The Guidepost Approach to Developing a Child's Mathematical Mind

Math is perhaps the most dreaded subject in conventional schools. Students everywhere bemoan that it’s boring, irrelevant, and utterly mystifying. In studying math, a student often finds that maybe he can memorize and obey the arbitrary algorithms and rules he’s taught—but he doesn’t understand why they work, when they should be applied, or how to apply them in new scenarios.

Math is a form of reasoning, according to Montessori, so every child has the potential to learn it at a high level. Beyond that, math is a way of reasoning about relationships in the world, so the subject should not feel disconnected and irrelevant. Indeed, it should feel rich with meaning for everyday life. Above all, she believed the child could learn math in a way that was deeply engaging and rewarding—it shouldn’t be boring and painful!

In a Guidepost Montessori environment, the child is first introduced to math in Children’s House (preschool and kindergarten). Throughout this program, the child joyfully builds a solid foundation of mathematical understanding, and, as a result, advances farther than commonly thought possible.

By the end of the capstone Kindergarten year, the child can perform all four operations with four-digit numbers as well as fractions—a feat which will not be accomplished by his peers in conventional programs until 3rd grade or beyond. But even more importantly, he isn’t just performing mysterious algorithms, he understands, from the ground up, why and how it all works.

Let’s take a journey through the Guidepost Children’s House math curriculum to discover the core materials and methods that make these achievements possible.

Step 1: Building an Analytical Mind by Training the Senses

Since math is a way of understanding patterns and relationships found in the world, sensorial exploration of the world is the foundation of the math curriculum at Guidepost. The child begins by actively experiencing those patterns and relationships, thereby refining her observational and analytical skills. At the same time, she cultivates an intuitive understanding that she can draw upon when learning math concepts more formally later.

The Pink Tower

The pink tower, for example, is an engaging puzzle scientifically designed to captivate the child and impart crucial math concepts. It consists of a set of pink, wooden blocks that increase in size by 1 cm along each dimension, so that the size of the first block is 1 cubic centimeter and the 10th is 1000 cubic centimeters.

The goal is to build a tower so that the biggest block is the foundation, and the smallest block sits at the very top. The child finds this activity interesting because of its challenge. It’s difficult for her to analyze and select the next smallest block at each step since the size difference is so minute. In the beginning, she sometimes builds the tower incorrectly because she hasn’t yet refined her ability to make the necessary precise observations. Eventually, however, she can build the tower from the ground up, moving progressively from the largest block to the smallest.

This is more than just your average block-building activity, however. Because every block is the same shade of pink, the same texture, and of the same construction, the difference in size is isolated for the child to focus on. In working with this material over time, the child automatically absorbs the foundation of certain math concepts. For example, she’s introduced to the algebraic series of numbers to the 3rd power, the decimal system, and the geometric ideas of volume and area.

Of course, she’s not at the stage to learn these ideas or work with them formally. But her early experiences provide her with a lens for viewing the world that will always remain with her—much as her native language and culture, absorbed at this same age, will always be a part of her. When she later studies these math concepts, at increasingly advanced stages in Children’s House, elementary, and secondary, she can relate what she’s learning to her intuitive understanding. This lifelong intuitive grasp of the world is the power of the pre-math sensorial curriculum, and what fuels the child’s accelerated learning.

Step 2: Learning the Numbers through Real Quantities

The transition to the formal study of math is as gradual and concrete as possible, giving the child the time he needs to fully internalize and build upon his observations and understanding. The first stage of this process, as in conventional programs, is to learn the numbers from 1-10.

The Number Rods

In a Guidepost classroom the child begins by relating these quantities to a skill he’s already had a ton of practice refining: measurement. The first material a child uses in this domain, therefore, is only a slight modification of an earlier sensorial material, the red rods. With the red rods, the child observed the differences in length, i.e. the measurements, in order to then precisely place each in order from shortest to longest.

The child’s first math material, the number rods, are identical to the red rods in their measurements, but instead of being solid red, they are divided into 10 cm sections that are alternating red and blue. The immediate goal is the same: to place them in order from shortest to longest. But now, the child places them in order by counting the individual sections. While he counts, he can see with full clarity that the 2nd rod is made up of two sections that are each the same length as the 1st rod, making it twice as big as the first, the 3rd rod has three sections that are the same length, making it three times as big, and so on.

With this material, the child learns, not only the quantities and their names, but their relationships to one another as well—all in a vivid and sensory-rich way. He experiences the facts, both visually and tactilely, so that his new learning always stays connected to the real world.

Once familiar with the number rods, the child will use sandpaper numbers to learn the print symbol that represent the numbers, following the same method used to teach the alphabet in the language curriculum. And, once the child is familiar with the printed numbers, he returns to the number rods to associate each quantity with its corresponding symbol, lining them all up in order, counting each section, and then labeling each with the appropriate card.

The Spindle Boxes

From there, the child works to see, in an increasingly explicit form, that each quantity is made up single units. He moves from the number rods to a material called the spindle box. This material consists of a set of spindles that the child must count, one-by-one, and sort into a compartment labeled with numbers 0-9. As he counts and reaches a new quantity, the child gathers the spindles and ties them all together with a bright green ribbon.

