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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.


INDEX

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Course Goals

Establish constructive student attitudes about math:

  1. interest in math
  2. value of math, and its link to the real world
  3. the likelihood of success and satisfaction
  4. the effective ways to learn math

Strengthen students' general academic skills:

  1. critical thinking
  2. writing
  3. giving clear verbal explanations
  4. working collaboratively
  5. assuming responsibility
  6. understanding and using technology

Improve students' quantitative reasoning skills:

  1. translating a word problem into a math statement, and back again
  2. forming reasonable descriptions and judgments based on quantitative information.

Develop a wide base of calculus knowledge:

  1. understanding of concepts
  2. basic skills
  3. mathematical senses (quantitative, geometric, symbolic)
  4. the thinking process (problem-solving, predicting, generalizing)

Improve instruction:

  1. more instructor commitment and interest
  2. increased student-faculty contact
  3. student-centered approach
  4. satisfaction with teaching
  5. ability to organize and maintain multiple methods of instruction

Improve student persistence rates

  1. students continue taking math and science
  2. more students become math majors

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Student Learning

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.


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Why a Team Approach to Mathematics Study?

Team homework is one of the most important features of the course. When a group progresses through deciding what to do, trying it, analyzing the results, and writing it up, the dynamics of the process greatly enhance their individual learning.

The following reasons help students appreciate the value of this approach. They are printed in each of the Student Guides.

    You can only use what you remember!!
    • 10% of what they read
    • 20% of what they hear
    • 30% of what they see
    • 50% of what they see and hear
    • 70% of what they discuss with others
    • 90% of what they teach someone else

    You can prepare for the 'real world' of work.

Here's what a principal aerodynamics engineer from The Boeing Company and members of the Washington State Software Association have to say.
    What do we look for in employees? We hire those who have demonstrated that they:
    1. Enjoy the process of learning & know how to learn independently
    2. Thrive on inteleectual challenges
    3. Are creative an flexible in how they solve problems
    4. Have a good understanding of the fundamentals (mathematics, science, economics)
    5. Can manage knowledge and information, as well as tasks and things
    6. Can operate effectively in a team environment
    7. Have good communication skills

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Organizing Homework Teams

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.)

Go to Appendix A: More About Homework Teams


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Math Lab

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.


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Gateway Tests

All instructors will periodically give gateway tests on basic skills, e.g. differentiation. The tests are already developed for your use. Students will be required to pass the gateway tests, but they retake the test as many times as necessary in order to pass. we have found that students like being able to retake the tests and it encourages them to continue studying until they know the material.

The first test will be given during class time. Each additional test will be given and monitored during your office hours or administered in the Math Lab.


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The TI-82 Calculator

Each student will be required to buy a calculator. Two workshops will be given through the Math Lab to help those who need it. Last year one of the biggest worries for some of the instructors was teaching students to use the calculator. It turned out to be a non-issue. Many students will come already knowing how to use the calculator, and they tend to help other students who need it. Encourage anyone who is having trouble to go to the Math Lab to get individual help.


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Classroom Activities

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."


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Student Attitudes and Typical Problems

This course comes as a shock to most students. This is not the way calculus is "supposed to be." Most University of Michigan students were admitted because they performed well in traditional courses; our approach may be unsettling and painful. You can sometimes head off student uneasiness by being very specific about how the course may be different from what they were accustomed to and why.

"This isn't calculus! I know calculus and this isn't it!"

This complaint usually comes from exposure to the traditional calculus high school course where they spent a lot of time with symbolic manipulation. They may be a little disgruntled because they hoped that taking calculus in high school would give them a head start and possibly lead to an easy "A." Assure them that it is calculus but that we are purposefully using a different approach -- that they will understand how calculus is used and gain some valuable skills that will help them through college. Tell them that it is hard, that it's supposed to be hard, and that they have what it takes to do it.

"My instructor isn't teaching; we have to teach ourselves."

Students are used to template learning. They think that a "good" instructor should simply lead them through each problem step by step. Real understanding, the kind that lasts, comes from struggling with the ideas. In this course we are tying to develop problem-solvers.

"The course is taking too much time."

The University's rule of thumb is that a student should allocate at least two hours of study time for each credit hour, so they can expect to spend 8-10 hours a week minimum on calculus. We are requiring new types of work; they may be inefficient in performing it. We have to help students learn to read, write, work, think, and cooperate without spending endless hours of wasted effort. Talk with students in or out of class; listen carefully to what they say about study habits and related matters. Remind them of the Math Lab, both as an environment conducive to getting work done and as a source of help with whatever is slowing them down.

"Why are we having to do all this writing? Writing has nothing to do with mathematics!"

Certainly traditional calculus courses do not emphasize writing. Many students adopt the strategy of writing as little as possible; their expectation (not necessarily a conscious one) is that the instructor will supply the right words to make sense of the mathematical symbols. "If I write something out, then I stand a better chance of being found wrong." You should make the case that writing is a crucial part of the thinking process, and that it will help them understand the material.

"Our homework problems are completely right and you're taking all these points off."

This stems from the common student view of grades as reward o punishment rather than feedback. Tell them that they are not competing against other teams. Explain what you mean by good work. They may think of a homework score such as 15/20 (which they convert to 75% -- a high school C) as well below average; whereas you probably consider it to be a score indicating good progress.

"I can't read the Book--it is too confusing and ambiguous."

