Elementary Mathematics

Overview

The study of mathematics offers gifts far beyond numeracy and calculation. It allows children the opportunity to develop and exercise their reasoning mind. It teaches them how to evaluate situations, test hypotheses, employ problem solving strategies, derive conclusions and articulate them precisely. These are critical skills that children will need in their future careers, in their personal lives and in every aspect of a world increasingly characterized by rapid change.

The reason we study mathematics is two-fold:

  • First, we want students to understand how numbers work and relate to each other, as well as the ability manipulate them quickly and easily.
  • The second reason is that mathematics is cognitive training that develops precise mental habits one can apply to all areas of life.

A student with a well rounded education in mathematics will gain:

  • Competence in everyday math concepts, like quantity, and an intuitive number sense
  • Ability to build on prior knowledge
  • The ability to organize and integrate knowledge
  • The ability to think laterally. Applying solutions from one problem to another
  • The ability to think about problem solving in a methodical way, considering all the steps required to complete work

The Montessori Method of Mathematics

The hand is the instrument of the intelligence. The child needs to manipulate objects and to gain experience by touching and handling.

Maria Montessori, The 1946 London Lectures

Our math curriculum begins with hands-on materials that help the children concretely experience quantity, place value, and geometry. By manipulating these materials students create mental models of abstract mathematical principles.

They aren’t shown examples of abstract ideas, they create abstract ideas from examples. They aren’t verbally told a “two plus two equals four” and then shown props that demonstrate this idea. Children work with beads and physically combine groups of two and discover, that no matter what, two items plus two items will always make a group of four.

Children learn by doing.

In traditional schools the child starts with the rules. In our schools, a rule is the final step taken by the child, comparable to putting a dot on the i. To us the rules are the most concise description of a long experience, and therefore the final step in the system.

Maria Montessori, Creative Development in the Child

The Montessori classroom is filled with beautiful and enticing materials. These materials are unique to Montessori education, and they are used in a unique way.

In a conventional classroom, a teacher might use materials to lecture to students. The teacher might hold up colored blocks and explain how they fit together, and then tell the students the formula for calculating a certain quantity about the blocks.

In a Montessori classroom, the materials are used to help students discover the formulas themselves. The materials are designed to be self-correcting, so that students can learn by trial and error. This allows students to develop a deep understanding of mathematical concepts.

After much practice with the materials, children develop an abstract mental model of the mathematical concepts. At this point, they no longer need the materials to help them solve problems. Only then do we show them the algorithm (provided, they didn’t find it themselves already!)

The best instruction is that which uses the least words sufficient for the task.

Maria Montessori, The Discovery of the Child

Subjects are ordered based on how easy it is for the child to understand not how easy it is to lecture on. This seems like it should be the norm, but in conventional schools, it’s not.

For example in conventional classrooms geometry is taught in middle school or high school, because the abstract mathematical representations of geometric concepts is difficult to teach without a grounding in higher mathematics.

But a very young child can distinguish the difference between a circle, triangle and square without the need of abstract representation.

With the hands-on materials we are able to sequence the curriculum to match a child’s developmental abilities. The approach is understandable at every level to the child and always firmly rooted in reality.

This then is the task of man- to think and to work.

Maria Montessori, Creative Development in the Child

The lessons are ordered in such a way that only one new difficulty is introduced at at time. These lessons are usually very short and given to students who are all ready for that new challenge. Thus, children learn each step precisely and thoroughly before being asked to add a new level of complexity.

The hands-on materials are designed to be enjoyable, so that repetition- a requirement for mastery- is fun. Practice becomes play.

Math lessons are often presented to individuals or pairs of students. When students need more time or support, they have as many lessons and as much time as they need. Students who excel are able to move forward without waiting for the group to “catch up” to them.

Time is variable, but mastery is fixed.

Lower Elementary Curriculum

Fraction arithmetic

Arithmetic

Great Stories
The Origin Story of Our Number System

The Decimal System
Composing Numbers
Place Value to the Millions

Intro to Math Operations
Addition
Subtraction
Multiplication
Division

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
Divisibility by 2
Divisibility by 5
Divisibility by 25

Multiples and Factors
Multiples through 100
Least Common Multiple
Factors

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 Numbers
Concept and Notation of Decimal Fractions
Decimal Hierarchies
Decimal Fraction Comparison

Word Problems
Strategies for Understanding
Applied to All Math Topics

Money
Counting Currency
Connection to Decimals
Adding and Subtracting Money

Temperature
Concept of Measuring Temperature
Reading Weather Reports, Tracking

Data and Graphing
Concept of a Graph
Collecting Data
Pictographs
Bar Graphs

Montessori geometric stick material polygons
Classifying kinds of quadrilaterals

Geometry

Great Stories
The Harpedonaptae and Measuring the Land of Egypt

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

Lines
Parts of a Line
Types of Lines
Relationships Between Two Lines

Angles
Parts of an Angle
Types of Angles
Pairs of Angles

Polygons
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
Congruence
Similarity
Equivalence

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

Transformations
Concepts of Reflection
Rotation and Translation
Tessellations

Length
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

SolidsSurface 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

Finding the cube of a binomial

Arithmetic

Great Stories
The Origin Story of Our Number System

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
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
Conversions
Base Systems in Time and Angles

Fractions
Fractions in Lowest Terms
Reciprocals
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

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

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

Montessori volume materials
Material for deriving the volume formula for a squared-based pyramid

Geometry

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?

Transformations
Reflection, Rotation, Translation
Transformations on a Coordinate Plane

Area
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

Education for life

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