Almost all mathematicians teach mathematics, but when they are asked the pedagogical question of how mathematics should be taught, their answers reveal many different viewpoints. Some lecture, give students traditional assignments, drill their students, and want students to take the initiative in the learning process (call them "traditionalists"). Others favor discovery, cooperative learning, use of technology, higher-order thinking skills, and downplay memorization and drill (often described as "reform"). Steven Krantz, after more than 25 years of teaching, describes himself as a traditionalist "who sees many merits in the reform movement" (p.xi). Since the first edition of How to Teach Mathematics the increasing maturity of both traditionalist and reform movements has given Krantz more insights into the teaching of mathematics. He shares these with us in this second edition.
The author describes this edition as a self-help book to be used for the teaching of mathematics at the post-secondary level. It contains advice and tips, based on his teaching experience, which can add to the readers' teaching methodology and philosophy. The book is intended primarily for the graduate student or novice instructor; however, the book is also valuable for others. Post-secondary instructors who have "been there, done that" may read it to reflect on their years of teaching mathematics. Mathematics department heads may read it out of concern with the teaching performance of their faculty. Teaching Development Centers, where the mandate is to support teaching and learning at the post-secondary level, will likely find the book useful. Finally, the book may be of use to university administrators that interview newly hired mathematics instructors. I have recommended How to Teach Mathematics to all these groups at the University of Regina.
Teaching is a personal activity where different instructors take different approaches to teach their students. The second edition of Krantz's book can be seen as a discussion between the traditionalist and reform camps which allows the reader to glean ideas from both camps to benefit his or her teaching. It is not necessary to pick a winning camp but rather to enjoy reading the book, reflect on your pedagogic beliefs, and "steal" the good ideas to support your teaching.
This edition is broken up into chapters on "Guiding Principles", "Practical Matters", "Spiritual Matters", "Difficult Matters" and "A New Beginning". The new Appendices, written by twelve distinguished scholars for this edition, help to balance out the book, and to demonstrate that any teaching question has many valid answers (p. xv).
The chapter on guiding principles discusses broad issues associated with teaching: preparation, respect, adjusting students expectations, time management, use of technology, clarity, demonstrating the applicability of mathematics, etc. For example, Krantz pre-warns the uninitiated to the proverbial questions, "When am I ever going to use this?" and "Will this be on the test?" The author later acknowledges that mathematics builds up and up, each new topic taking advantage of previous ones (p. 32). Possibly future editions could expand on teaching from a constructivist perspective in mathematics classes of various sizes. There also might be more discussion on how students can become more like experts, able to effectively retrieve past information to aid in the completion of a task. Such students, for example, would explicitly ask themselves "Where have I seen a problem with a similar underlying structure before?" when faced with a problem-solving situation.
The second chapter looks at practical matters associated with teaching: the instructor's voice, eye contact, blackboard technique, body language, homework, office hours, designing a course, handouts, syllabus, transparencies, choosing a textbook, tutors, etc. In particular, there is an informative section on how to interpret and benefit from student teaching evaluations. Among the good ideas suggested by the author is the use of mid-semester evaluations to indicate how the course is going. Also included are ideas that will help in setting exams, grading exams, and handling students' concerns regarding their exams. Readers will certainly enjoy the section that deals with how to handle large groups of students. One idea suggested is for the instructor to make his/herself available at the front of the classroom after class for a pre-determined length of time.
The third chapter deals with the spiritual matters or philosophical issues: teaching as a personal activity, one's attitude towards teaching and students, the necessity of mathematics teachers, math anxiety, how students learn, teaching with the Internet, etc. The author's objective for this chapter is to describe how instructors can interact with students, excite their intellectual curiosity, and help them discover ideas for themselves. The section on the inductive vs. deductive method could be expanded to include more examples to underscore the importance of students exploring and conjecturing on their own. Also discussed are the importance of the reform movement in helping traditionalists get students more involved in the learning process and some ideas to encourage more class participation.
The last chapters -- four and five -- deal with difficult matters in teaching, such as instructors for whom English is a second language, cheating students, late assignments, discipline, mistakes in class, etc. The section on "begging and pleading" contains some interesting reading on students who view the learning process as a passive one -- something like getting a massage (p.148). The author finishes his contribution to the book arguing that American academe is at a crossroads and that reading this book might give the reader food for thought regarding his or her teaching.
So far only one voice, the author's, has been heard from in the book. In the appendices twelve other mathematics teachers comment in some way on Krantz's text and give some insight into other approaches to teaching. These authors are not going to tell you how to teach but rather give you ideas so that you can make that decision yourself. The articles feature a neat proof of a trigonometric identity, an eclectic approach to teaching, ideas for assessment, educational experiences for students using Mathematica-based courseware, a discussion of "the joy of lecturing", etc. One article states that one can follow the prescriptions of How to Teach Mathematics without any real teaching ever occurring (p. 242)....
This book is a must read for instructors preparing their courses for next semester. To conclude this review I will borrow a phrase from Ed Dubinsky: Thank you, Steve and everyone who contributed to this book, for the opportunity to review your ideas. Let's get a cup of coffee and continue the dialogue.
Rick Seaman is Assistant Professor of Mathematics Education at the University of Regina in Regina, SK, Canada.