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ааааааааааааааа Student Feedback Teams in a Mathematics Classroom
ааааааааааааааааааааааааааааааа David Lomen
аааааааааааааааааааааааааа University of Arizona
Background and Purpose
The University of Arizona is a public institution with about 35,000 students.аа Each fall its Mathematics Department offers fifteen sections of second semester calculus (CalcII), with about thirty-five students per section.а Recently we noticed that students entering with credit on the AB Advanced Placement Calculus Examination do not seem to be well served by our
CalcII.а One possible reason is that the AB calculus syllabus does not mesh well with the way we split topics between our CalcI and CalcII.а The standard integration material at the start of the semester does definitely not challenge these incoming students, so instead of fully mastering the familiar material they relax.а By the time new material is introduced they are well behind their classmates and many never catch up.а Another possible reason is that in CalcII we emphasize written explanations along with numerical and graphical reasoning.
Most incoming students are not very proficient with this.а 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.а (These statements are partially based on comments received in student course evaluations.)
аааа During the academic year 1996-1997, we made an effort to remedy this problem.а We created a special yearlong course that uses differential equations to motivate topics from second semester calculus, which are not covered in the AB exam.а Enrollment was limited to incoming students who received a 4 or 5 on the AB Advanced Placement Examination.а To determine the course content, I consulted several high school teachers, the Advanced Placement syllabus and past AP examinations.
аааа 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.а The emphasis here is on numerical and graphical interpretation.а We then analyze simple ordinary differential equations using techniques of differential calculus.а Standard integration topics are covered as they occur in finding explicit solutions of differential equations.а Taylor series are introduced as a technique, which allows solutions of difficult nonlinear differential equations.а Here the ratio test for convergence of series is motivated by such series solutions. In this context, the need for theorems regarding differentiation and integration of power series is also apparent.
аааа The course was limited to thirty students, as that was the number of chairs (and personal computers) in our classroom.а 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.а The most immediate need was for assessment regarding the pace and content of the review topics from first semester calculus.а A different concern was how the class would adjust to my using visualization (usually using the computers at their desks) to motivate students to suggest a needed, or desired, result.а After a suggestion was given, we would discuss the relevancy of the suggestion in answering the proposed question, and then look for other suggestions for ways to determine the validity of the desired result.а As this was a departure for the students from the standard mathematics classes they had experienced previously, I was also looking for suggestions to help them acclimate to my teaching style.
ааа Another feature that was new to all of them was the emphasis on written explanations.а Both of our textbooks, Hughes Hallet et al. "Calculus" (1994) and Lomen & Lovelock "Exploring Differential Equations via Graphics and Data" (1996), often require such explanations and both include open ended exercises and ones with nonunique answers.а Having to write explanations caused a bit of anxiety in many of the students.
Method
аааа 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 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.а 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.а From the twenty volunteers I chose four who had different majors and had used different calculus books in high school.а (At our first class meeting I had all students fill out a one-page fact sheet about themselves which greatly aided in this selection.)а To facilitate communication among the class, I formed a LISTSERVE where I promised to answer all questions before going to bed each night.а I also handed out a seating chart, which contained each student's telephone number, e-mail address, and calculator type.а Comments, complaints, suggestions, etc., were to be directed either to me or to this committee, either in person or via e-mail.а 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.
аааа During the second semester of the course, these meetings of the four students were disbanded, and then resurrected and made open to any four students who wished to attend, with preference given to those who had not participated previously.а Also, during the second semester, I had the students work in groups for some of the more challenging homework exercises.а Included with their write-ups was an assessment of how the group approached the exercises, how they interacted, and what they learned.
