Is There a ‘Science of Math’ Too?
A small group of first-graders sits around teacher Rhianna Penner at Tomahawk Elementary School in Olathe, Kansas, whiteboards and markers in hand. They’re trying to find all the different ways numbers can add up to 19—ten and nine, or eleven and eight, or twelve and seven—and write them down.
This group of students is working to master a particular skill—adding and subtracting all the numbers between one and 20 with proficiency and fluency—considered foundational to success in math. In the past, when first-graders arrived in Penner’s class unable to add and subtract one through 20, she often recognized the gaps, but didn’t exactly know what to do to address them.
“I would just come up with something on the fly to try to help them,” Penner said. “But there was no structure there.”
Then, her district stepped in with support. Last year, the Olathe Public School District, located in a suburb of Kansas City, adopted a research-based framework to address skill gaps in mathematics, grounded in what cognitive science says are the most effective ways to teach and learn foundational skills. Teachers learned how to assess and diagnose gaps in learning, followed by how to teach those skills to a wide variety of students, some who are still learning English.
“It’s helping teachers help the students, instead of leaving a big question mark,” Penner said. “It’s easier to explicitly teach something when I explicitly know what needs to be taught.”
A Familiar Problem?
Does this sound familiar? A large number of students unable to reach grade level, missing foundational skills, needing more explicit instruction—might sound a lot like a recent reckoning in how reading gets taught in schools. A reckoning that revealed both the body of evidence on the most efficient ways to learn how to read—often referred to as the ‘science of reading’—and that most schools weren’t using it.
Experts say it’s time for districts to turn their attention to math instruction, now suffering through a parallel crisis—low student achievement, deeply entrenched philosophical battles over how to teach and thousands of teachers, like Rhianna Penner, never shown how to help struggling students.
Facets of the problem and the practices are so similar to reading instruction, say cognitive scientists and educational psychologists, that they are nearly interchangeable.
“Take every single thing that’s been written about the science of reading, and hit ‘find/replace’ for math,” said Sarah Powell, associate professor of special education at the University of Texas, Austin. She also leads a group of psychologists, cognitive scientists and math educators committed to evidence-based instruction, called, fittingly, The Science of Math. “Just as we know there are foundational skills in reading, there is the same thing in math. Schools have been swayed by sexy practices, but that’s not how people learn.”
Riffing off the science of reading, the group hopes to launch a movement, dispelling myths and clarifying misconceptions about math learning using decades of evidence on how to teach math to all students, especially struggling learners.
Poor Student Achievement Often Linked to Poor Instruction
Even before the pandemic, national test scores in mathematics had been basically flat for two decades. Pandemic school closures, hybrid schedules and Zoom learning made things worse. In 2022, the National Assessment of Educational Progress, the “Nation’s Report Card,” documented American students’ greatest loss in math proficiency since 1990.
American students also do poorly in math compared with international peers. American 15-year-olds rank 36th out of 79 countries in math proficiency, and only Turkey shows larger gaps between the highest scoring 8th graders and the lowest-scoring ones.
One big reason for decades of low achievement is inadequate math instruction, say the science of math experts, much of it at the foundational skills level. Understanding numbers and their relationship to one another, being able to add, subtract, multiply and divide fluently, and having the ability to do word problems are crucial to success at math’s higher levels. Those abilities also predict with astonishing accuracy which students go on to meet college readiness benchmarks.
But many classrooms don’t do a good job making sure kids have learned the foundational skills. As is true for reading, most current math instruction is not based on how the brain learns.
Research supports five main areas of math instruction that are linked to improved student outcomes: explicit, systematic instruction; visual representations and hands-on tools to help students “see” abstract concepts; teaching math language and vocabulary (nearly 500 terms by 8th grade); building fluency in math facts (like multiplication tables) and solving equations; and solving word problems.
Students gain vital fluency through practice, Powell said. Practice can often be the most important overlooked piece of math learning. “If you talk to computer scientists or expert mathematicians, they all suggest that learning mathematics is through modeling and practice,” she said. “Talk to someone about how they learned to code or do calculus—it wasn't individualized or nuanced. It was practice.”
Making more schools and teachers aware of this research on what works in math learning is urgent, the group says.
Psychologist Amanda VanDerHeyden, another science of math founding member, said that over the decades, American math education has lost its way. “What schools miss is the opportunity to do greater good for students,” she said. “If they actually understood the science of how people learn.”
The ‘Math Wars’
At the root of math’s instruction problems is a philosophical battle over what good instruction looks like. The current landscape looks remarkably similar to the so-called “reading wars.” One camp of educators embraces a “progressive” or “constructivist” orientation that emphasizes students learning something without much initial input from the teacher, sometimes called “discovery” or “inquiry” learning. Educators view this as opposed to more traditional methods where teachers provide explicit instruction, followed by student modeling and practice.
