Early math education can have a profound impact on a child’s cognitive development. Mathematical concepts are a gateway to understanding the world with clarity and precision. By introducing mathematics early on, we provide children with the tools to start decoding the patterns of the world around them. Unlike other approaches to early math instruction that rely heavily on memorization, Montessori Math places a premium on hands-on, experiential learning. Montessori’s immersive approach to math builds a strong foundation for a lifetime of numerical fluency and problem-solving skills. You can see full demonstrations of how to present all of the 3-6 Montessori Math materials in Trillium’s Montessori Math Curriculum Refresher course.

In this post, we’ll explore the fundamentals of mathematics in Montessori as well as the materials and curriculum used to present math at the Primary level. We will also share additional resources for adults looking to support the development of math skills in young children.

- Fundamentals of Montessori Math
- Montessori Math Resources
- Montessori Math in the Primary Classroom: Materials and Activities
- Extending Montessori Math Beyond the Primary Classroom
- Conclusion

**Fundamentals of Montessori Math**

**Defining Mathematics**

Mathematics is a broad and diverse field of knowledge studying the patterns, structures, and relationships of quantities, numbers, and shapes. Math also involves the use of logic, reasoning, and creativity to discover and prove new facts and concepts. Mathematics is sometimes described as a “universal language” that can be understood and appreciated by people from different cultures and backgrounds.

**Why Teach Math?**

Mathematics is an important and highly practical area of knowledge. Math skills are essential to scientific inquiry, critical thinking, and many basic activities in daily life. Even for young children, mathematics can become a valuable tool for expressing ideas and for problem-solving during shared activities. In a Montessori classroom, math is taught as a foundational pillar that equips children with skills vital for future success in academics and beyond.

**Montessori Math: Concrete to Abstract**

In Montessori, the mathematics curriculum follows a deliberate progression from concrete to abstract concepts. These lessons are presented using a unique set of educational materials. Montessori Math materials are carefully arranged in a specific order, with a purposeful flow from most preliminary/concrete to most complex/abstract. These materials are displayed on low shelves at child level, and are designed to be attractive and appealing to children. (*Looking for some Montessori Math shelf inspiration? Check out the tour of Lauren’s classroom here.*)

The Montessori Math curriculum is intended to build a solid understanding of foundational mathematical principles. In Montessori, the gradual transition to abstract representation (and, eventually, memorization) occurs only when the child has internalized mathematical concepts through hands-on, concrete experiences.

**A Curriculum Sequence Based on Mastery, Not on Age or Time**

Recognizing that each child progresses at their own pace, the Montessori math curriculum is not rigidly tied to age or predetermined timelines. Rather, the curriculum adapts to the child’s readiness and mastery of each concept. This applies to all areas of the curriculum, and math is no exception. For instance, in a Montessori Primary class, children are able to practice a particular math concept until they demonstrate mastery of that concept, whether such mastery comes after just a few presentations or after a period of extended practice. Because they are still in the absorbent mind stage, it’s much more important to allow young children (up to age six or so) ample time to thoroughly and accurately internalize preliminary math concepts than it is to rush them through the curriculum sequence. We want to avoid children absorbing the idea that math is confusing and difficult to understand!

**Assessing Montessori Math in the Primary Classroom**

As with many Montessori activities “assessment” is built into the activity itself rather than offered as a separate activity. Each lesson makes use of the “Three-Period Lesson” approach:

- The adult presents the activity to the child
- The adult helps the child practice the activity as presented
- The adult invites the child to practice the activity independently

During steps 2 and 3 of this process, the adult observes the child closely, watching and listening to check for understanding. If the child makes errors, the adult avoids correcting those errors in the moment and makes a note to re-present the lesson at a future time. Especially at the Primary level, Montessori Math does not involve timed assessments, quizzes, or other isolated assessments. This low-pressure approach helps the child focus on mastering the mathematical concepts being presented rather than on achieving a particular score or grade.

**Integrated Into the Curriculum**

**Integrated Into the Curriculum**

While there is a distinct and specific set of Montessori materials designed to isolate mathematical concepts, math is seamlessly integrated into various aspects of the Montessori Primary curriculum. Whether exploring geometric shapes in art, observing mathematical patterns in nature, or incorporating counting or measurement into literacy or practical life activities, taking such an interdisciplinary approach ensures that mathematical concepts are viewed as useful and relevant to real life.

