Our Montessori Bookshelf: Mathematical Thinking

As humans, we are predisposed toward order, exactness, and precision. With this tendency to abstract and imagine, we could be said to have a mathematical mind. Children, young and old alike, are drawn to numbers and mathematical ideas. 

For thousands of years, math has been a part of the human search for meaning. We have long tried to quantify our natural world. From carbon dating artifacts to analyzing voting trends in politics, from understanding traffic patterns to examining climate change, math continues to be an integral part of our search for understanding.

Learning to think in mathematical terms is an essential part of becoming a person adapted to our time and place. Math is such an integral part of our lives and we feel that it’s vital to ensure our children are not only in touch with mathematics but also captured by the beauty and wonder of math in our world. 

With this in mind, we pulled some of our favorite books that promote mathematical thinking for young children through early adolescence. 

Counting Is for the Birds

by Frank Mazzola Jr.

Written in rhyme, this picture book can be used in different ways with young children. Some may just enjoy the story and illustrations, others can clue into the counting aspect of the book, and older children might explore the ornithological details provided on each page. This is the kind of book that you can revisit again and again with your children!

4,962,571

by Trevor Eissler, Ruth Chung, Bobby George, June George

Written by a former Montessori parent, this picture book is a lovely introduction to and extension of the concept of place value. A young boy wants to see how high he can count, so he figures out ways to create groups of numbers so he can count to four million, nine hundred sixty-two thousand, five hundred seventy-one (and beyond!). Plus, anyone who has been in Montessori will appreciate the color coding of the numbers in the title!

How Much, How Many, How Far, How Heavy, How Long, How Tall Is 1000?

by Helen Nolan, illustrated by Tracy Walker 

Children at the end of their primary years or those who have recently transitioned into elementary will definitely appreciate this exploration of the quantity of 1,000. Full of thought-provoking questions, this picture book takes readers on a journey through how a 1,000 can be represented in so many different ways – and how that can change our impression of the size of the number. 

One Grain of Rice: A Mathematical Folktale

by Demi

This stunningly illustrated picture book provides both a moral tale and an example of the exponential power of multiplying by two. After a raja in India has hoarded rice for his own benefit, a young girl returns some spilled rice to him and as a reward requests only one grain of rice, as long as the raja doubles what he gave her the day before over the course of 30 days. By the end, she has more than enough rice to share with all the starving villagers, as well as the goodwill to support the raja in continued kindness.  

Anno's Mysterious Multiplying Jar

by Masaichiro Anno, Mitsumasa Anno

For those who love Anno’s Journey, this is a must-read, but this time the illustrations and text take the reader on a mathematical journey through factorials. Then to show what happened mathematically, the Annos (father and son) illustrate the multiplication in a graphic way that fits so well with what children experience with the Montessori math materials. 

Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Volumes 1 & 2

by Luetta Reimer, Wilbert Reimer

This collection of short stories dramatizes conversations and lives of mathematicians throughout history and can easily capture the imagination of elementary-aged children who love the power of a good story. The stories can stand alone or be jumping-off points for further mathematical or historical investigations. We love the glossary at the end, the short biography at the start of every story, and the fact that female mathematicians are fairly well represented in these two volumes. 

The Number Devil: A Mathematical Adventure

by Hans Magnus Enzensberger, illustrated by Rotraut Susanne Berner, translated by Michael Henry Heim  

This is the perfect book for older elementary-aged children who aren’t quite sure they want to still love math. A boy meets a number devil in his dreams who leads an exploration of all sorts of fascinating aspects of numbers. The wildly fun and irreverent approach (led by the devil) makes even complicated math feel accessible. The whimsical illustrations certainly help, too! And for those wanting to go back and reference helpful information, there is a “Seek-and-Ye-Shall-Find List” (aka index) at the end of the book. 

Doodle Yourself Smart . . . Math

by Helen Greaves, Simon Greaves

For elementary children and adolescents who like to play around with mathematical thinking, this is a fun activity-style book that appeals to mathematicians and artists alike. Each page offers beautiful space for playing around with the problems (and yes, there are answers in the back for those who just need to know if they got it right!).

The Man Who Counted: A Collection of Mathematical Adventures

by Malba Tahan

Those who like a good mathematical challenge, combined with a taste of the adventure that comes with travel, will love this series of chapters that form a bit of a novel. Each chapter of this book can stand alone or work as a cohesive whole as the narrator and the “man who counted” move through the Middle East. They encounter a slew of social problems that are solved with a sophisticated level of number sense that feels both mystical and matter-of-fact. 

Click here for a downloadable PDF of this booklist! As always you are also welcome to come visit the school and see how we support mathematical thinking for all ages.

Materials Spotlight: The Bead Chain Cabinet

Visit any primary or elementary Montessori classroom and you’ll immediately notice a stunning display of colorful glass beads hanging in an open cabinet. Children (and adults!) are drawn to the order and elegance of the bead chains. Beyond their initial aesthetic appeal, the bead chain material offers an amazing array of intellectual opportunities for young children to those in their elementary years. 

Color-coding

One of the brilliant aspects of the Montessori math materials is how they provide children with multiple ways to make neural connections. For example, each of the bead bars represents a quantity and each bead bar is also color-coded so that the quantity is connected to a color: a bar with two green beads represents two, a bar with five light blue beads represents five, a bar with eight brown beads represents eight, etc. This color coding allows the child’s brain to establish multiple quick ways to understand the quantity: the number of beads, the color, and the size. 

