Beyond the Right Answer

11/12/2026

Mike Bowler, Head of School

Beyond the Right Answer

Growing the mathematical mind at Harborlight

I can still picture Professor Starr at Amherst College, puzzling over a particularly interesting math problem, chalk smeared around his chin and mouth. As he thought, his hands would travel to his face, leaving traces of the work behind. He was a wonderful, enthusiastic teacher, visibly caught up in the pleasure of exploring an idea.

I wish I had known how to join him in that excitement.

I had arrived at college thinking I was good at math. Throughout school, I could compute quickly, remember procedures, and reliably get the right answer. My teachers noticed, and I enjoyed being seen as a strong math student. I made my way through Calculus in high school and took the next step in that work at Amherst. I even thought I might become a math major.

But in Professor Starr’s classroom, the familiar rules for success seemed to change. He wanted us to explore the ideas behind the mathematics, consider why something worked, and puzzle through a problem whose path wasn’t immediately clear.

I wanted him to just show me the procedure.

Looking back, I can see how much of my confidence rested on knowing what to do and doing it correctly. My college education had now moved me into abstract mathematics without developing the conceptual understanding that could help me make sense of it. I had learned to follow a path. I was much less comfortable finding one of my own.

And because being “good at math” had become part of my identity, uncertainty felt threatening. What if I couldn’t solve the problem? What would that say about me?

I earned a B- in that class. I never took another math course.

I return to that experience often as an educator. What would have helped me see an unfamiliar problem as something worth exploring? What experiences might have prepared me to share Professor Starr’s curiosity?

Those questions are part of what makes our mathematics work at Harborlight this year so meaningful to me.

Building on a strong foundation

One of the things I find most compelling about Montessori mathematics is that conceptual understanding comes first.

Children begin with something they can experience. They hold quantities, compare lengths, compose and decompose numbers, and explore relationships through carefully designed

materials. The materials give mathematical ideas a physical form that children can investigate. As their understanding develops, they move toward the symbols and abstractions that represent those ideas.

That foundation is something I was deeply missing in my own education. I could perform procedures, but I did not always understand the mathematics beneath them. Montessori offers children an opportunity to build that understanding from the beginning.

Our responsibility is to attend carefully to how each child makes that journey.

This year, we are working with Suzy Yakes, a current Harborlight parent and mathematics consultant and coach, and Katie Ibes, a longtime national Montessori curriculum consultant, to strengthen and revitalize mathematics across the school. Together with our teachers, we are examining our lesson sequences, the experiences children need, and the connections that support their understanding from one level to the next.

We remain firmly grounded in Montessori’s holistic understanding of the child. We are also responding to children’s lives today, the range of learners in our classrooms, and what current mathematical research can contribute to our practice.

Children arrive with different experiences of movement, materials, play, technology, and everyday problem solving. Following the child requires us to notice those experiences and needs, and to respond thoughtfully.

As a faculty, our guiding question is: How do we remain deeply faithful to human development while evolving our mathematical practices to meet the children in front of us now?

Making the connections visible

An important part of this work is helping children connect concrete experiences with abstract mathematical ideas.

Even with beautiful materials, those connections deserve intentional attention. How does a child connect what they have built with what they draw? How does that drawing connect to a written equation? Can they explain what each number and symbol means?

We are working number sense routines and opportunities to use and connect mathematical representations into our lesson sequences. A brief routine might invite children to consider how they see a quantity, find different ways to compose a number, or explain a mental strategy. Representational work asks them to connect materials, drawings, diagrams, spoken explanations, and symbols.

These practices give teachers opportunities to hear children’s thinking and help children make their understanding visible. They also help build a bridge from manipulating a material to understanding and using an abstract idea.

Across Harborlight, this work takes different forms:

In Infant and Toddler, we are looking closely at the embodied foundations of the mathematical mind. Long before children write numerals, they explore space, order, size, and relationships through movement and real experience. Our work includes recognizing the mathematical significance of those experiences and making it more visible to teachers and families.

In Children’s House, we are examining the foundations of numeracy, including subitizing: recognizing how many objects are in a small group without counting each one. Seeing five as a group of two and a group of three, for example, helps children explore how numbers are composed. Teachers are developing intentional experiences and games that support this understanding alongside Montessori materials.

Our Lower Elementary teachers are working with Suzy, an experienced math coach, to deepen their understanding of how children learn fractions and develop a thoughtful progression of lessons, beginning with our third-year students. Together, they are exploring how Montessori materials, visual models, and hands-on experiences help children understand what fractions mean, how they relate to one another, and why fractions that look different can represent the same amount—building a strong foundation for more advanced math.

The Upper Elementary team is exploring research-based best practices in mathematical problem solving and developing a thoughtful sequence that helps students build their skills over time. The goal is to strengthen students’ ability to approach both familiar problems and new challenges with confidence, flexibility, and independence, ultimately applying their mathematical thinking to increasingly complex, real-world situations.

In Middle School, Laura Davidson is introducing Socratic Math Seminars, giving students opportunities to deepen their understanding through mathematical dialogue. Students explain their reasoning, ask questions, consider different approaches, and examine one another’s ideas. These conversations encourage them to make their thinking visible, support their conclusions, and revise their understanding when a classmate offers a new perspective. Alongside number sense routines and number talks in the seventh-grade sequence, this work helps students grow as confident, flexible mathematical thinkers who can reason independently and learn with others.

Across all levels, we are also examining the transitions: what children bring into their next classroom, where connections are strong, and where we need to build them more deliberately.

This is a process of inquiry and design. Teachers, consultants, and school leaders are learning, trying ideas, observing children, and refining together. We expect questions and adjustments as we work toward a strong, coherent mathematical experience throughout the school.

Learning to stay with the question

When I think back to Professor Starr, I remember someone who found real joy in a problem that required thought. The puzzling was part of the pleasure. I wish I had understood then that uncertainty could be an invitation.

I wonder whether I could have shared more of that excitement if my mathematical foundation had been different. If I had developed a deeper understanding of the concepts beneath the procedures, perhaps I would have had more ways into an unfamiliar problem. Perhaps I could have stayed curious longer, without interpreting uncertainty as evidence that I wasn’t good at math after all.

That possibility is part of what draws me to this work.

For Harborlight children, I hope mathematical confidence grows from experience: I can make sense of this. I have ways to begin. I can try something, explain my thinking, and learn from what happens.

We want children to develop accuracy, fluency, and deep understanding, along with the willingness to keep thinking when an answer is not immediately clear.

Our anchor is the whole child. The mathematical experiences we offer should help children become increasingly capable, creative, and independent in their lives—ready to meet a worthwhile problem with curiosity, and perhaps even a little of Professor Starr’s delight.

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