**Montessori Math Resources**

Trillium offers two rich training options for adults looking to support mathematical development in young children.

**Montessori Math Curriculum Refresher**

For a deep dive into the world of Montessori Mathematics, check out Trillium’s Montessori Math Curriculum Refresher course. This highly practical self-paced course includes a demonstration of all the key materials and activities in the Primary math area, plus valuable discussion of prerequisite math concepts and the overall scope and sequence of the Montessori Math curriculum in Primary.

**Building Number Sense**

Adults working with toddlers or younger Primary students in a Montessori setting might also consider Trillium’s Building Number Sense course. This self-paced course is another fantastic option for adults who want to help younger children build a solid mathematical foundation in preparation for later exploration of more abstract math concepts.

**Montessori Math in the Primary Classroom: Materials and Activities**

In the Montessori Primary classroom, math lessons are presented using a set of educational materials curated by Dr. Maria Montessori. These materials are designed to introduce mathematical concepts to young children using a concrete, hands-on approach. When presented in a thoughtful, intentional sequence, the Montessori Math materials provide a robust and thorough foundation in early mathematics and typically require little supplementation.

Because these materials are central to the Montessori Math curriculum, it’s useful for adults working in a Montessori setting to be familiar with the names of each material and its intended use. In this section, for each core material (or activity) in the Montessori Math curriculum, you’ll find the following:

- a brief description of the material,
- the main purpose or aim of the material, and
- the age at which the material is generally presented to a child.

The materials are grouped by topic/purpose and presented in roughly chronological order. Later lessons build upon the concepts presented in earlier lessons.

Please note – and this is very important – the ages here are only a general guideline! Some children may be ready to grasp a particular math lesson at an earlier age than what’s listed here, and some may need more time to practice (more about this here).

**Topic: Numeration**

*These materials introduce the essential language of mathematics, especially the separate (but related) concepts of quantities and numerals*.

**Number Rods**

- Description: ten wooden rods, varying in length. Each rod is divided into equal sections and each section is painted an alternating color, with the most common pattern being alternating sections of red and blue. The number rods closely resemble the red rods found in the sensorial area – this is intentional. The two materials are intended to serve as a bridge between the sensorial and math areas of the curriculum. (This material also includes a set of wooden cars with numerals 1-10; the child is shown how to pair a number card with its corresponding rod only after the child is familiar with the process of arranging the number rods from shortest to longest.)
- Purpose: to introduce the concept of quantity and begin to associate the words for one through ten with the quantities they describe; to introduce the connection between quantity and symbol.
- Typical Age Introduced: around 3 years old.

**Sandpaper Numerals**

- Description: a set of ten cards depicting the numerals 0-9. Each card shows a single numeral, and each numeral is printed or coated in sandpaper (or another slightly rough material to contrast with the smooth background of the card). The rough texture provides a tactile experience when the child traces the numeral with their finger.
- Purpose: to introduce the numerical symbols associated with the words zero through nine.
- Typical Age Introduced: around 3 years old.

**Spindle Box**

- Description: a long wooden box divided into ten equal compartments. Each compartment is labeled with a single numeral, starting at 0 and ending at 9. Sometimes this material is presented as two shorter boxes instead of one long box, with the first box containing compartments labeled 0-4 and the second box containing compartments labeled 5-9. This material is accompanied by a set of 45 wooden “spindles,” wooden sticks about the size of a pencil.
- Purpose: to connect the concepts of quantity and symbol. The spindle box also helps children begin to understand the concept of zero.
- Typical Age Introduced: around 3-4 years old.

**Cards and Counters**

- Description: ten small cards, each displaying a single numeral from 1-10, and 55 small objects to be used as “counters” (the counters are most commonly wooden “coins” painted red). Some sets of this material include loose wooden numerals instead of number cards.
- Purpose: to reinforce the connection between quantity and symbol. Cards and Counters are also often used to introduce the concept of “odd and even” numbers.
- Typical Age Introduced: around 4 years old.

