My document of life-lessons spits out this (it has a focus on teaching children, but it aims high):

Key math insights of general value:

• What is a number really - Peano's sentence

• Equality (do the same to both sides, equivalence classes)

• Negation and inversion (reversing any relationship in general)

• Variables, functions, domains

• Continuous functions

• Limits, infinities (leads e.g. to real analysis)

• Postponing operations (fractions, 'primitive functions', lazy evaluation)

• Probability (enumerating paths that can are taken fractionally, Bayes rule)

• Tracking errors (dealing with two or more functions/results at the same time)

• Induction, proofs

• Transformation into another space (Fourier, dual spaces, radix sort)

• Representations of sequences and trees and graphs

• Decomposition of plans and algorithms (O-notation)

• Encoding of plans as numbers (Turing, Curry, Gödel)

The idea is to see the patterns behind the patterns (link in Einsteins Speed).