A student feedback team is a subset of the class which gives the instructor feedback on how the class is doing with new material.
Background and Purpose
The University of Arizona is a public institution with about 35,000 students. Each fall its Mathematics Department offers 15 sections of second semester calculus (Calc II), with about 35 students per section. Recently we noticed that students entering with credit on the AB Advanced Placement Calculus Examination do not seem to be well served by our Calc II. These incoming students are definitely not challenged by the standard integration material at the start of the semester, so instead of fully mastering the familiar material they relax. By the time new material is introduced they are well behind their classmates and many never catch up. Another possible reason is that in Calc II we emphasize written explanations along with numerical and graphical reasoning. Most incoming students are not very proficient with this. As a result, many of these students find their first mathematics course in college frustrating and unrewarding, even though they are often the most intelligent students in the class.
During the academic year 1996-1997, we made an effort to remedy this problem. We created a special year long course that uses differential equations to motivate topics from second semester calculus which are not covered in the AB exam. Enrollment was limited to incoming students who received a 4 or 5 on the AB Advanced Placement Examination. To determine the course content, I consulted several high school teachers, the Advanced Placement syllabus and past AP examinations.
The result was a course that starts with a very quick review of functions, limits, continuity, differentiation, and antidifferentiation, often within the context of some mathematical model. The emphasis here is on numerical and graphical interpretation. We then analyze simple ordinary differential equations using techniques of differential calculus. Standard integration topics are covered as they occur in finding explicit solutions of differential equations. Taylor series are introduced as a technique which allows solutions of difficult nonlinear differential equations. Here the ratio test for convergence of series is motivated by such series solutions. The textbooks for the course are Hughes-Hallett  and Lomen & Lovelock .
The course was limited to 30 students, as that was the number of chairs (and personal computers) in our classroom. With this agenda, a new type of mathematics classroom, and the nonuniform background of the students, it was evident that I needed some type of continuous feedback from the class.
My first assessment effort solicited their response to the following two statements:
"One thing I understand clearly after today's class is ____ ."
"One thing I wish I had a better understanding of after today's class is ____ ."
The students were to complete their responses during the last two minutes of the class period and give them to me as they left. (This is a particular version of the One-Minute Papersee David Bressoud's article in this volume, p. 87.) It had seemed to me that this was a very appropriate way to have them evaluate the material I was covering, especially during the review phase at the start of the semester.
The second method entailed my meeting with a committee of individuals once a week to assess their response to our classroom activities, their learning from the class, and their homework assignments. From the 20 volunteers I chose four who had different majors and had used different calculus books in high school. (At our first class meeting I had all students fill out a one page fact sheet about themselves which greatly aided in this selection.) To facilitate communication among the class, I formed a listserve where I promised to answer all questions before going to bed each night. I also handed out a seating chart which contained each student's telephone number, e-mail address, and calculator type. Comments, complaints, suggestions, etc., were to be directed either to me or to this committee, either in person or via e-mail. I emphasized that this was an experimental class, and I really needed their help in determining the course content and the rate at which we would proceed.
During the second semester of the course, these meetings of the four students were disbanded, and then resurrected (see below) and made open to any four students who wished to attend, with preference given to those who had not participated previously. Also, during the second semester, I had the students work in groups for some of the more challenging homework exercises. Included with their write-ups was an assessment of how the group approached the exercises, how they interacted, and what they learned.
This class was the most responsive class I have ever experienced as far as classroom interactions were concerned. However, there were so many unusual things about this class, it is impossible to attribute this solely to the use of feedback teams. For example, almost all of the students used e-mail to ask questions that arose while they were doing their homework, especially group homework. For example, in one assignment they ended up with the need to solve a transcendental equation. A reminder that there were graphical and numerical ways to solve such equations allowed the assignment to be completed on time. Students could send messages either to me or to the listserve. When the latter was used, several times other students responded to the question, usually in a correct manner.
