Montessori Basics: How Math Progresses Through the Levels

 Addition Strip Board

You know your four year old loves their classroom and their work.  You know their teachers are guiding them to learn early math skills.  But what, exactly, does that look like?  And how does it change as they get older?  Montessori math materials are nothing short of amazing.  While they look quite different than what we used growing up (pencil and paper?) there are intentional reasons for these methods.  Read on to learn more...

The Basics

Much of the Montessori curriculum is based on giving children exposure to concrete materials first, then giving them incremental opportunities to work to more abstract concepts.  This is no different when it comes to math.

What do we mean by concrete?  The children are able to hold a material in their hands.  The materials are symbolic or representative of something else (a number, perhaps), and that symbolism changes over time until children are ready to let go of the materials and find solutions on paper or even in their heads.  This idea of mastering a skill without the assistance of materials is what we refer to as abstraction. 

 Number Rods

What Does Primary (Early Childhood) Math Look Like?

At the primary level math starts out simple, but you may be surprised at how much preschoolers are capable of.  

Even before a child is able to count, they experience the skill using materials like the number rods, a series of blue and red colored wooden rods that are arranged in a stair-like pattern.  Children learn how to count using a variety of materials.  The spindle box is an early material with which children place the correct amount of wooden spindles in compartments labeled 1-9.  Sandpaper numbers (just like their letter counterparts!) teach children how to correctly form each number to develop readiness for writing them on paper.

When a child is ready to learn about basic operations, there are plenty of materials to support them.  Montessori math uses the golden bead material; first to build numbers into the thousands.  For example a single golden bead represents 1, a group of 10 beads are strung together in a straight line for 10, and 100 beads are affixed into a flat square.  The thousand cube is as large as 1,000 of the original single ‘1’ bead.  Once a child is able to build a visual representation of a number, the beads are used to teach basic operations.  Young children are able to add, subtract, multiply, and divide numbers into the thousands using this material.  They first learn with static problems - that is, with no exchanges - and then move on to more complex, dynamic problems.  They quickly learn that ten 1s is equal to one 10, and they do this by holding those numbers in their hands.

 Golden Beads

Montessori recognizes the importance of memorizing basic facts.  While when we were young we may have used flashcards to drill these facts into our heads, the Montessori approach begins by showing children why we manipulate numbers in different ways.  Young children appreciate the repetitive nature of the materials, which gives them plenty of opportunities to practice  (and memorize!) these facts.  The addition and subtraction strip boards show a child visually what is happening when we add numbers.  The same goes for the multiplication and division bead boards (which use small beads placed in divots on a wooden board to create an array).  

 Division Board

A Period of Overlap

Somewhere between kindergarten and the first year of lower elementary, children are taught to use new math materials depending upon their individual readiness.  The stamp game is a classic example.  

The stamp game material is a sectioned box with small colored tiles sorted inside.  There are labeled green, ‘one’ tiles, blue ‘ten’ tiles, red ‘hundred’ tiles, and green ‘thousand’ tiles.  Instead of holding a large cube that actually shows the relative size of one thousand as they did with the golden beads, they are now representing series of tiles that are all the same size, but are differentiated only by their color and number label.  Like the golden beads, the stamp game material is used to teach all four operations, with children adding, subtracting, multiplying, and dividing into the thousands.  Some children begin this work in their primary classroom and continue when they reach elementary, while others begin once they enter their lower elementary classroom.

 Stamp Game

It may be interesting to note that there are some Montessori materials that children spiral back to, over and over again, from ages 3 to 12!  The bead chains are a colorful, quintessential Montessori material.  In the primary classroom, children use them to learn how to count, and perhaps how to skip count.  In a lower elementary classroom they are used for skip counting and to help memorize multiplication facts.  In upper elementary children use them to solidify concepts like squaring and cubing, although they were indirectly preparing for that work for years previously. 

 Bead Chains

What Does Elementary Math Look Like?

Remember the green, blue, and red tiles of the stamp game?  Montessori refers to those as the hierarchical colors, and they are used to teach children about number series.  They first appear in the stamp game, but they continue to follow the child through lower elementary and into upper elementary until they have a firm grasp on the idea of the simple family of numbers (ones, tens, hundreds), the thousand family (thousands, ten thousands, hundred thousands), and so on.

After a child masters operations with the stamp game, they move on to use a material called the bead frame, which can teach addition, subtraction, and multiplication.  It looks a bit like an abacus, but with ten beads on each rod in the hierarchical colors.  After a child masters the bead frame, they are typically ready to add and subtract into the thousands (and beyond!) using just pencil and paper.

