Many people can remember a time when they were told, or they decided on their own, that they weren’t a “math person,” or they weren’t good with directions, or they couldn’t draw. Such thinking can stifle an as-yet-undiscovered skill or talent. The beauty of the human brain is its ability to continuously change and adapt.
Perhaps more than any other school subject, math seems to engender such misconceptions and beliefs. The subject often triggers self-doubt because learning it requires problem-solving, which inherently involves making and sorting out mistakes. There are also cultural myths to contend with that tie math ability to innate talent, speed, and even family history. In school, math insecurity can flare at any age, especially if a student is struggling with a particular unit or does poorly on a test.
At GFS, math teachers and learning specialists see all of this—anxiety, misgivings, apprehension—as normal parts of a student’s math journey. Strategies for coping with and overcoming these obstacles are exactly what they set out to construct for students in the critical formative years of Lower School. This math journey, with its ups and downs, and joys and struggles, through numbers, patterns, risk-taking, and reasoning, sets the stage for success as students progress to the more complex thinking needed for Middle School math.
“We look at the progression of learning math across a landscape,” said Katie Aument, Lower School Math Coordinator. “It's not a linear progression or a set of skills to check off of a list; it's really looking at how kids move from concrete to abstract thinking and from least efficient to the most efficient strategies. There is a path for everyone.”
Math Paths
A community of young mathematicians: that’s how the Lower School envisions the hands-on exploration of math that characterizes the K-5 years. This collective identity leaves less room for fear or negative thinking about math abilities and helps students forge their own unique places as math practitioners.
“Children’s brains are constantly rewiring and developing, and we explain this to them,” said Lower School Learning Coach Diana Gomez. “We believe everyone is a ‘math person.'”
Sometimes students doubt themselves when they observe a classmate who seems faster at math or appears to know more. Gomez explains that such speed or agility is usually due to more experience and exposure, which can be remedied through practice and atomizing.
Math calls for all kinds of talents and skills: reasoning, creativity, flexibility, organization, and imagination. There are many ways to be a good mathematician, which is why many Lower School teachers ask students at the start of the school year: What kind of mathematician are you?
Charting the Course
Lower School teachers and learning specialists collaborate to tap into each student’s unique math aptitude early on to get them on the right track.
They start by interviewing students each year. Instead of looking at how they perform on a math test or assessment, they dig into students’ understanding of concepts and what strategies they’re using. An inaccurate answer to a math problem might still illustrate understanding of the root concept.
Some students, for example, are able to reason abstractly at a very young age, while others need time and support to build meaning through visual models, contexts, and hands-on experiences. In these cases, teachers provide opportunities to work with manipulatives, drawings, representations, and meaningful mathematical situations that help students make sense of ideas. Their role is to carefully observe students' thinking, pose questions that deepen understanding, and support students in developing increasingly efficient and flexible strategies over time.
To accommodate all kinds of math learners, teachers employ several techniques and tools, including rich tasks, mental math practice, math congresses, and gallery walks.
How it Works
In Danita Knight-Paulis’ classroom, first graders measured picture frames to create a blueprint that would help organizers of the upcoming All-School Art Show set up the exhibits. While the measurements weren’t actually needed, this real-world scenario, known in math education as a “rich task”, gave students purpose. Rich tasks, which can also come from a book or story, also make math more accessible and pave the way for all kinds of learners, from those just beginning to build their math affinity to those who are ready to take on more advanced challenges.
To prime them for their Art Show calculations, Knight-Paulis warmed up the class with problem string exercises. And instead of the answer, she wanted to hear the students’ thought processes, which she then mapped out along a horizontal open number line on the whiteboard.
“What is 42 plus 30?” she asked. “Thumbs up when you know the answer. Make sure you’re doing some thinking.”
Speed was not the goal as the students settled into mentally working out their equations. “Think time” is provided strategically to allow students to work at different paces. In addition to embedding silence into their work, the Lower School math program integrates other Quaker practices.
“There's value in every voice, in every strategy. All the responses are valid, and we are really careful to recognize the rationale behind students’ calculations,” said Aument. “There’s also a lot of social discourse and collaboration.”
After using different colored blocks to measure out the frames, the students reconvened to report their findings and strategies in what is called a math congress. Doing things the “right way” or finding the correct answer were not the primary aims. If there were discrepancies in frame measurements, Knight-Paulis asked the students to share how they arrived at their answer.
