In mathematics pedagogy, mathematical maturity refers to the mastery of the way mathematicians think, operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it develops from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right.
Mathematical maturity has been defined in several different ways by various authors, and is often tied to other related concepts such as comfort and competence with mathematics, mathematical intuition and mathematical beliefs.
One definition has been given as follows:
A broader list of characteristics of mathematical maturity has been given as follows:
Finally, mathematical maturity has also been defined as an ability to do the following: It is sometimes said that the development of mathematical maturity requires a deep reflection on the subject matter for a prolonged period of time, along with a guiding spirit which encourages exploration.
Mathematician Terence Tao has proposed a three-stage model of mathematics education that can be interpreted as a general framework of mathematical maturity progression. The stages are summarized in the following table:
In overview, every mathematics student begins with more computational training as opposed to theoretical training, and that balance inverts as one progress through the second stage. At this point, Tao advises:
<blockquote> So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. </blockquote>
Mathematics students that have developed a solid skill set in rigor and theory then transition into the final stage as their perspective shifts to a more encompassing panoramic view of mathematics.