Our Curriculum

The curriculum developed by Dr. Gleizer and Dr. Radko has a foundation in the Eastern European Math Circles’ tradition. In the books by Dr. Gleizer and Dr. Radko, the curriculum has evolved to include many topics taught in the 21st century to US undergraduate math majors, redesigned for the cognitive level of younger students. We have five years of experience of teaching this curriculum which we believe is currently the best in the world. We have taught it live at various venues and are currently teaching remotely.

Breaking Numbers into Parts
- BNP -

1 and 2 Grade

Counting stories
The first lesson of the course is an ice-breaker centered around a non-trivial and fun problem. 

Parts of a number
They often ask in kindergarten and first grade to break a number into parts. For example, 5 = 2 + 3. In the standard curriculum, they always break numbers into two parts. In this lesson, we teach children to break a (positive integral) number into parts in all the possible ways, from one part on. For example, 5 = 5 is a way to break 5 into one part. It may seem trivial for the first look, but it is very important. One can also break 5 into two, three, four, and five parts. For example, 5 = 1 + 1 + 1 + 1 + 1

Partitions and subtraction
We used the technique of breaking a number into parts introduced in the second lesson to show students that addition and subtraction of whole numbers can be thought of as breaking numbers into parts.

The number line
We introduce the concept of the number line and show how to use the number line for addition and subtraction.

Digits and numbers
We show that numbers are made of digits the same way words are made of letters.

Young diagrams
We start calling the partitions introduced as "houses" in lesson 2 by their grown-up name, Young diagrams.

Harder subtraction problems
We show students how to use Young diagrams to solve harder subtraction problems.

Odd and even numbers
We use Young diagrams to introduce the concept of odd and even numbers.

Properties of add and even numbers
We use Young diagrams to study properties of odd and even numbers.

Commutativity of addition
We use Young diagrams to show students that x + y = y + x for any positive integers x and y. Most grown-ups are familiar with the fact, but very few can explain why it holds. The proof is given by means of playing the "build a house" game.

Adding more than two numbers
We use commutativity of addition to show students more efficient ways to add numbers. For example, 8 + 7 + 6 + 2 + 3 + 4 = 8 + 2 + 7 + 3 + 6 + 4 = 10 + 10 + 10 = 30

Operations
We introduce functions in an age-appropriate fashion.

Compositions of operations and commutativity
We use operations to show that commutativity is a big deal in mathematics and in real life. If you doubt the latter, change the order of operations "put on socks" and "put on shoes" you routinely perform dressing up in the morning and then look at yourself in the mirror.

Inverse operations
We introduce the notion of an inverse function in an age-appropriate way.

Plane reflections and other symmetries
We study reflections of flat solids in straight line and other symmetries.

Conjugation of Young diagrams
We introduce an operation that swaps rows and columns of a Young diagram.

Conjugation as reflection
We show that a conjugation of a Young diagram is a reflection in its main diagonal.

Beginners 1

3-4 Grade

Math Adventures Level 1 From Optical Illusions to Fighting Dragons

Advanced problem solving
Grown-ups usually solve problems we give our students in lessons 1 and 2 using variables and equations. The problems can be solved without employing the machinery of algebra. The more elementary approach requires visualization and deeper thinking, both replaced in the algebraic approach by the power of the machine.

Numeral systems
A large part of this mini-course is focused on giving students a better understanding of numbers. The numeral system currently in use by humanity is decimal place-value. The word decimal means that it has ten digits. The place value part means that the value of a digit depends on its position in a number.
For example, the digit 5 has the value five in a single-digit number 5, but it has the value five hundred in the three digit number 572.
To understand the workings and power of our numeral system, one has to take a look at a system that is decimal, but not place-value and at a system that is place value, but not decimal. In lessons 4 - 7 and 18, students study Roman numerals which are essentially decimal, but not place-value. In lessons 15 and 16, students take a deep look at the working of the decimal place-value system using the abacus as an efficient visualization tool. In lessons 20 - 23 and 28, we introduce students to binary numbers. Later on, we present a professional grade magic trick based on binary numbers that is totally mysterious to anyone not familiar with them.