To learn larger multiplication problems, children use a material called the checkerboard.  They begin small, but eventually work their way up to problems that have three or four digit multipliers.  For long division, children use a material that goes by different names at different schools: the racks and tubes, aka the test tube material.  Once children master the checkerboard and racks and tubes, they are able to multiply and divide large numbers without materials.

 Decimal Checkerboard

During the elementary years fact memorization continues.  In early lower elementary, many children continue to use the strip boards and bead boards of their primary years, but eventually move on to using finger boards and tables in which they place numbered tiles.  Children notice the patterns numbers make, giving them more tools to memorize their facts.

There’s More!

Of course, math isn’t just about operations.  Montessori students learn about geometry and fractions from an early age. 

 Geometric Solids

Did you know that primary children learn the names of geometric solids?  They can easily identify not just cubes and spheres, but square based pyramids, rectangular prisms, ellipsoids, and more.  As they move into elementary they learn about range of concepts, including studies of angles, triangles, polygons, and so much more.  A third grader can easily identify a right-angled, isosceles triangle.

 Fraction Insets

When it comes to fractions, first graders start out simple with an impressionist lesson involving an apple and a definition of fractions that includes how they must always be fairly divided (the connections between fractions and division are impressed early on).  They next move on to using fraction insets, which look a lot like the metal insets they used for handwriting preparation in their primary classrooms.  Before you know it, many third graders are learning to multiply and divide fractions.

Still Curious?

The best way to really understand Montessori math is to see the materials in action.  Schedule a visit to watch children using them in the classroom, or join us for our upcoming parent education session on Monday, March 26, 2018, 6:00-7:30.  Contact us for more details!  

Check out the stamp game in action:

Bursting the Montessori Bubble


What does "Bursting the Montessori Bubble" mean?

Our culture has long maintained the factory model of education and most of us were raised in that system. We experienced school in same-age classes and were taught the same subjects at the same time.

As children progress through the Montessori program, parents begin to feel the cultural pressures of traditional learning. Will my child be ready for testing? Will they be able to transition? They begin to feel that Montessori education is a protected bubble, not the ”real world” their child will be entering later on in their educational experience.

Have you ever had these same thoughts? Dr. Stephen Hughes, a Pediatric Neuropsychologist, has said in regards to this topic:

Which is the real bubble? Because the truth is, success in life is not built on a foundation of standardized tests, but on the freedom to make difficult choices and experience their consequences. Success in life is not built on grades and percentages, but on self-awareness and self-improvement. Success in life is not built on artificial competition among same-aged peers, but on the genuine collaboration between generations. Success in life is not built on cheating the system, but on having the wisdom and courage to transform it.
— Dr. Stephen Hughes

And Dr. Montessori said it herself:

If education is always to be conceived along the same antiquated lines of a mere transmission of knowledge, there is little to be hoped from it in the bettering of man’s future. For what is the use of transmitting knowledge if the individual’s total development lags behind?”  
— Dr. Maria Montessori

She was ahead of her time!

Kari Headington is Head of School at Hollis Montessori School.

Montessori and Peace Education

Our world is often a tumultuous and scary place.  How can we help our children feel safe and cared for, while preparing them to lead the way as adults?  How can we cultivate empathy, kindness, gratitude, and the sense of community that helps people work together?

Montessori education has been addressing these issues for over a hundred years.  Sometimes the lessons are direct; at other times they are more subtle.  The mission is always clear: we want children to have a wide view of the world.  We want them to appreciate the diversity of others.  We want them to have the tools to navigate this world peacefully.

How do we approach this critical task?

Montessori schools teach peace both directly and indirectly.  Sure, we talk about peace and its importance openly and frequently.  We talk about what it means and what it looks like and what children can do to become peacekeepers.  But, perhaps more importantly, we model.  Through our words, the tone of our voices, and with our actions, we show children what it means to be peaceful.  They watch our everyday actions and learn so much from them, so why not create constant learning opportunities?

Teaching a Global Perspective

Even from a very young age, Montessori children are taught geography through the lense of the whole world.  They learn about the continents when they are as young as three years old.  These studies often include learning about biomes, instead of an emphasis on political boundaries.  Teaching about the world in this way gives children a sense of the natural world and people as a whole as primary to different countries.

Elementary aged Montessori children enjoy many lessons with timelines.  They learn about the origins of humanity, and studying ancient cultures is fascinating for them. 

Embracing a Variety of Cultures

One important series of lessons in the elementary years teaches the fundamental needs of humans.  Children explore how groups of people around the world and across the ages meet and have met their needs.  Physical needs, such as food, shelter, defense, and transportation are considered, as well as spiritual needs like art and religion.