For decades, teachers have been taught that students must “construct” knowledge on their own through engaging activities and discovery, instead of “receiving” knowledge directly from the teacher.
Large groups like the National Council of Teachers of Mathematics (NCTM) appear to side with the discovery approach, even though it lacks evidence of large-scale success. NCTM president Kevin Dykema said the group advocates for “procedural fluency through exploration,” and find the science of math’s claims “troubling.” Many adults have grown up disliking math and feeling unsuccessful in the subject, Dykema believes, because math classrooms relied too much on memorizing algorithms and not enough on conceptual understanding through exploration.
“When I look at the science of math, I see the “‘I do, we do, you do’ approach,” Dykema said. “We have to break away from that approach.”
But cognitive scientists who study how the brain learns say there are consistent practices that are shown to be effective for most students, and there’s a strong evidence base for the kind of explicit instruction in “I do, we do, you do”—especially for those who are new to the subject or new to a particular set of math skills.
Explicit instruction is made up of three distinct pieces: modeling, which includes a clear explanation and planned examples; practice, whether guided by the teacher or independent student practice; and support, where teachers ask the right questions, students respond frequently and the teacher provides feedback.
“Direct and explicit instruction is what works best for reading and math in learners who are novices,” said Daniel Ansari, professor and Canada research chair in developmental cognitive neuroscience at University of Western Ontario in London, Ontario.
Learning through modeling and practice doesn’t mean teachers don’t provide conceptual understanding along the way—research shows that the two should go hand-in-hand.
Yet Ansari said that’s not what’s happening in many classrooms. It’s common for math curriculum and teacher training to focus heavily on students understanding concepts first, and putting less emphasis on procedures—otherwise known as ‘‘working problems to get the right answer.” But the research is clear, he said, that students need both.
Many classroom teachers, VanDerHeyden said, have been taught that “fluency” is a dirty word, and not the goal of teaching math, driving parents who can afford it to the billion-dollar tutoring industry of Kumons and Mathnasiums. Almost exactly like learning to read, in wealthier schools there is often a shadow education system of explicit instruction and practice happening outside the classroom, provided by tutors and tutoring centers using the research-backed methods.
“We lose sight of the fact that students are novices in math,” Ansari said. “Parents often think their child has a math learning difficulty, when in fact it’s lack of instruction, a lack of practicing math facts.”
The lack of a firm foundation in basic skills, Ansari said, acts as gatekeeper for many children who might otherwise move on to experience the creative, beautiful side of math—and the occupational opportunities it provides.
“We need to give them the basic tools, and then play with them in many different ways so they can explore the beauty of math,” Ansari said. “But first they need the basic building blocks.”
Sparking a movement
Just like the science of reading, Powell and VanDerHeyden are hoping to draw attention to the problem and the solutions, including improving teacher training and getting parents involved.
They emphasize the importance of giving teachers information on how the brain learns. Sometimes training involves undoing beliefs teachers have been taught about how children learn—like students “discovering” a way to add fractions on their own, or withholding how to solve a problem from students so they can engage in productive struggle.
VanDerHeyden recalled a presentation she made to a large group of international scientists and mathematicians a few years ago about American math education. The group of professionals, she said, was dumbfounded at what they heard. “‘What do you mean there are teachers who believe it’s a bad idea to teach students the algorithms?,’” she remembered one mathematician saying, describing the set of steps used to solve a computation. “‘I use them in my work every day!’”
Sometimes it’s teachers’ own math knowledge that needs shoring up.
Unlike middle and high school math teachers, who may enter the field because they love math and are good at it, many elementary school teachers don’t love math. Powell said teacher training often involves building the teachers’ own math knowledge. Elementary teachers benefit when they learn–or relearn–skills like regrouping numbers or tackling word problems, alongside how to teach those skills to students.
The science of math has yet to catch on in the same way the science of reading has, and so far the group is OK with that—innovations in math always trail reading instruction by a few years, they said. And they are used to the pushback from math teachers who say that the research isn’t conclusive, or much too complicated to arrive at a conclusion about which practices are most effective.
For first-grade teacher Penner, “the proof is in the pudding.” Though still early, the new framework and teaching techniques she learned based on the science of math have made a substantial difference, especially for her struggling students. Explicit teaching in foundational skills like composing and decomposing numbers, number bonds and relationships and fluency in addition and subtraction have pushed her classes to number one in math growth in the district from fall to winter, something that’s never happened in her seven years of teaching.
And for that reason, Powell and VanDerHeyden agreed, families can’t wait for the math wars to play themselves out. Taking a cue from Decoding Dyslexia and other parent groups who pushed the science of reading onto the national stage, VanDerHeyden said, “Parents need to get mad about this.”