Topology
Topology is a branch of math that studies properties of shapes preserved under continuous deformations, such as stretching, bending, and twisting, but not gluing, cutting, or making holes.
In lessons 9 and 10, we give students problems related to cutting solid cylinders and bagels. Answers to similar problems are different due the different topology of the objects cut.
In lesson 29, we study a one-sided surface, the famous Möbius strip, by comparing it to its two-sided cousin, a two-dimensional cylinder.

Combinatorics
Combinatorics is a branch of math that studies counting objects satisfying certain criteria. In lesson 12, we show children how to solve a highly non-trivial combinatorial problem of counting squares of all sizes, from 1 × 1 to 8 × 8 on a chessboard. We introduce graphs in various warm-up problems.
In lesson 22, we introduce a choice tree and (implicitly) permutations and combinations.

Game theory
In lesson 13, we introduce basic ideas of game theory and teach students how to always win playing a few cool games.

Coordinates and vectors
In lessons 30 and 31, we introduce coordinates in the plane as a tool to describe dragons and vectors in the plane as a weapon to fight them.

Review problems and questions
Lesson 32 is different from all other lessons of the book in structure and purpose. This lesson is a collection of problems similar to the ones given to students earlier in the course. Solving the problems reinforces understanding of the material covered during the year. 

Beginners 2

5-6 Grade

Math Adventures Level 2 From Ciphers to Nets of Solids

Intro to ciphers
A cipher is a tool for reading and writing secret messages. In this mini-course we study the reverse cipher frequented by Leonardo Da Vinci, the Polybius cipher used by Ancient Greeks,  
Caesar cipher, Pig Pen cipher and Rail fence cipher

Function machine and Mathematical notations for functions
When we introduce students to functions, we typically bring the concept to life through the idea of function machines. But functions will really begin to come to life as our students find uses for functions in the real world. FUNCTION MACHINES Students easily grasp the idea of a function machine: an input goes in; something happens to it inside the machine; an output comes out. Another input goes in; another output comes out. What's going on inside the machine? If we know the machine's function rule (or rules) and the input, we can predict the output. If we know the rule(s) and an output, we can determine the input. We also can imagine the machine asking, "What's my rule?" If we examine the inputs and outputs, we should be able to figure out the mystery function rule or rules.

A first look at the Geometry of Space - Polygons and solids
Students will learn basic facts about polygons. Then they will proceed to build some solids out of cubes and to draw their 2D projections.
A projection may be a 2D picture that shows us the "shape" of a side of a cube. Students will be given projections of the top, front, and left side of a solid that fits in the given dimensions of a cube. Can they identify the correct solid from this information? Or, is there more than one possible solution?
This mini-course will cover more complicated solids. we will be manipulating shapes in our heads and drawing their projections we are looking at 3D structures and their projections. 
Children build 3D solids based on 2D projections. As well as having build a 3D solid and sketch the 2D projections

Intro to Mathematical Logic 1
Knights and liars, The truth function and the and operation, The or and negation operations, Double Negation and Logic gates.

Here students will study the part of math called mathematical logic. Mathematical logic is one of the bedrocks of math and computer science.

Dimensions 
Lineland, Flatland
The space we live in is called Euclidean after an Ancient Greek mathematician known as the father of Geometry, Euclid. Our space is three-dimensional. Another example of a Euclidean space is a 2D plane, modeled by the surface of a flat table or a blackboard. One feature of a Euclidean space is that for any two different points of such a space, there exists a unique shortest path connecting them. We call the shortest paths in Euclidean spaces segments of straight lines. A sphere is an example of a non-Euclidean 2D surface. There are infinitely many shortest paths, called meridians, connecting the North Pole of a sphere to its South Pole.

Backward Reasoning
Backward reasoning problems teach students to trace back the information flow.
  
Traveling on a Cube
We teach students to find the shortest path from one vertex of a cube to it's opposite in three different situations. In the first case, the cube is 3-dimentional (3D) and the shortest path is a segment of a straight line. In the second case, we consider a cube as a collection of its 1D edges. In this case, there're many different shortest paths of equal length. In the third case, we consider a cube as a collection of its 2D faces. To show students how to find the shortest paths in this case, we teach them cubic nets first.

Clock-Face Arithmetic
We teach students the arithmetic of the face of the clock and the more general
mod n arithmetic. 

