Montessori materials make the Montessori method unique. Montessori mathematics education uses learning materials that are far more intuitive and superior to those in existing education. As a result, children develop their mathematical sense through self-education.
We’d love to hear your thoughts. Feel free to share your comments!
Profile
Yati Obara
Editor-in-Chief, Scientific Montessori
Based in Japan, born in 1989. CEO of Motherhand and Co-Director of the nonprofit think tank Polymath Research. Holds an M.Eng. and is an AMI-certified teacher. Focused on Montessori developmental theory and AI. Mother of two.
Hiro Obara
Publisher, Scientific Montessori
Based in Japan, born in 1990. CEO of StudyX and Co-Director of the nonprofit think tank Polymath Research. Works as a software developer and game designer. Father of two.
Reliving the history of human mathematics through Montessori math materials.
Hiro
Many people think the essence of Montessori education is the materials, but the first thing I need to say is that materials are just one element that makes up the environment. They are not absolutely necessary. For example, there is essential work for living, like cooking, growing plants, or caring for animals. The parts directly connected to daily life are the necessary and sufficient conditions. If you do those properly, you will naturally acquire the abilities you need.
Yati
Right.
Hiro
When cooking, some people do it by feel without looking at a recipe, but if you try to follow a recipe, you use quite a bit of mathematical ability. You measure quantities, and the ratios of seasonings determine the spiciness, saltiness, and sweetness. That can only be understood through the senses, and robots still cannot do it. But even small children can do such difficult things. They can slice bread, add their favorite fillings and ham, and eat it. So doing that kind of “work” is what is important first.
Yati
This does not change from ages 0 to 3.
Hiro
Materials are convenient tools for environments where such work is not possible, like when you want to cook but cannot, or when you live in the city and cannot touch plants. Now, with smartphones and tablets and lots of digital apps, I personally think Montessori materials are quite similar to them.
Yati
What do you mean?
Hiro
Materials are tools made of wood and metal that allow children to deepen their thinking and discover various things through play. But without technology, you cannot cut wood to the same length. They standardize natural wood, paint it all, and harden it. In a sense, that is industrially produced while ignoring the nature of the wood. It is quite digital. There is no drama of how that wood grew and was transported, so it ends up extremely inorganic. There is no handmade warmth; it is symbolic.
Yati
I see. But there is still an appeal to materials as concrete objects that digital apps do not have.
Hiro
The drawback of digital apps is that you do not use your hands. Just swipes and touches. Video game controllers also have low diversity in how hands are used. With Montessori materials, you can play by arranging large objects, picking up small things, and transferring liquids, using your hands in diverse ways. That is the fundamental point. There are not many systematically prepared things like that, so if you can use them, you should.
Yati
Right.
Hiro
But it is true that they are expensive. Too expensive to introduce at home. They cost a lot because they standardize specifications, ignore the nature of wood, and do metal processing and painting. They are forcing something that is not natural. Shipping is also difficult and not environmentally friendly. So please understand that materials are not perfect either.
Yati
Right.
Hiro
“Montessori education equals materials” is not correct. That said, compared to conventional kindergartens and elementary schools, materials provide much higher learning efficiency and essential understanding. So let’s examine that.
Yati
This time we will look at math materials, which are designed to repeat the history of human numbers. Stanislas Dehaene, the researcher who proposed “number sense,” said in his book “The Number Sense” that ideally every child should be able to retrace in their mind a greatly condensed version of the history of mathematics, along with the motivations that drove it. Why? Because the evolution of mathematics is also the evolution of the brain.
Hiro
What do you mean?
Yati
The latest research uses technologies like fMRI to roughly identify where in the human brain the primitive number sense that animals also have is located. It is thought that through evolution, humans developed from this primitive number sense brain region as a starting point, deepening connections with various brain regions governing vision, hearing, language, space, and time, eventually developing highly abstract mathematics like modern mathematics. In other words, ideally children develop their brains and number sense by repeating human history. And math materials make that possible.
Hiro
So it is aligned with brain development.
Yati
The first math material is called “Number Rods.” There are 10 wooden rods, ranging from 10 centimeters up to 100 centimeters in 10-centimeter increments, color-coded in red and blue every 10 centimeters. What is amazing about this is that it simultaneously provides children with two concepts at once: quantity, meaning how much, and ordinal numbers, meaning what position. There is often debate among math experts about whether to introduce quantity or ordinal numbers first, but Number Rods introduce both at once with a single material.
