2 comments about Symmetry: The Ordering Principle (Wooden Books).

1. David Wade makes the difficult concepts appears easy. The beautiful illustrations enhance the graphic presentation. If you have not explored these Wooden books and you like art with conceept, I recommend this one as a start.



2. I already crammed so much into the title of my review that the title almost says it all. This is a beautiful book to be sure. The book also is full of a variety of content. The price is certainly great for what you get. But being very knowledgeable about most (but not all) of this book, I can say that some of the explanations are too terse for a lay audience. Still there are great facts amidst the text too. I really would like a 4.5 rating, but I give the book a 4 because I think the book tends to make some readers want meatier text / explanation that it provides. But what is there is beautiful and varied in kinds of content.

5 comments about Sacred Geometry: Philosophy and Practice (Art and Imagination).

1. Reading Robert Lawlors book took me out of a classroom and into a discussion of the origins of mathematics. Just enough details to all the material covered, making it a breath of fresh air to others stodgy presentations.



2. Great source for a workbook; not an elementary book for the true beginner.



3. If you have been looking for the secrets of the Pythagorian Brotherhood, then look no futher - this is the book. Robert Lawlor takes you step by step into the realm of Hermetic Knowledge and connects it all together.



4. Very, very insightful intro to sacred geometry. If not familiar with this topic, I would suggest one first read the book to get your feet wet in a new way of perceiving what's around you. Then go back, re-read it slowly, and carefully do the math (which isn't difficult, really) and make the geometric constructs on graph paper like the author suggests. You have to do the exercises for it all to fully sink in, and achieve greater comprehension. This book is quietly profound. I only wish it was longer and for this talented author to get into the deeper end of the pool. Lawlor's commentary is often provocative and compelling.



5. Robert Lawlor's book presents itself in a "workbook" type form. It takes you through the process of understanding Sacred Geometry and how to look at, and understand these abstract principles in a very methodical process, by building and developing upon the simplest concept of One. Lawlor has written a classic.
   I have not seen many other books that take this approach as he does.
   He guides the reader to a deeper understanding of how the "unseen" universe works and helps one develop a better perception of that reality.
   The diagrams are easy to follow and the text is well written. It is presented in a simple format that anyone with a true desire to learn this subject will enjoy and understand, this is a great "starter" book and a wonderful reference book as well.

5 comments about Discovering Geometry: An Investigative Approach.

1. this book makes you think a lot... you have to figure out like EVERYTHING yourself... there isn't even a glossary... its hard to know if you got the answers right cuz there is no answer key... this book is made for really smart ppl...



2. I've used an older edition of this book in a high school geometry class. While the hands-on approach may be difficult to those who would rather have the concepts told to them, it allowed me to grasp the subject firmly. By allowing students to figure out different concepts, this book truly facilitates learning.



3. This geometry book has thought provoking problems, but that is all that is good about this book. There are many typos and awkward wordings to be found, and even incorrect answers in the teachers edition (my teacher has been correcting answers in his book all year)! This book is also useless without the only conjectures and vocabulary, something that should have been included in an appendix somewhere in this book! If you want to learn geometry, this is not the book to use.



4. Excellent condition. I used least expensive shipping so textbook took a while to arrive.



5. I had the misfortune of learning geometry from this textbook as a student, and now I have the misforture of teaching from it. I remember hating math as a high school student, and textbooks like these were the culprit. In high school, math was always presented as a set of problem-solving techniques that I had to learn and memorize. I was generally able to solve whatever problems came my way, but it always seemed like a trivial and pointless exercise. Luckily, I had some great college professors who made me realize that math was much more than memorizing algorithms, but a comprehensive logical system grounded in deductive reasoning.
   