He can see, even more clearly than with the number rods, that each quantity represents a grouping of individual units. He’s delighted to discover that they are bundled together as one thing, but they contain and represent an exact number of units!

Cards and Counters

In the final stage of this domain, the child goes the rest of the way to see that each quantity is made of individual units. He uses a material called the cards and counters, where he counts out a set of red disks and labels each quantity with a matching card. With this material, the child can clearly and explicitly see all the individual units that make up each quantity and how those units grow across the sequence.

Step 3: Ingraining the Decimal System

After the child has a solid understanding of the numbers from 0 to 10 and what they represent, she has everything she needs to understand bigger numbers, even without yet knowing their proper names. After all, every number—from one to 5 billion—is made up of just those digits!

Just as the child learned her first quantities using materials that clearly demonstrated their relationships, she now uses materials that ingrain the relationships and significance of place value in order to understand the decimal system.

She starts with perhaps the most iconic Montessori math material: the Golden Beads.

The Golden Beads

This material consists of a set of beads that are constructed to show the relationship between 1, 10, 100, and 1000. The first, called the unit, is a single bead, the second, called the 10, is a string of 10 beads fastened together, the third, called the 100, is 10 of those strings constructed to form a square, and the final one, called the 1000, is 10 of those squares constructed to form a cube.

In using this material, the child can clearly see that there is a stark change of shape that occurs at each order of magnitude and, through counting from 1-10, can recognize that this change significantly occurs each time she reaches the number 10 in a category. While playing a series of Simon-Says style games with her guide—placing the unit bead in a specific spot, handing the 10 to the guide, feeling the 1000 with her hands etc.—the child learns each quantity and its name in a way that feels completely grounded.

The Color-Coded Cards

Mirroring the progression with the number rods, the child then learns the printed symbols for each group—the tens, hundreds, and thousands—and eventually associates the two together. She learns the numbers from each category by relating them to the counting she is already familiar with: the numbers from 1 to 10. Twenty is just two tens and 500 is five hundreds, after all. Already primed with these quantities using the number rods, the child can immediately apply this to each category.

To help her even further, each category is color-coded. The units are green, the tens are blue, the hundreds are red, and the thousands, a unit in the next family of numbers, is also green.

Once the child has a physical understanding of these numbers from using the golden beads, knows the symbols using the color-coded cards, and has associated the two together, she begins a series of engaging activities.

Going to the Bank: Beads + Cards

First, she learns to make really big numbers. The size of the number is exciting to the child. Just a little bit ago she was counting from 1 to 10, where the 10th number rod was taller than her, and now she’s creating numbers as big as 85, 850, and 8500! She feels incredibly proud that such big numbers are within her grasp.

To create these numbers, the guide asks the child to go to “the bank”, a table that includes a big assortment of unit beads, strings of tens, squares of hundreds, and cubes of thousands. The guide may tell her she needs to get 6 tens and 4 units, for example, and the child will take her tray to the bank to gather 6 strings of tens and 4 unit beads. When she returns, together they will count what she brought, find the color-coded cards that match (60 and 4), and combine them to create a whole new number: 64.

From there, the child continues to use “the bank” and cards to learn the foundation of operations. The guide will work with multiple children at once, for example, asking each to bring a really big number, counting all the beads together, creating the new number with cards, and then, putting them all together to make really big new number, which they learn is called addition.

Over time, they learn how to “exchange” 10 unit beads for a string of tens, or to exchange 10 strings for a 100 square when the quantities they’re adding require them to carry. Using similar methods, they learn that subtraction means taking away some quantity from a bigger number, that multiplication means adding two or more of the same number together, and that division means creating equal shares of some big number.

The Stamp Game

Eventually, the child progresses to being able to add, subtract, multiply, and divide on her own. For this, the guide introduces her to the stamp game. Using the same color-coding system as the cards, the child is presented with a series of tiles that have 1, 10, 100, or 1000 printed on them. She uses these tiles to create big numbers and perform all four operations on them—including with carrying and borrowing.

Not only is the child now working independently (after a lesson introducing the material and methods), but she has progressed to working far more abstractly. She is not yet to the stage of doing math problems solely on paper, but she has moved away from the physical beads and cubes, and is working entirely with the symbols themselves, including the standard symbols used to represent the operations such as the equal sign and the plus sign. This is a big step in her journey, and, because of all the sensorial preparation she’s had up to this point, she traverses it with ease!

Step 4: Fueling the Passion for Counting

At some point along the way, as the child is learning big numbers and how to perform the various operations with them, he becomes obsessed with counting. He doesn’t just want to learn the names of all the numbers as he comes across them in practice. He is restless and passionate and wants to know the whole system, from start to finish. He wants to be able to count from 1 to 100 and from 100 to 1000!

The 100 and 1000 Chains

To fuel the child’s fire for counting, he is first presented with a series of beads and corresponding labels. There is the hundred chain, for example, which contains 10 sets of 10 beads which are connected together at the ends forming one long chain. There is also the thousand chain which contains 100 sets of 10 beads connected together.

