Discovering Geometry: An Investigative Approach [Hardcover]

Michael Serra (Author)

Book Description

ISBN-10: 1559534591 | ISBN-13: 978-1559534598 | Publication Date: August 1, 2002 | Edition: 3rd

Discovering Geometry: An Investigative Approach

--This text refers to the Kindle Edition edition.

Product Details

• Hardcover: 767 pages
• Publisher: Key Curriculum Press; 3rd edition (August 1, 2002)
• Language: English
• ISBN-10: 1559534591
• ISBN-13: 978-1559534598
• Product Dimensions: 11.1 x 8.7 x 1.4 inches
• Shipping Weight: 4.2 pounds
• Average Customer Review: 3.7 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #122,286 in Books (See Top 100 in Books)
  • #72 in Books > Teens > Education & Reference > Science & Technology > Mathematics

Customer Reviews

Most Helpful Customer Reviews

42 of 50 people found the following review helpful:

1.0 out of 5 stars A fine example of how NOT to teach geometry, June 28, 2008

By 

alacarte - See all my reviews

This review is from: Discovering Geometry: An Investigative Approach (Hardcover)

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 memorize and apply. 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; contrast this with algebra, where it would not be reasonable to expect students to understand a proof of Cramer's Rule or the Binomial Theorem. 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 too difficult to teach students this "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. Geometry 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 cover 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 the concept of similarity (Ch. 11) 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 traditional 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 (incorrect) definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter false because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler would 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 can proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

Addendum: Some commenters have asked about alternative geometry textbooks. While I (sadly) have yet to find a textbook that lives up to the promise, the study guide E-Z Geometry (I used an older edition which was called "Geometry the Easy Way") does an excellent job of presenting the material in a coherent, logical fashion, replete with proofs, and comes with a collection of excellent exercises. It's also cheap, although I'm not sure how it would fare as a classroom textbook, since all solutions are at the back. I have also heard excellent things about Harold Jacobs' Geometry: Seeing, Doing, Understanding, although I have not had the fortune to look over the book myself.

28 of 35 people found the following review helpful:

2.0 out of 5 stars Unacceptable, June 10, 2006

By 

ElBlufer - See all my reviews

This review is from: Discovering Geometry: An Investigative Approach (Hardcover)

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.

7 of 8 people found the following review helpful:

5.0 out of 5 stars Don't Listen to the NaySayers, January 28, 2011

By 

Brandon J. Porter - See all my reviews

This review is from: Discovering Geometry: An Investigative Approach (Hardcover)

Those above with negative things to say are only one viewpoint. How about a review from someone who has used this text (2nd, 3rd, and now 4th edition)?

I LOVE this geometry text. As a high school math teacher, I have used Serra's text for 10 years now, with great results. It teaches students how to think inductively, which is a greatly lacking skill among high school students. It also helps students to develop into independent thinkers and great problem solvers. In addition, it teaches students how to work with others, a skill that anyone who works with people can attest is lacking. It also develops students into better readers, as they have to read written instructions on a daily basis to succeed in the course. All of these are great benefits of the book. BUT...

If the facilitator/teacher doesn't use proper constructivist techniques in the classroom then this approach will fail. The teacher must be on his/her toes, must enforce cooperative problem solving norms in the classroom, must have rules for group work, social skills that students are to adhere to, etc... I suspect that those who have taught using this text and have had a hard time with it have experienced such because they aren't using it correctly.

This text will revolutionize your geometry class. It will transform your class from a traditional, 150 year old teacher-centered classroom to progressive, modern, discovery learning, student-centered classroom. It places the responsibility of the learning on the student--where it should be.

So what if there's no glossary? It's misleading of people to say that this is a bad text because there's no glossary. This is, once again, a product of using the wrong method with this text. Students discover and write their own definitions for terms, and keep a notebook to add definitions to each day. They also investigate geometric relationships and add conjectures (theorems) to their notebooks daily. If the students don't know terms or theorems, it's because they aren't doing the work (again, responsibility is theirs).

So what if there are no answers to the problems in the back of the book? This allows students time to explore and apply and critically think about problems without the crutch of the answer (most math books also don't offer solutions to problems, only answers, so they're not that helpful anyway). I make a copy of the answer key for each group in my class, and they sit as a group and discuss homework problems at the beginning of each class, so they have a chance to peer-tutor every day--another great skill to have.

Again, the nay-sayers aren't being fair. This is a great text, and certainly adds interest and fun to what is otherwise an often poorly taught, boring class.

One thing I love about this text is that it shows students that geometry is a mathematical system, and as the instructor you can facilitate some great discussions about the validity of Euclidean geometry being based on Euclid's assumptions. Obviously, they're not fool-proof as there are other types of geometry whose authors disagreed with some of Euclid's propositions.

Lastly, 99% of your students will not grow up to become mathematicians or engineers. Why do they have to engage in a class that forces proofs too early, asks them to think and act like mathematicians when they are not, and is often downright boring? Who cares if this text waits too long to introduce proof? How many of you did proof anywhere besides geometry? (By the way, the 4th edition has a proof strand throughout, and slowly allows students to get used to the idea of what a proof is and how to develop one.)

Traditional texts are too difficult for most students. Research suggests that more than 90% of students who take HS geometry never really understand proofs. Again, the nay-sayers need to read up on the recommendations from NCTM. This text is exactly what organizations like the NCTM recommend and endorse. They recommend more investigative, discovery learning because they realize that in the long run it is better for students and helps them to become better problem solvers. This text does exactly that.