What constitutes mathematical maturity?

As a learner transitions from secondary mathematics courses like algebra and easy Euclidean geometry to more difficult mathematical topics, how does the learner develop "mathematical maturity"? I am a coach of a homeschoolers math team, homeschooling parent of four children, and teacher in a supplemental afterschool class on mathematics for gifted students in a nearby public school district. Some of my students are going beyond my own level in mathematics, and I want to make sure they go in the right direction as they find other teachers and self-teaching resources.

What are some of the hallmarks of mathematical maturity? What are some of the signs when it is lacking in learners? What do you recommend for developing mathematical maturity? How much does age matter? Can a younger learner be much more mathematically mature than an older learner and, if so, how?

Thank you for your helpful suggestions and insights.

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Karl M. Bunday  P.O. Box 1456, Minnetonka MN 55345
Learn in Freedom (TM) http://learninfreedom.org/
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