When I began as a teaching assistant in the Department of Mathematics in 1991, the prevailing wisdom was that one learned to teach by teaching. While the personal consequences of the failure to adapt to the classroom quickly are not quite as tragic as drowning, the sink-or-swim mentality can have dire consequences for both TAs and the undergraduate student body. The Department of Mathematics, following a nationwide trend, has begun to focus on improving teaching, with particular emphasis on the preparation of TAs. As part of this effort, faculty in the Department of Mathematics won a grant from the University Teaching Initiative. In accordance with the terms of the grant, three TAs of the Department of Mathematics (Jim Bowling, Julie Theoret, and myself) produced a teaching handbook this past summer.

Nearly all graduate students in Mathematics teach for their entire term of enrollment at U.Va. These TAs serve both as leaders of discussion sections ("fourth hours") and as principal instructors, often performing these functions for different courses at the same time. To address the multiple roles they play, we included sections on:

•  examining the roles of a fourth-hour instructor and of a course instructor,
•  interacting with faculty as a TA for a course led by a faculty member,
•  interacting with a faculty member who is principal instructor for a coordinated course (i.e., a course directed by a faculty member who sets policies and teaches a section), and directing other TAs or undergraduates who may run fourth hours or grade homework for a class taught by another TA.

Other department-specific information includes a description of the duties of the secretarial staff ("who to see with what when") and special faculty roles, such as that of the teaching advisor and the department ombudsman. A section entitled "First Things First" directs new TAs where to find their teaching schedules and course materials, and how to get started in general.

In an attempt to make the handbook fairly self-contained, we included material regarding everyday teaching. This material is separated into two sections, "Administrative" and "In the Classroom." Since lecturing at the blackboard is still the predominant method of instruction in mathematics classes, we provided a list of hints for successful lecturing and discussed the techniques of problem solving.

The remaining sections are more general and may benefit seasoned TAs or new faculty. These include the use and availability of technology, outside agencies, resources, professionalism, and a discussion entitled, "When You Don't Want to Teach," which argues for the usefulness of developing teaching skills, along with suggestions for those times when the last wish on a TA's list is to set foot in the classroom.

Although time-consuming, preparing the handbook was a terrific experience, a chance to act upon our grousing as TAs regarding all we wished we had known when we began our graduate careers.  Future TAs will periodically revise and edit the handbook.  We hope our version serves as a great beginning and encourages TAs to join in the dialogue regarding the improvement of teacher preparation going on both in our department and in mathematics departments nationwide.