Mathematics

Montessori Theory Foundation

Maria Montessori believed that learning did not occur until the child did the work of mathematics for him or herself. So, in her theories for mathematics, she believed that the children needed to do the work of mathematics themselves. This is the path to self-construction. The introduction to mathematics begins with the story of how our ancestors needed to quantify and keep track of things and how the idea of numbers grew and developed through different cultures: the Sumerians, the Egyptians, the Phoenicians, the

Greeks, the Romans, the Arabians and the use of zero from India . We also discuss the role of the printing press in the stabilization of our numbers from A.D. 1500’s. We also tell stories of early mathematicians to show how both ordinary and extraordinary people have played important roles of our use of numbers to solve problems that have faced our species. The children are enticed to see themselves as part of this cultural evolution of one generation helping the next generation solve bigger and more complex world problems.

Mathematics in a Comprehensive Education Model

Montessori offers a comprehensive education to the child, not merely a mathematics’ curriculum. The foundation introduced to the child has many facets of mathematics, similarly to a gardener planting seeds. We plant the seeds of mathematical knowledge. A curriculum means to step through each point of the curriculum, test it, and move on whether they have mastered it or not. Montessori education means that we do not test each skill level, but rather we give them many opportunities to practice and to repeat the lessons until there is mastery. There is a moving forward as we plant the seeds for future work and there is also a moving backwards to repeat the lesson until there is mastery. There is a serious issue inherent in this. The child needs to know certain things in order to be part of the larger culture, but in the Montessori classroom, we are responsible to make sure that they are prepared to be an integral and functional part of their community. These elements include money, time, reading maps, schedules, and other skills that they need for life. So in the classroom, we set up the mathematics so that they can practice the isolated steps in the execution of a real task while also giving them real life applications to situations that they need to solve for their own activities. We prepare them for mathematics in real life.

We also give them repetitive experience with the more mechanical and difficult mathematical functions in order to foster the development of the logical and deductive reasoning. For example, the cube root of number and the non-decimal bases work are indirect preparations for algebra.

Mathematics Presented as Part of History

We are also planting the seeds of the history of mathematics. As noted previously, we tell the stories of mathematics. One example is the Story of Geometry. Geometry began with the Egyptian farmers who needed to survey their lands after every flood. There are numerous studies about the early use of geometry. There are many practical applications of mathematics, but there is also mathematics that are derived from people who had thoughts and worked them out. There are two characteristics of the second plane child that we utilize. They have both very active imaginations and are very prone to hero worship. We use their imagination to bring to life the heroes of our acquisition of mathematics.

Mathematics Stemming from the Human Tendency to Exactness

There is the human tendency to continually become more exact. This exactness and precision leads us, as a species, to seek better and efficient ways to address our problems and our relationship to the world around us. For example, there are the concepts of equivalence, congruency, and similarity; these are precise terms, but these simple concepts reinforce the idea that mathematics is not just an obtuse study for the ethereal thinker. It is a way of comparing, evaluating, and balancing qualities and quantities. There is a tendency to think mathematically by all human people, but there needs to be a foundational understanding before one can use these concepts in real life applications. Often there is a vocabulary that we need to understand before we can enter into that area. Mathematics in the Montessori classroom is for everybody. We do not teach higher math only to the students who are good at math. We plant the seeds in all of them for the majesty of our mathematical heritage.

We help them memorize their mathematical facts in order to free their minds for the more advanced concepts of mathematics. It has been shown through research that once children know their number facts there are able to solve problems at higher levels of reasoning. Subsequently we encourage the memorization of math facts.

Mathematics as an Abstraction

Mathematics, in human history and development, may mean simply the addition and subtraction of quantities. However, who was the first person to extract that 2a + 5a = 7 a? We start out with the children with symbolism only. We offer them a materialized abstraction. For instance, the number rods are a material that gives numbers a physical representation in an isolated way. Maria Montessori knew that the human mind needed to revisit the course of human history, how the human mind developed the mathematical process. The human mind abstracts from experience and then creates a symbol for that abstraction. You can see it in some sense with language. We see many types of chairs, but we came to know what is a chair from a table regardless.

Review of Mathematics in the Preschool

Tendencies and Sensitive Periods are Recognized: In the preschool class, everything is designed with the human tendencies and sensitive periods in mind. The tendency for exploration is inherent in all of the materials. Mathematics satisfy the child’s human tendency for order. Movement is accommodated by allowing them to move the materials around the room. Language is addressed in order to give the child the vocabulary of mathematics.

