Mathematical maturity

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Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts.

An illustrative example from common experience that may be more familiar to non-mathematicians would be high school geometry proofs. While most competent and interested students can follow a given proof and even determine whether or not it is correct, many still have trouble coming up with a proof. The ill-defined difference between those who can generate proofs and those who cannot is an example of a difference in mathematical maturity, in this case the ability to "see" how to proceed.

Students' initial resistance to proofs that 0.999... equals 1 is another, more specific example of lack of mathematical maturity, in this case the ability to accept the non-intuitive result of mathematical logic.

Mathematical maturity has been defined as:

... 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.

—Larry Denenberg

and as:

• 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

- Lyman Briggs School of Science