# At Proof School we begin the school year by studying a selection of mathematical topics that are not commonly addressed in the standard secondary school curriculum.

These topics, which are sometimes featured at math circles or summer programs, expose students to the breadth of this majestic subject. In the process, students come to understand that mathematics consists of so much more than the narrow sequence of algebra, geometry, precalculus, and calculus that typically defines the high school math experience.

Courses in block one generally fall into one of two categories. Our problem solving courses present a wide variety of engaging topics, with a focus on developing strategies for tackling difficult problems, learning methods of proof, and making progress in mathematical writing skills. Courses in discrete math address counting techniques, graph theory, and related areas. We are also offering one more utilitarian course this year, an Algebra 1 Lab to provide an on-ramp for students gearing up for Algebra 2 next block. We're pleased that the background of our math faculty, the structure of our days and year, and the enthusiasm of our students allow us to run such rich and thought-provoking math courses.

Algebra 1 Lab with John DeIonno
In this course, we will explore applications that rely only on skills learned in Algebra 1. We will solve geometry problems using equations of lines, fit curves to data, use piecewise-linear functions to compare insurance and cell phone plans, and use linear programming to find optimal production schedules. There will also be purely mathematical investigations, such as using visual means to divide polynomials and finding connections between Pascal’s triangle and n-dimensional cubes and simplexes. In addition we will reinforce algebra skills in areas of quadratic equations, graphing, and simplifying expressions with fractions.

Problem Solving 1 with Sam Vandervelde
This course is designed for students with minimal mathematical problem solving background. Students will learn foundational methods such as formulating a simpler question, finding a unifying pattern, or working backwards. They will develop strategies for making progress on problems when the best approach is not clear. Particular emphasis will be placed on working through a broad selection of carefully chosen problems. Specific topics will include basic probability, Pigeonhole Principle, logic, and tiling problems.

Problem Solving 2 with Susan Durst
This class will be an exploration of problem solving techniques, with an emphasis on proof writing. We will practice working through the complete arc of solving a math problem, articulating our reasons for believing that our solution is correct, and transforming our intuition into a rigorous proof. The particular problems that we work with will come from a variety of mathematical subject areas, including game theory, logic, and topology.

Enumerative Combinatorics with Sachi Hashimoto
This course is a formal introduction to counting techniques, with a twin emphasis on computational problems and the methods of combinatorial proof. Students will explore counting principles, binomial coefficients, permutations and combinations, combinatorial identities, inclusion-exclusion, finite probability, and graph theory. Along the way, students will hone proof techniques such as mathematical induction, recursion, and proof by bijection.

Topics in Algebraic Combinatorics with Austin Shapiro
In this course, students will add powerful tools from algebra to their repertoire of counting techniques. We will explore the many kinds of generating functions, including ordinary power series and exponential generating functions, which we will use to derive closed forms, identities, and asymptotics for a range of classic sequences. Along the way, we will learn about combinatorial objects such as integer partitions, set partitions, and trees. The second half of the course will cover group actions and Pólya theory, a suite of methods for counting in problems with complex symmetries. In addition, each student will complete a research project examining an application of generating functions.