This is very important when it comes to geometry, as research shows that it takes children a long time to be able to picture shapes correctly in their minds, and even longer to be able to mentally “rotate” them. Working with them in a hands-on manner helps immensely.

Where Did Geometry Come From?

Geometry is one of the oldest studies in human existence. The beginnings of geometry can be traced back to ancient Egypt, from around 3000 BC. (You might want to check out my Story of Geometry for a neat way to tell kids about these ancient beginnings). Since geo means “earth” and metria means “measure”, early geometry was all about measuring the earth to determine boundaries of ownership.

Geometry was originally a very practical science (as opposed to theoretical), involving measurements, areas, and volume. Important early discoveries include formulas for determining lengths, areas, and volumes; circumference and area of a circle; area of a triangle; volume of a cylinder, sphere, and pyramid. Quickly, geometry began to be applied to many other disciplines, most notably astronomy. The intersection of astronomy and geometry enabled scientists to determine the placement and movement of stars and planets.

Geometry through the Senses

In Montessori, we start working with shapes and patterns early. The Sensorial area of the 3-6 classroom is really the study of geometry; in elementary, this area becomes geometry. Early geometry materials include the triangle boxes, the geometric cabinet, and the geometric bases and solids. We call these early studies “Exploration of Forms”. (Thanks to Alison’s Montessori for the geometric cabinet picture!)

In these early activities, children are simply learning the names of shapes and beginning to associate the name with the shape. In true Montessori fashion, the early geometry activities also increase hand/eye coordination and fine motor skills as the child traces the outline of the shape with a fingertip or pencil.

Naming & Knowing

After the child has explored forms and shapes, they are invited to understand figures and their details. These are the nomenclature cards (or geometry folders) that comprise much of the 6-9 geometry curriculum. (See picture: Types of Angle Pairs). Additional materials include the geometry stick box and command cards. Now, the child is learning the different parts and types of lines, angles, triangles, and other shapes.

The difference between 3-6 and 6-9, as in any curricular area, is this: the child in 6-9 is discovering the properties of whatever it is they explored sensorially in 3-6. The child begins to understand simple definitions and express concepts in their own words. Besides the ready-made geometry materials (both wooden and printed), I find it helpful to keep small wooden sticks (found at craft stores) and string available in the geometry area for freeform exploration.

The Golden Elements of Geometry

In 9-12, the child is introduced to three great concepts of geometry: congruence, similarity, and equivalence. These concepts are referred to as the “Golden Elements” of geometry. The dynamic aspect of this level includes taking figures apart, moving them about, and then putting them back together in such a way that they show equivalency to another figure. For example, a child might reconfigure a triangle to show that it is equivalent to a particular rectangle.

The nomenclature cards at this level include the Golden Elements as well as terminology for figures, including such terms as height, width, and base, area, and volume. Children at this level will also enjoy researching and learning about famous names from geometry, including Pythagoras, Euclid, and Archimedes. I’m a big fan of using books to teach these sorts of concepts; you’ll find a few geometry titles listed here under the 9-12 age group.

Measuring Our Earth

Geometry should be fun! Shapes are all around us, and we use the concepts of geometry every day without realizing it. By introducing children to geometry step-by-step, we insure their firm grasp of concepts as well as enjoyment of the exploration that comes naturally when we measure the earth and everything in it.

Note: the Comprehensive Lists include a section for geometry, listing all the materials and concepts for geometry in each age group. Very helpful!

What makes them unique? First of all, the introductory materials are all hands-on rather than abstract. Second, they are sequential – each one builds upon the one before. Third, they contain many similar elements (colors for hierarchies, beads, etc.) that enable the child to master new work quickly.

Math begins in infancy. It can be something as simple as mom or dad counting baby’s fingers or toes. In early childhood, children learn to count and start to become aware of patterns and sequences. As they move into the elementary years, they begin to understand how numbers interact with each other, as well as notice how math is such a constant presence in our lives.

Sneaking Math into Sensorial

The very first Montessori math materials are actually in the Sensorial area of the classroom; they include the Pink Tower, Brown Stair, Red Rods, and Cylinder Blocks. At first glance, these materials may not seem to be mathematical in nature. A closer look reveals that each of the materials is based on the number 10: ten cubes, ten rods, ten cylinders.

This emphasis on base 10 (or the decimal system, which is the basis of modern math) is one of the less-apparent features of these materials. The obvious points of interest for the Sensorial materials are coordination, fine and gross motor skills, following directions, sequencing, and grading by size. The fact that Montessori decided to incorporate the decimal system into these materials is a brilliant way to introduce the idea of quantity early on.

