Mathematical maturity

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.

Definitions

Mathematical maturity has been defined in several different ways by various authors.

One definition has been given as follows:^[1]

... 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:^[2]

• 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:^[3]

• 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.

References