How did we, as humans, acquire mathematical abilities? How can the sense of grasping numbers intuitively and the ability to count be trained? We have discussed the important theme of mathematical thinking, which is also a key part of Montessori education, from a more scientific perspective.
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.
The world’s most advanced mathematics education begins at age zero.
Yati
Today’s topic is the “mathematical mind.”
Hiro
What does “mathematical mind” mean in Montessori education? Is it the ability to arrange things in size order with materials or to do matching?
Yati
No. The mathematical mind is involved in everything and is considered a characteristic that humans have from birth to death. In other words, it’s close to a sense. Moreover, it connects with all other senses.
Hiro
So it’s a sixth sense. That means it’s like any sense. If you use it a lot, you become able to use it; if you don’t use it, you lose the ability. The same can be said for all five senses.
Yati
Right.
Hiro
I’m not entirely sure whether “number sense” is a comprehensive ability of the five senses or something else. But since it’s connected to the five senses, I think what underlies number sense is, in a word, “pattern recognition.”
Yati
In the international course, measurement, comparison, contrast, calculation, reasoning, ordering, and sequential processing (executing procedures correctly) were given as examples of the mathematical mind. I don’t think it’s strange to consider that these are executed through sensory pattern recognition.
Hiro
Let’s proceed assuming it’s pattern recognition. Recognizing patterns by sight, by hearing, by smell, by touch. The same goes for taste. With taste, for example, “this is bitter,” “this is delicious,” “red foods are sweet,” “orange means sweet,” “yellow means sour.” That continuous process of recognition like this forms the foundation of mathematical ability. So in the end, using mathematical sense well means organizing within yourself the patterns of things you’ve experienced through various senses.
Yati
That’s exactly what Maria Montessori said about sensory education. During ages 0 to 3, you experience and accumulate various sensory experiences, and during ages 3 to 6, you organize them. Number sense is no exception.
Hiro
I’d like to think about this more scientifically, though.
Yati
I researched various studies and discussions about mathematics education. First, what Maria Montessori said doesn’t contradict the latest research. Babies just hours after birth already have number sense. Also, it’s becoming clear that all senses are connected with number sense in the brain. As for mathematics education discussions, they only go as far as elementary school and preschool. Children, without special education, naturally come up with various algorithms by around age 6.
Hiro
I see.
Yati
Like when counting objects by pointing, the total doesn’t change regardless of the order you point. Or when subtracting, for 8 minus 2 you count 8, 7, 6, but for 8 minus 6 you realize it’s faster to count 6, 7, 8. So the conclusion is that we should use this number sense in both preschool and elementary education. That’s why board games are being introduced in preschool, and Montessori materials for ages 3 to 6 are being reconsidered. But before all that, there’s no discussion about having lots of sensory experiences during ages 0 to 3.
Hiro
We might be talking about the world’s most advanced mathematics education here. For example, in preschool education, they might ask, “When the water level is the same in a tall thin beaker and a short wide beaker, which has more water?” This is completely unsolvable without sensory experience. If you haven’t poured water or juice into a cup yourself and experienced spilling it, you don’t understand what spilling means, and you can’t understand the volume or capacity of that cup.
Yati
I see.
Hiro
At that time you don’t know “this is capacity” or “this is volume,” but later you intuitively understand “which one has more.” So what’s important for ages 0 to 3 is using those senses a lot, and the more you use them, the more the mathematical mind will flourish in the future. Not “flourish” exactly. More like “develop further.” But even “develop further” is really just using abilities you already have.
Yati
Right.
Hiro
Basically, it’s pattern recognition. Various information comes in through the senses. For example, we perceive differences in light frequencies with our eyes and distinguish colors. But colors are something we create ourselves. How things look differs slightly from person to person. Some can’t see green, some have difficulty seeing red, and so on. As information, light reflects and enters our eyes. We perceive that something exists, that there’s a shadow, that there’s light. If light reflects back from every angle, we understand there’s something that reflects there. Confirmed by another sense, we might hear sound, or touching it, “there’s something here.”
Yati
I see. That’s interesting.
Hiro
This “exists or doesn’t exist” heavily emphasizes the tangible aspect of touch. Things you can touch “exist,” things you can’t touch “don’t exist.” But going further, there are things that can’t actually be touched but exist. Small things, for example, electricity, magnetic fields, electric fields. You can’t touch them, but they exist. That’s interesting and scary at the same time. You have to unlearn that once to reach a correct understanding of the universe. Does that make sense?
Yati
It does. Those invisible, untouchable aspects are dealt with at ages 6 to 12. After imagination develops and you can imagine, you explore the micro world and the universe.
