A few years ago we taught all of our calculus courses in lecture-recitation format: two 80-minute lectures each week (90 to 150 students per lecture), with each lecture dividing into recitation sections with 35 to 40 students meeting once a week for 55 minutes. We taught a very traditional course whose text was almost always the most recent edition of Thomas (and Finney and ...).

About eight years ago we introduced EXCEL (Math 153-154), our version of Uri Treissman's PDP project. EXCEL showed us that students could learn from "workshops" (working in small groups on non-routine problems, with problem solutions required to be written up in standard English format). EXCEL's workshops met for three 80-minute periods each week, in addition to two 80-minute lectures each week. The workshops were small and well-staffed, with about 15 to 20 students in each and with an instructional staff of one graduate student and one undergraduate. We were able to document increased GPA's and retention in technical majors of students who had taken EXCEL. The effort and expense of using this model more widely were forbidding. While continuing to offer a small number of sections of EXCEL, we decided to try some changes in one of our large calculus sequences.

We changed texts from Thomas and Finney to the third edition of Stewart's Text. We decided not to jump to a "reformed" calculus curriculum (e.g., the Harvard text) because it would have been too radical a change for us and because there were some doubts about the usefulness of certain aspects of "reformed" instruction.

We decided to introduce "technology": for us in the first two semesters, this meant the use of graphing calculators. (More recently, Maple has been introduced in the third semester of this sequence of courses.) The TI-82 was chosen to be the "supported" calculator, although any other instrument which did not have a QWERTY keyboard or symbolic manipulation capabilities was allowed on exams.

Students were allowed to bring one "cheat" sheet of notes and formulas to exams. Because we were unsure that students would really learn the algorithmic aspect of the course as well as they should, we also instituted several computational tests (one on limits and one on derivatives). The tests were graded for answers alone, no notes or calculators were allowed. At first the tests were set up to be infinitely repeatable---students could continue to take different versions until reaching a passing grade. In some cases this worked effectively, but it was not worth the effort in others, and in recent years the repeatability has been given up.

The courses were coordinated more closely than had been the custom at Rutgers, for instance with common final exams.

We opted for the "early transcendentals" version of Stewart for several reasons. First, it was a way to alert the faculty strongly and directly that change was taking place. Second, it seemed appropriate because the graphing calculators had a number of buttons whose use would remain undiscussed until the elementary transcendental functions were introduced.

Our greatest change lay in what had been the recitations. We changed to workshops. We decreased recitation size to at most 30 students. Each recitation section was staffed by one experienced graduate student and one undergraduate peer mentor. The recitation time was lengthened from 55 minutes to 80 minutes. The substance of the recitation was changed from passive question and answer to the workshop model. Students had to hand in weekly writeups of one problem (almost always chosen by the lecturer) of each workshop. They were not told which problem would be required to hand in until near the end of the workshop. Each workshop writeup was graded by either the teaching assistant or the lecturer for both mathematical content and expository style. Additionally we wanted every exam to have some questions which required written exposition for answers earning full credit. Peer mentors corrected some textbook homework problems for each section each week, which was also a departure from past practice when frequently little or no homework grading was done.

Since the old-fashioned recitations were abolished, there was always pressure in the workshops to spend some time discussing homework problems from the text. Instructors have made their own compromises, most frequently arranging to discuss such homework problems for, say, 20 minutes of the workshop time and using the remainder of the time for workshops per se. The department furthermore created a second format for the course, a small-section format, requiring a 55-minute "practicum" each week to replace the old recitation and in addition to the lectures and workshop. In this format the lecturer not only gives the lectures but also runs the workshops (again with a peer mentor), and the TA runs the practicum. Roughly 25% of the sections are offered in this alternative format.

The changes have met with some success. The peer mentors have generally enjoyed the experience; it is a useful educational opportunity for them and they are paid, to boot. Students have seemed to have fewer complaints. On the other hand, the workload of the instructional staff of the course has certainly been increased.

What is described above is the result of the leadership of Professors Amy Cohen, Stephen Greenfield and Michael Beals and the efforts of many faculty members and graduate students in teaching in the experimental formats, particularly Professor Roe Goodman. It took a great deal of administrative support, which came principally from Professor Charles Sims, Undergraduate Vice-chair of the Department of Mathematics. These changes were substantially supported by funding from the Rutgers-New Brunswick Learning Resource Center, the Rutgers-New Brunswick Teaching Excellence Center, the office of the Vice President for Undergraduate Education, Dr. Susan Forman, and the Rutgers-New Brunswick Faculty of Arts and Sciences.

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Math 151-152 Home Page, Fall 1998
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