Based on the belief that students should be working together, I have used "groups" many times for learning. The results were sometimes successful but I became frustrated by the lack of involvement of many of the student, the noise level was high, and the classroom was out of control. In essence, the classroom was out of control and I did not feel comfortable relinquishing power.

The push to improve mathematics education and make it more successful has ‘pushed’ me to improve my cooperative learning methods. I felt a strong need to learn more about this Vygotskian approach to learning. I had to make it successful in my classroom.

There is one fundamental reason that I have chosen to look at this method of learning, it is the critical situation in American education when it comes to mathematics. Academic standards and performance of US students in mathematics are low when compared with performances of students from other comparably advanced countries. Yet, the United States spends more on education than Japan and Germany. It is time to stop making excuses about how the other countries "this and that.." and time to start looking to solve the problem. I can only start in my classroom, but it is a start. I wanted to learn more about cooperative learning because I believe it holds the key to solving this problem, yet I do not know enough about it to implement it properly. My readings have helped me gain insight into the problem and how to go about incorporating cooperative learning in my mathematics classroom.

A major weakness of higher education in mathematics appears to be a dependence on traditional teaching methods, especially the lecture, which emphasizes the "sage on the stage" as a central source of knowledge. There has been little emphasis on motivating students to become interested in mathematics and technology, but rather, a great deal of emphasis on memorizing specific subject matter apropos to one field. It appears that the lecture, the traditional pedagogical approach to learning, rather than subject matter content, has had a significant dampening effect on student interest in mathematics.

Hand in hand with this goes the pursuit of technology. Reliance is heavy on the textbook, the use of technology being just a tutorial tool instead of exploring technology as a mechanism to further the educational method of cooperative learning. There is a lot of tension in relation to the use of multimedia and mathematics. An assumption is the limitless power of the technology seems to underpin such views as does an absolutist philosophy of education, based on a transference model of the learning of facts and skills. A result of such a perspective is that "I can be replaced by the machine". The use of technology to support cooperative learning and to support appropriate student interactions is necessary.

The more students invest in their own learning process, the more they will learn. The NCTM’s Curriculum and Evaluation Standards for Mathematics has called for changes in the teaching and learning of mathematics. "Mathematics teachers must involve students in their own learning." Math fear, math anxiety, apathy are predictable outcomes when students are not given greater responsibility for their own learning.

Vygotsky’s theory supports the emphasis on social interaction as a vehicle for learning. According to his work, students are able to solve problems cooperatively before they able to solve them on their own. Social interactions involved in group problem solving enable each student to extend his or her zone of proximal development. The opportunity to work with experts is one of the factors that contributes to the ability to solve problems. In a small group, the more expert members of the group often lead the novice members in the direction of gaining expertise.

The emphasis of my research brought up an underlining theory of how mathematics is to be learned. Basically it is a pedagogical approach based on a constructivist theoretical perspective .

According to this theory, students need to construct their own understanding of each mathematical concept. Hence, the primary role of teaching is not to lecture, explain, or otherwise attempt to "transfer" mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions. A critical aspect of the approach is a breakdown of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept.

Guiding Principles

• Research into how students learn is primary.
• Conceptual understanding is the most important, but calculations play a major role.
• Technology can be valuable, and some ways of using it can be more valuable than others.
• Cooperative learning is the right context for a mathematics course.
• Lecturing should be replaced by a task-oriented interactive classroom.
• Textbooks and course structure must support the pedagogical strategy.

Now that I have decided cooperative learning is thinkable in mathematics instruction, I need to find out how it can be properly implemented. Before my readings I came up with questions that I wanted to be answered by the research. So, I decided to break down my research into parts that help me identify what I wanted to obtain from my research.

Cooperative learning is an instructional strategy in which students work together in structured small groups. Myth: Just because students are working in groups doesn’t mean you are having cooperative learning. Group work is carefully designed to promote positive interdependence among students and individual accountability for learning and for contributions to the group task. Cooperative learning is a name used to describe a large group of instructional strategies in which students collaborate in order to reach a common goal.

