How Montessori Makes Long Division Make Sense

LakeCreek Montessori School • February 12, 2026

Montessori children experience long division in a concrete and meaningful way. This post shares how hands-on materials help children understand place value and build confidence with complex math.

For many of us, we remember learning long division as a confusing sequence of steps to memorize and repeat (bring down, divide, multiply, subtract), often without a real sense of why it works. In Montessori classrooms, long division unfolds very differently.


Through the Racks and Tubes material, children get to experience what division actually is.


Two Ways to Divide: Sharing and Grouping


Before introducing the material, we first clarify an important idea: there are two different kinds of division problems in real life.


One asks, “If I share this equally, how much does each person get?” This is partitive division, or division by sharing.


The other asks, “If I make groups of a certain size, how many groups can I make?” This is measurement division, or division by grouping.


The Racks and Tubes material focuses on partitive division. Children physically share quantities equally and discover what one share receives. Materials like the Stamp Game emphasize division of measurement. Together, these approaches give children a complete understanding of division and help them choose the strategy that best fits a given problem.


What Are Racks and Tubes?


At first glance, the material is impressive and a little mysterious. Children are often drawn to the material, both for its beauty and its seeming complexity. 


Racks hold test tubes filled with beads, carefully color-coded by place value: units, tens, hundreds, thousands, all the way up to millions. Matching cups hold the dividend (the number being divided). Boards and skittles represent the divisor (the number doing the dividing).


Every detail of the material reinforces place value. Each time children need to make an exchange, they trade in one bead of one category for ten of the next category (e.g. one hundred becomes ten 10’s). This process is visible and incredibly concrete.


This material takes intentional focus. It takes time. And it makes the steps of long division clear.


How Long Division Becomes Concrete


When children solve a division problem with Racks and Tubes, they follow a logical, embodied process:


  1. They build the dividend using the racks and cups.
  2. They represent the divisor with individual figures on boards.
  3. They share beads one at a time, equally, to each part of the divisor.
  4. They stop when sharing is no longer possible and then see what remains from that category.
  5. They then bring down the next category of beads to continue the sharing process. 


Each step answers a real question:


  • What does one unit get?
  • What happens when we run out?
  • What do we do with what’s left?


Instead of being told “bring down the next digit,” children literally bring down the next category of beads. When exchanges are needed, they perform them physically by trading beads. Remainders are not mysterious leftovers. They are beads still sitting in the cup.


Long division becomes a story children can follow.


From Material to Abstraction


One of the most beautiful aspects of this work is how naturally it leads into abstraction.


At first, children record only the quotient. Later, they begin recording intermediate remainders. Eventually, they discover that multiplying the quotient by the divisor tells them how much has been used at each step. This is the very heart of the traditional algorithm.


We don’t give abstract shortcuts. Instead we help children discover the pattern. This allows them to own the process.


By the time children are working abstractly on paper, the algorithm already makes sense. It matches what their hands have done again and again.


Why This Matters


The Racks and Tubes material does more than teach division. It teaches:


  • Deep place value understanding
  • Logical sequencing
  • Patience and precision
  • Trust in one’s own reasoning


Most importantly, it gives children confidence. Division is no longer something done to them. Instead, they can think through the process, step by step, with meaning and understanding. 


In Montessori, math is not about getting the answer quickly. It’s about building an understanding of why the process and answer makes sense. And with Racks and Tubes, long division finally does!


Schedule a visit to our classrooms in Austin, Texas  to see for yourself!


