In school, I was OK in math and even (somewhat) enjoyed it; yet, I didn’t have any interest in taking anything more than the required classes in high school and when I did complicated operations I HAD to write out every single step or else I would get confused. There were many things I really didn’t understand about math – I just memorized the steps one is taught to go through to solve the problem. I would often have a hard time wrapping my mind around what I was supposed to do or why this or that happened because I didn’t fully understand the WHY behind it.

Take, for example, the relatively simple, yet seemingly complex, aspect of “borrowing” in subtraction. Am I the only one who was eternally confused by borrowing? I “got” how you were supposed to do it – you slash through this one little number and write this other number above it, then continue with the problem – but I didn’t know what it meant. Never did a teacher try to explain it to me; I was shown how to do it and then told to practice. Because I didn’t really understand what I was doing, I disliked any problem that involved borrowing and used a calculator as soon as we were “allowed” to.

Fast-forward to my AMI Montessori training. One morning, our trainer discussed the Golden Bead material which is used in Montessori classrooms to introduce children to math operations (addition, subtraction, multiplication, division). Following the philosophy of always starting with the concrete, rather than the abstract, the Golden Bead material is a physical representation of numbers. There are single beads, representing units (1). Ten of these beads strung together in a straight line by a wire, is called a ten-bar and represents a unit of 10. It literally has 10 beads on it. Take ten of these ten-bars and string them together to form a square and you have a hundred-square (again, it literally contains 100 individual beads). Ten of these hundred-squares wrapped together make up a thousand-cube.

*As an aside here, isn’t it cool that the “square” of a number (10 squared = 100) is literally the shape of a square? And the “cube” of a number (10 cubed = 1,000) is, again, literally a cube? Why did no one point this out to me in school? *

When our trainer introduced subtraction, and borrowing, I cringed. He used the materials to work through the problem – I would write out the process here but that makes it seem long and complicated (which it isn’t!), so instead I urge you to find a Montessorian who can give you a lesson. It works much better in person and is *fascinating. *When he got to the borrowing part, he demonstrated and, more importantly, explained the process. It was like a light beamed down from the heavens. *I got it*. It *finally* made sense. In my defense, there was an audible “ooooh” of understanding in the room from all the trainees!. Actually seeing the problem worked out with physical quantities made it much, much easier to understand.

That discovery was followed by many others as I worked with the Montessori math materials. It still amazes me how much I didn’t understand about mathematics. As a student, I started doing math on paper with a pencil; in Montessori the abstract process of math is the* final step* of a long series of exercises. To me, and most traditional school students, numbers on the page are just that – symbols we are taught how to manipulate. To Montessori students, those symbols represent very concrete ideas that they have physically manipulated; they fully understand what they mean, how they work, and why.

Do you know many people who enjoy math? How many people do you know who cringe when doing simple addition on a piece of paper? Math is probably the most despised school subject… and it is no wonder. If all children were to learn math using the Montessori materials, almost every one of them would at least understand, if not actually enjoy, it. Imagine what that could do for technology, industry, society and our world!

*Marcy Hogan holds a Primary diploma from AMI. She lives in Sacramento, CA, along with her husband and two sons. She also writes about parenting and life in general on her blog, Life is Good.*

This lesson is one of my son’s favorites. I had the pleasure of having him teach it to me at one of our recent “parent days” at his Montessori school. I, too, found it fascinating.

I LOVE math….I also was a product of Montessori education through elementary. I can honestly say that I understood math concepts when others did not. Geometry was easy because I had held a sphere, cone, etc. (A cube is not really a cube on a blackline or in a textbook). Place value and fractions were easy beacause I experienced the hands on materials in the primary classes and retained the information. It was mastery and not memorization. Thanks to my Montessori parents, I got to experience math the Montessori way and still love it 38 years later!!!!

My extended day son’s teacher did demos for the parents at an open house night with these (and other) materials… I was floored that by the end of this school year my kindergartener will be doing long division! And beyond that, he’ll “get” it!

Seeing how math is taught in Montessori is what clinched our resolve to keep our children in Montessori as long as we possibly can. When my 6 year old, extended-day son came home with Stamp Game papers having done subtraction in the thousand units, I about fell on the floor. We love the idea of starting with large numbers so they are never “afraid” of them. Simply put, its all simply brilliant!

I almost cried when we started math during my Montessori education. It was incredible to finally understand! I have always been a math-imbesil, so to feel somewhat smart, was amazing 😀 I am still no math-wiz, but it is so much easier now.

I taught traditionally for 11 years before I began my Montessori training. During this presentation I literally began tearing up. Whenever I tried to teach this concept, my students had struggled. Finally I felt like I had been given the missing piece.

I just graduated from High school and I worked in a Montessori class for my senior year. I loved learning their math methods and every other method taught. I hope to be a Montessorian teacher after college.. Maybe having my own school. This should inspire all parents to look into the Montessori way.

