Discovering Geometry: An Investigative Approach

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.

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.