The essential elements of cooperative learning are:(1) positive interdependence; (2) face-to-face interaction; (3) individual accountability; (4) social skills; and (5) group processing. Positive interdependence indicates that group members can reach their goals only by working together. Group goals are the cornerstone of cooperative learning. and can be accomplished by having one product as the result of group work, or teachers may have students collectively find a solution to a problem. Another option is to have students research and seek consensus on a controversial topic. Giving a group grade or making certain that the group attains a minimum score also allows for mutual reliance. Positive interdependence can also be promoted through group rewards, assigning specific group roles, or sharing materials and resources.

The interdependence of group members must occur in a setting where students can discuss what they are learning and provide each other assistance and encouragement while students are working on the common goal. This doesn't mean that each time students are having a discussion that they are involved in a cooperative activity, but rather they cannot be collaborating without face-to-face interaction. This face-to-face interaction permits true teamwork--the give and take of a number of minds attempting to complete a task.

Students must be held individually accountable for their share of the work in a cooperative task. The task must be completed and the material learned. Individual accountability might be promoted by having students take separate tests. If all students in the group reach the criterion grade, a bonus to all group members can be provided. Students might also collaborate and produce their own work for an individual grade rather than produce a product for a group grade. Group grades can also be individualized by giving students a certain percentage for group work and a certain percentage for individual work.

Although other group members should share the responsibility for helping students reach the goals, we have all had experiences when someone in the group did not adhere to this principle. When this occurs, teachers determine the cause of the problem. Is it a lack of skill or a lack of will that keeps the student from participating and sharing the burden? Both of these problems can be addressed in the classroom. If students are deficient in an ability area that is necessary for the group task to be completed, teachers must help the student gain the missing skill. If a lack of will is involved, teachers must try to raise the level of concern that the student has for the task. The student must be able t o understand the importance and relevance of the task, why you are asking for the task to be completed, and why it is important to do the task in a group setting. It may also be possible to motivate the student by using positive, caring messages, by increasing the interest level of the tasks, by making certain there are many opportunities for success embedded in the cooperative task, and by providing students with quick, clear feedback so that they know what needs to be done and how they well they are carrying out their responsibilities.

Cooperative learning also requires good social skills. Cooperative skills will benefit students throughout their life and vary according to developmental and cultural norms. We all need to learn these skills; we are not born with them.

Cooperative learning has been used successfully at all educational levels, from preschool to postsecondary, and across a broad range of subjects. Research conducted to date shows positive effects of cooperative learning on student achievement, interpersonal and sociolinguistic skills, student attitudes about learning, student affect, and self-concept. Research shows that cooperative learning has been effective in meeting needs of culturally diverse learners, students with disabilities in the general education classroom, and regular education students as well. Studies show that students make progress in terms of academics, social skills, and acceptance of diversity. There is a corresponding improvement in the climate of the room as well.

Teachers must become a facilitator of learning and must make at least seven instructional decisions prior to implementing cooperative learning groups.

1. What are the content, objectives, and mastery criteria of the lesson?

2. What will individuals/groups be accountable for?

3. Will students contribute one at a time or will partners work at the same time?

4. Is the activity designed to form consensus, generate new ideas, or share information?

5. What will the structure of the activity be?

6. What will the size of the group be and how will you assign students?

7. What roles will students play?

Before planning a cooperative learning activity, teachers must model and teach critical social and group process skills to students. Without these skills, cooperative learning ventures will fail. These skills can be taught through direct instruction, modeling, simulations and exercises.

