This book is about the concept of mathematical maturity. Mathematical maturity is central to a mathematics education. The goal of a mathematics education is to transform the student from someone who treats mathematical ideas empirically and intuitively to someone who treats mathematical ideas analytically and can control and manipulate them effectively.
Put more directly, a mathematically mature person is one who can read, analyze, and evaluate proofs. And, most significantly, he/she is one who can create proofs. For this is what modern mathematics is all about: coming up with new ideas and validating them with proofs.
The book provides background, data, and analysis for understanding the concept of mathematical maturity. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration.
Table of Contents
1. Introductory Thoughts
2. Math Concepts
3. Teaching Techniques
4. Social Issues
5. Cognitive Issues
6. What is a Mathematician?
7. Is Mathematical Maturity for Everyone?
The Tree of Mathematical Maturity
Etymology of the Word “Maturity”
About the Author
Excerpt: 1.10 The Changing Nature of Mathematical Maturity (p. 18)
Mathematics is ummutable and unchanging. Mathematical facts—such as the Pythagorean theorem—that were established 2000 years ago are still valid (and useful) today. We can thank the Euclidean paradigm of axiomatic rigor for that stability and mobility of our subject. But the way that we see mathematics can change.
In the eighteenth and early nineteenth centuries, people believed that matehatmics was a symbolic representation of facts about certain features of the real world around us. In other words, mathematics had foundations in reality. THis is a very Platonic take on the subject. Today, however, we acknowledge the Platonic parts of the subject but often focus on a more Kantian feature—that much of modern mathematics is quite abstract (i.e., is cooked up inside our brains) and has little or no basis in reality.
About the Author
Steven G. Krantz was born in San Francisco, California in 1951. He received a B.A. degree from the University of California at Santa Cruz in 1971 and the Ph.D. from Princeton University in 1974. Krantz has taught at UCLA, Penn State, Princeton University, and Washington University in St. Louis. He served as Chair of the latter department for five years. Krantz has published more than 60 books and more than 160 scholarly papers. He is the recipient of the Chauvenet Prize and the Beckenbach Book Award of the MAA. He has received the UCLA Alumni Foundation Distinguished Teaching Award and the Kemper Award. He has directed 18 Ph.D. theses and nine Masters theses.