Emmett C. Dennis
 
 
 
 

After 13 years of teaching at various academic institutions of higher learning, I made a transition to high school teaching in 1991. It was here that I began the discovery of the learning style that I now have adopted. Until then, lecturing had been my primary method of instruction. It was the way I was taught, and it was all I knew. My student evaluations had generally been favorable, so I assumed that my teaching was effective and that students benefited by being in my classes. However, teaching in a high school system was completely different from college. The hands-off approach of college decorum did not work.

In the high school environment, a condition for success is complete involvement with your students in a more personal, emotional as well as professional manner. A teacher using the traditional lecture method has to maintain a more professional comportment, with little room for personal or emotional involvement. Hence in the high school setting, a traditional lecture method of teaching heightens the chances of inadvertently creating a scenario for mischievous behavior and disruption from the bored, hyperactive or restless students who refuse to be passive listeners. I discovered this problem when I began high school teaching.

While searching for and experimenting with innovative schemes to keep my students actively focused on the tasks at hand, I stumbled upon the method of using group work to get everybody involved in the learning and teaching of mathematics. I began reading about and experimenting with more discovery and cooperative learning in my classes, and soon realized I had found an exciting and energizing teaching and learning experience. This completely changed my views of teaching and my students’ views about learning mathematics, and I have not looked back since.

After three years of high school teaching, I returned to the college classroom in 1994 with a crusader’s outlook on what and how best college students could learn mathematics. I would like to share my experiences using cooperative learning techniques in college algebra classes that I have taught since returning to teaching at the college level.
 
 

FIRST DAY OF CLASS

My college algebra class consists of approximately forty students, meeting three times a week, for 50 minutes. The students are told that the section will operate in a nontraditional manner on the first day. A syllabus sheet for review and discussion is handed out. I then introduce the purpose and value of cooperative learning in the class, in an endeavor to sell the program to the students. An information sheet with general classroom policies as well as rules and procedures necessary when participating in cooperative groups is handed out. To create a sense of positive interdependence, roles specific to cooperative learning are introduced and defined, namely; facilitator, recorder, reporter, time keeper, materials manager, etc.

Following the discussion of the class procedure and after answering as many questions as are asked, the students are randomly placed into groups of between 4 to 5 members. I then pass out their first cooperative group worksheet. On this worksheet, the students engage in group interpretations of math conundrums and word problems for a limited time period. Then a whole class discussion of the results, as discovered in the various groups begin.

Also, while the students are in their groups, I pass around and solicit group names for the groups from the group members. This is an attempt to share ownership of the groups with the students. The smiles on their faces when the group names are announced (...the anonymous, the working minds, the can-dos, les miserables, the math meisters, the good fellas...) indicate some measure of success with the day’s work, an indication that they will be back the next day, more with curious anticipation than trepidation of mathematics. Following the discussion of the solutions on the worksheets, the class is dismissed.
 
 

GENERAL INSTRUCTIONAL MANAGEMENT

Instructional management consists of the following. The lesson begins with the whole class exploring certain key questions as introduced in the topic of the day. Next, the students get into cooperative groups and sort out the interpretations and explorations of the concepts using work sheets. The work sheets allow a more engaging interaction between the students, with hands-on applications and problem solving situations relating to the key questions introduced. For efficient usage of time, a maximum time limit is set for all group work done in the classroom.

Sometimes, before students begin their problem solving worksheets, I put the solutions of the problems on the board. Then I circulate around the room, observing each group’s progress and making suggestions to help them find the answers to their questions. If enough students appear to be having difficulty or generally seem to be making fundamental mistakes with a particular problem, I then facilitate a whole class discussion to resolve the problem.

I prepare and distribute solution sheets for homework problems from the previous class assignment. The students, in their groups, discuss and correlate these solutions with work they have done on the homework problems as written in their homework notebooks. Sometimes following these group discussions, the students take impromptu individualized or group quizzes relating to similar problems.
 
 

STUDENT REACTIONS

The goal of the cooperative group work is to enable the students to discover algebra for themselves by following the procedures on the handouts distributed in class. Thus, they are to "learn by doing" and be more heavily committed to the learning process. What I hope, naively as it turns out sometimes, is that by giving students the opportunity to think about the material and to work together, they will become accustomed to thinking and working cooperatively while producing positive results.

