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To understand what I mean, for three days I stared at a formula that looked Greek. Its semantics of relationship was unknown. The mathematics was just one long blur without logic and making no sense. I came back and forth to it. Then one day it was like a veil was lifted over my head and everything "clicked" and made sense.

I remember some of the best mathematics advice I received was "sometimes things click with repeated exposure and this happens a lot in mathematics." I never paid attention to this until I learned to appreciate this "click" phenomenon which I would like study in a more scientific manner but don't know the term(s) I should be looking for.

Thus why I'm here:

is there even a name for this "click" experience in mathematics?

There is a name for Déjà vu which even is studied, but what about the "click"?

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There are concepts like insight and the Aha moment which are similar to what you describe.

However, the aha moment is more aligned to flashes of insight to discrete problems. While this could be relevant to the experience you are describing of learning mathematics, perhaps you are also describing a more gradual process of learning. Mathematics can be quite complicated. With each iteration of practice, reflection and consolidation, an you typically learn more. At first you made need to go through the material just to get an organising framework. Then on a second read, you can start to apply the organising framework to your interpretation. The meaning of the text and symbols begin to be stored in long term memory, so you can start to focus on the broader meaning rather than spending substantial cognitive resources on the components.

So in general, what you are describing sounds a lot like the process of cognitive skill acquisition. If you want to learn more you could read something like Anderson (1982). Basically, this is a movement from effortful and error prone performance to fluent and relatively error free performance. Thus, at a certain point, you reflect on your performance and you realise that you have acquired the skill.

References

  • Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological review, 89(4), 369-406. PDF
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    $\begingroup$ I remember reading, in a book on intuition, that conscious problem solving / information integration is possible only up to a certain limit of complexity (the level of this limit depending on your intelligence), and that for problems with a volume of information that exceeds our conscious capacity unconscious processing does the job (the success depending on what you might call your "unconscious intelligence", which is higher than your conscious intelligence). This is Einstein dreaming of Relativity Theory, understanding once you stop focussing, sleeping over a difficult decision etc. $\endgroup$ – user3116 Jun 22 '13 at 8:48
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    $\begingroup$ So what I believe is that most of this cognitive skill acquisition is unconscious, and the "click" happens when the final understanding is "reported to" your consciousness. -- If I were going to become an experimental psychologist, this would be one of my areas of research. But statistics simply doesn't "click" with me :-) $\endgroup$ – user3116 Jun 22 '13 at 10:20
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A similar feeling is described by Poincaré and Hadamard about the mathematical invention. For instance, Poincaré recounts: "I turned my attention to the study of some arithmetical questions apparently without much success [...]. Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with [...] the [...] characteristics of brevity, suddenness, and immediate certainty."

You can find their studies in the book and the paper mentioned below:

Jacques Hadamard. An Essay on the Psychology of Invention in the Mathematical Field, https://press.princeton.edu/titles/5896.html

Henri Poincaré. “Mathematical creation”. The Monist, Volume 20, Issue 3, 1 July 1910, Pages 321–335, https://doi.org/10.1093/monist/20.3.321

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