Hiro
Two birds with one stone.
Yati
Furthermore, before these Number Rods, children become familiar with length using “Red Rods,” which is a version of the Number Rods that is all red. This is another excellent point.
Hiro
Excellent how?
Yati
Here, try to picture a small number, “1.” Now picture a large number, “100.”
Hiro
Okay.
Yati
Actually, we unconsciously have brains that map numbers to space. For small numbers, we become aware of the lower left, and for large numbers, the upper right. There is a number line in our heads. That is why at Paris Charles de Gaulle Airport, where higher gate numbers are arranged to the left, many people get lost.
Hiro
I see.
Yati
In short, “space” and “numbers” are closely related in humans. Getting back to the point, when the Red Rods used to introduce spatial length are upgraded and used to introduce numbers, it is smooth for a brain that is trying to connect space and numbers. It also looks like a number line extending. By the way, in some cultures the direction is reversed, with larger numbers on the left, but Number Rods can accommodate either culture just by changing the arrangement.
Hiro
So spatial perception is also being used.
Yati
And as an activity, children trace these long Number Rods with their fingers while counting “1,” then “1, 2,” then “1, 2, 3,” then “1, 2, 3, 4,” matching quantity with number names.
Hiro
Even someone who cannot see can understand numbers.
Yati
Indeed. The next material can also be understood without sight. It is called “Sandpaper Numerals,” which are wooden boards with numerals cut out of sandpaper and pasted on. Children trace them to learn numerals. After that, they match the numeral boards with the Number Rods, connecting numerals to quantities. This allows children to accurately match quantity, number words, and numerals from 1 to 10.
Hiro
That is well thought out.
Yati
Actually, this method of introducing numbers is also effective for chimpanzees. A chimpanzee named Sheba succeeded in matching quantities zero through nine with numerals over 2 years, according to Boysen and colleagues in 1996. Sheba started by placing biscuits one at a time on a tray divided into 6 sections. In the next stage, she learned to match the number of black dots on cards with the number of biscuits, and eventually could match the number of dots on cards with numerals. Another chimpanzee could even do fraction addition, and the tool used to express the answer was a divided disc, according to Woodruff and colleagues in 1981. This also matches the Montessori material for learning fractions. The fact that it is effective for non-humans too shows how user-friendly Montessori materials are.
Hiro
So they are monkey-friendly too.
Yati
Getting back to Number Rods. Because they are simple, they allow children to discover various number concepts on their own depending on their stage of number sense development. For example, combining the 5 rod and the 3 rod makes 8, so they can do addition. Finding combinations that add up to 10 is number composition. Making all combinations of 10 hints at the formula for the sum of consecutive natural numbers.
Hiro
Hmm. Like a universal seasoning.
Yati
After Number Rods, we use “Spindle Boxes.” This is an activity where you count spindles and bundle them with rubber bands to make a group. After bundling, you place them in the corresponding numbered compartment of the box. Defining 1 spindle as “1,” you learn by moving your hands that collecting, say, 3 of them into one bundle is “3.” Nothing goes in the compartment marked 0. In other words, you learn that “0” means “nothing.”
Hiro
Using small stones as tokens, matching 1 stone with “1” to count is human history itself. Representing 10 small stones with 1 large stone is acquiring the “base principle.” Combined with the “place value principle” of assigning positions, like ones place, tens place, and so on, the acquisition of these two major principles is extremely important in the history of human mathematics. For children to discover this themselves by moving their hands with materials is essential. If you realize that the “base” of a logarithm is just that “large stone” idea, college entrance exams get a lot easier.
Yati
Right. After Spindle Boxes comes “Cards and Counters.” This involves arranging small counter-sized balls in 2 rows corresponding to each numeral. It is a material for discovering odd and even numbers.
Hiro
Discovering parity, an important theme in number theory, around age 4 would make for an enjoyable life.
Yati
Next is the “Memory Game of Numbers.” It is a game for multiple children. You draw a slip with a number written on it and bring back that many of something. In other words, you let children define what counts as 1. This completes the introduction of the numerals 0 through 10.
Hiro
In human history, people counted and recorded wheat and cattle, but in modern times, we can freely count all sorts of things. How fun.
Yati
After the numerals comes the decimal system, then consecutive numbers, mental arithmetic, the process toward abstraction, and fractions. Materials in the three-to-six-year-old environment cover up to 7-digit arithmetic operations and fraction operations. There are finely graduated steps leading up to those 7-digit calculations.