**Topic: The Decimal System & Operations**

*The Golden Bead Materials allow hands-on exploration of place value and operations.*

**Presentation Tray**

- Description: a wooden tray containing one golden unit bead, one ten bar, one hundred square, and one thousand cube.
- Purpose: to introduce the component materials and vocabulary used in the Golden Bead Material.
- Typical Age Introduced: around 4-5 years old.

**Equivalency/Tray of Nines**

- Description: similar to the Presentation Tray, but with ten single Golden beads, ten Golden Bead bars, ten Golden Bead hundred squares, and one thousand cube.
- Purpose: to introduce the patterns of the decimal system used in the Golden Bead Material.
- Typical Age Introduced: around 4-5 years old.

**Number Construction**

- Description: using a sufficient supply of Golden Bead materials and a set of number cards, the child practices pairing a wooden number card with the quantity of beads that card represents. First, the child practices pairing single cards with their associated quantities (for instance, seven golden unit beads with the “7” number card, four bars of ten with the “40” number card, or three hundred squares with the “300” card). Later, the activity increases in complexity by layering the number cards to create more complex numbers and gathering and combining the component Golden Bead Materials that those cards represent. For instance, the numerals & quantities “3000,” “400,” 20,” and “8” can be combined to form the numeral/quantity “3428.”
- Purpose: to reinforce the patterns of the decimal system in preparation for golden bead operations; to reinforce the connection between quantity and symbol.
- Typical Age Introduced: around 4-5 years old.

**45 Layout**

- Description: a multi-step activity incorporating number cards and Golden Bead Materials. The child arranges the unit cards for 1-9 in numerical order on a large rug and places the correct number of unit beads next to each number. This process then repeats for tens, hundreds, and thousands. This activity is called the “45 Layout” because each “level” (units, tens, hundreds, and thousands) contains 45 of that particular type of item.
- Purpose: to reinforce the connection between quantity and symbol in preparation for Golden Bead operations.
- Typical Age Introduced: around 4-5 years old.

**Exchanging Activity**

- Description: a “game” using Golden Bead Materials. The child sorts a large, mixed quantity of Golden Bead Materials into units, tens, hundreds, and thousands. Once the sorting is complete, the child begins to count the materials starting with the units. Every time the child reaches “ten,” the child returns those ten units to the shelf, exchanging them for a single bar of ten. This process continues until the child is left with fewer than ten unit beads. The child then repeats this procedure with the tens bars and hundred squares until the entire group of materials is processed.
- Purpose: to reinforce the patterns of the decimal system in preparation for Golden Bead operations.
- Typical Age Introduced: around 4-5 years old.

**Intro to Operations**

- Description: a lesson introducing the procedure for using the Golden Bead Materials to work out arithmetic problems, starting with addition.
- Purpose: to reinforce the patterns of the decimal system; to become familiar with the procedure for using Golden Bead Materials to solve math problems.
- Typical Age Introduced: around 4-5 years old.

**Topic: Linear Counting**

*Additional bead materials (and the 100 board) support additional development and refinement of counting skills and pattern recognition.*

**Short Bead Stair**

- Description: a set of 55 colored beads threaded onto ten small metal rods in quantities of 1-10 (we call these rods “Bead Bars”). In the Short Bead Stair, each Bead Bar is a distinct color; this pattern of colors repeats consistently throughout the bead material. When arranged from shortest to longest, the ten Bead Bars create a triangular shape that resembles a set of “stairs.”
- Purpose: to practice accurate counting skills; to prepare for longer counting work with the Short Bead Chains and Long Bead Chains.
- Typical Age Introduced: around 3-4 years old.

**Teens Boards**

- Description: two wooden boards with the numeral “10” printed five times on one board and four times on the other. This material also includes nine small wooden cards each printed with a single numeral 1-9. The Teens Boards use a special set of Bead Bars: nine golden bars of 10, as well as one colored bar for each quantity 1-9.
- When the material is fully assembled, the zero in each 10 is covered by one of the wooden numeral cards, thus forming the numerals 11-19. One golden Bead Bar and one colored Bead Bar are grouped together next to each numeral, demonstrating the associated quantities 11-19.

