Mathematics Professional Series Cooperative Learning — Geometry   What Is Cooperative Learning? Cooperative Learning IS . . . Group members understanding that they are part of a team and all members of the team are working toward a common goal. Group members understanding that the successes or failures of the group will be shared by all members. Therefore, each member must contribute as much as he or she can to the group goal. All students learning to talk and discuss problems with each other in order to accomplish the group goal. Group success being dependent on, and being a direct effect of, the individual work of each member of the group. A process that capitalizes on the presence of student peers, encourages interaction among students, and establishes positive relationships among team members. A process that requires the guidance of a teacher who can help students develop the cooperative learning skills they need, understand group dynamics, and learn mathematics by working in groups. Students asking for help only after each one in the group has considered the question. Helping students to be individually accountable for their learning. This is discussed in detail by Spencer Kagan in his book Cooperative Learning. Cooperative Learning IS NOT . . . Separating students into small groups to work on a problem or a group of problems without direction or individual responsibility. Students sitting together in groups and working on problems individually without conversation or interaction regarding the method or process being used for problem solving. Students sitting together in groups and letting one student do all of the work while the others watch or listen. Learning to Work Together Checking homework daily is an excellent and efficient method to help teams learn to work together. It can be done in this way. The teacher displays correct answers on the overhead projector. Students in cooperative groups check their homework and discuss any differences with their team members. Many will find answers to their questions from members of their team. Each team must agree upon the problems that they cannot solve. The teacher then asks each team to indicate which problems they would like discussed. The teacher then guides students through a solution of those problems. This method will reduce the number of problems addressed by the teacher and thus allow for better use of class time. Students will have more peer pressure to complete homework fully and on time. Teachers may choose to have students keep their own weekly homework records by team, individually, or both. The teacher may choose to select a problem or two for discussion based on information gathered by circulating around the room while students are checking and talking about their homework. Calling on a random person in a random group can add responsibility to this method. Structures to Deliver Instruction to Students There are a variety of structures that can be used to deliver instruction to students via cooperative learning. One structure is called Numbered Heads Together. It is a simple four-step structure that is used to review basic facts and information. It may also be used for problem solving with problems of limited difficulty. Numbered Heads Together works well with knowledge and comprehension, and high-consensus type questions. Step 1 The students number off from one to four. Step 2 The teacher asks a question. Step 3 The students put their heads together to make sure everyone can answer the question and knows how to get the answer (Computation may be performed mentally, or with pencil, paper, or calculator as appropriate). Step 4 The teacher randomly generates a number from 1 to 4 (assuming maximum group size is 4), and those students with that number raise their hands or stand up to respond. Then the teacher randomly generates a number from 1 to x (x is the number of groups in your classroom). The person chosen from step 4 in group x is the one who is expected to respond. Variations on Numbered Heads After a student responds, the teacher can have the other teams agree or disagree with a thumbs up or thumbs down signal. In problems with multiple answers, the teacher can have students from different groups each give part of the response. All of the students can simultaneously give the answer. All students responding can write the answers on the chalkboard or on paper at the same time. The teacher can ask another student to add to the answer if an incomplete response is given. The following example problems could be used to practice using the structure Numbered Heads Together in a geometry classroom. Geometry Examples Lead molding is used to edge pieces of stained glass with sides of 1.5 inches. How many inches of molding are needed for 24 six-sided pieces and 14 square pieces? The length of a rectangle is 4 feet more than twice its width. Find the length and width of the rectangle. How many different rectangles can be drawn that have a perimeter of 12 units and side measures that are whole numbers? Find the fourth vertex of square ABCD if three vertices are A(-2, 2), B(2, 2), and C(2, -2). Beginning at the center R and moving only up, down, right, or left, how many different paths can you find to spell the word RAFT? T T F T T F A F T T F A R A F T T F A F T T F T T Show that the points A(l, l), B(-3, -11), and C(0, -2) are collinear. A triangle has three congruent sides. Two vertices of the triangle are located at (0, 0) and (8, 0). Find the coordinates of the third vertex. In a certain isosceles triangle, the third side is 3 inches shorter than either of the congruent sides. The perimeter is 102 inches. Find the length of each side. If the length of each side of a square is doubled, the area is increased by 36 square inches. Find the length of a side of the original square. The degree measure of an angle is one-fourth the measure of its complement. Find the measure of the angle. Answers: 306 inches Sample answer: 20 feet, 8 feet 3 rectangles D (-2, -2) 32 paths slope of = 3 and slope of = 3 35 inches, 35 inches, 32 inches 18° More Information For more information on cooperative learning, see "Cooperative Learning in the Mathematics Classroom", part of the Glencoe Professional Mathematics Series.