With intense delight and concentration, the child works to count each bead, labeling the first ten individually from 1 to 10, and then labeling the rest by tens, e.g. 20, 30, 40, and so on. And because, even this is not enough to satisfy him, he will not rest until he can count the whole chain backwards too!

Skip Counting with Short and Long Chains

Once the child is familiar with counting one-by-one, he is introduced to skip counting which enables him to count by twos, threes, fours… all the way to counting by nines! Not only is this an exciting new challenge for the child, but, with the help of new bead-chain materials, he gains a sensorial experience that prepares him for later memorization of the multiplication tables, and the concepts of squaring and cubing!

Step 5: The Adventure of Memorizing Math Tables

By the time the child reaches the capstone Kindergarten year, she’s excited to go beyond mere counting. She doesn’t want to add numbers together anymore simply by counting each number individually. She wants to know the crucial combinations and have them at-the-ready, always a moment away from being used. She’s ready to memorize her math tables!

The Snake Game

Like with the other math domains, she doesn’t start with the abstract tables themselves. She starts with materials that introduce and allow her to practice the basic concepts and moves progressively toward the more abstract skill. The most iconic activity at this stage is the snake game.

The snake game consists of a set of golden strings of beads, each with ten beads strung together, a set of colorful strings with 1 to 9 beads, each color-coded so all the strings with 2 beads are one color, all the ones with 3 beads are another, and so on, and a set of placeholder black and white strings with 1 to 10 beads each.

To play the game, the child first creates a fun zig-zag snake out of colorful beads on her work rug. Then, she is given the thrilling task of transforming that multi-color snake into a golden snake! To do this, she must replace the colorful beads, ten beads at a time, with the golden beads, until she has a completely golden snake.

So she begins counting the beads making up her snake, first counting 4 from a lavender string, for example, and then counting 6 from a following string made up of 9 beads. Since she’s counted 10, she now gets to create the first section of the golden snake! She removes the colorful beads, replacing them with one string of golden beads and a placeholder string of 3 beads to represent what was left over from the string of 9 beads she removed.

She continues counting, using placeholders, and replacing sections of the snake with golden beads until she has transformed the whole snake. Without it yet being explicit, she’s getting practice recognizing the addition combinations that come together to make 10!

Addition and Subtraction Strip Boards

From there, the child moves to a more explicit presentation of addition combination using the addition strip board. Mirroring the much larger number rods, this material consists of a set of wooden strips representing the numbers 1-9 that increase in length correspondingly.

With this material, the child works to make and memorize various combinations, from 1+1 all the way to 9+9. The strips allow the child to see and feel how the numbers combine together, as well as how different combinations relate to one another. She learns, for example, that there are many ways to make 10—from 5+5 to 6+4 to 7+3 and so on.

Both the snake game and the strip boards have corollary activities for subtraction, and help the child begin to memorize simple addition in a fun and sensory-rich way. Eventually, the child will move on to using real addition and subtraction charts, working her way to filling in a completely blank chart all by herself!

Unit Multiplication and Division Boards

In parallel with her work on addition and subtraction, the child begins memorizing her multiplication and division tables using the multiplication and division boards.

These boards allow the child to experience the multiplication chart in a sensory-rich form while also solidifying her understanding that multiplication is really just a special kind of addition. When she wants to know 3x3, for example, she will fill in three columns on her board with little red beads where each column contains 3 beads. Then 3x3 suddenly looks a lot like skip counting and the work with chains of beads from the year before! She can add by 3 and realizes with delight that 3x3 is 9!

Just like with addition and subtraction, the child works with these materials before progressing to the more abstract charts themselves. By the end, she knows her multiplication from 1x1 to 10x10 and can fill in a blank chart all by herself.

Step 6: The Mathematical Mind

In parallel and in progression with the child's work with math tables, he begins to work towards completing the operations in an increasingly abstract form. He not only knows, from the ground up, what math means, he is starting to progress to where he no longer needs the materials as physical reminders and supports—he can do math abstractly now without losing touch with the facts and relationships in the world that make it all possible.

When you introduce mathematics through the real, physical sensorial relationships that math-on-paper represents, you can get pretty advanced with surprisingly young children. For instance, it is absolutely possible, and not at all uncommon, to introduce factions to 5-year-olds in the Montessori classroom (typically not introduced until 3rd grade in traditional schools).

Math is ultimately about the real world. And when you introduce it through the senses—through a gradually-deepening process of exploration—it is grasped intuitively and joyfully. It is only when math is introduced as following a senseless, meaningless series of arbitrary manipulations of squiggles on a piece of paper that a child learns it is something he's supposed to dread and avoid.

In our program, children achieve advanced levels of abstract mathematical thought—building up to it, step by step, through a process of induction. They start with things they can touch and feel and see and understand, and everything always connects back to that early experience.

What makes the child's achievement and his joy possible is the scientifically designed sequence of materials that bring the concepts to life, ground them in reality, and guide him step-by-step to an increasingly abstract understanding. In short, what makes it possible is an environment that is especially designed to support the development of the child's mathematical mind.