Montessori math provides concrete, hands-on lessons to teach abstract concepts

Friday, May 19, 2017 10:24 PM
Randi Foster, Durango Montessori School

Photo credit: Thru The Lens Photography

“Montessori math” seems to have become a buzzword around Durango when it comes to elementary education. But what exactly is it?

Beyond the enticing, colorful wooden materials, it is an approach based on how children naturally learn best. Developed by Dr. Maria Montessori, an Italian scientist, it takes into account human development, emphasizes understanding over memorization and introduces concepts in concrete ways with hands-on teaching methods.

Montessori graduated from the University of Rome medical school as the first Italian female doctor in 1896. She spent most of her career developing teaching methods based on her careful observations of how children naturally learn. In an era where children were supposed to be “seen and not heard,” Montessori’s philosophy of supporting child development was groundbreaking.

Her philosophy was to tailor education according to each stage of a child’s development. Even her math teaching methods and materials reflect the needs of the elementary child.

To address physical needs, the shelves are all low enough for materials to be within reach, and the materials increase in difficulty in terms of fine motor skills. To illustrate, the “pink tower” requires stacking large cubes on top of each other while the division test tubes require placing tiny beads in skinny glass tubes. The math materials are designed to fulfill elementary students’ social needs: Tshe “multiplication bead frame” is taught so that one student manipulates the material while the other records the steps.

While learning division, students get to tap into their emotional need for justice by ensuring that each group gets an even number. Cognitively, elementary-age children need to learn concepts in a concrete way first because they have not yet developed strong abstract thinking skills. While adults can read or listen to an explanation of how something works, elementary children have to actually do it in order to develop a true understanding.

Montessori materials are designed so that students begin with concrete, hands-on work that allows them to develop a thorough, abstract understanding. Most of us learned long division in grade school, but how many of us – to this day – actually know how and why it works? Montessori students begin learning the concept of division around age 4: They understand it as equal sharing among people.

In elementary school, materials such as the “stamp game” allow them to practice it with larger numbers. For example, for the problem 3,429 divided by 3, they would first set up three “skittles” (wooden pieces that look like people). Then, they would share the three thousands among the skittles by giving each of them a wooden tile with “1,000” written on it. Next, each skittle would receive a 100 tile. However, there would be one 100 tile left over. That tile would be exchanged for 10 tiles each worth 10, and each skittle would get four 10 tiles. Finally, each skittle would get four 1 tiles. In order to find the answer, the child would answer the question, “What does one skittle get?” One 1,000 piece, one 100 piece, four 10 pieces, and three 1 pieces make 1,143.

Like all Montessori materials, this process gives children a concrete, hands-on understanding of how division works. Once children have mastered the process of long division with the stamp game, the teacher shows them how to record each step on paper as they do it. The on-paper steps are exactly the same as the traditional long division you and I learned as children, but these students have the physical representation of what it means to go with it.

After plenty of practice, the magical moment is when the child says, “Why do I need the stamp game anymore? I can just do it on paper!” This indicates that he or she has moved independently from concrete to abstract in terms of understanding long division. Because Montessori students construct their own knowledge of how it works and move to paper-and-pencil math on their own, they each gain a deep, conceptual understanding rather than memorizing the steps at the surface level.

As a teacher at Durango Montessori School, I have witnessed my students’ continued curiosity and eagerness to learn math that was presented in a hands-on, understandable way.

Effective Montessori math programs are expensive, difficult to implement, require immense teacher training and are tough to integrate into non-Montessori school settings. However, they are worth the deep understanding that children gain from Montessori’s hands-on, developmentally-appropriate learning methods.

Randi Foster is a fourth and fifth grade teacher at Durango Montessori School. Reach her at