Hiro
How does it start?
Yati
First, 1, 10, 100, and 1000 are provided with “Golden Beads,” where the physical size is proportional to the quantity. What is provided along with the Golden Beads is the material for the place value principle I called one of the two major principles. This is also interesting. They are numeral cards. For example, to express 2025, you stack the card for 2000, then the card for 20, then the card for 5. Together, they read 2025.
Hiro
I see, the cards literally line up.
Yati
In the stage after Golden Beads, the shape differences by number disappear, and everything becomes same-sized squares, called “stamps,” with 1, 10, 100, and 1000 written on them. The next stage represents 1, 10, 100, and 1000 with just dots. After that, the abacus is introduced. Step by step, from concrete to abstract, we trace the evolution of human numbers.
Hiro
The evolution of arithmetic technique from abacus to written calculation is inevitable, but you should not skip steps.
Yati
Also, though I cannot cover it all here, geometry is included in “Sensorial Materials.” For example, geometric solids like spheres, cylinders, and triangular pyramids start with lots of touching and feeling.
Hiro
Not with worksheets.
Yati
Getting back to math materials. Since numbers depend on language, there are stumbling points by culture. Another noteworthy aspect of Montessori materials is how brilliantly they avoid these.
Hiro
How so?
Yati
For example, in English, the words for eleven, twelve, and thirteen are irregular. Japanese is much easier to understand because it is based on the decimal system: ten plus one is “juu-ichi,” and ten plus two is “juu-ni.” So even in English-speaking countries, they introduce eleven and beyond with regular naming like Japanese. Using Golden Beads, when counting groups of 10, you count “one ten, two ten, three ten.” So 11 is understood in English first as “one ten and one.” After that, a different material called “Seguin Boards” is used to teach the correct reading, “eleven.”
Hiro
Interesting workaround.
Yati
There is actually research that asked American and Chinese children how high they could count. At age 4, Chinese children could count to an average of 40, while American children could only count to about 15. It was clear they stumbled on the irregular naming of 13 and 14, according to Miller and colleagues in 1995. Montessori education considers even these fine details.
Hiro
That is interesting. Because English speakers are at a linguistic disadvantage for calculation, they invented computers and actively use them. In the West, they have no hesitation in having AI solve calculation problems.
Yati
Similarly, Chinese and Japanese number names are short, easy to say, and easy to remember, so calculation is faster and memorizing multiplication tables is not that difficult. Furthermore, in China, the approach is to remove the “1 times” table, and by rearranging so smaller numbers come first, you only need to memorize half. If you memorize “two times three equals six,” you do not need to memorize “three times two equals six.” Such shortcuts in mental arithmetic are actively incorporated in Montessori education.
Hiro
That is rational.
Yati
Foreigners describe Japanese multiplication tables as being like poetry. I realized again that I love Japan’s number culture. For example, the square root of 5, which is two point two three six zero six seven nine, is remembered with the phrase “Fuji san-roku ni oumu naku,” which literally means “parrots cry at the foothills of Mount Fuji,” where the syllables encode the digits.
Hiro
That is a great mnemonic.
Yati
Learning that English speakers struggle greatly to memorize multiplication tables, and that recalling them takes more time and effort than for Japanese or Chinese people, really changed my view of brain training. For English speakers, it really is harsh training. In Japanese, “seven times eight is fifty-six” is recited as “shichi-ha gojuu-roku,” a quick rhythmic phrase. The English version, “seven times eight is fifty-six,” is much longer and harder to chant. That difference adds up over hundreds of multiplications.
Hiro
Just as parrots do not actually call at the foothills of Mount Fuji, Westerners do not like brain training. Though they do like strength training.
Yati
So as I said at the beginning, materials are not everything, but because they are symbolically organized, they are perfect for organizing the sensory experiences accumulated from ages 0 to 3, and they also compensate for disadvantages from cultural and language differences. I hope you have understood that they really are well designed.
Hiro
In the sense of advancing mathematical understanding more intuitively without depending on language, it would be great if the Montessori approach became standard. It would have very positive effects on preschool education, elementary education, and even higher education. There would be fewer people who hate math. I also found it convincing that the founders of technology companies like Google and Amazon, which handle astronomical volumes of data, came from Montessori backgrounds.
