Since the early 1970s, school-based research has been used to help educators identify and strengthen a number of traits or behaviors that have collectively become known as effective teaching. The focus of this paper is on the behaviors that I have identified as having an effect on me as a teacher which then have been translated into the performances and attitudes of students in my mathematics classes.

Joseph Lowman (1984) viewed the superior college teacher as one with two distinct sets of skills: speaking ability and interpersonal skills. He claimed that to become an excellent instructor, one must be outstanding in one of these sets of skills and at least competent in the other. Borich (1992) also identified three teacher behaviors that have consistently been supported by school based research. The first is lesson clarity, or the ability to present curriculum to students in a straightforward, understandable manner. The second is instructional variety, which refers to the use of a number of different materials, strategies, feedback techniques and questioning procedures during a given lesson. The third teaching behavior that supports student learning is task orientation, or the extent to which the teacher uses class time for curriculum instruction.

Based on the reflective thinking that I have made of my teaching effectiveness, I believe my performances as a mathematics teacher has been affected by the continual growth and development I have made to achieve lesson clarity, instructional variety and task orientation in my classes. John Dewey (1933) provided a comprehensive description when he wrote of reflective thinking as the "active, persistent, and careful consideration of any belief or supposed form of knowledge in light of the grounds that support it" (p.9) Skillfully elaborating on this definition, Eby(1992) indicates that an active, persistent, and careful thinker is one who acts responsibly, is committed to taking on difficult challenges, and is sincerely concerned about doing the best for herself or himself and the students. In addition, Eby suggests that Dewey’s definition points toward an open-minded, inquisitive thinker who does not accept knowledge or practice without questioning it in some manner.

My growth and development as a mathematics teacher came to a test in 1991 when I made a transition to high school teaching for three years, after 13 years of teaching at various academic institutions of higher learning. 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.

It is clear to me that an important part of being an effective teacher is to be a reflective teacher. By this I mean a teacher who is constantly listening to and acting on one’s inner voice, or being able to consider alternative actions during one’s teaching, usually as a result of any number of astute observations involving students. This semester, I took a position at Dillard University as a Visiting Lecturer of Mathematics and began another leg of my teaching journey. Because I had never been in a college classroom setting where all of the students were black, the initial experience for me was full of excitement and enthusiasm. I was now in an atmosphere where my students had the same skin color as me. Hence I was eager to find the positive relations of that. But the task at hand for us all was the teaching and learning of mathematics.

The only mode of instruction that I observed in the math department at Dillard was the lecture method. So as not to "rock the boat", I also used this mode of instruction with my classes. By the Midterm of the semester, through reflection, and constantly listening to my inner voice, reviewing student accomplishments and assessing their comprehension and knowledge of the mathematics concepts, I accepted as fact that my teaching method had to change or be modified. I realized that I was not making a major impact on the mathematical knowledge of my students that I had envisioned. It was then that I reverted to another style of teaching more in line with my beliefs, the cooperative group learning techniques. I believe the change made a difference and brought more life into my classes.

Using Dewey’s work on reflective thinking, Eby, Tann, and Pollard constructed a useful graphic model of reflective teaching that I believe is a model to be followed (Eby, 1992; see Figure 1 below).

Reflective Teaching

There is an overwhelming agreement that students of the 1990’s and beyond will develop mathematical competence only if they are actively involved in "doing" mathematics at every grade level. Mathematical competence here means a student’s ability to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods to solve problems. The American Mathematical Association of Two-Year Colleges, (AMATYC), (1995), in its Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, recommended that students should have an opportunity to read, write, and discuss mathematics problems and concepts. It also encouraged the following types of experiences in the college classrooms: cooperative learning, oral and written reports presented individually or in groups; writing in journals; open-ended projects; and alternative assessments strategies such as essay questions and portfolios.

