“Guided inquiry,” a term of art in the education community, is a currently popular idea that has recently been the subject of a number of papers and statistical studies. (There is even an online journal called the Journal of Inquiry-Based Learning in Mathematics, http://www.jiblm.org/, which makes articles on the method freely available.) The basic idea of guided inquiry, as in the familiar Moore method, is to replace the traditional lecture format by a program of active student involvement in class activities, designed to get the students to discover answers on their own. Many professors swear by it.

I am not, by any stretch of the imagination, an expert in the science of pedagogy, and I haven’t taught any classes using this approach, so I confess to not having enough first-hand experience to allow me to form any firmly-held opinions on the subject. The only time I ever saw the method used was in a mock classroom demonstration where the “students” were a collection of college professors, high school teachers, and high school guidance personnel, and the method was implemented by the professor giving some definitions, then posing problems for the students to solve (with collaboration encouraged), after which students presented their solutions at the blackboard (in this demonstration, two students were called to present solutions to each problem, and each student had to explain his or her work; if student #2 agreed with student #1, he or she couldn’t just say so, but still had to go through the steps of his or her thinking.)

This demonstration illustrated both positive and negative aspects to the method. On the plus side, many of the “students” really did seem to get interested in the problem, and there was a considerable amount of classroom interaction. It seems reasonable to believe that students develop greater insight into the material by discovering (or even attempting to discover) things on their own. On the negative side, the mechanics of guided inquiry are necessarily time-consuming: time is spent giving the students an opportunity to work on problems at their desk, and more time is spent giving them an opportunity to discuss their work at the blackboard. The upshot was that the amount of material covered in the hour-long classroom demonstration was about what I would have taken 15 minutes or so to cover in a class using the traditional lecture format.

I thought about this example as I read the book under review, which applies guided inquiry to basic Euclidean geometry. I have seen other geometry books over the years that use this method; two good ones are *Experiencing Geometry: Euclidean and non-Euclidean with History* by Henderson and Taimina, and David Gay’s *Geometry by Discovery*.

Clark’s book, however, has an entirely different focus than does either of those two. While the Henderson/Taimina text, as the title implies, focuses on the development of non-Euclidean geometries (and even includes directions for crocheting a hyperbolic plane) and Gay talks about an assortment of miscellaneous topics in Euclidean geometry (polyhedra, shortest path problems, symmetry, etc.), the book under review has as a stated goal an axiomatic development of Euclidean geometry. I am not aware of any other book which attempts such an undertaking via the guided inquiry method.

The choice of topics here, according to the preface,

follows the guidelines of Hung-Hsi Wu of U.C. Berkeley in ‘The Mis-Education of Mathematics Teachers’… Wu’s thesis is that, before we do anything else, our primary obligation to pre-service teachers is to give them a sound knowledge of the topics they will actually need to teach.

(The article by Wu that is referenced here appears on pages 372-384 of the March 2011 issue of the *Notices of the American Mathematical Society*.)

Accordingly, the topics covered in this very slim (about a hundred pages long) book are largely topics that are covered in a typical high school geometry course: congruence, parallel lines, area, angle measure, basic trigonometry of a right triangle, similar triangles. Using the guided inquiry approach, there may not be time available to cover much more than this.

And therein lies one of my principal problems with this book: it seems to convert a college course in geometry into a high school course. A statement on the back cover of this book says that it “covers all the topics listed in the Common Core State Standards for high school synthetic geometry,” a statement that might induce me to adopt the book for a high school course, but not, I think, for a course in college that presumes a high school education. The fact that most of the results proved in this book are results that the student has probably seen before in high school strikes me as troublesome, even granting that the approach here is different than in high school. Particularly at those colleges (which may be the majority of them) that offer only a single semester of geometry, a graduate who takes that course with this textbook will know little or nothing about non-Euclidean geometry and the wonderfully interesting revolution in mathematics that led to its discovery, and will not even know anything of the more advanced geometry of the circle or triangle (e.g., Morley’s beautiful theorem, or the nine-point circle) which lies just beyond the high school curriculum (and which could be used as enrichment material for a sophisticated high school class).