Giving Them Tools

Montessori teachers are equipped to give children skills to resolve conflicts.  We give children tools such as micro-mediation, and give them the words and actions to express their needs and feelings while listening to those of others.

In Montessori classrooms, children often learn a variety of self-calming strategies.  This might include mindfulness meditation, yoga, breathing exercises, or the use of small hand-held tools such as a finger labyrinth or polished stone. 

Taking the Time

In Montessori classrooms the day is not structured with rigid timed intervals.  For example, there is no predetermined time for math, reading, etc.  This flexible schedule lends itself to shifting courses and having discussions in the moment.

For example, if a group of children are experiencing difficulty resolving a problem together, the teacher is able to stop and sit with them.  Without feeling rushed, they can take the time to figure out what went wrong and how to make it right.  Instead of an adult doling out consequences, we have the time to sit and work through conflict authentically.  

Giving to the Community

As Montessori children get older, they are encouraged to give back to their community.  These acts of charity will often be inspired by the children’s ideas.  Children may collect food and supplies for a local animal shelter, read stories and sing songs to residents of a nursing home, or make and sell baked goods to benefit a cause they believe in.  

By supporting children with logistics, we can encourage them to learn how to be active and supportive members of their communities at a young age.  They learn the importance of volunteering and contribution to others.

Giving back is just one way a child begins their active role as a peaceful member.

March is Women’s History Month

This month we bring you a book list that will help families celebrate women’s history.  These are powerful times for women, and many people are turning to literature to celebrate their strength and accomplishments.  Children’s literature has played a large role in this movement as well.

Did you enjoy last month’s book about Wangari Maathai and want to learn more about the Nobel Peace Prize winner?  Do you love the lyrical writing and unique illustrations the Pinkneys bring to their books?  Just looking for a diverse collection of biographies to educate our littlest feminists?  We’ve got you covered...

(Click on the book image to go to the book's page on Amazon)


Amelia and Eleanor Go For a Ride by Pam Munoz Ryan, illustrated by Brian Selznick

Amelia Earhart and Eleanor Roosevelt were good friends.  They shared common traits of independence and fearlessness.  This story takes readers through an evening in which they snuck out of a fancy dinner and into a plane.  They flew off on a spontaneous adventure together, unaffected by what people thought they should be doing.


Seeds of Change: Wangari’s Gift to the World by Jen Cullerton Johnson, illustrated by Sonia Lynn Sadler

Not only was Wangari a powerful woman herself, but she was a champion for women’s rights.  Upon returning to her native Kenya and seeing the land destroyed by deforestation (ultimately negatively impacting the lives of many women and families), she worked hard to make change.  She taught the women to plant new trees, how to repair their land, and how to rebuild their lives.


Two Friends: Susan B. Anthony and Frederick Douglass by Dean Robbins, illustrated by Selina Alko

This book is a wonderful introduction to civil rights for younger children.  Based on the real, supportive friendship between Anthony and Douglass, readers will explore the history of the fight for equality in our nation.  At the time, women and African Americans found themselves fighting for many of the same rights, and this book explores the quiet, peaceful moments in between the hard work. 


I Am Jane Goodall by Brad Meltzer illustrated by Christopher Eliopoulos

Meltzer’s series of biographies are written and illustrated in the style of a graphic novel.  Readers will learn about Goodall’s fascination with animals as a child, following her on her journey toward her famed work with chimpanzees.  This book is perfect for children who love animals, adventure, and following their heart!


Georgia’s Bones by Jen Bryant

Even as a child, Georgia noticed things about her world that others didn’t.  She looked not only at the shapes of things, but the spaces between them.  She was always picturing what might lie beneath the obvious, and honed in on details that others missed.  It was this unique perspective of the world that led her to be one of the most celebrated artists in history.


Ella Fitzgerald: The Tale of a Vocal Virtuosa by Andrea Davis Pinkney, illustrated by Brian Pinkney

This story is narrated by fictional “Scat Cat Monroe”, a music appreciating feline that takes readers through the story of Ella Fitzgerald’s rise to jazz stardom.  The writing itself is organized into tracks on an album, rather than into chapters, and the illustrations are gorgeous.


Sonia Sotomayor: A Judge Grows in the Bronx/La juez que creció en el Bronx by Jonah Winter, illustrated by Edel Rodriguez

This book narrates Sotomayor’s childhood in the Bronx and her path toward becoming the first Latin American Supreme Court justice.  Readers learn how hard work and dedication can ultimately lead us to great things.  As a bonus, each page is written in both english and spanish.