   Geometry is the only math course in which rigorous deductive reasoning can be made accessible to high school students -- and not surprisingly, it was the first area of mathematics to be axiomatized (by Euclid). Unlike algebra or calculus, almost all of the theorems and formulas in geometry can be systematically obtained from postulates in a way that is intelligible to high school students; on the other hand, I have yet to see an algebra teacher attempt to prove Cramer's Rule or the Binomial Theorem to their students. The fact that geometry introduces students to a different, mathematical way of thinking is the only justification for maintaining geometry as a standalone math course, rather than integrating it into algebra courses. Otherwise, the "facts" of geometry are nothing remarkable in themselves. So what if opposite sides of a parallelogram are congruent? It wouldn't be that difficult to teach students that "fact" in an algebra class when they're learning about slopes of parallel lines. But what's important is that students understand and see how this fact derives systematically from already known facts.
   
   What does all this have to do with the book at hand? "Discovering Geometry" reduces geometry to the same collection of facts and algorithms that students have been doing in every math class since elementary school. While the problems that Michael Serra devises are occasionally interesting and even clever, he completely misses the point of geometry -- to understand WHY those "facts" are true.
   
   Unlike many critics of this book, I do not have any inherent qualms with the investigative approach to learning geometry. Investigation plays a central role in mathematics, and I applaud the author for giving inductive reasoning its fair shake in this book. But investigation has become more of an ideology than a pedagogical tool in this book. Even my weakest students groan at having to do some of the investigations, whose results they deem obvious. There are simply too many unnecessary investigations, many of which exist only to aggrandize the author's educational philosophy.
   
   As a student, I used the second edition of this book. The author has clearly made significant improvements for the third edition, but there are still serious pedagogical flaws. While Chapter 13 is a valiant attempt at introducing students to the deductive method of geometry, it is too little, too late. High school math classes rarely reach the last chapter, and separating the proofs from the theorems themselves feels artificial and contrived. The author makes another questionable pedagogical decision to area and volume into nonconsecutive chapters, Ch. 8 and 10 -- just so he can prove the Pythagorean Theorem using area in Ch. 9. But if he would only introduce similarity before the Pythagorean Theorem, he would be able to prove the Pythagorean Theorem using similar triangles in a much more elegant and motivated way.
   
   The unorthodox ordering of topics to which I have previously alluded creates problems for even the author. There are many practice problems that require concepts or techniques from later chapters. For example, students are asked to construct a square in Chapter 3 given a diagonal, before either the properties of quadrilaterals (Ch. 5) -- or even the properties of triangles (Ch. 4) -- have been introduced! How students are supposed to "guess" that the diagonal of a square bisects the angles -- I do not know. Furthermore, the first proof in the text is a paragraph proof that the perpendicular bisectors of a triangle are concurrent. I can only imagine the horrified looks on the faces of Serra's students. And these are supposedly students who are having too much trouble with the two-column proofs!
   
   There are outright mistakes in the textbook as well besides the usual typos. On page 333, Serra defines an irrational number as a number whose "decimal form never ends" and a transcendental number as a number whose "pattern of digits does not repeat." So according to his definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler might be expected to know -- much less an ostensibly expert math teacher!
   
   In his manifesto "Tracing Proof in Discovering Geometry," Serra attacks two-column proofs, saying that "so many students fail to master two-column proofs that some teachers are skeptical of claims that all students can learn geometry." While I agree that two-column proofs misrepresent mathematics and make proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

3 comments about How Round Is Your Circle?: Where Engineering and Mathematics Meet.

1. What this book shows you is that you can really understand Mathematics, when you try to build things, even something simple, like cutting a good circle from wood. Many areas of mathematics are discussed that people instinctively feel they understand, such as the roundness of a curve or circle, dividing an angle into 3 equal parts and other interesting Objects De Mathematica. You will find fascinating ways to really model the pythagorean theorem, or gather the sectors of a circle to make an equivalent triangle. There is much to discover between these pages, and Mathematics becomes concrete, objectified, and deeply understood. As another example: "what would a 3 dimensional object that has constant width throughout (based on the tetrahedron) Look like? You can see what this object looks like, when you read the work, and see the model. To add to your understanding, the Authors have constucted Models of the various mathematical principles and ideas, that you can see with your own eyes: such as "two-tip" polyhedrons, and summing the squares of numbers from 1 to n. Reading this book will improve your grasp of mathematics, as well as inspire you to study Engineering, if you havent already. Future Engineers, will be much smarter for having read this great book. Richard H. Pratt, Ph.D.