In the class I am scheduled to teach next semester on Euclidean geometry, using Isaacs’ book *Geometry for College Students*, I intend to spend only the first three weeks or so reviewing the basics of Euclidean geometry, covering in that review much of the topics discussed in this book (though not from an axiomatic viewpoint); then I plan on discussing new topics that could perhaps be understood by a bright high-school student, but are not generally taught there.

There are, admittedly, some topics in this book that do go beyond high school mathematics. The author uses geometric transformations when discussing similarity and congruence, so that, for example, two triangles are defined to be similar if one is the image, under a dilation, of the other. (There may be some precedent for doing things this way even in high school: this is the approach used in Lang and Murrow’s high school text *Geometry*.) In addition, the final chapter of the book is a brief introduction to some of the ideas of projective geometry, couched in the language of perspective drawing (the author refers to “perspective geometry” rather than “projective geometry,” “vanishing points” rather than “ideal points,” etc.) The account here is relatively brief: the axioms for a projective plane are not mentioned, and in fact the extended Euclidean plane, though essentially developed here, is not explicitly pointed out to be an interesting geometrical system which has a different parallel postulate than does familiar Euclidean geometry.

The selection of topics for a textbook is always a matter of individual taste, and although one might disagree with an author’s selections, one cannot really fault a book for not covering the topics you would have put in. In this case, though, in addition to the fact that I thought this text was too rooted in high school-level material, I was also very troubled by the way in which the “axiomatic method” was carried out. My problems began with the author’s definition of “axiom” as “a fact that we can experimentally verify by looking carefully at physical examples.” Not only is the injection of physical verification totally unwarranted, but one could also make a case that (even assuming the Euclidean parallel postulate could be “experimentally verified,” which is doubtful) this definition excludes the hyperbolic parallel postulate as an “axiom,” which of course it is.

The author, by the way, states in the Preface that Euclid “asked his readers to experimentally verify [the axioms] as thoroughly as they were able.” This is the first time I have heard this assertion, which is certainly not supported by my copy of *The Elements*, which contains no such request. As Greenberg points out in *Euclidean and Non-Euclidean Geometries: Development and History*, geometry is “a purely formal exercise in deducing certain conclusions from certain formal premises… it does not say anything about the meaning or truthfulness of the hypotheses.” *This* is the lesson we should be teaching students of the axiomatic method.

In addition, I was troubled by the actual choice of axioms. The author recognizes that a big pedagogical difficulty with a rigorous axiomatic approach is that students must spend a lot of time proving things that they regard as intuitively obvious; this is why, for example, Martin Isaacs made a conscious decision in *Geometry for College Students* to eschew the axiomatic approach altogether and develop geometry from what I would refer to as the kind of “naïve” approach used in high schools. Certainly, reasonable arguments could be made on either side of the argument as to which approach to use. What the author of this text does, though, is attempt to have his cake and eat it too by using a watered-down axiomatic approach that purports to be based on axioms but which relies on unspecified assumptions when the going gets tough. For example, the area axiom involves an assignment of area to every region enclosed by a “closed path,” but the word “path” is not defined. (In Moise’s book *Elementary Geometry from an Advanced Standpoint*, this problem is dealt with by using polygonal regions.)

More serious, I think, is the fact that the author relies not only on axioms but also on geometric “foundational principles” which run the risk of not making clear just what specific facts the reader can use to construct proofs. Instead of, for example, writing down specific axioms for the idea of “betweenness” (as is done, for example, in Greenberg’s book, or Wallace and West’s *Roads to Geometries*), the author expects the reader to use as a “foundational principle” the fact that properties of betweenness that “hold for all points and lines in the coordinate plane hold here as well.” The text then defines a point B to be between points A and C if “B actually lies on the straight line from A to C.”