Cooperative skills are a part of everyday classroom life and lead to social and academic success for all students. These skills should be consistently practiced and reinforced as students and teachers build a classroom community respectful of all learners. Both cognitive and interpersonal skills are critical for cooperative learning and the teacher needs to concentrate on these skills.

facilitating interaction evaluating ideas

respecting others analyzing

accepting differences justifying opinions

using humor appropriately summarizing

participating enthusiastically comparing and contrasting

disagreeing politely probing for deeper meaning

keeping self-control generating alternatives

describing feelings elaborating

caring for others applying

showing appreciation integrating ideas

including everyone checking for understanding

taking turns expanding others' ideas

sharing space and materials clarifying ideas

encouraging others verifying answers

avoiding put downs asking for clarification

praising estimating

quiet tones sequencing

being polite categorizing

using names describing concepts

- asking questions

- contributing ideas

- brainstorming

- recalling information

- staying on task

Group processing skills are also important to collaborative efforts. Students should be able to reflect on group goals and how, as a group, they are progressing towards these goals. Teachers should also provide feedback on how the groups are working. Groups and teachers may target specific goals for examination (e.g. including everyone in the discussion or sharing materials) and develop feedback/observation sheets to check on progress towards using the behavior consistently.

In cooperative learning students take active roles in their learning. Some of the suggested roles for students I collated are as follows:

1. Quiet Captains who remind group members to keep down noise levels.

2. Timekeepers who keep the team on schedule.

3. Cheerleaders who give encouragement.

4. Equalizers who attempt to equalize participation of members.

5. Recorders who write down significant information.

6. Presenters who report to the rest of the class.

7. Coaches who help peers to master the material.

8. Task Masters who keep the group on task.

9. Reflectors who summarize contributions and make positive group comments.

10. Question Commanders who attempt to answer group questions before going to the teacher.

All the research shows that students should be able to move into groups with a minimum of noise and complaining. They should be able to stay with the group and work quietly so that others can continue to work. These are the exact problems I encountered when I tried to start groups at the beginning of the year. These initial management skills also include group social skills such as encouraging everyone to participate, calling other group members by name, looking at the speaker, and eliminating put-downs. These are basic cooperative skills, and as teachers we cannot assume that children know and will be able to use these behaviors. Even adults may have to be taught how to work productively in groups. Although the content that we teach our children is critical, so are the skills that allow us to work cooperatively. It is this emphasis on good communication skills that distinguishes cooperative learning from group work. The student must take an active role in communication skills, group management skills, conflict resolution skills, and leadership skills.

Students need to develop their Communication Skills, some of which are:

complementing others, active listening, encouraging others, providing descriptive feedback, contributing ideas, encourage others to contribute their ideas, praising others, criticizing constructively, and paraphrasing and summarizing

Group Management Skills: staying with the group, encouraging equal participation, taking turns, time management skills, sharing materials, group observation and feedback skills, maintaining an appropriate noise level, and or staying on task.

Conflict Resolution Skills: stating a position, paraphrasing another's position, offering possible solutions, active listening, synthesizing diverse perspectives, and confirming others' personal competence.

Leadership Skills: explaining what needs to be done, keeping group members on task, making sure the materials are available, being a liaison with other, groups or the teacher, providing direction to the group, acquiring needed resources for the group, and being a liaison with other groups or the teacher.

Students also need the opportunity to be taught how to work in groups and how to be a productive group member. Group members should work on keeping the group on task and supporting individuals. This can be done by offering suggestions as to how tasks can be effectively completed. Group roles can be assigned during cooperative tasks which allow students to practice behaviors that will both benefit the group and allow for deeper understanding and retention of content.(long term memory). Assigning roles that both help to manage group behavior and include some important cognitive skills include:

recorder: documents the work of the group; takes notes, writes answers

reader: reads the written material, reads the answer for the group

summarizer: recapitulates what has been decided in the group, summarizes the ideas shared

courier: brings materials to the group, carries messages or assignments to their destinations, performs errands

checker: makes certain that everyone is on task, agrees with the answer, understands the assignment, discussion, or answer

interrogator: challenges group members to defend their answers and to avoid superficial responses or to explore a matter more deeply

manager: assures that directions are followed, organizes the group processes, makes sure the group is ready to report its answer or to turn in its work

time keeper: watches the time and keeps the group on task and moving forward.

voice control technician: monitors the noise level in the group and indicates when the students need to quiet down

equalizer: makes sure that all group members are treated fairly and courteously, that they have opportunities to participate and derive benefit from the groupwork.