Appealing

In Children’s House, young children are attracted to the beauty and fragility of these colorful, glass beads. Long before they are ready to use the bead chains, young children can learn how to dust and care for the beads. They develop a respect for the materials and understand how special they are. Often young children will watch in awe as their older classmates learn how to carry, lay out, count, and label the bead chains. 

Linear & Skip Counting

As they practice counting the bead chains, four- and five-year-olds solidify their understanding of teen numbers, as well as quantities from units, to tens, to hundreds, to thousands. Eventually the focus of work with the bead chains shifts from linear counting to skip counting, as children begin to focus more on the labels that indicate the end of each bead bar. For example, on the 100 chain, children label and name 10, 20, 30, 40, 50, 60, etc. Children can also layout, count, and compare the 100 chain and 1,000 chain side-by-side, providing the sensorial experience of the different quantities laid out in a linear fashion. 

Multiplying

As children move into their elementary years, they are solidifying their skip counting in relation to mastery of multiplication. They love to quiz each other by turning over some of the labels to see if their friends can figure out which of the multiples is missing. So for the short chain of multiples of seven, for example, they might turn over one label to see if their friend can figure out the missing multiple: 7, 14, 21, __, 35, 42, 49. 

The bead chains are also used to provide an impression of common multiples, which is critical for future work with fractions. Children lay out two long chains side by side, label the chains, and then find all the common multiples, and also begin to intuit the concept of the lowest common multiple.

Squaring & Cubing

The other fascinating aspect of the bead chains is how they geometrically represent the concept of squaring and cubing. Children learn how to fold up the bead chain so that it creates a square made up of four rows of four: 4 x 4 = 16. 

This work can continue with the long chains, each of which creates the cube of numbers one through ten.

In addition to the hanging chains of bead bars, the bead chain cabinet also contains beautiful squares of each number, which can be stacked to create cubes, as well as the actual cubes for each number. The squares and cubes can be used for explorations with powers of numbers as well as more advanced work when students start to explore working outside of the base ten number system. 

As children move into more advanced mathematics, they will often briefly revisit this foundational material to cue their memory when working with exponents.

Spiral Curriculum

The bead chain materials provide a perfect example of the spiral nature of the Montessori curriculum. From early linear counting, cycling into an understanding of multiples, to preparation for squaring, cubing, and base number work, children come back again and again to the beauty and breadth of the bead chains.

We invite you to visit our school to see the bead chains for yourself! 

Materials Highlight: Geometry From the Start

When many of us think of our geometry education, we have thoughts of identifying shapes in kindergarten, and then taking a class or two in high school.

The truth is, children are fully capable, and enthusiastically prepared, to learn so much more at a much younger age. While we can’t possibly cover our entire curriculum in one short article, we’ll share some of the highlights. It begins (perhaps unsurprisingly) in our primary classrooms.

Geometric Solids

Pictured above, the geometric solids are our students’ first direct exposure to geometry. The solids are displayed on a shelf and are contained by a basket or tray. The adult invites the child to a lesson and asks them to retrieve the shapes. They then look at each one. The lesson may go something like this:

1. The guide picks up the cube, feels each side, and hands it to the child. The guide states, “This is a cube.” The cube is placed on the work rug.

2. The process is repeated for each solid in the basket.

3. Depending upon the child’s readiness, they may continue, with the guide asking questions like, “Where is the cylinder?”

4. After the child has had some time to interact with the material independently for a while, the guide will again sit with them and assess their understanding. This is done by holding one sold at a time and asking the child to name it.          

There are many fun extensions associated with this material. One favorite includes putting the solids on a mystery bag or using a blindfold so the child has to guess and identify by touch alone.

Geometry Cabinet

The geometry cabinet is used in primary and lower elementary classrooms, although differently in each. What begins as a lesson in identifying basic shapes and discerning between their sizes evolves into complex identification and blending of skills. Some of the skills this material helps us teach our students aged 6-9 include:

• Types of triangles (acute scalene, acute isosceles, right scalene, right isosceles, obtuse scalene, and obtuse isosceles)

• Quadrilaterals (trapezoids, a rhombus, and a parallelogram)

• Regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, and decagon)

• Curved figures (curvilinear triangle, oval, ellipse, quatrefoil)

• Rectangles

• Circles

Constructive Triangles

Beginning during the primary years and continuing through lower elementary, the constructive triangle boxes are another child favorite. A series of boxes teaches a variety of concepts.

1. Triangle box: Used to show how different types of triangles can be combined to make other triangles, also indirectly teaches fractions concepts

2. Large hexagon box: Used to show how triangles can be combined to create other figures, including a hexagon, rhombus, and parallelogram

3. Small hexagon box: A continuation of the same basic concept as the previous box, but this time triangles are used to create rhombi, a trapezoid, and another configuration of a hexagon

4. Rectangle box: Triangles are used to create a square, rectangle, and other quadrilaterals

5. Blue triangles boxes: While the previous boxes utilized different colors for different types of triangles, they are all painted blue here. This is basically an extension of previous work and allows children an opportunity to rely less on previously helpful visual clues. There is also lots of opportunity to use the blue triangles to create more complex geometric figures.

Classified Nomenclature

Once some of the more basic skills have been mastered by the child, sometime during early lower elementary, they move on to engage with the classified nomenclature. As with all other Montessori work, this is a step toward abstraction; they are no longer relying heavily on the wooden materials they can hold and manipulate, rather they are using drawings, booklets, charts, and labels.

