Our Academic Goal
To foster creativity
We want to get kids excited about learning;
to foster their creativity and capacity for solving problems.
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 freedom of thought: "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."
We approach learning as a team effort
We approach math with problem solving, teamwork, and near-peer mentoring.
Our teams are structured with the aim of making the whole greater than the sum of its parts.
We start by giving students problems to solve. While each student must ultimately see for themselves how to approach a given topic, we support each other in the process of solving problems. 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. While we do have brief lectures based on Dr. D's textbook, the emphasis is placed on students engaging with the material and learning how to think logically to solve the problems they face.
We try to inspire our students
We agree with Albert Einstein that "The most valuable thing a teacher can impart to children is not knowledge and understanding per se but a longing for knowledge and understanding, and an appreciation for intellectual values, whether they be artistic, scientific, or moral.
It is the supreme art of the teacher to awaken joy in creative expression and knowledge."
We try to awaken joy in our students by sharing mathematical beauty with them.
This means that we don't just teach our middle schoolers a course concerned with adding two fractions, but we also teach them a course on adding an infinite number of fractions, using these infinite series to explore the beauty of ideas like partial sums and limits.
We do not just give our kids bread, but roses too.
Ordering and Operations
Intro to Calculus
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?
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.
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.
Integral and Differential Calculus
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.
In our introductory proof class we try to avoid asking our kids to follow directions. We ask our kids open-ended questions and give them full autonomy to find new directions for solving these problems. We aim to provide our students experience with the intellectual struggle and the ultimate joy of figuring something out. Emphasis is again placed on students making sound arguments. This course will vary from teacher to teacher, but will typically be based on the first half of Paul Lockhart's Measurement.
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.
Our proof class continues, focusing on the remaining material in Paul Lockhart's Measurement.
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.