Mathematical Foundations
February 24, 2025

Whether we embrace math or not, humans have a tendency to think in patterns, quantify, and make logical connections. Some even say we have a mathematical mind! Young children also have this innate ability and inclination, so in Montessori, we take advantage of this propensity for mathematical thinking and offer learning experiences that provide an in-depth understanding of math even at a young age. 


Organized into six general groups, the mathematical exercises in the Children’s House span from a foundational understanding of the numbers one to ten, to working with very large numbers and place value within our decimal system, to internalizing number facts, and even early work with fractions!


Numbers 1 to 10


Although learning the numbers 1 to 10 sounds simple, it actually involves the integration of several distinct concepts. The beauty of the Montessori materials is that they isolate each separate concept so that children’s understanding grows in progressive steps. This first group of exercises in the Children’s House aims to teach the names, symbols, quantities, and sequence for the numbers 1 through 10. 


The youngest children first use the number rods, which are color-coded with alternating red and blue sections to designate the quantities of one through ten. As children work with the number rods, they solidify the concept of quantity as a single entity while also practicing one-to-one correspondence.


The sandpaper numbers help children learn the symbolic representation of numbers. After children master the sandpaper numbers and can easily identify numerals, we help them connect the number cards with the corresponding number rods. Additional materials–like the spindle boxes, cards and counters, and the memory game–help children progress from understanding quantity as a single entity to grasping how quantity can be a set.


Throughout this progression, we offer experiential exercises to help children learn the concepts of addition, subtraction, multiplication, and division. At this stage, we don’t provide the vocabulary but rather give an impression of the concept of each operation. 


The Decimal System


Once children have mastered the numbers 1 to 10 activities, they are ready to begin exercises with the golden beads. Because there are never more than nine in a given place value, and children already know the quantities and the symbols, they are ready to learn the hierarchy of units, tens, hundreds, and thousands. 


As children set up the golden beads, the primary decimal system materials, they learn how to think through the process of each operation and the concepts of calculation. Are they putting quantities together or taking them apart and recounting the categories? 


The decimal system work gives children knowledge of the mathematical processes and solidifies their understanding of the mechanics of the decimal system. We focus on helping children understand the process of each operation. We aren’t looking for the right answer. We want the child to understand the operations and what adding, multiplying, dividing, or subtracting means. 


After children understand the operations with the golden beads, we offer other materials, such as the stamp game and dot game, to help them move away from a very concrete toward a more symbolic representation of the quantities. We offer word problems at the end of this series, which help verify whether children understand each operation's process.


Continuation of Counting 


Through this group of exercises, children learn the name, symbol, and quantity of the numbers from 11 to 100, and consolidate linear counting up to 1,000. The work starts with the teen boards (11 through 19) and tens boards (20 to 100). We also introduce the colored bead stair, which gives a color-coded sensorial representation for quantities of one through ten, a pattern replicated throughout the Montessori math materials. 


With the golden beads, children experience the difference between 100 and 1,000, by weight and geometrical representation. In continuation of counting, children lay out the bead chains and see the dramatic linear difference between 100 and 1,000. Children also use the bead chains to practice skip counting, which lays the foundation for further studies in multiplication. 


We also make sure there is lots of counting in the classroom! Children can be taught to count by rote, but mental maturity helps them take the leap from rote counting to understanding quantities represented by numbers. 


Exploration and Memorization of the Tables 


Through a variety of materials–the addition strip board, addition snake game, subtraction strip board, subtraction snake game, multiplication bead board, bead bars, and unit division board–children have lots and lots of practice with the essential math facts. This concrete exploration establishes a visual understanding of what something like 5 x 6 looks like and helps children establish early neural pathways for memorizing these math facts. 


After children use different forms of bead material to practice, they move into more abstract representations of quantity and eventually even use blank charts to test their memory. 


All of the repetition in this exploration phase leads to memorization as children become aware of which combinations they know and which they don’t know. Eventually, after a lot of practice and repetition, children realize that it is faster to do the calculations mentally rather than with the materials.


As children reach the end of their primary years, they want to be more efficient. At precisely this point, they are ready to begin the last group of exercises, called the passage to abstraction. 


Passage to Abstraction


In the exercises for passage to abstraction, children connect the skills and concepts from the previous strands of learning, and their knowledge base shifts from understanding the concepts experientially to understanding them logically. 


With the small bead frame, children revisit the process of addition and subtraction. With the wooden hierarchy material, children get a sensorial and symbolic introduction to the quantities ten thousand, one hundred thousand, and one million. This material also prepares children for the large bead frame and experiencing the multiplication of large numbers with a single-digit multiplier. The racks and tubes material allows children to begin dividing with larger divisors and dividends. With all of this concrete experience working with larger quantities, children are well on their path to abstraction. 


