Mathematics

Do you need to work more on your algebra? In Intermediate Algebra you will start by reviewing Algebra 1, with a focus on manipulating variables with confidence and modeling word problems with solvable equations. The course then turns to the start of Algebra 2 with a full study of quadratics, rational expressions, and conic sections. You will not just learn how to do algebra; you will also understand why it works as you develop your mathematical intuition.

What a surprise it was—and what fun—when I began to understand that math is about so much more than a bunch of calculations and equations!

More than two thousand years ago, Euclid wrote down the foundations of modern geometry in what is probably the most famous math book ever written, Elements. What did he get right? What did he get wrong? In Geometry you will study a modern form of Euclid’s assumptions and see for yourself what can be proven. This course not only teaches you what is true in geometry but also why it is true. The main course focuses on geometric facts and learning to write clear proofs. If you take Geometry Advanced, you will delve into more philosophical aspects of modern mathematics. What does it even mean in mathematics for something to be “true”? Studying hyperbolic geometry will challenge your preconceptions of how the world works and bring you closer to the beauty and mystery of theoretical mathematics. My freshman year Geometry class was inspiring. Suddenly homework became a ‘choose your own adventure story’—each student doing work in different ways. I learned how to plan an attack on a problem, and also how to restart and try again when I had gone down the wrong path.

Functions do all the heavy lifting in mathematics. In this course, you will learn to manipulate functions and represent them in different ways, from equations and tables of values to graphs. You will also study the basic panoply of common functions, from the steadfast polynomial to the transcendentals: exponential, logarithmic, and trigonometric functions. You will learn about these building blocks and how to transform and combine them into new and wonderful conglomerates. With these analytic tools in hand, you will also study conic sections, systems of equations, and (time permitting) matrices and probability.

Some of the most thorough answers can come out of disagreements leading to a clearer picture of what’s going on. How would a math class ever progress if it were not for the student willing to point out the calculation errors of a classmate—or of the teacher?

AP 

Algebra is the study of functions; Calculus is the study of how functions change. In Calculus you will learn about derivatives, what mathematicians call instantaneous rates of change: how fast an object is falling at any given time or how fast a radioactive substance is decaying (what is a half-life?). You will also study going in the other direction: if you know exactly what velocity you’ve been traveling at any given moment, how do you figure out how far you’ve gone?

Calculus 1 focuses on the theory behind calculus and real-world applications. It will prepare you for the Advanced Placement Calculus AB exam. Calculus 1 Advanced covers more advanced applications and shows proofs with more rigor.

Where the textbook provided convoluted explanations (under chapter names like ‘A Few Simple Remarks’), my math teacher brought organization to chaos. The homework provided a necessary backdrop, but class was where explanations came together and made me want to delve deeper.

Abstract Algebra is the purest of pure math courses. In this challenging, college-level elective, you will study the most fundamental of mathematical objects: groups, rings, and fields. Are there other “number” systems that mimic certain properties of the real numbers? In what ways are they the same, and in what ways do they differ? In this course, you will discover the solutions to some long-time geometric puzzles: Can one construct a cube of volume two or trisect a given angle using a compass and straightedge? You will hone your abstract mathematical skills and your ability to write clear and effective proofs.

By ‘engaging’ in your studies, I mean throwing yourself completely into the process of learning––allowing yourself to become fascinated by a certain subject (in my case, math) and then running away with it.

Axiomatic set theory provides a formal system in which all of mathematics can be constructed from a surprisingly simple set of axiomatic assertions, expressible in a beautifully concise formal language, about a collection of pairwise distinguishable objects known as “sets.” This course is devoted to a study of ZF, the axiomatic framework due to Zermelo and Fraenkel in which this program is most often undertaken. ZF and just one additional assertion, the axiom of choice, allow us to build a hierarchy of set objects rich enough to capture all of the mathematics that anyone but a small group of logicians would ever need to do. Each model of ZFC contains a set of “natural number” objects that behave as expected; surprisingly, the axioms of set theory allow us to extend these to the classes of ordinal and cardinal numbers—both too large to be sets. Transfinite induction and recursion, generalizations of techniques of the same name on the set of natural numbers, illuminate the workings of ordinal and cardinal arithmetic, transporting us quite literally to infinity and beyond. As time permits, we will begin exploring famous assertions such as the continuum hypothesis that mark the boundaries of what can be definitively known in a model of set theory.

AP 

Here’s your chance to take what you learned in Calculus, master all the details, and then extend those results to such applications as calculating arc length and the surface area of a solid of rotation and to calculate for parametric and polar functions. You’ll learn to anti-differentiate more complicated functions and then study infinite sums. The terms get smaller, yet there get to be more and more of them. How can you tell whether an infinite sum will stay bounded or explode? This course will prepare you for the Advanced Placement Calculus BC exam.

For students interested in science and engineering, a solid understanding of differential equations is crucial as they underlie nearly all of our mathematical models for mechanical systems. In this course we will solve differential equations with an eye toward their applications, such as computing the current in a circuit to understand resonance, damping, oscillations, and gain; or examining a system’s response to a sudden impulse input through use of delta functions. Early in the course we will see some exciting applications of complex numbers and Taylor series, which students examined in Calculus 2, before moving on to new techniques, including matrix manipulation.