By LakeCreek Montessori School April 1, 2026
When children struggle, Montessori asks: what's in the way? Explore how the prepared environment helps children find their way back to themselves.
April 1, 2026
Discover why Montessori teaches cursive first — and how neuroscience is confirming what Dr. Montessori observed about children's hands and brains.
By LakeCreek Montessori School April 1, 2026
More Than a Chart on the Wall: How Montessori Timelines Build History, Imagination, and Character
By LakeCreek Montessori School April 1, 2026
Discover how Montessori education nurtures children's deepest human needs — from exploration and meaningful work to belonging and spiritual growth.
By LakeCreek Montessori School March 6, 2026
Discover how peer learning, meaningful context, adult interaction, and order align Montessori with the science of how children learn best.
By LakeCreek Montessori School March 6, 2026
Montessori education has been in existence for over a century, but does it actually work? Dr. Angeline Stoll Lillard spent years researching this question, and her book, Montessori: The Science Behind the Genius, is a must-read. In her book, Dr. Lillard identifies eight principles at the heart of Montessori education. What’s key is that these Montessori principles align with what developmental science tells us about how humans actually learn. The remarkable thing is that Dr. Maria Montessori arrived at most of these insights through careful observation of children, decades before the research existed to corroborate how children learn. In this two-part blog post, we’ll examine these eight principles and the connected research. PRINCIPLE ONE: Movement and Learning Are Deeply Entwined In most traditional classrooms, children are still expected to sit still, as if stillness is a prerequisite for learning. In Montessori, we understand how movement and thinking are intertwined. And research backs this up. Studies have found that physical activity improves cognition, judgment, memory, and social reasoning. Moving the body isn't a break from learning. Rather, the movement is often the learning (and this is even more so for younger children!). Montessori materials are designed to be touched, carried, sorted, and manipulated. Children working with the knobbed cylinder blocks are actively perceiving, making judgments, and reasoning through their hands. The same is true when children sort fabric squares by texture, shake and compare sound cylinders, or lay out bead bars to represent quantities. Every material helps children integrate their minds and bodies. Practical life activities take this even further. When children learn to pour, button, fold, or prepare food, they are engaging in organized sequences of purposeful action that develop concentration and executive function skills. What the Research Shows A Milwaukee study found that high school students who had previously attended Montessori programs significantly outperformed peers on math and science assessments, subjects that rely heavily on the kind of reasoning that, in Montessori, is first built through hands-on materials. PRINCIPLE TWO: Choice Improves Both Learning and Well-Being The freedom to choose is at the heart of Montessori education, but this isn’t just about enjoyment. Having choice measurably affects how well children learn and how they feel about themselves. In a striking series of studies, children aged seven to nine were given anagram puzzles to solve. Those who chose their own category of puzzle solved twice as many as children whose category had been chosen for them, even though the actual puzzles were identical. Those who had a choice also spent far more time voluntarily working on puzzles during free time. The key finding is that the perception of control (even in small things) activates a fundamentally different relationship to the work. Children who feel in control tend to engage more deeply, persist longer, and take more ownership of their learning. In a Montessori classroom, children choose their own work throughout the day. Importantly, Dr. Lillard notes that this freedom is always paired with responsibility, and that too many choices can be as demotivating as none. The Montessori environment offers meaningful, bounded choice. Rather than an overwhelming array, each classroom has a selection of purposeful materials designed to match children’s developmental readiness. Choice and concentration are closely connected, too. When children choose work that genuinely engages them, they're far more likely to reach a deep state of focus, or what psychologists call a “flow state.” PRINCIPLE THREE: Children Learn Best When They're Genuinely Interested This sounds obvious, of course! It makes sense that we learn better when we are interested. However, think about this in terms of how classrooms are typically structured. If interest is one of the most powerful drivers of learning, then organizing a school day around a single curriculum delivered to the whole class at once works against almost every child in the room. Dr. Montessori understood children's interests as biological signals pointing toward what their developing minds most need to engage with at that moment in their lives. These windows of opportunity, or "sensitive periods,” are particular stretches of development during which children are uniquely primed to absorb certain kinds of learning. During these windows, learning that matches the child's inner readiness can be extraordinarily effortless and lasting. The role of interest is why Montessori materials are designed to be beautiful, engaging, and self-correcting. The sensorial materials, for example, aren't only teaching discrimination of size or color. They are designed to help children become more interested in noticing the world around them. The adult’s role is to observe carefully and offer new lessons at the moment a child's interest is most alive. PRINCIPLE FOUR: Rewards Undermine the Motivation They're Meant to Build Offering children external rewards (e.g., stickers, prizes, praise for being smart) for activities they already enjoy reliably reduces their intrinsic motivation to do those things later. What the Research Shows Researchers identified preschoolers who loved drawing with markers. They then told one group they would receive a "Good Player Award" for drawing (a fancy certificate with a gold star). Weeks later, the children who had expected the reward used the markers far less than they had before, and half as much as children who had never been offered a reward at all. Expecting a reward had turned something they loved into something they did for a prize. And when the prize was gone, so was much of the pleasure. Rewards like sticker charts, gold stars, and even grades and honor rolls, shift children’s relationship to learning from "I do this because it interests me" to "I do this to get the reward." When the reward is taken away, children’s inner drive has often already weakened. In Montessori classrooms, feedback comes through the work itself, which includes many self-correcting materials, so children discover their own errors without external judgment. The goal is to keep children's relationship to learning intrinsic, personal, and durable. This doesn't mean feedback is absent, though! What matters is the kind of feedback. Research by psychologist Carol Dweck found that praising children for effort (e.g., "you worked really hard on that”) produces dramatically better outcomes than praising ability (e.g., “you’re so smart”). Children praised for effort choose harder challenges, persist longer after failure, and actually improve their performance over time. Children praised for their intelligence begin avoiding challenges, fearing that failure will expose them as not as smart as they were told they were. In our following blog post, we’ll look at the next four Montessori principles outlined in Dr. Angeline Stoll Lillard’s book, Montessori: The Science Behind the Genius: Children Learn Powerfully from Each Other Meaningful Context Makes Learning Richer and More Lasting How Adults Interact with Children Shapes Everything Order in the Environment Supports Order in the Mind In the meantime, schedule a tour here in North Austin to see the principles in action! And let us know if you would like to borrow a copy of Montessori: The Science Behind the Genius by Dr. Angeline Stoll Lillard. It is one of the most research-based books on Montessori education, and we recommend it to anyone who wants to understand the deeper logic of Montessori!
By LakeCreek Montessori School March 6, 2026
When we lose our cool, repair matters most. Explore accountability, curiosity, and connection to break reactive cycles and parent with intention.
By LakeCreek Montessori School March 6, 2026
Explore the Montessori three-period lesson and how its quiet simplicity unites words and meaning during a child’s sensitive period for language.
By LakeCreek Montessori School February 13, 2026
Explore a curated list of children’s books about water, rivers, and watersheds. These stories invite curiosity, care for the planet, and meaningful reading at home.
By LakeCreek Montessori School February 12, 2026
Blog description: A child’s early learning is shaped by the presence of the adults around them. This post explores how mindful movement, language, and boundaries support security and independence in the early years.
Show More