All work of Preschool Contributes to Development of Mathematics: All of the child’s concrete experiences lead them to the development of the mathematical mind, because it allows them to experience the isolated quality, to name, organize it, classify it, and abstract it. All their work is indirect preparation.

The child is a sensorial explorer in the primary: The primary child is a sensorial explorer. It is the human condition that we receive information from the senses. We want to get every sense tuned to receive information. So when we speak of the preschooler’s development, we refer to the child’s ability to sensorially experience even if it is through pictures.

Preview of Mathematics in the Elementary

Mathematics for the six to twelve year old child requires that we plant the seeds for all of the mathematical functions and processes. We cater to the tendencies in the delivery of the mathematics to the child. We help them explore outside of the classroom. This is necessary, because their lives are not in a classroom and where else will they find out what is out there if we do not sow the seeds of what is out there. One interesting way is the use of class activities and outings. They require that the children plan, prepare, and predict. It also requires them to adapt in times of change. Necessity is the mother of invention. One of the psychological characteristics is to develop the reasoning mind. Mathematics is the greatest vehicle for the development of the reasoning mind.

Great work of the elementary: We present the big picture in order that the child sees the possibilities. If I can do this, then can I do that? The child naturally likes to take their information to an exponential level. We do not impede the child’s effort to do hard mathematics. We applaud their wonder even if they have a desire to do an amazingly hard task that they may not be completely ready for; we let them try it! If we see them waning and we know of a skill or a key that will help them, this is the time we give it. For example, the use of exponential numbers is often a great help in shortening a mathematical equation to a more manageable form.

Mathematics serves the goal of helping the child to develop: mathematics is at the core of the child’s self-construction. The child’s knowledge of mathematics is directly proportional to their ability to make numerical and quantitative decisions. They are able to measure cause and effect. They are able to predict outcomes more easily. The preparation of the mathematical mind is a necessary component of the child’s self-construction.

Mathematics Program Overview

The sequence of the presentations is important to the child’s success, as are the contents of each area. The sequence is set up to continually introduce one new layer of difficulty. The process is introduced sensorially first with manipulative materials. Usually the mathematical procedures are static at first, the child then proceeds one level at a time through dynamic mathematical processes. There is always repetition and variety within each topic. There are indirect preparations throughout. We also revisit skills and concepts continually. We teach x, and then we go back and look at again with negative numbers, powers, fractions, decimals, etc.

We also look at different types of numbers. For example, there are parts and wholes of numbers, the fractions and the decimals. We look at numbers in different bases. We look at signed numbers. We look at numbers with powers. We introduce numbers with signed powers and as with all of these types of numbers, we repeat the operations.

There is often further elaboration and information needed as we go through the study. For example, How do we determine the sign? Or, how do we determine where the decimal goes. There are connections between the sections of the math program. There are similar patterns that follow from one material to another.

Counting – The foundation

The foundation is the story of numerals. We help the child image the earliest of human beings need for counting and keeping track of things. Counting is the foundation of the math program, especially one to ten. Counting to ten is not enough if you don’t have a one to one relationship. It is important that the child realize that each count relates to a quantity.

Basic Math

There is a remedial component for the child who has not established a one to one correlation to quantity and symbol. These lessons are meant for the student who needs remedial work and not necessarily as an introduction. They are a shortened version of the math presentations given in the Montessori preschool.

The Decimal Systems

We bring to the child’s awareness the decimal system. We use the hierarchical materials and the Large Bead Frame. The children learn that there is a pattern in the categories. They learn that the pattern is repeated throughout math: units, tens, hundreds. Repeat. There are also families unit family, thousand family, millions family. There are color conventions that repeat throughout all of the materials.

The Operations

Based upon the work with the decimal system, we move into the operations. We do some preliminary work with the operations by giving the commutative and distributive laws. This pattern will recur throughout their work. There is a convention of the order. The multiplier is gray and the multiplicand is on white. The product is the answer.

Multiplication Proper: It includes all of the materials we use to teach the checkerboard. The large bead frame is for place value. It makes multiplying by ten easy. All of the families are grouped. The bank game is not sensorial, but it is still quantitative. The geometric form of multiplication allows children to see the patterns of multiplication as it begins to appear. The are links that connect the two beads line up in those same patterns. We move toward work on paper. Montessori has numerous ways to do multiplication so that the child is led to repetition on his or her path to mastery.

