In this Nautilus article by Barbara Oakley she does a great job explaining how she learned to became fluent in mathematics basing her method on how she learned to become fluent in Russian; thus her method works in other subjects.

There is a balance when learning between understanding concepts, memorizing and practice/application. Memorizing almost everything is bad because then normally you will not be able to apply it to new situations and problems. Learning concepts only will not lead you to becoming fluent.

Ms. Oakley explains how she addressed getting the right balance, which could be a little different depending upon the subject – “*in mathematics, students should gain equal facility in conceptual understanding, procedural skills and fluency, and application.*”

From her website:

“I love to bring fresh perspectives into my books by applying knowledge and experience from many different disciplines, as well as from “real world” experiences. Although I’m now a professor of engineering, I’ve also worked in lots of different places and doing very different things: serving as a Russian translator on Soviet trawlers up in the Bering Sea, teaching in China, going from US Army private to Regular Army Captain, and working as a radio operator at the South Pole Station in the Antarctic. (I met my husband there—I had to go to the end of the earth to meet that man!)”

Her book about the learning process:

Mindshift

Break Through Obstacles to Learning and Discover Your Hidden Potential

How we can overcome stereotypes and preconceived ideas about what is possible for us to learn and become.

Intro to the article in Nautilus magazine:

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.

Some excerpts from the article.

When learning math and engineering as an adult, I began by using the same strategy I’d used to learn language. I’d look at an equation, to take a very simple example, Newton’s second law of f = ma. I practiced feeling what each of the letters meant—f for force was a push, m for mass was a kind of weighty resistance to my push, and a was the exhilarating feeling of acceleration. (The equivalent in Russian was learning to physically sound out the letters of the Cyrillic alphabet.) I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it. Although I wouldn’t have put it that way at the time, the truth was that to learn math and science well, I had to slowly, day by day, build solid neural “chunked” subroutines—such as surrounding the simple equation f = ma—that I could easily call to mind from long term memory, much as I’d done with Russian.

Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their success. Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

As I discovered, having a basic, deep-seated fluency in math and science—not just an “understanding,” is critical. It opens doors for many of life’s most intriguing jobs. Looking back, I realize that I didn’t have to just blindly follow my initial inclinations and passions. The “fluency” part of me that loved literature and language was also the same part of me that ultimately fell in love with math and science—and transformed and enriched my life.