Humans learn best through repetition, and respond favorably to familiar concepts and objects. So, it makes sense to use one of the Sensorial materials as the first true Math material. By alternating blocks of blue on the red rods, the child becomes aware of the numerical properties of the work. The alternating colors enable the child to see how each rod increases by one length. Very simple math calculations can be performed with the red and blue rods, also. It’s very easy to see how the “2″ rod and the “3″ rod combine to equal the “5″ rod.

It’s the Concept Behind the Counting

Just as a young child can learn the Alphabet Song without having any idea of the sounds the letters make, a child often learns to count to 10 (or 20, or 100) without actually understanding what the words mean. There are actually three separate concepts that must be addressed: number, quantity, and the relationship of the two together.

One of the best ways for a child to learn all three of these concepts is with the cards and counters. This classic work is fun for kids to do, and has several further applications besides simply counting out the correct number of counters, for instance, learning about odd and even numbers. (See picture: cards and counters).

The golden bead material is extremely important, and indeed all the bead materials are versatile and can be used for everything from simple addition and subtraction to complicated long division. A child who works with the beads in 3-6 will happily find that the same beads, strung on wires, comprise the bead frames which make more complicated problem solving possible.

A Place for Rote Learning

What do I remember about math when I was growing up? Only two things: 1) that I didn’t like it, and 2) how boring it was to memorize the multiplication tables. I would definitely have benefited from the hands-on nature of the Montessori materials, and also from the focus on process rather than simply finding the correct answer.

Still, at some point children do need to commit their “math facts” to memory. There are several ways to do this that are fun and entertaining. Some favorites include extensions like picking up a handful of equations from one of the equation boxes and throwing them back in the box while saying the answer. Kids love to play this game together. What I’ve observed is that if children use the Montessori math materials on a regular basis, they often memorize their math facts without actually setting out to do so.

For some children, mastering math facts is a challenge even in Montessori. They may simply need more rote and repetition than other kids do. In these cases, flashcards and even learning tools like CDs that teach math facts set to music are very helpful.

What About Workbooks?

Is there are place for worksheets and workbooks in Montessori math? Surprisingly, I do believe there is. However, I think they should be used sparingly and only for concepts that aren’t covered in the traditional materials. For instance, time and money concepts, patterning and sequencing, and word problems are all areas where workbooks can be used to “fill in the gaps” that might otherwise occur.

Also, if children will be required to take standardized tests, I believe they should have the chance to use workbooks to become acquainted with testing conventions like multiple choice, fill-in-the-blank, and following written directions.

The Passage to Abstraction

Many times people ask me about the “path to abstraction” and whether or not kids are hindered by doing so many concrete math materials first. Every observation indicates the opposite: kids are better able to understand abstract mathematical concepts because they have first handled the actual materials.

When does the passage to abstraction begin? In my experience, it starts in stages during the 6-9 age period. A child who still needs the materials to do multiplication may already have begun to internalize the concepts of addition and subtraction and not need the materials for those processes anymore. In other words, there’s no one moment when a child moves from concrete to abstract.

It also starts gradually, and is built into the materials themselves. For instance, a child who has been using the small and large bead frames (with colored beads for units, tens, hundreds, and thousands) will then move to the golden bead frame, which is similar except that the color hierarchies are removed. (See picture: small bead frame).

Need More Info?

I highly recommend the math album authored by Sister Mary Motz, my Montessori trainer. It’s available through Nienhuis, and I won’t link to it because they change their links all the time. If you go to Nienhuis and search for “mary motz”, you will find the Montessori Matters Math album, as well as other albums authored by her. It’s just about the most thorough, best written Montessori math album out there.

Thank you to Alison’s Montessori for the pictures of the math materials!

I’ve written quite a bit about Montessori math; here are some helpful posts:

Using Stories to Teach Math Concepts

Yes, Children Can Love Math!

What Can You Do With the Elementary Math Basics?

All About Math Bingo Games

Working Montessori Math into Homeschooling

I got through high school and college with the absolute minimum in math requirements. In fact, joyfully, I had enough math credits in high school that I didn’t even take math my senior year. I took three history classes, though!

A few years later, I took the Montessori training. The training changed everything I had previously thought about learning and education, but no more so than in the field of mathematics. The Montessori math materials, in all of their golden-beaded glory, turned me into a lover of math and math concepts.

