Teaching the Grownups How to Count Like Kids

For at least the past three decades, children and mathematics and the question of who does well and why have fueled one of the great debates in education. While research suggests that some children may have a stronger innate “number sense”—an intuitive grasp of quantities which can predict early math skills—the most recent evidence also shows that no one is actually born “good at math.” Genetics play a role, yet depth and quality of early learning experiences are far more important than raw talent when it comes to developing high-level math ability that lasts a lifetime.

But what kind of learning gives elementary students that critical foundation—the kind that supports long-term success in math? 

That’s the million-dollar question that Colorado Academy Lower School Math Specialist Mary Singer has spent much of her professional career examining, as she has sought to align the school’s classroom practices with a growing body of data that shows confident, capable mathematicians are those who haven’t just memorized “math facts” and algorithms, but have also dug deeply into numbers and their relationships to hone their ability to reason mathematically.

 

“We talk about teaching ‘math for understanding,’” Singer explains, “because what we are doing is training students to wonder, to notice, and ultimately to make meaning from numbers—even when there’s no clear path to an answer. Successful problem solvers use what they know to help them figure out what they don’t know.”

 

Indeed, there is no single, well-marked path to teaching math that works perfectly for all children and teachers, notes Singer; if there were, there’d be no debate, no need for constant reevaluation. “Good schools look at their elementary math curriculum at least every five years,” she says. For someone as passionate about math as Singer, the cadence is even more frequent. 

“In my eight years as Math Specialist, I’ve spent weeks at a time every year learning from the nation’s best math instructors and researchers—within educational leadership programs at schools like Harvard and Stanford—about how the acquisition of math knowledge really happens in the brain. That’s just what we do at CA: We keep getting better for our students.”

Joined by many faculty members during those summer programs as well as in multi-week professional development sessions she has facilitated on campus, Singer attests, “I am constantly blown away at our teachers’ dedication.”

 

As a result, after a year’s worth of pilot testing and feedback from faculty, in fall 2025 the Lower School revamped its entire math curriculum from Kindergarten to Grade 5 to place even greater emphasis on the mathematical thinking and problem-solving skills that are ever-more-clearly seen as fundamental. 

For Kindergarten and Grades 1 and 2, i-Ready Classroom Mathematics delivers an instructional design that allows students to take ownership of their learning, engages diverse learners, and builds confidence. From Grade 3 to Grade 5, the inquiry-based Bridges in Mathematics curriculum focuses on developing mathematical reasoning and nurturing a classroom learning community where students gather evidence, employ multiple strategies, explain their results, and develop respect for others’ opinions and contributions.

Faculty and division leadership engaged in intensive evaluation and debate before settling on the new programs. That they chose two different curriculum models rather than a single “off-the-shelf” product, according to Singer, is a reflection of the fact that “our teachers and administrators did a fabulous job digging into the programs and the research and asking, what best fits our students’ needs?”

Mathematical thinkers

Still, it’s never simple to do something entirely novel; that’s as true for children learning to add and subtract as it is for faculty implementing a new curriculum. And that is why Singer spent last summer and this school year helping to support teachers and educate parents and guardians about the methodology and benefits of the new programs.

“Not every teacher feels comfortable teaching math for understanding,” explains Singer, “but I’ve just seen this philosophy grow and thrive in our Lower School classrooms. The work all of our faculty has done over the past years to become strong math educators has just been amazing.”

 

Assuaging teachers’ math anxiety parallels the process of turning students who might fear they’re “not good at math” into confident mathematical thinkers who truly understand the relationships between numbers. 

“If you think about how math is typically assessed,” Singer relates, “it’s whether the answer is right or wrong; as children get older, it’s how fast they can get the right answer. So for a long time, teaching math essentially meant drilling students with flashcards and lengthy problem sets to give them that rapid recall.”

But what is missing from that traditional, anxiety-provoking approach, she continues, is the flexibility and fluency that come when learners realize that while four plus four might equal eight, for example, it’s also possible to “decompose” the number eight into five and three, or six and two. Math isn’t just about digits and operations; it is also intuiting the multiple patterns and connections that underlie them.

