How I Rewired My Brain to Become Fluent in Math

How I Rewired My Brain to Become Fluent in Math
Sorry, education reformers, it’s still memorization and repetition we need.
Oct 2 2014

I was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague. So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and that crown jewel of mathematics, Euler’s equation. It’s hard to believe I’ve flipped from a virtually congenital math-phobe to a professor of engineering.

One day, one of my students asked me how I did it—how I changed my brain. I wanted to answer Hell—with lots of difficulty! After all, I’d flunked my way through elementary, middle, and high school math and science. In fact, I didn’t start studying remedial math until I left the Army at age 26. If there were a textbook example of the potential for adult neural plasticity, I’d be Exhibit A.

Learning math and then science as an adult gave me passage into the empowering world of engineering. But these hard-won, adult-age changes in my brain have also given me an insider’s perspective on the neuroplasticity that underlies adult learning. Fortunately, my doctoral training in systems engineering—tying together the big picture of different STEM (Science, Technology, Engineering, Math) disciplines—and then my later research and writing focusing on how humans think have helped me make sense of recent advances in neuroscience and cognitive psychology related to learning.

In the years since I received my doctorate, thousands of students have swept through my classrooms—students who have been reared in elementary school and high school to believe that understanding math through active discussion is the talisman of learning. If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must
understand it.

Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes memorization, repetition, and rote learning hand-in-hand with developing the child’s mastery over the material. This intense afterschool program, and others like it, is embraced by millions of parents in Japan and around the world who supplement their child’s participatory education with plenty of practice, repetition, and yes, intelligently designed rote learning, to allow them to gain hard-won fluency with the material.

In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science.

The latest wave in educational reform in mathematics involves the Common Core—an attempt to set strong, uniform standards across the U.S., although critics are weighing in to say the standards fail by comparison with high-achieving countries. At least superficially, the standards seem to show a sensible perspective. They propose that in mathematics, students should gain equal facility in conceptual understanding, procedural skills and fluency, and application.

The devil, of course, lies in the details of implementation. In the current educational climate, memorization and repetition in the STEM disciplines (as opposed to in the study of language or music), are often seen as demeaning and a waste of time for students and teachers alike. Many teachers have long been taught that conceptual understanding in STEM trumps everything else. And indeed, it’s easier for teachers to induce students to discuss a mathematical subject (which, if done properly, can do much to help promote understanding) than it is for that teacher to tediously grade math homework. What this all means is that, despite the fact that procedural skills and fluency, along with application, are supposed to be given equal emphasis with conceptual understanding, all too often it doesn’t happen. Imparting a conceptual understanding reigns supreme—especially during precious class time.

The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.

There is an interesting connection between learning math and science, and learning a sport. When you learn how to swing a golf club, you perfect that swing from lots of repetition over a period of years. Your body knows what to do from a single thought—one chunk—instead of having to recall all the complex steps involved in hitting a ball.

In the same way, once you understand why you do something in math and science, you don’t have to keep re-explaining the how to yourself every time you do it. It’s not necessary to go around with 25 marbles in your pocket and lay out 5 rows of 5 marbles again and again so that you get that 5 x 5 = 25. At some point, you just know it fluently from memory. You memorize the idea that you simply add exponents—those little superscript numbers—when multiplying numbers that have the same base (104 x 105 = 109). If you use the procedure a lot, by doing many different types of problems, you will find that you understand both the why and the how behind the procedure very well indeed. The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.



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