A Mathematician Comes of Age (Spectrum) by Steven G. Krantz (PDF)

Description

This book is about the concept of mathematical maturity. The goal of a mathematics education is to transform the student into a mathematically mature individual who treats mathematical ideas analytically and can control and manipulate them effectively. At the heart of the endeavour to solidify this loosely defined but widely acknowledged concept is the privileged position of mathematics as a pure science. The book provides background, data and analysis for recognising and nurturing aspects of mathematical maturity, such as the ability to analyse and communicate abstract ideas, criticise and construct proofs, and move from intuitive to rigorous reasoning. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration. This book will be of interest to anyone, at any level, who wants to be an effective mathematics teacher.

User’s Reviews

Editorial Reviews: Review 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. –Charles Ashbacher, Journal of Recreational MathematicsThe 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. –Johh H. Mason, Mathematical Reviews Book Description This book treats the maturation process for a mathematics student with description and analysis of how a student can develop into a sophisticated thinker who can understand abstract concepts and proofs with intellectual rigour. This book is an ideal skills development tool for graduate students and teachers of mathematics. About the Author Steven G. Krantz is a Professor of Mathematics at Washington University, St Louis, where he has served as Chair of his department. He has previously taught at the University of California, Los Angeles, Princeton University and Penn State University, having received a PhD from Princeton in 1974. Krantz has directed numerous PhD and Masters students and is the recipient of the UCLA Alumni Foundation Distinguished Teaching Award, the Chauvenet Prize and the Beckenbach Book Award. He was recently inducted into the Sequoia High School Hall of Fame. Krantz is currently the Editor of the Notices of the American Mathematical Society. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐My Dad, a math professor really wanted this book. Usually these specialized books take a really long time to get delivered. I got this earlier than promised and have it in plenty of time to give it to Dad for Father’s Day. Happy Dad and Happy daughter.

⭐Prof Krantz coins the word Math Maturity as the ability in Abstract Thinking for freshmen in undergraduates, a distinct capability which is lacking by some good high-school Math students who fail to do well in the university Math.The Math Tree contains all the necessary Math skills in undergraduates before able to transform from “Learning Maths of others” to “Creating your own Maths” in Grad schools.One mistake we face when learning Abstract Math is attempting (in vain) to match abstract ideas to concrete objects. It is not the way to learn. I observe those who are good in Abstract Math – they learn it like appreciating an abstract art, look at the whole picture, understand the mood, the idea, the beauty, without matching these pictures to a tree or a person or any object in real world.

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