Mathematical maturity

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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]


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


  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
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