Ways to Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers

The book is intended for teachers of mathematics in professional development and preservice settings. It is one part of a triptych that also includes the Facilitator's Guide and the Further Explorations CD-ROM. In the absence of the latter two pieces, the review relates only to the one book at hand.

In the preface the authors state that "the most effective professional development in mathematics for teachers is immersion in the mathematics itself, and the most engaging and ultimately useful mathematics is the very mathematics that the teachers use and teach every day." And later, turning directly to the book user, they say

While the goal is to help you become even a better teacher, the strategy of the materials is to get you to do mathematics.

The task of getting someone to do mathematics and teaching the person how to go about doing it is notoriously difficult. The authors base their pedagogy on solving and discussing problems (the Preface lists 29 of them) grouped into five chapters. The first chapter (by E. P. Goldenberg), "What is Mathematical Investigation?", sets the tone and provides a template for the remainder of the book.

The first chapter is an excellent demonstration of several problem-solving strategies. The authors deserve credit for including problem-posing strategies as an integral part of problem comprehension. Such strategies as restricting or relaxing a problem feature, altering feature details, checking for uniqueness of the solution are exemplified in the first chapter and consistently used throughout the book.

The authors avoid presenting mathematical facts directly, going to a considerable length to try to elicit the expected formulation or solution step from the reader by posing a multitude of questions. Virtually all sections in the book contain a "Reflect and Discuss" part with one or more questions of methodological character and a "Ways to think about it" part with suggestive questions related to the problems and questions posed and asked in the main body of a section.

Indeed, the number of questions in the book is staggering. This is a powerful teaching methodology that is bound to convey the spirit of problem-solving and I am inclined to believe the authors' claim that the materials in the book have been successfully field-tested, although they do not provide any supporting specifics or the criteria by which the success has been evaluated.

The downside of the approach is that not all questions get answered, not even the exercises, of which there is a plenty in the book. The authors state that the aforementioned CD-ROM contains the solutions to all exercises so that, I guess, a teacher should aspire to have the third part of the triptych and, perhaps, the "Facilitator's Guide" as well. A teacher who successfully mastered the problem-solving strategies must be able to pass the knowledge along to the students by donning the facilitator's hat in class. This is the idea, if I got it right.

The book, however, is planned as a standalone textbook for teachers. It even features a "Problems for the Classroom (with Solutions)" chapter. For this reason, I believe, the book might have benefited from a more complete set of answers.

The four content chapters cover topics in geometry (II: Dissections and Area, IV: Pythagoras and Cousins), algebra (III: Linearity and Proportional Reasoning) and combinatorics (V: Pascal's Revenge: Combinatorial Algebra). On the whole, the problems selected for discussion in each chapter are attractive, if not entirely novel. The discussions themselves are engaging and follow consistently the pattern set up in the first chapter. The reader is offered one or more problems, helped with questions about them in the "Reflect and Discuss" section, and is given hints at possible approaches in the "Ways to think about it" section. I was especially pleased that the section on the Pythagorean theorem presents, among other proofs, a shadow of Euclid's VI.31, which I believe is rarely done at the high school level. Unfortunately, the reference to Euclid isn't made and, instead, the approach goes under the rubric of Euclid's First Cousins. When it comes to suggesting improvements for the text, this is one thing I would change.

I have further suggestions, of course. One is rather peevish: the type faces used for section and subsection titles and their numbering are virtually indistinguishable. This does not interfere with sequential reading, but did affect my ability to locate a needed piece by skimming the book.

In a few places, the authors and editors of the book have allowed a certain laxity of expression that should have been avoided, especially in a book directed to math teachers. On the whole, I have a positive impression of the book and am sure that thoughtful teachers will find both the material and techniques useful and usable in class. But many a teacher may on occasion get confused, or perhaps miss the right idea, while working through the book.

Following are a few more:

The title might have been "Ways To Think Mathematically" and not the philosophically tinted "Ways To Think About Mathematics".

p. 34: A tangram depiction of a horse is accompanied by the question "Using the horse tail as a unit — a square of side 1 — find the area (the black parts) of the horse." Under the circumstances, speaking mathematically, it is only possible to estimate the area. A little later (p. 37) the authors sensibly warn, "When you're proving what shape a final figure is, pretend your cuts are exact, even though that's not really possible." I would put a comment to this effect right at the beginning of the section.

p. 77: The mediant fraction that emerges in solving mixture problems is quite unnecessarily labeled "strange addition". What for?

p. 86: Imagine a teacher who passes on to the students the following definition:

In mathematics, taking a variable to a certain value is called taking a limit. For example, the limit of x + h as h goes to 0 is simply x (provided that x and h are independent):

lim x + h = x.

p. 153. Why is Chapter V, dealing with elementary combinatorial problems, entitled "Pascal's Revenge"? There is no story line to justify the intriguing caption. What if a student asks this question?

Later in the chapter (p. 177) there is a marginal note:

This is also called Pascal's Law and turns out to be an exercise in very careful algebraic manipulation.

Mysterious and intimidating, is it not? But why not simply give a three line sketch of the proof?

In a book that depends so much on the reader/text interaction, one cannot expect the order in which the material is (eventually) presented to be conventional. Except for one time, I felt comfortable staying within the book's framework. In one place, though, I lost a thread of exposition for a few pages. It was all about p.

p. 57: The reader is asked, How would you respond to the following question from student in one of your classes: I know you told us that the area of a circle is the square of the radius times p. But how do we know that's right? The difficulty I encountered was in that p had not been defined up to this point. The question is followed by an estimation of the area of a circle based on an arrangement of equal circular sectors in the form of a curvilinear parallelogram with an increasing number of sectors. But this is a way of estimating p, not establishing the aforementioned formula. Later (p. 59), the reader is asked to reflect as to whether similar methods can be applied to the circumference of a circle, meaning of course an estimation of the circumference as the limit of the perimeters of regular polygons. On p. 62, it is brought to student's attention that the latter consists of isosceles triangles with half of the apex angle equal to p/n. And this is the first time where the reader comes to what may appear as an implicit definition of p.

There is no doubt that I would be more at home with a more conventional and a more explicit order: a definition of p, an estimation of p via approximation of the circumference, and a derivation of the area formula as a limit.

To sum up: the book is good in its intended setting, i.e. in the framework of guided instruction for student teachers, though a final decision about using it must be left to instructors with access to the Facilitator's Guide. As a standalone book, it will be useful to teachers with solid mathematical knowledge and problem-solving ability. The book itself and the methodology it promotes may conceal some pitfalls for less qualified and/or experienced teachers.

Alex Bogomolny is a business and educational software developer who lives with his wife and a little son in East Brunswick, NJ. His popular web site Interactive Mathematics Miscellany and Puzzles is poised to welcome its 20,000,000^th visitor.