Introduction to Geometry
We begin the systematic study of geometry with Euclid's axioms. We proceed to compass and ruler constructions and prove the SSS, SAS, and ASA congruences of  triangles. We then study the Pythagorean theorem. 

Powers and Exponents
We teach students some basics about the functions y = x^n and y = b^x.

Arithmetic and geometric sequences and series
We teach this material in more depth than schools usually do, including the famous problem about an arrogant shah, a wise man and rice grains on a chessboard.


Intermediate 1
7-8 Grade

Mini-Course Curriculum

The Hanoi Tower puzzle and the Sierpinski triangle
The course, designed for students in grades 4-7, introduces children to recursive algorithms and related fractal sets. The course was created and tested at LAMC, Los Angeles Math Circle, a free Sunday math school for mathematically inclined children run by UCLA Department of Mathematics. Hidden in the jungle near Hanoi, the capital city of Vietnam, there exists a Buddhist monastery where monks keep constantly moving golden disks from one diamond rod to another. There are 64 disks, all of different sizes, and three rods. Only one disk can be moved at a time and no larger disk can be placed on the top of a smaller one. Originally, all the disks were on the left rod. At the end, they all must be moved to the right rod. When all the disks are moved, the world will come to an end. (No worries here, it will take the monks a few hundred billion years to complete the task.) This tale was created by a French mathematician, Edouard Lucas, to promote the puzzle he had invented. Called the Hanoi Tower, it's a great way to get introduced to recursive algorithms. A fractal is a self-similar geometric figure. Many fractals have fractal dimensions, hence the name. One of the simplest, and most famous, fractals is the Sierpinski triangle, named after a renown Polish mathematician, Waclaw Sierpinski. The dimension of the triangle is 1.584962... Yes, it's less than two, but more than one! The Sierpinski triangle and the Hanoi Tower puzzle are very closely related. To take the course, you will need to purchase the Hanoi Tower puzzle. There are a plenty of them an Amazon.

Place-Value Numerals 2
To fully understand the working of our system of numerals, decimal place-value, one needs to do two things. The first is to take a look at the decimal system that is not place-value. In this course, it will be the system that was in use in ancient Egypt. The second thing is to look at place-value systems with bases different from ten. In this course these will be the binary, trinary, octal, and hexadecimal. The first and the last are very important for anyone interested in understanding of how computers work. On top of that, students will learn that BADDAD and C0FEEBABE are actually (hexadecimal) numbers!

Egyptian multiplication
The numeral system we currently use is decimal place-value. The word "decimal" means that we use ten digits, zero through nine. "Place-value" means that the value of a digit depend on its place in a number. For example, the value of the digit 5 is five hundred in the number 538, but it is fifty in the number 51. To understand our numeral system in depth, a student needs to take a look at a numeral system that is decimal, but not place-value and at a system that is place-value, but not decimal. Egyptian numerals are decimal, but not place-value. Egyptian multiplication is in fact multiplication in the binary system, a system of numerals that is place-value, but not decimal.

Venn Diagrams
A quick and visual introduction to set theory and probability.

Transformations and permutations
A hands-on introduction to group theory.

Introduction to Basic Game Theory Part I and Part II
Exploration of game theory by starting with the example of subtraction games of varying subtraction sets. In Part II we will look at games, this time with more piles and complications. 

Taxicab Geometry I and II
In Part I will be looking at distances with a twist. Normally we only consider the straight line distance (also known as Euclidean distance) or "as the crow flies" distance, but in this case we are looking at the Manhattan distance, where we are restricted. In Part II we will take a second look at non-Euclidean geometry and how shapes change under our new definition of distance.

Intermediate 2
9-10 Grade

The Number Line
In this mini-course, students will get some basic understanding of numbers, real, rational, and irrational, including geometric construction of rational numbers. To do the latter, students will need a compass (geometric, not magnetic), straightedge, a few pencils, pencil sharpener, and eraser. Students will also learn how to compute square roots with precision exceeding that of a standard calculator. For that, student will need a laptop with the programming language Python installed. Python is a free download. https://www.python.org/ The course was created and tested at Geffen Academy, a middle and high school on UCLA campus, by the Geffen Academy Mathematics Department Chair, Dr. Oleg Gleizer.