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. The topic is occasionally also addressed in literature in its own right.^[1]

Definitions

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

One definition has been given as follows:^[2]

... 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:^[3]

• 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:^[4]

• 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
• bring instinct and intuition to bear when needed

References

1. Lynn Arthur Steen (1983) "Developing Mathematical Maturity" pages 99 to 110 in The Future of College Mathematics: Proceedings of a Conference/Workshop on the First Two Years of College Mathematics, Anthony Ralston editor, Springer ISBN 1-4612-5510-4
2. Math 22 Lecture A, Larry Denenberg
3. LBS 119 Calculus II Course Goals, Lyman Briggs School of Science
4. A Set of Mathematical Equivoques, Ken Suman, Department of Mathematics and Statistics, Winona State University