Mathematics Courses

At Milton, we believe that mathematics requires both proficiency with procedures, as well as conceptual understanding. As such, in each Milton mathematics course, students will build confidence and fluency with mechanical skills and will also develop the necessary mathematical skills for deep and thorough comprehension. These skills are grouped into four categories:

• Student Skills: a range of skills related to organization, independence, self-knowledge, initiative, perseverance, and growth mindset
• Communication: constructing and presenting viable arguments about one’s work and the asking and answering of productive questions
• Novelty and Synthesis: finding structure in repeated reasoning, extending existing tools to new situations, and combining seemingly unrelated ideas
• Application: modeling real-world situations and interpreting the numerical results in context to make meaning from the numbers

Each course will involve the use and development of all these skills, though the emphasis may differ from course to course. Courses that come later in the sequence will require a higher level of development and mathematical maturity.

Proof & Problem Solving
Proof and Problem Solving is the course most Class IV students take; it is designed to set students up for success with our Milton math curriculum. This course asks students to identify the tools they possess when answering a question or producing a solution, solve novel problems, and analyze scenarios in context. The course aims to develop the art and skills of proof and mathematical problem solving. The course covers the following topic areas: problem solving, linear functions and angles, Java programming, similarity, formal proof and logic, polygons, right triangles and trigonometry, coordinate geometry, circles, and area and volume. Proof and Problem Solving will introduce students to all five pillars of math: mathematical fluidity, student skills, communication, novelty and synthesis, and application.

Algebraic Concepts (Honors)
Algebraic Concepts (Honors) is a course about recognizing, reinforcing, applying, and connecting algebraic structures for various algebraic families. The course covers the following topic areas: solving equations and inequalities, systems of equations, functions, quadratic equations, exponents and logarithms, non-right triangle trigonometry, and sequences and series. Additional topics may include rational equations, radical equations, and conic sections. Algebraic Concepts (Honors) focuses on the student skills and communication pillars, with an emphasis on the synthesis pillar. Compared to the Algebraic Concepts course, more independent mathematical thinking will be expected. Students will also be expected to find structure and patterns in repeated reasoning to apply familiar algebra procedures to unfamiliar situations. (Prerequisite: Proof & Problem Solving.)

Algebraic Concepts (Regular)
Algebraic Concepts is a course about recognizing, reinforcing, applying, and connecting algebraic structures for various algebraic families. The course covers the following topic areas: solving equations and inequalities, systems of equations, functions, quadratic equations, exponents and logarithms, non-right triangle trigonometry, and sequences and series. Additional topics may include rational equations, radical equations, and conic sections. Algebraic Concepts focuses most heavily on the student skills and communication pillars. Compared to the Honors Algebraic Concepts course, more time is allocated to repetition and practice to allow students to solidify mathematical fluency and build confidence in computation and problem solving to be successful in level 4 math and beyond. (Prerequisite: Proof & Problem Solving.)

Advanced Functions (Honors)
Advanced Functions (Honors) is a course about drawing connections between polynomials, trigonometry, and exponents. The course covers the following topics: circular trigonometry, functions and inverses, polynomials and rational functions, exponential and logarithmic functions, analytical trigonometry, complex numbers, and sequences and series. Advanced Functions (Honors) heavily focuses on the novelty and synthesis pillar. Students will be asked to find deep mathematical connections between topic areas that may not be obviously related and to use familiar tools in new and more sophisticated ways to build deeper understanding. Compared to the Advanced Functions (Regular) course, this course will have a greater emphasis on abstraction and theory than on application. (Prerequisite: Algebraic Concepts. May be taken concurrently with Statistics.)