The group roles are designed for cognitive purposes and to prepare students to become actively engaged in academic variance and discussion. The underlying purpose of these is not to help manage groups, but to help students improve cognitive abilities. These are designed for students who have already mastered the basic group process skills needed for cooperative e learning. By assigning roles the balance of the work is equally distributed.

A summarizer attempts to summarize material that has just been read or discussed. The corrector attempts accuracy by correcting a summary and/or adding important information. A memory helper can develop mnemonics and other memory aids and share them with the group so that basic information is easily accessible. The checker asks other students to explain their reasoning so that all ideas are open to correction and discussion and can be built upon by the entire group. As these roles are played out, students must also learn how to criticize ideas and not people, ask probing questions, and be able to extend a response by adding additional information. These behaviors are essential to successful cooperative learning and must be modeled and discussed in classrooms. We are not born with cooperative skills, and we live in a competitive world so time needs to be taken to nurture a cooperative environment.

Some concrete ways of setting up the physical structure of the group are:

1. Student Teams-Achievement Divisions require students to complete a common worksheet in groups of four or five but to take individual tests. The team's score is the result of individual students' improvement over past performance.

2. Think-Pair-Share involves three steps: Students first attempt to answer a question for themselves, then discuss their thoughts with partners, and finally share the combined effort with a small group or class.

3. Jigsaw uses teams of five or six. Each group member is given a piece of information and asked to teach it to the others. In Jigsaw 2, students obtain their own information to share. Students are then tested individually.

4. Team Accelerated Instruction combines individualized instruction and cooperative learning. Students are assigned materials at their levels and are assisted by peers. Group points are obtained through high achievement or improvement on individual tests.

5. Group Investigation is a higher level process in which students accept greater responsibility for their own learning. Small groups decide what to investigate, what contributions each member will make, and how to communicate what they have learned.

The greater the diversity of the group is, the more time may be needed to be spent on building a team atmosphere. One way to do this is to have team members play a game of "fact or fiction." Members can tell truths or fibs about themselves and other members will have to guess which statements are true. Or, members can be asked to interview each other. A group profile sheet can be made up to tell who has the largest family, who likes what music, who has the most unique pet, etc. Groups may even want to develop a team name, banner, hand shake, or secret code.

How Do You Prepare And Teach Cooperative Skills to Students?

In order for your cooperative learning activities to be successful, you must prepare your students to work as team members. In order to do this, teachers should do the following things.

1. Teach students about the roles they will play.

2. Teach students specific strategies to problem solve.

3. Discuss and even role play social skills, such as giving positive feedback.

4. Stress the importance of listening and being accepting of different ideas.

5. Build teams by spirit activities, establishing trust, and teaching members how to support each other.

How can cooperative skills be taught?

Cooperative skills must be taught, and the most successful way to do so is in a cooperative context. Teachers need to model the respect that successful group members have for each other with their students. Good communication skills should be used consistently from the time children enter school. Active listening, paraphrasing, and summarizing need to be so commonplace that once the behaviors are identified and practiced, they come almost naturally to children.

One method of teaching skills is through direct instruction. Teachers can actually teach what the skill is, why it is important, how it can be identified, and provide students the opportunity to practice the skill. A chart can be helpful during this kind of instruction. Students often are not certain what we mean when we ask them to perform certain skills. A chart can help students visualize and learn the skill under discussion. This may be a particularly useful technique when the skill is designed to extend critical thinking abilities.

Modeling and reinforcement are also useful ways to teach basic cooperative skills. In this case, new behaviors must be described, labeled, and discussed in a variety of settings. Students need the opportunity to identify the new behaviors when they occur and discuss the behaviors in an objective way (behaviors should be discussed, not personalities). The chance to practice these skills in defined settings is also important. Just being able to identify good group practices isn't enough, they need to be practiced and internalized. The reinforcement of these new behaviors is particularly notable, especially if the behavior is one that is complex.