This work can become rather in-depth and continues into upper elementary. A very broad overview of skills includes:

• Fundamental concepts (point, line, surface, solid)

• The study of lines

• The study of angles

• Plane figures

• Study of the triangle

• Study of quadrilaterals

• Study of regular polygons

• Study of the circle

These studies are not short lessons like the child has experienced previously. They are multi-layered and can take months to complete. For example, the study of angles may begin during the second year of lower elementary, but continue periodically through the years until the child reaches sixth grade. Concepts include:

• Parts of an angle

• Types of angles

• The measurement of angles

• Constructing angles

• Relationships between two angles

• Two lines and a transversal

• Constructing and copying an angle

• Bisecting an angle

• Operations with angles

Beyond all these amazing materials, it’s important to note that there is a lot of crossover when it comes to Montessori subjects. One perfect example is a favorite grammar work of third graders called the Detective Triangle Game. While its main intention is to practice using correct adjectives, this is done by way of sorting a multitude of triangles, with different colors, types, and angles.

Want to learn more? Please reach out if you have any specific questions and want to have a conversation. As always, we believe the best way to truly understand what goes on in a Montessori classroom is to sit quietly and observe in one. Contact us today to schedule a visit.

Materials Highlight: Memorizing Addition Facts

Ahh, memorizing one’s addition facts. Brings back great memories, doesn’t it? For some of us this was a boring and necessary part of our education, but for others it was downright dreadful.

It’s rare to find any sort of information required to be memorized in a Montessori school. We would much rather teach our students why various things are, then have them learn rote processes that mean nothing to them.

Math facts, however, are the exception to this rule...sort of.

We absolutely work with our students to ensure they memorize all their basic facts, we just do it a bit differently. It’s not your typical flashcards-and-timed-drills approach, but a series of strategies that appeals to the child while still reaching the ultimate academic goal.

Memorizing facts is essential to solid numeric understanding and as preparation for efficiently completing more complicated problems later on. As you might imagine, we start this process when children are young, and we use specialized materials to help them feel and envision what the numbers are doing.

Addition Strip Board

Pictured above, this is the first material intended to be used while teaching children to memorize their addition facts. It includes a wooden, gridded board with numbers across the top. Numbers one through ten are written in red, followed by a red vertical line drawn down the board, and numbers eleven through eighteen written in blue. The material also includes a box filled with wooden blue and red strips in varying sizes to be used on the board.

There are many ways to use this board, and a Montessori guide will gradually walk the child through a series of lessons to teach different skills. The basic concept involves the child laying out one wooden strip on the board, then laying another beside it. This allows them to clearly see something like 7+3=10. The strip board is also used in conjunction with the tables of addition (more on those below).

Addition strip board lessons may include:

• A first exercise introducing the child to the material and the basics of using it.

• Random selection and adding of numbers from a box.

• Combinations of a number (e.g. ways to make ten)

• Combinations with zero

• Doubling numbers

The addition may be introduced during the kindergarten year, but is used during the first year of lower elementary as well, or longer if a child needs it.

The Tables of Addition

At first used alongside the addition strip board and later used on their own, the tables of addition are another material that aids children on their path to memorization. The material actually includes four square working charts (one of them pictured above), two larger rectangular control charts, and a box of numbered tiles. To decode, the child will complete the work using one of the square charts and use the control chart to check their answers.

The first table of addition is what you might imagine having used on paper when you were a child. Numbers one through ten go across the top and also down the left side. Answers fill in the grid across the rest of the board, so that if you slide your finger down from the seven on top and right from the three on the left, you will arrive at the answer - ten - in the middle. Children can use this as a way to check their answers, too, as they use the addition strip board.

The second table (above) is essentially the same but with the center numbers left blank. Children can use this as they use the strip board, creating the problem on the strip board and then filling it in on the working chart, or they can use the working chart on its own, attempting to fill in all the blank spaces with the correct tiles.

The third table shows all possible combinations. This means there is significant blank space and children get an introduction to the commutative property. This particular visual will help the child see more patterns within the number facts than they may have in the past.

The fourth table is missing even more numbers, featuring each sum only once. The child is at this point required to complete even more independently, but as always, can rely on a control chart to check their answers if need be. These control charts are often nearby but flipped upside down by the child to self-encourage and figure out the answers on their own whenever possible.

The (Positive) Snake Game

There are actually multiple snake games used in the Montessori math curriculum, yet this is the first. It is used after children have had sufficient time to use the other addition memorization materials and have begun to memorize some of the facts. A major aim of this snake game is to revisit the concepts of making ten and exchanging for ten.

The material consists of three wooden boxes. One contains golden ten bead bars, another contains various colored bead bars for numbers one through nine, and the third contains black and white bead bars that will be used as place holders.

The child may lay out bead bars randomly, or they may follow along with cards given by the guide to complete a problem such as 4+2+8+1+7+9=. The colorful bead bars will be laid out in a zigzag formation, taking on a snake-like appearance. The child will start at one end and count beads until they get to ten, then, using the golden ten bars and black and white place holders to take the place of the colored bead bars. This continues until the snake has been all counted up, and the child can count by tens and the remainder to find the answer.

The educator in this video gives a clear demonstration of the process. You may notice him placing the used colored beads in a small glass bowl. Sometimes children will take these out afterward and count them up to check their answer.

Hopefully you have learned something new and interesting from this article. Want to see the materials in person? Reach out today! We would love to chat more.

