As teachers, you know math practice is critical to building your students’ understanding of concepts. But there’s more to effective math practice than memorizing facts and recalling answers. To develop foundational math skills, your students must also practice:
- Conceptual understanding—the “why” behind a problem
- Procedural fluency—how the math works
- Application—what happens when they apply the concept to a different problem
If your students only practice recall, they won’t develop a full understanding of mathematical concepts or be able to build on those concepts moving forward. It’s also unlikely they will perform well on high-stakes assessments.
Move beyond Recall for Greater Understanding
“Many teachers and parents learned math by rote memorization, so that’s what they’re comfortable with,” said Joe Flick, an elementary math interventionist in New York. But today's students are learning differently. “Even in Grades K–2, conceptual understanding has to be at the forefront of how we get kids to think about math. It starts at the concrete level, where kids use objects and visuals like ten-frames.” When students learn to add 5 + 1 for the first time, Joe asks deeper-level questions that go beyond the answer, like, “How do you know? Are you sure? Can you show me? Is there another way to do it?”
“Students often look for affirmation from their teachers that their answer is correct, which is why I believe we shouldn’t make a habit of confirming or denying student responses,” Joe added. “I want them to know they are correct because they understand the concept. When they do, their confidence grows.”
Quality Practice versus Quantity
We often equate practice quantity with quality. But “more” doesn’t mean “better”—it’s just more. “'Drill and kill’ turns children away from mathematics at an early age,” said Joe. In an EdWeek article, Homework and Higher Standards: How Homework Stacks Up to the Common Core, researchers found that “much of the homework students are asked to do . . . overwhelmingly focuses on rote learning.” Even the youngest students need more than repetitive recall to learn foundational mathematics principles. Fewer, more thoughtful questions that offer a range of practice opportunities and require students to think deeper and apply what they learn can build upon the rigor of standards.
“Math fluency for first graders shouldn’t be about how quickly they know that 9 + 8 = 17,” Joe said. “That’s just recall. When they understand that 9 + 8 is the same thing as 10 + 7, they’re using their number sense. That’s conceptual thinking.” He often asks his students to apply a concept to a real-world situation. “If you can make math relevant to a 5-year-old, they will care more about it and understand it more deeply.”
But students need time to get there. When they’re put under time pressure, it can backfire. “The timer beeps, and they feel like a failure if they haven’t finished,” Joe said. The National Council of Teachers of Mathematics (NCTM) notes that despite what many believe, timed tests do not assess fluency. Developing automaticity of math facts has its place, but only after students understand the concepts and have developed a high level of accuracy of math facts.
Make Practice Personal and Fun
To effectively reinforce students’ mathematical understanding, it helps to offer a variety of rich practice opportunities. Exploratory play can be an engaging way to foster deeper mathematical skills, particularly for reluctant students. In an NCTM article, Why Play Math Games? educator Kitty Rutherford wrote, “Math games give students opportunities to explore fundamental number concepts. Games encourage students to explore number combinations, place value, patterns, and other important mathematical concepts,” which ultimately helps students deepen their understanding.
Math games also make learning fun. Because everyone learns differently, at different paces, digital interactive resources that individualize math practice can be game changers. “I have a lot of middle school students, and several are scoring on the third and fourth grade levels in math,” said Candace Sanders, a Grade 6 math teacher in Tennessee. “It’s very challenging for me as one person to meet so many different needs within a classroom, so I rely on a digital math program to help.” Her students work independently, yet each of them gets individualized instruction and a breadth of practice opportunities designed for their needs. “If they get stuck, they receive help right on the screen. And they get fun little Brain Breaks when they need them. They’re really engaged and challenged,” she added.
Cross-Train Our Brains
Math practice has changed dramatically since most of us were students, and we now recognize that our brains need to be cross-trained, the same way we cross-train our bodies. A holistic, comprehensive approach to math practice that goes beyond rote memorization and incorporates critical thinking ultimately turns students into more independent thinkers and problem solvers—something the world needs more of these days.
For more on the value of deeper mathematical practice, check out this podcast: Making Math Magical with Liz Peyser.