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. Instead, it 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, lack of mathematical maturity manifests itself as the inability to accept non-intuitive results of mathematical logic.

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.

—Larry Denenberg

A more broad 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

—Lyman Briggs School of Science

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,
• generalize,
• remain focused, and
• bring instinct and intuition to bear when needed.

—Ken Suman, Department of Mathematics and Statistics, Winona State University

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