Math in Montessori Elementary classrooms utilizes a unique hands-on approach to mathematical education and discovery. Rather than simply memorizing facts, formulas, and procedures, Montessori students are encouraged to explore math through games and activities that allow children to develop a deep understanding of mathematical concepts, and to learn to make connections and to see patterns.

**Principles of **Montessori Elementary Math

In Montessori Elementary, the principles of math are based on the understanding that children learn best through hands-on experiences and exploration. The Montessori Method emphasizes the use of concrete materials to teach abstract concepts, allowing children to develop a deep understanding of mathematical principles.

One of the key aspects of Montessori math is the use of manipulatives. Manipulatives are physical objects that children can touch, move, and manipulate to explore mathematical concepts. These manipulatives include objects such as beads, rods, and cubes, which are used to teach:

- Numeration and Linear Counting
- The Decimal System
- Arithmetic: Addition, Subtraction, Multiplication, and Division
- Geometry
- Fractions
- Measurement
- Pre-Algebra

Montessori guides incorporate the use of the “three-period lesson” to teach new concepts. In these lessons, the guide introduces a new concept, demonstrates it to the child, and then allows the child to practice the concept on their own. This three-step process helps children to internalize new concepts and develop a deep understanding of mathematical concepts.

By providing students with a solid foundation in mathematical concepts, the Montessori method helps children to develop a lifelong love of learning and a deep understanding of the world around them.

### The Mathematical Mind

In Montessori Elementary, the focus is not only on teaching mathematical concepts but also on developing a mathematical mind. The mathematical mind is a way of thinking that involves logical reasoning, problem-solving, precision, and critical thinking skills. One of the key methods of developing a mathematical mind is through the use of Montessori manipulative equipment.

These materials allow students to visualize and manipulate mathematical patterns in order to make connections, and build a strong foundation for future learning. It is through repetition and practice with the materials that students build confidence in their mathematical abilities and move on to more abstract concepts.

Montessori guides rely on the use of open-ended problems that require students to use their critical thinking skills to find solutions. By working through these problems, students learn to approach mathematical concepts in a creative and flexible way.

To take a deeper look into the importance of building a solid mathematical foundation, check out this course by Betsy Lockhart : Concrete Foundations in Algebra

## Montessori Elementary Math: Classroom Materials and Curriculum

Montessori is known for the beautifully crafted manipulatives used in its lessons and the mathematics equipment is arguably the shining jewel of the classroom. Not only are the materials visually appealing, they are supremely intelligently designed. These materials give children the chance to experience abstract mathematical concepts through sensorial exploration.

Each piece of equipment is used to isolate and demonstrate one concept at a time. Through repetition of use the child masters mathematical concepts and moves onto the next at their own pace. The curriculum builds upon itself from the most concrete to the abstract realms of mathematical study.

**The Great Lesson – The Story of Numbers**

Each year, the study of mathematics begins with a Great Lesson, “The Story of Numbers” which introduces students to the history of numbers and the concept of infinity. This lesson sets the stage for the rest of the curriculum, providing a context for the study of mathematics and inspiring students to explore the subject in depth.

Montessori’s The Story of Numbers is one look at how this story may be told:

**Numeration and Linear Counting**

Students begin their mathematics journey with an exploration of numeration and linear counting. They learn to count and represent numbers, small to large, using a variety of materials, including beads, rods, and cubes. This hands-on approach helps students develop a deep understanding of the base ten system and prepares them for more advanced mathematical concepts.

**Materials Used for Numeration and Linear Counting Lessons**

**Teens Board**– Explores numbers 11–19 by creating quantities from the 10 bars and colored bead stair. Ages 5-6**Tens Board**– Explores numbers 11–99 by creating quantities from the 10 bars and colored bead stair. Ages 5-6**Hundred Board**– Teaches number sequence from 1 to 100. The material consists of a wooden board printed with 100 squares and chips printed with the numbers 1–100. Ages 5-6**Bead Chains**– Used in the exercises of linear and skip-counting, the quantities of the squares and cubes of the numbers 1-10. It prepares the child for later activities in multiplication, squaring and cubing, as well as base number work. Ages 5-6

**The Decimal System**

Once students have a solid grasp of numeration and linear counting, they move on to the decimal system. Using the golden bead materials, they learn to represent numbers using place value and explore the relationships between numbers. A thorough understanding of place value sets the stage for the rest of the mathematics curriculum. Each new introduction of an operation begins again with the golden beads.