Marcy, great post!!! Math in Children’s House is beautiful, and the concepts “carry over” (pardon the pun) so well into the Elementary curriculum! We just learned how to present addition with negative numbers this week, and my jaw DROPPED at the simplicity and ENJOYMENT the children experience by working with the materials and reaching abstraction effortlessly!

My children, (both in Montessori for the last 3 years) absolutely LOVE math. My daughter thinks “racks & tubes” and “long chains” are the bees knees. And it makes me so very happy. Because I have always been so afraid of math, but more worried that I’d pass that anxiety of numbers on to them. When I think of the potential this understanding and love opens for them… it’s just amazing.

Once upon a time, I was a “Montessori bambini” …

And suddenly I was led into a traditional school. I really do not understand why all my classmates hate math – but now I do!

I had the same experience as you. When it came to subtraction and borrowing, I just followed the steps with do idea why until I learned it in my training to be a Montessori teacher (American Montessori). Talk about a light bulb turning on.

Patty

I enjoyed this and feel the same way.

As an aside here, isn’t it cool that the “square” of a number (10 squared = 100) is literally the shape of a square? And the “cube” of a number (10 cubed = 1,000) is, again, literally a cube? Why did no one point this out to me in school?

The simplicity is incredible.

Someday, somewhere I hope to see Montessori golden beads set up to show 10 to the 4th, 5th, 6th, 7th, 8th, 9th, 10th . . .

Does anyone know of this being done?

Hi Marcy!

I was just googling Montessori and mathematics in order to try to explain to someone how the Montessori method differs from standard maths teaching, and loved your post. It was only when I got to the bottom that I realized that I knew you — we met in Geneva.

Anyways, great post!

hi marcy

thanks for such useful work!

i found your pic of the golden beads very clear and would like to use it for a book i have written on education… called ‘cracking the education code: an introduction to curriculum analysis … to be published by pearson…

are there any copyright issues …

best

wayne

I see how this benefits those who are not otherwise math-inclined. However, I am not yet convinced this program is ideal for kids who are gifted in math. My five-year-old daughter, who learned to add, subtract, and multiply on paper at home before entering school (K) at all and had no trouble grasping the concepts, found the manipulates in her Montessori kindergarten to be annoying and called them “baby math.”. I waited for teachers to let her zoom ahead to pencil-and-paper math, but that never happened. I ended up enrolling her in a math class for gifted kids at our own expense so that she would have a chance to learn more weighty math concepts. She finished third grade accelerated math this way but is still forced to do the hands-on baby math at school in first grade. The method is really neat but not needed for every kid. Btw, our school is first-rate and I love very other thing about it, but the math is striking out for us.

HIgh school hire special teachers with degrees in math to teach students. If you want your child to go into sciences and engineering, I believe the public school, which has more monies to fund math teacher salaries is the way to go.

Well, as a former high school science teacher, including calculus-based physics, chemistry, and environmental science, I beg to differ. My experience was that even the most accomplished students relied primarily on memorization to slog through the formulas, which I tried very hard to make applicable to their lives. Now, as Coordinator of a Montessori Adolescent Program, I see the value of the Montessori approach to mathematics as much greater than the traditional approach taught in the public schools. These kids really get the mathematics because they have had experiences since an early age with manipulating objects that make the math concrete. By the time they become elementary students, they’ve internalized this concrete understanding and are motivated to move to more abstract understandings. By the time they become adolescents, they spend their time seeking more knowledge, rarely being satisfied with doing the minimum necessary to get the A.

I can’t speak to the unique characteristics of the classroom in which Jennifer’s daughter, referenced above, was a student. I do know that there is a wide variety of experience, training, and attitude among people claiming to be Montessorians, and the type of training DOES matter. I’m AMI trained, and, as a result, tend to favor their more rigorous training protocol.

As to salary, public school teachers do make a lot more money, but in exchange they are often shackled by pre-packaged curricula, standardized tests that start as early as kindergarten, and teaching loads that prevent them from individualizing instruction for kids who are “above the norm” as well as those who need additional help. In addition, there seems to be a need to ‘reinvent the wheel’ when it comes to the latest fad in educational circles, and the influence of politics must also be considered when it comes to the quality of your child’s educational experiences. It is hard to resist the temptation to ‘teach to the test’ when that is an essential element of your evaluations!

Montessori education still retains its unique character even after more than 100 years, and my own daughter has blossomed in the time that she has been in a Montessori school. In fact, her educational experiences, when compared to mine, gave me the impetus to switch careers and become AMI certified!

I am a volunteer at a Montessori preschool. I was fortunate to be good at, and enjoy, math all my life. I am thrilled to see the preschool children enjoying their work with all the number activities, and I can see they are a a real understanding of how numbers work. When I read here how concepts are taught to older children I realize these children will understand their math, and many will enjoy it.

I would love to see an older child who has ‘played’ with the binomial and trinomial cubes start working with algebra. There has got to be a wonderful ‘aha!’ There.