“Cooperative learning” (i.e., jigsaw, learning together, group investigation, student teams-achievement divisions, and teams-games-tournaments) is a generic term that is used to describe an instructional arrangement for teaching academic and collaborative skills to small, heterogeneous groups of students (Rich,1993; Sharan,1980). Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation, and to promote academic achievement and positive interrelationships. A benefit of cooperative learning, therefore, is to provide students with learning disabilities (LD), who have math disabilities and social interaction difficulties, an instructional arrangement that fosters the application and practice of mathematics and collaborative skills within a natural setting (i.e., group activity). Thus, cooperative learning has been used extensively to promote mathematics achievement of students both with and without LD (Slavin, Leavey, & Madden, 1984; Slavin, Madden, & Leavey,1984).

According to the National Council of Teachers of Mathematics (NCTM; 1991), learning environments should be created that promote active learning and teaching; classroom discourse; and individual, small-group, and whole-group learning. Cooperative learning is one example of an instructional arrangement that can be used to foster active student learning, which is an important dimension of mathematics learning and highly endorsed by math educators and researchers. Students can be given tasks to discuss, problem solve, and accomplish.

Cooperative learning activities can be used to supplement textbook instruction by providing students with opportunities to practice newly introduced or to review skills and concepts. Teachers can use cooperative learning activities to help students make connections between the concrete and abstract level of instruction through peer interactions and carefully designed activities.

Finally, cooperative learning can be used to promote classroom discourse and oral language development. Wiig and Semel (1984) described mathematics as “conceptually dense.” That is, students must understand the language and symbols of mathematics because contextual clues, like those found in reading, are lacking in mathematics. For example, math vocabulary (e.g., greater-than, denominator, equivalent) and mathematical symbols (e.g., =, , or ) must be understood to work problems as there are no contextual clues to aid understanding. In a cooperative learning activity, vocabulary and symbolic understanding can be facilitated with peer interactions and modeling.

Research (e.g., Johnson & Johnson, 1986) supports cooperative learning as an effective approach for including students with LD in classroom group work and promoting peer acceptance. However, some researchers (e.g., Andersen, Nelson, Fox, & Gruber, 1988; Rivera, Gillam, Goodwin, & Smith,1996; Slavin, Madden, & Leavey,1984) caution teachers to integrate direct instruction and cooperative learning. Thus, in cooperative learning groups with proper instruction and preparation (i.e., previous direct instruction on skills), students with math disabilities can benefit from peer interactions to learn mathematics skills and concepts. The purpose of this article is to discuss the components of cooperative learning and to present an example of how cooperative learning can be used to teach mathematics skills.

## Cooperative learning components

The literature (e.g., Johnson, Johnson, & Holubec,1994) is replete with descriptions of cooperative learning; therefore, only a brief overview of the components of cooperative learning are described to serve as a foundation for the remaining section of this article. Cooperative learning consists of three major components: “lesson preparation,” “lesson instruction,” and “lesson evaluation.” Each component is briefly described.

### Lesson preparation

During the “lesson preparation” component, teachers (a) select the mathematics and collaborative objectives to target for instruction and cooperative learning groups, (b) plan the math activity, (c) identify ways to promote the elements of cooperative learning, (d) identify roles, and (e) establish groups. To identify mathematics content area objectives for instruction, teachers can examine a variety of resources, such as curriculum guides, textbooks, the Curriculum and Evaluation Standards (NCTM, 1989), and students’ Individualized Education Programs (IEPs). Additionally, teachers can use assessment information obtained from clinical interviews, criterion-referenced tests, and error analyses (Enright, 1995; Rivera & Bryant, 1992) that reflects students’ knowledge of prerequisite mathematics skills and concepts (see Bryant’s article in this series).

Collaborative objectives, in turn, can be drawn from curriculum guides, IEPs, and other references (e.g., Jackson, Jackson, & Monroe,1983; McGinnis & Goldstein,1984; Walker et al.,1983). Additionally, it is recommended that student group behaviors and interactions be observed to identify those collaborative skills (e.g., listening, sharing, taking turns, asking questions, using self-control, compromising, contributing ideas) that require intervention to enable students to work successfully as a group towards task completion.