Materials Highlight: The Fraction Insets

For this month’s Materials Highlight we bring you the fraction insets; a beautiful set of metal templates resting on slanted wooden trays. As you can see in the photo above, the insets range from one whole through tenths, and each piece has a small knob allowing children to move them easily.

But before we get to the insets, perhaps we should back up just a bit.

Prior to an introduction to fractions, the child has had extensive instruction and experiences with numeration being based on the unit. One unit (or one, one whole, etc.) has been the basis by which they have learned to count, skip count, add, and subtract. As the child enters lower elementary, they are ready to learn who we may divide a unit.

This work often starts with an apple. The Montessori guide sits the children in a small group and tells them the apple will be divided for them to share. They then proceed to cut the apple without any regard to straight or even lines, creating small chunks and larger ones. The children quickly realize the injustice in distributing such apple slices, so the guide takes out a second apple to cut it evenly and impart the importance of equal slices being fair. The stage is set for learning about fractions.

An Introduction

It’s important to note that while the most commonly used and popular fraction insets are circular, there are also triangular and square fraction insets. It’s important for guides to refer to this at times so the children have an understanding that anything may be divided, not just circles.

The first time children use the insets they are encouraged to observe what they notice, and they develop the concept that each inset is a family of sorts. “These are the thirds, these are the sixths, etc.”

The guide will make a point to use intentional language to create a firm basis in understanding: “This circle is divided into four equal parts. We call them fourths.” The guide will write out “fourths” as well as “/4” as children are able to verbally express their understanding.

The Numerator

During the course of this lesson, the guide doesn’t actually use the term numerator just yet. What is emphasized is that while the children previously learned the family names of each inset, the focus will now shift to individual pieces. Examples will be shown using the material, and both verbal and written expressions will accompany each.

For example: “This is one third, or ⅓.” “This is four fifths, or ⅘.” This may be the end of the lesson, or, if the children seem to grasp the concept quickly and easily, it may be combined with the third presentation.

The Third Presentation

The third presentation is essentially a culminating review of what has been covered so far. The children may take turns matching labels with fractions to show their understanding. The critical piece is that the guide will now formally name the numerator and denominator. 

There is a lot of opportunity for practice and extension work at this point. Children may trace and label fractions, make booklets or charts, work together to match labels, and so on. This work typically happens during the first year of lower elementary.

Equivalence

This is an exciting lesson for children. Once they have a firm grasp on naming fractions, the guide will again sit them down in a small group. The one whole circle will be removed from its frame and the two halves will be put in its place. The guide will show the children how one whole is equal to two halves. This will be repeated with similar equivalencies: 3/3=1, 4/4=1, etc.

Next, smaller equivalencies will be discovered. The guide will try and fit a piece into a number of different spots, proving where it does and does not fit. Children will learn several simple equivalencies, such as 2/6=1/3 .

As with the previous skill, there is plenty of opportunity for exploration and extension in regard to equivalencies. This is arguably the most important fractions skill of lower elementary.

Operations with Fraction

Once a child has a firm grasp of fraction basics, they are ready to learn operations. This will likely begin in lower elementary and extend into upper elementary, and are taught initially using the fraction insets material. Another material often used is called the fraction box, which includes small plastic replicas of the red circular fraction pieces. Skills include:

• Addition and subtraction using the same denominator

• Multiplying fractions by whole numbers

• Dividing fractions by whole numbers

• Addition with different denominators

• Addition with more than two addends

• Subtracting with different denominators

• Multiplying whole numbers by fractions

• Multiplying fractions by other fractions

• Dividing whole numbers by fractions

• Dividing fractions by fractions

Moving to Abstraction

Use of materials when teaching fractions is critical; we believe Montessori students excel later in life with more complicated math concepts because they have such a strong foundation in the basics. Rather than memorizing rote procedures they are physically manipulating numbers with their hands, giving them a deeper understanding of why we do what we do.

One cannot rely on materials forever, though, and there comes a time when the child is prepared to move onto abstraction.

This is often achieved by the teacher again showing an operation with the material while also writing out the pencil and paper process simultaneously. In fact, children will often come to this learning independently. They are able to make the connections as they master skills. If not, the guide is there to show them the way. There comes a point during the upper elementary years when a child no longer needs to rely on the materials to determine the answer to a problem. In fact, using the materials becomes cumbersome, and they are eager to put them behind.

Want to learn more? We encourage you to reach out and schedule a virtual tour.

Montessori Materials: The Stamp Game

This post is the first in a new series we are so excited about. Each month, throughout this school year, we will share information about a different Montessori material. Doing so will help parents who are curious about what goes on in our classrooms, but it will also give unique insight into Montessori principles and how the method was developed in the first place.

Today we talk about the stamp game. A beloved math material that is used by children sometime between their kindergarten and second grade years (depending upon their readiness), it allows young children to add, subtract, multiply, and divide using numbers into the thousands. Using a material such as the stamp game allows children to learn a concept in concrete terms, rather than abstractly (which is what they will be doing when they eventually complete the same types of problems with just pencil and paper).

Think back to when you learned basic math computations: it was very likely done abstractly, and you memorized what must be done when your numbers added up to more than ten in one column, or you needed to borrow some from the next column, and so on. Being taught that way certainly gets the job done, but what we are effectively doing then is teaching children to memorize the process.

Using a material like the stamp game? This allows children to physically manipulate the numbers in a way that provides a deeper and richer understanding of mathematical processes. Instead of feeling tedious and confusing, the stamp game provides a stepping stone that makes them excited to discover the secrets of numbers and operations. Many Montessori children have reported that as they get older and find themselves working on much more complicated math concepts, they still picture the movement of the tiny stamps in their minds. They have a lasting visual image of what the numbers are doing as they work; it’s not just rote memorization.