Hiro
Right. First, you experience “exists or doesn’t exist” with the five senses, and you make various assumptions, including misconceptions. “This is this,” “this is this,” deciding on your own. At ages 3 to 6, you get a bit cleverer. “Since it’s always like this, this should be like this,” “Since this is like this, it should be like this.” You start being able to calculate a bit: “1 and 2 together make 3.” And “There were 3 but I ate 1, so there are 2 left.” The next stage is that what you processed only through senses, you can now extend in your mind and do something like simulation. You can imagine and act as if “something is there even though it’s not in front of you.”
Yati
Yeah, that’s right.
Hiro
But that’s actually an ability that transcends the limits of the senses, and senses have limits. No matter how much you improve your vision, there are things you can’t see. Small things or distant things are invisible. Stars at the edge of the universe. I want to see them but can’t. But if you increase resolution, you can see more. So “developing imagination” means “imagining things that don’t exist and developing that,” but also externalizing the senses as tools, and if you develop that ability, what you imagined turns out to be correct. That’s the “imagination” trained from ages 6 to 12 onward. Imagination means becoming able to imagine more in accordance with the reality of the universe.
Yati
So, the power of imagination is growing.
Hiro
Ages 3 to 6 come before that. Recognizing patterns and using those patterns to estimate a bit. Being able to calculate. Starting to organize. Ages 0 to 3 aren’t about organizing, calculating, or estimating at all. It’s completely about experiencing various patterns: “If I do this, this happens,” “If I touch here, this happens,” “That was hot,” “That was cold,” “That was spicy,” “That was bitter,” “That was sweet.” Experiencing those sensations and patterns is what’s important.
Yati
For example, cookie-making can start from around 18 months, and this involves truly various elements of the mathematical mind. When measuring, large numbers with units like “150 grams” appear, and “2 tablespoons” might come up too. You mix and the dough becomes one mass. When you rest it in the refrigerator, you feel the cold temperature of the dough. With cookie cutters, you handle various shapes, and while cutting out shapes, you experience counting how many cookies you’ve made. When baking, you experience temperature and time like “180 degrees for 10 minutes.” When they’re baked, children think “I want to share equally with everyone,” so they naturally start doing division. When they repeat such experiences many times, children understand this sequence of steps and start planning and executing on their own.
Hiro
I see.
Yati
It’s actually very important that this is something like cookies that you can actually eat and enjoy. In one experiment with young children, it was found that children could compare numbers more accurately with chocolate than with abstract objects.
Hiro
That’s interesting.
Yati
Our 3-year-old second daughter, who usually struggles to count coins accurately beyond 3, can count caramels accurately up to 15.
Hiro
[laughs]
Yati
Come to think of it, when our daughters were 2 and 6 years old making a tart, the recipe said “roll out the dough to 30 centimeters diameter.” The younger one was rolling with a rolling pin while the older one measured with a tape measure, saying “It grew 1 centimeter. It’s 25 centimeters now. 5 more centimeters to go!” They were doing it together on their own, completely absorbed. They were doing addition, subtraction, and units intuitively. I’m the type who doesn’t worry about those little details in recipes, so it was just when the older one had learned to read and knew how to use a tape measure that she started spontaneously. That’s why she could get so absorbed, I think.
Hiro
Yeah.
Yati
There’s a program called “Rightstart” developed by American educational psychologists. Using board games and such, children of low-income immigrants surpassed the math scores of children in conventional education in just one semester. It’s research that’s like a rediscovery of Montessori education. They had children count the movement of game pieces, calculate the distance to the goal by subtraction, and compare who was likely to win. This is the same thing as “if the tart stretches 5 more centimeters, it’ll be 30 centimeters.” So if you cook and do daily household tasks, you don’t need to prepare special toys or materials.
Hiro
Right.
Yati
I think you can easily include children when adults are casually using their mathematical minds in everyday life. When counting things, point and slowly recite “1, 2, 3.” Show how to count by folding fingers. Show that you can represent “1” with one finger up and “2” with two fingers. In the elevator, say “Let’s press 2” while pressing the button together. Read the calendar or clock together. The same goes for babies. When shopping, you might be carrying them in a baby carrier or pushing a stroller, but you can narrate “We’re buying 2 carrots” while showing them each carrot going into the basket one by one.
Hiro
It’s fascinating that babies can grasp numbers.
Yati
Research has shown that 10-month-old babies, after watching 1 to 3 crackers being put into each of two boxes, can choose the box with more. Don’t underestimate them thinking “they’re still small.” If you properly show them what you’re doing, they understand more than you’d expect.
Hiro
Right. Intuitively.
Yati
I’d be happy if I’ve conveyed that you don’t need to prepare some contrived numerical environment. Our lives are overflowing with numerical elements, and adults just need to naturally introduce interactions with numbers in daily life.
