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Planning and Change: The Michigan Calculus Project

Morton Brown (University of Michigan, Ann Arbor)

The Years Leading Up To Reform

At the University of Michigan, Calculus I has a fall enrollment of some 1900 students in 55 sections (class size about 35, mostly entering freshmen) meeting four times per week. Calculus II has a fall enrollment of about 1000 students, many of whom are also entering freshmen but with advanced placement credit. The instructors for these courses are also largely new to Michigan. They are new graduate student teaching assistants (TA's) or new Ph.D.'s. Many of these are foreign-educated. Only a very few senior faculty members had ever taught in this standard first-year calculus sequence.

For some years the Department has held a week-long training program for new TA's prio to the fall semester. In addition, the University requires all its international TA's to participate in an intensive three-week summer training workshop co-sponsored bythe Center for Research on Learning and Teaching (CRLT) and the English Language Institue. Beginning junior faculty got little orientation; a single one-hour introductory session a day or two before classes began.

Often the TA's primary concern was (and is) with their own graduate programs, and the beginning faculty were (and are) deeply concerned with their research programs. Faculty instructors got a very small amount of undergraduate grader assistance. The TA's got no assistance with grading. Effectively, this meant that very little homework was collected and graded, and what was graded was seldom throroughly corrected. Teh students' uniform exam grade largely determined their final grades. As a result, instructors often found themselves teaching to the uniform exams.

Before reform, a senior faculty mentor and a part-time TA designed the syllabus, coordinated the course, wrote the unifrom mid-term and final exams, and met weekly for an hour with the 55 or so instructors of Calculus I to discuss course-related matters. The trainers (who had worked in the training program) and the mentors themselves visited classes on an "as needed" basis.

For several years proceeding the reform, the Calculus I course (but not Calculus II) progressively de-emphasized symbol manipulation and increased emphasis on concepts and geometrical visualization, all within the confines of a traditional text.

The Department operates a "Math Lab" which offers some self-study courses, but whose primary function is to provide tutoring for all elementary courses.

1991-1992: Early Pilots

In the fall 1991 term, we taught three pilot sections of Calculus I in which we introduced a number of innovations: (unstructured) team homework, graphing calculators (Texas Instruments loaned us 100 TI-81 calculators for the year), and a reform textbook in which we covered the same basic topics as the regular sections. At this stage, our thinking centered around the new technology, and the opportunity of the instructors to correct homework themselves. The team homework idea originally developed as a means of reducing the number of homework appers to be corrected. Only after the pilot began did we realize the powerful possibilities of team homework as a cooperative learning excercise.

Since a goal of the pilot was to anticipate eventual universal adoption, the first instructors were chosen to represent a spectrum of the teaching staff. One experienced TA, one junior, and one senior faculty member volunteered. Students were not told when they registered that they would be in a pilot section because we wanted a reasonable cross section of the student population. But we did not want students to feel like "guinea pigs" in a course which was as important to them as calculus, so they were allowed to switch to a traditional section later if they desired. In fact, only a few did so, primarily because they recognized that the pilot would be more demanding.

Our impression, by the end of the term, was the student performance on examinations did not seem much different from that in the traditional sections. An independent assessment by CRLT found that the students tended to be enthusiastic about both the calculators and the small group homework sessions involving their use. All the instructors found the course more interesting to teach, and felt that the students were more engaged.

The same three instructors tried repeating the pilot again (Calculus I) in the winter term, but this time using the textbook used in the standard sections. There was much less success. One of the instructors reverted to a traditional course, and the other two could not reconcile the technology with the traditional text.

Spring/Summer 1992: Developing the New Program

During the winter and spring of 1992, we drew up a plan to completely revise first-year calculus in both content and in style. The plan was developed by the senior professor who had been involved with the program for the earlier pilot (i.e., a "zealot") and the senior lecturer who generally ad the responsibility for the first-year calculus course. An instructional consultant from CRLT was involved from the beginning. The period from pilot to general adoption was to be three years. The Department, the College, and the NSF all committed to help set up the program. The chair supported the program enthusiasticaally. The attitude of the general math faculty initially, and aat least for the next two years, was generally supportive but tempered by an appropriate "wait-and-see" attitude.

The College, the Department, and CRLT all contributed support for the initial stages of planning. We negotiated with Texas Instruments for them to donate 200 used TI-calculators and several view screens while the College paid (as matching funds to the NSF grant) for 200 more. We planned to loan these calculators to the students each term but give one to each instructor to keep. It would have been extremely difficult, if not impossible, to embark on this major innovation without these considerable extra resources. We anticipated that a great deal of support would be needed to help the instructors who were eventually to teach in this program. Also, we would have to help the students who would find themselves in a course different from what they had been expecting. We were planning a completely new course: new teaching and learning paradigms.

These are the main features of the program as it actually began in September of 1992.

  • Incorporation of graphing calculators into the curriculum
  • A "reform" text (the decision of which book to adopt came just two months before classes began)
  • Team homework; three or four students submitted one paper (our graders converted to Math Lab tutors)
  • Cooperative learning in the classroom
  • Mid-semester assessment (the staff from CRLT visited each instructor's classroom around mid-term, and then met with the instructor to give feedback and discuss what to change in response to students' suggestions)
  • A short, but specialized, training program (the instructors were to all be experienced faculty and TA's) covering:
    1. use of the graphing calculator, how to help students get comfortable with it, and its pedagogical opportunities
    2. an introduction to the philosophy of the course and the new syllabus
    3. a workshop on classroom cooperative learnin techniques using problems from the text as examples
    4. some discussion of anticipated student reactions to the course
    5. a workshop on the CCH syllabus run by an invited Consortium member who had used the Harvard text
  • We prepared a pamphlet for instructors which commented on all the homework problems in the text and sorted them according to difficulty. (A version of this idea was later incorporated inwo a publisher's instructor handbook relieving us of repeating this task for future textbook editions.)

We anticipated the more widespread changes to come by restructuring the Department's week-long TA training program. It was renamed "Instructor Development Program." and all the faculty who were new to Michigan were included in the same program as the incoming TA's (an immediate benefit of this was the social interaction between junior faculty and TA's). The new program emphasized student-focused teaching. The instrucotrs who had volunteered to teach the reform sections were encouraged to attend those sessions relevant to the new course as well as the sessions developed especially for them.

1993: The New Program Begins

In the first semester we had ten sections. Three senior faculty, three junior faculty, and four experienced TA's all volunteered to teach the course (and, or course, attend the training program). The senior faculty all had strong research credentials. Furthermore, one had been an associate chair, and one was to become the new Department chairman a few years later.

Students were not told when they registered that they would be in a section with a new format, and they were discouraged from changing sections to retain a valid comparison between the new course and the traditional course.

Our weekly meetings focused on issues of classroom management when one was using cooperative learning, team homework, student moreale, and student understanding of the goals of the course. This last was particularly importnat. We had not fully anticipated the strong studnet reaction to the new pedagogy. (This kind of reaction is now well-known and characteristic of almost all calculus reform programs.) Fortunately, the instructors were quick to realize that we needed to devote some class time frequently during the first few weeks, and consistently during the semester, to discussions of the goals of the course, and reminding the students that they were learning how to think about math rather than just follow recipes. On the instructors' side, we definitely under-estimated the need for morale building among ourselves, particularly the TA;s, whose own pride required them to perform successfully in a style of teaching for which they had never seen a role model.

That first year, we learned a lot about using cooperative learning. For instance, we only began to introduce structure into the formation and activities of the "homework teams" after the third week of classes. i.e. we were learning on the job. The mid-semester feedback sessions carried out by CRLT helped us understand many of the frustrations and anxieties that the students were experiencing. Their fears were compounded by seeing peers in the traditional sections who seemed to be covering more material. However, the feedback also confirmed that the students were largely supportive of what was going on in the course. Naturally, there were many students who didn't buy into the program, and who criticized our changes. After a few complaints of this sort, any unsympathetic department chair or Dean might have quickly squelched the new program without a fair trial. Fortunately, we had full support from both our chairman and our Dean.

We made sure to document our new teaching styles. During the term each instructor was videotaped far at least one class hour. The videotapes were first viewed by the instructor for self-assessment, and some tapes were incorporated into future training programs and dissemination materials. (At this writing, some of the instructors who were taped are applying for jobs next year, and are using their tapes in the application process.)