Often this type of complaint comes from the fact that when they read the book they cannot find a "formula" for answering the problems at the end of the chapter as they are accustomed to doing with math books. Help them understand why the book is written the way it is and that the problems are meant to be hard and require sustained thinking. Once they get used to thinking hard and develop some problem solving skills; they will find everything they need to work out the answers is provided in the text. Let them know that one of the reasons for having homework teams is to help them learn the skills they need to succeed. Make sure they understand that reading mathematics is not like reading the newspaper. It is unlikely they are going to get everything the need the first time through. Help them learn how to read the text. Make sure they are trying the embedded exercises; encourage them to mark up the text, and ask to see their copy when you hear this complaint. Encourage them to discuss their reading with other students in the class (teammates or not), to ask questions in class, and to use the Math Lab.

"We never know if our answers are correct."

There is a fundamental problem here. Students are conditioned to believe that the only way to know that an answer is correct is to see if it agrees with the back of the book or if the teacher says it is right. They need to be encouraged to look at problems in alternate ways, to see if the answer is consistent with intuition. Spend some class time on checking procedures. For example, it is not enough to tell students that derivative calculations can be checked by doing one or more difference quotients on their calculators--make them actually do it and share the results with each other.

"It isn't fair for my grade to depend on the work of others."

Group work is a new idea; cooperation is a new idea. Remind students that there is research evidence that even the best students can improve if their group is working together properly. Many students' normal mode of operation is cutthroat competition. Remind them that there is not a preordained number of A's, B's, etc. Point out that when they go to work in the real world, their performance will be judged on how their group works. Also, if you adopt some form of evaluation of individual efforts, this may help students accept the grading scheme as "fair."

"Someone in my group is not doing enough work."

One of the skills students need to learn is how to work together. Part of their responsibility is to ensure, as much as possible, that everyone is contributing. In most jobs they will work in later, they will be told to get together with certain people and do something. Their boss is not going to be impressed if they are not able to work together. This said, it still may be the case that one or more members of the group is (are) not doing enough work. Remind the students that teams will change. If a member of the team does no work on the homework or a project then his or her name need not appear on it. Students may find it difficult (especially at the start) to leave off the name of a non-contributing teammate, but this is one aspect of making the students responsible for what the group produces..

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Instructor Attitudes and Typical Problems

Giving up lectures.

Lecturing is a very satisfying activity for us. We can organize the materials, display the logical structure, and introduce just the right examples. When the period is over, we feel that we have given a good account of ourselves, a good performance. It is hard to accept that this may not be the best way for students to learn. However, respectable educational research suggests that prolonged lecturing is the classroom activity least likely to result in learning.

Giving up control.

When you move to a classroom mode that is more student-directed, you will feel as if you are giving up control of the class. You cannot necessarily tell what students will want to discuss, what suggestions they will make. It is likely that students will ask questions for which you do not know the answer. This is painful the first couple of times it happens. However, it is very enlightening to the students. When you say, "I don't know what happens if we let b = 10,000 ; let's try it and see," they realize that it is not a personal failing when they have to admit often that they don't know the answer. To become good learners, they need a good learner as role model.

Listening to students.

One of the things you will learn as you read student papers and hear them discuss topics in their groups is that they are not thinking what we thought they were thinking. They have some surprising ideas fixed in their minds; a new, more useful concept cannot take the place of a faulty one until the old notion has been dealt with. This course has taught us to work harder at listening to what the students are telling us, not assuming that we are going to hear or read one of the most "five common errors." One of the instructors shared that he always thought that his students learned from an organized, interesting informative lecture. He even asked a few questions to be sure that the students were understanding. When he started to use cooperative learning in the classroom (after giving one of his thoughtful, informative lectures) and got a chance to listen in while students expressed to each other what they thought they understood. He was amazed at how little of his careful, clear lecture they understood.

Dealing with writing.

Many mathematicians admit that they do not like to deal with student writing. One of the reasons they went into mathematics is that they did not like to write a lot of papers! The type of writing we are expecting of the student, expository writing, is not particularly hard as writing goes. Previous instructors have found themselves to have the ability to improve their students' writing level dramatically simply by stressing the importance of carefully written solutions. When you are working with student writing it helps to be it helps to be very explicit about what you want students to do -- what you expect an assignment to look like. You will find that the more writing the students do the better they will get -- this is especially true if you have various students read what they have written out loud to each other in small groups and to the large group.

Working too hard.

It is hard teaching a new course for the first time; this course is newer than most. Instructors commonly report that teaching this way requires more emotional energy than teaching traditional math classes. If you actually know whether or not students are learning, you tent to maintain a higher level of personal engagement - you worry about them. There is a tendency in dealing with things we are unsure of (grading student writing, for example) to compensate by being too conscientious, spending too much time on the work. At the beginning, it is helpful to allow a limited amount of time for dealing with each paper. Nothing terrible happens if you do not make a comment that you might have made if you had more time. The students understand time pressure. You will find it useful to set a goal for the average time you will spend on each paper or set of homeworks, and then push yourself to keep up with that pace. This will vary somewhat with the nature of the assignment, but if you are taking more than two to three hours per week, your students are making you work too hard. The better their work, the easier it will be for you to grade it. Be very explicit about what kind of papers you will accept. Tell the students that carefully done homework will always lead to a higher grade.

Go to Appendix A: More About Homework Teams

Go to Appendix B: Team Evaluation Form

Go to Appendix C: Cooperative Learning in the Classroom

Go to Appendix D: Guideline for Analyzing A Cooperative Learning Exercise

Copyright © 1997 University of Michigan Department of Mathematics