Like math congresses, gallery walks help students value everyone’s contributions and graciously accept feedback. Students create a poster about a math task they’ve grappled with and their strategy or big idea. Other students then walk around with sticky notes with drawn-on question marks, exclamation points, or checks. Translation: This is confusing to me, or This is what I agree with, or This is a really interesting idea. The gallery walk then concludes with a discussion.
“By fifth grade, they're writing really thoughtful wonderings, and offering specific suggestions,” Aument noted. “Students then make refinements based on the notes and share their thoughts with one another.”
In Danita Knight-Paulis' first-grade classroom, rich tasks (real-world problems)—such as measuring frames for student artwork—help give students purpose.
Curriculum Coordination
GFS’ math curriculum is carefully formulated with a student’s entire educational journey in mind. Aument and Math Department Head Anne Ross partner together regularly to ensure the coherence of the curricular and philosophical approach in math instruction from Early Childhood all the way up through twelfth grade.
In Lower School, teachers consider what's developmentally appropriate at each grade level when evaluating and evolving the math curriculum. For the K-5 math learning progression, the consensus is that it’s most important for students to be able to communicate their math thinking, read and decipher someone else's thinking, reason abstractly, and try out more nuanced thinking.
Ultimately, teachers want students to develop a deep sense of numbers by understanding the relationships—through factors, multiples, and parts—which, in turn, helps to reveal patterns. At this age, they are eager to interpret and explain the world around them. Math gives them a framework.
A major milestone in Lower School is developing a strong understanding of the place-value system, the foundation of algebraic reasoning, which will lay the groundwork for algebra and algorithms that students tackle in Middle School.
“Algebraic thinking shows up in Lower School because we don't show problems in a static way,” said Aument. “Students regularly work with missing values—whether that’s a start, change, or result—and they build understanding of equivalence and the magnitude of numbers.”
Beginning in Kindergarten and first grade, kids count and organize objects, group them into tens and hundreds, and then look for patterns. In a second-grade pre-algebra unit, students are given equivalencies to determine the value of foreign coins. By beginning to look at equivalents, substituting, and canceling out, they are practicing algebraic strategies.
In third and fourth grade, students progressively strengthen their place-value understanding so that when algorithms are presented to them, they can skillfully explain how the numbers are operating. In fifth grade, the story of a frog-jumping contest is part of the algebra unit. Students come armed with some knowledge: how far the frogs jumped and how many jumps, but not the size of their jumps. To solve the problem, they need to conceptualize the unknown.
About Math
Middle School math teacher Matt Greenawalt ’02 starts the year with his eighth-grade class by giving them a math conundrum with a pattern, but not enough information for a solution. Students spend the entire period working on the problem. They can talk to their neighbors and learn from them. As the period passes, students make interesting observations and organize their information in various ways.
“Especially with math, there's often a belief that there's one correct answer and one way to go about finding it,” said Greenawalt. “But at GFS, there is an understanding that math is about the pathway and the problem-solving that comes along the way. I don't care if you get the right answer as much as how you're thinking about it, organizing your understanding, and processing.”
Middle School math teacher Matt Greenawalt '02 at the 2026 Pi Day Assembly.
In Middle School, students delve into more advanced mathematical concepts, such as algebra, geometry, and statistics. This level-up is fortified by the number sense, adaptability, and problem-solving stamina they developed in Lower School.
New students are able to flow easily into Middle School math classes because teachers craft paths for every kind of learner. In addition to a variety of support structures, including specialists who drop in on classes as needed, teachers offer easier entry points for students who need it, while not slowing down those who are more advanced.
In Upper School there are multiple paths laid out for math learners; calculus and statistics tracks can be taken on their own or mixed; some students take both. Most GFS students go on to take advanced math by senior year which is uncommon in many high schools. Ross attributes this to the school’s belief that math is for everyone.
“Because we design a curriculum that’s highly engaging for every single student and create all different kinds of math experiences, the practice becomes both fulfilling and fun, which has really stirred interest in math across the grades,” she said.
Seventh-grader Charlotte Schumacher reports steadily improving at math as she’s gotten older. She now puts more work into it by completing all the homework, asking questions if she needs help, and preparing for quizzes. As a result, this year’s math class has been the most fun yet.
“I like that as you get older, you are taught new things and new ways to find answers, and you can build on your skills based on previous skills you've learned,” she said.
What kind of mathematician is Charlotte? “One that doesn’t give up and always keeps trying.”
—Eryn Jelesiewicz