Fractions


The final material in the mathematical progression is fractions. In the Children’s House, we have already helped the youngest children explore fractions sensorially. They return to this material from a mathematical perspective to understand quantities smaller than one unit. In addition, children begin simple operations with the fraction material and start to explore the equivalence of fractions. 


In the Children’s House, the progression we use for all of these exercises helps children deeply solidify their mathematical understanding. We first ensure children have strong concrete experiences, followed by more symbolic representations. After children associate the concrete and the abstract, we offer lots of activities for practice, repetition, and verification.


In the process, we support children’s growth in classification, comparison, and reasoning. We invite you to come see children’s mathematical minds in action. Schedule a tour today!



October 27, 2025
Since the pandemic, we can find ourselves carrying new or heightened worries about our children’s social lives. We can have concerns about whether our children “lost time” with peers during critical developmental years. Or maybe it’s worry about shyness, reluctance in groups, or how screen time may have replaced face-to-face play. Even now, long after schools and activities have reopened, it’s common for anxiety to surface: Is my child making enough friends? Are they socially behind? Why We Feel This Way The truth is, these concerns are natural. COVID disrupted our social connections, and children being with other children is an important part of childhood. For months (and for some even years), daily opportunities to share space, work through conflict, and experience the joy of play were limited. Add in the reality of increased screen use, the culture of comparison amplified by social media, and children who may show hesitations around groups, and it’s easy to see why we are carrying this extra layer of worry. What Montessori Reminds Us Montessori provides us with a helpful perspective: growth is not lost, it is simply unfolding in its own time. Children are incredibly resilient. Social development isn’t a race, and there isn’t one perfect timeline. Just as each child learns to walk or read at their own pace, children find their own paths into friendships and community life. How We Can Support Children (and Ourselves) So, what can we do to ease our own (and our children’s) anxiety while supporting their social growth? First, remember that depth of connection matters more than numbers. A child with one or two genuine friendships has a strong foundation. In Montessori classrooms, we often see that meaningful, sustained interactions carry more value than a large peer group. Second, provide opportunities for natural interaction rather than overscheduling. Time at the park, family gatherings, or small playdates often offer more authentic growth than tightly managed activities. In the classroom, children practice collaboration daily, from sharing materials to working on group projects. Third, support independence. After COVID, it’s natural to want to step in quickly if our child seems uncomfortable socially. But small moments of trust—like letting our children introduce themselves, ask to join a game, or resolve a conflict—help them build confidence and resilience. Finally, model calm and social grace. Children absorb how we approach social situations. When they see adults showing openness and curiosity, they are more likely to approach peers with the same ease. Partnering with Teachers Your child’s teacher is also a valuable partner. Montessori guides closely observe children’s social interactions and can offer concrete reassurance: “I noticed your child inviting a friend to join her work,” or “He showed patience while sharing materials today.” These insights often show growth that may not be as apparent outside the classroom. Supporting Social Growth at Home Simple, practical activities in our children’s lives can make a big difference. Hosting a “work together” playdate, for instance, shifts the focus from entertainment to shared purpose. Children might bake bread, build with blocks, or create art together. These collaborative activities naturally encourage conversation, cooperation, and problem-solving, giving children structured opportunities to navigate relationships. Practicing grace and courtesy at home also builds confidence. Small role-plays, like how to join a game or politely declining an invitation, help children learn the social scripts they need. Learning how to insert themselves into play: “What are you playing? I’ll join too!” or “No, thank you. I’d like to work alone right now,” are examples that may seem small but make a significant difference when children encounter peers in real settings. Family game nights are another excellent tool. Cooperative board games, memory games, or card games teach turn-taking, handling winning and losing, and joyful connection. Games provide low-stakes opportunities to practice critical social skills like patience, negotiation, and flexibility. Finally, spending time in community spaces offers invaluable real-world practice. Visits to the library, farmers’ market, or nature center allow children to engage in everyday interactions, such as greeting a librarian, asking a vendor a question, or navigating shared space with peers. These experiences help children build confidence and fluency in social settings. Moving Forward with Confidence It’s understandable to feel anxious about our child’s social life after such a collective disruption. But rest assured: children are not permanently “behind.” With trust, time, and supportive environments, children continue developing the skills of friendship, collaboration, and community. Montessori reminds us that growth is steady and ongoing. Our role is to provide the space, opportunity, and confidence children need to thrive socially. When we step back and trust the process, we discover that children are, in fact, developing social competence. Come schedule a tour of our classroom environments in Oak Park, IL to see how Montessori fosters a lasting sense of confidence and a deep form of belonging.
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October 6, 2025
From curiosity to self-control, Montessori aligns with the human tendencies that help children grow, adapt, and flourish.