Designing math activities for cooperative learning groups requires consideration of both the instructional objectives and the purposes for having children work in a cooperative instructional arrangement. Teachers should design activities to promote math understanding by having students practice, experiment, manipulate, reason, and problem solve. Such math activities may help students make connections across math skills and concepts, and other disciplines. Kagan (1989/90) identified ways to structure group activities to foster group interactions. Table 1 presents four examples of activity structures, their definitions, and applications to math activities.

According to Johnson et al. (1994), there are five basic elements of cooperative learning: positive interdependence, face-to-face interaction, individual accountability, group behaviors, and group processing. Positive interdependence means that students see the importance of working as a team and realize that they are responsible for contributing to the group’s effort. face-to-face interaction involves students working in environmental situations that promote eye contact and social space so that students can engage in discussions. Individual accountability suggests that each person is responsible to the group and must be a contributing member- not someone who lets others do all of the work. Group behaviors refer to those interpersonal, social, collaborative skills needed to work with others successfully. Finally, group processing is a time after the cooperative learning task is finished when team members analyze their own and their group’s abilities to work collaboratively.

Structure | Definition | Math Example |
---|---|---|

1. Categorizing | 1. Students analyze and classify objects based on specific criteria. | 1 a. Categorize based on attributes. 1 b. Categorize numbers in various ways: odd, even, multiples. |

2. CO-OP | 2. Each student studies a part of a topic and then presents his or her information to group teammates. | 2 a. Study classmate preferences on certain topics and construct graphs. 2 b. Learn algebraic formulas and solve equations together. |

3. Numbered Heads | 3. After each team member numbers off, students discuss the answer to a question. Then, in a large group the teacher calls a specific number and group to answer the questions. | 3 a. Discuss the answer to a mental computation problem. 3 b. Apply the definition of a rule previously introduced to problems; explain the application of the rule. |

4. Round the Table | 4. Students work on problems jointly by passing the problems around the table for each member’s response. | 4 a. Pass a worksheet with multiplication facts for each member to answer a problem. 4 b. Pass problems for each member to compute the next step of an algorithm. |

Adapted from B. Andrini, (1991) Cooperative learning & mathematics. San Juan Capistrano, CA: Resources for Teacher, Inc. |

These five elements can be structured to promote team work and collaborative skills. They can be facilitated in various ways, for example, by (a) asking students to be responsible for certain duties (e.g., record keeper, spokesperson, encourager); (b) providing limited materials thus necessitating sharing; (c) providing bonus points for demonstrating collaborative behaviors; (d) asking students to self-evaluate after-task completion, (e) assigning a group grade to the math activity, and (f) arranging the environment so students interact in small groups (see Johnson et al.,1994 for a thorough discussion of the five elements and activities to promote them).

Roles with specific responsibilities can be assigned to each group member. Examples of roles include materials person, spokesperson, writer, encourages and timekeeper. Roles should be taught and practiced prior to placing students in cooperative groups; students need a good understanding of the responsibilities associated with each role.

Groups should contain various ability levels. By limiting group size to four to six students, each member should be able to have an active role and access materials within a reasonable amount of time.

### Lesson instruction

The “lesson instruction” component of cooperative learning refers to the time in which cooperative learning activities occur. Students should engage in cooperative learning activities after they have received direct instruction in the mathematics and collaborative skills objectives targeted for the group activity. Asking students to perform math activities and collaborative skills for which no previous direct instruction has occurred puts students with LD (as well as other students) at risk for failure and group frustration. Inevitably, the lack of direct instruction prior to cooperative learning may result in numerous questions requesting clarification and assistance. Therefore, “lesson instruction” consists first of direct instruction, and then the cooperative learning activity. Cooperative learning can be used as the “guided practice” time when students engage in tasks to practice introduced skills. Cooperative learning can be used at the onset of math instruction as a means of reviewing skills and concepts or after the presentation of subject matter where new material is practiced within the context of previously taught material. For example, if the math objective is to teach students how to solve story problems using a strategy, then the strategy steps should first be taught directly. Students could then work in a cooperative learning activity that requires the use of the strategy to solve story problems.