Use of Findings
It turned out that the students required no urging to express their opinions. At the inaugural meeting first semester, the committee of four very clearly stated that they KNEW my first means of assessment was not going to be effective. They reasoned that the Calc I review was being presented and discussed in such an unusual manner that they had no idea what they understood poorly or well until they had some time to reflect on the material and worked some of the assigned homework. After much discussion, they agreed to a modification of this procedure where these two questions were answered AFTER they completed the written homework assignment. This was done, and provided very valuable feedback, including identifying their struggle in knowing how to read a mathematics textbook. To help with this reading problem, I had the students answer a series of True/False questions (available from http://www.calculus.net/CCH/) that were specific to each section of the calculus book. I would collect these as students came to class and use their answers as a guide to the day's discussion. While not all the students enjoyed these T/F questions, many said they were a big help in their understanding the material.
One other item this committee brought to my attention was the enormous amount of time it took them to work some of the word problems. To rectify this situation, I distributed the following seven point scheme, suggested by a colleague of mine, Stephanie Singer: 1) write the problem in English, 2) construct an "English-mathematics dictionary," 3) translate into equations, 4) include any hidden information, and information from pictures, 5) do the calculations, 6) translate the answer to English, 7) check your answer with initial information and common sense. This, together with some detailed examples, provided a remedy for that concern.
As the class continued on for the spring semester, I thought that the need for this committee had disappeared. However, after three weeks, three students (not on the original committee) wondered why the committee was disbanded. They had some specific suggestions they wanted discussed concerning the operation of the class. My reaction to this was to choose various times on Friday for a meeting with students. Whichever four students were interested and available at the prescribed time would meet with me. The first meeting under this format was spent discussing how to improve the group homework assignments which I instituted Spring semester. Because several homework groups had trouble arranging meeting times, they suggested these groups consist of three, rather than four students, and that any one group would not mix students who lived on campus with those living off campus. This was easily accomplished.
A meeting with a second group was spent discussing the "group homework" assignments which were always word problems, some of them challenging or open ended. These students noted that even though these assignments were very time consuming, they learned so much they wanted them to continue. However, they requested that the assignments be more uniform. Some assignments took them ten hours, some only two hours. The assignments given following this meeting were more uniform in difficulty.
At a another meeting the student's concern shifted to the number of exercises I had been assigning for inclusion in their notebook (only checked twice a semester to assess their effort). These exercises were routine, and for every section I had been assigning all that were included in a student's solution manual. They said there was not enough time to do all of these exercises and they did not know which they could safely skip. In response, I then selected a minimal set which covered all the possible situations, and let those students who like the "drill and practice" routine do the rest.
For larger classes, the student feedback teams could solicit comments from students before or after class, or have them respond to specific questionnaires. Occasionally they could hold a short discussion session of the entire class without the instructor present. In an upper division class for majors, this team could work with the class to determine what prerequisites needed to be reviewed for rapid progress in that class, see . Another aid for this process are computer programs (available from http://math.arizona.edu/software/uasft.html) which help in identifying any background weakness.
Other instructors have used "student feedback teams" in different ways. Some meet with a different set of students each time, others meet after each class. A more formal process usually used once during a term involves "student focus groups" (see the article by Patricia Shure in this volume, p. 164). The essence of this process is for a focus group leader (neither the instructor nor one of the students) to take half a class period and have students, in small groups, respond to a set of questions about the course and instructor. The groups then report back and the leader strives for consensus.
Other suggestions for obtaining feedback from large classes may be found in .
 Hughes-Hallett, D. et al. Calculus, John Wiley & Sons Inc., 1994.
 Lomen, D. and Lovelock, D. Exploring Differential Equations via Graphics and Data, John Wiley & Sons Inc., 1996.
 Schwartz, R. "Improving Course Quality with Student Management Teams," ASEE Prism, January 1996, pp. 19-23.
 Silva, E.M. and Hom, C.L. "Personalized Teaching
in Large Classes," Primus 6, 1996, pp. 325-336.