Please bear in mind that prior to being introduced to this material, the child will have a solid understanding of place value, as well as the basic concept of adding numbers. These skills will have been gained through other Montessori materials that were carefully developed and intended to be used in a specific sequence.

So let’s get down to the important part: how the stamp game is used.

A child or small group of children will be seated on the floor across from their guide. A work mat will have been unrolled, and the guide will stand up, walk across the room, and carefully select the material from the shelf, carrying it carefully to the work mat. Even this small action has purpose: the guide is wordlessly teaching the child where the material can be found and where it is expected to be returned, as well as modeling how it should be carried around the room.

When the box is opened the guide may ask the children what they notice. They may comment on the colors of the wooden tiles: green, blue, and red. They will notice the small numbers printed on each tile: ones (units), tens, hundreds, and thousands. They may even notice little pegs that they will use much later when they use the same material for division.

The first step is to learn how to make numbers using the stamp game. The guide will either have a pre-printed card or perhaps a dry erase board to write a number such as 3,721. They will then demonstrate by taking one green unit stamp out of the box and lining it up neatly on the rug in front of the compartment it came from. This will be followed by two blue tens tiles, seven red hundreds tiles, and three green thousands tiles. (The color pattern begins to repeat because much later, using a different material with the same colors, the child will learn about number series and why we separate larger numbers with commas. The idea is being introduced indirectly long before it is expressively taught.)

Children will then take turns making numbers. If the concept takes some effort, this may be their work and their practice for several days or weeks. If they seem ready for more, the guide will move on to the next step.

An addition problem will be presented. The children will learn the terms addend and sum, and will make each addend, separated by a space or perhaps a pencil, on the work mat. The guide will then slide the bottom of each column of tiles upward, creating a single line for each place, as illustrated in the photo above. Starting at the bottom of the units, the tiles will be carefully counted and recorded in the proper place wherever the problem is written down. This will continue with the tens, hundreds, and thousands.

At first, the problems will have been carefully selected so that there is no need for exchanging. Once the child is ready for more of a challenge, they will learn that if there are ten unit tiles, they will need to be gathered up, deposited back into their compartment, and exchanged for one tens tile. They will learn to say aloud to themselves, “Ten tens is equal to one hundred,” and so on.

The lesson will end, and if the guide feels the children have grasped the concept well enough, they will be expected to use the material regularly and independently (or with a friend) to complete problems. The guide will periodically check in to observe and determine when mastery is achieved.

Whenever the child is ready (which could be weeks or even months), they will learn how to use the stamp game for subtraction. Again, the first problems will not involve any exchanging of numbers and will simply be a way to understand the basic process. You can begin to imagine the many steps and complexity of each Montessori material. When subtracting, the child will lay out the minuend, slide down the subtrahend, and find the difference.

Multiplication comes next. Children learn that all multiplication is making the same number a specific amount of times. They will see its connection to addition, as the process is very similar.

As for division, the guide will introduce tiny wooden cups - one for each place value. Children look at the dividend and put the correct amount of tiles into the cups. They will then use the wooden pegs mentioned earlier (called skittles as they resemble bowling pins), to mark the divisor. Rather than lining the tiles up beneath the compartments, they now learn to line them up beneath the skittles. They learn that division is about being fair, and that it is the only operation in which we start by using the largest number available rather than starting with the units (ones).

It can take an entire year (or longer) for a child to move through each of the steps described. The guide will keep a close watch on each individual’s progress, and provide them with more challenge as soon as they are ready. When a child has fully exhausted their learning with the stamp game, they are ready to move on to a slightly more abstract math material: the bead frame.

We hope you enjoyed this article. Want to see the stamp game in action? Contact us to learn more.

Book List: Math Books for Children of All Ages

In Montessori schools, children tend to really love math. It’s no wonder, with gorgeous materials created to appeal to their senses and developmental stages. One fun way parents can support their children’s love of math is by reading to them about it. This month we have curated a list of books that will appeal to children of all ages, whether they are learning to count or discovering the magic of Pi. Check out these titles and let us know what you think!

Numbers by John J. Reiss

There are some iconic early counting books out there, but if you’re looking for something fresh to inspire a love of numbers, check out this beautiful little book. With bold illustrations and sturdy pages, children can learn to count to ten and beyond (there’s even a page for 1000!).

How Much Is a Million? by David M. Schwartz, illustrated by Steven Kellogg

Is your child ready to expand their understanding of place value? Are you looking for a way to help them conceptualize larger numbers? This charming book is both fun and educational, giving kids concrete ways to visualize one million. How tall are a million children? How long does it take to count to a million? How much water would a million goldfish need? Find the answers to these questions and more!

The Grapes of Math by Greg Tang, illustrated by Harry Briggs

Adding, subtracting, multiplying, and dividing are important skills for all young children to learn. Finding different ways to look at groups of numbers can make operations work a breeze. The Grapes of Math uses clear illustrations and catchy rhymes to challenge children to find new ways to count larger numbers. Your child is sure to have fun figuring out these math riddles with you.

One Hundred Hungry Ants by Elinor J. Pinczes, illustrated by Bonnie MacKlain

Everyone loves a picnic, including ants! In an attempt to get to the food as soon as possible, the ants keep rearranging themselves into different groups. Two lines of 50 is surely faster than one line of 100, right? What about five lines of 20?

