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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