The goal of this course is to lay a formal foundation for the study of continuous, differentiable, and integrable functions on higher-dimensional Euclidean spaces, allowing a rigorous development of the content of multivariable calculus as is done in Theoretical Calculus for the calculus of functions of a single variable. We begin by studying the topology that arises on n-dimensional Euclidean space from its structure as a normed linear space, then introduce the derivative as a linear transformation from one higher-dimensional space to another. After establishing properties of differentiation and proving the implicit function theorem, we introduce integrability of functions on higher-dimensional spaces and define integration on chains and manifolds with a view toward improving the classical theorems of multivariable calculus: Green’s theorem, the divergence theorem, and Stokes’s theorem. 

Parabolas, circles, ellipses, and hyperbolas are known as conics because they arise as cross sections of planes with so-called extended cones—a fact that is mentioned in modern textbooks, but seldom proved. Another conic property routinely cited without proof is the “optical property” of the parabola that underlies the use of parabolic satellite dishes to collect incoming signals at their focal points. Other conic sections have optical properties of their own, and all can be derived using vector calculus. Surprisingly, there are geometric proofs of these facts, due to ancient Greek geometers, that are elegant, beautiful, and accessible to students with no more background than a first course in geometry. In this course, we will explore the conics using purely geometric methods. Besides establishing their familiar properties, we will encounter a veritable trove of little-known results. Were you aware, for example, that if a triangle is inscribed in a circle, every point on that circle is the focus of a unique parabola that is tangent to each of the three lines determined by the sides of the triangle? That for a given ellipse, the locus of points that are intersections of pairs of lines perpendicular to one another and tangent to the ellipse is a circle containing that ellipse and concentric with it? These and other delightful surprises await!

By the time you are ready to take this course, you will already have learned what vectors and matrices are and how to manipulate them. We follow an axiomatic approach to arrive at a deeper understanding of how and why they work. What is the determinant, and how is it characterized? What does a “change of basis” mean (what is a “basis,” anyway?), and why would you want to change one? This class is for budding mathematicians and those curious to know what the world of theoretical math is all about.

I’m good at math, and it’s always fun, though before Commonwealth I’d always felt that I was essentially learning it on my own, with my textbook and homework. But being in this classroom added more than I could possibly have learned working by myself.

What is truth? How do we formalize the notions of truth and provability as objects within a formal mathematical system, and what conditions on a mathematical assertion guarantee the existence of a proof of that assertion within the system?

These are just some of the questions to be addressed in this course, a rigorous introduction to mathematical logic. We will begin with a discussion of propositional logic, including truth assignments, induction on formulas, and unique readability. We will then move on to first-order languages and predicate logic, introducing models as interpretations of languages and establishing connections between the syntactic (proof-theoretic) and semantic (model-theoretic) approaches to truth. Questions of cardinality, or size, will occupy our attention throughout the course. For instance, which theories are finitely axiomatizable? And under what conditions does a theory have models of all infinite cardinalities?

As time permits, we will examine other areas of pure mathematics, such as group theory, number theory, axiomatic set theory, and analysis, through a model-theoretic lens. Important theorems to be studied include the compactness theorem and the Lowenheim-Skolem theorem.

AP 

After a review of single-variable calculus, you’ll cover infinite sequences and series. As in Calculus 2, you’ll learn to distinguish those that come to a limit from those that “diverge” to infinity. In the second semester, you will encounter multivariable calculus. Instead of derivatives, you’ll learn about partial derivatives. From the single-variable integral, you’ll turn to vector functions, line integrals, and double—and even triple— integrals. If time permits, you’ll learn about the mysteries of Lagrange Multipliers and Green’s and Stokes’ Theorems. This course also prepares you for the Advanced Placement Calculus BC exam.

Commonwealth teaches what I call real math. And knowing what real math is like before I start college is probably going to make me a math major.

How can you tell that a phenomenon is random, and what does that mean? When the newspaper reports a poll with +/- 3% after it, what exactly does that +/-3% signify? When drug companies claim that a drug will help you, how do they know, and why do scientists often change their minds later? Statistics is the study of random phenomena. In this class, you will learn how to design a study that will give you good data, how to describe the data accurately, and how to use inference to derive appropriate conclusions. The half-credit elective course moves quickly, while the full-credit course covers the same material but at a more deliberate pace.

One of the things that surprised me the most during my project month at Harvard Medical School’s Department of Global Health and Social Medicine was the importance of having a solid expertise in statistics. Statistical analysis is a fundamental part of doing research in social medicine.

As members of communities and participants in democracy, we often feel strongly about the equitability (or lack thereof) that we see in systems of decision making. Sometimes, though, our intuition and feeling are not enough, and concrete mathematical metrics are needed to quantify and qualify just how fair or unfair mechanisms may be. This course will explore these ideas in the realms of voting, gerrymandering, apportionment, and allocation:
we'll quantify the power of committee members with probability, measure the fairness of districting maps with ideas from geometry, and seek out fair divisions of shared goods with algebra, all with a focus on hands-on applications to real world systems.

AP 

Knowing how calculus works isn’t enough? What do the real numbers have that the rational numbers don’t? Theoretical Calculus develops all the theorems of calculus from the axioms of the Real Numbers, including the elusive Completeness Axiom. Enter the world of suprema and infima, follow the partition definition of definite integrals, and revisit old friends such as the Fundamental Theorem of Calculus.

When you’re done with the basics, you’ll use these same methods to investigate infinite sequences and series of numbers and infinite polynomials. You’ll end with an introduction to the beautiful theory of complex power series, including a quick proof of the famous Euler equation, e^iπ+1=0. Theoretical Calculus is equivalent to an Introduction to Real Analysis course in college. Some of the easier material will prepare you for the Advanced Placement Calculus BC exam.

I never had to think so hard on an exam in my life! It was awesome.