An Introduction to Montessori Math: Place Value as Soulcraft

It's easy to underestimate the power of math, especially for those of us who did not enjoy it when we were at school. But learning math the Guidepost Montessori way not only improves your critical thinking and problem-solving skills—it shapes your character and judgment for the better. VP of Pedagogy Matt Bateman claims the Montessori way of learning math is a form of "soulcraft" (an activity that shapes and nourishes the soul). Read our interview with Bateman to find out how.

Introducing Matt Bateman: Dr. Bateman earned his Ph.D. in Philosophy in 2012 from the University of Pennsylvania, where he studied the history of thought in philosophy and psychology, as well as conducted research in cognitive science. He works closely with the programs team at Guidepost Montessori to ensure depth, quality, and consistency across all our school and at-home programs.

Let's start by talking about some of the key differences between the way math is taught in the Montessori environment versus traditional schools.

Matt Bateman: If you look at the way Maria Montessori approached math and compare it with the way math is taught in traditional schools, you will see a plethora of differences. Here’s one: the starting point of math in the Montessori environment isn't didactic. The child explores various mathematical concepts through a range of scientifically designed learning materials, and starts to independently notice mathematical patterns and relationships. This is because Montessori education approaches math in a way that grasps what math actually is. We were told at school that mathematics means numbers and formulas and methods and tests, but when you really get down to it, mathematics is the study of the measurable structure of the actual, observable universe. Which means you can seed mathematical knowledge with observation and exploration, especially if you design learning materials that perceptually highlight mathematical relationships.

A particular element of the Montessori approach to math which really stands out to me—and strikes me as pure genius—is the way in which children master the concept of place value in the Children's House, by way of a series of carefully constructed presentations and exercises. This not only helps the child become proficient in math, it also helps him develop strength of character; something he will take with him well into adulthood.

Before we take a look at the Montessori approach to "place value", can you explain what "place value" means? (Not all of us were paying attention at school!)

M.B.: So, in the decimal system—which is the system we use on a regular basis in almost all common math—the position of a number determines its order of magnitude. This is place value. For example, $1000 is more expensive than $100, and $100 than $10, because of the position of the digit "1". Those "ones" all mean different things. If you move the digit "1" three spaces to the left, you get "1000", which is a bigger number than when you move "1" only two spaces to the left, which gives you "100". 1000 is bigger than 100.

This is one of the most important aspect of everyday math. It’s a major innovation in the history of math notation. Imagine being alive in the Roman era and having to learn the different names for 4, 40, and 400 (IV, XL, and CD). There’s a structure there, but it doesn’t map very cleanly onto something that matters in mathematical operations. Conversely, place value maps notation onto orders of magnitude. It’s simple, useful, and important. And it’s something that is hard for young children to understand.

In the article, The impact of place value on mathematics, Professor of Education Helene J. Sherman, and other professors, argue that place value is perhaps "the most fundamental concept embedded in the elementary and middle school mathematics curriculum." Do you agree?

M.B.: Yes. Place value is the key that enables foundational mathematical operations (addition, multiplication, and so on) to be done with large numbers. It’s not that hard to add or divide small numbers—but this doesn’t generalize very far. When a child understands place value, they gain increased fluency in operations, because they can do the simpler thing with smaller numbers in a recursive, orderly way with larger numbers.

As the child becomes ready to perform more complex operations, like algebra, fluency in basic operations becomes increasingly important. Otherwise, you constantly get stalled any time a large number arrives on the scene. Moreover, if you do place value right, the child will also gain an implicit conception of “order of magnitude”, which is valuable, in life and in math.

This brings us onto place value in the Montessori environment in particular. How do children master place value in the Children's House? And how do you go from the idea of place value being useful in math to place value being something that helps human beings flourish?

M.B.: Introducing the concept of place value in the Children's House takes the form of the Golden Beads, which are presented to the young child once he has mastered numbers 0 to 10. (Earlier materials which introduce numbers 0 to 10 do themselves hint at a place value within the decimal system, but this is the first time the child sees categories change, from one unit, to one ten, to one hundred, to one thousand.)

The child sees and physically holds the beads in her hands, noticing that one unit is a single bead, a bar of ten beads is a line, a set of hundred beads is a square, and a thousand beads form a cube. Any time you get up to a new “10” of something, the form changes. Beads become one line, lines become one square, squares become one cube. Along with the quantity itself, the shape and weight of the object changes in front of the child's eyes. The child can handle these objects in her small hands, exploring them, manipulating them, feeling their weight and parts.