- Purpose: to learn the vocabulary and numerals for 11-19; to associate numerals 11-19 with their corresponding quantities, to introduce place value concepts.
- Typical Age Introduced: around 4-5 years old.

**Tens Boards**

- Description: two wooden boards with the numerals 10, 20, 30, 40, and 50 printed on one board and 60, 70, 80, and 90 printed on the other. This material includes nine small wooden cards each printed with a single numeral 1-9. The Tens Boards use a special set of Bead Bars: 45 golden bars of 10, as well as one colored bar for each quantity 1-9.
- Purpose: to learn the vocabulary and numerals for 11-99; to associate numerals 11-99 with their corresponding quantities, to introduce place value concepts.
- Typical Age Introduced: around 4-5 years old.

**Short Bead Chains**

- Description: a special set of colored Bead Bars linked together and organized into ten chains, demonstrating quantities from 1-100. The shortest chain contains one bar of one (a single bead), the next longest chain contains two bars of two (four beads), the next chain contains three bars of three (nine beads), and so on, ending with a chain containing ten bars of ten (100 beads). Each chain is accompanied by a set of small numerical labels corresponding with the quantity of beads in the chain.
- Purpose: to practice linear counting skills and provide an early introduction to multiplication and skip counting concepts.
- Typical Age Introduced: around 4-5 years old.

**100 Board**

- Description: a square wooden board with a blank 10×10 grid, accompanied by a set of 100 square wooden tiles with the numerals 1-100.
- Purpose: to reinforce number sequencing up to 100 and to provide additional practice with place value concepts.
- Typical Age Introduced: around 5 years old.

**Long Bead Chains**

- Description: similar to the Short Bead Chains but longer – these chains represent quantities up to 1000. The shortest chain is 1x1x1 (1 bead), the next longest chain is 2x2x2 (8 beads), the next is 3x3x3 (27 beads), and so on, ending with a very long 10x10x10 chain (1000 beads). Each chain is accompanied by a set of small numerical labels corresponding with the quantity of beads in the chain, as well as a cube consisting of the same color and quantity of beads in the chain, providing a concrete demonstration of the mathematical concept of “cubing.”
- Purpose: to practice linear counting skills and provide an early introduction to multiplication and skip counting concepts.
- Typical Age Introduced: around 5 years old.

**Topic: Operations & Memorization**

*The versatile Golden Bead Materials provide hands-on experience with operations in preparation for memorization of basic math facts*.

**The Golden Bead Material**

- Description: a set of loose single Golden Beads as well as Golden Beads arranged in bars of ten, squares of 100, and cubes of 1000, representing units, tens, hundreds, and thousands respectively. (In some cases the hundreds and thousands are represented by wooden squares and cubes printed with golden circles to represent beads.) The Golden Bead Material is typically accompanied by wooden numeral cards with symbols for 1-9, 10-90, 100-900, and 1000-9000.
- Purpose: to provide a concrete understanding of the decimal system and arithmetic operations. Initially, these versatile materials facilitate extensive practice with place value concepts. Later, the Golden Beads help students explore mathematical operations including addition, subtraction, multiplication, and division.
- Typical Age Introduced: around 5-6 years old.

**Topic: Abstraction & Fractions**

*After lots of practice with manipulating quantities and exploring mathematical relationships, the next step is to build fluency with the language of these relationships – in other words, memorizing basic arithmetic facts.*

**Snake Game**

- Description: in this activity, the child builds a long “snake” using colored Bead Bars. The child then “transforms” the colored bead snake into a Golden Bead snake by counting the colored Bead Bars and exchanging sets of these Bead Bars for bars of ten from the Golden Bead Material. A more advanced extension of this material incorporates a special set of black and white colored Bead Bars.
- Purpose: to reinforce the patterns of the decimal system; to reinforce linear counting skills; further exploration of math facts.
- Typical Age Introduced: around 5-6 years old.

**Addition with Bead Bars**

- Description: this activity uses Bead Bars for hands-on addition exercises.
- Purpose: concrete practice with addition; supporting fluency with math facts.
- Typical Age Introduced: around 5-6 years old.

**Addition Strip Board**

- Description: a wooden board specially printed with a grid and numerals 1-18, accompanied by colored wooden “strips” of varying lengths labeled with numerals 1-9.
- Purpose: concrete practice with addition, builds fluency with math facts.
- Typical Age Introduced: around 5-6 years old.