Advanced Functions (Regular)
Advanced Functions is a course about extending algebraic reasoning into functional thinking, which serves as the basis for future learning in statistics or calculus. The course covers the following topic areas: circular trigonometry, polynomial functions, power functions, exponential and logarithmic functions, rational functions and limits, analytical trigonometry, and statistics. Advanced Functions focuses most heavily on the application pillar. Students will have the opportunity to connect the new mathematical content to real-world situations of their interest and to hone their communication and presentation skills along the way. Compared to the Advanced Functions (Honors) course, this course will have a greater emphasis on applications than on abstraction and theory. (Prerequisite: Algebraic Concepts. May be taken concurrently with Statistics.)

Calculus (Honors)
Calculus (Honors) is designed to cover the content of a typical college-level Calculus I course, extending prior knowledge about limits to study rates of change (derivatives) and accumulation of change (integrals). The course covers the following topic areas: derivative fundamentals, limits and continuity, derivative techniques and applications, integral fundamentals, integral techniques and applications, and differential equations. Calculus (Honors) focuses most heavily on the novelty and synthesis pillar. Students will be asked to discover patterns from concrete examples and repeated reasoning, to combine multiple skills within the context of larger problems, and to extrapolate from previous learning to attempt problems without having been explicitly taught the methods beforehand. (Prerequisite: Advanced Functions. May be taken concurrently with Statistics.)

Calculus (Regular)
Calculus is designed to expose students to concepts and techniques related to rates of change (differential calculus) and accumulation of change (integral calculus). The course covers the following topic areas: limits and continuity, tangent and secant lines, derivatives, optimization, and definite and indefinite integrals. Calculus focuses most heavily on the application pillar. Students will be asked to use prior algebraic, trigonometric, logarithmic, and exponential knowledge in the new contexts of derivatives and integrals. Compared to the Calculus (Honors) course, more time is allocated to repetition and practice to allow students to solidify and build confidence in computation and problem solving. (Prerequisite: Advanced Functions. May be taken concurrently with Statistics.)

Calculus (Accelerated)
Calculus (Accelerated) is a fast-paced course covering three semesters of calculus content within a single year; students in the course will be prepared to take the AP Calculus BC exam in May. This course covers the following topic areas: polar and parametric functions, limits and continuity, differentiation rules and techniques, derivative applications, definite and indefinite integrals, integral applications, and infinite series. Calculus (Accelerated) focuses most heavily on the student skills pillar. The course will move between topics rapidly without as many opportunities to practice, meaning that students will be expected to self-advocate to reach out for help and to solidify material on their own without review time being allocated in class. Compared to the Calculus (Honors) course, topics are introduced at a much faster pace and not as much depth; there are also fewer opportunities during class to discover patterns inductively. (Prerequisite: Advanced Functions (Honors) and permission of the department. May be taken concurrently with Statistics.)

Calculus & Applied Economics (Honors)
Calculus & Applied Economics (Honors) is a course designed to cover the essentials of single-variable calculus through applications of economic principles. This course covers the following topic areas: differentiation and optimization, the production-possibility frontier, trade, elasticity, supply and demand models, and integration. Calculus & Applied Economics focuses most heavily on the application and communication pillars. Microeconomics concepts are taught alongside calculus concepts to motivate the need for these mathematical techniques. Students will have the opportunity to model consumer and producer behavior, optimize situations given constraints and objectives, and justify their analysis within a specific context. This course uses economics as the concrete motivation for the calculus material, whereas in comparison, the Calculus (Honors) course begins with the theoretical foundations and extends that to applications afterward. (Prerequisite: Advanced Functions. Students may not take this course if they have taken or are planning to take Microeconomics. May be taken concurrently with Statistics.)

Statistics (Honors)
Statistics (Honors) is a course designed to build data analysis skills and data literacy, focusing on the principles of statistical inference. This course covers the following topic areas: sampling, experimental design, exploratory data analysis, probability and random variable distributions, binomial and normal distributions, confidence intervals, and hypothesis testing. Statistics (Honors) has a particular emphasis on novelty and synthesis. As the course progresses, the statistical analysis skills above are developed as students are given opportunities to find, display, and analyze data, often utilizing statistical software. The course uses real-world data to connect students to areas of interest and allow them to uncover historical and current trends. Statistics challenges students to engage as critical consumers of data and statistics in all settings, learning to assess any information before them for validity, significance, and connections. Compared to the Statistics course, this course will have a greater emphasis on the mathematical calculations and underpinnings of our analysis. (Prerequisite: Algebraic Concepts. May be taken concurrently with Advanced Functions or as a standalone math class for those who have completed Advanced Functions.)

