Elementary Mathematics


Students who study math gain the ability to better form and deploy complex knowledge — in general. Whether its solving a difficult engineering problem or solving a thorny personal problem, one needs to be able to: identify and separate different operative factors, organize these factors, bring different elements of knowledge to bear on the problem in a systematic way, and pressure-test one’s thinking for mistakes, problems, and weaknesses. These are the skills that our elementary programs impart when teaching mathematics.

And so the key outcomes of our math education are two-fold:

  • A deep understanding of the way numbers work and relate to each other, as well as the ability to manipulate them quickly and easily
  • The acquisition of cognitive powers and habits that can apply to all other areas of one’s life.

Our students, in mathematics, gain:

  • Explicit competence in foundational domains of mathematics (see below for a content list), along with an intuitive number sense.
  • The ability to build new knowledge from previous knowledge (by rigorous mathematical inference).
  • The ability to organize knowledge into an integrated, hierarchical structure (e.g. logical chains).
  • The ability to deploy highly abstract knowledge in solving particular problems (e.g. word problems, heuristics, models).
  • The ability to participate in discussion and debate with others using clear, step-wise reasoning and truths (such as measurements and prior axioms).

Program Elements

  • Hands-On

    Our math curriculum begins with hands-on materials that foster an intuition of quantity, place value, and geometry. By manipulating these scientifically designed examples of abstract ideas, students build a library of experiences that develop into mental models of mathematical principles. A set of developmentally and mathematically refined hands-on Montessori learning materials, along with unique learning materials developed by our pedagogy team, forms the backbone of the math curriculum.

    Each learning material not only concretizes but isolates an aspect of mathematics. The cleanliness and objectivity of math is manifest in the learning materials, and reinforced in the more abstract presentations. Definitions are clear and concise, statements are correct and precise, and problems are solved by laying bare every step. The precision and isolation in the materials allows students to practice one skill or concept at a time, repeated as often as needed for mastery.

  • Concrete to Abstract
  • Intentionally Sequenced
  • Mastery Based

Lower Elementary Curriculum

Numbers and Operations

Great Stories
The Origin Story of Our Number System

The Decimal System
Composing Numbers
Place Value to the Millions

Intro to Math Operations

Memorization of Math Facts
Strategies for Mental Math
Fact Fluency

Intro to Multi-Digit Multiplication
Concept of Pace Value in Long Multiplication
Geometric and Cross Multiplication

Laws of Multiplication
Commutative Law
Distributive Law

Long Division
Concept of Place Value in Long Division
Distributive Division

Divisibility by 2
Divisibility by 5
Divisibility by 25

Multiples and Factors
Multiples through 100
Least Common Multiple

Squares and Cubes of Numbers
Concept of Squares and Cubes
Notation of Squares and Cubes
Operations using Bead Material
Building the Decanomial

Intro to Fractions
Concept and Notation of Fractions
Connection Between Fractions and Division
Fraction Equivalence and Comparison
Operations with Like Denominator

Intro to Decimal Fractions
Concept and Notation of Decimal Fractions
Decimal Hierarchies
Decimal Fraction Comparison

Word Problems
Strategies for Understanding
Applied to All Math Topics

Counting Currency
Connection to Decimals
Adding and Subtracting Money

Concept of Measuring Temperature
Reading Weather Reports, Tracking

Data and Graphing
Concept of a Graph
Collecting Data
Bar Graphs


Great Stories
The Harpedonaptae and Measuring the Land of Egypt

Geometric Concepts
Plane Figures
Geometric Constructions
Designing with Insets
Point to Solid Symmetry

Parts of a Line
Types of Lines
Relationships Between Two Lines

Parts of an Angle
Types of Angles
Pairs of Angles

Parts of a Polygon
Polygon Classification
Regular and Irregular Properties of a Triangle, Quadrilaterals

The Circle
Parts of a Circle
Relationship of Triangles and Circles

Geometrical Relationships

Equivalence Study
Triangle and Rectangle
Rhombus and Rectangle
Trapezoid and Rectangle
Regular Decagon
Triangle with Same Base and Height