In the spring of 1992, seven professors of higher education, Hagelgans, Reynolds, Schwingendorf, Vidakovic, Dubinsky, Shahin, and Wimbish (1995), who had tried a variety of approaches to implementing cooperative learning in their classrooms, identified several characteristics for a course or classroom setting using cooperative learning groups. They stressed student involvement or participation in a significant amount of group work to maintain regular and often communication with each other. The aim was to encourage the students to become involved in the kind of discussion that would lead to multiple approaches to solving mathematical problems..

American researchers agree that students in a cooperative learning environment must be tested individually and held accountable for mastering the assigned material, but the focus is on improvement rather than mastery. Robert Slavin (1990) stresses the need for group rewards and the idea of equal opportunity for success. Equal opportunity for success for Slavin, however, means not just that the students feel they are not penalized by being in a particular group, but that each student can contribute to his or her team by improving over past performances. Slavin would structure group learning so that students begin with a base score and any improvements over that base score would increase the score of the whole group. This type of equal opportunity gives an incentive to all learners to achieve and help one another.

I have used the cooperative learning method in my math classes for the past seven years and have found this process of teaching and learning to meet the needs of the majority of my students. 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.

The goal of the cooperative group work is to enable the students to discover mathematics for themselves as they interpret 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 a cooperative learning 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.

Three or four problem quizzes were taken by students individually or in their groups. These quizzes were given at most once a week. On some group quizzes, one paper was selected by the group members to be graded by me. All group members got the same grade from this paper. The goal here was to create some form of positive interdependence. On other group quizzes, the students were individually accountable for their work, but the average of the individual grades served as the group grade. All group members got the same average group grade. Members who were absent did not get any credit for the group quizzes. They had to do it on their own. Make-up group quizzes were never given. Take-home quizzes or problems were given for individual assessment. However the student groups were 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 problems were given on a regular basis. Students did the homework problems and turned in homework folders for inspection and grading on test days. The grades were awarded for neatness and completeness in doing the problems with minimal emphasis on accurate answers.

Topics were introduced for journal writing on a regular basis. Students wrote at least a one paragraph (5 sentences) essay in class, or turned it in on the following class day. This depended on the nature of the journal entry. Grades were awarded for completeness and accuracy of thoughts, writing skills and neatness.

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

Group participation evaluation was done on the day of a test. Each student got the opportunity to rate the performance of their group mates (anonymously), with regards to their productivity during all group activities. The evaluation sheets were 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 were added to the test grade as bonus points.

The students did several group projects in class based on special topics covered in the course. They collected, measured or analyzed data and solved problems. Group reports were turned in, one per group, for group grades. Individual projects were also assigned to analyze data and solve problems for individual grades.

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

In order to create a smooth flow from one assessment method to another, 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.

As far as these college 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 been in where they had to work 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.

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. 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. It is fair to assume here that 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. Student input into group membership should be encouraged and given careful consideration. Students should also be allowed to give regular evaluations and assessments of their group membership to determine whether changes are necessary.

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.

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.

Americal Mathematical Association of Two-Year Colleges. (AMATYC) (1995). Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus. Memphis, Tn.

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

Borich, G. D. (1992). Effective teaching methods (2^nd ed.). New York: Merrill.

Dewey, J. (1933). How we think: a restatement of the relation of reflective thinking to the education process. (rev. ed.). Lexington, MA: D.C. Heath.

Eby, J. W. (1992) Reflective planning, teaching, and evaluation for the elementary school. New York: Merrill.

Hagelgans, N. L., Reynolds, B. E., Schwingendorf, K., Vidakovic, D., Dubinsky, E., Shahin, M., Wimbish, G. J. (1995). A Practical Guide to Cooperative Learning in Collegiate Mathematics. MAA Notes 37., Washington, DC.

Lowman, J. (1984). Mastering the techniques of teaching. San Francisco: Jossey-Bass.

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.

Slavin, R.B., (1990). Cooperative Learning: Theory, Research, and Practice. Englewood Cliffs, N.J.: Prentice-Hall.

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.