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Geometry has been an essential element in the study of mathematics since antiquity. Traditionally, we have also learned formal reasoning by studying Euclidean geometry. In this book, David Clark develops a modern axiomatic approach to this ancient subject, both in content and presentation. Mathematically, Clark has chosen a new set of axioms that draw on a modern understanding of set theory and logic, the real number continuum and measure theory, none of which were available in Euclid's time. The result is a development of the standard content of Euclidean geometry with the mathematical precision of Hilbert's foundations of geometry. In particular, the book covers all the topics listed in the Common Core State Standards for high school synthetic geometry. The presentation uses a guided inquiry, active learning pedagogy. Students benefit from the axiomatic development because they themselves solve the problems and prove the theorems with the instructor serving as a guide and mentor. Students are thereby empowered with the knowledge that they can solve problems on their own without reference to authority. This book, written for an undergraduate axiomatic geometry course, is particularly well suited for future secondary school teachers. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
- Sales Rank: #795267 in Books
- Published on: 2012-08-12
- Original language: English
- Dimensions: 9.75" h x 7.00" w x .50" l, .64 pounds
- Binding: Paperback
- 127 pages
Review
An interesting and singular approach of the Euclidean geometry is contained in this book . . . [The] book covers all the topics listed in the common core state standards for high school synthetic geometry . . . [T]he didactical approach of the large collection of problems, solutions and geometrical constructions is very important to consider it as a good textbook for teaching and learning synthetic geometry. --Mauro Garcia Pupo, Zentralblatt MATH
About the Author
David M. Clark is at State University of New York, NY, USA.
Most helpful customer reviews
3 of 3 people found the following review helpful.
It does not fit the Mathematical Circles Library
By ScienceThinker
This is the third book for which I use the same title in my review. The other two are: Geometry: A Guide for Teachers (MSRI Mathematical Circles Library) and Integers, Fractions and Arithmetic: A Guide for Teachers (MSRI Mathematical Circles Library). As I wrote in one of these reviews, AMS and MSRI could have started another series (say Teacher's Mathematical Training Library or Inquiry-Based Learning Library) and place the three books there. None of these books is directly related to mathematical competitions except from the fact that they are written on topics included in the curriculum of all mathematical competitions. But then, there are thousands of books which fulfill this criterion.
Overall, I like this book better than Geometry: A Guide for Teachers (MSRI Mathematical Circles Library). I am less negative and, if it was not for its wrong placement in the wrong series, I would probably had given it a higher rating despite some problems as I explain below.
Assuming some primary notions and some definitions (which are not always stated) and using 10 axioms motivated by measure theory, the author `derives' many of the results of Euclidean geometry without the `painful' methods of synthetic geometry. (For those not familiar with Euclidean geometry, these axioms are neither Euclid's nor Hilbert's improved axioms.) The approach is interesting and worth knowing but, for those of us who have studied synthetic geometry, it eliminates a lot of the appeal and `magic' of synthetic geometry. It also minimizes the extend of how much someone can be trained in proofs and reduces the opportunity to help someone build deductive reasoning, two of the most important benefits of synthetic geometry. Fortunately, these benefits are not completely absent since the approach is still kind of synthetic. However, the author provides no proofs to the theorems and propositions, expecting the reader to discover all of them by himself/herself. This is a little extreme for a reader who has never seen geometry and, hence, he/she must study the book under the supervision of an experienced person. In this regard, I will be a little harsh to the author: `Inquiry-based' does not mean `Discover everything on your own.' If that was the case, no math book would have any proof claiming to be helping the reader to discover mathematics by inquiry. Inquiry-based learning means discovery through experimentation and try-and-error activities; kind the same way geometry was discovered by the Babylonians and Egyptians in the first place and eventually was formulated by the Greeks as a logical framework. No such activities are given. Therefore, I would argue that this book is not an inquiry-based book but just learn-on-your-own-if-you-can one.
The author has included three appendices. The first collects the axioms used in the book which are otherwise scattered in the various chapters. The third appendix summarizes Hilbert's axioms. Amazingly, the author separates these two appendices with an appendix entitled "Guidelines for the instructor". This fact makes one wonder how much the author cared about the logical order of the topics!
In any case, my conclusion is that although this is not the book to use to get training for mathematical competitions and although some improvements can be made in a second edition, it can prove useful to students who do not want to spend numerous hours to study synthetic Euclidean geometry using Hilbert's axioms. Or it can just be used as a quick reference book that contains a list of the most important theorems and propositions.
10 of 13 people found the following review helpful.
An intellectually dishonest book
By Michael Weiss
This is a truly terrible book. Things start off badly in the preface, when Clark claims that Euclid "asked his readers to experimentally verify [the axioms] as thoroughly as they were able". Euclid says no such thing. Anyone who knows beans about ancient Greek mathematics and philosophy will appreciate the sheer historical wrongness of this assertion. (Hint: Platonism.) Statements about the curvature of space are equally incorrect.
But the central problem with this book is its shoddy treatment of the axiomatic method. In some ways, Euclidean geometry is hard soil for the axiomatic approach: Euclid falls well short of modern standards of rigor, mainly because of the lack of so-called betweenness axioms. Hilbert of course (and others) showed how to remedy this early in the 20th century.
There are two intellectually honest ways to deal with this. You can bite the bullet, include the axioms, and go through the tedious steps of verifying things that seem intuitively clear (or insist that your students do the same). You tell your students, hey, I know it gets boring, but that's just how it goes sometimes. That's the price of rigor.
Or you don't claim total rigor. You tell your audience that it's just too painful to prove all the visually obvious statements about betweenness. You will (or they should) just assume them. Of course there are pitfalls with this approach, but it can work.
I should also mention hybrid approaches. You could do one or two proofs with full rigor, but then adopt more relaxed standards. You could illustrate full rigor with a toy axiom system, say just the basic incidence axioms. Whatever you do, though, you should not pretend you're being rigorous when you're not.
The author's approach: he introduces the "foundational principle" that properties of betweenness that "hold for all points and lines in the coordinate plane hold here as well."
Problem is, how does one know that something holds in the coordinate plane? Actually PROVING this almost always involves messy algebra, as bad as the tedious steps one faces with the rigorous axiomatic approach.
Clark doesn't expect the student to give any such algebraic verifications, nor does he do so himself in his sample proofs. He gives not even a hint of what the algebraic verifications would look like. The words "foundational principle" are employed like magic fairy dust: instead of saying, "this is visually obvious, so we'll assume it", the student is supposed to say, "this holds in the coordinate plane, so it holds by the foundational principle". The book seems to claim this amounts to a huge improvement, far more faithful to the spirit of the axiomatic method.
For a gentler review that makes many of the same points, plus some others, see the MAA:
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2 of 9 people found the following review helpful.
Euclidean Geometry
By Azadeh
Very interesting book, I learned so much from it , I suggest it to people who likes to learn geometry
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