# All About Montessori Math

Let’s talk math, shall we? I believe that Montessori math is just about the best thing going. Sure, there are lots of other educational methods that have wonderful ways of teaching language, science, history, art, and music, but for me the Montessori math materials are truly unique.

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

Anne Ssaid at May 26th, 2008 at 6:30 am :I’m really enjoying your blog Lori! I’ve sent the link to so many people who are interested in Montessori too. Thanks!

Lori Bournesaid at May 26th, 2008 at 6:48 am :Thank you so much, Anne! I appreciate your kind words. Thanks for stopping by!

AR_Teachersaid at May 29th, 2008 at 11:44 am :This is very interesting. I didn’t realize that there were ten of each of the rods, and cylinders and such. I counted them at my son’s school and thought, “How ingenious!” Everytime I learn something more about the way that Dr. Montessori set up her teaching and learning experiences, I become more and more impressed. I only hope that I can find an elementary school to send my son to when he’s old enough.

Thanks for your blog too, Lori. It is always thought provoking and interesting. I always send it to my wife and usually send it to a few education colleagues who find it interesting.

Lori Bournesaid at May 30th, 2008 at 1:09 pm :Thank you so much for your comments! I really appreciate it. I hope you find an elementary school for your son, also. What a great experience that would be for him!

Anonymoussaid at June 3rd, 2008 at 10:49 am :There is a large gap in my son’s (7 yo) understanding of simple math problems. While he understands things conceptually and has no problem with addition, subtraction, etc. boards, he is having difficulty moving from the visual quantitative learning to 1+1=2 type problems. I have decided to teach him by memorization, which is not what I thought I would do. He has a fantastic memory. So, I’m making flash cards.

Do you have any other suggestions as we move into a more main stream curriculum with him this summer and fall? How do Montessori children typically make this transition? I have a number of friends who also home school and teach Montessori who have found this to be a difficult transition for some children.

Thank you,

Kolein

Lori Bournesaid at June 3rd, 2008 at 12:55 pm :I don’t think the transition needs to be difficult – but it is important to expose your son to traditional-type materials ahead of time, like workbooks and worksheets. A child going straight from Montessori to traditional without any help making the transition will probably struggle.

Buy a few of workbooks for 1-2nd grade and work on them this summer. Come fall, he should be fine.

Also, keep in mind that in Montessori, we honor the child’s pace. We don’t talk about a child being “behind”, because we know that every child develops differently. His traditional teachers might try to use words like that, but you’ll just have to smile and nod and know that he’s simply developing at his own pace.

Anonymoussaid at June 6th, 2008 at 4:37 pm :thank you thank you thank you

Zinnadasaid at March 3rd, 2009 at 5:00 pm :I’m so glad that I found your website. I plan to look at this often for Montessori ideas.

Lori Bournesaid at March 3rd, 2009 at 5:58 pm :I’m glad you found me, too! Thanks for stopping by.

Kerrysaid at October 16th, 2009 at 12:52 am :Lori,

Your site is extremely helpful and I appreciate the time you put into it. Perhaps you’ve already answered this question but I didn’t find it in searching for it.

Are small cubed blocks a sufficient replacement for the beads or are the beads absolutely essential? It does appear that tens of thousands of beads are needed to teach the math so I assume that many blocks would be needed also? I’m not stuck on the blocks but was wondering if it would be a suitable replacement or not. Also, do you know exactly how many golden and other colors are needed?

Thank you!

Lori Bournesaid at October 16th, 2009 at 7:27 am :Hi, Kerry! I’ve had people ask me if they could substitute Cuisinaire rods for the beads, since they are a similar idea. Is that what you’re referring to? I think those would work fine for anything using the bead stair (1-9). Generally I recommend buying a box of 55 each, since there are so many things we use them for.

Keep in mind that the bead stair and the golden beads are two different materials – the bead stair is colored beads strung in groups from 1-9. The golden beads are all golden and found in groups of 1, 10, 100, and 1000.

For work with tens, hundreds, and thousands there’s really no substitute for the golden beads. It’s easy enough to buy golden beads inexpensively on eBay or from a supplier like Alison’s Montessori. Ideally you’d want 10 of each (units, tens, hundreds, and thousands) to be able to use them with presentations.

Kerrysaid at October 21st, 2009 at 3:24 am :Lori,

Thank you for the info. I was actually thinking of the Base 10 blocks or something similar. I’m not familiar with the Cuisinaire rods but I’ll read up on them.

Thanks,

Kerry

Lilisaid at July 8th, 2010 at 6:59 pm :Gracias siempre estoy pendiente de tu blogs saludos

Lori Bournesaid at July 8th, 2010 at 7:18 pm :¡De nada!

Louellasaid at August 1st, 2010 at 6:40 pm :Hi Lori! I would like to bring the montessori concept into our home because I found it to be a very good approach especially in helping my son appreciate math. I send him to a montessori school but he tells me the students are not allowed to touch them and are collecting dust. Could you please give me an advise as to where I could get a montessori guide that I could use for homeschooling him as well as used Montessori materials or DIY sites for making these materials? I saw how much they cost and they’re beyond what I can afford. Plus I live in Asia – most suppliers are in the US.

Lori Bournesaid at August 1st, 2010 at 7:16 pm :Sure, you can find albums (including math albums) here: Ultimate Post About Montessori Albums. And that post includes links to free online albums, so you wouldn’t have to buy them.

For making your own materials, you can print many of the boards and charts instead of buying the expensive wooden ones: Math Materials.

I Googled “montessori math DIY” (for “do it yourself”) and found some great results. I’m guessing if you continued Googling similar phrases (and looked through all the results for “montessori math DIY” you’d find tons of other sites with good suggestions.

Most manufacturers of Montessori materials are in Asia, so if you could somehow find a way to order directly from them rather than a distributor, that could work out great for you.