It’s the end of math class at Forest Hills Elementary School in Sidman, Pennsylvania, and after a short break, fourth-grade teacher Dawn McCall’s students are getting down to business—with an additional 20 minutes of math work.
McCall’s students know the routine: They pair off with a partner—usually a higher-performing student and a lower-performing one—and work through short sets of problems together that target weak or missing skills, like multidigit subtraction or multiplication. One student solves the problem while describing their thinking aloud; the other student offers support and highlights mistakes. Then they switch roles.
This is SpringMath, a short and intensive math intervention that McCall’s school is implementing for all students, regardless of their achievement levels. The targeted daily practice is designed to strengthen foundational skills, and it’s rapidly putting grade-level math mastery within reach for a lot more students, according to McCall. “To succeed at fractions and decimals, you have to know your multiplication and division facts,” McCall said. “If you know those multiplication and division facts, things like reducing fractions is going to be a lot easier. It’s that little extra bit of practice every day.”
Since Forest Hills Elementary began implementing the math intervention at the beginning of 2018—slowly adding a few grades at a time, beginning with fourth through sixth grade—student test scores for some participating groups have steadily improved, according to assistant superintendent Dr. Rob Dill. In 2022, 56 percent of fourth graders at the school scored proficient or advanced on the Pennsylvania System of School Assessment state test; by fifth grade, 76 percent of those same students scored proficient or above. Meanwhile, math achievement across the nation is headed in the opposite direction. Only 36 percent of fourth graders scored proficient in math on the 2022 National Assessment of Educational Progress (NAEP) after a historic five-point drop following pandemic school closures. For the decade before that, scores had hovered around only 40 percent proficiency. Eighth-grade achievement dropped even more—down eight points from 2019, with only 26 percent of students scoring proficient; 40 percent scored below basic.
Now, a growing number of researchers, school leaders, and educators are calling attention to research-based solutions to combat slippage in math, including the crucial role that adequate practice time plays in student performance. “They are considered dirty words, but drill and practice, and explicit instruction on how to procedurally solve math problems, are evidence-based strategies that work,” said SUNY University at Albany math researcher Ben Solomon. There’s often a misconception that rote math practice is dull and uninteresting, but “think about how adults thought phonics was boring, but kids actually feel awesome when they can be successful at learning to read,” said University of Winnipeg professor of mathematics Anna Stokke. “It’s the same with math. Teaching foundational math skills does not need to be mindless and boring. Adults who have already mastered a topic don’t view things with the same sense of excitement that kids do. Kids get excited about math when they feel successful, regardless of the topic.”
The Fluency Question
A key missing ingredient in higher math achievement is math fluency, especially in the basic operations of addition, subtraction, multiplication, and division. Research shows that fluency in these foundational skills is critical to math performance and achievement, but experts and educators say there’s a disconnect between how much practice students need to achieve fluency—especially students who struggle with math—and how much they often get in classrooms. The National Mathematics Advisory Panel includes the importance of practice in basic operations among its top findings and recommendations—”Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall,” the authors write—yet schools and teachers say that, for a variety of reasons, students aren’t getting enough practice to solidify foundational concepts.
Researchers like Solomon say that memorizing multiplication tables, timed practice with feedback, and the dreaded worksheet of problems actually matter to long-term math achievement because they help students build the kind of domain knowledge in foundational skills needed to perform more complex math—and that some kinds of math practice are better than others. Evidence shows that procedural and conceptual understanding go hand in hand and are mutually reinforcing, but experts say that in many classrooms an overcorrection has been made that puts too much focus on conceptual understanding at the expense of procedure. “Conceptual understanding really has been overemphasized to the point that students don’t have procedural skills,” Stokke said.
Lack of procedural skill often keeps students from ever reaching the creative, open-ended side of math. “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,” said cognitive scientist Daniel Ansari, professor and Canada research chair at the University of Western Ontario in London, Ontario. “But first they need the basic building blocks.”
Obstacles to practice time
Math practice gets shortchanged for a constellation of reasons—an overstuffed curriculum and the drive to cover an overwhelming number of standards; test-prep pressure; and instructional philosophies that consider repetitive practice, sometimes negatively referred to as “rote” or “drill and kill,” to be uniformly harmful and to be avoided at all costs.
When Lee Williams, mom to two middle school students in Williamson County, Tennessee, checks in on their homework, she said, the most notable thing is that they’re moving through math at lightning speed—a common refrain among parents interviewed for this story. Both kids are tackling complex concepts in math class, yet they haven’t mastered the fundamentals of arithmetic that make learning more challenging topics possible. “They move to a new topic every week in elementary and don’t ever master the basic functions,” she said. “They start with algebraic concepts before they even have multiplication and division facts at the ready—in third grade.”
In classrooms, teachers say there’s just not enough time for extra practice. Pressured to cover every state standard with equal emphasis, they often can’t spend extra time on fundamentals like memorizing multiplication tables, or even on the standards themselves. “It’s a huge problem,” said Sarah Powell, associate professor at the University of Texas at Austin, who researches math instruction and trains teachers. Some districts go so far as to put out a calendar dictating what every teacher should be working on every week of the year—regardless of how far along (or behind) their students are in the curriculum, Powell noted. “We have trapped ourselves with having so many expectations; at each grade level there are these standards, and you are supposed to cover these 30 things and not let anything go,” she said.
But when it comes to standards, it’s important to recognize that they’re not all of equal importance, said Phil Daro, who helped write the Common Core math standards. Some of the fundamentals, such as single-digit addition, subtraction, multiplication, and division for the youngest students, are critical to spend ample time teaching and have kids practice. “Students need to know foundational math facts fluently,” Daro said. “Practice needs to focus on the foundational skills. If you fill that practice bucket with more and more stuff, that will dilute the amount of time and effort available for practicing the real foundation.”
Adding to the problem, researchers say that popular math curricula, and even some teacher training programs, de-emphasize practice’s crucial role in math learning. “Many of the newer textbooks don’t provide enough practice problems,” said Doug Rohrer, a professor and math education researcher at the University of South Florida.
Where does practice belong?
A few years ago, SpringMath founder Amanda VanDerHeyden overheard a tutoring session where a student was trying to divide 148 by 3. Instead of doing long division, the student skip-counted by 3s all the way to 148, which didn’t work out evenly. After several more attempts, the student finally arrived at the correct solution. “The student has the horsepower, he conceptually understands,” she thought to herself at the time. “But he has been deprived of the most efficient way to get there.”
Like learning any skill, after students learn a particular concept like long division, in order to consolidate that learning and reproduce it in the most efficient way possible, they have to practice until it becomes second nature. But because learning isn’t a linear process, students should regularly be moving between foundational skills and application, with teachers frequently planning tasks that encourage retrieval of that material in a variety of ways. In most math classrooms, says VanDerHeyden, students are required to stop working on a skill too soon, cutting short valuable opportunities for repetition and limiting their ability to lock the learning into long-term memory.
SpringMath is based on a learning framework called the instructional hierarchy, a framework for how students acquire new skills that includes four distinct stages, including acquisition, fluency, generalization, and adaptation. According to this model (and there are a variety of learning frameworks), teaching looks different in the acquisition stage when students are brand-new to a concept, compared with when students can take a skill like long division and adapt it to other things, like complex story problems. A deeper awareness of how learning works, said University of Florida school psychology assistant professor Kathrin Maki, makes it easier to use techniques like inquiry learning at optimal times—but not when kids haven’t yet had enough practice to adapt new skills to new problems.
“Inquiry learning isn’t necessarily bad,” said Maki. “It’s that those techniques are only helpful when kids have basic, prerequisite skills. Something we struggle with in math instruction is trying to implement those [inquiry-based] techniques too soon. Kids who are in the acquisition and proficiency stages, those are kids who need a lot of repeated practice and explicit instruction.”
The goal of practice is to move foundational skills into long-term memory so they become quick and automatic. Since math constantly builds new skills on top of existing ones, it’s “relentlessly hierarchical,” said Maki, and once foundational skills like addition and multiplication facts are locked into long-term memory, learning new, more complex math skills like multiplying fractions gets easier; working memory doesn’t have to work so hard. “Long-term memory is effectively limitless; that’s our superpower,” said researcher and teacher Greg Ashman. “Things that are extremely hard to do when you’re a novice become easy when you’ve got lots of stuff in your long-term memory.”
Best practice for practice
Improving math fluency doesn’t necessarily require endless hours of drilling, research shows. Short but frequent bursts, using a variety of retrieval practice strategies—think flash cards and brain dumps, for example, that get students to identify what they know about a concept, thus strengthening long-term retention—are more effective than longer, less frequent practice. “Teachers tend to focus on getting info into students’ heads,” said cognitive scientist Pooja Agarwal, coauthor of Powerful Teaching: Unleash the Science of Learning. “But a big component of learning is getting the information out of our heads.” Frequent low-stakes quizzes, entry and exit tickets, and some digital games, like FactFreaks, can offer effective retrieval practice.
Retrieval strategies like spaced practice, where students revisit concepts over a period of time with breaks in between, and interleaved practice, where mixing up different types of math problems helps students remember information better than when they work through long blocks of one type of problem, are strongly backed by the science of learning. Interleaving is so effective, University of Florida researcher Rohrer said, because students must decide which problem-solving strategy to use before solving. “In math textbooks, students don’t have a chance to choose a strategy because a typical math assignment is devoted to a single concept,” he said. “Which means that they usually know the strategy for each problem before they even read it.” Interleaving, however, “gives students a chance to learn what they need to know.”
In the Forest Hills school district in Pennsylvania, assistant superintendent Dill said a series of evidence-based reforms, including a new curriculum, have rewired how the district teaches math from kindergarten to high school—but the importance of the extra daily practice can’t be understated. “Kids who are now fourth- and fifth-graders, they’ve been doing this for a few years,” Dill said of the intervention. “And their skill development is far better than we’ve ever seen before.” And in McCall’s classroom, she’s seeing overall math confidence blossom. “They know they can do it,” she said. “They’ve learned those math facts.”