Middlebury

 

Courses

 


 

Courses offered in the past four years.
indicates offered in the current term
indicates offered in the upcoming term[s]

MATH 0100 - A World of Mathematics      

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.

DED

Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014

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MATH 0109 - Mathematics for Teachers      

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. (Not open to students who have taken MATH/EDST 1005. Students looking for a course in elementary school teaching methods should consider EDST 0315 instead.)

DED

Spring 2014

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MATH 0116 - Intro to Statistical Science      

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 0210 or PSYC 0201 previously or concurrently.) 3 hrs. lect./1 hr. computer lab.

DED

Spring 2011, Fall 2011, Fall 2012, Spring 2013, Fall 2013, Spring 2015

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MATH 0121 - Calculus I      

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. 4 hrs. lect./disc.

DED

Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

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MATH 0122 - Calculus II      

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. (MATH 0121 or by waiver) 4 hrs. lect./disc.

DED

Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

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MATH 0190 - Math Proof: Art and Argument      

Mathematical Proof: Art and Argument
Mathematical proof is the language of mathematics. As preparation for upper-level coursework, this course will give students an opportunity to build a strong foundation in reading, writing, and analyzing mathematical argument. Course topics will include an introduction to mathematical logic, standard proof structures and methods, set theory, and elementary number theory. Additional topics will preview ideas and methods from more advanced courses. We will also explore important historical examples of proofs, both ancient and modern. The driving force behind this course will be mathematical expression with a primary focus on argumentation and the creative process. (MATH 0122 or MATH 0200) 3 hrs. lect.

CW DED

Spring 2014, Spring 2015

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MATH 0200 - Linear Algebra      

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. (MATH 0121 or by waiver) 3 hrs. lect./disc.

DED

Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

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MATH 0217 - Elements of Math Bio & Ecol      

Elements of Mathematical Biology and Ecology
Mathematical modeling has become an essential tool in biology and ecology. In this course we will investigate several fundamental biological and ecological models. We will learn how to analyze existing models and how to construct new models. We will develop ecological and evolutionary models that describe how biological systems change over time. Models for population growth, predator-prey interactions, competing species, the spread of infectious disease, and molecular evolution will be studied. Students will be introduced to differential and difference equations, multivariable calculus, and linear and non-linear dynamical systems. (MATH 0121 or by waiver)

DED

Fall 2010, Fall 2013

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MATH 0223 - Multivariable Calculus      

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. (MATH 0122 and MATH 0200 or by waiver) 3 hrs. lect./disc.

DED

Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

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MATH 0225 - Topics in Linear Alg & Diff Eq      

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 0200 or by waiver) 3 hrs. lect./disc.

DED

Spring 2011, Spring 2012, Spring 2013, Fall 2013, Fall 2014

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MATH 0228 - Intro to Numerical Analysis      

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. (MATH 0200)

DED

Spring 2015

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MATH 0241 - Elementary Number Theory      

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. (MATH 0122 or by waiver)

DED

Fall 2010, Fall 2011, Fall 2012, Spring 2015

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MATH 0247 - Graph Theory      

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. (MATH 0122 or by waiver) 3 hrs. lect./disc.

DED

Fall 2011, Spring 2014

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MATH 0250 - Ethnomathematics      

Ethnomathematics: A Multicultural View of Mathematical Ideas and Methods*
What are the cultural roots of the mathematics we study and use today? Even though it has been developed by individuals from widely varying cultural contexts, we take the verity, consistency, and universality of mathematics for granted. How does the western tradition stand in comparison to the mathematics developed by indigenous societies, labor communities, religious traditions, and other groups that can be studied ethnographically? By examining the cultural influences on people and the mathematics they practice, we shall deepen our understanding of mathematics and its relationship to society. 3 hrs. lect/disc.

CMP DED

Spring 2011

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MATH 0261 - History of Mathematics      

History of Mathematics
This course studies the history of mathematics chronologically beginning with its ancient origins in Babylonian arithmetic and Egyptian geometry. The works of Euclid, Apollonius, and Archimedes and the development of ancient Greek deductive mathematics is covered. The mathematics from China, India, and the Arab world is analyzed and compared. Special emphasis is given to the role of mathematics in the growth and development of science, especially astronomy. European mathematics from the Renaissance through the 19th Century is studied in detail including the development of analytic geometry, calculus, probability, number theory, and modern algebra and analysis. (MATH 0122 or waiver)

CMP DED

Spring 2014

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MATH 0302 - Abstract Algebra I      

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. (MATH 0200 or by waiver) 3 hrs. lect./disc.

DED

Fall 2010, Spring 2011, Fall 2011, Fall 2012, Spring 2013, Fall 2013, Fall 2014, Spring 2015

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MATH 0308 - Mathematical Logic      

Mathematical Logic
Mathematicians confirm their answers to mathematical questions by writing proofs. But what, exactly, is a proof? This course begins with a precise definition specifying what counts as a mathematical proof. This definition makes it possible to carry out a mathematical study of what can be accomplished by means of deductive reasoning and, perhaps more interestingly, what cannot be accomplished. Topics will include the propositional and predicate calculi, completeness, compactness, and decidability. At the end of the course we will study Gödel’s famous Incompleteness Theorem, which shows that there are statements about the positive integers that are true but impossible to prove. 3 hrs. lect. DED (D. Velleman)

DED

Spring 2014

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MATH 0310 - Probability      

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. (concurrent or prior MATH 0223 or by waiver) 3 hrs. lect./disc.

DED

Fall 2010, Fall 2011, Fall 2012, Spring 2014, Fall 2014

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MATH 0311 - Statistics      

Statistics
An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, 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. (MATH 0310) 3 hrs. lect./disc.

DED

Spring 2012, Fall 2013

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MATH 0315 - Mathematical Models      

Mathematical Models in the Social and Life Sciences
An introduction to the role of mathematics as a modeling tool and an examination of some mathematical models of proven usefulness in problems arising in the social and life sciences. Topics will be selected from the following: axiom systems as used in model building, optimization techniques, linear and integer programming, theory of games, systems of differential equations, computer simulation, stochastic process. Specific models in political science, ecology, sociology, anthropology, psychology, and economics will be explored. (MATH 0200 or waiver) 3 hrs. lect./disc.

DED

Spring 2011, Spring 2013

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MATH 0318 - Operations Research      

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. (MATH 0200 or waiver)

DED

Spring 2012, Fall 2013

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MATH 0323 - Real Analysis      

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. (MATH 0223) 3 hrs. lect./disc.

DED

Fall 2010, Fall 2011, Spring 2012, Fall 2012, Fall 2013, Spring 2014, Fall 2014

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MATH 0325 - Complex Analysis      

Complex Analysis
An introduction to functions of a complex variable. Mappings of the complex plane, analytic functions, Cauchy Integral Theorem and related topics. (MATH 0223 or by waiver) 3 hrs. lect./disc.

DED

Spring 2011, Spring 2013, Spring 2015

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MATH 0335 - Differential Geometry      

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.

DED

Spring 2011, Spring 2013, Fall 2014

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MATH 0345 - Combinatorics      

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. (MATH 0200 or by waiver) 3 hrs. lect./disc.

DED

Fall 2010, Fall 2012, Spring 2015

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MATH 0402 - Topics In Algebra      

Topics in Algebra
A further study of topics from MATH 0302. These may include field theory, algebraic extension fields, Galois theory, solvability of polynomial equations by radicals, finite fields, elementary algebraic number theory, solution of the classic geometric construction problems, or the classical groups. (MATH 0302 or by waiver) 3 hrs. lect./disc.

DED

Spring 2012, Spring 2014

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MATH 0410 - Stochastic Processes      

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. (MATH 0310) 3 hrs. lect./disc.

DED

Spring 2011, Spring 2013, Spring 2015

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MATH 0423 - Topics in Analysis      

Topics in Analysis
In this course we will study advanced topics in real analysis, starting from the fundamentals established in MA401. Topics may include: basic measure theory; Lebesgue measure on Euclidean space; the Lebesgue integral; total variation and absolute continuity; basic functional analysis; fractal measures. (MATH 0323 or by waiver) 3 hrs. lect./disc.

DED

Spring 2012, Spring 2015

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MATH 0432 - Elementary Topology      

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 by waiver) 3 hrs. lect./disc.

DED

Fall 2011, Spring 2014

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MATH 0500 - Advanced Study      

Advanced Study
Individual study for qualified students in more advanced topics in algebra, number theory, real or complex analysis, topology. Particularly suited for those who enter with advanced standing. (Approval required) 3 hrs. lect./disc.

Fall 2010, Winter 2011, Spring 2011, Fall 2011, Winter 2012, Spring 2012, Fall 2012, Winter 2013, Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015

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MATH 0704 - Senior Seminar      

Senior Seminar
Each student will explore in depth a topic in pure or applied mathematics, under one-on-one supervision by a faculty advisor. The course culminates with a major written paper and presentation. This experience emphasizes independent study, library research, expository writing, and oral presentation. The goal is to demonstrate the ability to internalize and organize a substantial piece of mathematics. Class meetings include attendance at a series of lectures designed to introduce and integrate ideas of mathematics not covered in the previous three years. Registration is by permission: Each student must have identified a topic, an advisor, and at least one principal reference source. 3 hrs. lect./disc.

Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015

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MATH 0710 - Advanced Probablility Seminar      

Advanced Probability Seminar
This course is a tutorial in Probability Theory for students who have completed work in Probability and Real Analysis. Starting from elementary results about random walks, we will explore the fundamental mathematical ideas underlying measure theoretic probability, martingales, the Weiner process, and the Itô stochastic calculus. Working independently and in small groups, 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 0310, MATH 0323, and by approval). 3 hrs. sem.