The Librarian of Basra: A True Story From Iraq by Jeanette Winter

Alia Muhammad Baker was a librarian in Iraq.  The onset of war led her to seek any means possible to protect the valuable resource that was her community’s collection of library books.  With Baker’s determination, courage, and the support of some friends and neighbors, she was able to do just that.


Celia Cruz, Queen of Salsa by Veronica Chambers illustrated by Julie Maren

“She looked like a girl and talked like a girl, but everyone who met her agreed, she sang like a bird.”  This lovely book tells the story of Cruz’s childhood in Havana, and how she became one of the most recognized salsa singers of all time.  


The Girl Who Ran: Bobbi Gibb, The First Woman to Run the Boston Marathon by Kristina Yee and Frances Poletti, illustrated by Susanna Chapman

Sometimes, for the sake of progress, rules must be broken.  Sometimes listening to our hearts is more valuable than listening to the expectations of others.  This tale reminds readers to question why we have rules that hold back some people, and how the bravery of one can change the rules for all.


May this month’s reading be filled with bravery, adventure, and a healthy dose of reverence for women who have paved the way!

What Your Child Will Really Get Out of Montessori


Is it really worth it?

I mean, why should you spend the time, effort and money to find an authentic Montessori program for your child? Wouldn’t it be easier to just find a good, basic preschool?

For me, the answer is easy - enrolling my children in a Montessori program was one of the best decisions I made as a parent. But, then again, I am a bit biased!

For most parents, however, the question remains: “What will my child really get out of attending a Montessori school?”

Why Your Background Matters

I’m going to go out on a limb - my guess is that you did not attend a Montessori school when you were a child.

This isn’t a problem, of course. You turned out just fine. But, as you consider early education options for your child, your own educational experiences can make the decision a bit difficult.

At first glance, Montessori classrooms don’t look familiar to most people. There are no rows of desks, no blackboards and no teaching to the entire class. Parents are often intrigued by the peaceful, calm environment and the hum of activity, with young children choosing their own activities and concentrating deeply for long periods of time.

Montessori is so different, however, from traditional programs that it’s natural to leave a bit perplexed. I mean, what is really going on here?

How Other Preschools have it So Easy

I’ll admit it: When I was the Admissions Director I was a bit jealous that most preschools didn’t have to spend a lot of time explaining what they do to parents. The goal of the program is clear - to prepare children for a conventional kindergarten program. 

In a conventional school, your child will need basic academic skills, so they focus on “pre-reading and pre-math” with workbooks, flash cards and rote memorization. Think ABC and 1-2-3.

In a conventional school, your child will need to adjust to the schedule of a traditional school, so they have activities where everyone does this same thing at the same time. Think group art projects where your child will learn to color within the lines.

In a conventional school, your child will need to learn how to pay attention to one teacher speaking to the group. Think long circle times with one teacher talking to all the children together.

There is nothing wrong with this, of course. In a Montessori classroom, however, we believe your child deserves an education that focuses on all aspects of him as a human being.

How Montessori is Different: A Three Word Answer

Education for life.

Rather than just preparing your child for the next step in school, we seek to support his academic, social, emotional, intellectual and spiritual development.  We want him to be successful at life in the future, not just in kindergarten.

Take a second to imagine your child twenty years from now. What skills will he need to be successful in college, his chosen profession and in life in general?

Here is a primer. He will need to:

  • Know how to regulate his behavior
  • Control his impulses 
  • Learn to plan and strategize
  • Hone the ability to problem solve
  • Learn to be flexible and course-correct when necessary
  • Learn to take initiative 
  • Develop responsibility
  • Engage in depth-based thinking requiring long periods of concentration
  • Work collaboratively with peers on projects  

Researchers who study the traits of successful adults coined the term for these skills: “executive functions”.

These executive function skills, that are so important to life’s success, must be continually developed, day in and day out, or else they will not materialize. They result from the way an activity is done and the time spent doing it – pushing oneself to do better and better.

The Link Between Montessori and Executive Functions

Research comparing children attending inner-city Montessori schools with those attending traditional schools was conducted by University of Virginia professor, Dr. Angeline Lillard, and was published in the prestigious journal, Science, in September 2006.  

The result?

Montessori students rated higher on “executive function skills”- skills like selective attention, self-control, problem solving, reasoning and not getting into trouble.

On behavioral and social tests, 5 year old Montessori children scored higher than their peers from conventional schools, showing that they had a greater sense of fairness and justice; out on the playground, they were more likely to engage more in emotionally positive play with their peers and less in rough housing.