Division: Two main materials are the racks and tubes and the stamp game. They represent two different ways to look at division. One way is to look for the size of the share and the other is how many shares can be made. There are all these factors that link these works together. They learn what a factor and a multiple are. Divisibility links in here as well. Be on the lookout for how you set things up.

Numbers Less Than One

These include fractions and decimal fractions. They repeat patterns learned earlier. They are introduced sensorially. There is also an ability to relate percentages, ratios and proportions to this.

Other Number Systems

They are also introduced sensorially and then familiar patterns are used.

Measurement and Work

This links all of the lessons to practical experiences and applications to the children’s work. Have this link to all areas of the math work.

Algebra

Algebra appears as information that allows you to solve for unknown information. The child learns that we can solve anything using equations. There is also the side of algebra where it is a just an academic process. We solve the equation for the pleasure of solving the equation.

THE MONTESSORI SYLLABUS FOR MATHEMATICS

GREAT STORY OF MATHEMATICS

The Story of our Numerals

THE DECIMAL SYSTEM: CATEGORIES AND NUMERATION

Wooden Hierarchical Material

Large Bead Frame

Laws of Multiplication

Commutative Law for Multiplication

Distributive Law: Multiplication by one digit

Investigating the Distributive and Commutative Laws

Distributive Law (multiplication by one sum)

Distributive Law: Passage to Abstraction

Operation signs & products expressed in cards

Bead Bars for Partial Products Only

Cards Only

Paper Only

Distributive Law: Application to the Decimal System

Introduction

Cards Only

Paper Only

Notes on the Distributive and Commutative Laws of Multiplication

MULTIPLES AND FACTORS

Multiples

Concept and Language of Multiple

- Short Chains

- Bead Bars

Investigation with Multiples

- ‘Multiples of Numbers’ Paper

- Tables A and B

Concept and Language of Common Multiple

Investigation of Common Multiple

- Table C

- Leading to Concept and Language of Prime Numbers

Concept, Language, and Notation for Least Common Multiple(LCM)

Factors

Concept and Language

Factors using the Pegboard

Common Factors

Prime Factors using Table C

Investigation of Prime Factors using the Pegboards

Prime Factors to find the Least Common Multiple (LCM)

Concept and Language for the Greatest Common Multiple (GCF)

DIVISIBILITY

Divisibility by 2, 5, 25

Divisibility by 4, 8

Divisibility

By Prime Factors

By Products of Prime Factors

Divisibility Chart

Divisibility by 3, 9, 6

Divisibility by 11

LONG MULTIPLICATION

Checkerboard

Reading Numbers: Single to Multi-digit

Multi-digit multiplier: No number Facts, No Writing

Multiplication

Using Number Facts and Recording Answer

Writing Partial Products

Writing Products Directly

Geometric form of Multiplication

Flat Bead Frame

Multiplication

Multiplication by a 2 to 4 digit multiplier:

Writing Final Product Only

Writing partial Products

Bank Game (no writing)

Multiplication by 2-digit multiplier

The Multiplication Algorithm

LONG DIVISION

Racks & Tubes with Boards

Single digit divisor (record problem, quotient, remainder)

Multi-digit divisor (up to 4 digits)

