Since the Tulane conference [42], substantial amounts of time and resources have been put towards developing materials for non-lecturing, student-centered classroom techniques. Some techniques that are used by many instructors are cooperative learning activities, student presentations, laboratory activities, discovery learning, and the Moore method. In our classrooms we extensively use the technique of having students work in cooperative groups on mathematical problems. In this chapter we focus on developing the instructors' facility with implementing and managing in-class cooperative activities. If your program includes student- centered teaching methodologies not addressed by one of the meetings in this book, Chapter 16 provides support for developing new training meetings.

Cooperative learning is a popular pedagogical device in elementary collegiate mathematics classrooms. There are many reports that suggest that when cooperative learning is implemented effectively in a mathematics class, there can be dramatic gains in what students are able to accomplish in terms of solving non-routine problems, in students' appreciation of and enjoyment in mathematics, and in students' willing- ness to persist in their attempts to solve and develop meaningful understanding of mathematical concepts that go beyond simple imitation of “worked examples” from the textbook. Although some investigators have reported some gains in terms of students' scores on standardized tests, the reports are not usually of dramatic gains [35]. A typical report is of the form “The students in the cooperative learning group performed at least as well as students in a control group.”

“The Torch or the Firehose?” is the Math and Physics recitation guide published by MIT and reproduced in [23], on page 185. It notes that,

In our experience, and as reported in a survey of graduate student instructors' experiences, [121], most instructors who teach a significant number of students will eventually find themselves in an “idiosyncratic” situation.

We distinguish this meeting from the meeting titled, “Establishing and Maintaining Control in Your Classroom” with the observation that in this meeting we are interested in discussing encounters, or more likely a series of encounters, between instructors and particular students. The earlier meeting can be understood as a meeting about gaining and maintaining students' cooperation with the instructor's plans for the class and how those plans are to be executed. In some sense, that meeting was about overcoming difficulties that may be thought of as in some way inherent to the structure and opportunity that an interactive classroom offers to students. That meeting was about addressing difficulties with an entire class of students. The present meeting focuses on addressing difficulties with particular students. Naturally, there may be relationships between the two phenomena here. For example, a particularly difficult student may cause classroom-wide problems when given access to the opportunities for self-directed behavior that the interactive classroom offers.

Observing textbooks on introductory collegiate mathematics, and instructors who use these textbooks, one notices that there is little consensus on the role or function of a mathematics textbook. Some textbooks attempt to provide a narrative that tries to access concepts through intuition, application, and explanation. Others attempt to provide an exhaustive resource and reference. Yet others function primarily as a repository of worked examples. In addition, there are the ubiquitous tensions between rigor and accessibility, and between conceptual understanding and skill acquisition.

In our experience, many students use mathematics textbooks as little more than large collections of homework problems, templates for the “types” of problem encountered in the homework, and lists of answers (usually to the odd-numbered exercises). In some of the classes that we have taught, students clearly believed that their textbooks were dry, boring, expensive, and not at all to be read!

We both base our philosophies of the role of the textbook in a mathematics course on the assumption that such an expensive investment ought to, in fact, be read. Much more than economic considerations drive our intent to have our students read their textbooks—we feel that the textbook can and should be a vital part of the course that contributes to the learning goals that we have set for our students. Many of these specific goals are mentioned in this meeting.

Introduction

In developing this book, we have tried to identify a collection of experiences for training college mathematics instructors that is comprehensive and well integrated. This is appropriate, as teaching is a highly complex activity, especially in the setting of a student-centered interactive classroom, and it is somewhat artificial to treat teaching issues as isolated from each other.

The range of experiences that we have described in this book is consistent with our experience of the needs of graduate students and new faculty instructors at the University of Michigan. Naturally, other programs with different goals and methods may find that there are teaching issues that we have not substantially addressed, but which are nonetheless highly relevant. In this chapter, we hope to provide a framework for adapting materials to the specific needs of other programs.

A Process for Designing Your Own Meetings

We believe that instructor training meetings should be as lively as possible, and should actively involve the instructors as much as is possible. In a sense, the training meetings can, and perhaps should, serve as a model for how the instructors' classrooms should function. With this in mind, the “Lesson Planning Worksheet” included in Chapter 12 serves as a useful framework for planning a training meeting.

The major steps described in Chapter 12 are recounted below.