The first reaction by many students is excitement with the novelty of the class and freedom from long, boring lectures. Taking quizzes in groups is an immediate success for the majority. However, there are always some students who find this approach frustrating. Their previous impressions of a mathematics class have been based on the teacher doing all of the talking and teaching and they, all of the learning by listening. They have no interest in "teaching themselves" within the classroom. They usually have no idea how to approach new material or how to undertake and solve unfamiliar problems. The traditional lecture method of instruction has left them untrained in thinking systematically and critically about problems without first seeing a step-by-step example problem.

Some high achievers reacted negatively to the low achievers in their groups who do not understand an explanation the first time around. Groups with no outstanding high achievers give up easily on complex or challenging problems. In some groups the social interaction of the students with each other improve quickly, but not so quickly in others.

By circulating among the groups I am able to answer some questions before things get out of hand. But sometimes I find my time limited in getting to all the groups and answering all of the questions that come up in the allotted time for group work. The group problem solving time then has to be extended, to the demise of the whole class discussion time. I find myself spending more time lecturing than planned, hoping that a thorough introduction by me at the beginning of the class period, will make the work move along smoothly. It is safe to note here that because of my emotional and personal interaction with the students, I am able to go with the flow of the lesson plan as dictated by the immediate feedback from my students ‘ reactions.
 
 

GROUP MODIFICATIONS

The drop/add period at the beginning of the semester create some unwanted headaches. Because groups are organized on the first day of class, there is a considerable readjustment of group membership when some groups lose members who drop the course and new students add into the course. Unfortunately this does not happen evenly across the groups. So there are groups with fewer members than others because I am limited in the amount of shifting I can do and still maintain a sense of team cohesion.

Following the first test, the students have a journal assignment where they are asked to analyze their respective groups and group mates. The assignment is to identify the strengths and weaknesses of their groups and indicate whether they are pleased with their groups or not. This is the time to ask for reassignment to another group. Based on the requests, the groups are modified or maintained accordingly. However, having spend a couple of weeks together, the social interaction creates more positive bonding between the students and less reasons for wanting to be reassigned to new groups. Nevertheless, the same journal assignment procedure is followed after every test.
 
 

STUDENT ASSESSMENT

Quizzes

Three or four problem quizzes are taken by students individually or in their groups. These quizzes are given at most once a week. On some group quizzes, one paper is selected by the group members to be graded by me. All group members get the same grade from this paper. The goal here is to create some form of positive interdependence. On other group quizzes, the students are individually accountable for their work but the average of the individual grades serve as the group grade. All group members get the same average group grade. Members who are absent do not get any credit for the group quizzes. Make-up group quizzes are never given. Take-home quizzes or problems are given for individual assessment. However the student groups are given a maximum of 10 minutes at the beginning of class to discuss and compare answers on the take-home problems before turning it in.

Homework Folders

Homework problems are given on a regular basis. Students do the homework problems and turn in homework folders for inspection and grading on test days. The grades are awarded for neatness, completeness in doing the problems and minimally for accuracy.

Journals

Topics are introduced for journal writing on a regular basis. Students write at least a one paragraph essay to be turned in on the following class day. Grades are awarded for completeness and accuracy of thoughts, writing skills and neatness.

Tests

Individualized comprehensive test are given as topics are covered. On the day before a test, students in their cooperative groups take a practice test. Correct solutions are displayed on the board. For every test after the first, if the group average on that test exceeds the group average on the preceding test, every member of that group gets 4 bonus points added to that test grade. Make-up tests are not given, but the next test counts twice as much to replace the points missed.

Group participation evaluation is done on the day of a test. Each student gets the opportunity to rate the performance of their group mates (anonymously), with regards to their productivity during all group activities. The evaluation sheets are then returned to the students for reflection and hopefully positive growth and development in their interpersonal skills. The rating scale range from 1 to 5 points. These points are added to the test grade as bonus points.

Projects

The students do several group projects in class based on special topics covered in the course. They collect, measure or analyze data and solve problems. Group reports are turned in, one per group, for group grades. Individual projects are also assigned to analyze data and solve problems for individual grades.

Portfolios

A portfolio of the student’s work throughout the semester, (test, quizzes, etc.) is presented during the last week of classes along with a self-evaluation cover letter. In the cover letter the students assess their class performance and indicate what grade they would give themselves for the course based on this performance.
 