-Problem, Quotient, Remainder

-Intermediate Remainders, Quotient, Final Remainder

-What’s used, Intermediate Rem., Quotient, Final Remainder

Special Case

Zero in the middle of divisor

Zero as last digit(s) of divisor

Stamp Game

One digit divisor

Multi-digit divisor

Special Cases: Zero in the quotient

Division Algorithm

FRACTIONS

Concept and Equivalence

Introduction to Fractions

- Quantity/Language

- Symbol, Notation, Further Language

Other Representation for Fractions

Equivalence

Nomenclature for Equivalence

Operations – Simple Cases

Addition, Same Denominator, Reduction, Sensorial

Subtraction, Same Denominator, Reduction, Sensorial

Multiplication by a Single Digit, Whole Number, Reduction, Sensorial

Division by a Single Digit, Whole Number, Reduction, Sensorial

Operations – Beyond Simple Cases

Addition and Subtraction – Different Denominators

Multiplication by a Fraction

Division by a fraction – Measurement/Group

Division by a Fraction – Partitive/Sharing

Operations with Fractions – Passage to Abstraction

Addition/Subtraction

Finding Common Denominators using transparencies

Finding Common Denominator using graph paper

Raising/Reducing a Fraction Arithmetically

Addition/Subtraction: Using only tickets

Multiplication of fractions using only tickets

Addition/Subtraction

Find Numerators Raising/Reducing a Fraction,

Given a Common Denominator

Other methods of finding a common denominator

Multiplication Using Graph Paper

Fraction as Part of Sets

Abstraction of the Rules for Operations with Fractions

Addition/Subtraction – Finding the Least Common Denominator

Sensorial Exploration: Rule for Multiplication of a fraction by a fraction

Multiplication of Mixed Numbers

Sensorial Exploration of Rule for division of a fraction by a fraction

Applications

DECIMAL FRACTIONS

Introduction

Introduction to Quantity and Language

Introduction to Symbolic Notation for Decimals

Formation in Cards and Reading of Multi-Digit Decimals

Operations, Simple Cases

Addition and Subtraction using Decimal Board

Addition and Subtraction on Paper Only

Multiplication by Unit Multiplier Using Decimal Board

Division by Unit Divisor

Multiplication with Decimals, Beyond Simple Cases

Introduction to the Decimal Chequer Board

Multiplication on Decimal Chequer Board

The Decimal Felt Squares

Multiplication/Division by Powers of 10 Using the Decimal Board

Multiplication on Paper Only

Division with Decimals, Beyond Simple Cases

By a Mixed Number or by a Decimal Using a Skittle

Algorithm for Division of Decimal

Introduction to Percentage with the Centesimal Frame

Concept (hundredths), language, notation

Conversion of fraction insets to percentage using the Centesimal Frame

Handouts Extending the Exploration of Decimals

Relative size of terms when Multiplying, Dividing in Decimals

Rounding with Decimal Numbers

Conversion of Common to Decimal Fractions and Vice Versa

Percentages

SQUARES AND CUBES OF NUMBERS

Concept and Notation of Square of a Number

Concept and Notation of Cube of a Number

The Decanomial

Building the Decanomial Using the Distributive Law

Finding Squares and Cubes in Decanomial – Building the Tower of Jewels

Numerical Decanomial

Operations with Numbers Written as Squares and Cubes

Finding Squares in Multiplication Bead Bar Layout

SQUARING

Arithmetic Passages

Transformation of the Square of 10 (numeric)

Into square of a binomial:

Bead squares of 10

Other bead Squares

On Graph Paper

Building a larger square from a Smaller Square

Squaring a Sum (one-digit terms)

Application to Decimal Numbers ( Square of Whole Numbers </= 31)

With Golden Beads

Application to Decimal Numbers (Square of whole Numbers >31)

Squaring a binomial representing a 2-digit number

From the Real Square to the Symbolic Square (Preliminary)

On Graph Paper

Squaring a trinomial representing a 3-digit number

Algebraic Passages

Derivation of formulas and subsequent application of formula

For square of a binomial using binomial cube

For square of trinomial using Trinomial Cube

For square of a polynomial (children’s Work)

Derivation of formula

Applicable to decimal system and subsequent application

SQUARE ROOT

Extracting a Square Root: Sensorial Passages

Concept, Language, Notation for Square Root (Bead Squares)

Extracting a Square Root for Numbers <225

Square Root Board and Loose Units

Extracting a Square Root for Numbers <9999

Golden Bead Material

Extracting a Square Root for Numbers <99,999,999

Pegboard, Hierarchical Pegs/Cups

4-category number, 2-digit root

5-category number, 3-digit root

5-category number, 3-digit root, zero in middle of root

6-category number, 3-digit root, zero in dividend

6-category number, 3-digit root, zero at the end of root

7-category number, 4-digit root

Extracting a Square Root: Passages to Abstraction

Building the square by category and writing the amount used by category

4 category number, 2 digit root

4 category number, 2 digit root, backtracking example

Building the square by category and writing amount used plus an analysis

8-category number, 4-digit root

Calculation on paper using Guide Squares and N, N2 chart

4 – category number, 2 digit root, (4 examples)

5 – category number, 3 digit root

8 – category number, 4 digit root

6 – category number, 3 digit root, zero in middle of divisor

5 – category number, 3 digit root, backtracking example

Verbalization of the rule and application of the algorithm (handout)

CUBING

Arithmetic Passages

Preliminary: From a given cube to the cube of the next size (52 to 62)

Preliminary Extension: From a given cube to a larger cube (43 to 73 )

