Montessori Mathematics - The Glen Montessori School

Why is Montessori Math different?

“This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child’s special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to write out the operation. They can thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations.”

-Dr. Maria Montessori, The Discovery of the Child

Studies have shown that children in Montessori schools often excel academically, particularly in math. A comprehensive meta-analysis of 32 studies found that Montessori education had “strong and clear effects on math, literacy, general academic ability, and executive function.” pmc.ncbi.nlm.nih.gov

At The Glen, Montessori education follows a continuous, individualized approach from birth to 6th grade, allowing students to progress at their own pace. Some children advance quickly, while others benefit from repeated practice to achieve mastery. At The Glen, Montessori math materials support this progression, though not every child uses the same tools, nor does this list cover all available resources.

Sorting, Classifying,
and Identifying Patterns

The Montessori Math materials are carefully designed to introduce each concept in isolation, allowing the student to practice the skill in a manner that combines physical activity with mental tasks.

The work of math begins even before “math” materials are introduced. In our Birth to Three classrooms, our youngest students start Sorting, Classifying, and Identifying Patterns. These early discrimination skills are the foundations of a mathematical mind.

Red and Blue Rods

Children are introduced to counting, the names of numbers, and their sequence. As they progress, they learn that a written symbol, can represent a quantity. They can also compare two quantities and use words like shorter and longer.

To the right, the child is working with the Red and Blue Rods. This is numeration work, assigning a written number to quantities one through ten. How the rods are painted indirectly prepares the child for odd and even numbers. It also demonstrates greater than and less than comparisons; because of how the rods line up against one another, the quantity is easily visible.

Spindle Boxes

As children become more skilled and hone their abilities, the Montessori math materials become increasingly challenging and require the refinement of fine motor skills.

The student pictured here is working with a counting work, the Spindle Boxes. These boxes have spaces labeled for spindles from 0 to 9. The child must count the correct number of spindles in each space. The work, like many Montessori works, is self-correcting. There are only the exact number of spindles, so if a mistake is made, the child will often find it themselves.

Golden Bead One Tray

Children love big numbers. Before introducing those numbers, we must help students comprehend the relationship between units, tens, hundreds, and thousands - and learn that vocabulary.

This child is working with the Golden Bead One Tray to learn the names of the place values and observe the quantity differences.

Teen Boards

As children are exploring different place values, they are also counting beyond ten. The hands-on, concrete materials require significant concentration and focus for practical use.

To the right, the student uses the Teen Boards to create teen numbers using a ten-bead bar and then the appropriate number bead bar. She is also building the number on the board with number tiles.

Hundred Board

Focus and independence are two hallmarks of a Montessori education. They can be observed when a child is working for a sustained time on challenging work. Errors can be noticed, and the child can self-correct when a control is present. This encourages independence and allows the student to set their own pace. This process allows the child to refine their problem-solving skills as well.

The child pictured is completing the Hundred Board. This work counts the numbers from one to a hundred and is organized in tens. The counting is accomplished by placing wooden tiles on the board. There are no extra tiles, so the child can see if something doesn’t seem correct - and revise their work.

Long Chain

Once children have mastered counting, Montessori introduces skip counting. This lesson uses the same bead chains children have used to count linearly since their days in Children’s House. Now, those beads are counted swiftly and establish the basis of multiplication, squaring, and cubing. The benefit of this approach is that the student has practice and experience to draw upon rather than relying solely on memorization.

The student pictured is recording their work from the Long Chain of Six. This hands-on material helps students reinforce counting and skip counting. It is self-correcting (it has tags placed after every set of six beads) and, when folded correctly, becomes the cube of six, so students have a concrete understanding of what six three ways look like.

Golden Bead Material

Montessori place values have assigned colors. Units are green, tens are blue, hundreds are red, thousands are green, and so on. This color system aids children in keeping their work organized, recording tallies, and saying the numbers correctly. Materials like the Golden Beads provide a concrete understanding of the decimal system and can be used for the basic mathematical operations: addition, multiplication, subtraction, and division.

This student is pictured working with Golden Bead Material. This material shows the quantity concretely, providing a visual and sensorial experience to aid in understanding the concept. He has his problem laid out on a place value mat and combines the addends below the line at the bottom to find the sum. He will exchange ten of those in his bank if he discovers he has more than nine in any of the places on his mat.

Stamp Game

As the child’s knowledge grows, the visual and tactile differences in the materials reduce, as the student no longer requires the same types of support. The materials also change in size - requiring a greater degree of dexterity, fine motor control, and concentration to manipulate them successfully.

The student to the right is using the Stamp Game for multiplication. She has moved her multiplicands into sets; the multiplier determines the number of sets she needs. Once each set is laid out, she will combine those to find the product.

Large Bead Frame

The Montessori math materials become less and less concrete as they progress, encouraging more abstract calculations and less physical demand from the student.

The Large Bead Frame that the student uses is a Montessori math material that bridges the concrete and abstract for many students. The beads are moved from left to right and top to bottom, with the very top line of beads representing the units. This material can be used for addition, multiplication, and subtraction operations.

Checkerboard

Montessori math materials also include materials specifically designed to aid in the practice of complex operations with large numbers.

The Checkerboard is a Montessori math material specifically designed to assist children in practicing multiplication with numbers up to 100,000,000. It is color-coded in the familiar green, blue, and red colors, helping the students stay organized and compute the numbers in their equations.

Test Tubes

Manipulating numbers through Montessori math materials continues with increasingly more detailed materials as the students progress. As the fine motor, focus, and mental demands of Montessori math materials grow, it becomes clear that the goal is not just for the child to increase their skill with the material but for the child to outgrow the material and transition to abstraction.

The child pictured uses the Test Tube Division material also called Racks and Tubes. This material requires students to place their divisors on wooden boards with divots and then divide their dividends--the beads in the cups--between them equally. This material helps to strengthen the student’s understanding of the division process.

Algebraic Peg Board

Dr. Montessori understood that giving children the chance to develop skills with materials that support their understanding during a period in their development when their curiosity is at its peak is the foundation of a love of learning. The hands-on experience that the Montessori materials provide is a new way to problem-solve and think critically.

Here, a student is building squares using the Algebraic Peg Board. This work strengthens the student's knowledge of multiplication, provides a visual understanding of the binomial, and prepares for future work of extracting square roots.

“Montessori often used to remark ‘present the child with materialized abstraction.’ By this she meant, ‘present the idea in a material or concrete form’ and always combined with an activity. As the child works for a long time with this material day after day…at his own pace, unperturbed, unhurried, gradually there comes off from the materials the very essence of the operation…’ Always the child works the operation…’ Always the child works the operation first in the concrete until the very essence of the rule becomes absolutely clear to him.”

-E. M. Standing, Maria Montessori: Her Life and Work