Measuring Penny by Loreen Leedy

Lisa’s teacher asks the class to go home and measure something. What could be more fun than measuring your pet dog? This book discusses multiple standard units, nonstandard units, comparisons, and a myriad of ways to measure. Lisa and Penny have fun at the park, where there are plenty of other dogs to measure, too!

The Lion’s Share: A Tale of Halving Cake and Eating It Too by Matthew McElligott

An ant attends the Lion’s annual fancy dinner, along with a number of other animals. The sweet ant is shocked by the terrible behavior of the other guests. Beyond their awful table manners, they divide a cake with no regard for fairness (although they do give readers a great lesson about fractions along the way!). When the ant offers to bake the king a cake, the other animals try to outdo her, with each doubling the offer of cakes. Kids will be amazed and amused!

Math Curse by Jon Sciezka, illustrated by Lane Smith

What would you do if your math teacher told you that basically everything in life could be a math problem? If you’re the main character in this book, you might feel like you have a math curse. When life is suddenly numbers, fractions, and word problems, how can you possibly think about anything else? When even the character’s dreams become math problems, they realize they have to find a solution.

Sir Cumference and the Dragon of Pi by Cindy Neuschwander, illustrated by Wayne Geehan

When Sir Cumference has a stomach ache, his son Radius runs off to find some medicine. He accidentally turns his father into a dragon, and finds a mysterious mathematical poem to help him undo the magic. While searching for answers, Radius sees circles everywhere, but he needs to figure out how to measure circumference in order to save his dad. He does, of course, and what better way to celebrate than with some pie?

The Boy Who Loved Math: The Improbable Life of Paul Erdos by Deborah Heiligman, illustrated by LeUyen Pham

Young Paul did not live his life like most people. He struggled with seemingly simple tasks, yet he was always thinking about math. As a four-year-old, he enjoyed mentally calculating how many seconds old people were. This story highlights the fact that there is a place in this world for all of us. The genius of this one boy was realized as he traveled the world collaborating with other mathematicians.

What’s Your Angle, Pythagoras? by Julie Ellis, illustrated by Phyllis Hornung

While we don’t actually know anything about the childhood of the famed Pythagoras, Ellis imagines the types of mischief (and learning!) he may have gotten up to. Curiosity can lead us in many different directions, and Pythagoras goes on a voyage with his merchant father. His passion for buildings leads him to learn about angles, and how a knotted rope helped early builders calculate precise measurements. He explores some more, and uses patterned tiles to discover the concept of squaring.  

Preventing Math Anxiety

Researchers from the University of Cambridge recently released a report following their study of math anxiety in primary and secondary students.  [https://www.repository.cam.ac.uk/handle/1810/290514] Their findings illustrate interesting characteristics of children who experience math anxiety, and suggest a potential connection to interactions with teachers and parents.  The interview-based study included 2,700 children in primary and secondary school in the United Kingdom and Italy.  

What researchers discovered

Four general themes emerged from the research:

1. Girls were more likely than boys to experience math anxiety at both the primary and secondary levels.

2. An overall perception that math is more difficult than other subjects contributed to developing anxiety.  Children spoke about comparing themselves to others and receiving poor grades in classes and on tests.  This led to a decline in confidence and seemed to contribute to development of math anxiety.

3. Interactions with adults is a factor.  Children in the primary grades spoke about feeling confused by the variety of methods used to teach certain skills.  Older children felt that negative interpersonal interactions with teachers and parents contributed to their anxiety.

4. Children in secondary school felt overwhelmed by the transition from primary school.  They indicated increased pressure in regards to the difficulty of math content along with more testing and homework.

What might this mean for Montessori classrooms?

The approach and structure of Montessori classrooms is already so different from that of conventional settings; this may serve as a benefit to students learning about math.  As educators it is critical, however, to be open to new research and dedicated to creating an environment that will nurture our students and their learning in the best ways possible.

We might question: why are girls experiencing greater levels of math anxiety?  It could be beneficial to pay close attention to the girls in our classes and be ready to intervene when markers of anxiety appear.

The children in the study expressed frustration as a result of comparisons with peers.  Montessori strives to create an educational environment that downplays competition and focuses instead on intrinsic motivation.  Not asking students to take tests, not giving grades, and not having a sticker chart on the wall that displays who has memorized their multiplication facts can all help with this.

In Montessori schools we recognize that learning is not a steady, linear progression, nor is it the same for different children.  Students work through a series of materials at their own pace; teachers teach small groups or individuals and reteach as necessary, for as long as necessary, without any pressure to move along a predetermined pace.  

It can be challenging at times to compete among schools that take on more traditional methods.  Montessori schools can feel obligated to offer standardized testing and homework.  It may behoove us to recall the success Montessori has had for over a century without tests or homework.  Most importantly, even while finding a balance, we need to keep our children’s development in the forefront of our decision-making.  One question to ask while implementing something new might be, “Is this new structure affecting our students’ attitudes toward math?”

Of course, as children get older we have a responsibility to prepare them for whatever setting they will transition into.  How might we do this without compromising our ideals?  How can we present homework and testing to Montessori adolescents in such a way that they understand what will be expected of them, while continuing to support them in a supportive and non-competitive learning environment?

What might parents do to help prevent math anxiety?