Statistics
This course serves as an introduction to statistics, focusing on applying the principles and process of data collection and analysis. This course covers the following topic areas: single variable statistics, normal distribution, two-variable statistics, probability, experimental design, sampling distributions, and hypothesis testing. Statistics focuses most heavily on the communication and application pillars. Students will use statistical software to find, display, and analyze real-world situations of interest and communicate their results. Students are challenged to be critical consumers of data and to assess any information for validity, significance, and connections. Compared to the Honors Statistics course, this course will have a greater emphasis on the interpretation of our analysis and less emphasis on the mathematical calculations. (Prerequisite: Algebraic Concepts. May be taken concurrently with Advanced Functions or as a standalone math class for those who have completed Advanced Functions.)

Advanced Statistical Methods (Honors)
This course will begin with a brief review of the four stages of the statistical process that are learned in Statistics: producing data, exploratory data analysis, probability theory, and statistical inference. From there, students will explore more advanced statistical topics, including linear regression, multiple regression (inference and variable selection), logistic regression, one-way and multi-factor ANOVA, nonparametric methods, bootstrapping, and time series analysis. The learning of these concepts will be accompanied by hands-on exploration, including using the free statistical software program R. Throughout the year, students will conduct a variety of research projects and will be encouraged to engage in cross-curricular exploration and utilize real-world data in their analysis. Students should be interested in collaborating with their peers, working on long-term projects, and grappling with serious inquiries about the world around them. (Prerequisite: Statistics and Calculus, which may be taken concurrently.)

Calculus 2
We offer two parallel, second courses in calculus that apply and extend the concepts and techniques of calculus in different contexts. Both will cover topics of Accelerated Calculus that have not been covered in other first-year calculus courses, such as power series. Calculus with Differential Equations (Honors) explores applications and extensions of calculus within deterministic systems, whereas Calculus 2 with Probability (Honors) focuses on applications and extensions in a probabilistic context.

Calculus 2 with Differential Equations (Honors)
This explores applications and extensions of Calculus within a deterministic lens. We focus on some historically significant masterpieces of applications encapsulated in the form of curves, exemplifying profound questions, ingenious ideas, powerful tools, and the recurring theme of discovery. These specific curves include but are not limited to, conic sections, logarithm and exponential (with real and complex exponents), cycloid, catenary, bell curve, Weierstrass curve, Bezier’s curve, space-filling curves, rainbow and caustic curves, the blackbody curve, Koch curve, and Lorenz attractor, each of which bears rich mathematical or physical meanings and is often governed by differential equations. Among them, the ellipse captures a fascinating journey from Ptolemy’s system of epicycles (essentially Fourier Analysis) through Kepler’s three empirical laws of planetary motion to the final triumph of Newton, proving Kepler’s Laws through the formulation of calculus. Techniques of integration, including integration by parts, trig substitution (change of variables), and Feynman’s trick for integration, will be discussed in depth, and more advanced topics, including power series, transforms (Laplace, Fourier, and Legendre), and calculus of variations will be introduced as needed, within the context of formulating and solving differential equations behind these curves. This course promotes all five pillars of math: mathematical fluidity, student skills, communication, novelty and synthesis, and application. (Prerequisite: Calculus. May be taken concurrently with Linear Algebra and/or Statistics.)

Calculus 2 with Probability (Honors)
This course is about the calculus underpinnings of the math of uncertainty. In this way, the realms of Calculus and Statistics are intertwined, using infinite series and integrals to explore distributions of discrete and continuous random variables. This course covers the following topic areas: combinatorics, conditional probability, elementary set theory, discrete random variables and infinite series, and continuous random variables as an application of integral calculus. Students continue their work in calculus and are introduced (or reintroduced) to the idea that mathematics can be playful and is sometimes unintuitive when dealing with risk. There is an emphasis on advanced ideas, and we take the time to think deeply about problems that may take a week to solve. We practice statistical thinking and develop calculus skills simultaneously. Students in this course are expected to work together, play with ideas, pose their own questions, and explore their solutions. (Prerequisite: Calculus. May be taken concurrently or after Statistics.)

Multivariable Calculus
Multivariable Calculus is designed for students to integrate and deepen their previous skills and conceptual understanding of algebra, calculus, and geometry through abstraction and application. In the fall semester, we will introduce vector algebra (addition, scaling, dot, and cross-products), linear transformations, and matrix multiplication and then apply this to generalize differentiation by constructing partial derivatives, directional derivatives, the gradient, linear differentials, and the Jacobian. In the spring semester, we will cover double and triple integrals, change of variables, and generalize the fundamental theorem of calculus to Green’s theorem, Stokes’ theorem, and the divergence theorem. Additional topics may be included at the discretion of the instructor. Multivariable Calculus will emphasize the following two pillars: novelty and synthesis, and application. Through an axiomatic treatment of vector algebra and calculus, in concert with elementary geometry, algebra, trigonometry, and single-variable calculus, we build deep interconnections and fluency of proof and problem-solving, enabling novel treatment of applications arising from, but not limited to, 3D geometry, mechanics, and E&M. (Prerequisite: Accelerated Calculus. Alternatively, Honors Calculus or Calculus with Economics, with departmental permission. May be taken alongside Linear Algebra and/or Statistics.)

Linear & Abstract Algebra
Designed as a first course in abstract mathematics, the goal is to introduce students to the language of abstract mathematics, abstract ways of thinking, helpful habits of mind, and essential procedural techniques that will lay a foundation for future work in any number of upper-level undergraduate courses. A proof-oriented introduction to the study of vector spaces and the field of linear algebra, as well as other concrete categories, such as sets, groups, and abelian groups (depending upon time and interest), the course centers around the linear algebra topics of span, linear independence, coordinate vectors, dimension, matrix representations of linear transformations, the rank-nullity theorem, change of basis, determinants, and eigenvectors. Within abstract algebra, possible topics include Lagrange’s Theorem, Cayley’s theorem, the Isomorphism theorems, and Sylow’s theorems. In the spring semester, we will apply linear algebra to the solving of linear differential equations. Interested students may also study nonlinear differential equations and/or other applications of linear (and/or abstract) algebra. This course will emphasize the following two pillars: mathematical fluidity and novelty and synthesis. Students will be expected to work independently and collaboratively, take intellectual risks, lean into confusion and discomfort, and assimilate abstract concepts and complex procedures through repeated reasoning and reflection. (Enrollment by permission of the department. With departmental permission, this course may be taken concurrently with Calculus.)

Advanced Topics
We offer two advanced topics courses, in alternating years, designed to provide further study in math for students who have exhausted the rest of the curriculum. Students may take one or both of these advanced courses in either order, depending on their individual journey.

Advanced Topics: Probability
(This course is running in 2025-2026 and offered in alternating years)
This is an advanced course, and so it blends pace and depth in a way that is different from other courses. We cover basic ideas, such as discrete random variables, elementary probability and set theory, and spend the majority of our time diving into continuous random variables and their applications, including joint distributions, generating functions, Markov chains, renewal processes, homogeneous and nonhomogeneous Poisson processes, as well as Bayesian updating. Students are not required to have covered any formal statistics prior to taking this course, and there is little overlap between this course and other statistical leaning courses we offer. There is an emphasis on independent project work later in the course. (Prerequisites: Multivariable Calculus and Linear Algebra. May be taken alongside or after Advanced Statistics.)

