ways migration systems in the United States changed during the COVID-19 pandemic. We will draw on data collected from mobile phones recording each phone’s monthly place of residence at the census tract level. The dataset includes monthly observations from January 2019 through December 2021 allowing the analysis to compare migration systems pre-pandemic with those during the pandemic.  INTD 1230A: Data is a powerful tool for improving health outcomes by making programmatic choices to support justice. In this afternoon section of Data Across the Disciplines, students will be working with Addison County Restorative Justice (ACRJ) on understanding patterns in the occurrence of driving under the influence. ACRJ has over 1,000 cases and would like to better understand their data and come up with ways to access information. We will explore how identity, geography, and support impact outcomes from DUI cases. Using statistical analysis and data visualizations, along with learning about ethical data practices, we will report our findings.   INTD 1230B: Let’s dive into the minutes and reports of local towns to develop an accessible news and history resource. Could this be a tool for small newspapers to track local news more easily? Can we map this fresh data for a new look across geographies? Do you want to help volunteer town officials make decisions and better wrangle with their town’s history and data?  In this course we will develop a focused database of documents produced by several municipal boards and commissions. We will engage in conversation with local officials, researchers, and journalists. This course aims to introduce students to making data from real world documents and the people that make them to generate useful information that is often open but frequently difficult to sift through.  PSCI 1230: How do candidates for U.S. national office raise money? From whom do they raise it? In this section we will explore these questions using Federal Election Commission data on individual campaign contributions to federal candidates. Our analysis using R will help us identify geographic patterns in the data, as well as variations in funds raised across types of candidates. We will discuss what implications these patterns may have for the health and functioning of democracy in the U.S.

Alex Lyford

Becky Tang

Bill Peterson

Daniel Velleman

David Dorman

Emily Malcolm-White

Emily Proctor

Frank Swenton

Gyula Zsombok

Ivy Zhang

Izabel Aguiar

Jeanne Albert

Jennifer Crodelle

John Schmitt

Joseph Crodelle

Kara Karpman

Katherine O'Brien

Michaela Kubacki

Michael Czekanski

Mike Olinick

Pete Schumer

Priscilla Bremser

Rose Morris-Wright

Siegfried Bleher

Steve Abbott

Read 470 reviews for Mathematics courses at Middlebury College. Find the best Mathematics professors and courses.  Discover your new major today!

I love Steve! He was incredibly enthusiastic and cared a lot about the students' learning. He was clearly nervous about teaching this class for the first time, but I feel like he did well nonetheless and prioritized student engagement in class. The exams were long and hard, but if you studied you'd usually do okay. 

The professor is great and super approachable.  Easy to ask questions and provides lots of examples for each student.  Puts a lot of effort into making sure each student understands the material.  Lots of homework but the exams are very reflective of the material covered.

Stats inference is very useful, but the professor's reverse classroom style of teaching made this kind of impossible. The textbook is good but not the most straightforward, and the content was hard if you don't remember much from MATH 0310