As it turned out, the previous summer's extensive planning was largely successful. The year ended with a general feeling on the part of students and instructors (yes, we have evaluations) that the new program was quite successful. Students, with some reservations were particularly enthusiastic about the positive aspects of the calculators and the team homework. During the term, to help with student morale, students were asked to name the program. Eentually the name "New Wave Calculus" was adopeted. The student newspaper, after expressing reservations about New Wave Calculus in the fall, gave it a strong endorsement in the spring.

    There were problems:
  1. the classrooms with strip tables or arm-chairs and a general teacher-in-front-of-the-classroom architecture were unsuitable for cooperative activities
  2. the workload was much higher due in part to greater preparation time and homework grading, but mostly reflecting the emotional drain that occurs when instructors get to know their students as individuals, and care for them
  3. the classes were too large for most of the instrucotrs to get seminar type discussions going successfully.

It was still unclear whether such a program could be successfully instituted across the board. At the very least, we knew that a much more extensive instructor training and professional development program would be required.

Writing Our First Materials

An Instructor's Handbook was developed to assist instructors. The curernt handbook (1994-1995) familiarizes the instructor with the goals and features of the program. There is a section on student learning (how students learn), discussion of the setting up and monitoring of homework teams, cooperative learning, information about Math Lab and other program features, a section on student attitudes and typical problems associated with teaching, and detailed suggestions for each day of the first few weeks. Of course, there are also a daily syllabus and suggested individual and team omework assignments for the term. We also produced a brief (ten-page) Student Handbook, introducing the students to the new syllabus and pedagogy, and illustrating some examples of acceptable and unacceptable homework writing.

Setting Up the Instructor Development Program

Anticipating that there would be about sixteen instructors teaching twenty section of New Wave Calculus I in the fall term, we felt that we should continue to keep the weekly staff meetings small, so we divided the instructors into two groups which were to meet weekly with a mentor. The mentors were a junior faculty member and a senior TA. Copying the cooperative pattern for the students, we formed self-support teams of four instructors who were to visit each other's classes, discuss teaching problems, prepare tests and class exercises, and commiserate with each other. This turned out to be sporadically successful. Some groups operated very productively. Most became dysunctional.

1993-1994: Operating Traditional and New Wave Courses in Parallel

This year we ran two large calculus courses, traditional and New Wave, side by side. They used different texts and different teaching styles.

By now, the staff was getting stretched pretty thin; coordinating both courses, training the forty new instructors, and meeting them on a weekly basis. the operation was becoming too large and complicated. We assigned jobs so that within the New Wave, we had one senior faculty to coordinate the sections, while a TA and junior faculty "mentored". The traditional sections were coordinated and mentored by the senior lecturer who also taught a New Wave section.

I quickly became impossible to keep the courses as distinct as they had been in the previous year. We slowly began to add components from the New Wave to the traditional calculus course. Starting in the fall of 1993, all students in all the introductory calculus courses were required to buy calculators, and in the winter of 1994, we went across te board with the reform textbook. Additionally, the director of the Math Lab had revised the pre-calculus program over the summer so that the twenty fall sections began using the TI-82 and a reform pre-calculus text which emphasized more writing, discussions, and problem solving. We were also seeing some changes in the third-semester calculus which reflected the same general spirit. (These changes were beig developed by a different cohort of faculty.)

In the New Wave calculus program the emphasis had refocused from technology issues to teaching and learning issues. The discussions in the regular mentor meetings developed our ideas. Near the end of the first semester. CRLT held a focus session for the New Wave instructors to get some feedback from them. This session was not only informative for us in preparing to go across the board, but helped the instructors focus on important issues concerning their own attitudes and needs.

All of us were concerned that there might be a weakening of the "basic skills" normally developed in Calculus I, e.g., differentiating functions, so we agreed to have a number of such questions appear on botht he traditional and New Wave final exams for the fall term. An analysis of performance showed little difference between the two groups, with the New Wave students scoring slightly higher. This made us more comfortable about continuing to make changes.

Beginning in the winter term, we switched all section to the reform textbook. The meant that the only remaining differences from the student point of view were that not all sections used cooperative learning or team homework. We noticed that using a single text for all the sections made the students muc less anxious about whether they were covering the "right" material. We found, however, that most of the instructors in the "traditional" sections spontaneously tried the cooperative learning techniques from the New Wave it was clear that these instructors needed much more guidance than they had gotten in their first few days of training. We began developing far more specific instructional guides, and we had one of our experienced instructors videotaped each day during the frist week of classes. These tapes were to be used in later terms to help instructors set the tone of the course.