**Materials Used for the Decimal System Lessons**

**Golden Beads****Hierarchy of Numbers**– This material represents the hierarchy of numbers for: 1, 10, 100, 1000, 10,000, 100,000 and 1,000,000. It is used to compare by size the concept of decimal value and the families of numbers. Ages: 7 – 8**Small Bead Frame**– The Small Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 6-7**Large Bead Frame**– The Large Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 7-8**Flat Bead Frame**– The Flat Bead Frame activities are the final step in the passage to abstraction with arithmetic. Ages 7-8

**Arithmetic**

Arithmetic is a central component of the Montessori Elementary Math curriculum. Students learn to perform calculations using a variety of materials, including the bead frame, stamp game, and golden beads. Through repetition with a variety of materials, students gain mastery of math facts and build confidence in their ability to be mathematical thinkers.

**Materials Used for Arithmetic Lessons**

**Addition: **

**Addition Strip Board****Addition Charts**– Aids the child in the practice and memorization of the unit addition combinations. Includes 2 control charts, 4 working charts and 1 box of wooden answer chips. Ages 5-6**Golden Beads**– Introduces children to the decimal system and explores quantity and place value through addition, subtraction, multiplication, and division activities. Ages: 5-6**Stamp Game**– Provides individual practice in the operations of addition, subtraction, multiplication and division. The quantity and symbols of the decimal system are combined and are represented by each small square ‘stamp’. Ages 6-7**Small Bead Frame**– The Small Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 6-7**Large Bead Frame**– The Large Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 7-8**Flat Bead Frame**– The Flat Bead Frame activities are the final step in the passage to abstraction with arithmetic. Ages 7-8

**Multiplication:**

**Multiplication Bead Board**– used for practice with the multiplication tables 1 x 1 through 10 x 10. Ages 5-6**Multiplication Charts**– Aids the child in practice and memorization of the multiplication tables 1 x 1 through 10 x 10. Includes 2 control charts, 3 working charts and a box of wooden product chips. Ages 6-7**Bead Chains**– Used in the exercises of linear and skip-counting, the quantities of the squares and cubes of the numbers 1-10. It prepares the child for later activities in multiplication, squaring and cubing, as well as base number work. Ages 5-6**Golden Beads**– The Golden Bead Material introduces children to the decimal system and explores quantity and place value through addition, subtraction, multiplication, and division activities. Ages: 5 – 6**Stamp Game**– Provides individual practice in the operations of addition, subtraction, multiplication and division. The quantity and symbols of the decimal system are combined and are represented by each small square ‘stamp’. Ages 6-7**Decanomial Beads**– Provides practice with multiplication. Tables from 1-10 may be constructed in geometric form using the beads. Includes 55 of each of the colored bead bars from 1-10. Ages 7-8**Small Bead Frame**– The Small Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 6-7**Large Bead Frame**– The Large Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 7-8**Flat Bead Frame**– The Flat Bead Frame activities are the final step in the passage to abstraction with arithmetic. Ages 7-8**Checkerboard**– Provides practice with short and long multiplication with the hierarchies in the decimal system up to 100,000,000. Ages 7-8**Bank Game**– Reinforces the concept of long multiplication as the children assume different roles as they perform the game exercises. Includes plastic number cards and role designator cards. Ages 8-9

**Subtraction:**

**Subtraction Strip Board**– Practice for the equations of the subtraction tables 1–18. Natural wooden strips corresponding in length to the squares 1–18 are used to indicate the number to subtract from (minuend) and 1 set of blue strips used to subtract with (subtrahend). A set of red strips is included for further explorations. Ages 5-6**Subtraction Charts**– Aids the child in practice and memorization of the subtraction combinations. Includes a control chart, 2 working charts and a box of wooden answer chips. Ages 6-7**Subtraction Snake Game**– Introduces the child to the unit combinations of subtraction. Includes one box containing 5 sets of Negative Bead Stairs, one box containing 5 sets of Colored Bead Stairs, one box containing 23 Golden Bars Of 10, and one box containing 1 Black And White Bead Stair. Ages 6-7**Golden Beads**– The Golden Bead Material introduces children to the decimal system and explores quantity and place value through addition, subtraction, multiplication, and division activities. Ages: 5 – 6**Stamp Game**– Provides individual practice in the operations of addition, subtraction, multiplication and division. The quantity and symbols of the decimal system are combined and are represented by each small square ‘stamp’. Ages 6-7**Small Bead Frame****Large Bead Frame**– The Large Bead Frame forms the bridge to abstraction for the decimal system exercises of arithmetic. Ages 7-8

