Challenging All Campers

Epsilon Camp is for kids who are captivated by math. We provide a rich environment, exposing them to some of the most substantial mathematical ideas of the last two thousand years. We have developed a curriculum that is both accessible and challenging to the brightest of young minds. We hire professional mathematicians as faculty because we want our campers to experience the true beauty of higher math.

The camp activities are intended to provide engagement in many different ways—both through serious study and serious play. Most days, campers attend three classes. Time is set aside for self-directed learning and fun, including puzzles, math games, and hands-on activities. Our curriculum emphasizes three fundamental mathematical skills (ACE):

Abstraction — Modern mathematics is developed abstractly from axioms, with logical deduction of theorems. This is the common thread in such disparate subjects as Euclidean geometry, set theory, abstract algebra, and topology. Campers will learn to trace mathematical ideas back to axioms, distinguishing steps which were logically precise from axioms which need to be assumed.
Communication — Mathematical communication requires good writing skills, sound logic, and knowledge of proof techniques. Campers learn to communicate verbally in class and gain experience reading and writing mathematical proofs.
Exploration — Progress in mathematics comes from exploration and experimentation, recognizing patterns, making conjectures, and finally proving theorems. Campers will gain valuable experience in this exploration process.

Campers are organized into small classes in five levels. The number of groups per level varies from year to year, depending upon enrollment. New campers enter at Pythagoras, Pythagoras Transitional, or Euclid level (ages 7-10) and take core curriculum classes. Returning campers may reach Gauss and Conway levels (ages 10-12), where they take classes that vary each year.

Note: All curriculum changes slightly from year to year.

Pythagoras Level, Ages 7–8

We design the academic program for ages 7–8 to introduce campers to a wide variety of deep ideas that they will continue to see throughout their mathematical careers. Topics include:

Number theory: prime numbers, divisibility tests, arithmetic in different bases, and more.
Computational mathematics: recurrences for figurate numbers, generalizations, and binary information encoding.
Geometry: polygon angles, plane tilings, polyhedra, and more.
Counting: permutations and combinations, Pascal's triangle, inclusion-exclusion, and other counting methods.

The material is chosen so that various themes of the camp weave together, and expose campers to both what’s to come in the later years’ curricula, and the history of our subject. The Pythagoras Level curriculum incorporates hands-on learning, play-based activities, structured lessons, and more, for a well-rounded, yet age appropriate, introduction to what deeper mathematics has to offer.

Pythagoras Transitional Level, Ages 8–9

We design the academic program for ages 8-9 to extend the ideas of the Pythagoras level, while also working on skills to prepare campers for the coming year’s Euclid level. At the Transitional level, campers explore a wide variety of mathematical areas (graph theory, game theory, combinatorics, and more) via interesting problems, often working together on hands-on and embodied activities. Campers work to creatively solve problems, in teams, and begin to see the connectedness of the mathematical ideas and topics that they learn in the Pythagoras level, as well as in their various other contexts for learning. Transitional level campers also practice asking mathematical questions, making and exploring conjectures, formalizing mathematical ideas, and forming generalizable arguments and strategies. Introductions to proof, mathematical induction, precise mathematical communication, algebraic reasoning, and logic further prepare them for the Euclid level of Epsilon Camp.

Euclid Level, Ages 9-10

Euclid Level is open to new and returning campers age 9-10. This is the first of three levels of structured courses covering advanced topics. Euclid curriculum is standardized so that campers will acquire the skills necessary to excel in Gauss and Conway Levels. All Euclid campers take foundational courses Axioms in Geometry, Communication and Proof, and Explorations in Number Theory. Each course encapsulates one of the core ACE skills taught at Epsilon Camp: Abstraction, Communication, and Exploration. While the courses and included content vary slightly depending on the instructor, the overall outlines given below touch on the key aspects of each of these three courses.

Axioms in Geometry
In this course, campers will focus on rebuilding Euclidean geometry from the ground up, with an emphasis on rigorous axiomatic thinking and the importance of Euclid’s fifth postulate (the parallel postulate). Campers will study straightedge and compass constructions and will focus on rediscovering statements and proofs of many basic theorems. They will learn the importance of working from the axioms only, without relying on prior knowledge or intuition, and discover which familiar results do or do not depend on the parallel postulate. One of the main goals of the course is to prepare campers for future courses investigating non-Euclidean geometries, such as hyperbolic, spherical, or projective geometry. Campers will also be prepared to learn about axiomatic mathematics outside of geometry, such as abstract algebra and set theory.
Communication and Proof
In this course, campers will learn to write clear and rigorous mathematical proofs. Campers will investigate propositional logic and first-order logic, including predicates and quantifiers. They will gain experience reading, interpreting, and writing mathematical statements and they will learn to construct valid arguments using techniques such as direct and indirect proof, proof by contradiction, proof by cases, proof by mathematical induction, and the Pigeonhole Principle.
Explorations in Number Theory
In this course, we use the division algorithm in order to introduce many of the basic ideas in number theory. The emphasis is on using exploration to rediscover and provide context for important concepts through one central story. Therefore, this course is suitable for those who are new to number theory as well as those who have had some coursework before. Along the way, campers investigate different positional number systems, the divides relation, greatest common divisors, solutions to linear Diophantine equations using the extended Euclidean algorithm, modular arithmetic, and divisibility rules. Afterwards, campers investigate several concepts in modular arithmetic such as Fermat's little theorem, Wilson's theorem, and Euler's theorem as well as finding roots modulo n. These concepts lay the groundwork to investigate the RSA public-key cryptosystem, and we finish by discussing the Chinese remainder theorem. Along the way we will introduce various related puzzles and tricks.

