Many years ago, I did a lot of traveling to foreign countries. My first experience was unplanned; I had the names of a few places on the back of an envelope, and no other information about transportation, sites, museums, restaurants, or hotels. It went OK, but then I was lucky. The next time I had a chance to travel, I purchased a guide book, *Let's Go Europe*, which turned out to be extremely useful. This book helped me find my way around, introduced me to many beautiful cities, mountains, beaches, and cultures, helped me find my own style of traveling (more on that later), and helped me stay in one piece. It significantly improved the quality of my trip.

The book under review is a guide book too. It gives information for those beginning to teach mathematics at the college level, the kind of information a new TA might find useful. (A new Ph.D. who has never TAed might also find the book useful.) In character it is a *Let's Go* — directed at the novice traveler, giving the nuts and bolts, easily used. The book has three main sections ("First Steps," "More Advanced Topics," "Professional Questions") along with a section of brief case studies and appendices which give things such as a sample syllabus.

In a travel book, there is a checklist of things to do before you go. On page 5 of this book one finds "Before You Teach: A Checklist". Just as in "Let's Go," one starts with the basics. The first item: "Do you have keys for your office or classroom?". Later: "Can you get a desk copy of the textbook for your course?"; "Can you get old syllabi for your class? How about last year's exam?"; "What is a typical workload for a new T.A.?". At this point the traveler, equipped with a brand new passport and a still shiny backpack, is ready to board the plane.

What do you do when you land? Read "Day One", on page 7. This gives basic suggestions on what to do on the first day (hint: don't dismiss them after calling roll!). There is a model of a discussion one might have with a beginning Calculus class, and another of a more active discussion with a pre-Calculus class. Here, as elsewhere in this book, the guidelines are precise: spend about 20 minutes on the logistics, then turn to content. At this point, we've landed and gotten some local currency. Time to load the camera.

Now follow a number of sections about the basic sights. "What Goes On in Recitation." "What Should be in a Syllabus." "Lesson Planning: Survivalist Tactics." "Grading Issues." "The Semester in Five Minutes." These sections describe basic aspects of these topics. The discussions are concise, intelligent, and prescriptive. Thus, for your syllabus, "Name the required and recommended texts and readings, including edition numbers, specifying which texts are required and which are recommended." Also, "As far as grading is concerned, offer a general statement like.... In this way, you offer students a framework, while at the same time allowing yourself some leeway." Like most of the book, the sections here are brief, only a few pages, and filled with cheer and good sense. We've been to a few of the main sites. What next?

A travel book might suggest some day trips outside the city. In this book, next follow sections on "Cooperative Learning", "Technology", "Writing Assignments", "The Active Classroom", "Motivating Students", and "Course Evaluations". The sections are still filled with clear prescriptive advice and thoughtful commentary. For example, the author offers 5 Rules for using technology in the classroom. "Rule Two: Make sure your examples justify the technology you are using." "Rule Five: Be prepared for total system meltdown." These sections raise more advanced topics than the previous ones; these topics are developed further later in the book. Just as one might balance day trips with further explorations of the city itself, the sections just mentioned are interspersed with further sections on basic activities which are part of TAing: "Making Up Exams and Quizzes", "What Was That Question Again?" (answering student questions), and "How to Solve It". In this last section the author outlines Polya's model of solving problems, and offers a few brief suggestions for implementing it and otherwise presenting problems in class. At this point, the new T.A. reading this book is probably getting the feel of this travel (I mean TAing) thing, thanks to all the great maps (and wise advice).

What about those jerks at the Youth Hostel who kept everyone awake playing music until 3 a.m.? No guide book can stop them--but it can do its best to stop you from turning in to one of them. The sections "Get Along With Colleagues", "How to Get Fired", "What is a Professional?" address aspects of proper professional behavior. It's impossible to overstate this reviewer's appreciation for these sections. Graduate students with a sense of professional responsibility become faculty with a sense of professional responsibility, the kinds of colleagues we all appreciate.

At this point we've traveled around. Things aren't so new and scary, and we can also see that there are a lot more options than we imagined. Perhaps it's time to visit some beaches and mountains. The second section of this book is "More Advanced Topics." They're still short. The book discusses "Teaching Methodologies for Various Types of Classrooms", "Problems of and with Students" (what do you say to someone who can't take the exam because "my goldfish died"?), "Student Types", and "Advice to International TAs". This last has a sensible and very reasonable discussion of cultural aspects of teaching, along with some suggestions about how to cope. (At 3 1/2 pages, it is one of the longer sections.) Next follows a review of the Checklist, and a discussion of disagreements with the course instructor. The author offers some sample situations and advice about how to deal with them, based on common sense and a sense of professionalism. Then it's time to site the nearby peaks. Here we find the three most advanced sections of the book: "Using Cognitive Models to Make Appropriate Problems" (co-authored with Mary Ann Malinchak Rishel), "The Perry Model", and "Finding Voice Through Writing in Mathematics". These sections offer information and ideas which may not be at the fingertips of even experienced mathematics faculty. These trips need not be for everybody, but it is nice to have them in the book.