Advanced Topics: Differential Forms and Calculus on Manifolds
(This course is not running in 2025-2026, offered in alternating years)
Differential Forms and Calculus on Manifolds dovetails nicely with multivariable calculus and linear algebra and helps provide students with a necessary bridge between relatively concrete mathematics topics and more abstract concepts and ways of thinking. The main topic in this class, differential forms, plays an important role throughout modern mathematics and physics. For example, differential forms provide a beautiful way to rewrite Maxwell’s equations. Calculus on manifolds will be discussed, with 2D and 3D Euclidean spaces being the primary manifold examples. Tangent and cotangent spaces and bundles, the wedge product, exterior differentiation, pull-backs and push-forwards, integration of differential forms, and the generalized Stokes’ theorem will be covered. The students will be able to relate these more abstract ideas to concepts they already know from multivariable calculus, such as div, grad, and curl, as well as the fundamental theorem of line integrals, Stokes’ theorem, and the Divergence theorem. This provides students with a firm grounding on calculus on manifolds necessary for a range of more advanced topics, including the proofs of the Theorema Egregrium and Gauss-Bonnet theorem. Differential Forms and Calculus on Manifolds will help students in two pillars in math: mathematical fluidity and novelty and synthesis. (Prerequisites: Multivariable Calculus and Linear Algebra.)

Semester and Half Course Electives

Mathematics of Elections
(Semester 1)
Classes I, II, & III
This course will examine the mathematical basis for how elections can be run and connect voting theory to real-world implementations. Students will rigorously define voting systems and draw conclusions about their properties in order to understand why there is no perfectly fair democratic system for more than two candidates. Concurrently, students will also learn about real-world implementations of voting, selection, and apportionment, such as the U.S. Electoral College, ranked-choice voting, and parliamentary systems. These theoretical and historical perspectives will be used to preview and predict the outcomes on Election Day and then analyze the results afterward. (This course may be taken concurrently with any class beyond Algebraic Concepts.)

Mathematics & Social Justice
(Semester 2)
Classes I, II, & III
This course will encourage students to explore issues of equity and justice through a mathematical lens. We will discuss our intersecting identities, learn to question our assumptions, and think critically about how bias influences the presentation of information. Students will then work with the instructor to design and complete mathematical analyses of social issues that interest them. The specific topics and mathematical tools used by an individual will depend on that student’s interests and knowledge. For example, one student might use geometry to investigate gerrymandering, while another might use calculus to analyze mass incarceration. (This course may be taken concurrently with any class beyond Algebraic Concepts.)

Discrete Mathematics Seminar
(Half Course)
Classes I, II, & III
Students will study introductory graph theory and combinatorics, which are the foundations for understanding a wide range of problems in probability, computer programming, and discrete applied mathematics. Students will use specific motivating questions to direct topic exploration. Motivating questions include: What is the fewest number of colors necessary to color a map of the United States so that any pair of neighboring states are different colors? If a five-card poker hand is chosen randomly, what is the probability of obtaining a flush? Can a knight move around a chessboard, landing on every square exactly once? These questions are limited to the use of discrete number systems (e.g., the counting numbers and the integers). Specific topics may include planar graphs, Euler cycles, Hamilton circuits, coloring theorems, trees, permutations, combinations, and recursion. Classwork will include numerical problems, as well as introductory logical proofs. (This course may be taken concurrently with Algebraic Concepts or Advanced Functions only.)

Mathematics & Art
(Half Course)
Classes I, II, & III
In this course, we will consider some of the myriad connections between art and mathematics, providing students with opportunities to study concepts beyond a traditional high school scope and sequence. Students will work in a range of media to explore mathematical connections to architecture, engineering, the geometry of materials, textile creation, transporting items into space, and other topics. (This course may be taken concurrently with any class beyond Algebraic Concepts.)