I cannot help but feel that this approach is fundamentally flawed. For one thing, it appears to put the cart before the horse: if you’re going to develop Euclidean geometry from axioms, you ought not rely on unspecified facts *about* Euclidean lines in your proofs. I will grant that this circularity issue can perhaps be logically avoided, but only at the cost of assuming facts about real numbers and solutions of equations that seem, if anything, more difficult than the subject of the axioms themselves. However, whether logically circular or not, I think that a real problem with this approach is that it does not really teach the essence of the axiomatic method: if a student is to be taught the method, he or she should, I think, be taught what it really involves — after all, it is the desire to avoid the dangers of reasoning from diagrams or implicit assumptions that motivates the introduction of the axiomatic method in the first place.

This “foundational principle” is not the only problematic one used by the author: another is that points of intersections of lines and circles exist if these objects, constructed by ruler and compass in a coordinate plane, necessarily cross. Thus, the axiomatic development in this book assumes as known not only the coordinate plane but also unspecified facts about ruler and compass constructions, a subject that is usually taught as part of a geometry course. Surely some students, particularly the brighter ones, are going to be disturbed by the fact that facts about Euclidean geometry are being used in an axiomatic analysis, the goal of which is to develop Euclidean geometry in the first place.

The author’s use of these “foundational principles” (which he himself states in the Preface are different from axioms because they are “essentially subjective” and not “precisely specified”) is in my view wholly inappropriate for a textbook that purports to be teaching students the axiomatic method. The last thing we want to do is teach such students that it is a good idea to construct proofs from facts that are not “precisely specified.”

There is, of course, nothing wrong with doing Euclidean geometry, even in a college text, from a non-axiomatic approach; as I said, this is the way Isaacs does it, and I enjoyed his book. Isaacs, however, did not purport to be doing an “axiomatic approach,” and in fact used the first section of his book to explain the difference and apologize for his less formal treatment. The problem here is that this book claims to be doing things axiomatically.

To give other examples of possible confusion: one of the axioms is a “distance measure” axiom; a student may reasonably wonder why, if knowledge of the Cartesian plane is available for our use, we cannot simply assume without an explicit axiom these basic facts about distance. Likewise, another axiom states that the image of three collinear points under a dilation are again collinear, and that dilations preserve betweenness; these are also facts which can be proved in the Cartesian plane, and therefore should not require an explicit axiom.

Using ten axioms (one of which — the statement that through a point not on a line, there is at most one line parallel to the given line — is incorrectly described by Clark as “Euclid’s famous Fifth Postulate”; in fact, this is Playfair’s postulate, which is logically equivalent to Euclid’s fifth postulate but completely different from it) and three “foundational principles” (the one that has not already been discussed says that two points determine a line, that every line contains at least two points, and that there exist three non-collinear points; this is not objectionable, but I wonder why the author makes this a “foundational principle” rather than an axiom), the author then develops the various topics in Euclidean geometry referred to previously, stating many theorems and giving lots of problems for the student to do.

In view of the “guided inquiry” nature of the book, neither solutions to the problems nor proofs of the theorems are provided in the book (except for theorem 35, which is proved in the Preface as an illustration of what the author thinks a proof should look like). There is an Instructor Supplement on the AMS webpage for this book, but it contains hints and teaching suggestions rather than detailed solutions and proofs. It is also (at least as of this writing) not password-protected, but probably should be.

In summary, the book under review represents an interesting effort to do something genuinely novel, but, for the reasons expressed above, does not really meet my needs. However, people who would like to try a guided inquiry approach and who are not as troubled as I am by the selection of topics or axioms that appear in this book, should certainly give it a look, and might also enjoy looking at the jiblm.org website cited earlier, which identifies Professor Clark as an editor and contributor.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.