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.
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 concept
- the capacity to handle increasingly abstract ideas
- the ability to communicate mathematically by learning standard notation and acceptable style
- a significant shift from learning by memorization to learning through understanding
- the capacity to separate the key ideas from the less significant
- the ability to link a geometrical representation with an analytic representation
- the ability to translate verbal problems into mathematical problems
- the ability to recognize a valid proof and detect 'sloppy' thinking
- the ability to recognize mathematical patterns
- the 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 attitude
Finally, mathematical maturity has also been defined as an ability to do the following:
- make and use connections with other problems and other disciplines,
- fill in missing details,
- spot, correct and learn from mistakes,
- winnow the chaff from the wheat, get to the crux, identify intent,
- recognize and appreciate elegance,
- think abstractly,
- read, write and critique formal proofs,
- draw a line between what you know and what you don’t know,
- recognize patterns, themes, currents and eddies,
- apply what you know in creative ways,
- approximate appropriately,
- teach yourself,
- remain focused,
- bring instinct and intuition to bear when needed.