Real Understanding

Statistician John Cook recalls the following from his graduate school coursework under a mathematician who recently won a prestigious award:

Passing a test on something is like having a bulb. Understanding something is like having it plugged in. (Image by TeroVesalainen, via Pixabay, license.)

I had a course from Karen Uhlenbeck in graduate school. She was obviously brilliant, but what I remember most from the class was her candor about things she didn't understand. She was already famous at the time, having won a MacArthur genius award and other honors, so she didn't have to prove herself.

When she presented the definition of a manifold, she made an offhand comment that it took her a month to really understand that definition when she was a student. She obviously understands manifolds now, having spent her career working with them.

I found her comment ... extremely encouraging. It shows it's possible to become an expert in something you don't immediately grasp, even if it takes you weeks to grok its most fundamental concept. [bold added]

That's a good point, and I think it works the other way around, too: Respect this material, and you can succeed.

But that can be tricky if one does not appreciate what it means to understand something. For example, my weak point during my college education was physics, for which I partially blame the way it was taught. To this day, I find myself finally, actually understanding things from physics that I aced tests on decades ago without real understanding. This usually happens when I have to focus on some matter dealing with physics, such as recently, when I was helping my daughter prepare for a simple classroom demonstration about electricity from batteries. (Oh! That's why this works. I'll think, apparently from nowhere, at some connection my subconscious will make.) That's a very simple example, but there can be a wide gulf between being able to pass an exam and knowing what one is talking about.

Although tests can be helpful for the purpose of measuring progress in younger students, it would be a good thing if they could be improved or repurposed in some way to help older students gauge whether they have achieved understanding. Or better yet, educators could focus more on helping students realize what it actually means to understand something at an appropriate point in the transition from childhood to adulthood.

-- CAV