Naturally, I’ve wanted my own children to love math and not hate it as I did. I’ve worked hard to keep a positive attitude towards math both as a teacher in a classroom and a homeschooling mom. It must be working, because my kids love math. Nothing we’ve done remotely resembles the stuffy, leaden math curriculum of my own youth.

The Beauty of Montessori Math Materials

What makes Montessori math so different? Much of the difference is because of the hands-on, concrete nature of the materials. Beautiful in their simplicity, they take the most complicated of math concepts and break them into easy-to-understand bits of information. (If you’re not familiar with Montessori math materials, please see the links at the bottom of this post for more info.)

In traditional education, we expect children to be able to mentally envision math operations long before they possess the neurological power to do so. No wonder they struggle to understand and end up frustrated and upset. Montessori materials also contain control of error and clarity of purpose – both characteristics that empower children rather than making them dependent on teachers or textbooks.

How Should Math Look in Montessori?

Mull over this quote from Susan Stephenson of The Michael Olaf Montessori Company as she talks about math in the primary (3-6) classroom:

“Many people misunderstand, at first, what it means to learn math at this age. They remember how they learned the multiplication tables for example—tedious and boring, hours of painful repetition that was certainly not the first choice of activities.

In the 3-6 class, children love to learn the quantities and symbols for numbers in the thousands. They often learn addition, subtraction, multiplication and division with the decimal system and with fractions, simultaneously.

None of this work is required of the children, but it is offered, presented with manipulative materials to one child at a time – by the adult and sometimes another child. There are no teachers lecturing to a group of children who are required to sit still and listen. The children choose this work, and repeat each step with joy and enthusiasm until they are ready to move to the next step.”

Several things she mentions jump out at me. One of the most crucial is that the child chooses their own math activities rather than having the entire class do the same work at the same time. This freedom ensures that children will work as long as they need to on any given material, only moving on when they are thoroughly ready to do so.

Another point she makes is that children can work on several math materials simultaneously. This parallel work brings out the natural symmetry in mathematical operations – what child isn’t delighted to discover that subtraction is the opposite of addition, and that division and multiplication are flipped versions of each other?

If you fear math, or are afraid of some of the more complicated math materials, take a deep breath and see them in a new way. Practice them by yourself as needed, and as you prepare presentations for the children, let the Montessori math materials wash away the unpleasant memories you might have from your own schooling experience.

A Real-Life Example of Montessori Math in Action

Recently, a traditional homeschooling friend of mine was shocked that my just-turned-seven-year-old son had taught himself the multiplication tables. “How did he do that?” she asked incredulously. How did he do that, indeed! Well, he’s been using the Multiplication Bead Board since he was five. After he had done each number several times with that material, he started setting out 15-20 multiplication equations on his rug, putting out the answer tiles, and then checking his work with the Multiplication Finger Charts. He used both materials for about a year.

When he was ready to practice the multiplication facts without needing any concrete material, I put out the Multiplication Wheels and he worked on those on and off for another year. This past fall, he got out multiplication flash cards (on his own) and worked with them until he knew every answer with no mistakes. Not only does he really, truly understand multiplication (2 x 3 is two sets of three), but he did it all of his own volition. I never once told him to choose multiplication materials. I just made them available and helped when he needed me.

The verdict is in: let the child set the pace, and they will choose how and when to learn. The Montessori math materials are just tools, but they are powerful tools that can transform a child – and us right along with them.

Helpful Montessori Math Resources:

Math Materials from Alison’s Montessori
Shu-Chen Jenny Yen’s Online 3-6 Math Album
Montessori Teacher’s Collective Online 3-6 Math Album
Montessori Teacher’s Collective Online 6-9 Math Album
Math Activities, Operations, and Word Problems & Equations.

All of these books are all available through Amazon or other retailers; a quick Google search should turn up many options (including other books by the same authors). I’ve categorized them loosely, but due to the nature of the stories, they can probably move up or down age-wise based on the child’s interests. Most of the books in the 6-9 category, for instance, could overlap with 9-12.

