Theory and practice are both important. As Donald Knuth put it,

If you find that you’re spending almost all your time on theory, start turning some attention to practical things; it will improve your theories. If you find that you’re spending almost all your time on practice, start turning some attention to theoretical things; it will improve your practice.

However, there’s an implicit third factor: integration.

We can have theoretical knowledge and practical experience, but keep them isolated in different parts of our brain. It’s possible to do well in a probability class, and play poker well, but not be able to think about probability *while* playing poker, to be no better at poker for having taken probability.

It may not be immediately obvious that someone excels at integration. It comes out in off-the-cuff remarks, little comments that show someone can quickly apply abstract theory to a concrete problem.

It’s hard to teach integration. A course can have lectures and labs, but that doesn’t mean students will fully connect the two in their minds. Maybe the connection forms later after some experience and some contemplation.

A very timely post. I’m tutoring 2 students on AP Physics and Math, and found they are compartmentalize what they’ve learned, i.e., not integrating them. So after starting on a new unit, they let go of the prior units, without integrating them. I wondered what learning techniques they should use to help them better integrate what they have learned and what they now learning.

Thanks for this post!

In my experience this integrating factor is where a lot of opportunity lies.

I have observed many examples where integrating relatively simple levels of theory has much more positive impact than increasing ether theory or practice.

Those off the cuff comments you reference reminds me of what John Allen Paulos calls having a mathematical accent