Research shows that an emphasis on memorization, rote procedures and speed impairs learning and achievement
In December the Program for International Student Assessment (PISA) will announce the latest results from the tests it administers every three years to hundreds of thousands of 15-year-olds around the world. In the last round, the U.S. posted average scores in reading and science but performed well below other developed nations in math, ranking 36 out of 65 countries.
We do not expect this year's results to be much different. Our nation's scores have been consistently lackluster. Fortunately, though, the 2012 exam collected a unique set of data on how the world's students think about math. The insights from that study, combined with important new findings in brain science, reveal a clear strategy to help the U.S. catch up.
The PISA 2012 assessment questioned not only students' knowledge of mathematics but also their approach to the subject, and their responses reflected three distinct learning styles. Some students relied predominantly on memorization. They indicated that they grasp new topics in math by repeating problems over and over and trying to learn methods “by heart.” Other students tackled new concepts more thoughtfully, saying they tried to relate them to those they already had mastered. A third group followed a so-called self-monitoring approach: they routinely evaluated their own understanding and focused their attention on concepts they had not yet learned.
In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment. Further analysis showed that memorizers were approximately half a year behind students who used relational and self-monitoring strategies. In no country were memorizers in the highest-achieving group, and in some high-achieving economies, the differences between memorizers and other students were substantial. In France and Japan, for example, pupils who combined self-monitoring and relational strategies outscored students using memorization by more than a year's worth of schooling.
The U.S. actually had more memorizers than South Korea, long thought to be the paradigm of rote learning. Why? Because American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.
To help change that, we launched a new center at Stanford University in 2014, called Youcubed. Our central mission is to communicate evidence-based practices to teachers, other education professionals, parents and students. To that end, we have devised recommendations that take into consideration how our brains grapple with abstract mathematical concepts. We offer engaging lessons and tasks, along with a wide range of advice, including the importance of encouraging what is known as a growth mindset—offering messages such as “mistakes grow your brain” and “I believe you can learn anything.”
The foundation all math students need is number sense—essentially a feel for numbers, with the agility to use them flexibly and creatively (watch a video explaining number sense here: https://www.youcubed.org/what-is-number-sense/). A child with number sense might tackle 19 × 9 by first working with “friendlier numbers”—say, 20 × 9—and then subtracting 9. Students without number sense could arrive at the answer only by using an algorithm. To build number sense, students need the opportunity to approach numbers in different ways, to see and use numbers visually, and to play around with different strategies for combining them. Unfortunately, most elementary classrooms ask students to memorize times tables and other number facts, often under time pressure, which research shows can seed math anxiety. It can actually hinder the development of number sense.
In 2005 psychologist Margarete Delazer of Medical University of Innsbruck in Austria and her colleagues took functional MRI scans of students learning math facts in two ways: some were encouraged to memorize and others to work those facts out, considering various strategies. The scans revealed that these two approaches involved completely different brain pathways. The study also found that the subjects who did not memorize learned their math facts more securely and were more adept at applying them. Memorizing some mathematics is useful, but the researchers' conclusions were clear: an automatic command of times tables or other facts should be reached through “understanding of the underlying numerical relations.”
Additional evidence tells us that students gain a deeper understanding of math when they approach it visually—for instance, seeing multiplication facts as rectangular arrays or quadratic functions as growing patterns. When we think about or use symbols and numbers, we use different brain pathways than when we visualize or estimate with numbers. In a 2012 imaging study, psychologist Joonkoo Park, now at the University of Massachusetts Amherst, and his colleagues demonstrated that people who were particularly adept at subtraction—considered conceptually more difficult than addition—tapped more than one brain pathway to solve problems. And a year later Park and psychologist Elizabeth Brannon, both then at Duke University, found that students could boost their math proficiency through training that engaged the approximate number system, a cognitive system that helps us estimate quantities.
Brain research has elucidated another practice that keeps many children from succeeding in math. Most mathematics classrooms in the U.S. equate skill with speed, valuing fast recall and testing even the youngest children against the clock. But studies show that kids manipulate math facts in their working memory—an area of the brain that can go off-line when they experience stress. Timed tests impair working memory in students of all backgrounds and achievement levels, and they contribute to math anxiety, especially among girls. By some estimates, as many as a third of all students, starting as young as age five, suffer from math anxiety.
The irony of the emphasis on speed is that some of our world's leading mathematicians are not fast at math. Laurent Schwartz—who won math's highest award, the Fields medal, in 1950—wrote in his autobiography that he was a slow thinker in math, who believed he was “stupid” until he realized that “what is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant.”
A number of leading mathematicians, such as Conrad Wolfram and Steven Strogatz, have argued strongly that math is misrepresented in most classrooms. Too many slow, deep math thinkers are turned away from the subject early on by timed tests and procedural teaching. But if American classrooms begin to present the subject as one of open, visual, creative inquiry, accompanied by growth-mindset messages, more students will engage with math's real beauty. PISA scores would rise, and, more important, our society could better tap the unlimited mathematical potential of our children.