[remote_phone]

In college, I was always the first one out of my friends to “get” a concept, like fast Fourier transforms, anything with signal processing or even coding or any labs we had to do etc so I would spend time teaching them in the library. However I never did any of the exercises, mostly due to laziness and not arrogance. They would get A’s and I would get C’s and D’s.

I emphasize this story to my kids because knowing isn’t important because everyone eventually figure it out. It’s the ones who can do the problems and get good marks that succeed in the end.

[javajosh]

I think this shows a lack of self-doubt, which can be deadly. Those problems acted as verification to yourself that you understood the theory and its application. If you truly understood the material, then the problems would be zero effort. However, if you struggled with them it's a signal that you don't know what you think you know.

[TheCleric]

As someone who had an a similar experience to whom you are replying to, this was definitely not the case. The problems were easy, but were not "zero effort". Even if it takes you only a few minutes to do the steps and show the work per problem, then that could still take you 30-60 minutes to complete the assignment. That was time I'd spend doing things I wanted to do (fun in the short term, a nightmare in the long term).

[wholinator2]

I think usually the easy problems are just the "burn-in" time to solidify understanding, but there's usually a couple hard problems that take way longer to work through and those will teach you the intuition. Doing simple calculation is different than being forced to conceptualize the entire path from starting information to system to evolution to result.

[froggit]

This is kind of interesting to hear because I was the other way around. I found the best way to understand something was to teach it to others. That way I took what I already understood and was able to see what other people misunderstood, which was often something I'd never expected to be an issue, and add their experience in learning the topic on to what I already knew which expanded my overall understanding.

Then again, in the process of teaching I always found myself teaching people to work problems, which required me to be able to work the problems myself. In a way, it's kind of impressive you managed to avoid doing that.

[j7ake]

Especially these days, the bar for “knowing” Fourier transforms is simply a 7 minute video from 3blue1brown.

The gap between knowers and doers will only get larger as math explainers improve their content.