Make Division Easy With Montessori Materials

When most of us think about learning division, we remember staring at a strange-looking symbol and hearing something about splitting pizza. Division is typically not introduced until the middle of elementary school in a traditional classroom. In a Montessori school or Montessori homeschool, division concepts are presented as early as age five, using hands-on materials that make the concept concrete, visual, and genuinely enjoyable.

🧮 What Montessori says about division and concrete-to-abstract progression: Maria Montessori introduced division to young children through physical sharing activities long before any symbolic notation was used. She believed that the concept of division, distributing a quantity equally among groups, is naturally intuitive to children and that the difficulty most students experience comes from being introduced to the symbol before the concept. Her golden bead materials allowed children to physically divide quantities, see remainders, and understand exchange, all through direct manipulation. Research on concrete-to-abstract mathematics instruction (Bruner, 1966, Toward a Theory of Instruction) established that children learn mathematical operations most durably when they move through three stages: concrete (physical manipulation), pictorial (visual representation), and abstract (symbolic notation). A 2016 meta-analysis in Journal of Educational Psychology confirmed that students taught division through concrete manipulative-based instruction showed significantly stronger conceptual understanding and procedural fluency than those taught through symbolic methods alone, with the largest gains for students in grades 1 through 4.

Montessori Makes Math Concepts Easy to Grasp

Division does not have to be hard or intimidating for the homeschool parent to teach. With the Montessori method, division can be presented in a non-threatening, enjoyable way that allows the child to thrive. By using manipulatives and hands-on multisensory learning, we engage the child in multiple ways that reinforce concepts in a child-friendly manner. This approach is especially effective for learners with special needs.

The ShillerLearning decimal material is the starting point for addition, subtraction, multiplication, and division. It is used in counting, sorting, balancing, and much more throughout preschool and elementary education. Because children are already familiar with these materials when division is introduced, new concepts feel far less daunting. The decimal material provides a visual, tactile, and kinesthetic way to learn division. By engaging the auditory system through songs and listening to directions, division becomes a whole-body experience that sticks.

Montessori decimal material for division

What Is Montessori Decimal Material?

Traditionally in a Montessori school, golden bead sets are used for decimal material. While these sets are beautiful, they are also large and quite costly. The ShillerLearning decimal material is a more cost-effective option that takes up less space as well.

The materials are kept in a wooden tray on the shelf. The set includes 1 plastic 1,000 cube, 16 cardboard 1,000 cubes, 27 hundred flats, 27 ten rods, and 100 unit cubes. These materials form the basis for understanding number size and serve as manipulatives for solving equations throughout preschool and elementary education.

See what these materials look like on a homeschool shelf:

Beginning Division With Young Students

The decimal material is introduced very early. First, unit cubes are introduced as children learn numbers 0 through 9. Then ten rods, hundred flats, and thousand cubes are introduced in quantities 0 through 9. Once students have a solid understanding of individual place value components, they work with mixed multi-digit numbers. Materials are then combined to make larger numbers, first through addition and then subtraction. Children discover that addition and subtraction are inverse operations. They then play the exchange game, learning to exchange 10 units for a ten rod, 10 tens for 1 hundred, and 10 hundreds for 1 thousand cube. Once a child has mastered the exchange game, they are ready to add and subtract multi-digit numbers with exchange. By the time multiplication and division are introduced, these materials are already familiar.

A student's first introduction to division comes through building numbers with unit cubes. Here is an example of how that first lesson might look:

🧰 A first division lesson with unit cubes

The parent asks the student to make a number with unit cubes. Using eight as an example, the student takes out eight unit cubes and places them in a row at the top of the work area.
The parent shows the student how to break the group of eight into two even groups of four.
Groups of two are made next.
Then the eight cubes are separated into eight groups of one.
Finally, 32 unit cubes are taken out to demonstrate all the different groupings of eight that are possible.

This is basic division made concrete and accessible. The activity can be repeated with any number that divides evenly.

A second early division lesson begins similarly. A three or four digit number is created with number cards, ideally one that divides easily by two. The student builds that number using the decimal materials. The parent then says, "Now we will divide this number by two," and demonstrates how to divide the materials into two even piles. Together, the number representing each group is determined and made with number cards. The materials can then be recombined and divided further.

Division with Montessori decimal materials

Moving Away From the Decimal Material

Once a student feels comfortable and confident using the decimal materials for basic division, more advanced skills are introduced. Students work on division with remainders, division with exchange, and work through many division problems with the materials.

🎯 The ultimate goal in Montessori is for a student to move on to solving equations without the materials at all. The first transition is division using number tiles only. The process mirrors the process of learning division with the decimal material. Around third grade, students typically begin moving away from decimal materials, though they may use them for as long as they need.

The decimal materials are used in over 60 lessons in Math Kit I for pre-K through 3rd grade. They provide the foundation for mastering 4-digit division and beyond.