Findings and Use of Findings
аааа It turned out that the students required no urging to express their opinions.а 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.а They reasoned that the CalcI 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.а After much discussion, they agreed to a modification of this procedure where these two questions were answered AFTER they completed the written homework assignment.а This was done, and provided very valuable feedback, including identifying their struggle in knowing how to read a mathematics textbook.а To help with this reading problem, I had the students answer a series of True/False questions (available from HYPERLINK "http://216.33.240.250:80/cgi-bin/linkrd?_lang=EN&lah=c98e4398816dd3472103f0483ed56310&lat=975965914&hm___action=http%3a%2f%2fwww%2ecalculus%2enet%2fCCH%2f" \t "_blank" http://www.calculus.net/CCH/) that were specific to each section of the calculus book.а I would collect these as students came to class and use their answers as a guide to the day's discussion.а 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.а To rectify this situation, I distributed the following six point scheme: 1) Write the problem in English, 2) Construct an "English-math dictionary", 3) Translate into equations, 4) Include any hidden information & information from pictures, 5) Do the calculations, 6) Translate the answer to English, 7) Check your answer with initial information and common sense.а 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.а However, after three weeks, three students (not on the original committee) wondered why the committee was disbanded.а They had some specific suggestions they wanted discussed concerning the operation of the class.аа My reaction to this was to choose various times on Friday for a meeting with students.а Whichever four students were interested and available at the prescribed time would meet with me.а The first meeting under this format was spent discussing how to improve the group homework assignments, which I instituted Spring semester.а 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.а This was easily accomplished.
аааа 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.а These students noted that even though these assignments were very time consuming, they learned so much they wanted them to continue.а However, they requested that the assignments be more uniform. Some assignments took them ten hours, some only two hours.аа The assignments given following this meeting were more uniform in difficulty.
аааа At 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).а These exercises were routine, and for every section I had been assigning all that were included in a student's solution manual.а They said there was not enough time to do all of these exercises and they did not know which they could safely skip.а 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.
Success Factors
аааа I was thrilled with the success of the weekly meetings with the group of four students.а The feedback obtained in this timely manner allows the instructor to implement modifications to enhance the learning by the students. It is a far better mechanism than waiting for the course evaluation at the end of the semester, which can only impact future classes.а This class was the most responsive class I have ever experienced as far as classroom interactions were concerned.а However, there were so many unusual things about this class, it is impossible to attribute this solely to the use of feedback teams.а For example, almost all of the students used e-mail to ask questions that arose while they were doing their homework, especially group homework.а For example, in one assignment they ended up with the need to solve a transcendental equation.а A reminder that there were graphical and numerical ways to solve such equations allowed the assignment to be completed on time.аа Students could send messages either to me or to the listserve.аа When the later was used, several times other students responded to the question, usually in a correct manner.
аааа For larger classes, the student feedback teams could solicit comments from students before or after class, or have them respond to specific questionnaires.а Occasionally they could hold a short discussion session of the entire class without the instructor present.а 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 Schwartz (1996).а Another aid for this process is computer programs (available from HYPERLINK "http://216.33.240.250:80/cgi-bin/linkrd?_lang=EN&lah=6519e989c06c3920f3e65f01eed1c6ed&lat=975965914&hm___action=http%3a%2f%2fmath%2earizona%2eedu%2fuofasoftware%2ehtml" \t "_blank" http://math.arizona.edu/uofasoftware.html), which help in identifying any background weakness.
аааа 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.а A more formal process usually used once during a term uses "student focus groups" as described in Redmond and Clark (1982).а 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 the following questions:
What do you like about this course?
What do you think needs improvement?
What suggestions do you have for change?
The groups then report back and the leader strives for consensus.а Abbott et al. (1990) report that this significantly affected student satisfaction and the faculty was rewarded with some course-specific suggestions.
Other suggestions for obtaining feedback from students may be found in Angelo and Cross (1993) or Silvia and Hom (1996).а Whatever the size of the class, I would recommend some type of ongoing feedback during the duration of the course.а Student teams provided a simple, versatile, and effective means of achieving such feedback in my class.а I highly recommend you consider trying them in yours.
REFERENCES:
Abbott, R., D. Wulff, J. Nyquist, V. Ropp, and C. Hess, (1990) "Satisfaction
With Processes of Collecting Student Opinions About Instruction: The Student
Perspective", Journal of Educational Psychology, V. 82, p. 201-206.
Angelo, T. A. $ K. P. Cross (1993), "Classroom Assessment Techniques: A
Handbook for College Teachers", Second Edition, Jossey-Bass Publishers, San
Francisco.
Redmond, M, and D. Clark, (1982) "Small Group Instructional Diagnosis:
A Practical Approach to Improving Teaching", AAHE Bulletin, V. 34(6), p. 8-10.
Schwartz, R., (1996) "Improving Course Quality with Student Management
Teams", ASEE Prism, January, p. 19-23.
Silva, E. M., and C. L. Clark, (1996) "Personalized Teaching in Large Classes",
Primus Vol VI, p. 325-336.
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