How Montessori Makes Long Division Make Sense
March 23, 2026

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 Miami Beach, Florida, to see for yourself!


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As parents, we quickly learn that young children hold ultimate control over three main areas of daily life: eating, sleeping, and toileting. When we lean into power struggles over these routines, it can feel exhausting for everyone involved. Instead, the Montessori approach invites us to step back and help children develop the intrinsic skills and body awareness they need to achieve true mastery over these essential functions. In our infant and toddler communities at The Montessori Academy at St. John’s, children naturally learn to prepare snack, rest when their bodies are tired, and dress themselves. Alongside these milestones, they are also developing the neurological and physical control needed to stay dry. We intentionally refer to this journey as "toilet learning" or "toilet awareness," rather than "toilet training." After all, we aren’t training our children through external conditioning. Rather, we are walking alongside them as they discover the joy of physical independence and become masters of their own bodies. While this process takes time, it does not have to feel daunting. Here is how we approach toilet learning at MASJ, and how you can mirror these practices at home. 1. Establish Body Awareness from the Beginning Toilet awareness does not start at two or three years old—it begins at birth. From the earliest days of life, we can speak respectfully to our infants about their bodies during diaper changes. Saying, "Your diaper is wet from your urine," or "I am wiping your bottom," helps them connect words to their physical sensations. To support this awareness, we highly recommend using cloth diapers or moving into cotton underwear as they enter toddlerhood. Modern disposable diapers are remarkably effective at wicking away moisture. While convenient, they often prevent a child from feeling the actual sensation of being wet, which delays the brain-to-body feedback loop. Phrases to try: "Does your diaper feel wet or dry right now?" "Your underpants feel heavy. You had a lot of pee come out." "I see you are squatting down and concentrating. Your body is working on a poop." 2. Form a True Collaboration When a child is wet or soiled, we view the diaper change or bathroom visit as an opportunity for connection, not an interruption to the day. We invite the child to actively collaborate in the process. Even young toddlers can assist by pulling down their own pants, grabbing a clean pair of underwear, or placing their soiled clothes into the laundry basket. This active participation satisfies their developmental need for functional assignment and autonomy, reducing the urge to resist. Phrases to try: "You can hold your shirt up while I help you pull down your pants." "You open this side of the diaper tabs, and I’ll open this side." "I will take a turn to wipe, and then it will be your turn." 3. State the Facts Safely and Plainly In our MASJ classrooms, our Guides stay entirely matter-of-fact throughout the toileting routine. Elimination is a completely natural bodily function, and we treat it with calm neutrality. If a child experiences an accident and wets the floor, we avoid expressions of disappointment. Similarly, when they successfully use the toilet, we avoid excessive clapping or rewards. Over-celebrating can accidentally introduce performance anxiety or make the child dependent on external praise. Instead, we acknowledge the direct reality. Phrases to try: "Pee and poop go into the toilet." "Your underpants are wet. Let’s change into dry ones. Do you remember where your clean clothes are kept?" "I see pee on the floor. Please bring me the clean-up cloth from the bathroom so we can dry the puddle together." 4. Keep the Attitude Light and Friendly Children are incredibly sensitive to adult emotions and can easily internalize our underlying stress or impatience. Maintaining a light, relaxed, and encouraging attitude ensures that children feel safe and comfortable with their normal bodily processes. We want to entirely eliminate any sense of shame, frustration, or disgust around toilet learning. Phrases to try: "We can always change into dry clothes, it's no big deal!" "Everyone used to wear diapers when they were small—even Mommy and Daddy. Now we use the toilet, and you will learn to use it too." 5. Step Away from Tricks or Treats At MASJ, we offer the toilet based on the consistent rhythms of our daily routine, but we never force a child to sit. We also tend to avoid asking open-ended questions like, "Do you want to use the toilet?" as a toddler's natural default answer is often a resounding "No!" Instead, find the comfortable middle ground by clearly stating that it is time to visit the bathroom as part of a transition (e.g., before going outside, or right after waking up). We intentionally avoid sticker charts, treats, or bribes. While these tactics can produce short-term compliance, they interfere with the long-term goal of developing genuine self-assurance and internal motivation. Trust the process and trust your child's natural drive toward mastery. Phrases to try: "It is time to walk to the bathroom and try sitting on the toilet." "You listened to your body and peed in the toilet. You did it all by yourself." Respecting the Journey When caregiving routines are rushed, children miss the opportunity to understand their own physical identity. Taking the time to slow down, communicate clearly, and invite collaboration shows the utmost respect for your child's developing personality.  By guiding your toddler through toilet learning with patience and consistency, you are helping them take a monumental step toward becoming a confident, fully independent young person. If you are currently navigating this milestone at home, remember that our AMI-trained Guides are always here to support your family. Feel free to reach out to discuss how we can beautifully bridge the routines of our school environments with your home.