 

STUDENT OBSERVATIONS

As far as these college algebra students were concerned, the following observations seem to hold, based on their journal writing assessment of the course. Working in groups was generally held to be a good way to learn mathematics. In particular, for weaker students in the more diligent groups, this approach was highly successful. Some students seemed to develop more confidence in their ability to write and talk about mathematics. The social interaction help make mathematics a fun subject to learn for the first time in all of their experiences.

Many indicated that this was the first math class they had worked hard for a goal that they knew could be achieved. Most high achieving students got reassuring and positive feedback on their abilities as leaders from their fellow group mates, and eventhough their level of understanding did not improve as drastically as the low ability students, they did gain a wholesome comprehension of the subject. Complaints were associated with groups having class attendance problems. They could not do good jobs with the classwork and projects out of class because some group members did not show up to contribute.
 
 

COMMENTS AND OBSERVATIONS

It is now clear that to be a more effective facilitator of the groups one needs to be in constant contact with all group leaders. A brief meeting of at most once a week to discuss any difficulties within the group is important. The teacher should provide more materials to the groups which will help guide them through the learning process, and work with them more on developing appropriate group social interaction skills. Office visits by entire groups should be encouraged to discuss group work assignments.

In a cooperative group class, the group assignment should be controlled by the teacher. However student input into group membership should be encouraged and given careful consideration. Both academic and social strengths should be used in determining group composition, so as to give every student, at some time, the experience of being the most able member of a group. Students should be allowed to give regular evaluations and assessments of their group membership. Remember each student has different strengths and weaknesses. Membership should be based on several different variables, including but not limited to; math ability, sex, race, language, and perspective

A teacher as facilitator of a cooperative learning class must be in complete control of the class at all times, monitoring the groups’ progress, offering advice, and demonstrating how to behave as a contributing member of a group. The teacher must be prepared to instruct students in behaviors such as how to ask for help, how to listen and probe, and how to give clear explanations that allow the listener to follow the thought process.

The handouts or worksheets must be well constructed, brief and easy to follow. This will cut down on the unnecessary questions coming from the groups as they do their work. The projects or group activities must also be very relevant to the course. Real-world applications of the concepts being studied should be identified in the activities. And when students are assessed individually with tests, the teacher must be sure to use the same or similar style of questioning that was applied in the group activities. This is an important consideration to insure a smooth transition from cooperative learning to competitive or individualistic learning.

REFLECTIONS

When mathematics is taught as dry, disembodied knowledge to be received, it is learned, and forgotten or not used, in that way. The activities in the classrooms must reflect and foster the understanding that doing mathematics is an act of sense-making. However, in too many classrooms, students are required to do little more than listen passively. The assumption is that they need more teacher explanations and modeling of problem solving in order to help them relate what they already know to the present instruction. But the importance of students’ active role in the learning process must be stressed. In particular, students’ interactions - with one another, with the learning material, or with the teacher - are significant activities for effective learning (Bishop, 1985).

One of the great pioneers of cognitive theory, Jean Paiget (1926, 1932), believed that human beings learn about an idea from social interaction and experience and not merely because someone tells them about it. He emphasized that in the process of social interaction, the individual contributes as much as he or she receives. Vygotsky’s theory (1982) supports the emphasis on social interaction as a vehicle for learning. According to his work, students are able to solve certain problems cooperatively before they are ready to solve the same problems on their own.

Teaching students the social skills to be productive group members takes time--and this is time they are not using to learn the curriculum. It is true that the class may cover less material, especially when students are first learning their social skills. But there is some evidence that the social aspects of cooperative group work may actually reinforce the academic ones, and whatever is lost in quantity is more than made up for in the quality of the work done.

REFERENCES

Bishop, A. (1985). The social construction of meaning - a significant development for mathematics education? For the Learning of Mathematics, 5, 24-28.

Piaget, J. (1926). The language and thought of the child. (M. Gabain, Trans.). London: Routledge and Kegn Paul, Ltd.

Piaget, J. (1932). The moral judgment of the child. (M Gabain, Trans.). New York: Harcort, Brace and World, Inc.

Vygotsky, L. S. (1982). Thought and Language [Mishlenie I rech]. In L. S. Vygotsky, The collected works of L. S. Vygotsky [Sobranie sochinenii, t.s.] (pp. 5-361) Moscow: Pedagogika.