Cubing a binomial, numeric

Starting from the Square

Starting from the Cube of the First Term

Cubing a trinomial, numeric

Starting from the Square

Starting from the Cube of the First Term

Algebraic Passages

Cubing a Binomial, Algebraic

Cubing a Trinomial, Algebraic

Application to the Decimal System

Preliminary:

The Story of the Three Rulers: Introducing the Hierarchical Trinomial

Arrange the Procession

Revolt (Change to the decimal System)

Label the Pieced of the Algebraic Trinomial (a, b, c)

and change the decimal labels (h, t, u)

Calculate the decimal value (h3=1 000 000 to u3 = 1

Introduction of the hierarchical Trinomial Cube

Cubing a decimal number (3digits, eg 2373) using the hierarchical cube

CUBE ROOT

Concept, Language, Notation for Cube Root

Finding Cube Root of Cubes in the Bead Cabinet

Finding Cube Root of 1-3 digit #’s using Chart

Finding Cube Root using 2 cm Cubes

Finding Cube Root of 4-6 digit #’s with Wooden Cubing Material (by category)

Calculate by first extending 3 dimensions of cube one unit at a time

Consolidating the calculations of identical groups of prisms

Finding Cube Root of 7-9 digit #’s using Hierarchical/Decimal Trinomial Cube

Writing the calculations form the cube to the prisms

Special Case: Backtracking

Special Case: Zero in the middle of the Root

Special Case: Zero at the end of the Root

Writing the calculations from the decimal terms of the formulas

Rule for extraction of a Cube Root

NEGATIVE NUMBERS

Addition of Signed Numbers

Subtraction of Signed Numbers

Multiplication of Signed Numbers

Division of Signed Numbers

Word Problems Using Signed Numbers

POWERS OF NUMBERS

Notation and Numerical Value for Powers of 2

Unit can be any Size

Any decimal number has powers

POWERS OF NUMBERS: OPERATIONS

Operations (+, -, x, ¸) with Numbers Having Exponents

Negative Exponents for Decimal System

Special Cases of x or ¸ of powers of numbers having Same Base

Operations with Multi-Digit Numbers Written in Expanded Notation

(given after Negative Numbers)

Note on Scientific Notation

MEASUREMENT

Length

History of the Measurement of Length

The concept of Measurement

Small non-standard Unit of Measurement for Length

Larger non-standard Unit of Measurement for Length

Standard Unit of Measurement for Length

The Metric System

Other Measurement

Area (Geometry)

Angles (Geometry)

Temperature: the Farenheit Scale

Volume (Geometry)

Weight

Time (History)

The Story of the Measurement of Time

Conversion Between Metric System Units and English System Units

Force

NON-DECIMAL BASES

Counting in a non-decimal Base

Operations in different bases

Conversion of notation from one base to another

INTRODUCTION TO ALGEBRA

Concept of Equation and Balancing an Equation

Solving for the unknown

RATIO AND PROPORTION

Ratio

Concept, Language, Notation

Ratio can be expressed as a Fraction

Ratios are equal if they are equivalent fractions

Problem Solving Using Ratios

Proportion

Concept, Language, Notation

Cross Multiplication (key concept for solving equations)

Ratio and Proportion Word Problems/Other Activities Using Ratio and Proportion

WORD PROBLEMS

Introduction

Distance, Rate, Time Problems

Solving for Distance (Sensorial, Arithmetic, Algebraic)

Solving for Time (Sensorial, Arithmetic, Algebraic)

Solving for Rate (Sensorial, Arithmetic, Algebraic)

Principal, Interest, Rate, Time Problems

Solving for Interest (Sensorial, Arithmetic, Algebraic)

Solving for Rate (Sensorial, Arithmetic, Algebraic)

GRAPHING

Notes to the Teacher on Graphs

Interpreting Graphs

Constructing Graphs

Types of Graphs

Pictograph, Bar, Line, Circle: Uses

FUNDAMENTAL MATHEMATICAL CONCEPTS WHEN TAUGHT IN ELEMENTARY

For children who need one to one correspondence between quantity and symbol: counting:

Counting and Numbers to 10

Spindles or Counters

Number Rods

Number Rod Cards

Number Rods and Cards

Decimal System

Symbols and Names Beyond 10

Introduction to 2-digit numbers 11-19

Numbers 20-100

Numbers 100-1000

Changing between Categories (Trading, Regrouping)

Smaller to Larger Categories

Larger to Smaller Categories

Reading Composite Numbers

Operations

Addition, Subtraction, Multiplication, Division

Memorization of Facts

Addition and Subtraction on Paper