 

“As math gets more complicated,” says Singer, and students move from counting strategies in Pre-K and Kindergarten to additive, multiplicative, and proportional reasoning through the elementary grades into Middle School, “children cannot simply rely on memorization; understanding numerical relationships is key.”

 

But for parents who hear words like “mathematical reasoning,” it’s tempting to wonder: Does that mean memorization and mastery of foundations like the “stacking” algorithm for multiplication don’t matter anymore?

Not at all, according to Singer, who offered a series of workshops throughout the fall and winter to give parents firsthand experience with the new curricula. The sessions allowed her to demonstrate how the programs carefully balance the acquisition of foundational math knowledge with the development of deeper reasoning skills required to solve novel problems—and the power of combining the two.

 

In one workshop on Third through Fifth Grade math, Singer had parents try to quickly multiply 99 by 5 without resorting to using the traditional stacking algorithm. They saw that while the two numbers could be challenging to handle as-is, it was much easier to think of multiplying 100 by 5 and then subtracting the difference of five.

“It can be hard for adults to try to ‘back out’ of their years of math knowledge and experience to put themselves in their kids’ shoes,” Singer emphasizes, “but when they do, they realize that understanding numbers empowers young learners.”

 

In another workshop on very early elementary math, parents had to “learn” counting like a Pre-Kindergartner. Instead of the numerals 1 through 10, Singer had them memorize a sequence of nonsense words and then try to solve problems like “nudge plus drift.”

The exercise, she says, forced them to relearn the three elements of a number—cardinality (how many), ordinality (place in a sequence), and nominal or symbolic use (name/label)—and made clear “what a huge accomplishment number sense represents for a child.”

A math community

When Singer first came to teach at CA 20 years ago—she had fallen in love with the school as the parent of two CA students and was invited to consider joining the school’s teacher-internship pathway after stints in corporate finance and marketing—she had her own Fifth Grade homeroom. “As a teacher early in my career, I thought the way we taught math at that time—it was math for understanding even then—was ridiculous. Why not just have kids memorize the facts and formulas and move on?”

But as she learned more about the way children acquire math knowledge—witnessing firsthand the lightbulbs go off as they perceived the deeper connections between numbers—and as the school tapped her to guide the evolution of the Lower School’s math program, she knew there must be a better way to teach math. She was inspired by the subject in the way others are moved by music.

“Everybody has that innate ability to make and appreciate music, just like everybody has the ability to ‘mathematize.’ And to be able to look at a student and see where they are in their journey and nudge them to the next stage and watch them grow—that’s fascinating.”

 

Working with students as Math Specialist for the past eight years—as well as nudging fellow teachers and the parents and guardians of her students along their own math journeys—Singer has seen plenty of evidence that math, like music, indeed has something for everyone.

Though COVID-19’s isolation, the ubiquitous nature of digital screens, anxiety, and ever-shorter attention spans have brought new challenges to the classroom, Singer says, CA students’ performance on standardized math testing has held strong. “We’re not teaching math in a vacuum—we’re very conscious of state and national standards. Just like every other school, by the end of Fifth Grade we expect our students to have mastered their math facts through twelve and the standard algorithm for multiplication.”

Beyond the data, though, is where Singer is tracking the biggest gains. 

“When I interact with kids in my math lessons, I’m actually seeing a strong math community developing. Especially when you get to Third, Fourth, and Fifth Grade, the classroom conversations about math are so much richer than they’ve ever been; our students’ ability to think and discuss different ways of solving problems is so much more nuanced.”

 

She’s noticed many other benefits. Whereas in the past students who were well ahead of or well behind their peers might find limited “entry points” into each day’s lessons, both of the new math programs offer many more ways to either scale up or scale down the depth and complexity of learning tasks. 

“We can look at individual students and see where they need more support or more challenge, and we can now offer that across the board,” explains Singer. The result, she says, is deeper understanding for every learner.

“We want our mathematicians to own their knowledge and to think of themselves as a problem-solver. At CA, we don’t merely teach math; we teach people.”