In order to help things run smoothly, or as smoothly as possible, the following tips may help the teacher experience success as they implement cooperative learning groups. Review rules, as posted in your room, prior to group work. Review and display the goals of the activity and the outcome expected. Decide on systems for distributing materials and turning in materials. Develop a hand signal to request quiet if students become too noisy. AT a conference I went to, the presenter used a small drum. When she wanted to attention of all the students , she banged once on the drum. Along with this, you could employ some type of cost system which results in the loss/addition of a privilege for the amount of time compliance takes.

It is obvious that cooperative learning requires an extra effort from traditional learning. But my research indicates that this is a worthwhile effort.

Cooperative vs. Traditional Groups

COOPERATIVE TRADITIONAL

positive interdependence no positive interdependence

individual accountability no individual accountability

cooperative skill instruction no cooperative skill instruction

concern for peer learning little concern for peer learning

heterogeneous groups homogeneous groups

teacher selected groups student selected groups

student reflection and goal setting no student reflection and goal setting

teacher observation and feedback no teacher observation and feedback

equal opportunity for success uniform standard for success

The above shows that cooperative learning is not simply putting students in a group in order to accomplish a task, but a well-planned, sophisticated instructional strategy designed to promote content learning, trust in others, critical thinking skills, friendships across gender, racial and ethnic differences, and respect for diversity.

The work of Vygotsky has been underlining all my readings. Soci-cultural factors are essential in the development of the mind. The learner develops intellectually in terms of memory, attention, thinking, perception, and consciousness which evolves from the social( interpersonal) to the individual(intrapersonal). The primary development is the social with the individual dimension being secondary.

Cooperative learning is the future direction mathematics instructors will need to incorporate in their classrooms to increase the level of success of all American students. We need to move from traditionally being involved in pencil-and-paper based activities and independent situations to learning distributed among coparticipants. Given the increase in use of technology, the challenges that are likely to be presented will involve rethinking our notions of what it means to do mathematics, what it means to be a learner of mathematics, and how we might begin to address issues of assessment when mathematics learning is distributed among the group.

Bailey, Clare and Jack Chambers. "Interactive learning and technology in the US Science and Mathematics reform movement." British Journal of Educational Technology 27(May 1996):123-133.

Clements, Douglas H. "(Mis?) Constructing Constructivism." Teacher Children Mathematics 4(Dec. 1997):198-200.

Denning, Rebecca and Philip J. Smith. "Cooperative Learning and Technology." The Journal of Computers in Mathematics and Science 16, (no. 2-3,1997):177-200.

Hudson, Brian. "Group work with Multimedia: The Role of the Computer in Mediating Mathematical Meaning-Making." The Journal of Computers in Mathematics and Science 17 (no. 2-3, 1998): 181-201.

Johnson, David W., Roger T. Johnson, and Karl A. Smith. "Maximizing Instruction through Cooperative Learning." ASEE Prism 7(Feb. 1998):24-29.

Leikin, Roza and Orit Zaslavsky. "Facilitating Student Interactions in Mathematics in a Cooperative Learning Setting." Journal for Research in Mathematics Education 28(May 1997):331-354.

Lemme, Barbara S. "Putting Mathematics into Routine Classroom Tasks: Some Ideas for Teams in Cooperative-Learning Structures." 4(Jan. 1998): 250-253.

Lindauer, Patricia and Garth Petrie. "A Review of Cooperative Learning: An Alternative to Everyday Instructional Strategies." 24(Sept. 27):183-187.

Mancini, Bob M., Richard H. Hall, Maureen A, Hall, and Beverly Stewart. "The Individual in the Dyad: A Qualitative Analysis of Scripted Coooperative Learning." Journal of Classroom Interaction 33(no. 1 Winter, 1998):14-22.

Odafe, Victor. "Students Generating Test Items: A Teaching and Assessment Strategy." Mathematics Teacher 91(March 1998):198-202.

Williams, Kimberly. "Cooperative Learning: A New Direction." Education (Chula Vista, California) 117(Fall, 1996):39-42.