**Division:**

**Unit Division Board****Division Charts**– Aids the child in practice and memorization of the division combinations. Includes 2 working charts, 1 control chart and a box of wooden answer chips. Ages 7-8- Golden Beads – The Golden Bead Material introduces children to the decimal system and explores quantity and place value through addition, subtraction, multiplication, and division activities. Ages: 5 – 6
**Stamp Game**– Provides individual practice in the operations of addition, subtraction, multiplication and division. The quantity and symbols of the decimal system are combined and are represented by each small square ‘stamp’. Ages 6-7**Racks and Tubes Division**

**Geometry**

Geometry is another important component of the Montessori Elementary Math curriculum. Students explore geometric concepts using a variety of materials, including the geometric solids, plane figures, and construction materials. Through multisensory learning experiences, students develop a foundational understanding for basic geometric principles and are prepared for more advanced lessons, such as trigonometry and calculus.

For a more detailed explanation of the study of geometry in a Montessori classroom be sure to check out this article written by Cynthia Brunold-Conesa A New Look at Psychogeometry: The Construction of Geometric Knowledge from Absorption to Abstraction.

**Materials Used for Geometry Lessons**

**Geometric Cabinet**– Introduction to plane geometry. A six-drawer cabinet containing 35 geometric insets and frames. Ages 5-6**Geometric Solids**– Introduction to solid geometry with ten Geometric Solids. Ages 5-6**Constructive Triangles**– Concrete experiences with plane geometry. The set consists of: two rectangular boxes, one triangular box, one small hexagonal box, and one large hexagonal box, each containing triangles of differing size, shape and color. Ages – 6-7**Geometric Stick Material**– Used for the study of lines, the measurement of angles, and the construction of plane geometric shapes. Consists of a series of color-coded sticks, plastic curves, a plastic measuring angle, a plumb line, tacks and brad fasteners. Ages 7-8**Nail Board**– Used to study geometric shapes, angles, fractions and other patterns. Consists of a box with lid (lid serves as the nail board), elastic bands, a mirror, and a protractor. Ages 9-10**Equivalence Insets**– Provides experience in studying the relationship of equivalent figures that leads to the discovery of area and theorems. The material consists of 13 green metal frames with bottoms and red insets. Ages 9-10**Yellow Triangles for Area****Volume Cubes**– Used for calculation of volume and with the Cubing Material. The wooden box with lid has a hinged side and contains 1000 wooden, 1 cm cubes. Ages 9-10**Yellow Prisms**– Used in calculating the volume of a rectangular prism. The material consists of 5 wooden rectangular prisms painted yellow and ruled in lines and squares. Ages 9-10

**Fractions**

Fractions are introduced to students after they have a solid foundation in arithmetic and place value. They learn to represent fractions using concrete materials like the fraction circles and fraction strips. Students continue to use these same manipulatives to add, subtract, multiply, and divide fractions.

**Materials Used for Fractions Lessons**

**Fraction Skittles**– Represent divisors of whole, 1/2, 1/3, and 1/4. Ages 7-8**Fraction Insets**– Introduction to the concept of fractions. Ages 7-8**Fraction Circles**– Used for practice with fractions. The plastic set includes 10 whole circles and 5 sets of labeled fraction pieces for each of the values from halves to tenths. Ages 7-8**Decimal Fraction Board**– Introduction to decimal number formation and for performing the operations of addition, subtraction and multiplication with decimal and whole numbers. Ages 9-10**Decimal Stamp Game**– For practice with short and long multiplication problems of whole and decimal numbers. This material is made up of forty-nine squares, colored in the hierarchical shades of green, blue and red, that represent the hierarchies from millions to millionths and differentiate decimals from whole numbers. Ages 9-10

**Measurement**

Students learn to measure length, volume, weight, and time using a variety of materials, including rulers, scales, and clocks. An open ended, hands-on approach helps students to understand these abstract concepts through concrete experiences. Students are presented with opportunities to apply these lessons in practical situations in order to see how measurement will serve them in life.

Here is An Introduction to Measurement that showcases a way a teacher may present measurement to their students:

**Pre – Algebra**

Algebra is introduced to students after they have a solid foundation in arithmetic and fractions. Students learn to represent algebraic equations using a variety of materials, including algebraic tiles, algebraic pegboards, and cubes. Students build upon the mathematical knowledge they have gained so far to make connections, and to see the patterns that form algebraic expressions.

