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Instead of starting from the mathematically "rigorous" notation, I am able to jump to a more familiar idea and proceed to link it back to the notation and rigor described in the definitions and associated theorems. This is a much more "efficient" way, in my opinion, than staring at the notation and trying to decipher it for hours. Having examples makes it easier to understand the context of the symbols and rigor involved: by having a specific example of a general form.

Hmm are you in Applied Math? Or by "graduate" you mean perhaps some kind of terminal (non-PhD track) Masters program? Because "staring at the notation and trying to decipher it for hours" (when lucky; sometimes it could be days, or even weeks) is precisely the bulk of the Math graduate experience IMO --especially after quals...

Also, do know that after a certain point, the whole "examples then general form" approach is not just barely applicable (e.g. the 'example' alone often requires so much setup that it ends up being harder than the formal statement itself!) but is also a serious handicap to your capacity for thinking 'syntactically' from formal statements alone. Terry Tao has an excellent post on this[1]; although it might or might not be exactly applicable to your situation...

[1] https://terrytao.wordpress.com/career-advice/there’s-more-to...

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