The second year proved to be another successful one for the program. Once again the instrucotrs and students alike felt they were getting a better course.

Spring/Summer 1994: Getting Ready to Take the Plunge

Our plans were to complete the changeover. We did not have the resources to direct two types of Calculus I and two types of Calculus II in each of the fall and winter term, so we went across the board with the new program. In the coming fall we diecided to have all the sections of both Calculus I and Calculus II operating in the new way. (This did not include 10 sections of Calculus II which were reserved for advaced placement students who wanted to learn Maple.) We also began referring to the course as "Michigan Calculus" to avoid the connotation for students that we were trying something "new" on them.

Total Training

It appeared that we would have to train about 50 souls in the fall: new junior faculty, new TA's, and graduate students who were not new but had not taught this brand of calculus before. Some instrucotrs would be teaching in the idea setup; overhead projectors (and view screens), classes of 24 students with the 3' x 3' tables which made group work in class easy, but others would still be looking at inappropriate classrooms with 32 students. We decided that everybody should be trained in the new methods. The beginning-of-term training program, which now encompasses the seven working days before classes began, again seemed successful. This impression was later confirmed in several focus sessions held with instructors during the following winter.

1994 Fall: A Rocky Road In the 1994 fall term there were eighty sections of Calculus I and Calculus II, and the vast majority of the instructors had not taught before in the program. This once-only situation resulted from going across the board for the first time. We saw the immediate need for continuing instructor development. Several of the new junior facutly had little or no teaching experience. Furthermore, a number of faculty had brought with them attitudes aout teaching that reflected the values of mathematicians in countries that have a very different prevalent philosophy from what we were trying in our program. Also, even in ordinary years, a percentage of our new TA's and faculty need a great deal of extra attention. Since this was the first time that the new program was not voluntary for the instructors, the were some (actually not many) who did not "buy into" its philosophy or want to put in the extra effort the new program required of its first-time teachers.

To run the fifty-nine sections of Calculus I, we divided the instructors into six groups and had each group meet weekly with one of three mentors. The mentors themselves met regularly, and tried to coordinate their activities. Syllabus matters were dealt with by e-mail, so there wer few large required "staff meetings". The instructors clearly would have benefitted from some teaching "clinics", but we were hesitant to increase the time commitment for the instructors who were already being asked to more than they had done (or seen their peers do) in previous terms.

With so many sections, the experienced CRLT staff person who had been doing the mid-semester feedback sessions in previous terms, now needed to have a new group of assistants. Thus, there was an extra layer of bureaucracy that slowed and somewhat distorted information in both directions between the directors of te program, the instructors, and the students.

As it turned out, because of extensive construction on campus, a very large number of the classrooms were dramatically unsuitable for any kind of teaching and frequently much too small for the number of students. Finally, to compound the logistical problems, the overall university enrollment was larger than expected, so quite a few classes remained at size 35.

Was the new program a success under these somewhat difficult conditions? Certainly there was less cooperative learning going on in the classes than we had hoped for, but still there was quite a lot. Many of the instructors did "buy into" the program. Some were very successful, many were struggling with the new teaching style, and many needed more help than they got. The student mid-semester feedbakc, the instructor focus sessions, and the students end-of-term evaluation forms suggest that the students generally liked the team homework, although many expressed concern that their grades were being affected by the work of others. This third year of the program brought a downturn in students' evaluation of the course. The single number expressing "this was an excellent course" on their evaluation forms fell byu aobut a third of a point on a scale of one to five. This drop can be partially explained by our problems with logistics in going across the board.

1991-1995: Logistics In Hindsight

We did not foresee that the amount of planning we would need would be about five times what we had expected. The centralized University structure made even small changes quite difficult. For example, getting the College to dedicate two classrooms for the program, was a major endeavor. The we had to get the rooms fitted with locked cabinets to hold the overhead projectors and the calculator demo view screens. There were lots of details, for example, even low-tech equipment needs attention: the demo calculator may need repair (or even new batteries)

In the summer of 1992 alone, we had to

  1. get the rooms ready
  2. be sure the textbook would be published and delivered on time (the actual textbook choice was not made until the end of June, and the book was at the time in a preliminary edition)
  3. arrange for enough calculators for both students and instructors
  4. explain our program to the College academic counselors, and to the counselors in the many other departments and colleges
  5. design a more elaborate evaluation program
  6. prepare the training program for instructors and tutors
  7. plan, write, publish, and distribute Instructor Handbooks
  8. plan calculator clinics for the beginning of the term.

