Combining Individual and Group Evaluations

One frequent concern of faculty members who have not yet tried cooperative learning is that giving the same grade to a whole group will be unfair both to the hard workers and the laggards. This article addresses this issue.

Background and Purpose

Assessment of student learning in a mathematics course with cooperative learning groups includes assessment of group work as well as assessment of individual achievement. The inclusion of a significant group component in the course grade shows the students that group work is to be seriously addressed. On the other hand, assessment of individual work is consistent with the goal of individual achievement. Such an assessment plan promotes learning through group interaction and also holds students accountable for their individual learning.

Ursinus College is an independent, coeducational, liberal arts college located in suburban Philadelphia. Over half the graduates earn degrees in science or mathematics, and biology is one of the largest majors. About 75% of the graduates enter graduate or professional schools. Mathematics classes are small, with a maximum of 20 students in calculus sections and even fewer students in upper level courses. In recent years, a majority of the mathematics majors has earned certification for secondary school teaching.

My first experiences with extensive group work occurred in the fall semester of 1989 when I took both my Calculus I sections to the microcomputer laboratory in order to introduce a computer algebra system. Throughout the semester, the students worked in pairs on weekly assignments, which counted about 20% of their course grade. I continued this arrangement for the next three semesters in my sections of Calculus I and Calculus II. Then, after the summer of 1991, when I attended a Purdue Calculus workshop, I started using the cooperative learning aspects of that program in many of my courses. Subsequently I was a co-author of the book, A Practical Guide to Cooperative Learning in Collegiate Mathematics [2], which includes a full chapter on assessment. I now use cooperative learning and assess the group work in all the mathematics classes that I teach. The assessment method is applicable to any mathematics course, including first-year classes and majors' courses, in which cooperative learning groups are assigned for the semester. I have used similar methods in Calculus I, Calculus II, sophomore Discrete Mathematics, Topology, and Abstract Algebra.

Method

The assessment plan for classes with cooperative learning includes evaluation of most of the students' activities. I assess quizzes, tests, a final examination, group assignments and projects, and "participation." The participation grade reflects each student's involvement with all aspects of the course, including submission of journal entries each week, preparation for and contributions in class, and group interaction during class and outside class. A typical distribution of total points for the various activities assessed is shown Table 1.

With this scheme about 45% of a student's course grade is earned in the group since the first of the three hour exams given during the semester is a group test and about half the participation grade is related to group work. The quizzes, two tests, and the final examination all are individual tests. At times I give a few extra points on an individual test or quiz to all members of a group that has excelled or greatly improved. I give each student a midsemester report in which I allocate up to half the points for group projects and for participation. In this midsemester report, with one group test and two quizzes given, 275 out of a total of 350 points (about 79%) are related to group work. The final examination is a challenging comprehensive examination on the entire course.

The group test is given during a class hour around the fourth week of the semester. The goals of the group test include promoting group interaction early in the semester, reducing tension for the first test of the course, avoiding early withdrawals from the course (especially in Calculus I), and learning mathematics during a test. A group test is somewhat longer and harder than an individual test would be, and it certainly is too long for an individual to complete during the hour. A typical problem on an abstract algebra group test is: Prove: If x=x^-1 for every element x of a group G, then G is abelian. Seven of ten problems on one discrete mathematics group test asked students to determine whether a statement is true or false and to provide either a proof or counterexample with a full explanation; one such statement was: The square of an integer has the form 4k or 4k + 1 for some integer k. The group test in any course is designed with completely independent problems so that students can start anywhere on the test. Each student receives a copy of the test questions and blank paper for the solutions. Each group may submit only one solution to each problem.

Preparation for a group test includes the usual individual studying. In addition, I frequently hold a review problem session in the class hour just before the test, and the groups continue to work on these problems outside class as they study for the test. Also, I instruct the groups to plan their test-taking strategies before the day of the test, and we discuss the possibilities in class. For example, a group of four students could work in pairs on different questions, one pair starting with the first question and the other with the last question; after all problems have been solved, students can check each others' work and discuss the solutions. I mention that individuals need time to think quietly before talking with other group members about a problem.

The group assignments, which vary with each course, comprise challenging problems that are submitted and graded almost every week during the semester. The calculus classes solve problems using a computer algebra system; the laboratory book is chosen by the Department. The discrete mathematics students implement the fundamental concepts with some problems that I have written for the same computer algebra system [1]. The abstract algebra and topology students solve traditional problems without the use of technology, but with an emphasis on writing clearly.

The journal is a diary in which the students write about their own experiences in the class. On some weeks I ask them to concentrate on certain topics, such as their group meetings, the text book, the department's open tutoring service, or how they learn mathematics. I respond to the journals without determining a grade. Each week I give any reasonable submission a check to indicate credit toward the participation grade.