1. Read the section to be covered.

2. Identify the goals for the session.

3. Decide on a strategy to address each goal.

4. Begin time management.

5. Plan the student activities.

6. Write scripts for the mini-lectures.

7. Make an assessment plan.

8. Put it all together.

The chapters in this book provide detailed information for orchestrating training meetings. Trainers are frequently faced with situations that the meeting descriptions do not anticipate. This appendix is included as a resource for dealing with the inevitable surprises and problems that arise when running effective training meetings. This is a helpful section to read before running any training meetings at all. It is also intended to serve as a reference for dealing with recurrent problems. This appendix is intended to be helpful, but not encyclopedic.

To Meet or Not To Meet

When asked how to improve meetings, many people will supply one or more of the following suggestions.

• Make the meetings shorter.

• Have fewer of them.

• Make the meetings more relevant to me.

• Get rid of them altogether.

The unfortunate reality of the situation is that many people have no love for meetings, and many consider meetings to be a complete waste of time. Part of this stems from the fact that many meeting participants have difficulty organizing their time and priorities, but some also stems from the fact that meetings can be long, boring, and unnecessary. Before planning or announcing a meeting, trainers must ask themselves the important question,

“Do we really need to have a meeting at all?”

If the honest answer to this question is “No,” in other words, if there is an equally effective and reliable way to convey information to instructors or help them make sense of their experiences, then trainers should seriously consider not holding a meeting at all.

Introduction

The purpose of this chapter is to describe a session suitable for introducing new instructors to your mathematics program. As mentioned in Chapter 1 of this book, we envision a training program that includes intensive pre-semester training as an integral and important component. Throughout this book we have made reference to the pre-semester training materials developed by Shure, Black and Shaw [103] at the University of Michigan. Our experience suggests that such a pre-semester training program provides new instructors with a set of foundational experiences that the meetings described in Chapters 4–15 build upon.

Unfortunately, circumstances such as budget, availability of training personnel, and time sometimes make an intensive pre-semester training program a luxury that departments feel that they cannot afford. Recognizing this reality, we include this chapter that describes a very basic form of pre-semester training. The orientation sessions that we have described here raise many of the issues that will be expanded upon by in-semester training sessions, such as grading student work, using cooperative learning and teaching with technology. However, the sessions that we describe in this chapter should not be seen as a “stand alone” training program. They are intended to help instructors to recognize the issues that they will likely have to cope with on the first day or during the first week of class.

As all instructors are aware, teaching opportunities are not confined to the classroom. Office encounters provide one outlet for additional, and in this case more focused and personalized, instruction. Unfortunately, some instructors see office hours as an annoying burden and a waste of time that, in their eyes, could be more fruitfully spent on research or other duties. In addition, many students are reluctant to utilize office hours, because of various beliefs and perceptions that they hold.

Most instructors readily recognize that a primary function of one-on-one teaching in office hours is to provide students with individualized instruction and help. However, few may have reflected upon the other possibilities that such encounters afford—alleviation of mathematics anxiety, motivation, nurturing, retention, feedback (for the instructor as well as the student), etc. For this reason, we begin the meeting with a discussion of the purposes of holding office hours.

Students, on the other hand, often see office hours as a means to get “unstuck,” especially in a mathematics class. Their primary motivation for coming to office hours will then be to enhance their performance on a current homework assignment, or an upcoming quiz or exam. Unfortunately, there are many motivational obstructions that hinder students from coming to office hours for help. Thus, we have included a section of the meeting on ways to encourage students to come to office hours.

Arguably the most important component of one-on-one teaching in a mathematics class is teaching problem solving.

This book is intended to be a handbook for implementing three of the components of an integrated professional development program described in the previous chapter. Chapter 3 details a short pre-semester training course, although those departments that are able to commit to a longer pre-semester training pro- gram may use the model provided by Shure, Black, and Shaw [103]. Chapters 4 through 16 are devoted to the weekly in-semester training meetings. Chapter 17 is devoted to class visits and feedback. This chapter explains how to use the rest of the handbook to implement these three components. More detailed advice on running effective meetings is provided in Appendix A. Detailed discussions of implementing an educational issues seminar and using portfolios to integrate a professional development program are not included in this book.

The Orientation Session

Chapter 3 describes a session suitable for introducing new instructors to their mathematics department, and for getting all instructors ready to teach the first week of class. Much of the format of that chapter is similar to the format of Chapters 4 through 15, which detail the twelve in-semester training meetings. Therefore, most of the instructions for using Chapter 3 can be gleaned from those given below.

