The first thing you need to read is actually the introduction to the workbook your students will be using, but we have made that easy by reproducing it below. Incidently, the reading of it should be non-optional for your students. We suggest that you make it part of your first assignment. To give more weight to the assignment, ask the students to turn in the answer to some question along the lines of "What sounds best to you about this format? What sounds worst?"
Here, in any case, is the Workbook Introduction:
An educational theory which is currently widely accepted is that knowledge cannot be directly transmitted by one person to another. What you are told has nothing to stick to unless you yourself have constructed a framework to which to attach it. The framework may need reconstruction from time to time, but it must be there. That theory is at the base of this course.
Also at the base is a belief about mathematics itself: the beauty and the usefulness and the longevity of the field all have the same source. That source is the fact that mathematics makes sense.
Those two beliefs combine to define this course. An extremely careful structuring of the material provides the wherewithal for you to build yourself a solid framework. An equally careful choice of use of class time provides you with the opportunity to build and test this framework as you go along. In fact, it requires you to do so. This workbook then supplies the chance to solidify the framework with further applications of the ideas and concepts you have worked out in class.
What does this mean in concrete terms? For a start, it means that in class you will need to be an active participant. It is your responsibility to ask questions when you are feeling bewildered, and to help your classmates when they are feeling bewildered. It is your responsibility to try to answer questions the instructor asks you even when you feel unsure of your ground. It is also your responsibility to keep an eye on what goes onto the blackboard, and to speak up if you think there may be an error, whether produced by a fellow student or by the instructor.
Note that the last two items carry with them two contracts about the behavior of your instructor. One is that you will never be ridiculed or lose credit or come to other grief as a result of giving an incorrect answer or of asking what you fear may be a foolish question. The other is that your instructor will leave to you, as far as possible, the correcting of mistakes on the board. This is not laziness on the part of the instructor (in fact it is a remarkably hard thing to do!) It is an absolute necessity if you are to be left free to do your own constructing.
Another concrete consequence is the form of this book. Since it is intended strictly as a support tool for the learning that goes on in the classroom, it does not contain explanatory passages attached to each set of problems. On the other hand, at the end of each chapter you will find a fairly exhaustive summary. These are designed as an aid to tying the material together after you have worked all of the way through it, but could upon occasion be of use in the course of figuring out the material as you go.
That covers the basic structure of the course, and the philosophy behind it. It might be worthwhile, however, to discuss briefly two of the situations that our experience over the two decades we have been teaching this particular course at the University of Washington tells us that you will encounter. One is a feeling almost of betrayal when you discover that the instructor leaves mistakes on the board for you to correct, or refuses simply to hand you the solution to a problem. The reason for this behavior should by now be clear to you, but you may have to remind yourself of it from time to time.
The other is a question of how to deal with the chunks of prior knowledge and procedures which will inevitably turn up in your mind as you go through the course. Are you supposed to run the old technique in and use it on whatever problems you are now working on? Not instantly, because using freely an undigested lump of mathematics in the form of a technique is roughly akin to juggling with a time bomb. It may go fine for a while, but eventually it is going to blow up in your face! Does this mean, then, that you must permanently jettison a beloved old technique and replace it with one which is unfamiliar and sometimes even rather cumbersome? Definitely not--in the long run. What you need to do is ask yourself the question: "How can I explain the fact that this technique leads consistently to the correct solution to this type of problem?" When you can answer that, you are perfectly free to go back to using it. Until you can do so, however, you need to exercise extreme restraint and stick with the techniques introduced by your instructor. They may be clumsy at times, but they work, and they hang directly on that framework we were talking about before. They may even be part of it!
One final admonition:
The crucial thing to remember where attitude is concerned is that the major objective is not the conveying of information. Clearly, there is a certain amout of material‹in fact, a fairly large amount of material--of which the students must be in control by the end of the course. On the other hand, before a student can possibly handle a body of information, she must have faith in herself, a faith that she can, indeed, handle mathematical concepts competently and correctly. To most students in the type of course for which this text is designed, such an idea is a novel one. The instructor must therefore be prepared to spend a great deal of time replying to "I can't", with "Sure you can"--or, better yet, forestalling the "I can't" by giving the class as a whole and each student individually the experience of successfully producing solutions and arguments and proofs. The instructor must foster a class atmosphere in which students help and encourage each other (this is not difficult to achieve, and is a tremendous source of satisfaction for all concerned). One of the trickiest, and most important, elements in producing the desired atmosphere is that the instructor must play down her inevitable role of god. There is no getting around the fact that ultimately she is the authority, but heavy use of this position of authority will annihilate the class' self-reliance. She must control her instinctive desire to nip false arguments in the bud and to give students' questions a prompt, slick, and generally incomprehensible answer. Sometimes it is necessary to step in to prevent a class from bogging down, or careening off in a profitless direction, but to as great an extent as possible, students should monitor each other's arguments, and answer each other's questions.
