This is a superb book. If you only buy one book for your own or your departmental library this year, this would be an excellent choice...I thoroughly recommend this book to teachers, mathematics educators, students and anyone interested in the visual side of mathematics. --Mary Coupland, Australian Mathematics Teacher
This wonderful book is a fitting edition to the Classroom Resource Materials series. Indeed any teacher of mathematics at the high school level or above should have a copy. Let me take that a bit further: anyone with an interest in mathematics should have a copy! While I ended up reading it pretty much straight through, this is a wonderful reference book which can be consulted whenever one is stuck for a way to make a concept come to life or for an activity to get students involved in mathematics.
The first two-thirds of the text consists of a wonderful set of examples of how visualization can aid understanding and inspire exploration. Each section ends with a set of challenges for the reader. These problems would make wonderful projects for pre-service high school teachers -- many of them can be implemented in Geometer's Sketchpad. The final section of the book consists of hints for solving these challenges. Sandwiched between these two sections is a short section providing suggestions as to how these ideas can be used in a classroom. While technology would certainly help, many of the hints involve simple paper folding and cutting. Geometer's Sketchpad would certainly suffice to create most all of the 2-dimensional figures.
Part I consists of 20 short chapters each with several related concepts. These chapters are only loosely related to one another and assume minimal mathematical background on the part of the reader. They seem designed with browsing in mind. If you are stuck for a way to explain a concept or for nice examples spend a few minutes with Math Made Visual. Here are a few samples. I am not going to attempt a summary as the book has no 'plot' -- just a wonderful collection of short stories!
Chapters 1 through 4 demonstrate ways to represent numbers (and their sums and products) using geometric elements (triangular numbers), line segments, areas, and finally volumes. The formula 1 + 2 + ... + n = n(n + 1)/2 is derived in several interesting ways, all of them, I suspect, more convincing to students than the standard proof by induction. This formula and others (sum of the odd numbers, sum of the first in squares, etc) are then illustrated using line segments, areas, and volumes in subsequent chapters.
As one might expect, the Pythagorean Theorem shows up early and often. There is also a very nice proof of Herron's formula for the area of a triangle. We also encounter proofs of several of the standard trigonometric identities including Ptolemy's Theorem.
...A review of this sort can't do this book justice. To appreciate it you have to see it. The visuals (and the wonderfully clear text which accompanies them) are wonderfully conceived and masterfully executed. This is a book you will find yourself picking up again and again. --Richard Wilders, MAA Reviews
The object of this book is to show how visualization techniques may be employed to produce pictures that have interest for the creation, communication and teaching of mathematics. The authors describe methods to visualize mathematical ideas, with applications to concrete cases, and practical approaches for making visualizations in the classroom.
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