Course subject: Mathematics (MATH)

This course will explore the basic properties of probability focusing on both discrete and continuous random variables. Topics include: Laws of probability, discrete and continuous random variables, probability distributions, mean, variance, generating functions, Markov chains, problem solving, history of probability.

This course explores the foundations of linear algebra and some of its applications. An emphasis will be placed on the development of mathematical thinking and the importance of proof in mathematical teaching. Topics include: matrices, linear mappings, vector spaces, determinants, diagonalization, inner products, the Fundamental Theorem of Linear Algebra, and the method of least squares.

This course explores the intersection between mathematics and computer science by examining general methods for solving problems efficiently. We will study a variety of algorithm design techniques for problems in diverse application areas and use mathematical models to compare algorithms. Students will be introduced to "big ideas" from computer science, such as the generalization and reuse of previous solutions, and the notion of limits to the power of computation.

This course will use mathematics to study the foundations of problem solving and computation. A decision problem (answering "yes" or "no") can be characterized as a set of strings of characters encoding inputs that yield "yes" answers, and a computer can be characterized as a simple mathematical machine model that processes strings of characters. By studying properties of classes of problems of increasing complexity, we will be able to establish relationships among classes, membership in classes, and hard problems for classes. The course will culminate in the examination of classes and models that correspond to the power of computers, and the demonstration that there exist problems that cannot be solved.

This course explores the foundations of integral calculus and the use of series in approximating the basic functions of mathematics. Topics include: Understanding the Riemann Integral and its flaws, the idea of Lebesque, the geometric meaning of the Riemann-Stieltjes integral, the Fundamental Theorem of Calculus, numerical integration, numerical series, uniform convergence of functions and the extraordinary nature of power series, Fourier Series.

This course explores the Euclidean geometry of triangles and circles from elementary to advanced settings. The course also briefly discusses three-dimensional geometry. An emphasis is placed on proof, on problem solving, on problem creation in a geometric context, and on aspects of teaching geometry. Where possible, problems that combine multiple areas of mathematics are used.

We explore the who, where, when and why of some of the most important ideas in mathematics. Topics include: William T. Tutte and Decryption, Euclid and the Delian Problem, Archimedes and his estimate of Pi, Al Khwarizmi and Islamic mathematics, Durer and the Renaissance, Descartes and Analytic Geometry, and Kepler and Planetary Motion.

This course examines the history of mathematics through the prism of a particular person working on a particular problem in a particular geographical and temporal place. Topics may include: Egyptian and Babylonian mathematics, Chinese mathematics, mesoamerican mathematics, Euclid and the Delian Problem; Archimedes and an estimate of Pi; Apollonius and conic sections; Al Khwarizmi and Islamic mathematics; and the transmission of works of antiquity to Europe.

This course provides the opportunity to connect and collaborate with other students, to synthesize key takeaways from other courses, and to bridge the gap between educational theory and practice. Course activities may include problem-solving, discussion, analysis of mathematics education research, presentations, and peer review. The course includes a limited number of synchronous meetings, and may include optional in-person participation in a CEMC conference for mathematics teachers.

Students will undertake a reading, research and writing project on a mathematical topic of interest to teachers.