A modern Red Riding Hood on her way to Grandma’s house can meander through the woods, or drive down the interstate, or even climb a mountain to come around the back way. All those routes will get her where she wants to go, provided she knows the directions for the different routes and has the appropriate tool, be it a car or a crampon.
But she might pick the best path based on how quickly she needs to get there or how many wolves lurk in the bushes.
For mathematical fluency, students need to quickly recall math facts and procedures—the directions and tools, so to speak. But procedural fluency is more than just memorizing and recalling math facts; it’s choosing the correct path, understanding when and how to use procedures, and doing so “flexibly, accurately, and efficiently.”
That’s where many U.S. students fall short. In comparison with global peers, American teenagers can solve relatively shallow and simple calculations and equations but “have particular weaknesses in demanding skills and abilities, such as taking real-world situations seriously, transferring them into mathematical terms, and interpreting mathematical aspects in real-world problems,” concluded Janina Krawitz, a math education researcher for the Organization for Economic Cooperation and Development, in an 2016 analysis for the Program for International Student Assessment, or PISA.
“These are tasks where the well-known superficial classroom strategy, ‘Don’t care about the context, just extract the numbers from the text and do some obvious operations,’ is definitely bound to fail,” she wrote.
The whole of math achievement is greater than the sum of its parts. Students need to understand the what (factual knowledge), the how (procedures), and the why (conceptual understanding) to progress in the subject.
And concepts from spatial reasoning to algebraic thinking build on each other over time. Research suggests it can be harder for students to develop deep understanding of why math rules and procedures work if they are taught those procedures first, before being taught underlying reasoning.
That can be tricky to get across. In a nationally representative survey by the EdWeek Research Center this spring, a majority of math teachers said they found mathematical reasoning somewhat or very challenging to teach, and more than a third found abstract algebra somewhat or very challenging. (Eleven percent said they don’t teach mathematical reasoning, and 51 percent indicated they don’t teach abstract algebra.)
There’s strong evidence that students need both explicit and systematic instruction to develop automaticity in using basic math facts and to learn how foundational concepts connect.
How students build procedural fluency
Students develop better procedural fluency when they get opportunities to compare and contrast problem-solving approaches and justify the approaches they use in different situations, suggests an ongoing series of studies by Bethany Rittle-Johnson, a professor of psychology and human development at Peabody College in Vanderbilt University.
“To solve problems, we have to figure out what strategy to use when—and that tends to get too little attention,” Rittle-Johnson said. “We ask kids to think about similarities and differences and think about when is each a good strategy?”
It’s important, she said, for students to be able to explain the concepts that make a particular strategy correct or incorrect.
“Just because you teach kids correct ways of doing things, that doesn’t mean the incorrect strategies disappear,” she said. “Students really need help thinking and reasoning through why those are wrong.”
To build deeper fluency, students need:
- Demonstration. For a new concept, teachers should provide clear models of how to solve the problem type, why it works, and how it relates to other types of problems. (For example, draw connections among ratio, fraction, and percentage problems.) Provide examples at different levels of difficulty and complexity.
- Practice. Students work through new strategies and skills in ways that help them understand the bigger picture. For example, as students memorize foundational facts for quick recall (such as basic addition, subtraction, multiplication, and division in elementary grades, or integrals and derivatives in high school calculus), they should have opportunities to discuss how networks of facts relate to an underlying concept or procedure. (For example, understanding the connection between 6x7 and 6x9.)
- Reflection. Students should practice talking through why they choose a particular approach and the steps they take to solve the problem. Group work can help students reflect on different ways of approaching a problem and justify the differences in solutions used for different contexts. Involving the whole class in such discussions, as opposed to calling on a few students, can help reduce math anxiety.
- Feedback. Beyond learning the correct answer, students benefit from understanding common misconceptions and thinking through why they came to wrong answers. However, students learn more when they dig into problems with a clear goal and sufficient detail and highlight the upcoming steps needed to solve them.
Data analysis for this article was provided by the EdWeek Research Center. Learn more about the center’s work.