Fall 2014

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MATH 1004 - The Shape of Space      

The Shape of Space
We know that the earth we live on is a sphere, but consider the three-dimensional shape of the universe. Does it go on forever, or could it wrap back on itself in some way? In this course we will consider the shape of space. We will learn how topologists and geometers visualize three-dimensional spaces, with a goal of learning about the eight three-dimensional shapes that form the building blocks of all three-dimensional spaces. In the process, we will learn about the celebrated Poincare Conjecture. The ideas we encounter will be deep, but we will study them in a hands-on way.

DED WTR

Winter 2011

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MATH 1005 - Mathematics for Teachers      

Mathematics for Teachers
What mathematical knowledge should elementary and secondary teachers have? We will investigate recent research that describes specialized mathematical content knowledge for teaching. Participants in this course will also strengthen and deepen their own mathematical understanding in a student-centered workshop setting. Readings include Knowing and Teaching Elementary Mathematics by Liping Ma as well as readings from the Journal for Research in Mathematics Education. Anyone interested in mathematics education at any level is welcome. (Not open to students who have taken MATH 1003)

DED WTR

Winter 2012

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MATH 1006 - Heart of Mathematics      

The Heart of Mathematics
Wrestling with the infinite, tiling a floor, predicting the shape of space, imagining the fourth dimension, untangling knots, making pictures of chaos, conducting an election, cutting a cake fairly; all of these topics are part of the landscape of mathematics, although they are largely excluded from the calculus-centric way that the subject is traditionally presented. Following the acclaimed text, The Heart of Mathematics, by Ed Burger and Michael Starbird, we will dive headfirst into ideas that reveal the beauty and diverse character of pure mathematics, employing effective modes of reasoning that are useful far beyond the boundaries of the discipline.

DED WTR

Winter 2012

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MATH 1007 - Combinatorial Gardner      

The Combinatorial Gardner
It has been said that the Mathematical Games column written by Martin Gardner for Scientific American turned a generation of children into mathematicians and mathematicians into children. In this course we will read selections from three decades of this column, focusing on those that deal with combinatorics, the “science of counting,” and strive to solve the problems and puzzles given. An example problem that illustrates the science of counting is: what is the maximum number of pieces of pancake (or donut or cheesecake) one can obtain via n linear (or planar) cuts? (MATH 0116 or higher; Not open to students who have taken FYSE 1314).

DED WTR

Winter 2013

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MATH 1009 - Discovering Infinity      

Discovering Infinity
"Infinity" has intrigued poets, artists, philosophers, musicians, religious thinkers, physicists, astronomers, and mathematicians throughout the ages. Beginning with puzzles and paradoxes that show the need for careful definition and rigorous thinking, we will examine the idea of infinity within mathematics, discovering and presenting our own theorems and proofs about the infinite. Our central focus will be the evolution of the mathematician’s approach to infinity, for it is here that the concept has its deepest roots and where our greatest understanding lies. In the final portion of the course, we will consider representation of the infinite in literature and the arts. (Not open to students who have taken FYSE 1229). 3 hrs. lect.

DED PHL WTR

Winter 2014

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MATH 1015 - Philosophy of Mathematics      

Philosophy of Mathematics
Mathematics is one of humankind’s greatest cognitive endeavors, yet it raises many puzzling questions. Unlike much of our other knowledge, most mathematical knowledge is not established by gathering empirical evidence. So how is mathematical knowledge possible? Unlike most other things we consider to be real, mathematical objects are not physical objects. So in what sense do mathematical objects, such as numbers, exist? What are the foundations of mathematics? Do some mathematical proofs provide greater understanding than others? No prior knowledge of mathematics or philosophy is required.

DED PHL WTR

Winter 2015

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MATH 1038 - Combinatorial Games & Puzzles      

Combinatorial Games and Puzzles
Games and puzzles with a combinatorial flair (based on counting and arrangement) have entertained and frustrated people for millennia. Mathematicians have developed new areas of research and discovered non-trivial mathematics upon examining these amusements. Students will play games (including nim, hex, dots-and-boxes, clobber, and Mastermind®) and be presented with puzzles (including instant insanity and mazes) in an attempt to develop strategy and mathematics during play. Basic notions in graph theory, design theory, combinatorics, and combinatorial game theory will be introduced. Despite the jargon, this course will be accessible to all regardless of background.

DED WTR

Winter 2011

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MATH 1139 - Statistics with Randomization      

Understanding Uncertainty: Exploring Data Using Randomization
In this course we will use computer-intensive methods to explore the randomness inherent in a data set and to develop the scientific logic of statistical inference. We will introduce randomization methods as a basis for framing fundamental concepts of inference: estimates, confidence intervals, and hypothesis tests. The capabilities of computers to draw thousands of random samples and to simulate experiments will replace theoretical approximations grounded in mathematical statistics, especially the normal theory methods like t-tests and chi-squared analyses. Students will use the R programming language to implement the analyses. Much of the course development will proceed through independent and collaborative computer investigations by students using real data sets. No prior experience with statistics and with computer programming is necessary.

CW DED WTR

Winter 2014

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