When it comes to parental worries, there’s something special about math. Maybe it’s because many of us have recurring dreams about a math test we forgot to study for, or because we fear a child’s ninth-grade math level will determine their entire future. Both of these fears are figments, of course, but the anxiety remains.
For all the attention math draws, we can often lose track of the true purpose of learning it. Yes, we all need to understand the principles of mathematics, but within that act, students have a chance to learn the most critical skill of all: the ability to persist.
To raise a child who is actually sturdy in a world of instant answers, we have to protect their right to struggle.
A Healthy Relationship with Challenge
In an era where AI can solve almost any equation in seconds, the value of a human being isn’t in our ability to calculate; it’s in our ability to persevere through complexity. To build that internal drive, we move away from the “unhealthy” obsession with speed and toward what Jo Boaler, author of Mathematical Mindsets, calls a true mathematical mindset.
During my recent parent coffee (which you can listen to below!), I asked the audience to look for Boaler’s six pillars in our math curriculum. These aren’t just academic targets; they are the markers of a healthy relationship with challenge:
- Mistakes: Seeing errors as brain growth rather than failure.
- Struggle: Understanding that the “hard” part is where learning happens.
- Creativity: Approaching math as an open, creative subject.
- Beauty: Appreciating the elegance and patterns in relationships.
- Flexibility with Numbers: Seeing 15 as 10+5 or 3×5.
- Rich Mathematical Tasks: Deep, open-ended problems that invite collaboration.
The Hidden Foundation: Practical Life and Sensorial
Humans are born with mathematical minds. We see it in infants, figuring out how much force they need to exert to get a fist to their mouth, or what angle they need to kick a mobile above their bed.
It is a common misunderstanding that math begins with numbers. In reality, it begins with experiencing and interacting with our surroundings. That’s why, in our Montessori Primary classrooms, before a child ever calculates a sum, they spend time washing tables or polishing silver. By naming each tool and following a consistent, multi-step process, the child builds the precise vocabulary and logical discipline required for later mathematical operations.
These Practical Life activities prepare the child in two vital ways: they require the child to follow a sequence of steps in precise order, and they encourage deep concentration. When you wash a table, if you dry it, then scrub with soap, then clean the crumbs, you will not be left with a clean table. This is the first exposure to “ordered tasks,” which means it’s also the scaffolding for mathematical tasks that work out only when done in the correct sequence.
Following this, our Sensorial Materials prepare the mind more directly. Because they are all made in sets of ten, they provide a physical impression of the decimal system. These are “mathematical mind materials” designed to demonstrate specific relationships through size, dimension, and sequence. They offer experience with and language for the relationships they observe in their environments.
Mathematics: Quantity, Symbol, and Sequence
Every Montessori math material follows a specific logic: we always introduce the physical quantity first (like three beads), then the written symbol (the numeral 3), and, finally, we compare the two to ensure the child can match the abstract number to its concrete reality.
Beyond this sequence, two qualities make these lessons unique:
- Control of Error: The materials themselves indicate when a mistake is made—such as a puzzle piece not fitting or a quantity not coming out even—so children are encouraged to pivot and problem-solve on their own rather than relying on an adult for correction.
- Isolation of Difficulty: Each lesson isolates exactly one new skill or challenge, surrounding it with familiar concepts. By changing only one variable at a time, we allow the child to focus intently on the new hurdle while also allowing teachers to pinpoint exactly where a child is snagged, so we can make targeted adjustments.
The Elementary Transition: The Reasoning Mind
As children move into the Elementary years, their characteristics change. They transition to a Reasoning Mind hungry for intellectual exploration, imagination, and “Big Work, “which is what we call collaborative projects that span multiple disciplines.
The passage toward mastery is well illustrated through Multiplication. It begins in Primary with the “miracle” of the Golden Beads, moves through Memory Work with the Multiplication Bead Bars, and moves toward abstraction with the stamp game and small bead frame. In Elementary, the study of multiplication continues with a number of materials designed to illustrate how multiplication works. Perhaps the most iconic of these is the Checkerboard.
The Checkerboard is a bridge that allows a child to perform multiplication into the billions. But here is where we observe the most important tension in our classrooms: the withholding of the shortcut.
I am often asked: “If they know the answer, why are they still using the beads?” The answer is that we are waiting for abstraction. We want the child to discover the rule for themselves. If I hand them the “trick” for how to do the problem on paper, I rob them of the intellectual victory. We want them to wrestle with the material until the logic becomes so undeniable that they no longer need the wood and glass to see it.
The Geometry of Algebra
To keep the children engaged, multiplication is practiced with many materials and many different ends in mind, including eventually Cubing, which is a complex task most of us didn’t encounter until high school. By physically building a binomial cube, Elementary students don’t just memorize a formula; they hold what a formula describes in their hands. As they do so, they also build their own framework for understanding.
The sequence of lessons in the cubing curriculum starts with a lesson that introduces the materials themselves and, via those materials, the concept of the cube of a binomial, and it ends with the children devising the algebraic formula for binomial cubes.
However, our goal is not to rush through to the formula. The goal is to give the children richer and richer experiences with the multiples, and squares, and cubes that make up a binomial cube – to deepen their understanding of these relationships. We give them opportunities to struggle through the work (and practice lots of multiplication along the way!).
Knowing that they will someday be able to figure out the formula for the cube of a binomial is the carrot that keeps them doing the hard work.
Assessment: Knowing When It “Clicks”
How do we know when they’ve got it? We don’t mark every paper with red ink. Instead, we use:
- Peer Correction: In the Elementary, the “social development” of the age means children often check each other’s work, sparking rich dialogues about why 4×7 is 28 and not 27.
- Observation: Watching how they interact with the materials or with the numbers on the page.
- Skills Checks: Evaluating progress during individual meetings, during lessons, or via benchmarked assessments.
The Long-Term Gift
Our goal isn’t to produce human calculators. It is to raise adults who are intellectually brave. When our students encounter a challenge, they don’t think, “I can’t do that because I don’t know the rule.” They think, “How am I going to figure this out?”
By protecting the struggle today, we ensure they have the confidence to soar tomorrow.
Listen to the full audio of the session here.
As the Director of Education for Elementary and career-long Montessori educator, Minnie relishes the gift that a Montessori education offers to children, families, and society.