**Materials Used for Pre – Algebra Lessons**

**Algebraic Pegboard**– Used for demonstrating the mechanism of finding the square root, the lowest common multiple and the factors. Ages 8-9**Patterns for Square Roots**– Patterns guide when extracting the square root on the Algebraic Pegboard. Ages 9-10 Bead Cabinet**Binomial Cube**– Focuses on patterning and ordering, and making connections as to the relationships between the different cubes. Ages 9-10**Trinomial Cube**– Focuses on patterning and ordering, and making connections as to the relationships between the different cubes and prisms regarding dimension. Ages 9-10**Cubing Material**– These exercises prepare the child for abstract analysis of the binomial and trinomial cubes along with cube roots. The material consists of 1 cube and 27 squares for each of the numbers from 1–9. Ages 9-10**Base Number Material**– Hierarchical and geometric representation of numbers in bases 2 and 3 for counting forwards and backwards, and for converting numbers from base 10 into these bases. Ages 10-11

## Montessori Elementary Math: Teaching and Assessment

### Individual Learning

In Montessori Education, individual learning is a cornerstone of the teaching method. Each student is allowed to work at their own pace, with the teacher acting as a guide and facilitator. This individualized approach is also applied to math instruction.

Students are given a variety of math materials and activities to choose from, which helps to keep them engaged and interested in the subject matter. Through repetition, students gain mastery of mathematics skills and are able to move onto the next lesson. The teacher observes each student’s progress and provides guidance and support as needed.

To further your understanding of mastery of mathematical concepts within Montessori Elementary, you can read this article written by Letty Rising, How to Ensure Students are Repeating Math Materials and Concepts to Attain Proficiency.

### Problem Solving and Logical Thinking

Rather than just providing answers, the Montessori Math curriculum begins with questions that encourage students to think critically. This process teaches students to break down complex problems into smaller, more manageable parts, and to use a variety of strategies to find solutions. This approach helps students develop creativity, perseverance, and the ability to think outside the box.

This curriculum emphasizes the use of logic and reasoning to solve problems and make sense of complex concepts. Students learn to identify patterns, make connections between different ideas, and use deductive reasoning to arrive at conclusions. This approach develops strong analytical skills and a deep understanding of mathematical concepts.

### Abstract Reasoning

Abstract reasoning is the ability to think conceptually and understand complex ideas without relying on concrete examples. This skill is essential for success in higher-level mathematics and science courses. Montessori math helps develop abstract reasoning by introducing concepts in a concrete, hands-on way before moving on to abstract representations. This approach helps students learn to apply their math skills in a variety of contexts.

Here is a video by the Edinburgh Montessori Arts School that explains how all of these methods work together:

**Assessment and Progression**

Assessment in Montessori Elementary Math is an ongoing process that helps children understand their own progress. The Montessori approach to assessment is focused on the individual child’s development, and not on comparisons with others.

Observation is a primary tool used by teachers for assessment in the Montessori classroom. Guides observe children as they work with materials and interact with others. This allows them to identify areas where children need additional support or challenge.

Self-assessment is another tool used in the Montessori classroom. Students are taught to reflect on their own progress and set goals for themselves. This helps them take ownership of their learning and develop self-motivation.

Understanding how assessment works in a Montessori classroom can be challenging. Be sure to check out the Trillium P2P library where Letty Rising thoroughly explains how to use formative assessment strategies in her video, “Formative Assessment Part 1” (See Month 14). Learn more about Principles to Practice (P2P) for Elementary and join here.

**Montessori and State Standards**

A very common question asked about Montessori, especially Elementary Montessori, is how does the Montessori Curriculum work with state learning standards? And the answer is that Montessori aligns quite nicely with state standards.

Each state has its own laws regarding the learning standards schools must follow but here is a link that shows how the Montessori Elementary Math curriculum lines up with Common Core Standards: AMI Montessori and the Common Core State Standards

**Conclusion**

Growing and developing a mathematical mind is a key component of a Montessori Elementary education. By focusing on hands-on materials, open-ended problems, and repetition students gain mastery of the skills needed for higher level mathematical learning.

The focus on individual learning, goal setting, and self assessment allows students to take ownership and find joy along their path of mathematical education.

Madeline has been a Montessori educator for 16 years, with experience in both Children's House and Elementary levels. Her favorite Montessori lessons to give are the Grammar lessons. Madeline lives in Virginia and loves to travel with her family and trying out new recipes at home!