Well, we've had a good trip. Just a few more days and the Eurailpass will run out. What now? The third section of this book concerns Professional Questions. "Letters of Recommendation" (how to write them for your own students, that is!). "Jobs" (a brief page and a half, but mentioning creating teaching and research portfolios, and attending AMS and MAA meetings). "Mathematical Talks" (both job-talks and 10 to 20 minute meeting talks). "University Governance". TA and Faculty Evaluation. "Becoming a Faculty Member" (manage your time!). And a summary section — "The Essence of Good Teaching". The author, who has clearly put years into thinking about this topic, offers some appropriate concluding remarks. "Good teaching...is teachable." And "great teaching comes in all forms, but...mainly it comes from the delicate interaction between two personalities: that of the instructor who somehow conveys a love of learning and the student who comes ready to absorb and apply what the instructor has to give". May all of us be such instructors... and have such students!

This ends the book's prose discussion of teaching. However, there are several other sections. Next is one giving about 15 very brief Case Studies (the longest is 11 lines). Case studies are conundrums for the reader to analyze. To give a flavor of this section, here is Case Study VII (the shortest): "You are conducting a review session for tomorrow night's exam. One of the students asks you to solve a problem that you know will be on the test. What do you do?" No answers are provided; rather these case studies are meant to provoke a discussion in the "classroom or corridor". The reviewer is the lead author of a volume of longer case studies meant for TAs which complement the ones given here, and will offer a few comparative remarks later in this review. Following this section is the book's bibliography, listing 30 references. Appendices (unfortunately not listed individually in the table of contents) give a questionnaire, a sample student assignment, some sample syllabi, and the syllabus for the author's course in College Teaching for Graduate Students.

There are travel books for many audiences, from novice travelers to those with more experience. Similarly, this book aims at a specific audience: individuals with a minimal amount of teaching experience, who are looking for some basic guidelines to help them serve as TAs. It is frequently prescriptive. For example, in "The Active Classroom", we read "Show up early, maybe by five minutes. Say hello. Cheerily. Start handing out old homework or new handouts. Hand them out by calling names; this will help you remember students' names. Ask a general question, like "How's it going?"... If the response is '...', say '...'. If it's '...', then tell them '...'." Like this sample, the writing is informal; one feels that the author is having a friendly conversation with the reader, as he must have done with many a new TA during his years as TA-development specialist with the Cornell math department. The sections are short, most under 2 pages. For some TAs (and the faculty working with them) this will be perfect, while others may prefer a different type of guide, a less prescriptive approach to teaching or a narrative which is less informal or is more comprehensive.

There are two other books which complement this one. The first is Steven Krantz's How to Teach Mathematics, a personal perspective (AMS, 1993). Prof. Krantz's book (the reviewer apologizes — he has two copies of the first edition so hasn't bought the second) is aimed at the beginning faculty member, someone who will organize and be responsible for his or her own course. The prose is more formal, the list of topics necessarily more advanced, including for example sections on choosing a textbook and on designing one's own course. There is also a somewhat richer discussion of aspects of good lecturing. Though there is overlap, my guess is that a new TA would be more likely to read and digest the book under review, while someone with TAing experience who is about to become a junior faculty member might appreciate the focus of Prof. Krantz's. Let me emphasize that both texts are potentially useful, and that they share common ground.

It is not necessarily easy for a new teacher to take all the sage advice, reformulate it to fit his or her own approach to teaching, and implement it in his or her own classroom. To address this difficulty, there is Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom (CBMS Issues in Mathematics Education Volume 10, American Mathematical Society in cooperation with the Mathematical Association of America, 2001), by the reviewer et al. This book, also directed at TAs (and new faculty), is rather different from the two books above; in particular, it omits the prescriptive advice. Here is some context. Case studies have been widely used in university-level teacher development in other disciplines, in work pioneered by Roland Christensen of the Harvard Business School (see Barnes, Christensen and Hansen, Teaching and the Case Method, third edition, Harvard Business School Press, 1994). Christensen's cases are complicated, difficult, teaching situations, situations which may involve multiple issues (including some which may not be apparent) and individual personalities. The goal of considering them is the development of judgement — a goal which is second nature at any good business school. The reviewer's book offers similar case studies for mathematics graduate students. In our context, judgement means equipping TAs and faculty, for example, to successfully explain to undergraduates not just what the difference quotient is, but why it is important, and not just 'the hard way of taking the derivative'. The judgement comes in figuring out how to do this so that the students buy in. The case studies of the volume under review are shorter than those of Friedberg et al; the ones of *Teaching First* typically describe a single problem, very briefly. This brevity and focus make them good places to start the process of talking about teaching. The reviewer's book would then be a natural follow-up. It would also naturally complement Prof. Krantz's book.

In conclusion, let me reminisce a little more about travel. As I got more experience, I found some of the information in *Let's Go* useful, but much more of it superfluous. By then I had developed my own sense of what I liked about travel, what I liked to do (talk to the locals) and not do (go to discos). *Let's Go* no longer served my needs. But earlier on, it certainly had; indeed it had launched me on an enjoyable pastime, mentioned places where I met friends with whom I still keep in touch, and guided me to some remarkable adventures. My affection for *Let's Go* is everlasting, and my appreciation for the ways it enhanced my travels is real.

Solomon Friedberg (

friedber@bc.edu) is Professor of Mathematics at Boston College. His research concerns number theory and representation theory, especially the area of automorphic representations and L-functions. He is also the founder and director the Boston College Mathematics Case Studies Project, devoted to promoting the development of the teaching skills of university level mathematics instructors.