2. This book is in the tradition of the famous book "Mathematical Models," by H. Martyn Cundy and A. P. Rollett. It shows how to create models that illustrate particular mathematical laws, and in fact Cundy was consulted, while he was still alive, by the author. It is a worthy successor to Cundy & Rollett's book, concentrating mainly in two areas: linkages to draw straight lines and curves, and constant-breadth shapes, though entering a few other areas.
   
   An example of the type of problem this book considers is: How would you construct "the first" protractor or ruler, if there were none already existing?
   
   The spirit of the book is the kind of practical thinking that is thought of as engineering, but the mathematics discussed is fundamental. This is a highly recommended book.



3. This fascinating book flags the spot where engineering and mathematics meet. Each chapter essentially covers a different subject: from linkages to vernier scales to slide rules to balancing dominoes to suspension bridges and so much more. The authors combine the rigidly theoretical approach of mathematics to the very real, practical and physical problems faced in engineering. The result is an amazing romp through various subject areas where the two meet. Very few mathematical derivations are presented here; instead, appropriate references are given throughout (but the reader may feel the urge to attempt some of the derivations him/herself). Some of the results are truly amazing, e.g., stacking a leaning tower of dominoes; some are very ingenious, e.g., the vernier scale and the slide rule; and some chapters I found rather disappointing, e.g., the chapter on suspension bridges - a subject dear to my heart that somehow I felt was lacking. The writing style can be a model of clarity for many chapters while, unfortunately, others seem rather cloudy by comparison; for example, I would place the first (Hard Lines) and seventh (Follow My Leader) chapters in the second category. But overall, the reader is bound to find this book very much worth the read. Those who are likely to relish this book the most would include mathematicians, engineers and serious science buffs. This book could also be used as a supplementary text for related university courses.

4 comments about Painless Geometry (Barron's Painless Series).

1. Since I'm homeschooling my high school sophomore this year, I've been spending time looking at math books. "Painless Geometry" seemed like a good bet. Profusely illustrated (albeit with silly monkey pictures) and written in plain English, it looked like just what we'd want.

   That's until I started actually using the book. First of all, who ever heard of a 300-page reference book with only three pages of index? How are you supposed to find things that way? It's missing things like the base of a triangle (the index has neither "base" nor "triangle:base") and how to label an angle. The information's in the book, but you certainly can't find it using the index. Not only that, but the pages aren't labeled like a normal book, with the name and number of the chapter at the top or bottom of each page. You can't find your place in a book that way!

   There's little depth to the book. There are experiments with pencil and paper, but no real-world examples of where you'd use geometry. Area is calculated in "square units" with no discussion of real units of measure. Pi is introduced with a single paragraph. No explanation is given of its rich history, how it's calculated, or applicability throughout mathematics.

   The oversimplifications in this book may make life difficult later. The book states that all angles are measured in degrees, and the degrees symbol is generally omitted. Whatever happened to radians? In one of the problems, she asks for the area of a circle with diameter of ten. The correct answer is 100 times pi. The book states the answer as 314. That's an approximation, not an answer!

   Then we started finding the mistakes. Typos like "Computer the area of a circle" (page 184) I can live with. It's hard core mistakes like these I can't tolerate:

   The reader is asked to identify what type of triangle has angles of 120, 35, and 35 degrees (page 101). The answer says it's isosceles and obtuse. In reality, it's not a triangle at all, as the angles don't add up to 180 degrees!

   How's this for a statement of the Side-Angle-Side postulate (page 126)? "If two sides and the included angle of one triangle are congruent to two triangles and the included angle of a second triangle, then the triangles are congruent." Huh?