An important aspect of the “lesson instruction” component is the teacher’s role. The teacher must (a) have students transition quickly after direct instruction, (b) have activities and materials ready, (c) monitor student progress in groups, and (d) reinforce the occurrences of collaborative behaviors.

During cooperative learning activities, teachers should circulate among groups monitoring the students’ ability to complete the assigned mathematics activity and demonstrate the targeted collaborative skills. The teacher can facilitate group work by asking questions to help students redirect their work, by providing additional instruction to some students who may be struggling with the task, and by reinforcing students’ efforts for working collaboratively and seeking solutions to problems.

### Lesson evaluation

The purpose of the “lesson evaluation” component is to assess student mastery of the math objectives and the group’s ability to work collaboratively. Teachers can conduct such evaluation by (a) observing students during the cooperative learning activity, (b) having students complete individual tasks following cooperative learning activities, and (c) asking students to engage in group processing (self-evaluation).

Teachers can assess students’ mathematics abilities during the group activity by addressing evaluation questions, such as those listed in Table 2. Group and/or individual responses and needs can be recorded on a clipboard to determine if additional instruction and group work are necessary for students to achieve mastery. Answers to the evaluation questions may suggest further direct instruction in a math skill with some or all of the students.

When the cooperative learning activity is finished, teachers may want to administer an individual posttest to determine how well each student has mastered the mathematics content. This is a common form of pupil evaluation that typically yields some type of permanent product, which can then can be graded. The purpose of this evaluation is to ascertain whether students are capable of performing the mathematics objectives independently at mastery level.

Students also should be given the opportunity to evaluate their ability to be team players; this is called group processing. Johnson and Johnson (1986) recommended that, following any cooperative learning activity, students should have time to discuss how their group performed in completing the math activities. Their responses could be recorded and discussed with the teacher to determine pupil-teacher agreement on the group’s ability to work collaboratively.

With careful planning, implementation, and evaluation cooperative learning activities can be achieved successfully by most students. The next section provides an example of using cooperative learning to teach mathematics.

## Teaching mathematics using cooperative learning

Below is an example of using cooperative learning to teach a math lesson based on the three major components of cooperative learning: “lesson preparation,” “lesson instruction,” and “lesson evaluation.” In this example, five students with LD attend a third grade general education classroom for most of the school day and receive special education resource remedial assistance for mathematics skills. The cooperative learning activity in this example is taking place in the general education setting where the general and special education teachers plan and teach cooperative learning math activities collaboratively twice a week.

### Lesson preparation

During “preparation” the cooperative learning math activity is designed; a description of “preparation” activities follows.

Establish objectives. In this example, the instructional objective for mathematics is: “Students will solve two-step story problems containing extraneous information with 90% accuracy.” The collaborative objective is: “Students will encourage and support teammates and share materials when requested.” The objectives are based on (a) school district special education curriculum guides, (b) students’ Individualized Education Program goals for mathematics and social skills, (c) curriculum-based assessment of whole number computation, and (d) observations of group behaviors and interactions.

Structure the activity. In whole group instruction, the instructional objective will be addressed by reviewing with all students the steps of a story problem-solving strategy that was learned the previous week. Students will recite the strategy’s steps using cue cards. Using the strategy, two story problems will be solved by the teachers who will recite the steps and verbalize their thinking processes as they work through the problems. Then, students will solve two story problems with the teachers. Next, students will review cooperative learning role responsibilities and explain ways to encourage and support each other. Rules about sharing also will be reviewed.

In the cooperative learning group, “numbered heads” will be used as the activity structure. Students will use their strategy cue cards to solve four story problems. Teachers will facilitate group work and interactions. Time will be allowed for group processing and students (when called on by group and number) will explain how their group solved a particular story problem.

Promote the elements of cooperative learning. Student roles will be assigned and bonus points will be distributed intermittently based on each group’s demonstration of encouraging and supportive behavior. One strategy cue card will be distributed to each group, thus necessitating sharing of the card. A posttest will be individually administered containing four story problems to determine if students can solve the story problems independently using their cue cards. The reading level of the story problems will be controlled for different ability levels in the classroom.

Identify the roles and groups. Each group will include a timekeeper to monitor the time and keep the group on task, a materials person to manage the cue card, a writer to record the group’s problem-solving responses and answers, and a spokesperson to lead the group during group processing time and to share the group’s results with the teacher. The groups will consist of four students; only one student with LD will be a member of each group.

1. Language/Vocabulary - Are students using new vocabulary words properly? - Do students possess prerequisite vocabulary? - Can students provide explanations in their own words for cooperative learning math activities, such as solving word problems and algorithms? |

2. Rules - Can Students explain to each other the rules that were taught during direct instruction, which must be applied in the cooperative learning activity? - Can students apply the rules to the cooperative learning math activity or do they require teacher assistance? - Can students use manipulatives to demonstrate rules? |

3. Strategies and Algorithms - Have students learned the strategies and algorithms? - Can students explain the strategies and algorithms to each other? - Do students require visual cures for remembering the strategies an algorithms? - Can students apply strategies and algorithms to a variety of problems? - Do students require teacher prompting and questions to help remember the strategies and algorithms? |

4. Connections - Can students explain how the new information relates to previously mastered math skills and concepts? - How do students explain the relevance of learning new math skills and concepts to everyday life? - How do students apply the new knowledge to activities that involve other disciplines (e.g., science, social studies)? - Can students depict math information using visuals, graphics, manipulatives, and abstract symbols? - Can students make connections between concrete-semi-concrete-abstract representations? |

### Lesson instruction

Implementation of the math lesson, in this example, requires direct instruction followed by the cooperative learning activity. The instructional steps are described below.

Provide an advance organizer. Explain the purpose of the lesson and the instructional and collaborative objectives. Describe the lesson’s activities and the teachers’ roles in the lesson. Remind students that they worked on a story problem-solving strategy last week and ask for a definition of a strategy.

Present the lesson. Have students refer to their strategy cue cards and repeat the strategy steps. Ask individual students to recite the steps, then ask students to repeat the steps without referring to the cue card, if possible. Next, model solving a story problem using the strategy cue card and verbalizing the steps. Have students imitate this process solving another problem at their desks. Ask for answers and explanations of how the problem was solved.

Explain the cooperative learning activity, using the “numbered heads” structure. Remind students that they can use a cue card to solve their four story problems. Review students’ roles and responsibilities and ask for explanations of how students encourage and support one another. Provide directions for transitioning into cooperative learning groups, set a time, distribute materials, and review the task. Once students are in groups, serve as a facilitator by guiding students with questions (e.g., “What are the steps in the strategy?” “What do you do first?” “How do you determine extraneous information?”) or providing further instruction if necessary. Reinforce groups for demonstrating appropriate collaborative behaviors. Provide time for group processing, and call on students by number and group to provide answers to the story problems.

### Lesson evaluation

Evaluating the students’ mastery of the instructional and collaborative objectives is critical. As mentioned earlier in this article, there are three types of evaluation. In this example, the first evaluation can be done during the cooperative learning activity: note evaluative comments that may assist in planning additional lessons or document individual student difficulty. For instance, evaluation questions like those in Table 1 can be used to determine mastery or potential trouble spots solving story problems. The second evaluation is individual and can be done following the group activity by administering a posttest. This can help teachers determine students’ ability to solve story problems on their own and to apply the strategy. Finally, have students evaluate themselves during group processing to determine their abilities with the designated collaborative skills. This evaluation should be shared with the teacher to be sure that teacher and student perceptions of abilities match.

## Conclusions

Cooperative learning is a popular instructional arrangement for teaching mathematics to students both with and without LD. Coupled with direct instruction, cooperative learning holds great promise as a supplement to textbook instruction by providing students with LD opportunities to practice math skills and concepts, reason and problem solve with peers, use mathematical language to discuss concepts, and make connections to other skills and disciplines. Carefully constructed lessons, using the “lesson preparation,” “lesson instruction,” and “lesson evaluation” components can offer students with LD rich learning opportunities in mathematics instruction.