As most of you know, there has been a national movement to "reform" the teaching and content of calculus. Michigan's Calculus is being used as a model for colleges and universities throughout the country. The syllabus and text provide a fresh and innovative approach to teaching and learning calculus. The main goals of the course are to help students learn how to think about mathematics and to broaden their experiences with the symbolic, numeric, visual, and vebal components of mathematics.

The syllabus emphasizes problem solving, geometric visualization, and real world applications. It concentrates on the three basic concepts of calculus: the derivative, the integral, and the fundamental theorem which connects them. It de-emphasizes formal arguments, exceptions, and symbolic manipulation skills.

The teaching style incorporates cooperative learning in the classroom as well as outside in homework teams. The emphasis will be on learning rather that teaching. The University is encouraging us in this endeavor by refurbishing some of our classrooms to make them more conducive to cooperative learning and by adding fudning to make smaller classes possible.

You may find that the text, the syllabus and the teaching style are quite different from those you are familiar with. This will be a challenge for you, and for the students. We will do everything we can to assist you in this new way of teaching and learning calculus.

The purpose of this guide and our professional development program is to help accustom you to the new program. Throughout the term all the instructors teaching this course will have a weekly staff meeting to share ideas on what is working and what isn't working. Since this course is still fairly new, it evolves with each year's infusion of new teachers; we will be counting on you to help us make it successful.

*Learning versus teaching.*

When we think about teaching, many of us imagine ourselves at the front of the classroom with all eyes on us. We think in terms of syllabus, coverage of material, and lecture notes. Teaching Michigan Calculus is different and we'll talk about what you, the instructor, should be doing later; now we want you to imagine that you are sitting at a table with a small group of students working on a problem or watching over the student's shoulder as he or she experiments with a new concept or struggles to apply old ideas in a new setting.
*Learning versus training*

Learning is not the same as training. Training emphasizes rote learning, speed, efficiency. The Army is expert in training: "Take the rifle apart and put it back together--blindfolded--in less than thirty seconds." Training has a place in this course. We want students to know that the derivative of the sine function is the cosine function; this aspect of the course is addressed in the homework and the gateway tests. In this section we consider how students
*learn*.
*
How students learn.*

Students learn by *thinking* and *doing*, not by watching and listening. Learning is an active process; it is something the students must
*do*, not have done to them. The Calculus course is structured around student activities--in the classroom and outside the classroom in homework teams. They are encouraged to experiment and conjecture, to describe and discuss.
Students learn by working on *real-world problems* in which they have an interest--or at least in which they can see that others might have an interest. They are motivated by the visible or tangible, and they use this to "anchor" the more abstract concepts. Most of our students will not become mathematics majors; few of them share our interest in the subject for its own sake. Even those who will become mathematicians benefit by seeing how the ideas of calculus can be applied. Students learning calculus in a real-world context will gain a deeper understanding because the concepts are often presented numerically and graphically as well as algebraically.

Students learn by *working together*. They are encouraged when they see their classmates struggling as they are struggling; they are rewarded when they have a good suggestion or a sudden insight. Problems seem less daunting when there is someone else with whom to work. Even dealing with the graphing calculator is more rewarding and fun with a partner.

Students learn by *talking* about what they are doing--by explaining what they have discovered, by discussing a common strategy to attack a problem, by asking questions. Students have little or no experience talking about mathematics; this takes time and practice. The results can be rewarding. The first year of the project we knew it was working when an entire homework group would show up together to the Math Lab and then proceed to hash the problem out themselves without any help from the tutor. They were learning much more than if someone had explained how to work the problem.

Students learn by *writing*. Writing forces students to organize their ideas and experience. Often, real learning begins only when the students begin to write the meaning of a particular problem or function. The more students are required to write and the more they see other students' writing the better they get in expressing their ideas and understanding of mathematics.

Students learn by *reading*--when they are *actively engaged* in the reading. Early in the course you will need to discuss with the students how to go about reading a chapter and how to best learn from the reading. This will help the students get the idea of what they are supposed to be doing when they read. This can head off the massive frustration that is likely to result from just turning them loose with what may be the first math book they have ever had to read. Don't try to
*cover* everything in the book in class. This just discourages students from reading the book on their own and does a disservice to the students. Early on we learned that if the instructor tried to lecture in detail over everything in class, the students objected to the book--they found it confusing, ambiguous and hard to read. In those sections where the instructors didn't "overcover" the material and trusted the students to read the text (always giving them hints on what to look for and any problems they might run into) the students liked the book. They found it readable and helpful.

Almost all of these activities--cooperative learning, talking about mathematics, writing about mathematics, reading mathematics--are foreign to our students. They are used to learning how to solve template problems by appropriate symbol manipulation, and they have been encouraged to consider the task "done" when they circle the "answer." You will have to encourage them in and repeatedly justify these new and difficult activities. Just letting them know that "It is hard, it is supposed to be hard, and you have what it takes to do it!" can go a long way in supporting your students.

As a reward you will see students blossom, including students who never before liked or did particularly well in mathematics. You will see creative students succeed who never developed the discipline to master algebra and trigonometry. And, as a consequence of this success, you will see them begin to work on those skills. You will see good students take off on their own, exploring ideas and connections you never imagined. You will see your students as individuals with different strengths and weaknesses, not just as points on a normal distribution.

Instructors in previous years have found that teams of three or four worked best, because groups consisting of only two students working together failed to generate enough ideas. Some instructors felt that four was the ideal number to assign just in case one student missed a particular session.

We generally change the homework teams two or three times during the semester, because we want each student to get used to working with a range of other students. Also, mixing groups helps in dealing with problem students or with groups where one or two students tend to dominate. Some instructors have kept the same teams throughout the semester, feeling that the advantages of learning to work together and establishing times for joint work outweight the disadvantages. Last year in a couple of sections where groups had not been changes by midsemester, students asked to have them changed.

It is the instructor's responsibility to assign teams. A reasonable strategy for assigning the first teams, when you know little about the students, is to group students who live near each other -- preferably in the same dorm. This minimizes logistical problems concerning where the group will meet.

Later in the term when you change teams, you may want to use a mixture of criteria for groupings. For example, there is evidence that having a mixture of levels in a group helps it function better. In this case, if everyone is working, both strong students and weak students grow stronger. It is also useful to have a mix of gender and race. There is evidence that when students have a good experience working together it leads them to respect the opinions of others. (For more details on homework teams, turn to Appendix A.)

Encourage your student to go to Math Lab if they have quesitons about how to use the graphing calculator or need help with the material. If a team is having difficulty with a homework problem, please suggest that they go to the Math Lab together.

When students have questions about the substance of the homework, we want the students to figure out as much as possible for themselves. Sometimes they need a hint, sometimes they need to reread the instructions or the relevant section of the book, sometimes they need to be given direct help, and sometimes they need to be told to go back and work on it. When a whole team comes in to get help, it is a good time to help them learn to work together and it may only take a few hints or steering-type questions to get the group unstuck. For the tutor (or the instructor) knowing how much and how little to say is an art that takes experience and restraint.

Activities in the classroom generally fall into the following categories: lectures, class discussions, group activities, etc. (Sample lesson plans for the beginning of the term are available.) Often several modes are mixed into one class period. The proportions of the mix will depend on the ideas to be investigated, the size and maturity of the class, and the personality and style of the instructor.

*Lectures*

We are convinced that giving frequent long lectures are, in general, one of the least effective uses of classroom time. There will be many times when you sould lecture, but there is no need to systematically go through all the material step by step. The book is very readable and students can learn the content by reading the book. In some sense, the better the lecture is, the worse it is for the students, that is, the less they have to work. Our rule of thumb has been to lecture no more than fifteen minutes as a time. The guiding principle is to tell the students something they want to know at the exact time they want to know it.

Short "bursts" of lecture are useful in the midst of a cooperative activity. If a large fraction of the students have run into the same difficulty, it may be time to intervene. You can spot this kind of moment by listening carefully to the conversations going on within groups and to the kinds of questions directed at you

Lecturing also may be useful to clarify a particular concept when you know that the students will have a hard time understanding the concept without additional assistance.

Another good use of a short lecture is to give guidelines to the students on what to expect in the next readings and any pointers on how to learn the material. This type of lecture should not last more than 10-15 minutes.

*Cooperative Learning Activities*

Using cooperative learning in the classroom can be the most rewarding and, for some instructors, the most difficult to do. In Appendix D you will find tips and advise for using cooperative learning gleaned from the experiences of previous Instructors. In Appendix E you will find a check list that you can use as you develop cooperative learning activities.

*Quizzes*

Students need quizzes to evaluate their individual performance and to practice their test-taking skills. Many instructors include a problem from the individual homework to encourage their students to do all the homework. You should not let quizzes take up too much valuable class time. Try to have them no longer than 20 minutes at the most.

*Motivational Talks*

Students (and instructors!) need frequent encouragement in this course. There is a substantial payoff if you take a few moments to talk to your class about the progress they have made, for example, "Your writing has really come a long way", or "Look how well you've learned to handle these long word problems."