   There's a "super brain tickler" on page 163 which indicates, according to the answers in the book, that for squares, rhombuses, rectangles, and parallelograms, all four sides are parallel! No. Four parallel line segments wouldn't ever meet. Those four shapes have two sets of parallel sides, not one set of four parallel sides!

   .... That tends to leave us with drek like "Painless Geometry."

   All in all, I found this book to be poorly proofread, ridded with errors, badly indexed, oversimplified, and disconnected from the real world. It may be good as an adjunct for a student having trouble with a real geometry book, but only if there's someone around to explain what "Painless Geometry" omits or misstates.




2. On page 16, it is stated that the area of a circle is pi times the diameter. Is there anybody out there who DOESN'T know that the area of a circle is pi times the square of the radius? That error wouldn't such a big deal, except that there are plenty more to come. I don't recommend this book to anyone.



3. I've been meaning to write a review to respond to those on this page for a while. I guess I have used so many math books that contain an error or two that I just can't possibly throw away such a good book over that.
   
   The fact is that we homeschool and my son LOVED this book which we picked up at the library. It is full of wonderful, hands-on work and SIMPLE explanations that make geometry easier to understand than most other books we tried - yes, truly understand because you not only had it explained well, but also "did" something on paper or folding paper to experience it.
   
   He enjoyed it so much that when I picked up another Painless book at the used book store, he wanted to start it that day, rather than waiting 'til next semester.
   
   So I don't know if y'all just glanced at the book or really tried it, but this family tried it and loved it - and I own a red marker so I can cross out the one incorrect answer I found in my edition!



4. Great tool for young mathematicians new to Geometry. Supplements school textbooks -- exercises in the book facilitate review of concepts learned in class.

5 comments about Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.

1. Wauw, never thought the prime principles and theories behind it could be explained so well and most of all so easy to understand.
   With this book, the writer makes one of the most mysterious and complex theories in mathematics easy to understand for the common man.
   Simply great!



2. The Prime Number Theorem and its related consequences is a fascinating subject. This book however is a long winded, very poorly written attempt. The author never seems to be able to make a point clearly and succinctly. Instead we are treated to a ramble that confuses more than elaborates. For example the presentation on functions is very poorly done. I have seen basic algebra texts explain the idea of functions far better than in this book. To make matters worse the author takes about ten pages to explain the beautiful idea of a function thus killing a sublime truth.
   
   Sorry to say but far better general math books abound. Take a look at Journey Through Genius.



3. This is a fine book that lays out clearly Riemann's life and times and the mathematical work he did, but I'll explain more about that later in the review.
   
   The author certainly knows his subject matter and it thoroughly researched, which I'll touch on a little later. The only thing that drove me to distraction was that at least 2 times in every chapter he would make a point, then say something like "I'll explain more about this in just a bit" or "More about this later" or "In chapter 12 I'll touch on this again". I'm not kidding, it happened EVERY chapter and after a while it got annoying....but I'll show you more about that in my other review.
   
   Still, a great read and required reading for anyone into mathematics and number theory.



4. There are a number of authors who have tried to esplain difficult scientific or mathematical problems to non-expert readers. I have read and enjoyed many such books. Thus far, Derbyshire is the master.



5. Very good book, easy to read, no previous knowledge on math is needed but just secondary.

3 comments about Mummy Math: An Adventure in Geometry.

1. This is a picture book about the Zills family. The family is off on an adventure to hidden burial site of an ancient pharaoh. The twin brothers in the family get stuck inside a pyramid with their dog. They must use math to figure out a way to escape the pyramid. They must use skills to figure out the riddles written on the wall and locate the hidden burial chamber. Will they ever figure it out? Read Mummy Math to find out!
   
   It's one of those special books which children will read without realizing they are learning a mathematical concept.
   
   This is a recommended book for those teacher or parents that want to make math a little more exciting for children.



2. I used this book for two separate lesson plans. I made a game board and added game cards based on the book. This quickly became an engaging way to count faces, vertices, and edges. I also used it in 7th grade social studies to add questions about history. BONUS: my 13 yr. old and 10 yr. old thought it was an engaging book!



3. Illustrations are done well, the story line is engaging for children. It keeps them involved in the mystery.

5 comments about Euclid's Elements.

1. I received my copy of the softcover euclid's elements, and it looked glued to me, not sewn. This was my error, and indeed it is both sewn AND glued, hence it took a little more knowledge than I had to detect the stitches, but not much.
   
   I would like to raise my rating to 5 stars, but it seems that is one aspect of a review that cannot be edited. My apologies to the publisher. This is top rate work.



2. Euclid's Elements is a beautiful classic, and anyone with a serious interest in mathematics should look at it at some point.
   
   This is a nice edition: the paper and binding are high quality, and diagrams are repeated if necessary so that you never have to flip pages between a diagram and the argument that is referring to it.
   
   There is no commentary except for a brief introduction (some of which consists of quotes from older commentaries). Many readers may find this preferable to the Heath edition, where Euclid is almost drowned out by overwhelming amounts of commentary.
   
   If you haven't read Euclid before, you might be surprised to find that the proofs are full of gaps. Many proofs use arguments which are not justified by the axioms (for example the proof of the first proposition takes it for granted that a certain pair of circles intersect), and there are many examples of "proof by special case". These issues don't really detract from the beauty of the work (and one can argue that putting everything on an absolutely rigorous foundation wasn't the original goal anyway), but are probably part of the reason why this book has been so heavily commented over the millennia. For a modern discussion of the mathematical issues in Euclid, and some related subsequent developments, I recommend "Geometry: Euclid and Beyond" by Hartshorne.



3. This edition of the Elements is not a new translation -- it's Heath's classic translation that has stood as the authoritative English version for almost a century. Rather, it's a repackaging of Euclid so that all the books are in a single volume (unlike the Dover edition) and with some marked improvements in typesetting and binding. Unlike most repackaging efforts, which provide little to no value added to the work, the folks at Green Lion Press have made simple changes that provide great improvement.
   
   First, the front matter of the book is extremely helpful for first-time readers. There are detailed descriptions of the idioms used by Euclid in his proofs, especially some of the more esoteric terms such as "same ratio" (which Euclid uses but never really clarifies). The explanations draw upon classical and contemporary scholarship in Greek mathematics, and they elucidate the text greatly.
   
   Second, the binding is outstanding and sure to stand up to robust reading and annotating (which is what books are for, right?). The paper is not glued to the binding, so you can open it flat without tearing the pages from the spine. I found it a little hard to open the book flat due to the sheer size of it, especially in the first few pages. But that's why God made paperweights.
   
   Third, the figures in the text -- which are obviously crucial to any text on geometry -- are reproduced whenever they are referred to, to minimize having to flip back and forth to view the same diagram and thus saving wear and tear on the book. Again, this is not high-tech here, just a simple change that makes the book a lot more usable.
   
   I am reading through this book right now and it's like being a kid in a toy shop. I'm looking forward greatly to sharing the book with my geometry students next year. I think any person studying geometry or just interested in digesting some of the greatest intellectual works of all time, as this book is, would do well to reinvest and purchase this volume.



4. No doubt this is a precious edition, but the recent one by Richard Fitzpatrick in Greek and English is really astounding (and affordable), especially the version in 4 volumes, where a full page (and more than one if needed) is generously given to each proposition.
   
   OTH I find the classical Heath/Dover edition an essential reference, precisely for the illuminating Heath's interspersed commentary. In fact Heath's "A History of Greek Mathematics" and the shorter "A Manual of Greek Mathematics" are just as essential to a better understanding of Euclid, and of course Greek Mathematics.
   
   As a complementary/companion reading I would heartily recommend Benno Artmann's excellent "Euclid - The Creation of Mathematics", priceless, if only, for its bibliography.
   
   An indispensable and brilliant mathematical "guided reading" based on the development of Greek Mathematics, fundamental for an overall understanding of the Elements, is B. L. van der Waerden's "Science Awakening" (1954, out of print, but still possible to find).
   
   A philological approach, incredibly rich in its revelations about the Elements and early Greek Mathematics in general is Arpad Szabo's "The Beginnings of Greek Mathematics" (very expensive!, but worth every penny).
   
   An then there's David Fowler's "The Mathematics of Plato's Academy", an up-to-date comprehensive account on Greek Mathematics of the time, and Euclid of course.
   
   Just a final, more general comment. Through its many layers Euclid's Elements is a key to the better understanding of our deepest roots. I myself discovered it late in my life and cannot believe what I missed. I now understand that the fundamentals of Ancient Greece - and consequently ours - cannot be fully grasped without this book. It would be no exaggeration to affirm that Plato and Aristotle are not fully intelligible without Euclid and that, in general, without knowledge of the Elements Ancient Greece would be just a "shadow of the past", devoid of the sparkling light cast by its mathematics and science.



5. This book, a translation from Greek, is one of the most printed and used books
   with the exception of the Bible being greater in both actions. After Gutenberg
   processed his bibles and a few other works - he printed Elements for the craftsmen
   of Europe. Those builders of Churches and massive buildings.
   This is a fine translation and all 13 books of Elements are contained within.

5 comments about The Geometer's Sketchpad (R): Student Edition.

1. This product is really good for working through classroom problems. It gets a little tougher, if you're not used to the program, for more advanced problems. But it does help even by just using it to mess around with when having to do any geometry problem.



2. The cheapest, before shipping, I could find this on the Key Curriculum website was $39.99. This is a much better price.



3. This is a great tool when studying Euclidean Geometry. It works just like a straight edge and compass, except the diagrams can be manipulated afterwards so that one sketch can be used to test multiple cases (acute vs obtuse, for instance) which helps in the observation of geometric properties. The interface is easy to use and quick to learn. On the down side, the program offers very little help, which can be frustrating when you're trying something new. (I have yet to make the Construct Locus function work.)



4. This sketchpad is a great way to supplement any geometry curriculum. I bought the student version (the only downfall I can find) and you have to put in the cd every time you run the program.
   
   I will use it to demonstrate the Pythagorean theorem and construct basic geometric figures for my middle school curriculum. Very easy to use too! It does link you with additional lessons through NCTM. I will probably purchase the full version next year.



5. Loved it and have been happy with this software. It helps me decide whether to make orders of that for my school where I am teaching Geometry so I will do that. Thanks.

5 comments about Geometry (Cliffs Quick Review).

1. This is a helpful tool for practicing concepts of geometry, and stimulation of memory. However, it may not be helpful if you never had any experience with geometry.



2. After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."



3. As a long-time teacher of mathematics at the college level, I am always trolling for additional/better materials to help students learn mathematics. Since I am not a fan of the Cliffs Notes series, I hesitated before purchasing this book in a used book store. However, once I started looking through it, I realized that it is a very good review of basic geometry.
   It begins with the fundamental postulates and immediately goes to some basic theorems, although no proofs are offered. The chapters are:
   
   *) Fundamental ideas
   *) Parallel lines
   *) Triangles
   *) Polygons
   *) Perimeter and area
   *) Similarity
   *) Right triangles
   *) Circles
   *) Geometric solids
   *) Coordinate geometry
   
   There are a small number of exercises at the end of each chapter and a summary exam at the end of the book. Solutions to all exercises are included. If you need a fast, complete review of geometry, then this is an excellent selection. However, it has little value if you are trying to learn geometry.



4. Overall this isn't bad for a rapid summmary
   but there are some subtle errors which may
   undermine the reader's confidence in the
   material (e.g. note altitudes and areas
   of the triangles on page 76).



5. I am involved with tutoring and haven't looked at geometry for MANY years. I am using Cliff's for review and to supplement more in-depth material. I am finding it great for bringing back the material.