Concepts of Reflection
Rotation and Translation

Concept of Measuring by Unit
Metric and US Customary Systems

Perimeter and Area
Concepts of Perimeter and Area
Relationship of Area and Multiplication
Deriving the Area Formula for the Rectangle and Parallelogram

Solids, Surface Area, and Volume
Parts of a Solid Solid Classification
Three Important Dimensions
Concept of a Net
Equivalence with Liquid Volume

Volume and Weight Measurement
Concept of Measuring Liquid Volume
Measuring Spoons, Cups, Graduated Cylinders
Concept of Measuring Weight

Upper Elementary Curriculum

Numbers and Operations

Multi-Digit Multiplication
Accurate Computation on Paper
Strategies for Mental Math Estimation

Multi-Digit Division
Group Division: Stamp Game
Computation with and without Remainders
Strategies for Mental Math Estimation

Divisibility by 2, 5, and 25
Divisibility by 4 and 8
Divisibility by 3, 6, and 9
Divisibility by 11 and 7

Multiples and Factors
Least Common Multiple
Factors and Prime Factors
Greatest Common Factor

Negative Integers
Comparing Negative Integers
Negative Integer Operations

Non-Decimal Base Systems
Historical Basis
Counting and Operations
Base Systems in Time and Angles

Fractions in Lowest Terms
Fraction Operations with Unlike Denominators
Mixed Number Operations

Decimal Fractions
Calculation with Materials
Centesimal Frame Conversion of Fractions to Decimals
Fraction and Decimal Conversion on Paper

Concept and Definition
Conversions of Fractions, Decimals and Percents
Percents in Area, Scale, and Finance

Concept and Connection to Fractions
Ratios in Unit Rates, Figuring Simple Interest, and Scale Drawings


Geometric Concepts
Constructions Using a Straight Edge and Compass
Geometry in Art (Perspective, Symmetry, Transformation)
Geometry in Engineering (Bridges, Arches)

Advanced Study of Geometric Figures
Construction and Notation
Angle Measurement and Estimation
Sum of the Degrees of Interior Angles of a Polygon
Circumference and Pi

Congruence, Similarity, Equivalence
Side Lengths and Angle Measures of Congruent Figures
Side Lengths of Similar Figures
Similar Figures and Indirect Measurement

Theorems and Proofs
Pythagorean Theorem Statement and Concept
Euclid's Proof
Formula for the Length of the Hypotenuse
Converse of the Pythagorean Theorem: Is it a Right Triangle?

Reflection, Rotation, Translation
Transformations on a Coordinate Plane

Relationship Between Perimeter and Area
Apply Fractions and Decimals to Figuring Area
Area of a Triangle, Trapezoid, Rhombus Area of Compound Figures

Solids, Surface Area, and Volume
Platonic Solids, Polyhedra
Surface Area of Cubes, Prisms, Pyramids, and Cylinders
Volume of Cubes, Prisms, Pyramids, and Cylinders

Applied Mathematics

Research and Timelines
Timeline and Research of the History of Human Mathematical Discovery
Building Models of Great Geometric and Mathematical Concepts

Operations with Money
Cash Ledgers

Customary and Metric Estimation
Customary and Metric Conversion

Length, Weight, Volume and Temperature
Applications to Cooking, Science Labs, and Building

Data and Statistics
Line Plots
Frequency Charts
Stem-and-Leaf Plots
Box Plots


Square and Cube Root
Concept of a Root with Bead Materials
Concept of a Root with Peg Boards

Operations with Powers
Negative Exponents
Exponential Notation

Numerical Expressions
Parenthesis in Numerical Expressions
Expressions with Mixed Operations

Squaring and Cubing
Squaring and Cubing Sums
Binomials and Trinomials
Orders of Magnitude

Concept of Balance
Input-Output Tables
Solving One-Step Equations with Whole Numbers and Decimals
Tables and Graphs of Two-Variable Equations