For Preschool (3-6):

Inch by Inch by Leo Lionni

Busy Bugs: A Book About Patterns by Jayne Harvey

Counting Caterpillars and Other Math Poems by Betsy Franco

The Greedy Triangle by Marilyn Burns

The Go-Around Dollar by Barbara Johnston Adams

For Lower Elementary (6-9):

The Coin Counting Book by Rozanne Lanczak Williams

Can You Count to a Googol? by Robert E. Wells

The Great Graph Contest by Loreen Leedy

How Much is a Million?, Millions to Measure and If You Made a Million by David M. Schwartz & Steven Kellogg

Apple Fractions, Count to a Million, and One Hundred Ways to Get to 100 by Jerry Pallotta & Rob Bolster

The Count Your Way through… Series (11 different countries) by Jim Haskins

The Grapes of Math by Gregory Tang

For Elementary (9-12):

The Sir Cumference Series and Mummy Math: An Adventure in Geometry by Cindy Neuschwander

What’s Your Angle, Pythagoras? A Math Adventure by Julie Ellis

The Great Number Rumble: A Story of Math in Surprising Places by Cora Lee

Fractals, Googols, and Other Mathematical Tales and The Adventures of Penrose the Mathematical Cat by Theoni Pappas

“I have a question for you, and I wonder if you can help me out when you have some time. As you know, I am homeschooling my daughter. We use Montessori materials for language and science, but for math I have purchased a homeschooling math curriculum to use. My daughter is in first grade, and understands the concepts well (addition, subtraction, patterns, etc.), however, she is reluctant to do math work. She finds it “boring and too long” (her words).

The curriculum includes practice sheet and assessments that she needs to complete, as well as a lot of repetition of the math problems. I have also tried some games with her to teach her the concepts, but none of these have worked. Math has become a bad experience for the two of us. She complains, whines, and refuses to cooperate. I feel frustrated and helpless. What can I do?”

My answer:

“I’d be happy to try and help. First, let’s step back and look at all the dynamics that are going on. When you say things like “practice sheet and assessments”, little bells go off for me. Both in Montessori and my own experience, a first grader shouldn’t need to be completing those kinds of things. This does lead to the kind of behavior you speak of – complaining and whining – but that is the only way a child this age is able to tell you: “Mom, I don’t want or need to do this!”

Children resist textbooks, workbooks, and worksheets because they sense an inherent irrelevance in the materials. A page of math problems with no real-life application is just busy work, and they know it. In a classroom situation, a child might not verbalize these feelings; they see their classmates doing the worksheets, which makes them feel like “everyone is doing it”, or they might not feel comfortable enough with the teacher to state their true feelings.

With mom, however, it’s a different story! As you’ve seen, they feel no social pressure to comply just because you’re telling them to. As I’ve browsed homeschooling websites, I found a quote by one homeschooling mom (her kids are now grown) who said: “The more I tried to make it similar to real school, the worse it was. It wasn’t until I switched gears completely and let them do the leading that things fell into place”.

Maria Montessori felt that children needed a strong foundation in concrete operations before moving to abstract. By concrete operations, I mean using objects, beads, or tiles to solve math problems rather than just doing them on paper. In this way, children begin to understand the why and how of the math they are doing. Most children (with a few exceptions) respond really well to hands-on math materials, especially if they are given the freedom to choose when, how, and where to work with them.

I’m not familiar with the curriculum you’re using, but if you take 10 different math curriculums, I’m sure that there’d be 10 different ways to teach addition, 10 different timelines for when each skill should be learned, and on and on. In other words, the “requirements” of many curriculums are mostly arbitrary. In Montessori, the math curriculum starts with a hands-on approach to the decimal system and slowly builds off of that.

I would recommend that you lighten up on math with your daughter – put out some hands-on math activities and let her be the one to choose to do them. Trust her to know what she needs to work on. If you can get a copy of Maria Montessori’s book “The Secret of Childhood”, I think you would be fascinated. She never imposes her own ideas on the child; she only observes and sees what they teach her about themselves. It’s awesome.

It sounds like this could be a pivotal moment for you as you decide what to do in the next few months. At a certain point, all Montessori teachers and homeschoolers have to relinquish some control back to the child in order for there to be unrestricted learning. It’s something that I’ve struggled with too.

The “proof is in the pudding”, as they say, and rather than worrying if your daughter gets certain test scores, or completes a certain number of worksheets, focus more on helping your daughter become creative, intelligent, thoughtful, and curious. After all, if she completes a worksheet, it begins and ends there. If she is allowed to learn freely, she can teach herself for the rest of his life.

I don’t want to be preachy, and I know there’s a lot to take in. At the very least, hopefully this will give you more options to pursue as you continue on your homeschooling journey.”

Any thoughts, readers? I’d like to know what you think about this.