Chapter 3 Scaffolding & Cooperative Learning
Both major teaching techniques topics, scaffolding and cooperative learning, have been shown to be effective for developing students' higher level thinking strategies and abilities to work independently.
Some cooperative learning techniques were presented in earlier in this text. Reciprocal teaching for reading comprehension was discussed in chapter 2 while the 6PQ method of discovery learning, pair problem solving and I DREAM of A are discussed in chapters 3, 4 and 10.
Scaffolding
Teachers can aid intellectual development in students by providing them with information and temporary support which can be gradually decreased as the students' competence increases. The goal of providing scaffolds is for students to become independent, selfregulated thinkers who are more self sufficient and less teacher dependent. Using a scaffolding approach in teaching is comparable to the scaffolding of a building which is gradually removed as its structure becomes better able to support its own weight. Scaffolds are like training wheels on a bicycle which provide temporary support while the rider learns to maintain balance. Once the bike rider is secure about maintaining balance, the training wheels are removed and the rider selfbalances. Scaffolding involves providing support (models, cues, prompts, hints, partial solutions) to students to bridge the gap between what students can do on their own and what they can do with guidance from others. Teachers use scaffolding as a strategy for shifting instruction from others' (teacher's) control to student selfregulation. The teacher's role shifts from being a model or an instructor to being a manager, who gives prompts and corrective feedback.
At the beginning, the teacher (expert) completely guides the student's activity, modeling how to perform the task. The student observes the teacher and does little independent thinking during this phase, other than reading the material and observing the expert's behavior. Once internalized, the student can copy the expert's thinking/learning strategies and apply them to his/her own academic work.
Next, the student attempts to do the task with the teacher providing supportive cuing, assistance, and additional modeling, as needed. If the student has trouble using the strategies, then sometimes the teacher has to model or demonstrate again how to think about and use them. This gives the student another opportunity to observe the thinking and behavior that is appropriate for the situation. Gradually the student plays a greater teaching role and assumes more responsibility for self instruction and for teaching peers.
Scaffolding often involves the following basic components: 1. Present the new cognitive strategies, 2. Regulate difficulty during guided practice, 3. Provide varying contexts for student practice, 4. Provide feedback, 5. Increase student responsibility, and 6. Provide independent practice (Rosenshine & Meister 1992). Eventually the student learns to do all the thinking applying the content, skills, and strategies without the teacherexpert's assistance. The teacher plays only a supportive role at this point.
Can you give your own examples of scaffolding? Compare with others and critique.
Cognitive Behavior Modification (Meichenbaum 1977) is a method of gradually changing behavior based on scaffolding instruction through five stages:
1. Cognitive Modeling: the model (teacher) performs the task while talking out loud to him/herself.
2. Overt, External Guidance: the student performs the same task under the teacher's direction.
3. Overt, Self Guidance: the student performs the task while instructing him/herself aloud.
4. Faded, Overt SelfGuidance: the student performs the task while whispering instructions to her/himself.
5. Covert Self Instruction: the student performs the task while guiding her/his performance through silent self talk, private speech, or nonverbal selfdirection.
The following example shows cognitive behavior modification for scaffolding the use of self questions while reading in order to monitor comprehension and clarify misunderstanding. The procedure starts with teacher direction and leads to student selfdirection.
1. Cognitive Modeling: The teacher reads a section of the text aloud. While reading
aloud the teacher asks and answers comprehension monitoring and clarifying selfquestions aloud. For example, the teacher says, "Does this all make sense to me? Well, some of it does and some doesn't. Maybe I should reread the parts that are unclear." Then the teacher rereads the unclear parts aloud and says, "That makes more sense now. I skipped over some key words when I read it the first time."
2. Overt, External Guidance: This time the student reads a different portion of the text aloud. The teacher says to the student "What question will you ask yourself to check up on your understanding?" Then the student asks and answers a selfquestion such as, "Is there anything in here I don't fully understand?" If the student finds there is something unclear, the teacher says, "What can you do to clarify your understanding?" The student then uses a clarification strategy, such as looking a context clues.
3. Overt, Self Guidance: The student reads another section of text aloud, asks a comprehension monitoring question aloud, and seeks clarifying information as needed. At this stage, the teacher listens actively to make sure the student asks a comprehension monitoring selfquestion and clarifies, if needed. If the student forgets to ask a comprehension monitoring question, or has trouble doing it, the teacher prompts or assists the student.
4. Faded, Overt SelfGuidance: The student repeats the procedure in step three, but this time whispers while reading aloud and self questioning. The teacher listens to the whispering and tries to tell if the student asks and answers self questions. If the teacher isn't sure because the whispering made it hard to hear what the student was saying, the teacher asks the student about it when the student has finished the section of text.
5. Covert Self Instruction. The student reads a section of text silently and silently asks and answers self questions to comprehension monitor and clarify as needed. The teacher watches the student, and when the student is finished, asks what selfquestion was asked and what, if any, clarification occurred and how. At this point the student has become selfdirected in the use of selfquestions to monitor comprehension and clarify confusion.
Cooperative Learning
Cooperative learning involves students working together towards a common goal in a teachinglearning situation. There are three basic forms of cooperative learning: tutoring (peer or crossage), in which one student teaches another; pairs, who work and learn with each other; and small groups of students teaching and learning together. Not all groupwork is cooperative learning. When using cooperative learning approaches teachers need to make sure that all students are actively involved in the process working towards a common goal.
There are good reasons for the old saying, "The best way to learn something is to teach it." Teaching requires considerable depth of knowledge, understanding, organization and memory of important concepts and skills. Cooperative learning provides situations for students to teach each other. When students explain and teach concepts to each other, retention of these concepts improves Explaining also helps students connect their prior knowledge with new information. Contrary to some current speculation, cooperative learning is more than an educational fad.
It has been successfully used in the Netherlands since the early 1970's. Now word is out internationally that cooperative learning is a powerful instructional method for developing content knowledge and higher level thinking skills across the curriculum .
Academic work is usually much more fun and exciting to students when they work together cooperatively.
The social context and active involvement make it more motivating to learn. Research has shown that cooperative learning increases confidence in students' abilities. It improves selfesteem as well as feelings of competence in specific subjects. Research has also documented the positive effects of cooperative learning on improving social relations with students of different ethnicity and cultural backgrounds (Hartman, 1996).
It is useful to have a repertoire of teaching strategies. Even the most effective instructional technique does not work in all situations. Teachers need to have multiple techniques available to allow them to be flexible and shift as the situation requires. In addition, variety is necessary to prevent boredom. There is an increasing amount of ethnic and linguistic diversity in classrooms in the U.S.A. Cooperative learning has been demonstrated to be an especially effective method of teaching in settings characterized by such diversity. Cooperative learning can be done at almost any age and often with teachers' existing instructional materials. It helps improve achievement from elementary grades through graduate school.
Teachers should carefully observe group so they can serve as effective resources and assess performance. They should evaluate at least three aspects of group performance: 1). How students approached their tasks, e.g. What strategies were considered? Which approaches were rejected? Why? Were there any careless mistakes? Did students check and evaluate their work? 2). the ideas students generated, and 3). How well did the groups function as cooperative learners? This is called "group processing" and it is discussed later in this chapter.
What experiences have you had working in cooperative learning groups?
How would you assess your experiences with cooperative learning?
What is your opinion of how students working together and arriving at consensus on how to revise a paper can help students emotionally? What is your opinion of how can it help them intellectually?
Helps Emotionally
Helps Intellectually
Group Roles for Structuring Student Participation
If you are having trouble getting all students in a teaching group to take an active part, then it may be useful to assign each group member a specific assignment or role to fulfill. Jigsaw (Aronson etal. 1978) is a method of assigning each student responsibility for solving a particular part of the problem, structuring material and activities so students become experts on their parts, and teaching their parts to the rest of the group. Problems need to be carefully selected so that students can learn their own parts relatively independently.
Roles can be focused on specific subject matter e.g. a problem restater in mathematics, or can be more general and used across subjects e.g., an encourager of participation. The chart of group roles below is based on Johnson and Johnson's work on cooperative learning. These models are to stimulate your thinking about different ways to help students take an active role in their own learning. Experiment!
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Sample Group Roles, Reasons & Responsibilities
What _______ When/Why How
Facilitator To structure and maintain Orients group to
effective group functioning task, raises issues, calls on people,
keeps group on task, pushes for
decisions, initiates ideas for solution
____________________________________________________________________________
Encourager To make sure everyone Gives own ideas, asks for others'
participates and no one ideas, reacts to others' ideas, asks
dominates the discussion for reactions to others' ideas, stops
anyone from dominating _____________________________________________________________________________
Checker To make sure everyone Asks: "Does anyone
understands, agrees, and have a question or
completes work in allotted want clarification?"
time "Does everyone agree?"
Reminds group of
time and amount
of work remaining
_____________________________________________________________________________
Praiser To recognize positive Says: "That's a good
contributions from group idea". "You're doing a
members and make them good job as checker".
feel good about their
participation
_____________________________________________________________________________
Recorder/Reporter To preserve group's ideas, Writes down ideas
and present group's work contributed, writes up
to the teacher and/or work to be turned in,
the rest of the class gives oral reports
_____________________________________________________________________________
Observer To improve individual and Takes notes on how
group performance of roles group members perform
roles, gives feedback
based on observations
_____________________________________________________________________________
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Training Students in Cooperative Skills
Johnson and Johnson are well known for their contributions to cooperative learning. They emphasizes facetoface interaction as students work in heterogeneous groups with individual accountability on tasks that require positive interdependence. They believe that students need to be taught cooperative skills so that the groups function effectively. Their guidelines for training students' cooperative skills are:
1. Make sure students see the need for the skill.
2. Make sure they understand what the skill is and when it is used.
3. Provide opportunities to practice and master the skill.
4. Give students feedback on their use of the skills and adequate time for skill development.
5. Make sure students practice the skill until it is internalized.
6. Have students process (evaluate) their use of the skills.
In order for students to develop and use skills effectively, they need to know what the skills are, and when, why and how to use them. Finish the following chart and compare you ideas with others.
Strategic Knowledge about Cooperative Skills
What When/Why How
Asking for clarification When recognizing something "What does...mean?"
is unclear and it is worthwhile "Does it mean...?"
getting it clarified
____________________________________________________________________
Providing clarification
____________________________________________________________________
Paraphrasing
____________________________________________________________________
Coming to consensus
______________________________________________________________________
Providing encouragement
______________________________________________________________________
Criticizing an idea,
not the person
who presented it
____________________________________________________________________
Johnson and Johnson's work on cooperative learning has important implications for teaching, especially for tutors working with teaching groups. In both cooperative learning and teaching groups, students help each other with academic work. The tutor has to make decisions, plan instruction, monitor students' performance, evaluate students' progress and provide feedback. Johnson, Johnson & Holubec's (1986) "The Teacher's Role in Cooperation" follows for useful suggestions for managing teaching. They characterize the information below as some of the" Essential Elements of Cooperative Learning Groups".
The Teacher's Role in Cooperation
Planning I:Make Decisions
Specify Academic and Collaborative Objectives. What academic and/or collaborative skills do you want students to learn or practice in their groups? Start with something easy.
Decide on Group Size. Students often lack collaborative skills, so start with groups of two or three students; later advance cautiously to fours.
Assign Students to Groups. Heterogeneous groups are the most powerful, so mix abilities, sexes, cultural backgrounds, and task orientations. Assign students to groups randomly or select groups yourself.
Arrange the Room. The closer the students are to each other, the better they can communicate. Group members should be "knee to knee and eye to eye."
Plan Materials. Materials can send a "sink or swim together" message to students if you give only one paper to the group or give each member part of the material to learn and then teach the group.
Assign Roles. Students are more likely to work together if each one has a job which contributes to the task. You can assign work roles such as Reader, Recorder, Calculator, Checker, Reporter, and Materials Handler or skill roles such as Encourager of Participation, Praiser, and Checker for Understanding.
Planning II: Set the Lesson
Explain the Academic Task. Prepare students by teaching them any material they need to know, then make certain they clearly understand what they are to do in the groups. This might include explaining lesson objectives, defining concepts, explaining procedures, giving examples, and asking questions.
Structure Positive Interdependence. Students must feel that they need each other to complete the group's task, that they "sink or swim together." Some ways to create this are by establishing mutual goals (students must learn the material and make certain group members learn the material), joint rewards (if all group members achieve above a certain percentage on the test, each will receive bonus points), shared materials and information, and assigned roles.
Structure Individual Accountability. Each student must feel responsible for learning the material and helping the group. Some ways to ensure this feeling include frequent oral quizzing of group members picked at random, giving individual tests, having everyone in the group write (pick one paper at random to grade), or having students do work first to bring to the group.
Structure Intergroup Cooperation. Having groups check with and help other groups and giving rewards or praise when all class members do well can extend the benefits of cooperation to the whole class.
Explain the Criteria for Success. Student work should be evaluated on a criterionreferenced rather than a normreferenced basis. Make clear your criteria for evaluating the groups' work.
Specify Expected Behaviors. The more specific you are about the behaviors you want to see in the groups, the more likely students will do them. Make it clear that you expect to see everyone contributing, helping, listening with care to others, encouraging others to participate, and asking for help or clarification. Younger students may need to be told to stay with their group, take turns, share, ask group members questions, and use quiet voices.
Teach Collaborative Skills. After students are used to working in groups, pick a collaborative skill they need
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to learn, point out the need for it, define it carefully, have students give you phrases they can say when using the skill, post the phrases (praise, bonus points, stars), and observe for and encourage the use of the skill until students are doing it automatically. Then teach a second skill. Consider praising, summarizing, encouraging, checking for understanding, asking for help, or generating further answers.
Monitor and Intervene
Arrange FacetoFace Interaction. The beneficial educational outcomes of cooperative learning groups are due to the interaction patterns and verbal exchanges that take place among students. Make certain there is oral summarizing, giving and receiving explanations, and elaborating going on.
Monitor Students' Behavior. This is the fun part! While students are working, you circulate to see whether they understand the assignment and the material, give immediate feedback and reinforcement, and praise good use of group skills.
Provide Task Assistance. If students are having trouble with the task, you can clarify, reteach, or elaborate on what they need to know.
Intervene to Teach Collaborative Skills. If students are having trouble with group interactions, you can suggest more effective procedures for working together or more effective behaviors for them to engage in. You can ask students to figure out how to work more effectively together. If students are learning or practicing a skill, record on an observation sheet how often you hear that skill, then share your observations with the groups.
Evaluate and Process
Evaluate Student Learning. Assess how well students completed the task and give them feedback on how well they did.
Process Group Functioning. In order to improve, students need time and procedures for analyzing how well their group is functioning and how well they are using collaborative skills. Processing can be done by individuals, small groups, or the whole class. To start, have groups routinely list three things they did well in working together today and one thing they will do better tomorrow. Then summarize as a whole class. Group processing is discussed in the next section of this chapter.
Provide Closure. To reinforce student learning you may wish to have groups share answers or paper, summarize major points in the lesson, or review important facts.
Is there anything you don't understand in the section above? Self test your understanding by creating some of your own examples.
Group Processing
The social dynamics of cooperative learning can often be improved through systematic "group processing". Group processing refers to a group systematically reflecting on and evaluating its social interactions. Johnson and Johnson consider group processing an essential element of well functioning groups.
One example of group processing involves group members discussing their communication patterns, for example, "Did everyone in the group participate about equally? Did anyone dominate the discussion? To what extent did people patiently listen to each other without interrupting? To what extent did group members give and receive useful feedback? To what extent did group members maintain eye contact during discussions? Instead of focusing on just one area, groups can evaluate a few different issues such as communicating, sticking to assigned roles, budgeting time effectively, and staying on task.
In the approaches above issues are addressed at the group level. Another approach to group processing starts with thinking about issues on an individual level. This approach involves completing a rating scale, like the one below, which is an elaboration and adaptation of Johnson and Johnson's "Student Checklist: Cooperation". Individuals fill it out and share their responses with the group. The group discusses individual ratings within the context of the overall functioning of the group. Then the group arrives at consensus about its own overall functioning. Yes No or openended formats may be used instead of the 4 point scale. For example, YesNo: I talked or I didn't talk or Openended: To what extent did I talk?
Group Process Rating Scale
Group Behaviors very little very much
1 2 3 4
1. I talked. 1 2 3 4
2. I shared materials. 1 2 3 4
3. I gave feedback. 1 2 3 4
4. I received feedback 1 2 3 4
5. I interrupted others. 1 2 3 4
6. I was interrupted by others. 1 2 3 4
7. I asked for help when I needed it. 1 2 3 4
8. I encouraged others to participate. 1 2 3 4
9. I helped others learn. 1 2 3 4
10. Others helped me learn. 1 2 3 4
11. I maintained an open mind. 1 2 3 4
12. I tried to understand others' ideas. 1 2 3 4
13. I made sure others understood my ideas. 1 2 3 4
14. I clarified when needed. 1 2 3 4
15. I summarized our ideas/information. 1 2 3 4
16. I maintained eye contact. 1 2 3 4
17. I listened actively and carefully 1 2 3 4
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Reciprocal Education
Reciprocal teaching is already established as a powerful technique for improving reading comprehension. Students and a tutor or teacher alternate roles leading text dialogues structured around modeling the strategies of predicting, clarifying, questioning and summarizing. This teaching procedure is based on a set of instructional principles that have practically unlimited application potential. Fantuzzo , King and Heller (1992) have successfully used a related instructional method, "reciprocal peer tutoring" (RPT), for elementary students in math computation. RPT is based on cognitive theory and research showing the academic benefit of explaining material to other students. In this strategy two or more students work together cooperatively and follow a structured format in which students teach, prompt, monitor, evaluate and encourage each other. Students alternate between teacher and student roles and engage in peer teaching, peer choice of rewards, and peer management. Fantuzzo etal. emphasize that it is the combination of these components (peer teaching, peer choice of rewards, and peer management) that produces greater academic and motivational gain than using them in isolation.
Reciprocal questioning involves students taking turns asking and answering questions about the material after a lesson or presentation. Students learn to ask questions through a scaffolding procedure in which the teacher provides question stem prompts such as "What do you think would happen if...?, What is a new example of...? and What are the strengths and weaknesses of...? Eventually students can create their own questions without the teacher's stems (King, 1990).
Reciprocal tutoring is a model in which all tutors first get experiences as tutees as part of their apprenticeship for becoming a tutor. This model provides tutors with an experiential basis for tutorcentered learning (Gartner & Riessman, 1993).
There are other variations of this type of approach. The pairproblem solving method of Whimbey and Lochhead (1982) and I DREAM of A methods in this book involve reciprocal teaching types of activities. The varieties of reciprocal teaching types of procedures have a core of two common principles: students work with other students and students take on roles of both teachers and learners. Instructional models that share these basic elements can be called "reciprocal education". The term "reciprocal" is used to reflect students taking turns, especially with other students. The term "education" is chosen to represent participation in both teaching and learning activities. Reciprocal education may be adapted for use in virtually any subject area (Hartman, 1994).
How could you use reciprocal education to develop students' study skills? What do you see as the possible advantages and disadvantages of reciprocal education methods?
How might you apply the reciprocal peer tutoring approach or some other form of reciprocal education to your teaching?
Why Use Cooperative Learning ?*
*Adapted from Cooperative Learning Approaches to Mathematical Problem Solving Hope J. Hartman (1996) The Art of Problem Solving: A Resource for the Mathematics Teacher. Alfred S. Posamentier (Ed.). Corwin Press, Inc. Newbury Park, CA
There are good reasons for the old saying, "The best way to learn something is to teach it." Teaching requires considerable depth of knowledge, understanding, organization and memory of important concepts and skills. Cooperative learning provides situations in which students are required to teach each other. When students explain and teach concepts to each other, retention of these concepts improves. Explaining also helps students connect their prior knowledge with new information. There are many good reasons for using cooperative learning and at least 16 advantages to using cooperative learning in high school . Contrary to some current speculation, cooperative learning is more than an educational fad.
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As a professional educator it is useful to have a repertoire of teaching strategies. Even the most effective instructional technique does not work in all situations. Teachers need multiple techniques available to allow them to be flexible and shift as the situation requires. In addition, variety is necessary to prevent boredom. Many classes have students of widely ranging achievement levels. There is also an increasing amount of ethnic and linguistic diversity in classrooms in the U.S.A. Cooperative learning has been demonstrated to be an especially effective method of teaching in classrooms characterized by such diversity. Cooperative learning can be done at any age, often with teachers' existing curricular materials. It has been demonstrated to be successful for improving achievement from elementary grades through college.
Students can learn to solve problems and can improve their understanding of concepts without lecture or other forms of direct instruction. ^{ } In math, through studying workedout examples of factorization and learning by doing their own problems, students learn to recognize when the procedures are applicable, and to apply them. Cooperative learning is another alternative to direct instruction and lecture. In comparison to lecture and direct instruction, cooperative learning requires students to be more actively involved in the learning process. The social nature of cooperative learning, and the relative emphasis on student rather than teacher control over instruction, makes it fun and highly motivating for most students. Because the teacher is more of a manager of instruction than a transmitter of information, cooperative learning promotes student self regulation as students make, detect and correct their own and each other's errors.
The National Council of Teachers of Mathematics supports the use of cooperative learning for teaching mathematics because it enables students to discuss and learn to listen to each others' ideas, ask questions, make mistakes and offer constructive criticism. Students help each other connect new information with what they already know and discover their own meaning as they explore ideas that arise in groups.
Cooperative Learning and Achievement
By comparing problem solving processes and outcomes with other students, they can learn to differentiate between appropriate and inappropriate strategies to use with different problem types and in specific cases. They can learn what strategies not to apply. Learning this from other students rather than from the teacher can help demystify the problem solving process and reduce anxiety. Anxiety and misconceptions about the learning process can contribute to poor self concepts and can inhibit success in thinking and problem solving. Too often students feel dependent upon the teacher's expertise for learning. Problem solving is one component of higher level thinking. To think effectively, students also need to develop their metacognitive skills of planning, monitoring and evaluating so they can plan, monitor and evaluate their academic work Cooperative learning is an especially effective method of spontaneously activating metacognitive aspects of thinking, learning and problem solving.^{9}
Studies comparing cooperative learning with competitive and individualistic learning have demonstrated that cooperative learning promotes higher achievement than the other two methods. Not only do students solve problems more successfully and learn and retain concepts, but cooperative learning also results in more use of higher level thinking, more frequent discovery, generating new ideas and solution strategies, and more transfer of what is learned about problem solving in groups to individual problem solving situations. These benefits are a result of students internalizing concepts and problem solving through their discussions and explanations of problem solving strategies and approaches with their peers. Giving explanations to other students requires deeper understanding than just putting an answer on a worksheet.^{11}
There are numerous effective strategies students can learn, such as working backwards and trying extreme values, to help them become effective mathematical problem solvers. A major benefit of using cooperative learning in mathematics is that it gives students a chance to compare the strategies they use and discuss the advantages and disadvantages of using different strategies, depending upon the specific problem. When structuring groupwork around instruction in such problem solving techniques, make sure groups of students ask and answer key questions about their strategy use. These questions include: What is the strategy? Why is it a good strategy to use in this situation? How will it be applied? What other strategies could be used? What are the advantages and disadvantages of each? Through dialogue with other students about the use of problem solving strategies, misconceptions can be clarified and strategy use improved.
Students can work cooperatively on significant, interesting and complex tasks. Through working with others they can enhance their ability to communicate about mathematics, to understand it, and to think critically about it ^{12}. Discussing problems helps students become aware of what they know and what they do not know and what they understand and do not understand. This awareness leads to control over the problem solving process as students begin to see the need for specific problem solving strategies and the limitations of other approaches. When students find their approaches and outcomes for the same problem differ, this discrepancy can stimulate reorganization and development of their thinking to a new and higher level.
Scaffolding
Cooperative learning can be used as a strategy for scaffolding instruction from other (teacher) direction and control to student self regulation. Scaffolding means providing support (models, cues, prompts, hints, partial solutions) to students to bridge the gap between what students can do on their own and what students can do with guidance from others. The goal of scaffolding is for students to become independent, selfregulating problem solvers who are more selfsufficient and less teacher dependent. Scaffolding is an especially effective teaching approach for developing higher level cognitive strategies, such as those involved in problem solving. Based on his experiences with small group mathematical problem solving, Schoenfeld believes it is valuable for giving teachers a chance to give students support and assist them while they are actively engaged in the problem solving process.
Many educators believe that to learn, students must construct and reconstruct concepts, relationships and procedures in their own minds and within the contexts of meaningful situations. Teachers can start with thinkaloud modeling and direct instruction, then shift to cooperative learning formats for extensive and varied practice with feedback. Students can then guide each other to effective thinking and problem solving. As students generate and evaluate alternative approaches, they learn that there are different ways of approaching tasks and solving problems, and that some ways are better than others. Students internalize feedback and gradually learn to detect and correct their own errors, enabling them to think effectively and solve problems on their own. ( See p. 150, Vygotsky and the Zone of Proximal Development.)
Personal and Social Aspects of Cooperative Learning
Learning and problem solving are usually much more fun and exciting to students when they work together cooperatively. The social context and active involvement make it more intrinsically motivating to learn. Research has shown that cooperative learning increases confidence in students' abilities, improves self esteem as well as feelings of self efficacy in approaching . Other research has documented the effects of cooperative learning on improving interpersonal relations with students of different ethnicity and cultural backgrounds ^{14}.
Students using reciprocal peer tutoring in mathematics not only improved their achievement, but also improved in their adjustment to school. School adjustment measures included student self perception measures of: behavioral conduct, scholastic competence, social acceptance and global self worth, and teacherreported conduct.^{15}
Specific affective objectives targeted by cooperative learning include: positive attitudes,
confidence in one's thinking, willingness to take risks and try various strategies, accepting frustration and persevering when solving difficult problems, and attributing failure to not using the right strategy yet rather than to lack of competence.^{ } Studies have documented that cooperative learning in mathematics classes improves students' attitudes towards mathematics as a subject and towards math instruction. Students feel more confident in their ability to do mathematics and are therefore less anxious about it. Cooperative learning enables students to get and receive help in a relatively nonthreatening context.^{17}
How Does Cooperative Learning in Mathematical Problem Solving Work?
The following is a simple cooperative learning segment of a lesson from a senior honors mathematics class. The class was studying relatedrate word problems. The problem was:
A boat is pulled in to a dock by means of a rope with one end attached to the bow of the boat, the other end passing through a ring attached to the dock at a point 4 ft. higher than the bow of the boat. If the rope is pulled in at the rate of 2 ft./sec., how fast is the boat approaching the dock when 10 ft. of rope are out?
Students were assigned to heterogeneous groups of 34 students to work on the problem. Their task was to: a) individually generate questions for solving the problem, b) share their questions with the group, c) as a group decide on the best questions for this problem, d) individually solve the problem using the group's questions, and e) share and compare individual solutions and explain how they were obtained from applying the questions selected. Students were taught to use self questioning as a strategy for thinking through the problem solving process. Initially the students found it strange to be asked to write questions for a mathematics class. They learned how to use questioning to help them plan, monitor and evaluate problem solving. Examples of questions were modeled by the teacher thinking aloud how to use them when solving a problem. Then students generated and used their own questions. Questions generated for this problem were:
Group 1
Student 1) What should the diagram look like? Where do the values belong?
What do I want to find?
Student 2) Where do I start? How do I find the desired answer? Where do the numbers belong in the formula? Which number goes to which part?
Student 3) What does the diagram look like? What variables should I use?
Where does the 2ft./sec. go? What derivatives do I have to find?
Student 4) How do I draw a picture to represent what the problem says?
What parts of the diagram get labeled? What is the unknown?
What equation do I use to get the derivative?
This group discussed their questions and made the following list for their group to use when solving the problem:
1. What should the diagram look like?
2. How should it be labeled?
3. What do we have to find?
4. What equation do we use to find the derivative?
Group 2
Student 1) What does this look like? What formula do I use? How do I approach it? What do I want to find?
Student 2) Are they on the water? How long is the rope? Is this a controlled area with no waves and no current? How heavy is the boat?
Student 3) What do I have to find? What speed, velocity, and rate will help me solve the problem? Why can't I figure this out? How do I differentiate the problem with respect to time?
Student 4) What am I given? What must I find?
After discussing their questions, this group made the following list:
1. What is given and what must be found?
2. What variables must be considered and how?
3. How do we approach this?
Group 3
Student 1) What do I find? What does the diagram look like? How do I draw this? Is it similar to something we've done? What is the equation? Is my algebra correct? Am I using the right formula? Is the diagram correct?
Student 2) What type of diagram will this be? What are the dimensions of the diagram? What is the rate of the rope being pulled in? What is the problem looking for? What is given? What formula will be needed?
Student 3) Is the rate 2ft/sec. horizontal or vertical? Where does the equation come from? Is 10 ft. used here right now or after the derivative? How is the Pythagorean Theorem used if the hypotenuse isn't there?
This group's discussion led to the following set of questions:
1. What is given?
2. What should the diagram look like?
3. Is the diagram correct?
4. Is this similar to something we've done before?
5. What formula will be needed?
6. Is this the right formula?
7. What do we do with the speed and distance of the rope?
8. Is the algebra right?
While the groups worked on their questions and used them to solve the problem, the teacher walked around to watch and listen to each group to make sure they were on task and making reasonable progress. As she checked up on each group she saw that some students still could not solve the problem. She checked the individual and group lists of questions and realized they were incomplete, so she decided to have the groups share their questions, evaluate them as a class, and come up with a composite list. She guided the discussion to make sure the class generated questions for all three phases of the problem solving process (planning, monitoring and evaluating). The following is the composite list that emerged:
Planning:
1. Does this problem resemble a problem already done?
2. How should I diagram this problem?
3. What do I have to find?
4 . What equation must I differentiate?
Monitoring
1. Is my algebra correct?
2. Am I using the correct formula?
3. Is my diagram labeled correctly?
Evaluating:
1. Does the answer make sense?
2, Did I find what I was supposed to find?
3. How can I check my answer?
Students then returned to solving the problem with the new set of questions. Individuals within the group shared their answers with each other, decided on the correct answer, and raised their hands to let the teacher know when they were finished so she could check their solutions. She randomly asked
students to explain their solutions to make sure everyone in the group understood the problem
and solution process. Then she had students who had solved the problem help those who had difficulty. At the end of the lesson the class looked at how the questions related to each part of the problem solution. Constructing, comparing, discussing and evaluating problem solving questions, individually, in small groups and with the entire class enriched students' understanding of what questions and strategies were best suited for the particular problem. Some of the students said that in the past, they had been so concerned with getting the right answer that they had never given as much thought to the thinking process.
When each student has just his/her own knowledge, thoughts and questions, the perspective on problem solving is much more narrow and shallow. Mathematicians frequently discuss their solution strategies and outcomes with others . They know that others can sometimes detect limitations, suggest alternative approaches to and applications of problem solutions. By discussing problem solving with others, students learn to think more like mathematicians.
The Math Solution (Marilyn Burns, 1987)
The Math Solution is an elementary school inservice program for teaching mathematics as a tool for solving problems and as a way of thinking. It is based on the assumptions that for learning mathematics to occur, students need maturity, physical experience, and social interaction. Communication between students should be encouraged so that all children get experience explaining and clarifying their thinking and can move from subjective to more objective views. This program recommends teaching mathematical problem solving in heterogeneous groups of four, seated together. Three rules govern groupwork:
1. Students are responsible for their own work and behavior. 2. Students must be willing to help any group member who asks, and 3. Students may ask the teacher for help only when everyone in the group has the same question. Encouragement, practice and discussion are needed for students to learn to work together successfully.
Three stages of instruction comprise a Math Solution lesson. Lessons may vary in length, lasting from a class period to a week or longer. The stages are: introduce, explore and summarize. Whole class instruction occurs during the introduction stage, and teachers: 1. Review or present concepts that are needed, 2. Pose a similar or smaller problem or part of the problem for students to try.
3. Present the problem to be solved, 4. Discuss to make sure students understand what to do. Cooperative learning begins with the exploration stage. As students work on the problem, the teacher: 1. observes student interaction, listening to each group's strategies, procedures and ideas 2. assists only when needed (if hands are raised or the group is not working), and 3. provides an extension activity for groups that finish early. The teacher may pose several problems at once so groups can continue working at their own pace. The third stage, summarizing has three goals. First, groups share their problem solving processes, procedures and strategies. Students critically evaluate their approaches and consider alternatives for future application. Second, groups present solutions, showing their work whenever possible. During this stage it is recommended to ask, "How did you decide if your findings made sense? How can you check the solution? "^{19 } Third, generalize from the solutions. During this stage students abstract what they did in solving a particular problem to think about how they would approach related problems. Questions to ask include: "Are there patterns or relationships you can see from your solution? Does the problem remind you of another problem you have solved? How are they alike or different?"^{20}
Finding Out/Descubrimiento (DeAvila & Duncan, 1980)
This mathematics and science curriculum was developed specifically for bilingual, elementary school students. It is designed to develop thinking skills and improve the academic and linguistic performance of children in culturally and socially heterogeneous classrooms. Since 1979 research has demonstrated this curriculum to be successful in promoting student achievement in mathematics and science. The key features of the approach are: differentiated tasks, delegation of authority, student interaction, and the treatment of status.
Proponents of the method argue that "the development of thinking skills requires increased amounts of taskrelated interaction, through which students have the opportunity to develop problem solving strategies" ^{22}. Students work together, continually communicating about their handson work at learning centers where they learn to use problem solving strategies and explore the world around them. Complex, multiple ability tasks are carefully selected for cooperative, discoveryoriented learning. Two instructional features are built into this model to address the inequalities of heterogeneous groups: Multiple Ability Treatment and Assigning Competence. The former involves a direct statement by the teacher that many abilities are needed to perform each task. No one is good at absolutely all of them, and everyone is good at some of them. The teacher emphasizes this point repeatedly and points out specific skills and abilities needed for various tasks. Teachers assign competence to lowstatus students to help equalize their interaction because low status students are often perceived as unable or unwilling to help others and are generally less influential. Observing and recognizing low status students being competent is the essence of this technique. The teacher looks for low status students making valuable contributions to the group and then publically identifies the particular skill reflected by the contribution. For example, a low achieving student may be excellent at cutting out the shapes needed for a group task, which reflects good visual, spatial and motor skills. This person gains status as the group's official cutter. As a result of public recognition, low status students often raise their expectations for their own performance.
Real Maths/Maths for All (The Netherlands  National Curriculum Development Foundation)
This curriculum for students ages 1216 is based on the theory of Van Hiele, an internationally known Dutch mathematics teacher and researcher. Van Hiele's level theory identifies three levels:
Zero or perceptive level: students look at the whole problem without analyzing the parts.
First or descriptive level: students describe the parts and their characteristics, have an intuitive understanding, but there is no reflection on fundamental ideas,
Second or theoretical level: students' intuitive concepts are formulated more explicitly and students reflect on concepts and on the relationships between problem parts and the whole.
Van Hiele recommends a five stage teachinglearning process:
1. Information: students get materials (e.g. objects, graphs, papers...) to use in exercises.
2. Structured Orientation: Students are assigned specific tasks. Each task is designed to teach students one characteristic of the material they are using.
3. Expliciting: Students describe the characteristic verbally.
4. Free Orientation: Students are given general tasks that require them to find their own way in a network of relations.
5. Integration: Students reflect on different solutions, explore relationships between them and formulate laws of a new and higher level structure.
The Dutch were pioneers in using cooperative learning in mathematics. Reallife settings are essential features of all mathematics problems in this approach. The lesson design model for this curriculum consists of three stages:
1. Introduction. Working with the whole class, the teacher introduces the problem, explores aspects of it, and may give hints about the solution and place it in an everyday life context.
2. Group Work . Students work in groups as the teacher observes and manages their cooperative problem solving. When necessary, the teacher deals with individual problems.
3. Reflection and Evaluation. Students discuss several topics related their group process and results. The discussion involves: identifying of all the different solutions and strategies groups used to solve the problem ; teacher questioning to explore other possible solution strategies; reformulating and summarizing solutions; and generalizing about solutions.
Sample lessons involve using newspaper ads, deciding which video shop to become a member of, and selling badges as a small business enterprise. The Real Maths curriculum has accompanying teachers manuals as resources for using real life situations for mathematical problems and for teaching with heterogeneous, small groups. It also has student materials and videotapes.
Team Accelerated Instruction ^{24}
Team Accelerated Instruction, (TAI), formerly called Team Assisted Individualization, involves specially developed curricular materials for cooperative learning in mathematics. The major components are: Teams, Placement Test, Curriculum Materials, Team Study, Team Scores and Recognition, Teaching Groups, Fact Tests, and WholeClass Units. TAI has been used at elementary, middle and high school levels. This approach combines individualized instruction with cooperative learning, and individual accountability with group rewards. Slavin suggests that mathematics, in particular, requires individualization because students often show a wide range of individual differences in ability or achievement because learning in mathematics is so dependent on prerequisite knowledge and skills. TAI was developed to address the high level of student heterogeneity in mathematics classes. A middle school teacher made the following comments about using TAI, " The teacher has flexibility to vary from group to group or individual to individual. TAI lets students process a lot of the paperwork that ties a teacher down. Students check each other's work as they progress through the units. This is important because it provides immediate feedback students need and it identifies problems that often can be handled in the group or answered by the teacher if further help is needed." ^{25} A fifth grade teacher reported that individual accountability and group rewards are especially effective for students who dislike mathematics. TAI enables students to work at their own level and achieve success, thereby eliminating frustration and boredom. Student success is translated into team points, which motivates students to do well. The students gets self satisfaction; the group gets a reward and positive reinforcement from the teacher. A high school special education teacher who used TAI noted that working toward a team score led students to encourage each other to complete class assignments and resulted in a faster rate of mastery than did individualized mathematics instruction. Mastery of 80% is required to move to the next level, so the quality of work remains high.
Teaching and Pair Methods
Reciprocal Peer Tutoring
Reciprocal peer tutoring (RPT) is for elementary students learning mathematical computation. In this approach two or more students work together cooperatively and follow a structured format in which they teach, prompt, monitor and evaluate each other. Students alternate between teacher and student roles and engage in peer teaching, peer choice of rewards, and peer management. Variations of RPT have been used successfully in urban, low income public schools with minority and white elementary students. Students are trained to use the procedures in two or three sessions of 45 minutes each. Modeling the procedures and using instructional prompts enables students to use these procedures without assistance. Training includes discussing with students the value of working with peers, teamwork, partnership and cooperation. A unique aspect of this approach is the emphasis on student involvement in the reward structure. The use of group rewards has been found to improve conduct in the classroom. It is the combination of these components (peer teaching, peer choice of rewards, and peer management) that produces greater academic and motivational gain than using them in isolation.
What is the Teacher's Role in Cooperative Mathematics Problem Solving Lessons?
One of the key features about cooperative learning is that the teacher delegates considerable responsibility to the students. Both teachers and students must adopt roles that are quite different from whole class instruction. "The essence of good problem solving is selfcorrection "... "teachers should become facilitators of learning, not sole dispensers of truth."... "Teachers must relinquish the safe seat of authority and step into the classroom"...^{35}.
An important task in making cooperative learning successful is attitude change. The teacher's own 53
attitude has to shift from teacher as transmitter of knowledge and center of attention and authority to teacher as manager and facilitator of learning. Attitudes towards noise in the classroom sometimes must be changed. Many people equate a "noisy classroom" with disruption, chaos and learning not taking place. In cooperative learning, noise in the classroom can reflect high level thinking and learning about mathematics!
Both teachers' and students' attitudes about learning and how it occurs must change for cooperative learning to be successfu. Teachers attitudes that require change were just described. Several student attitudes must change. Student attitude change is needed so students show interest in finding solutions, confidence to take risks and try various strategies, willingness to be wrong, accepting frustrations from not knowing, persevering when solutions are not immediate, and understanding the difference between not having found an answer yet and not knowing it. Students must shift from depending on teachers for the answers to becoming independent thinkers and learners. Students need to become responsible for their own and for each other' learning. Teachers as role models must emphasize to students the importance of being good thinkers, learners and problem solvers. Students must become active learners and seekers who are willing to take risks and make errors. Mistakes should be treated as learning opportunities and student must understand that important aspects of problem solving lie beyond the correct answer. Teachers and students alike must learn to value the learning process  not just its products, such as grades and test scores.
A "Pretty Good List" of nine steps for teachers to follow in implementing cooperative learning is: 1. Ensure a successful experience the first time; 2. Decide what to watch for, 3. Decide on a grouping strategy, 4. Prepare the materials, 5. Prepare yourself, 6. Explain the rules and expected behaviors, 7. Do it!, 8. Debrief the class and 9. Debrief yourself. ^{37}
Planning for Cooperative Learning
Training Students for Cooperation
Although many everyday life activities require cooperation, seldom are students taught how to work together cooperatively. For cooperative learning to work effectively, some educators recommend giving specific training in skills needed for cooperative learning. These skills include: careful observations, reasoning, asking key questions, being supportive and helpful to others, explaining clearly, thinking visually, reasoning spatially, recording data, exploring new solution strategies, understanding the problem, being persistent and using ideas of other students.^{38 }
Teacher modeling of social skills and role playing are effective strategies for training students for cooperative learning. Johnson and Johnson's six guidelines for training students for cooperation were described on p. 41. They identify four categories of skills to be developed for cooperative learning: forming skills, functioning skills, formulating skills and fermenting skills.
Forming Skills :
* moving without noise
* staying within the group
* using quiet voices
* encouraging participation by all
Functioning Skills:
*using names
* looking at the speaker
* being respectful of others
* directing the group's work
* expressing support
* asking for help or clarification
* offering to explain or clarify
* paraphrasing other people's work
* energizing the group
* describing feelings when appropriate
Formulating Skills :
* summarizing out loud
* seeking accuracy by correcting and/or adding to summaries
* seeking elaboration
* seeking clever ways of remembering information
* demanding vocalization
*asking other members to plan out loud
Fermenting Skills:
*criticizing ideas not people
* integrating ideas
* asking for justification
* extending other students' answers
* asking indepth questions
* generating further answers
*checking the group's work
Group processing is recommended as the final stage of a cooperative learning group activity. Group processing refers to students reflecting on, analyzing and evaluating how the group functioned. This can be done in at least two different ways. The group as a whole can evaluate its overall performance by systematically discussing issues such as: Did everyone participate? Did anyone dominate? To what extent did group members stick to their assigned roles? To what extent did students listen to each other carefully without interrupting each other? To what extent were students courteous and respectful of each other? How well did the group use its time? To what extent did group members maintain eye contact while communicating? What were the group's greatest strengths? How could the group work better together next time? Another way group processing can be conducted is to have a group discussion about the issues just described after students have individually assessed their own performance in the group. Once each group has evaluated its performance, the teacher can ask groups to share their results, list each group's strengths and weaknesses on the blackboard, and as a class discuss whether and what additional training in cooperative learning may be appropriate.
Structuring Groups
Most proponents of cooperative learning in mathematics recommend student pairs or heterogeneous groups of 35 students but there is not total consensus on group size or structure. Some teachers begin with students working in pairs and gradually shift to 3, then 4, and finally 5 students working together. Many people recommend groups of four. Some of the advantages of groups of four are: they are large enough for generating ideas and discussing solutions of challenging problems; they are small enough for all students to participate; they can be conducted without a leader; and they can be split up into pairs for occasional practice.^{ 40} Groups may stay together for a single class period, a week, school term or the entire school year. Many proponents of cooperative learning recommend changing groups often enough that eventually all students in the class have the opportunity to work together. Some teachers allow students to choose their own groups; many assign students to groups to ensure heterogeneity. Assignment can be done on the basis of students' achievement records, randomly, or their regular seats in the class. Research supports the value of gender, racial and ethnic diversity in cooperative learning groups.
Deciding how to arrange groups in the classroom is another decision teachers have to make. Groups can be structured into small circles, squares/rectangles or triangles to facilitate communication that will not disturb other groups. Moveable desks or, tables and chairs are important but not necessary. Pairs can easily be used even in large lecture halls with no moveable furniture or room to sit on the floor.
There are numerous ways to use cooperative learning for teaching mathematical problem solving. Some methods are relatively informal in structure but involve students working together cooperatively to achieve a problem solving goal. Other methods are highly structured in terms of student roles, instructional techniques, materials and assessment strategies. Wellstructured cooperative learning groups can increase the chances that all students will examine alternative solution strategies, observe peers engaged in problem solving, and will formulate, analyze, and interpret problems and solutions. Positive interdependence and face to face interaction are guiding features behind some approaches to cooperative learning in mathematics. These are facilitated through assigning distinct roles to perform within groups. ^{42}
Assigning Student Roles (See p. 40 for examples)
Teachers sometimes have each group member fulfill a distinct role to ensure all students participate in the process and to achieve a variety of goals. However, specific roles are not necessary. For example, the "SmallGroup Discovery Method" of using cooperative learning in mathematics does not involve student roles.^{43} Some teachers prefer to assign students to their roles, other teachers prefer for students to choose their own roles in the groups. Students should understand that their participation in the group is not limited to the roles they perform. Their roles are just one aspect of their participation in the group's problem solving activities. Students need to see and be part of the "big picture" of problem solving. Many teachers have students change roles after one class period or after one group project. Regardless of which method is used, teachers should make sure that eventually all students get a chance to function in each of the group roles. There are several models of roles that can be use for problem solving groups.
Model 1
* Problem Restater: paraphrases the problem and says what information is given and what must be found,
* Elaborator: asks group members whether the problem is similar to others they have solved before,
* Strategy Suggester/Seeker: makes suggestions about possible strategies that could be used to solve the problem, and/or asks groupmates for alternative strategies,
* Approximator: gets the group to estimate what the answer will be before they begin actually solving the problem,
*Reviewer/Mistake Manager: has the group figure out how they can learn from whatever mistakes may have been made and when the group is successful, has the group determine how their solution could be even better next time
*Confidence Builder: ^{ } encourages the group to keep going because they will succeed if they persist and work together effectively.^{ 44}
Model 2
* Initiator: gets the group started and keeps them on task,
*Idea Person: gives mathematical ideas to the group, e.g. how to solve the problem
*Challenger or Rebel: does not passively accept the approaches selected and the answers, but questions whether they are correct,
* Synthesizer: resolves differences by reconciling opposing views, the peacemaker ,
*Ego Builder^{ }: builds pride in the group by praising its members.
Model 3
* Jigsaw^{ } is a method of assigning each student responsibility for solving a particular part of the problem, structuring material and activities so students become experts on their parts, and teaching their parts to the rest of the group. Problems need to be carefully selected so that students can solve their parts relatively independently.
Model 4
* Facilitator : leads the group in the problem solving process ,
* Comprehension Monitor/ Clarifier: checks up on group members' understanding of the problem and how they are solving it; clarifies and/or asks for clarification as needed,
*Checker for Accuracy and Direction: checks for mistakes, makes sure the approach they are using is leading in the right direction,
* Encourager: praises group members and invites their contributions,
*Summarizer:, reviews progress the group has made and what still needs to be done,
* Recorder/Reporter: writes down the group's strategies, calculations, and answers, and presents this information to the class.
Model 5
* Accountant : makes sure all group members perform mathematical operations, *Architect: ensures all group members contribute to the overall group product, and *Elaborator: relates current work to prior mathematical problems. ^{47}
There at least two different types of group roles that can be assigned: management and instructional roles. Examples of management roles include facilitator and recorder; examples of instructional roles include chief investigator and advisor. Some controversy exists about the value and effectiveness of different types of roles. Management roles have been found to be especially important for younger children because they help prevent confusion, wasted time and conflicts, while instructional roles sometimes create problems for both older and younger children.^{48}
Selecting or Preparing Materials
Although some teachers find they can adapt their existing mathematics curriculum for cooperative learning, others find that more appropriate materials are needed. There are several resources specially developed for implementing cooperative learning in mathematics. One especially good resource is Get It Together: Mathematics Problems for Groups Grades 412^{ 49}, developed by Project Equals at Lawrence Hall of Science, University of California  Berkeley. The National Council of Teachers of Mathematics handbook on cooperative learning in mathematics is a clear and concise guide with carefully constructed, rich problems, clear instructional objectives, problem sheets, activities and teaching notes.^{50} Others include the Real Maths ^{51} curricular materials in the Netherlands, the Finding Out/ Descubrimeinto ^{ }curriculum from the University of California  Stanford and Focus on Developmental Mathematics.^{53 } The final chapter of the outstanding book by Davidson, Cooperative Learning in Mathematics,^{ 54 } identifies additional resource materials. In general, problems should be rich, meaningful to students, and often have more than one solution. Designing Groupwork contains excellent descriptions of several cooperative learning methods and especially useful materials and procedures for training students for cooperation. ^{ }
Monitoring and Evaluating Groupwork
As a manager of learning, teachers using cooperative learning must monitor groups as they are problem solving and evaluate their performance. Monitoring involves checking up on students' performance of the assigned task. While cooperative learning is in progress, the teacher circulates from group to group observing and listening to students, and interpreting the results. It is best for the teacher to think about what will be looked for in the groups before the lesson begins. Johnson and Johnson^{ }describe monitoring as "the fun part" of cooperative learning for the teacher. Students can become enthusiastic problem solvers, and this is exiting to observe in progress. Behaviors to check up on include: seeing if students are on task, if they understand the assignment, if they are really working together cooperatively, if they are functioning in any roles that may have been assigned, and if there are any problems that should be dealt with. Intervention in groupwork by the teacher should be avoided unless absolutely necessary. If intervention is called for, it should come in the form of questions that guide students to resolve their own problems or that steer them in the right direction. Teachers should resist the temptation to tell students answers or how to solve problems. During this
phase the teacher can provide feedback and praise students for good performance. While observing groups the teacher may discover that students need additional information to help them solve a problem or additional training in working together cooperatively. Therefore, part of the monitoring process is to evaluate groupwork and to plan for followup instruction.
Teachers need to evaluate at least three aspects of group performance: 1). how students solved their problems, e.g. What strategies were considered? Which approaches were rejected? Why? Were there any careless mistakes? Did students verify their solutions? 2). the solutions students generated, and 3). groups processes, i.e. how the groups functioned as cooperative learners. Some teachers offer rewards to groups for good performance. Competitions between groups are sometimes used as an incentive. Praising groups for good performance is usually an effective motivational tool.
An important decision teachers need to make is how they will evaluate individual student achievement of the specific mathematical problem solving objectives that were targeted by the lesson. Both individual and group accountability are important to emphasize. To ensure individual accountability, teachers can give tests (inclass or takehome) , quizzes, or randomly ask group members to explain problem solutions. Additional measures include: classwork (attendance, participation, cooperation), group projects, homework, self evaluation and peer evaluation. When evaluating the group product, the teacher can give each student a grade for her/his individual contribution. Students should know the specific evaluation criteria that will be used to assess their performance.
What are Potential Pitfalls of Cooperative Learning?
Like other teaching methods, cooperative learning can be ineffective if it is not handled right. Not all groupwork is cooperative learning. Students can sit side by side in a group and do their work completely independently without cooperating. Potential problems implementing cooperative learning in high school mathematics classes may be studentoriented or teacheroriented Studentoriented problems include: a group of students may become bored with each other, there may be inadequate leadership within a group, students may feel abandoned by the teacher, difficult problems may cause feelings of defeat while easy problems may be boring, and students may need a change of pace or more praise. Teacheroriented problems include: teachers may feel uncomfortable not being the center of the classroom, they may not have explained the task adequately, and they may get mixed feedback about what students have learned^{ 58}.
Although many students prefer working cooperatively to working independently, some students would rather work alone. Such students can inhibit effective group interaction. The teacher's role as observer and supervisor is important in this type of situation. Depending upon the particular class and curriculum, teachers may decide to use cooperative learning as an option for students rather than as a requirement.
Another problem is that one or two students can do all the work solving problems while the others do not. To prevent this, individual accountability is essential. Groups must be structured to foster cooperation between students. Assigning roles, sharing materials, requiring a group product and using group incentives can be used to structure effective cooperation.
Time can be a problem when implementing cooperative learning and sometimes lessons end without summarizing what was learned and assessing the group process. One way of handling this problem is to assign roles of summarizer and leader of group processing. Otherwise the teacher can lead the summarizing and group processing at the end of each lesson (or set of lessons).
Cooperative learning can lead to incoherent presentation and interruptions while working so that students need more time reviewing and practicing. If initial training of students to work cooperatively is not adequate for some students or groups, followup training may be needed. Sometimes problems arise if teachers set only academic goals rather than specifying both academic and social goals in advance.^{59}
58
Summary and Conclusion
Cooperative learning has many advantages for teaching. It improves achievement and higher level thinking Working in pairs or small groups is highly motivating for most students and improves students' attitudes about themselves as learners and problem solvers. Improved interpersonal relations between students of different ethnic/cultural backgrounds is another benefit of cooperative learning.
There are many different approaches to using cooperative learning just for teaching mathematical problem solving. Some methods involve small groups of 35 students learning together; others involve pair of students working together. Small group methods include: The Math Solution, Finding Out/Discubrimiento, Real Maths, and Team Accelerated Instruction. Teaching/Pair Learning methods include: Reciprocal Peer Tutoring, the 6PQ Method of Discovery Learning, Pair Problem Solving, and I DREAM of A. The Reciprocal Teaching method of using cooperative learning to improve reading comprehension was described in Chapter 2.
The teacher's role in cooperative learning is different from whole class instruction. In cooperative learning, the teacher is more of a manager and facilitator of learning, or a coach, than a transmitter of knowledge. Major teacher responsibilities include: training students for cooperation, structuring groups, deciding whether/how to assign roles, selecting and/preparing instructional materials (planning) and monitoring and evaluating student performance. Teachers can develop personal action plans to design cooperative learning lessons that meet the needs of their specific students and curriculum. Resources are available for cooperative learning lessons in many subjects.
Cooperative learning is not just a fad. It has over a twenty year history of success as a technique for developing students content knowledge, thinking and problem solving skills. There are some potential pitfalls to using cooperative learning for teaching problem solving. Some problems are studentoriented, others are teacher oriented. The benefits of using cooperative learning as a technique for teaching have been well documented. The advantages far outweigh the disadvantages. There is not a "right" way of using cooperative learning; there are numerous options. Cooperative learning can make teaching and learning more lively and fun for both teachers and students.
Group Investigation
Sharan and Sharan's Group Investigation method has six stages and involves conducting research through cooperative learning in small groups and cooperatively making a presentation about that research to the class. One advantage of this method is that it gives students choices about what they will investigate and how. It can be implemented in as short as onetotwo weeks or as long as several months, depending upon the particular research, students and teacher. Research on the effectiveness of Group Investigation has been conducted with elementary and secondary school students in the Middle East and Western countries. The results generally show Group Investigation to result in higher academic achievement and better performance on questions evaluating highlevel when compared to whole class instruction. Group Investigation helps develop positive social interactions between ethnically diverse students and promotes helping and cooperation whereas whole class instruction tends to foster competition between students. Research on teacher reactions to Group Investigation indicates that after using this method teachers had better attitudes about their schools work (less need to control students) and their schools (more positive environment). Comparing the nature of teaching under Group Investigation and whole class instruction, research shows this method of cooperative learning promotes students' communication, support, feedback, and intimacy whereas whole class instruction tends to be characterized by long lectures, questions requiring short answers, and giving students orders The six stages, guidelines and a worksheet for Group Investigation follow. (Sharan and Sharan, 1989/1990).
GROUP INVESTIGATION: OVERVIEW*
STAGE 1: IDENTIFYING THE TOPIC & ORGANIZING STUDENTS
a. Students scan resources, propose topics & categorize
suggestions.
b. Students join the group studying the topic of their
choice.
c. Group composition is based on interest & is heterogeneous
d. Teachers aids information gathering & organization.
STAGE 2: PLANNING THE LEARNING TASK
Students plan together:
a. What do we study?
b. How do we study?
c. Who does what? (division of labor)
d. Why do we want to investigate this topic? (What are our purposes/goals?)
STAGE 3: CONDUCTING THE INVESTIGATION
a. Students gather information, analyze data, & reach conclusions.
b. Each group member contributes to the group effort.
c. Students exchange, discuss, clarify, & synthesize ideas.
STAGE 4: FINAL REPORT PREPARATION
a. Group members determine the essential message of their project.
b. Group members plan what they will report and how they will make their class presentation.
c. Group representatives form a steering committee to coordinate plans for the class presentation.
STAGE 5: FINAL REPORT PRESENTATION
a. The class presentation is made in a variety of forms.
b. Part of the presentation should actively involve the audience.
c. The audience evaluates the presentation's clarity and appeal based on preestablished criteria.
STAGE 6: EVALUATION
a. Students share feedback about the topic, the work they did, and about their affective experience.
b. Teachers and pupils collaborate in evaluating student learning.
c. Evaluate higherlevel thinking.
*based on R. Slavin (1990). Cooperative Learning
60
An Example of Group Investigation Guidelines
Your Individual Investigation
This work is to be done on your own time, outside of class. Each student should prepare a 23 page report on her/his topic. Make sure your individual report is consistent with the group goals and essential message.
Your Group Report
On November 19 your group will discuss and clarify the essential message of your project and begin preparation for the class presentation (see section below).
Your Group Report Presentation
On November 19 your group will have 2030 minutes to discuss, clarify, synthesize ideas and plan its presentation. The plan should include what each group member will do and how your group will present what it learned, including how it will actively involve the audience (the rest of the class) in the presentation. On November 28 your group will meet briefly (about 20 minutes) to update and continue the prior discussion and synthesis. A representative of your group will meet with a representative of the other groups in a steering committee to coordinate the class presentation. The class presentation should involve a variety of formats (e.g. poster, boardwork, discussion, lecture, modeling etc.). The class will have approximately
an hour and fifteen minutes (75 minutes) for all group presentations. The steering committee will allocate time for each group.
On December 3 your group will make a class presentation.
The rest of the class will evaluate your group's presentation according to the criteria outlined below.
CRITERIA FOR EVALUATING FINAL REPORT PRESENTATION
The audience will evaluate your group's presentation in terms of the following criteria:
Group Project Evaluation
Excellent Good Fair Poor
1. clarity
2. brevity
3. usefulness
4. presentation form /variety
Comments:
GROUP INVESTIGATION WORKSHEET
STAGE 2: GROUP PLANNING OF THE INVESTIGATION
OUR RESEARCH TOPIC:
NAMES OF GROUP MEMBERS:
WHAT WE WANT TO INVESTIGATE:
OUR RESOURCES:
HOW WE WILL DIVIDE THE WORK:
JIGSAW
In this original version of the Jigsaw method, developed by Aronson, each student becomes an "expert" on a topic and teaches it to the rest of her/his group. All group members are responsible for being good teachers and good listeners. The group members must depend on their teammates to learn about material which they will later be quizzed on. In Slavin's adaptation, Jigsaw II, all group members read all the material.
Example: From faculty development on cooperative learning
For today's class, just read the section to which you are assigned. For homework, read the rest of the material assigned, as well as the rest of chapters 2 and 5. These page numbers refer to Slavin’s book  not our text.
Directions
1. Set up heterogeneous groups, with four people per group.
2. Each group member will assume responsibility for learning and teaching one of the following in the Slavin book:
a. Chapter 1 p. 12 & Chapter 2: Cooperative Learning and Student Achievement pp. 1317 (through "Pitfalls...")
b. Chapter 3: Cooperative Learning and Outcomes Other than achievement pp. 3446
c. Chapter 3: Cooperative Learning and Outcomes Other than Achievement pp. 4753
d. Chapter 5: TAI and CIRC pp. 7992
3. Read the section you are responsible for teaching to the rest of your group. Focus on topics identified on your "Jigsaw (Expert) Worksheet".
4. Meet in "Expert Groups" with people from different groups who have the same assignment to discuss the material and how to teach it for 2030 minutes. Appoint a discussion leader to call on people and make sure everyone participates.
5. Return to your group and teach your teammates. Make sure they take notes. Each group member has 510 minutes to teach what they have learned from their reading and their discussions in the expert groups.
6. When all group members have finished teaching, take turns quizzing your teammates on the material until you are satisfied that they all have mastered the material and are ready to take a quiz. (Optional).
7. Quiz (with equal number of items from each unit).
8. Scoring: The group score is the average of each of the team members. Each person also has an individual score. If you have prior quiz grades for students, you can use them as "base" scores, and grade students for improvement over their past quiz performance.
9. Reward high scoring groups.
Jigsaw Quiz
1. Which of the following is not true of cooperative learning methods?
a. Groups have cooperative goal structures.
b. Task specialization methods can help prevent the "free rider" effect.
c. Individual accountability is not recommended.
d. Scoring methods can help ensure that all students have an equal opportunity to contribute to their teams.
2. How do cognitive theories contribute to our understanding of cooperative learning?
a. They show the importance of peer norms which promote academic success.
b. Some developmental theories show the value of social interaction in learning
c. They show that adult guidance is the only way students can learn.
d. Developmental theories concentrate on individual restructuring of the material to be learned.
3. What does research show about the effects of cooperative learning on intergroup relations?
a. Cooperative learning does not affect students' friendships.
b. Traditional instruction is better than cooperative learning from improving intergroup relations.
c. Cooperative learning groups are best when students are from the same ethnic group.
d. Cooperative learning promotes crossracial friendships.
4. What does research show about the effects of cooperative learning on students' self perceptions?
a. Cooperative learning has a positive effect on students' self esteem.
b. Cooperative learning has no effect on students' self esteem.
c. Cooperative learning increases students' external locus of control.
d. Cooperative learning has no effect on peer proacademic norms.
5. Which of the following has not been found in cooperative learning research?
a. Many studies show it increases timeontask.
b. Cooperative learning has noncognitive benefits.
c. Cooperative learning can increase positive affect among students.
d. Research consistently shows that students prefer cooperative learning to other methods of instruction.
6. What does research suggest about the relationship between cooperative learning and altruism?
a. It clearly has no effect on altruism.
b. Competitive goal structures enhance the development of altruism.
c. Cooperative goal structures enhance altruistic behavior.
d. Cooperative goal structures inhibit altruistic behavior.
7. Which of the following is true about TAI and CIRC?
a. They are contentbased approaches to cooperative learning.
b. They emphasize competition more than cooperative learning.
c. They do not focus on particular curriculum areas.
d. They are task specialization methods of cooperative learning.
8. The TAI method was primarily developed for which purpose?
a. to integrate reading and writing instruction.
b. to adapt instruction to students' individual differences.
c. to avoid the need for specific instructional materials.
d. to improve intergroup relations.
Jigsaw Worksheet
Assignment A
1. What are the characteristics of cooperative learning methods?
2. Why does cooperative learning facilitate student achievement?
3. What are some pitfalls of cooperative learning?
Jigsaw Worksheet
Assignment B
1. How does cooperative learning affect intergroup relations?
2. What does research show about cooperative learning and mainstreaming?
3. How does cooperative learning affect students' self perceptions?
Jigsaw Worksheet
Assignment C
1. How does cooperative learning affect timeontask?
2. How does cooperative learning affect how students' schoolrelated attitudes?
3. What does research suggest about the relationship between cooperative learning and being cooperative?
Jigsaw Worksheet
Assignment D
1. What is the TAI method of cooperative learning?
2. What is the CIRC method of cooperative learning?
3. How do TAI and CIRC differ from generic forms of cooperative learning?
name______________________________ 65
Part 1: Graphic Organizer
Using your own words, briefly summarize what the most important ideas you want to remember from this chapter why they're important and how you might you use them.
WHAT is the idea? 
WHY is it useful? 
HOW might I use it? 





Part 2 Do a readerbased summary of 100 words or less on the section of this chapter on scaffolding p. 37 and ending on p, 38 at the end of the information on Cognitive Behavior Modification.