Customer Reviews

Proofs Without Words II: More Exercises in Visual Thinking (Classroom Resource Materials) (v. 2)

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Most of the "proofs" in this book will be of more
use and interest to math teachers -- how many
people want to see 10 different diagrams showing
trigonometric identities? Still, much is of great
use to teachers; the standard textbooks are often
redundant in their figures and thus bore students
in their predictability. As well, students are in
need of stretching their mathematical intuitions
and understanding -- when all right triangles are
shown with the legs parallel to the pages' sides,
when all variables are either x or y, people think
that math is a matter of grinding through standard
procedures. PROOFS WITHOUT WORDS II links
subjects that are usually treated disparately:
geometry is connected to combinatorics (a fancy
name for counting), calculus, and linear algebra.
Many of the most technical figures are accompanied
by equations and words explaining the use. The
back cover blurb does admit that many of the
proofs aren't actually wordless, but I'm sure no
one will sue them for false advertising -- these
problems would be difficult to interpret
otherwise.

However, the math enthusiast will be most rewarded
by the figures with the least amount of words, for
they provide mini-mysteries to be solved. The
book starts with six different figures proving the
Pythagorean Theorem -- and only one has any kind
of "language" on it (a few variables and their
products). These examples' author range from a
10th century Arab mathematician, a 3rd century
Chinese mathematician, and Leonardo da Vinci to
current contributors to the journals from the
Mathematical Association of America. A few of
these proofs are easier to interpret, as they
involve cutting up squares into various pieces and
rearranging them (but how to prove these
various pieces are congruent?), one uses similar
triangles, and the da Vinci one still has me in a
fog. The Arabic proof was very elegant and
involves a tiling pattern that looks like it would
work well on a modern kitchen floor. The most
elegant of all the proofs in the book is the one
on the front cover: a geometric series represented
by stacked equilateral triangles, fitting inside a
larger triangle. The combinatorics proofs are
also very elegant ways to visualize special sums
and numbers.

None of these figures would be considered proofs
by most people because one needs to have various
parts explained; however, all crucial parts of the
proofs are in the figures. As the editor writes,
what makes these proofs good is that they show
=why= a statement is true. Many
mathematicians discover new theorems by playing
around with figures representing already known
objects -- some of these figures can show how
certain relations were discovered. Unfortunately,
we are usually shown a cleaned-up, perfectly
deductive version of theorems in school, making it
difficult for us to make the leap to new
mathematical discoveries and understandings. The
figures in this book make for a good course of
mental calisthenics, and they provide inspiration
to one to find one's own visualizations.

I recommend this book for high school and college
math teachers, particularly those who teach
trigonometry, calculus, and discrete math. If you
know a student gifted in math, this book is
appropriate for any student who is familiar with
some basic geometry (Pythagorean Theorem, area of
triangles and rectangles, similar triangles); they
will be able to figure out a few of the counting
and geometry proofs, and will grow into the other
figures in time. For the intelligent child who
enjoys math, this provides an extra challenge, and
as they learn the math various proofs refer to,
the pieces will fall into place.

While the entries in this book are not proofs in the classical sense, they are so in the effective sense. The simple diagrams show the way to the solution and in most cases, all that is needed is to include the words that justify the inclusion of the various features of the diagram. Humans are visual creatures, there is no better way to teach and understand than by the presentation of an effective diagram.
Teaching mathematics is largely an exercise in giving the students an explanation as to why something is true, initially rigor is beyond them. In fact, maintaining formal mathematical rigor early in the introduction is generally counterproductive. The sensible and understandable diagram is the best approach and this book contains diagrams that will allow the student (and teacher) to make sense of many mathematical topics.