The Meeting Chapters

This book contains chapters detailing twelve training meetings. Each chapter is divided into the following sections.

Description and Purpose of the Meeting: This is a brief description of the purpose of the meeting, and what is likely to take place if the meeting is run as recommended.

As we mentioned in Chapter 4, cooperative learning is gaining popularity as a pedagogical device in introductory collegiate mathematics courses. Instructors are encouraging students to learn in teams in or out of the classroom, and many instructors, ourselves included, have their students work cooperatively in and out of the classroom. Chapter 4 was primarily concerned with issues of the implementation of cooperative learning in the classroom, whereas in this chapter we explore issues of implementation for cooperative learning outside of the classroom.

In this chapter we discuss the management of “homework teams,” because we require our students to meet regularly outside of class to work on difficult, non-routine homework problems. Other instructors require their students to meet in teams outside of class to work on long-term projects, prepare presentations, do laboratory activities, etc. The issues discussed in this chapter arise in any setting where team efforts are required, especially outside of the watchful eye of the instructor. This meeting is easily adapted to address the kinds of activities mentioned above, so as you read consider the phrase “homework teams” to be more broadly defined.

In the introduction to Chapter 4, we pointed out that cooperative learning can have positive or negative effects, depending on how it is implemented. Much of the literature on the implementation of cooperative learning in a college mathematics class is connected with the implementation of cooperative learning in a classroom environment where the instructor is physically present.

A training program for new instructors will naturally begin with attention to day-to-day concerns. It is only after some familiarity and facility is achieved in these areas that a novice instructor has the opportunity for reflection on what is working in the classroom and why. One area for consideration at that point is the subject of student motivation.

An instructor's approach to student motivation will likely depend upon where he or she is situated on the pragmatist-idealist continuum. A total idealist would believe that the inherent beauty and power of mathematics should lead to intrinsic student motivation to learn the subject matter. A total pragmatist would maintain that the extrinsic motivation of grades alone determines a student's approach to mathematics. It would clearly be a superior situation if all students were intrinsically motivated to learn. This, unfortunately, is not the current reality of college mathematics courses, nor is it likely ever to be. In the absence of the ideal, an instructor must give careful consideration to the interplay between intrinsic and extrinsic motivators and the role that the instructor plays within each. Care must be taken, because under certain conditions extrinsic rewards can damage intrinsic motivation.

Student motivation is a complex and varied issue for which no one has all the answers. However, much can be learned by defining terms and comparing and contrasting possible approaches.

The introduction to this chapter is slightly longer than is typical. The idea of this introduction is to provide instructor trainers with a basic idea of what issues may be described under the title of “Control in the Classroom.”

Instructor trainers are usually naturally talented and effective instructors who have a clear vision for what they want to make happen for the students in their class. Likewise, instructor trainers will often have a natural talent for clear communication, and will instinctively take steps to ensure that their classroom is an ideal environment for this communication to take place. As such, instructor trainers may have had little direct, personal experience with classrooms that are out of control. This was certainly the authors' experience when visiting the classrooms of some novice instructors for the first time. We found that in order to help instructors, we needed to reflect on exactly what we understood by “control” in the interactive classroom. The introduction to this chapter presents some ideas of what “control” in a mathematics classroom can mean, and areas where difficulties can arise.

The word “control” evokes a variety of responses from mathematics instructors. Some instructors respond that control is essential—without control over the classroom, students will create so many distractions that no learning is possible. The following short account of such a classroom (related by Steven Krantz [69]) vividly illustrates this possibility.

Class Visits and Professional Development

Overview of Class Visits

An important component of a professional development program is a system of visits to new instructors' classes. These visits take a substantial amount of time and effort, but they afford many benefits that cannot be gained from other aspects of a professional development program. Through class visits, instructors can get concrete, focused, and personal feedback. Through the sharing of sensible and helpful advice, the course coordinators can gain the respect and trust of their instructors. Visits keep the coordinators connected with the classrooms: the nature of the students, the classroom layout, and developing problems course—wide or with individual instructors. Finally, visits are a valuable tool for assessing and improving the professional development program.

The bulk of the chapter is an explanation of three types of class visits. The types of visits are observational visits, student-feedback visits, and peer visits. The observational visits are conducted by a course coordinator, and are detailed in Section 17.2. The student-feedback visits are also conducted by a course coordinator, but a similar less effective method can be conducted by an instructor in his or her own class. These types of visits are detailed in Section 17.3 and Section 17.4 respectively. Peer visits allow instructors to give and receive feedback, and don't involve the course coordinators.

