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.

Montessori Basics: Geometry from the Start

Perhaps it happens one day when your four-year-old comes home from school one day, excited to show you their work for the day.  They proudly show you a perfectly traced pentagon with elaborate, colorful patterns inside that they have created.

Maybe it’s when your eight-year-old casually references acute-angled scalene triangles.

Regardless of when it happens, as Montessori parents, there comes a moment when we become acutely aware (pun intended) of our children’s interesting knowledge of geometry.  We may recall our own study of the subject beginning much later - likely sometime during our high school years and typically not as exciting as our own children depict!  We notice that our children seem to be really ready for the information, which can feel surprising.  Not only are they ready, but the work seems to fill them with joy and satisfaction.

What, exactly, is going on?

As with so many things, Montessori discovered that young children are fully capable, and in fact developmentally primed, to learn about subjects that have traditionally been reserved for much older children.  Geometry is a perfect example.  Read on to discover what this portion of a Montessori education can offer your child.

The Primary Years

From ages 3-6 much of children’s geometry instruction in Montessori classrooms is indirect.  That is to say that while they are practicing crucial developmental skills, they are often doing so through the lens of geometry preparation.  One obvious example, as mentioned above, is with the metal insets.  Children trace a variety of geometric figures including squares, triangles, circles, curvilinear triangles, and quatrefoils, among others.  The main objective of this work is to prepare the child’s muscles for proper pencil grasp and handwriting.  When they have mastered tracing they work to create intricate designs within the figure.  

Primary children are also given a number of simple geometry lessons that allow them to begin naming figures and exploring shapes.  Wooden geometric solids are held and named by the children (cube, sphere, square-based pyramid, etc.).  The geometry cabinet is composed of drawers of related figures; small wooden insets are organized into a polygon drawer, curvilinear figure drawer, triangle drawer, and so on.  Children also use constructive triangle boxes to manipulate triangles in order to form larger triangles and other geometric figures.  The key during these early years is to give children early exposure to geometry and allow them to use their hands to explore these concepts.

The Elementary Years

During the elementary years the Montessori geometry curriculum expands significantly.  Teachers often begin by reviewing content taught during the primary years, but 6-year-olds are ready and eager for more.  This begins with a detailed study of nomenclature.  Using a series of cards and booklets that correspond with lessons given by the teacher, children explore and create their own nomenclature sets.  Topics include basic concepts such as point, line, surface, and solid, but go on to teach more in-depth studies of lines, angles, plane figures, triangles, quadrilaterals, regular polygons, and circles.  For example, when children learn about lines they begin by differentiating between straight and curved lines, but go on to learn concepts such as rays and line segments, positions (horizontal and vertical), relational positions of lines (parallel, divergent, perpendicular, etc.)

Throughout the second plane of development (ages 6-12) the study of geometry continues to spiral and go into more and more depth.  Children as young as seven learn about types of angles and how to measure them.  Eight-year-olds explore regular and irregular polygons, as well as congruency, similarity, and equivalency.  In lower elementary children begin learning about perimeter, area, and volume.

In upper elementary, children begin to learn about the connections between the visual aspects of geometry and numerical expressions.  They apply what they’ve learned about perimeter, area, and volume to measuring real-life objects - including Montessori materials they’ve seen in their classrooms since they were three years old.  They learn about things like Fibonacci numbers and Pythagoras which appeal to their sense of number order and geometric patterns.

Now, when your child comes home with surprising knowledge about geometry content, we hope you have a better idea of where they’re coming from.  If you have any questions or would like to see this type of work in action, please give us a call.

Montessori Basics: How Math Progresses Through the Levels

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. 

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.

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).  

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.

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. 

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.

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. 

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.

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: