One of the most widely used yet ill-defined terms in mathematics is "mathematical maturity." It is used to describe everything from the background needed to understand a textbook, to succeed in a course, to explain a person's success in the field. In this book, Steven Krantz gives a more formal explanation of mathematical maturity and he succeeds in that endeavor.Not only does he give an excellent explanation of what it is but he also explains how one gets there. Acquiring mathematical maturity is something developed over a long period of time by subjecting yourself to a series of partial successes, significant failures and a willingness to keep trying. While there may be a few "Aha!" moments in achieving high levels of mathematical skills, they never appear in isolation. A single light bulb may suddenly appear brightly lit, but only after hours, weeks and maybe years of steadily building the supporting power plant.Solid arguments can be made that one of the reasons for so much math anxiety is that actually understanding mathematics takes a great deal of internal intellectual ferment. While you may have done all the problems for a class and gotten the right answers that does not mean that you understand it. Krantz makes those points in a book that could be read in the first week in the life of a math major or much later in their career. In the first case it will help prepare them for the road ahead in college and in the second case help you either stay on the road or understand why you are there.

The author's intention is to investigate and make as precise as possible the notion of mathematical maturity, a phrase often heard when discussing the behavior of students who might, for example, go on to graduate work. The author, who has written cogently about his approach to teaching mathematics at the university level describes his own process of maturation as a mathematician. The style is a mixture of auto-mathography, anecdotes about colleagues, and essays and comments about various factors that contribute to, or are associated with, mathematical maturity. ...For me, the most important point, made in several different ways, is that in order for students to appreciate what it is like to think mathematically, they need to be in the presence of someone more experienced who is behaving mathematically. Students lucky enough to be taught by a mathematically mature thinker who is sensitive to what it is like to be learning to think mathematically have a real advantage over students who are subjected to a constant diet of previously digested and honed mathematics. Seeing others make mistakes, specialise, modify their conjectures, extend, generalise and abstract is much more likely to foster mathematical maturity in the novice than going without this experience. So what is mathematical maturity? Krantz lists 13 things to work on learning in order to develop maturity. The book ends with a tree of topics, partially ordered with respect to maturation of the student, and a partial etymology suggesting a basis for 'maturity' in the notion of 'ripeness.' Presumably the mathematically mature teacher offers students nutrious fruit and potent seeds for continuing the species.