Findings

With the assessment described above, the students actively participate in their learning both in class and outside class every week of the semester. They interact with other students as they work together in long, problem-solving sessions, when they learn to discuss mathematics. They encourage each other to persist until the problems are solved. During the frequent group visits to my office for help, the students fluently talk about the mathematics, and they show that they have thought deeply about the problems. The students are motivated to complete the group assignments by the assignments' part in determining the course grades as well as by the responses I write on the work.

The test grades on the group test are quite high, and no group has ever failed a group test. The lowest grade ever was 68 in one Calculus I class, and one year all four groups in Discrete Mathematics earned grades above 90. This test forestalls a disastrous first testing experience for first-year students in Calculus I and for majors enrolled in a first abstract mathematics course. Few students have complained that the test grade lowered their individual grades, since, as a matter of fact, most students have a higher grade on the group test than on subsequent hour exams. On the other hand, some weaker students have written in their journals about their concern that they cannot contribute enough during the test, and no student has depended entirely on other members of the group. Since it is to the advantage of the group that all members understand the material, the group members usually are very willing to help each other study for the exam. The group exam does reduce tension and foster the team spirit. Many students claim that they learned some mathematics during the test, and they frequently request that the other tests be group tests. I see evidence that students have edited each others' work with additions and corrections written in a different handwriting from the main work on a problem.

Table 1 two tests, and the final examination all are individual tests. At times I give a few extra points on an individual test or quiz to all members of a group that has excelled or greatly improved. I give each student a midsemester report in which I allocate up to half the points for group projects and for participation. In this midsemester report, with one group test and two quizzes given, 275 out of a total of 350 points (about 79%) are related to group work. The final examination is a challenging comprehensive examination on the entire course.

The group test is given during a class hour around the fourth week of the semester. The goals of the group test include promoting group interaction early in the semester, reducing tension for the first test of the course, avoiding early withdrawals from the course (especially in Calculus I), and learning mathematics during a test. A group test is somewhat longer and harder than an individual test would be, and it certainly is too long for an individual to complete during the hour. A typical problem on an abstract algebra group test is: Prove: If for every element x of a group G, then G is abelian. Seven of ten problems on one discrete mathematics group test asked students to determine whether a statement is true or false and to provide either a proof or counterexample with a full explanation; one such statement was: The square of an integer has the form 4k or 4k + 1 for some integer k. The group test in any course is designed with completely independent problems so that students can start anywhere on the test. Each student receives a copy of the test questions and blank paper for the solutions. Each group may submit only one solution to each problem.

Preparation for a group test includes the usual individual studying. In addition, I frequently hold a review problem session in the class hour just before the test, and the groups continue to work on these problems outside class as they study for the test. Also, I instruct the groups to plan their test-taking strategies before the day of the test, and we discuss the possibilities in class. For example, a group of four students could work in pairs on different questions, one pair starting with the first question and the other with the last question; after all problems have been solved, students can check each others' work and discuss the solutions. I mention that individuals need time to think quietly before talking with other group members about a problem.

The group assignments, which vary with each course, comprise challenging problems that are submitted and graded Students in all the courses have time to become acclimated to the course before the individual work counts heavily, and no students have withdrawn from my Calculus I classes in recent years. This assessment plan is much more successful than a traditional plan that involves only periodic assessment of performance on timed tests. The plan promotes student behavior that supports the learning of mathematics, and it allows me to have a much fuller understanding of the current mathematical knowledge and experiences of each student.

Use of Findings

By reading the groups' work each week, I am able to follow closely the progress of the students throughout the semester and to immediately address any widespread weaknesses that become evident. For example, in one proof assigned to the abstract algebra class, the student groups had twice applied a theorem that stated the existence of an integer, and they had assumed that the same integer worked for both cases. During the next class I discussed this error, later in the class I had the groups work on a problem where it would be possible to make a similar mistake, and then we related the solution to the earlier problem.

I use the information in the weekly journals in much the same way to address developing problems, but here the problems usually are related to the cooperative learning groups. I offer options for solutions either in a group conference or in my response written in the journal. In addition, when students write about their illnesses or other personal problems that affect their participation in the course, we usually can find a way to overcome the difficulty.

Success Factors

I discuss the assessment plan and cooperative learning at the beginning of the course to motivate the students to fully participate and to avoid later misunderstandings. The course syllabus includes the dates of tests and relative weights of the assessed activities. The midterm report further illustrates how the grades are computed. At that time I point out that the individual work counts much more heavily in the remainder of the semester.

References

[1] Hagelgans, N.L. "Constructing the Concepts of Discrete Mathematics with DERIVE," The International DERIVE Journal, 2 (1), January 1995, pp. 115-136.

[2] Hagelgans, N.L., Reynolds, B.E., Schwingendorf, K.E., Vidakovic, D., Dubinsky, E., Shahin, M., and Wimbish, G.J., Jr. A Practical Guide to Cooperative Learning in Collegiate Mathematics, MAA Notes Number 37, The Mathematical Association of America, Washington, DC, 1995.