Mathematical maturity is an informal term used by mathematicians to refer to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics student's erudition in mathematical structures and methods. The topic is occasionally also addressed in literature in its own right.
Mathematical maturity has been defined in several different ways by various authors.
One definition has been given as follows:
... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas.
A broader list of characteristics of mathematical maturity has been given as follows:
the capacity to generalize from a specific example to broad conceptthe capacity to handle increasingly abstract ideasthe ability to communicate mathematically by learning standard notation and acceptable stylea significant shift from learning by memorization to learning through understandingthe capacity to separate the key ideas from the less significantthe ability to link a geometrical representation with an analytic representationthe ability to translate verbal problems into mathematical problemsthe ability to recognize a valid proof and detect 'sloppy' thinkingthe ability to recognize mathematical patternsthe ability to move back and forth between the geometrical (graph) and the analytical (equation)improving mathematical intuition by abandoning naive assumptions and developing a more critical attitudeFinally, mathematical maturity has also been defined as an ability to do the following:
make and use connections with other problems and other disciplinesfill in missing detailsspot, correct and learn from mistakeswinnow the chaff from the wheat, get to the crux, identify intentrecognize and appreciate elegancethink abstractlyread, write and critique formal proofsdraw a line between what you know and what you don’t knowrecognize patterns, themes, currents and eddiesapply what you know in creative waysapproximate appropriatelyteach yourselfgeneralizeremain focusedbring instinct and intuition to bear when needed