M.B.: The child is then given enough time and motives for activity by her teacher in order to gain experience with the Golden Beads, and to grasp the different categories before being introduced to more challenging work. The child moves through different materials, such as the Collective Exercises, Stamp Game, the Dot Game, Snake Game — all of which offer her different ways to practice this new concept.

This sensorial approach to learning is geared towards developing a cognitive, habitual, almost physical understanding that going past 9 means that something changes. From 1 to 9 you are piling things. But add one more and you get something new, a new form, a new unity, and new “one”. The child also gets to experience the change on a deep, almost spiritual level, as she feels the change occur. She notices the emptiness of the unit category when it moves from 9 to 10. Zero is empty, zero means nothing.

The Montessori approach to math echoes ancient views of knowledge, which tell us that if you can parrot it—or even if you can do it—it doesn't necessarily mean you know it.Educators need to scaffold a system that approaches learning in a physical, emotional, and even spiritual way, which stirs the learner's soul and becomes a part of her. That sounds mystical, but what it really means is that the child is interested in it and understands it well enough to do the work of fully internalizing it. The foundational concept of place value, which the child learns in the Children's House while her mind is still growing, forms a part of herself. It becomes something that is unerringly true, and so enriches her understanding of the intelligible world.

Not only that, discovering this rule—that going past 9 means that something changes—is something that children delight in! They experience pleasure not only from the activities that lead them to discovering the rule, but from opening up a whole new area of discoveries they can make in the future. For example, in the elementary classroom, when children multiply numbers by multipliers with two digits or more (say, 525 x 30), they demonstrate a clear understanding that 525 is being multiplied by three tens. We don't teach them the shortcut of 525 x 3 and then add a zero—they find it out themselves, and it ends up meaning more to them precisely because the discovery was their own.

So far, you've explained how the Montessori approach to math helps young children gain mastery over place value while experiencing immense joy along the way. How is this a form of soulcraft?

M.B.: The mastery of math is actually just the tip of the iceberg in terms of the profound effects the Montessori Method can have on a child's soul (or "psychology" or "character"—whichever term speaks to you, and means the distinctive qualities a person needs in order to live a good life).

Let me try to explain why Montessori math is a form of soulcraft by telling you what it isn't. If you went to a traditional school, where you learnt how to do column addition, you probably had to learn a set of rules involving carrying the one over when something adds up to more than nine, and so on. (That's the way most of us learnt it.) What do you think that did to your soul, as it were, while you were growing up? You internalized a set of rules about math that was pretty mysterious, that you weren't really sure why they worked, that you might have even been confused by. The point I'm making is that, from an early age, you learnt to follow seemingly arbitrary rules, and probably later discovered that you would be tested on your knowledge of those rules—and even that the rules seemingly really worked. And so it didn't matter why the rules worked, just that you could remember them. That’s one of many, many such lessons that forms the implicit pattern: don’t question the path laid out for you, just trust that it works.

Fast forward to today, visit any number of our 90 schools across the US and Asia, and you will see young children learning how place value actually works. They learn about how numbers fit into the decimal system, and so when they add large numbers together, they can see that each figure falls into a particular category in the decimal system. The sense of order the children gain from understanding the decimal system allows them to see the mechanics of column addition with more perspective than those of us who just wrote out the numbers and carried over the ones. That's a very powerful thing for young children to experience. The children who attend Guidepost schools therefore don't have to just accept the rules of the knowable universe. They can explore them and understand them. So, you can see how the traditional approach to math makes you dependent—you’re expecting someone to deliver to you an algorithm—while the Montessori approach makes you independent—you’re expecting to do work yourself to understand the world. This is what makes Montessori math a form of soulcraft. You become more autonomous by studying mathematics using the Montessori Method. 

M.B.: Furthermore, by studying math using the Montessori Method, children develop a growth mindset and increased ability to gain clarity and understanding in any area of life, not just math. The reason we think this is because we believe that if you can train your mind by studying mathematics, thinking more fluently and precisely with abstract concepts, you will make better decisions and become a better person. To think of math as numbers is to underestimate its role in human progress.

And it's not just Montessorians who believe that learning mathematics helps children develop into adults with strong character. Mathematician Allen Paulos wrote in Innumeracy that the knowledge of mathematics can influence one's judgement in everyday life for the better. In fact, mathematics has always been thought of as a model for human knowing and thinking, since ancient times. Montessori participates in that tradition, and also in the traditions that say that mathematics can and should be learned experientially. 

Certain subjects like place value remain mysteries to most people, but are explored and understood at Montessori schools, as children explore and understand the world — and decide what their place will be within it.

This interview has been edited and condensed for clarity.