Curriculum

The Jason Horowitz Academy of Mathematics opens its program with kindergarten and continues through the end of secondary school. Each grade is proof centered and assumes rapid mastery. Weekly seminars, student presentations, and original problem writing begin in kindergarten and grow in sophistication each year.

Kindergarten introduces algebraic thinking. Children learn to treat the unknown as a genuine object, use colored blocks to represent variables, and balance simple linear equations such as red plus two equals five. They also practice place value up to one thousand and explore counting in different bases through games.

First grade formalizes whole number operations and examines prime factors, greatest common divisors, and least common multiples. Students encounter negative numbers on a tactile number line and use them to solve two step linear equations with integer solutions. Reading sessions introduce classic counting proofs such as proof by pairing.

Second grade extends arithmetic to rational numbers, builds early set theory language, and proves properties of fractions with manipulatives. Geometry enters through congruent triangles, angle chasing, and symmetry arguments, always accompanied by short written proofs.

Third grade develops combinatorial reasoning. Pupils enumerate paths on grids, prove the handshake lemma, and begin generating function techniques for small counting problems. Algebra expands to include quadratic equations with integer roots, introduced through visual area models.

Fourth grade moves to real analysis foundations. Students construct the rational numbers formally, study decimal expansions, and give epsilon style proofs of limits for simple sequences. Concurrently, they meet introductory abstract algebra through permutation puzzles, using them to prove group laws.

Fifth grade treats Euclidean geometry axiomatically and introduces projective geometry through homothety and cross ratios described in words rather than symbols. Complex numbers appear as ordered pairs, and pupils prove algebraic identities using geometric diagrams.

Sixth grade unifies earlier work with a semester of elementary number theory that ends in proofs of quadratic reciprocity using Gauss sums written out in line by line logic. The spring focuses on basic graph theory, including Euler circuits, planar graphs, and the proof of Kuratowski theorem for small cases.

Seventh grade covers real analysis rigorously: construction of the real numbers by Dedekind cuts, convergence tests for series, and a first approach to the Riemann integral. Students write full proofs of the intermediate value theorem and mean value theorem. A parallel laboratory introduces algorithmic number theory such as fast modular exponentiation.

Eighth grade offers abstract algebra with rings and fields, culminating in a proof of the fundamental theorem of algebra via topological argument phrased without special notation. An elective in combinatorial game theory lets students analyze impartial games and prove the Sprague Grundy theorem.

Ninth grade turns to complex analysis and special functions. Pupils prove Cauchy theorem through homotopy ideas expressed in plain language, study the Gamma and Zeta functions, and present an exposition of the prime number theorem based on elementary methods. Geometry continues with differential geometry of curves and surfaces using curvature computed from limits.

Tenth grade includes algebraic geometry without schemes. Students learn about projective varieties, Bezout theorem, and work through classic Diophantine problems using these tools. A concurrent course in spectral graph theory treats eigenvalues of graphs, expander constructions, and Cheeger style inequalities.

Eleventh grade introduces topology and manifolds. Learners define the fundamental group, explore covering spaces, and use differential forms to prove Stokes theorem. They also complete an expository paper aimed at publication in a high school mathematics journal.

Twelfth grade is entirely capstone. In the fall each senior enrolls in two graduate level university courses of their choice such as Lie algebras, Riemannian geometry, or additive combinatorics while meeting weekly with academy mentors. The spring is given to an original research thesis that requires a literature review, new results or computational experiments, and a formal defense before a faculty committee. Many theses evolve into peer reviewed publications.

Student giving a presentation