And, yes, your child will still be ready academically for elementary school, whether in Montessori or any other program. The same study found that among the 5 year olds who were studied, Montessori children were better prepared to enter first grade with stronger reading and math skills than children in traditional schools.

The Choice is Yours

Is Montessori worth it? You decide. You, as the parent, are charged with raising your child in the best way you see fit. What do you value? What kind of adult do you want your child to become? There are a multitude of options available to you. Think carefully and choose well.

Camille Campbell is one of the founders and the former Admissions Director at Hollis Montessori School. Her two daughters are Hollis Montessori graduates and are now thriving in high school and college.

Why Encourage Self-Directed Play?

Perhaps you’ve heard about self-directed play, also known as open-ended play.  Maybe you haven’t.  It may seem like a recent trend, but the truth is the concept is nothing new.  Plus, the benefits are extensive.

Once you understand the reasons for encouraging our children to engage in self-directed play, and you have a basic understanding of how to try it at home, it’s simple!

Bonus: self-directed play embraces many Montessori ideals.

What, exactly, is self-directed play?

If your child is using simple toys in creative ways with no adult-directed outcome, there’s a good chance they’re already engaging in self-directed play.

Many of the toys available today are intended for a specific purpose.  Let’s consider, for example, a doll.  Sure, a child can embark on some imaginative play with it, but a doll will always be a doll.  The same goes for a small toy train or a plastic dinosaur.  This is not to say there is anything wrong with these toys, but the ways in which children can use them are limited by their nature.

Now let’s consider a cardboard tube.  The possibilities are endless!  The tube could be a telescope one minute and a megaphone the next.  It could be a log, a bridge, or something to guide a ball through.  Materials we offer children for self-directed play are simple.  Think balls, cardboard tubes, sticks, scarves, playdough...the list goes on. 

When children embark on self-directed play, it’s important for adults to remember that the children are the ones calling the shots (within safe boundaries, of course!).  It is our natural tendency to have pre-determined ideas of what the outcome of a certain activity should be.  We often, instinctually, feel the need to jump in and teach children the “right way” to do things.  Give yourself permission to step back.  When we observe the the way in which children discover their own outcomes, it can be magical to see the process from a new viewpoint.  

How can self-directed play benefit children?

  • It builds self confidence.  By exploring on their own, children realize there is so much they can do for themselves.  They make their own games with their own rules, and they feel successful.  
  • It encourages independence.  Isn’t our ultimate goal for children that they might be able to get along just fine without us?  Self-directed play lets them experience independence from a young age, all while in a safe, prepared environment.
  • It stimulates imagination.  Children can’t help but be creative during self-directed play.  By giving them these opportunities, we are allowing them to flex their creative muscles; they will see possibilities no one else has imagined, and they will develop their own story lines as they play.
  • It teaches problem-solving.  Coming up with one’s own rules naturally leads to problem solving.  Children will have to figure out how to make something work the way they want it to.  
  • It allows children to learn at their own pace.  With self-directed play, there is no timeline and there are no academic benchmarks to meet.  Kids have the opportunity to build on their own knowledge, day after day, in ways that make sense to them.
  • It cultivates internal motivation.  Without adults defining the success of an activity, children will be compelled to find the innate joy in their play.  They will naturally tend to challenge themselves to try new, innovating ideas, and they will find their own personal delight in doing so.

Getting started at home

If you’re feeling ready to give self-directed play a try in your home, consider these tips to get you started:

Materials/Toys - Remember, these should be simple.  As an added benefit, simple toys tend to be much easier to obtain and far less expensive (and often free!).  If possible, toys should be made of natural materials.  Think wood, fabric, and items found in nature; avoid plastic if possible.  As mentioned above, collect toys that can be used for any number of possibilities.  Things like balls, scarves, blocks, boxes, sticks, or clay are great.  Some people like to collect trays of loose parts to leave out for children.  Loose parts trays might include pebbles, seashells, buttons, bits of string, pieces of tree bark...whatever looks (and feels) interesting!


Prepare the Environment - Make sure children have a safe, open space in which to play.  Depending on your home and the weather, this could be your living room, backyard, or whatever space works for your family.  It’s important to make sure children have flexibility in their movement though, so make sure they can sit, stand, jump, roll, and explore!

Sit Back and Enjoy! Another great benefit to self-directed play is that because children can engaged on their own, you are free to spend time checking off your own to-do list.  But feel free to sit nearby or even alongside your child if you wish.  Just remember to let them take the lead and explore their world and imagination.