In this recent blog post the author lists several interesting concepts he has discovered pertaining to learning. One thing he mentions is the following:

Mastery is more important than passing grades. You will reach limits in learning when you have a "Swiss cheese" foundation full of misconception holes. Master each concept before building on top. ~Salman Khan

(This is actually from Salman's TED talk which you can view here.) My questions is as follows: I graduated about a year ago with a Master's in Applied Physics from a reputable university. I've been a B student throughout my academic career and I feel like I started building on un-mastered concepts somewhere towards the end of high school. Despite obtaining a relatively tough engineering degree, I cut corners whenever possible. If I could pass an exam by being able to do all the exercises, I would get to that level and stop there. I would rarely - if ever - go for complete mastery. As a result I feel like I have Swiss cheese holes in my understanding that date back to high school.

If Albert Einstein's quote is anything to go by:

If you can't explain it simply, you don't understand it well enough.

I'm afraid I can't simply explain a whole lot. For instance I would not be able to explain simply:

  1. What is the proof of Euler's identity?
  2. Why can you express a wave as an imaginary exponential?
  3. How does a photon have momentum, but not mass?
  4. What is the intuitive understanding of eigenfunctions, eigenvalues and their physical manifestations?

With the wealth of knowledge available today on the internet, I could look up any one of my knowledge gaps, and with some effort I feel confident I would be able to fill them. My question is, would that be worth it?

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    What do you mean by "would that be worth it"? For a career in academia? In any case, it seems like it bothers you, so why not try to fill the gaps? Just don't be too critical of yourself - some concepts can take a long time to digest and really understand deeply. – Bitwise Aug 8 '13 at 20:59
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    During my PhD I always felt I was learning new things at a much lower rate that I was forgetting what I learned during my undergraduate studies. Upon questioning, everyone around in the office seemed to be going through different versions of the same process. So I guess the message to convey here is that your Swiss cheese holes will keep growing at the "expense" of hardening the interstitial matter that, in the end, is what keeps it all together... – Miguel Sep 1 '14 at 19:44
  • Focus on your strong sides. And find teammates who fill out the holes with their strong sides. – Moritz Feb 9 '15 at 18:10

First, for context, I'd disagree strongly with a too strict notion that "mastery" is required before "moving forward", for several reasons. It is easier to understand the purpose of something after one sees how it is used. It is all too easy to acquire a fake mastery that is not functional, but only refers to some artificial tasks created for a textbook, and, all the worse, may be wastes of time. And, finally, the kind of "mastery" we are often led to think we should attain is basically unattainable... That is, unless one keeps ones list of things-to-master really short, very elementary, and uncomplicated, this "mastery" is just unreachable on the terms a novice or even journeyman conceives.

So... don't over-interpret "mastery".

On the other hand, _of_course_ it is good to go back and fill in. And one will find oneself doing this many times, as one discovers, upon "stressing" one's knowledge, that what one thought one knew "well enough" was not quite good enough.

For that matter, even with an excellent memory, things slip out of a person's head.

My bottom line recommendation is to maintain (at least) two "threads", one to review and backfill, the other to move forward.

The latter is surprisingly useful at helping review make more sense! :)

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  • My bottom line recommendation is to maintain (at least) two "threads", one to review and backfill, the other to move forward. – CoderInNetwork Aug 17 '15 at 17:40

The ancient Hebrew wisdom of Solomon applies here: "Of making many books there is no end, and much study is wearisome to the flesh." (Ecclesiastes 12:12b) There is no end of making books, so we can never completely master anything. Mastery is an ideal that no one ever really reaches; it is more like a compass to steer by. So if those holes are bothering you, why not fill them in as you run into them? You'll never reach a place where there are no holes left, as paul garret implied in his answer. This fact should not stop us from trying, though. Enjoy the journey....

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When you have collected discouraging quotes from authorities, it is a good idea to find other authorities who have held the opposite point of view. I suggest that the arch-experimentalist Oliver Heaviside would be a good match for you.

Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?


The prevalent idea of mathematical works is that you must understand the reason why first, before you proceed to practice. That is fudg e and fiddlesticks. I know mathematical processes that I have used with success for a very long time, of which neither I nor any one else understands the scholastic logic. I have grown into them, and so understand them that way .

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