Preface; 1. Introductory thoughts; 2. Math concepts; 3. Teaching techniques; 4. Social issues; 5. Cognitive issues; 6. What is a mathematician?; 7. Maturity for everyone; The tree of mathematical maturity; Etymology of the word 'maturity'; Bibliography; Index.
This book describes and analyses how a mathematics student can develop into a sophisticated and rigorous thinker.
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.
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
Mathematics
""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."" - Johh H. Mason, Mathematical Review
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