The mathematics course that we envision utilizes a variety of assessment techniques, including team home- work, projects, student presentations and writing assignments. A common element of these assessment practices is the clear and precise communication of mathematical ideas.

In many cases, students are expected to work cooperatively – to produce a single piece of work representing the collective efforts of three or four students. In order to provide sufficient challenge for a group of students and sufficient incentive for the students to work cooperatively (instead of simply dividing the work among themselves, and later compiling their individual contributions for submission) the problems assigned are more complicated, and are sometimes more “open-ended” than exercises typically assigned in traditional mathematics classes. The problems assigned typically require students to make appropriate assumptions, to try alternative avenues of inquiry, to try to understand the mathematics more thoroughly by recognizing its application to otherwise unfamiliar situations, etc.

The work that students produce on these more complicated assignments is not simply pages of algebraic manipulations with boxed answers at the end. Instead, students are encouraged (and helped) to exhibit their understanding of the mathematics in multiple ways (such as graphs, written accounts of their assumptions and reasoning processes), instead of simply recording the algebraic steps that they performed.

As may be expected, assessing and grading student work of this kind can be radically different from assessing and grading pages of algebraic manipulations with a conveniently highlighted answer at the end.

The widespread availability of affordable graphing calculators and computers running powerful computer algebra systems (CAS) is changing the way elementary mathematics is taught. Research in mathematics education (see the suggested readings at the end of this chapter for examples) indicates that the careful use of appropriate technology can help students to achieve learning outcomes that are rare in more traditional mathematics courses. Graphing calculators and CAS have been likened to the microscopes of a biology lab. Just as students can use microscopes to directly observe the working of organisms and other biological systems, calculators and computers can be used by students to explore mathematics. Just as personal observation of the behavior and structure of biological systems can make basic principles of physiology and anatomy more concrete for students, observation of the behavior and structure of mathematical phenomena can make abstract mathematical concepts more concrete for the students.

Nationwide, some instructors have reservations about incorporating such technology into the classroom. Critics of the use of technology in mathematics classes include that the net effect will be that students will fail to learn any mathematics, instead they will simply become adept at using the technology without developing any substantial understanding of any mathematical concepts. Another frequently heard criticism (particularly of calculators) is that the technology will become a “crutch” for students.

These are potentially valid criticisms of the use (perhaps “misuse” is more accurate) of technology in mathematics courses.

Overview

We believe that it is important for a mathematics department to be actively involved in the professional development of its faculty and graduate students. By professional development we mean the training, assessment, and improvement of teaching skills and practices. Professional development that comes from within the mathematics department, rather than through non-disciplinary channels, is more likely to be meaningful and credible to the instructors who are involved. For example, sensibilities about teaching mathematics are sometimes quite different from those in other disciplines. Compare what is meant by “read the textbook” in a mathematics class to what is meant by the same statement in a literature or history class. In addition, the types and standards of argumentation and precision are different in the mathematics department than they are elsewhere. Moreover, there are certain obstacles that are more prevalent in mathematics than elsewhere. For instance, innumeracy and mathematical illiteracy are far more socially acceptable, or even favored, than is an inability to read or write. For these and other reasons, we feel that the best place for training and supporting mathematics instructors is within the mathematics department.

We believe that an effective way for a mathematics department to train its inexperienced instructors, and to support and develop its other faculty members and graduate students is through an integrated professional development program that includes four components.

One of the “nuts and bolts” issues that all teachers must get to grips with is proctoring students during tests and exams, and taking appropriate action if they observe students cheating. Much of the advice and many of the precepts set out in this meeting will be obvious to experienced and capable instructors. In our experience, many beginning instructors are aware of the possibility of cheating and want to prevent it. However, most do not have clear ideas of exactly how to go about this. We have worked with instructors who sit at the front of the exam room, obviously engrossed in a book, and subsequently complain about the high number of “suspicious similarities” between exam papers. Occasionally, we have run into instructors who seem to be a little too zealous in their desire, and too severe in their methods, to eliminate cheating. We have found it important to try to be very clear about the limits of the proctor's authority. Finally, most universities have set procedures and official channels for dealing with cases of suspected cheating. Instructors who are new to teaching or new to a mathematics department need to be informed of these processes, and need to be made aware of the appropriate contacts within the department, should a situation arise.

Description and Purpose of the Meeting

This meeting is intended to be a short session (thirty minutes would probably be adequate).