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
- ^ 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
- ^ Math 22 Lecture A, Larry Denenberg
- ^ LBS 119 Calculus II Course Goals, Lyman Briggs School of Science
- ^ A Set of Mathematical Equivoques, Ken Suman, Department of Mathematics and Statistics, Winona State University