Saturday, May 14, 2011

The Long, Slow Slide

I teach only geometry.

I've thrown myself into the subject over the past four years. Only now do I believe that I've begun to understand what it is and how it should be taught. (Of course the former should determine the latter. Pedagogy must follow the contours of content. Geometry is often ruined by the insistence that it be taught in accordance with one or another pedagogical theory.) What is it? What is this thing that we call "Geometry"? We must say first that it is a branch of mathematics. It is, if you like, the mathematics of shape. But what then is mathematics? I don't pretend to have a complete answer. But this I know: mathematics is of its essence systematic and deductive. (The application of mathematics need have neither of these qualities, and unfortunately what often goes by the name of "mathematics" in our schools is simply unsystematic application.) By "systematic" I mean that it begins in a small set of first principles and that all else grows out of them. By "deductive" I mean that all of its conclusions must be true - not just might be, but absolutely must be true - if the first principles are assumed.

The core concepts within mathematics - and so in geometry too - are axiom, proof and theorem. The axioms are the first principles, those fundamental assumptions, with which we begin. From them, we construct proofs of certain propositions that, once proven, are no less certain than the axioms on which they are based. A conclusion once proven comes to bear the name "theorem".

Since this is what mathematics in general and geometry in particular are, this is how geometry must be taught. It is a proof course. It is, moreover, the very first (and alas likely the very last) proof course that students will ever take. In this lies the importance of geometry. We all wish our students to learn how to think. This is a teacher's first goal, the goal that all others serve. But to think is to reason, and to reason is to reason from one thing to another. Thought is a process, a process that begins with what we might call a data set (whatever precise form this might take) and from that data set draws a set of inferences. In geometry, we shed all extraneous concerns and do just this. We think. We reason. Thus in geometry we focus to the exclusion of almost all else on the most important of skills that can be imparted to students.

On this basis I form my judgment of texts. Is a geometry text primarily about application of theorems whose proofs are at best an afterthought? I despise such books. Of these there are many. It seems that this is the only sort of text that the major publishers - Pearson, Glencoe and all the rest - can produce.

Over these past four years, I've built up a collection of texts, some old some new. The good ones are the ones that teach proof and teach it well. The bad ones are the ones that only occasionally touch on proof and teach it poorly when they do. Here's a sample:

1. Euclid's Elements. A reviewer at Amazon called it "the best book ever written by a human being". An overstatement, perhaps. But I share the reviewer's enthusiasm. It always has been, and always will be, the model of how mathematics should be done. More than any other book every published, it has shown us how to think.

2. Kiselev's Geometry. Published in the late 19th century, this was the standard text in Russia for almost 100 years. I finally yielded to a more Soviet text but was still widely used. Verdict: absolutely superb. I'd be delighted if I could use it.

3. Bartoo and Osborn's Plane Geometry. This was the text used by my grandmother when she first began to teach in the 30's. (It has her notes scribbled in the margins.) The date of publication was 1939. I know little about its authors. I don't know how widely it was used. But I do know the quality of the text. It isn't as good as the Kiselev, but it is still excellent.

4. Lewis' Geometry: A Contemporary Course. First published in 1964, the text maintains a standard of rigor that is matched by very, very few texts in publication today. I find the discussion sometimes less than crystal clear, but the problem sets are quite good. They're proof after proof after proof, just they they should be.

5. Harold Jacobs' Geometry: Seeing, Doing, Understanding. First published in 1974, this text is now in its third edition. I've had a fascination with it for some time. To my dismay, it was not the one chosen by my corporation. It is by far the best text currently in use. If I were king, it would be the only text ever used. Explanations are always clear, but Jacobs truly excels in the problem sets. They are always creative, always a challenge for students, and always fun. (I grin through every problem set that I do.) Moreover, Jacobs knows his history. The authors of most geometry textbooks seem to have a knowledge of the subject that extends back only to the prior edition of their text. Not so Jacobs. Jacobs' text seems to have gone out of print. It seems that demand was not great. Indeed it was I gather the least used of the texts currently available. Only a brave few took its challenge. Verdict: an extraordinary achievement and by far the best of its era.

6. Glencoe's Geometry 2004 . We now have a text written by committee. Coherence has been sacrificed. We do not have a single vision throughout. We have multiple visions. This creates a kind of content schizophrenia. Upon occasion it's good, but the quality is never sustained. The good and the bad (and the completely irrelevant) coexist in a random mix. Proof is done upon occasion, but problem sets are dominated by simple-minded application of results whose proofs are sometimes given, sometimes not.Verdict: mediocre at best.

7. Glencoe's Geometry 2011. Here the bad has become worse. The book is a travesty. Many of the decent problems from the 2004 edition - ones that had the potential to challenge students - have been sacrificed. All that's left is an endless profusion of problems that require little more than simple-minded application of basic results. Verdict: a multicolor abomination.

8. McDougal's . Equally as bad as the 2011 edition of the Glencoe text. Unfortunately the McDougal and the Glencoe texts are in wide use. Most of our students learn their "geometry" from texts such as these.

Note that my list was chronological. Moreover, each is representative of the era in which it appeared. A certain conclusion appears inevitable: we're on a long, slow slide downhill. Content has been, and continues to be, watered down. Hard problems have been cut. Postulate sets have become bloated. Proof is often an afterthought. Irrelevant little side-topics abound. The systematic construction of a system of geometry has been sacrificed. Most of what remains requires only the mindless application of basic algebra to a set of monotonous problems.

Your children have been cheated. Their course in geometry (unless that are lucky enough to find themselves with a teacher who can correct these many problems) has been so dumbed down that it has become a waste of their time. If I were a parent of a child in such a course, I'd be as mad as hell.

What to do about this sad state of affairs? I say that we need a return to rigor, that we should make our classes much, much harder than they are at present. We need a return to a genuine geometry, the sort given to us by Euclid, Kiselev and Jacobs. The new breed of text produced by corporate behemoths like Glencoe and McDougal should be consigned to the flames. I do not doubt that more students will fail. But so be it. Some will rise to the challenge; those that do will have learned something worthwhile.

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