Gauss Level, Ages 10-11

After their Euclid year, campers who are invited to return are placed in the Gauss Level. In this year, our standardized course is Sets and Functions.

Sets and Functions
Campers investigate set theory, the foundation of modern mathematics, which brings together two skills they learned at Euclid Level the previous year — proof techniques and the axiomatic method. They learn about fundamental ideas in set theory including set connectives, the subset relation, the power set, and relations. They learn how functions themselves can be represented as sets and investigate interesting properties functions can have. These ideas lead to a discussion of equinumerosity and different sizes of infinities. Along the way, we discuss some of the major set theory axioms and discover some important theorems and paradoxes.

Campers attend additional courses based on faculty interest and expertise. Sample courses include:

Algebraic Curves
We will study the geometry of zeroes of the polynomials in two variables over the reals. After studying linear polynomials we will learn how the quadratic polynomials can be classified. Useful methods like factorization or substitution of variables will be discussed and used to study higher degree curves. We will prove the Nullstellensatz for some simple cases. As an application we decide if Watt's curve includes a straight line segment or not, and how the situation can be improved by using 6-bar or 8-bar planar linkages.
Enumerative Combinatorics
Enumerative has the same Latin root as the word number, and combinatorics is derived from the German and English words meaning combinations. This class will focus on the counting of combinations. Enumerative combinatorics questions are often in the form "How many different ways can you select a subset of objects from a larger group?" Characteristics of the desired subset and properties of the group of objects can vastly change the difficulty and complexity of the questions. We will explore counting in the contexts of probability and graph theory. We will use the context of the board game Pandemic to motivate counting in a discrete mathematical modeling context.
Visual Group Theory
We will focus on symmetry groups from the view of group actions by studying the symmetries of objects such as games (for instance Spinpossible and the Rubik's cube), rearrangements of cards, regular polygons, and light switches. After playing with Cayley diagrams and letting our intuition guide us, we will see the need for a more rigorous algebraic definition of symmetry groups, which will lead us to the study of groups more formally. We will then focus on properties of groups generalizing familiar arithmetic/algebraic laws and investigate Cayley tables. We will study cyclic groups, dihedral groups, direct products of Z_2, and symmetric groups (as well as the hyperoctahedral groups), and finish with Cayley's theorem. Along the way, we will discuss one-to-one and onto functions, groups created by generators and relations, directed graphs, and the notion of subgroups and group isomorphism (these last two informally).
Projective Geometry
We introduce homogeneous coordinates of points and lines of the Euclidean plane; then we add the ideal line to construct the real projective plane. We study a little linear algebra to use it as a tool. We prove the theorem of Desargues, and if time permits, the theorem of Pappus in the real projective plane. Then we change gears: we start the axiomatic study of projective planes. We discover the existence of finite geometries, and we prove many properties, including relations to Graeco-Latin squares.

Conway Level, Ages 11-12

After their Gauss year, campers who are invited to return are placed in the Conway Level. In this year, our standardized course is the Immersive Research Experience.

Immersive Research Experience
Conway campers will participate in a "research deep dive," where they will be provided an open-ended problem to explore with a team. This course, which aims to simulate an “REU-style” experience, will serve as a capstone for Conway campers at the end of their time at Epsilon Camp. The key elements of this course include:
- Deep mathematical exploration
  Campers are provided with a single focal problem with supporting documents and ramp up questions. They ask questions, make conjectures, and learn to utilize resources including research articles, textbooks, and more.
- Reflection
  Campers learn how research differs from their usual coursework. Class discussions center around readings, videos, and campers’ experiences of being stuck and overcoming difficulties in their own research projects. Themes of productive failure and perseverance are emphasized.
- Communication
  Campers actively participate in sharing the results of their research in verbal and written formats. The second week of class culminates in a poster session, where campers are able to discuss their ideas and proofs with visitors in real time, around a poster that they created to showcase the solutions to their team's hard work.