More research needs to be done to determine how parents can help stave off negative feelings about mathematics.  We have a few ideas to share:

1. Make a conscious effort to not emphasize your own math anxiety.  It’s certainly fine to share your experiences with children; this allows them to see that we can be successful in the face of adversity and challenge.  The key is to not dwell or allow any residual math anxiety to affect their own perception.  Make sure to avoid saying things like, “I’m bad at math.”  If you hear your child saying something along those lines, you might encourage them by reminding them that they’re not bad at it, it’s just an area that might feel a little more challenging right now.

2. By all means, engage in math activities with your child!  Just remember to keep them light and fun.  We may have grown up with math drills and rewards for achievement, but a growing body of research is showing us that external rewards are not usually effective.  Math at home should be a fun way for children to see how we use numbers in our everyday lives.

3. Trust the Montessori process.  This one can be hard.  If you are the type of parent who is actively engaged in your child’s education, you are likely to want to teach them whatever you can.  This is great!  Unfortunately, it can be hard to know exactly when a child is developmentally primed to learn a particular skill.  One common example is parents wanting to teach their child how to add larger numbers.  We believe they could grasp the concept of carrying and doing it all on pencil and paper.  While the child may be able to, Montessori curriculum utilizes materials that allow the child to arrive at such a discovery without the assistance of an adult (and often much earlier than they would typically be taught in a conventional school).  By learning first with the hand and figuring it out authentically, a child is able to understand the whys of number manipulation while simultaneously feeling a sense of empowerment and confidence.  Showing children how to use these specialized materials requires extensive training that all Montessori teachers must complete to become credentialed.  

It will be interesting to see what future research learns about math anxiety in children and how we, as adults, might support them further. 

Have you ever dealt with math anxiety?  What do you think might have made your experience different?

As always, please feel free to reach out with any questions, ideas, or to schedule a tour.

Math on the Go!

You already know that reading aloud to your child daily can have a huge impact on their literacy development.  Did you know that doing math together at home is also important?  By integrating math into your daily lives at home, you as parents are teaching your child not only that math really is applicable to our daily lives, but that you value it as an area of study.  Finding a variety of ways to work through problems together prevents children from developing the self-narrative of “I’m not good at math” before it ever starts.  

Looking for tips to get started?

In the Kitchen

While there are likely nights you need to whip up a quick dinner, get everyone fed and off to bed, it can be nice to find ways to invite your children to cook with you sometimes.  Doing so has a host of benefits, including the development of practical life skills, confidence building, and family bonding, but there are also plenty of opportunities to learn about and practice math skills.

Consider what it takes to make a meal.  From reading a recipe, to combining ingredients, thinking about cooking temperatures, and even how long to cook a meal, there are a wide variety of skills your child can experience first-hand: 

• Reading written fractions in recipes

• Comparing differences in volume while adding measured ingredients

• Adding fractions or utilizing fraction equivalencies

• Using multiplication or division when halving or doubling a recipe

• Calculating elapsed time while waiting for a treat to bake

• Understanding units of measurement concerning temperature

At the Store

Shopping is one of those frequent life necessities, and we often have our children in tow.  Turn this family chore into a fun learning experience by incorporating math.  Here are some ideas for a variety of ages:

• Counting specific items

• Identifying numbers on signs

• Estimating costs of items

• Rounding costs of items to the nearest dollar and adding mentally

• Identifying coins and their values

• Comparing price and quantity to determine product value

• Weigh produce on the scale

• Use addition or multiplication to determine cost when buying multiples on an item

• Determine how much change will be received from the cashier

In the Car

Whether you’re making the quick drive to school in the morning or settling in for a lengthy family road trip, it’s possible to incorporate math skills along the journey.  The key is to make it fun and not work!

• Notice numbers on signs.  Talk about place value.

• Similar to the alphabet game, play the number game.  Look for numbers outside and call them out in order.  “I see a 1 on that sign!’  “I spotted a 2 on that license plate!”

• Play a shape-finding game.

• Clue kids into mileage information.  Have them figure out how far you’ve traveled or how much further you have to go.

• Keep track of time.  Solve problems similar to the mileage ones.

• Make your real-life word problems multi-step: ask your child how their answers might change if you need to drive a certain number of miles or minutes out of the way to make a stop.

• Estimate fuel costs, both before you arrive at the pump, and guessing how much the tank will need to fill.

• Skip count together in silly voices.  Count by 2s, 5s, 10s, and more!

The Backyard

Believe it or not, your own backyard is likely full of real-life math opportunities.  Whether you’re gardening, making repairs, or building something together, keep an eye out for things like:

• Size comparisons: which tree is taller? Wider?

• Notice the temperature.  If you’re really motivated, keep track over a week and make a graph.

• Measure everything!  Younger children can stick to non-standard units.  “How many ‘mommy feet’ long do you think this piece of wood is?  Now let’s try your feet!”

• Kids love to use adult tools, so show them the correct way to use a measuring tape.  Start with length, and explore perimeter and area with older children.

• Kids always seem to be collecting small objects.  Use these rocks, acorns, or sticks to count, add, or subtract.

• With older children, use seeds for math before planting.  Show them an array and how it relates to multiplication and division.

• Estimation opportunities are everywhere.  How many leaves are on that branch?  How many insects might we find under this log?  How many dandelions are blooming right now

No matter where you are or what you’re doing, your children love just spending time with you.  Finding simple ways to incorporate mathematical thinking can be a fun way to squeeze a little bit of learning out of an already enjoyable experience

Remember to ask your child lots of questions, but don’t feel like you need to give them the answers right away.  When we discover something for ourselves, the information is so much more powerful.  Of course, if they seem confused or ask for help, it’s okay to model and teach!

