Given my efforts in “How to Turn Every Child into a ‘Math Person’" to figure out something helpful to say about math education, I was delighted not too long after to run across Barbara Oakley's Wall Street Journal op-ed

How We Should Be Teaching Math: Achieving 'conceptual’ understanding doesn’t mean true mastery. For that, you need practice.

(As with all Wall Street Journal articles, if you hit the paywall, just google the title. The Wall Street Journal lets you jump the paywall if you come from Google.) Here is the key passage:

True experts have a profound conceptual understanding of their field. But the expertise built the profound conceptual understanding, not the other way around. There’s a big difference between the "ah-ha” light bulb, as understanding begins to glimmer, and real mastery.

As research by Alessandro Guida, Fernand Gobet, K. Anders Ericsson and others has also shown, the development of true expertise involves extensive practice so that the fundamental neural architectures that underpin true expertise have time to grow and deepen. This involves plenty of repetition in a flexible variety of circumstances. In the hands of poor teachers, this repetition becomes rote—droning reiteration of easy material. With gifted teachers, however, this subtly shifting and expanding repetition mixed with new material becomes a form of deliberate practice and mastery learning.

I also especially like her conclusion:

Understanding is key. But not superficial, light-bulb moment of understanding. In STEM, true and deep understanding comes with the mastery gained through practice.

For anyone learning math, the key to learning is patience–patience with both with the mental training needed to become good at working through details and with moments of confusion that come along the way. Believe me, those who do math for a living (as I do in important measure) face many moments of confusion in our work. What makes a mathematician is patience and persistence through those moments–and often hours or days, and sometimes months–of confusion, as well as the hours of honing skills for getting mathematical details straight.