Our Curriculum
Our curriculum is split into Bread and Roses. We want to give our students the basics that they need to survive. But we also want them to do more than just survive.
Bread
The goal of the Bread curriculum is to ensure that our students:
• are numerate and proficient in algebra by the end of middle school, and;
• are exposed to the intuition, formal theory, and applications of calculus while in high school.
These courses follow the textbook Math As Flexibility of Mind, written by The Math Movement's Director Dionissi Aliprantis.
Roses
The goal of the Roses curriculum is to expose our students to beautiful math.
Course offerings tend to be idiosyncratic and driven by the passions of a given summer camp's instructors.
Flexibility of Mind
As described by the great physicist Richard Feynman, we see our subject less specifically in terms of math and more generally in terms of developing habits of mind: "What we have been doing in the past is teaching just one fixed way to do arithmetic problems, instead of teaching flexibility of mind - the various possible ways of writing down a problem, the possible ways of thinking about it, and the possible ways of getting at the problem." Our goal is therefore to teach "a new subject in a sense - an attitude of mind toward numbers and toward mathematical questions which is precisely that attitude of mind which is so successful later in technical applications of mathematics."
Teams
We approach math with problem solving, teamwork, and near-peer mentoring. Our Bread courses give students many problems to solve. While each student must ultimately see for themselves how to approach a given topic, we support each other in the process. Teams of middle school students work together, with high school students serving as paid mentors who work closely with their students. College students lead teams of students, tracking students' progress and working with mentors to create an individualized curriculum for each student. Our Roses courses are more open-ended, but always approach material with a team attitude.
Courses
The Bread Curriculum
Example Roses Courses
7th Grade
Ordering and Operations
Infinite Series
We begin with the idea of ordering objects. We discuss this idea with arbitrary sets, and then lean on the number line to look at the ordering of the natural numbers, the integers, and the rationals. Because ordering some fractions is not clear using the number line, we introduce the operation of multiplication. We study how to multiply fractions and how this allows us to get a common denominator that helps when ordering fractions.
We reinforce concepts and provide inspiration for students to learn about fractions by also teaching students a course on infinite series that introduces students to concepts in integral calculus. We explore ideas like: What is an infinite sum? What is a limit and what does it mean for a sequence to converge? Can an infinite sum have a finite limit? What idea(s) might help us find the limit of an infinite sum? Can we find common patterns in different infinite sums?
8th Grade
Operations and Equations
Statistical Decision Theory
We begin by introducing the operation of addition. Armed with both operations, we study how to add and simplify fractions. Then we turn toward solving equations. We begin by studying the order of operations. We then use the additive and multiplicative inverses, along with the order of operations, to solve equations. We conclude with a look at how to translate words into equations and how to solve inequalities.
We discuss how to make predictions about uncertain outcomes, starting with flipping a coin. This leads us to discussions about statistical statements about samples we have observed and experiments yet to be realized. We conclude with a look at how statistics can play a role in our decision making, and how to think about model-based predictions about uncertain outcomes.
9th Grade
Functions
Geometry
We introduce students to functions as a subset of the Cartesian plane and define specific functions using their domain, range, and rule. We examine the graphs of functions and, focusing on linear and polynomial functions, we study how to find roots and unknown values of functions. We study conic sections, separating curves, and define a continuous function. We conclude with a look at asymptotes and fixed points.
We take students on a guided tour of Euclid's Elements. The emphasis is placed on beauty and demonstration. We want our kids to learn how to provide logical arguments for why something is true, and we want them to see the beauty of geometry. Two textbooks used in this course are Lang and Murrow's Geometry and Posamentier and Lehmann's The Secrets of Triangles.
10th Grade
Introduction to Calculus
Graph Theory
We begin our study of Integral Calculus by asking our students the question Archimedes and others studied thousands of years ago: How do we find the area of a circle? We generalize this answer to think about finding the area under a function. We discuss relevant notation for the study of sequences, series, and limits before introducing students to the Riemann Integral. We then look at a few examples of the Riemann Integral. We introduce the derivative in terms of describing motion, and use notation from the integral for our formal definition. We discuss sketching and characterizing functions, and conclude with a look at optimization. This course is focused on building intuition.
We define what a graph is and then let the students do their own research on a pursuit-evasion game with a chaser and a runner. We also discuss other topics including coloring planar graphs.
11th Grade
Calculus
Game Theory
We begin with an overview of the topics covered in the last class by refreshing students' understanding of The Fundamental Theorem of Calculus. We derive rules for computing derivatives and integrals and practice applying those rules. We study how to differentiate a linear function and derive the sum, product, power, and chain rules. We give students plenty of practice in computing various derivatives before then showing them how to compute anti-derivatives in the process of computing integrals. We conclude with a look at convex sets and how first and second derivatives can be used to understand convex functions and optimization.
We define choice sets and strategies and then start searching for optimal strategies in several games. We consider how to formulate a winning strategy via planning ahead in a "Stones" game and consider isomorphisms between a "Sum to 15" card game and Tic-Tac-Toe. We use a "Greedy Pig" game to illustrate risky and risk-averse strategies. We introduce students to the idea of equilibrium strategies and use multiple-player games to study notions of Nash equilibrium.
12th Grade
Applications of Calculus
Set Theory
We give students a taste of how calculus is often used in applications. We start with a look at some introductory topics in statistics. We build up from frequency distributions to Probability Mass Functions, and introduce the Cumulative Distribution Function as an integral. We then examine the link between discrete and continuous distributions. We use these ideas to look at data, beginning with a look at income inequality. We conclude with a look at numerical optimization.
The crown jewel of the our curriculum is a course on transfinite set theory. In this course students explore ideas related to the cardinality of infinite sets, with some goals being for students to struggle a bit with the Continuum Hypothesis and to write a version of Cantor's diagonalization proof for themselves. A strong emphasis is placed on students learning to formulate their own arguments.