Let us know what you learn together!

Math Fact Memorization: Montessori & Current Research

When someone starts talking about memorization of math facts, people tend to have strong opinions.  We all had a variety of experiences as children ourselves, and those experiences coupled with notions of best practices in education can cause for heated debates.  In today’s post we would like to share the results of recent research on the topic.  At first glance, the results may seem at odds with Montessori theory, but upon further examination this couldn’t be further from the truth.  Read on to find out why our materials and methods can provide children with exactly what this “new” information suggests they need.

The Findings

This summer Paul L. Morgan, Ph.D. published an interesting article for Psychology Today.  Morgan works at Penn State as a professor in the education department as well as Director of the Center for Educational Disparities Research.  He and several colleagues (George Farkas and Steve Maczuga) conducted research to investigate instructional practices and their effects on student achievement.  Their specific focus was on first grade classrooms in math.  

The researchers observed a variety of students and classrooms and determined that the only teaching method that had a positive effect on student achievement was teacher-directed instruction, as opposed to student-centered.  [This is the part where those of us who adore Montessori will audibly gasp, but keep reading!]

Morgan, Farkas, and Maczuga define teacher-directed as the following: “Teachers initially demonstrate specific procedures for solving problems, and then provide students with repeated opportunities (e.g., worksheets, routine practice and drills) to independently practice these procedures. Teacher-directed practices should help students increase their procedural fluency in applying explicitly taught and repeatedly practiced sets of procedures to solve mathematics problems, which should result in more effective use of higher order thinking and problem-solving skills.”

What Do Montessori Teachers Do?

According to the researchers’ definition of teacher-directed learning, this is exactly the approach that is taken in regards to students learning their basic math facts in Montessori classrooms.  Our initial reaction is simply a result of semantics and misconceptions.  Do Montessori teachers utilize worksheets and drills?  Well, not exactly, but we still meet the definition in other ways.

One of the most common misconceptions about Montessori education is that the children are let to do whatever they want all the time.  Some people think that choice is the driving force (it is, at least in part) and the students run amok.  Anyone who has spent any time learning about Montessori or observing in a classroom knows the opposite to be true.  Montessori is really all about choice within limits.  Teachers create an environment rich with materials that call to the children, and while they do have lots of decision-making opportunities, they are only provided with options that will lead them to meet desired outcomes.  The same is true for math fact instruction and practice.  

Let’s get down to the facts: in Montessori classrooms, the teachers provide direct fact memorization instruction.  At the lower elementary level in particular, one will find shelves stocked with materials that were designed to aid the process of math fact memorization.  Typically a teacher will give a child a lesson on a material to explain how it is to be used, and then detail their expectations to the child.  Children are generally going to be practicing and recording their facts on a daily basis.  Montessorians believe that while understanding the concept of why we manipulate numbers and having a visual representation helps children in the long run, we agree that when it comes down to it those basic facts really need to be memorized with an emphasis on speed and accuracy.  This is one of the reasons our children are able to solve larger complex operations problems at a younger age than children in many traditional settings.

Bead Cabinet

Taking a closer look at specific materials, how they are used, and their intended outcomes may help to refine these points.

• The bead chains/bead cabinet: Children begin using this material as early as 3 years old.  While the initial purpose is for children to learn to count, during the elementary years that skip counting translates into speedy memorization of multiplication tables.  The transition tends to be fairly smooth, as they’ve already been practicing for years!  Bonus: the same material will help them understand squares and cubes just a little further down the road in their school career.

• Addition and subtraction strip boards: While a student initially uses these materials to explore the concept of adding and subtracting, they quickly notice patterns and build speed as they gain confidence.  These materials are typically introduced at age 5 or 6.

• Multiplication and division bead boards: Like the strip boards, these are initially used by students to gain a basic understanding of the concept.  As they master individual facts, they naturally start to create shortcuts for themselves and a trained Montessori teacher will observe that they are ready to move on to more challenging materials.

• Finger boards: Created for each of the four operations, these materials are essentially wooden versions of the classic fact charts.  Children are slow to fill them in the beginning, but after repeated practice they build speed and accuracy.  Control charts are readily available for immediate feedback, ensuring that even when a child is practicing independently they will be able to know whether they are answering correctly or not.

• Blackline masters: Montessori teachers have blackline masters, which are essentially paper copies of the finger boards, that are available for children.  Children are expected to complete the material and then record the information on the paper version, thus providing another layer of repetition.  

Room for Improvement?

If there is one area that many math teachers (Montessori and otherwise) could stand to improve upon when it comes to math facts, it would be taking the time to target math fact instruction.  Making sure our students are practicing daily, quickly, and accurately is critical, but might we also help them practice smarter?

Sitting down with each individual child to gather a quick assessment periodically can make a huge difference in progress.  While gathering data in this fashion is time-consuming, we may find it well worth the effort.  Teachers can sit with a child and quickly run through a chart of facts, asking the child for answers orally.  Highlight the facts the child can answer quickly and confidently, leaving the others blank.  Teachers make a copy of this sheet so that the student can keep it to reference.  When it’s time to practice math facts, children can focus on the ones they haven’t yet memorized, rather than eating up time going over the facts they’ve already mastered.

If you are interested in reading more about Dr. Morgan’s work, follow the links to his article and the research findings:

https://www.psychologytoday.com/us/blog/children-who-struggle/201808/should-us-students-do-more-math-practice-and-drilling

http://journals.sagepub.com/stoken/rbtfl/J2BxFXoAWRPSo/full