id: 06340819
dt: b
an: 2015b.00762
au: Johnson, Craig M.
ti: Exploring mathematics. Investigations with functions.
so: Burlington, MA: Jones \& Bartlett Learning (ISBN 978-1-4496-8854-7/pbk).
xxi, 656~p. (2015).
py: 2015
pu: Burlington, MA: Jones \& Bartlett Learning
la: EN
cc: I15 M15 A80
ut: textbook; mathematics for non-mathematicians; popularization of
mathematics; methodology of mathematics; general mathematics
ci:
li:
ab: What should be taught in a one- or two-semester course in mathematics for
college students majoring in education, languages, music, art,
philosophy, or any discipline in humanities or liberal arts? How to
demonstrate the utility of mathematical thinking in various areas of
society to those students, and how to illustrate the many aesthetic
features of mathematics to rouse both their attention and their
interest in such a course? The book under review is a very successful
attempt to meet these pedagogically demanding challenges. The
author’s unique strategy is to give a versatile sampling of
mathematical topics in the form of explorations of concrete
applications, with the concept of function serving as the central
unifying aspect. The general approach is rather informal and utmost
student-friendly, thereby focussing on an abundance of examples from
the humanities as the basic educational tool. As for the precise
contents, the book comprises ten chapters, each of which comes with
several sections. Chapter 1 develops the concept of function via
accustomed examples, concluding with the interpretation of graphs of
functions of one real variable. Chapter 2 showcases the use of
functions in personal finance matters by discussing interests and
effective yields, annuities, and ammortization of loans. Chapter 3
illuminates concepts from logic and computer science, with statements
and connectives, truth and consequences, tautologies and syllogisms,
and computer programming structures being main themes of discussion.
Chapter 4 provides a historical excursion through some of the
developments of astronomy as a driving force in mathematics, together
with the biographies of some of the main contributors. Chapter 5 gives
a leisurely introduction to graphs and circuits, thereby varying
somewhat from the central theme of functions. Among the topics touched
upon in this chapter are Euler and Hamilton circuits, complete graphs,
and the traveling salesman problem. Chapter 6 briefly describes the
mathematics underlying problems of social choice and voting methods in
social sciences, whereas Chapter 7 turns to elementary probability
theory and the functions occuring there. This chapter also gives a
quick introduction to naive set theory. Methods of statistics are
explained in Chapter 8, including the representation of data, the
dispersion of data, the normal distribution, and confidence intervals.
Again the continued emphasis is on concrete examples rather than on
rigorous abstract definitions. Another distinguishing feature of this
elementary textbook is Chapter 9, where mathematics in music and in
cryptology is depicted, mainly through sections on arithmetical residue
classes and music notes, the circle of fifths in music theory, and
modular arithmetic in cryptology. Finally, Chapter 10 gives a brief
introduction to mathematical modeling by means of linear interpolation,
prediction in linear regression via the so-called Wald function, and
further examples of nonlinear modeling functions and transforms of
data. Each section ends with its own exercise set, and each chapter
contains a so-called “chapter review test” for self-control. Also,
there are an extra section providing solutions to all odd-numbered
exercises, a carefully compiled glossary, and a list of references for
additional and further reading. As a mathematical textbook for
non-mathematicians, the present primer fulfills its task completely and
perfectly. Being the outcome of about twenty years of the author’s
thinking and teaching, this book is a lovely invitation to mathematics
as a whole, with numerous illustrating pictures, instructive examples,
and quite a bit of non-standard material covered. No doubt, this book
will appeal to a wide audience of students and instructors likewise.
rv: Werner Kleinert (Berlin)