**Multiplication with Bead Bars**

- Description: a large quantity of colored Bead Bars used for hands-on multiplication exercises.
- Purpose: concrete practice with multiplication; builds fluency with math facts.
- Typical Age Introduced: around 6 years old.

**Multiplication Board**

- Description: a wooden board and a special set of colored beads, used together for hands-on multiplication exercises.
- Purpose: concrete practice with multiplication; builds fluency with math facts.
- Typical Age Introduced: around 6 years old.

**Division Board**

- Description: a wooden board and a special set of colored beads, used together for hands-on division exercises.
- Purpose: concrete practice with multiplication; builds fluency with math facts.
- Typical Age Introduced: around 6 years old.

**Finger Charts**

- Description: wooden boards printed with grids and numerals demonstrating basic math facts for addition, multiplication, subtraction, and division.
- Purpose: allows the child to check their own work during hands-on practice; building fluency with math facts.
- Typical Age Introduced: around 5-6 years old.

**Fraction Circles (Introduction to Fractions)**

- Description: a set of ten small circular puzzles, often colored red. The first circle is whole and undivided. The second circle is divided into two equal portions, the third into three equal portions, and so on, with the tenth circle being divided into ten equal portions.
- Purpose: concrete practice with fractions, including the concept of equivalent fractions (i.e. 1/2 is equal to 3/6).
- Typical Age Introduced: around 5-6 years old.

**Extending Montessori Math Beyond the Primary Classroom**

**Supporting Early Math Development at Home**

Montessori math isn’t just about memorizing numbers; it’s about understanding the fundamental principles that underlie mathematical concepts. At home, families can engage their young children in practical, hands-on activities that reinforce these fundamental skills.

Here are some simple ideas families can try outside of school with their children:

- practice counting everyday objects during grocery shopping or while on a walk outdoors
- make comparisons about quantity – for instance, are there more blue cars or green cars in the basket?
- notice out loud when things are “empty” – this vocabulary supports the concept of zero
- work together to count items or measure ingredients while cooking a meal or packing lunch
- sort household items like toys or laundry into distinct groups before putting them away
- discuss sequences or routines using vocabulary like “first” “next” and “last”

Using strategies like these, families can create a math-rich environment that supports their children’s cognitive development in a holistic way.

Looking for more ideas for supporting math development outside of school? Simone Davies of The Montessori Notebook offers some fantastic suggestions here (*when viewing her guide, try searching for the word “mathematics” to locate activities designed to promote math skills*).

**Montessori Math in the Elementary Years and Beyond**

The Montessori Elementary curriculum for math builds upon the strong foundation laid during the primary stage. Montessori Elementary students work on more advanced mathematical concepts using a combination of hands-on materials and more abstract printed or written work. This combined approach provides a tangible way to explore multiplication, division, and more complex mathematical operations.

In Montessori Elementary mathematics, the emphasis begins to shift from concrete to abstract understanding. This shift helps students gain fluency with basic arithmetic facts – an important component of real-world math applications. Montessori Elementary classrooms continue to integrate math across the curriculum. Students at this age may use math skills to calculate the number of supplies needed for a project, measure attributes of specimens in nature, or create graphs, charts, and other visual representations of data.

**Conclusion**

The Montessori approach to mathematics empowers young minds to grasp abstract concepts through a carefully crafted sequence of tangible, hands-on materials and activities. The benefits are profound: a deep understanding of mathematical principles, the development of critical thinking skills, and the cultivation of a lifelong love for learning.

In Montessori, mathematics is a dynamic and integrated part of a child’s holistic education. The Montessori approach to mathematics goes beyond shaping proficient mathematicians; it molds inquisitive, confident, and creative thinkers ready to navigate the complexities of the world with confidence and enthusiasm.

Holly Earnest is a content writer on the Trillium team. After eight years in the classroom and another 2.5 as a center director, Holly transitioned from full-time campus life to focus on supporting Montessori educators and caring for family. She is AMS-credentialed at the primary level, and enjoys coaching Montessori guides, creating Montessori training materials, and presenting at Montessori conferences.