Montessori Basics: Math Activities Through the Levels

Your four-year-old loves their classroom and their work. Their teachers are guiding them to learn early math skills. But what, exactly, does that look like? And how does it change as they get older? Montessori math materials are nothing short of amazing. While they look quite different than what most folks used growing up (pencil and paper?) Montessori’s approach to math is very intentional, developmentally appropriate, and truly genius!

The Basics

Much of the Montessori curriculum is based on giving children exposure to concrete materials first, then giving them incremental opportunities to work toward more abstract concepts. This is no different when it comes to math.

What do we mean by concrete? The children are able to hold a physical material in their hands. The materials are symbolic or representative of something else (a number or quantity) and that symbolism changes over time until children are ready to let go of the materials and find solutions on paper or even in their heads. This idea of mastering a skill without the assistance of materials is what we refer to as passage to abstraction.

What Does Primary (Children’s House) Math Look Like?

At the primary level math starts out simple, but preschoolers are capable of so much more than is generally expected.

Even before a child is able to count, they experience this skill using materials like the Number Rods, a series of blue and red colored wooden rods that are arranged in a stair-like pattern. Children learn how to associate quantity with the correct symbol using a variety of materials. Sandpaper Numbers (just like their Sandpaper Letter counterparts) teach children how to correctly form each number to develop readiness for writing them on paper. The Spindle Box is an early material with which children place the correct number of wooden spindles (think smooth, long sticks) intoseparate compartments labeled 1-9. There are also activities to help a child practice linear counting (past 10): Teens Board and Beads, Tens Board and Beads and the Short and Long Chains.

Once a child has mastered quantities 1 – 10 and recognizing the numerals they are ready to learn about basic math operations, there are plenty of materials to support them. Montessori math uses the Golden Bead Material to build numbers into the thousands. For example, a single golden bead represents 1 unit, a group of 10 beads are strung together in a straight line for a bar of 10, and 100 beads are affixed into a flat square equaling a square of 100. The thousand cube is made of 1,000 of the original single ‘1’ unit beads. A child will work through the stages of activities with the Golden Bead Material for 2 – 3 years. Once a child can build a visual representation of a number, the beads are used to teach basic mathematical operations. Young children are able to add, subtract, multiply, and divide numbers into the thousands using this material. They first learn with static problems - that is, with no exchanges - and then move on to more complex, dynamic problems with “changing” or carrying over. They quickly learn that ten 1 units is equal to one 10 bar, ten 10 bars are equal to one 100 square and so on. Children are able to do this by physically counting and manipulating those quantities in their hands with the Golden Bead Material.

Dr. Montessori recognized the importance of memorizing basic facts. Traditional educational methods may use flashcards to memorize the math facts, the Montessori approach begins by showing children why we manipulate numbers in different ways. Young children appreciate the repetitive nature of the materials, which gives them plenty of opportunities to practice (and memorize!) these facts. The Addition and Subtraction Strip Boards show a child visually what is happening when we add numbers. The same goes for the Multiplication and Division Bead Boards. There are also activities called the Addition Snake Game, Subtraction Snake Game, and the Dot Game, which aid the child in memorizing the math facts in a fun way which leads to lots of repetition and practice.

A Period of Overlap

Somewhere between kindergarten and the first year of lower elementary, children are taught to use a new math material, The Stamp Game, depending upon their individual readiness. The Stamp Game is an example and one of the most used Montessori materials.

The Stamp Game is a sectioned box with small colored tiles sorted inside. The tiles are color coded and labeled: green, ‘1’ tiles, blue ‘10’ tiles, red ‘100’ tiles, and green ‘1,000’ tiles. Now instead of holding a large cube of beads that shows the relative size of one thousand as they did with the golden beads, quantities are now represented by a series of tiles that are all the same size but are differentiated only by their color and number label. Like the Golden Beads, the Stamp Game material is used to teach all four operations: adding, subtracting, multiplying, and dividing into the thousands. Some children begin this work in Children’s House and continue when they reach elementary, while others begin once they enter their lower elementary classroom.

It is interesting to note that there are some Montessori materials that children spiral back to, repeatedly, from ages 3 to 12! The Bead Chains are a colorful, quintessential Montessori material. In the Children’s House, children use the chains to learn how to count linearly and to skip count. In a lower elementary classroom, they are used for skip counting and to help memorize multiplication facts. In upper elementary children use them to solidify the concepts of squaring and cubing numbers, although they were indirectly preparing for that work for years previously in Children’s House!

What Does Elementary Math Look Like?

Remember the green, blue, and red tiles of the stamp game? Montessori refers to those as the hierarchical colors, and they are used to teach children about number series in the decimal system. These colors first appear in the Stamp Game, but they continue to follow the child through lower elementary and into upper elementary until they have a firm grasp on the idea of the simple family of numbers (ones, tens, hundreds), the thousand family (thousand, ten thousand, hundred thousand), and so on.

After a child masters operations with the Stamp Game, they move on to use a material called the Bead Frame, which can be used for addition, subtraction, and multiplication. This is a step closer to abstraction from the Stamp Game. It looks a bit like an abacus, but with ten beads on each rod in the hierarchical colors. After a child masters the Bead Frame, they are typically ready to add and subtract into the thousands (and beyond!) using just pencil and paper.

To learn larger multiplication problems, children use a material called The Checkerboard. They begin small, but eventually work their way up to problems that have three- or four-digit multipliers. For long division, children use a material called The Racks and Tubes. Once children master The Checkerboard and Racks and Tubes, they can multiply and divide large numbers without materials.

During the elementary years math fact memorization continues. In early lower elementary, many children continue to use The Addition & Subtraction Strip Boards and Bead Boards that they used in Children’s House, but eventually move on to using finger boards and tables in which they place numbered tiles. Children notice the patterns numbers make, giving them more tools to memorize their facts.

There’s More!

Of course, math isn’t just about operations. Montessori students learn about geometry and fractions from an early age.

Primary children (3 – 6 years) learn the names of Geometric Solids? They can easily identify not just cubes and spheres, but square based pyramids, rectangular prisms, ellipsoids, and more. As they move into elementary, they learn a range of concepts, including studies of angles, triangles, polygons, and so much more. A Montessori third grader can easily identify a right-angled, isosceles triangle.

For learning fractions, they start out simply with an impressionistic lesson involving an apple and a definition of fractions including how they must always be fairly divided (the connections between fractions and division are impressed early on). They next move on to using Fraction Insets, which look a lot like the Metal Insets they used for handwriting preparation in their Children’s House classrooms. Before you know it, they are learning to multiply and divide fractions.

The Montessori math materials, or curriculum, are some of the most utilized and loved materials in our school. The approach Dr. Montessori took when creating these materials was truly incredible. She created a sequence that is developmentally logical, comprehensive and materials that are interesting to children. If you’d like to know more, please reach out to your child’s Guide, we love to talk about our math materials!