Each of these jobs proved more complicated than we had imagined. Take, for instance, the evaluation program. The effort of hiring a principal evaluator, setting up the evaluation program, and then budgeting it, overseeing the budget, and coordinating this budget with the budgets of both the College (which helped fund the evaluations) and CRLT turned out to be a major headache. During the 1993-1994 year we met with the evaluation team for about 3 hours twice a month, and it actually took about two years to get the logistics of the evaluation running smoothly.

If the project operates on a large scale, even simple problems can be time-consuming to resolve.

Observations About Teachers and Teaching in a Reform Course
  1. For many instructors, a fully-invested teaching effort requires so much intellectual and emotional energy that it interferes with their graduate, postgraduate, or research programs. Certainly first-year graduate students should not usually be given primary responsibility for teaching a calculus course. They could apprentice, work in a Math Lab, or perhaps have supervised recitation section responsibility.
  2. Be sure that the people in charge of the curriculum teach the course.
  3. Be sure to attend to instructor maorale. Everyone will need frequent reminders of the goals of the course.
  4. Sudents will take responsibility for their learning if
    • they are given the opportunity, and
    • they are helped in learning how to take this responsibility.
    The teacher's responsibility is to help the students learn and help the students learn how to learn.
  5. Since many students will not willingly accept the new ways and demands of reform calculus, they will want the "old way" back. Often instructors will use this attitude to justify their own temptation to return to a traditional teaching style.
  6. We need to think hard about the differences between what we (or the text) say and what the students (differentially) hear. For example, many of the concepts that instructors may think of as "trivial", are perennial stumbling blocks for students.
  7. Students can have ideas (correct and incorrect) about solutions to first-year calculus problems that we could never have guessed. This actually explains a lot of apparently nonsensical (to us) statements that come from reasonable students.

Obersvations About the Student Response to Reform
  1. The students need constant encouragement. This is a big change.
  2. Most students like the strengths of cooperative learning, but many are concerned about its perceived downsides (particularly in its relation to grades and scheduling problems).
  3. All tutoring shoudl be restructured so that the tutors learn how to guide students in problem solving rather than simply answering the team homework problems. Our undergraduate tutors usually could not solve the problems anyway which led to many complaints from our students, e.g., "how can we be expected to answer homework problems when the tutors can't even do them?"
  4. Expect student evaluations generally to go up during the pilot stage, and then generally go down for a few years when you go across the board.

The Politics of Reform
  1. Reform efforts must be accompanied by evalusation, and evaluation requires the setting of goals.
  2. Get the client departments on board. Many are concerned with the same pedagogical issues, don't know what you are doing, and are often unfamiliar even with your old program. They can be a great source of support with the higher administration, and within your department.
  3. Get the academic counselors behind you. Explain the goals of the course so that when disgruntled students come to them, the counselors can help defuse the situation.
  4. Pasting on technology is not calculus reform, but it can be a doorway to reform. Practically, without an ingredient of technological innovation, administrators and funding agencies are not likely to support a reform effort.
  5. The reform textbook (i.e., the content of calculus reform) is an instrument of reform; it is not reform in itself. The real reform will be in how we teach and how students learn.
  6. We had difficulty getting senior faculty to volunteer to teach the New Wave course. In the first two years about eight senior faculty did so. On the other hand, that's a lot more than did so before the new program began. Also, as part of its bargain with the College, the Department pledged that a far greater number of senior faculty would teach a first-year course.
  7. Because calculus (and pre-calculus) is such a large aprt of the teachin role of a Mathematics Department, serious reform will affect the whole operation of the department: hiring of faculty, support of TA's, the reward system, prioritizing course assignments, and the involvement of senior facutly in the elementary mathematics courses. Reforming calulus will require a considerable increase in resources. This must be done through an increase in resources for the department as a whole and a reprioritizaiton of resources within the department.

Copyright © 1997 University of Michigan Department of Mathematics