Before proceeding to a discussion of techniques, let us insert a comment on the technique of learning techniques. There are an enormous number of techniques involved in discovery teaching--sufficiently many so that no one person could possibly make use of all of them. The question of who will profit from which ones depends on the individual personality of the instructor and the group personality of the class. It is therefore essential that besides reading this introduction prior to the beginning of the course (which is necessary because some of the techniques are basic and universal), the instructor should re-read it several times in the course of the quarter, as he becomes more familiar with the class, and more conversant with his own fortes and foibles as a discovery instructor.
The category of techniques is a good deal more massive and more diffuse than that of attitude. We shall therefore attempt to divide it roughly according to the goal each technique is attempting to achieve.
Regular attendance: This may seem rather an elementary consideration, but it is absolutely crucial. Obviously, it is a side benefit of any effective technique for livening up class and keeping it interesting. There are, however, a few slightly cruder methods which are highly effective. One is to call the roll, especially at the beginning of the quarter (this is also helpful in the highly necessary enterprise of learning everybody's name) and let it be known that attendance may influence grades. Another is to pick the day of the week most subject to heavy cutting (generally Friday!) and schedule a weekly 20-minute quiz. This again has many benefits, notably helping the students to keep their knowledge of the subject matter up to date, and the instructor to keep his knowledge of the state of knowledge of the students up to date.
Participation:: Having achieved the physical attendance of the students,
the instructor is next faced with the problem of bringing about their mental
attendance. This means persuading them to abandon their traditional role of
transmitters of information from blackboard to notebook and become active
contributors to the class. Again, the structure of the manual is such as to encourage
this development, but some further tactics and suggestions might be useful. In the
first place, although calling for volunteers to give answers will get a portion of the
class into action, there are bound to be at least a few who, either from laziness or
timidity, will attempt to creep quietly into the woodwork. This is demoralizing for the
class as a whole and disastrous for the student in particular. The method of preventing
it is simple: a minimum of two or three times per day, call on each student
specifically to answer some question. If the student is indeed timid, or is a slow
learner, make most of his questions easy ones‹but do not miss him.
One situation which ofter brings a marginal participator into action is an opportunity
to correct an error. This is particularly effective if the error is made by the instructor.
For most of us, these occur sponteously, but an instructor who is inordinately
infallible should make an effort to produce an occasional convincing mistake. .
An extremely basic tactic for keeping students tuned in is the one referred to in this
manual as a "paper problem." When a concept has been introduced and a couple of
examples have been worked out at the board, the instructor produces a highly similar
one, with standard instructions: "Work this one out on your paper. When you finish,
see if you agree with your neighbor. If you get stuck, get a neighbor to help." The
examples on which the paper problem is modeled should remain on the board. Timing
is determined by pencil motion and noise level‹when enough comparisons with
neighbors have happened, take them swiftly through the solution
together. This should happen many times per class (how many depends on how bulky
the problems are). Also useful on occasion is "Here's a slightly tougher one‹get to
work with at least one neighbor to solve it!"
Further devices for encouraging participation:
One other factor that encourages participation, good spirits, a lively class and just about every other desirable feature is the development of a spirit of community within the class. There are many ways to foster this. The most basic one is the simplest one--when a student asks a question, get another student to answer it. At first the second student will address you with the answer--in which case you say at the end of the response, "Joe, did that answer your question?." As the quarter wears on, they will begin to address each other and feel responsible for each other's comprehension. .
Another very helpful trick is to pass around a list as soon as class attendance settles down on which everybody interested in doing so writes his or her name and phone number. You then make and pass around copies, so that anyone who bogs down on a particular homework problem can phone a classmate and get help, or at least moral support. .
Now that you've read through the course philosophy as explained in the Workbook introduction for the students, and have seen some of the specifics on how we apply this philosophy to promote student learning, let me point out a parallel with your own learning. We make a constant effort to present students with a certain amount of information, a supportive structure and atmosphere, and enough freedom so that they can construct their own knowledge of the mathematical content and concepts. Then we encourage them to think and work not just on their own but with their fellow-learners in so constructing. We have now presented you with a certain amount of information, and the text of the manual provides a supportive structure and as much atmosphere as the printed word permits. The remaining ingredient we have no technique whatever for supplying, but it is enormously important: in whatever way, shape or form you can do so, try to work with others who are teaching this course. The absolute best is to observe each other's classes and discuss them afterwards--an activity which benefits the observer and the observee in equal measure. Failing that, just getting together and discussing the ups and downs of the week can smooth many a situation . And finally, we are hoping to set up an electronic discussion group. For information on that, send e-mail to
warfield@math.washington.edu.Our last admonition to you is not merely parallel to that to the students, it is identical: