Mastering Mathematical Fundamentals: A Guide for Montessori Teachers

The mathematical mind is a mind which is especially interested in mathematics. Instead of finding mathematics idiotic and absurd, it finds them interesting and absorbing. It is a fact that most of the children in our Montessori Schools do achieve great enthusiasm in doing mathematics. It is the preparation of their minds which enables them to derive this pleasure.

—Maria Montessori
The 1946 London Lectures, p. 87.

Providing children with a solid foundation is the basis of Montessori early childhood math. However, even experienced early childhood and elementary Montessori teachers sometimes struggle with the precise terminology and fundamental concepts underlying basic arithmetic operations. Understanding and using these key terms will increase your confidence when working with Montessori math materials and presenting Montessori math lessons.

Understanding the Individual Components of the Four Basic Operations

Let's begin by breaking down the parts of equations for the four basic operations.

Operation	Equation Format	Parts
Addition	3 + 4 = 7	Addend + Addend = Sum
Subtraction	8–5 = 3	Minuend - Subtrahend = Difference
Multiplication	6 x 2 = 12	Multiplicand × Multiplier = Product
Division	15 ÷ 3 = 5	Dividend ÷ Divisor = Quotient
Division with Remainder	16 ÷ 3 = 5 R1	Dividend ÷ Divisor = Quotient Remainder

Let's explore each operation in more detail:

Addition

• Addends: the numbers being added together.
• Sum: the result of the addition.

Subtraction

• Minuend: the number from which we are subtracting.
• Subtrahend: the number being subtracted.
• Difference: the result of the subtraction.

Multiplication

• Multiplicand: the number that is being multiplied.
• Multiplier: the number by which the multiplicand is multiplied.
• Product: the result of the multiplication.
• Multipliers and multiplicands may also be referred to simply as factors. A factor is a number that divides into another number without a remainder. For example, both 2 and 4 are factors of 8.
• The definitions for multiplier and multiplicand are not entirely concrete and are seemingly interchangeable throughout the math world. This is probably because of the commutative property of multiplication, where 2 x 4 is equivalent to 4 x 2 (see below).

Division

• Dividend: the number being divided.
• Divisor: the number by which we are dividing.
• Quotient: the result of the division.
• Remainder: the amount left over if the division is not exact.

Basic Properties of the Four Operations

Now that we've clarified the components of each operation, let's examine some fundamental properties of each. Understanding these properties will help you guide students in developing a deeper comprehension of mathematical relationships.

Commutative Property

• Applies to Addition and Multiplication only.
• Definition:Changing the order of the numbers does not affect the result.
• Examples: Addition: 3 + 4 = 4 + 3; Multiplication: 2 × 5 = 5 × 2

Associative Property

• Applies to Addition and Multiplication only.
• Definition: Grouping numbers differently does not affect the result.
• Examples: Addition: (2 + 3) + 4 = 2 + (3 + 4); Multiplication: (2 × 3) × 4 = 2 × (3 × 4)

Distributive Property

• Applies to Multiplication over Addition or Subtraction
• Definition: Multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.
• Example: 3 × (4 + 2) = (3 × 4) + (3 × 2)

Identity Property

• Applies to all four operations.
• Definition: There is a number that, when combined with any other number using the operation, leaves that number unchanged.
• Examples: Addition: The identity is 0 (e.g., 5 + 0 = 5); Multiplication: The identity is 1 (e.g., 5 × 1 = 5); Subtraction: The identity is 0 (e.g., 5 - 0 = 5); Division: The identity is 1 (e.g., 5 ÷ 1 = 5)

Zero Property of Multiplication

• Definition: Any number multiplied by zero equals zero.
• Example: 5 × 0 = 0

Inverse Operations

• Addition and subtraction are inverse operations, as are multiplication and division. This means they "undo" each other.
• Examples: If 3 + 4 = 7, then 7 - 4 = 3; If 3 × 4 = 12, then 12 ÷ 4 = 3

Applying These Concepts in the Montessori Classroom

Montessori teachers can use these properties to help their students develop a deeper understanding of mathematical relationships. Here are a few suggestions:

• Use concrete materials: Montessori materials like the Number Rods, Bead Bars, and the Checkerboard demonstrate these properties sensorially.
• Encourage exploration: Allow students to discover these properties for themselves through hands-on activities and guided exploration.
• Connect to real-life situations: Use everyday examples to illustrate these properties, making them more relatable and memorable for students.
• Introduce terminology naturally: Using this terminology when presenting materials and activities helps children incorporate it into their mathematical lexicon.
• Emphasize patterns and relationships: By using materials repeatedly, students discover the connections between operations and how these properties create predictable patterns in mathematics.

By mastering these fundamental concepts and properties, we can better guide our students in developing a strong mathematical foundation. Remember, in the Montessori approach, our role is to provide the environment and tools for discovery. As students explore these concepts concretely, they will naturally begin to internalize these mathematical truths.

Learn more about the fundamentals of Montessori mathematics in NAMC's Montessori Diploma Programs:

NAMC’s Montessori Early Childhood Diploma Program

NAMC’s Montessori Lower Elementary Diploma Program

NAMC’s Montessori Upper Elementary Diploma Program

Michelle Zanavich — NAMC Tutor & Graduate

As much as possible, NAMC’s web blog reflects the Montessori curriculum as provided in its teacher training programs. We realize and respect that Montessori schools are unique and may vary their schedules and offerings in accordance with the needs of their individual communities. We hope that our readers will find our articles useful and inspiring as a contribution to the global Montessori community.
© North American Montessori Center - originally posted in its entirety at Montessori Teacher Training on Wednesday, July 10, 2024.