Saturday, February 4, 2012

Mathematics - Table of Contents

We focus on the Keys of learning - and when we do, children learn through their hands. 

Once a child has mastered the concept of 1-9, the concept of 0 and the basic idea of putting groups together and separating groups (adding and subtracting), they are ready to work with 4 digit numbers. 

Mathematics begins at or close to the child's 4th birthday: 

Numbers 1-10 - 4.0-4.5
Number Rods
Sandpaper Numerals (ciphers)
Number Rods and Cards
Spindle Boxes
Zero Activity
Cards and Counters
Memory Game of Numbers (Chit Game)

Decimal System - 4.5-5.5
Introduction to the Decimal System
Introduction to the Bead Material
Introduction to the Card Material
Formation of Numbers
Change Game (with golden beads - exchanging; the Bank Game is an elementary work with colored cards)
Addition with the Golden Beads
Subtraction with the Golden Beads
Multiplication with the Golden Beads
Division with the Golden Beads
Long Division with Bows
Stamp Game Addition
Stamp Game Subtraction
Stamp Game Multiplication
Stamp Game Division
Dot Game
Word Problems

Counting - 4.5-5.0
Teens Stage I: Teen Boards 11-19
Teens Stage II: Teen Boards 11-19
Teens Stage III: Teen Beads and Boards 11-99
Tens Stage I: Tens Boards 10-90
Tens Stage II: Tens Beads and Boards 10-90
Tens Stage III: Tens Beads and Boards 11-99
Linear Counting: Stage I – Hundred Chain
Linear Counting: Stage II – Thousand Chain
Skip Counting

Memorization - 5.0-6.0
Addition Snake Game
Addition Strip Board
Addition Charts
Subtraction Snake Game
Subtraction Strip Board
Subtraction Charts
Multiplication with Bead Bars
Multiplication Board
Multiplication Charts
Unit Division Board
Division Charts

Passage to Abstraction - 5.5-6.0 (leads into elementary as well)
Small Bead Frame
Wooden Hierarchical Material
Large Bead Frame
Division with Racks and Tubes

Fractions - 4.5+ 
Introduction to Fractions
Fraction: Simple Operations
(there is also design work with the fraction materials, leading to concepts of equivalencies - design work begins shortly after introduction to the metal insets)

Elementary - covering ages 6-12 years: 

I. Introduction
        5th Great Lesson: The Story of Numbers
II. Early Work - Numeration
        Wooden Hierarchical Material
        Large Bead Frame: Introduction
                Exercises I-IV
        Commutative and Distributive Laws of Multiplication
        Least Common Multiples
        Last Steps of Least Common Multiples to Abstraction
        Divisibility: by 2, 5, 25
                By 4, 8
                By 3, 9, prime numbers, 7, 11
III. Operations
        Long Multiplication
                Large Bead Frame
                Bank Game
                Flat Bead Frame
                Checker Board: Exercises 1 & 2
                Geometrical Form of Multiplication
                Category Multiplication
        Long Division
                Distributive Division with Racks and Tubes
                Group Division
                Group Division & Word Problems w/ the Fraction Material
IV. Squares and Cubes of Numbers
        Prerequisites to Squares and Cubes of Numbers
        Notation of Squares
        Notation of Cubes
        Games 1-3
        Game 4: Decanomial Square
        Paper Decanomial
        Sums Using Squares and Cubes
V. Fractions
        Introduction to Fractions (review if received in primary)
        Equivalence of Fractions
        Simple Operations
        Addition & Subtraction of Fraction w/ Different Denominators
        Exercises Leading to Abstraction of the Above                
        Multiplication of a Fraction by a Fraction
        Division by a Fraction
        Notes on the Fraction Charts
VI. Decimal Fractions
        Symbol Linked to Quantity
        Formation and Reading of Quantities
        Conversion of Common Fractions to Decimal Fractions
        Effects of Multiplying or Dividing Numbers by Powers of 10
        Multiplying a Decimal Fraction by a Decimal Fraction
        Introduction to the Decimal Checkerboard
        The Decimal Checkerboard
        Relative Size of Terms in a Multiplication Problem
        Relative Size of Numbers When You Are Dividing
        Division of a Decimal Fraction by a Decimal Fraction
        Leading to Abstraction of Multiplication of Decimal Fractions
        Division of a Decimal Fraction on Paper
VII. Squaring and Cubing
                Transformation of a Square, Ex 1-6
                Passing From One Square to Another
                Squaring a Sum
                Squaring with a Hierarchical Value
                Passing From One Cube to Another
                Cubing a Binomial
                Cubing a Trinomial
                Cubing a Quadrinomial
                Cubing a Trinomial having Numerical Value
                The Story of the Three Kings
                Cubing a Number with Decimal Value
        Square Roots
                Introduction to the Concept
                Exercises 1-3
                Passages to Abstraction
                Special Cases
        Cube Roots
                Exercise 1
                Exercise 2: 3 digit roots
                Exercise 3: Last Steps Towards Abstraction
VIII. Other Topics
        Signed Numbers
        Powers of Numbers
                Powers of 2
                Powers of 3
                Powers of 10
                Operations Using Exponential Notation
                Expanded Power Notation
                Operations Using Expanded Power Notation
        Other Number Bases
        Ratio and Proportion
        Word Problems
                Introduction to Word Problems
        Introduction to Algebra

UPDATE: See a math topic missing? Browse our other Montessori Nuggets, as I have upcoming Nuggets on this topic; then post in comments and I'll reply with where it is found within the albums --- not all Mathematics topics are found in the math album ;)

Be sure to check the reaction boxes below this post - this provides me some feedback of what you'd all like to see! 


  1. Excellent post. Thank you for putting it all into a concise outline. Hope to see more "Table of Contents" for other areas of study.

  2. More coming :) I am looking to do each one as a combination of primary and elementary to show the complete flow from one plane of development to the next.

    If you (or anyone) has a particular area you'd like to see addressed that would fit with the theme here, please let me know. In particular along the lines of "top questions for new Montessori parents or teachers - or those seeking information on Montessori". :)

  3. This is so useful post! Thank you!

  4. Still learning here... we started Montessori "late" (always parented in the 'style' but haven't formally homeschooled Montessori until my son was about 5 ish). Does that mean I'm destined to constantly catch up with him? Even now at 5.5 y.o. he resists academics (sitting down to read on his own and is still struggling with memorizing numbers - 0 thru 9 is pretty good, but is just now starting to get a handle on teens & up recognition). I'd like to continue Montessori, but I worry that if the tools I have will no longer work since he's out of the sensitive periods, I may need to seek out an alternative.

  5. By the end of your first year of elementary, it won't be "playing catch up" so much - you'll see :)

    There is a point where you have to let go of everything he COULD have had in primary because he is moving into the second plane and the quality of his needs change.

    At 5, they like humor, they like games. I had 5 and 6 year olds during my student teaching who were "behind" so to speak; I had them start over with the sensorial album and focused on the games - they ate it up! Especially when there was a little trick or catch for us all to laugh about in the end.

    For mathematics materials - the core materials continue into elementary (golden beads, colored bead bars, chains/cubes/squares, fraction insets); and while there is a remediation section, again, at some point he just moves on to the next thing and he will be totally FINE.