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URING OUR EXPERIENCES AS HIGH SCHOOL MATHEMATICSteachers, we have discovered these three things.First, many high school students do not like to takemathematics courses. These students sometimesfind that mathematics is boring and believe that itwill be of no use to them after they graduate fromhigh school. Stuart (2000) states that many peoplethink of mathematics as something that causesstress and is unpleasant. Such students have highanxiety about learning mathematics and trying tosucceed. Second, students have difficulty express-ing their thoughts on paper or in front of theirmathematics class. This phenomenon may occurbecause many traditional mathematics classroomsfoster a competitive atmosphere among students(Johnson and Johnson 1989). Third, the studentsare not accustomed to taking an active role in learn-ing mathematics. In light of these discoveries, wewanted to find a method of teaching high schoolmathematics classes that would help our studentsunderstand and enjoy the mathematics. In particu-lar, we asked the following question: Would our stu-dents understand and enjoy mathematics more if wetried a cooperative learning approach rather thanthe traditionally taught teacher-centered method?

UNDERSTANDING COOPERATIVE LEARNINGRoger T. Johnson and David W. Johnson (2000, p. 1)define cooperative learning as a relationship in agroup of students that requires positive interdepen-dence (a sense of sink or swim together), individualaccountability (each of us has to contribute andlearn), interpersonal skills (communication, trust,leadership, decision making and conflict resolu-tion), face-to-face promotive interaction, and pro-cessing (reflecting how well the team is functioningand how to function even better). Cooperativelearning is important for developing communica-tion skills and students abilities to work togetherin a mathematics class. When cooperative learningoccurs in the classroom, students not only learnmathematics but build with their peers personal

relationships that they might not be able to form ina traditional classroom (Johnson and Johnson 1989).

When planning to implement cooperative learn-ing in the classroom, a teacher must consider thefollowing: individual accountability and group re-wards, student preparation, and common difficulties.

Many researchers state that successful cooperativelearning requires that students have both groupgoals and individual accountability (Leiken andZaslavsky 1999). Many problems can be minimizedif the teachers reward system is structured so thatstudents are individually accountable for theirwork while they collaborate with their peers for thesuccess of the group (Lindauer and Petrie 1997).The use of group goals (particularly social goals) incooperative learning provides the students with areason to work together (Johnson and Johnson1989). Requiring individual accountability ensuresthat each student benefits from the experience byincreasing his or her mathematical understanding.It is a great way to make sure that everyone in thegroup does learn the new concepts. The success ofthe group in achieving its goals then depends onthe success of each group member.

When forming groups, teachers must establishgroup goals, as well as a system that rewards success.Teachers can do so in many different ways. For

Barbara Edwardsedwards@math.orst.edu. Oregon State UniversityCorvallis, OR 97331-4605

Margaret Kinzelkinzel@diamond.boisestate.edu. Boise State UniversityBoise, ID 83725

Angela Walmsley, awalmsley@maryville.edu, teaches atMaryville University, St. Louis, MO 63141. Her teachingspecialty is in mathematics education and curriculumand instruction. Joe Muniz, jmuniz@nvl.1cusd10.madison.k12.il.us, teaches at Collinsville High School,Collinsville, IL 62234. His interests include the use ofcooperative learning in a mathematics classroom.

D

MATHEMATICS TEACHER

Many students

are notaccustomed

to taking anactive role in

learningmathematics

Angela L. E. Walmsley and Joe Muniz

Connecting Research to Teaching

Cooperative Learning and ItsEffects in a High School GeometryClassroom

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright 2003 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

Vol. 96, No. 2 February 2003 113

example, team members can earn points or otherrewards for their team by performing well on a test(Stevens, Slavin, and Farnish 1991). Since the suc-cess of the team depends on the learning of eachstudent, this method reinforces the value of helpingeach group member achieve success (Stevens, Slavin,and Farnish 1991; Posamentier and Stepelman1999). Cooperative environments, which reinforcegroup goals and individual accountability, help stu-dents care about the success of their fellow students,become better listeners, and value alternativemethods for solving problems (Stevens, Slavin, andFarnish 1991).

Students must be prepared to participate in co-operative learning, and teachers must set the stagefor successful cooperative learning experiences intheir classrooms. Many people mistakenly believethat cooperative learning takes place wheneverteachers have students work in groups during class.However, cooperative learning is only successfulwhen members of a group perceive themselves aspart of a team that together must reach a goal(Posamentier and Stepelman 1999). Since coopera-tive learning is based on the premise that studentswho work together are responsible for one anotherslearning as well as for their own (Lindauer andPetrie 1997), students must learn to listen to oneanother and to value the notion that more than oneway of approaching a problem may be possible.Cooperative learning is a great learning strategy,but it does not happen without some preparation.

A teacher may experience some difficulties whenimplementing cooperative learning, and it will pro-duce successful results only if teachers learn how toemploy it in the classroom (Slavin 1990). Indeed,cooperative learning can be detrimental to studentslearning. For example, weaker students may copythe work of better students in their group, and theresult may be a decrease from what the weakerstudent would have been able to learn in a tradi-tional classroom. Another potential difficulty isthat teachers must be prepared to give up some ofthe traditional control that they may have once hadover the activity in the classroom. While ensuringthat students are on-task is necessary in cooperativeclassrooms, students working together do createmore noise. A teacher may perceive this noise as anindication of a loss of control. Some teachers believethat they are losing control of the direction or paththat the students take to reach a solution, sincestudent inquiry and questioning often take a differ-ent direction than the one that the teacher hasplanned.

BENEFITS OF COOPERATIVE LEARNINGResearch shows that the benefits of cooperativelearning include increased academic achievement,better communication skills, and successful social

and academic group interactions. Student achieve-ment in cooperative-learning environments is high-er (Slavin 1991; Stevens, Slavin, and Farnish 1991;Whicker, Bol, and Nunnery 1997). The effects ofcooperative learning on student achievement arevery impressive. Students achieve in a cooperativelearning setting for many reasons. In those settings,students see a variety of other students in variousstages of mastery of cognitive tasks, and peers pro-vide support and assistance to one another. Whenstudents interact cooperatively, they like to explaintheir strategies to one another in their own words(Stevens, Slavin, and Farnish 1991). The studentswho are explaining can then understand the mater-ial more clearly. When students are required toexplain, elaborate, and defend their positions toothers, they may be forced to think more deeplyabout their ideas. However, students who are lis-tening to the explanations of others are exposed to,and must think about, other approaches to a giventask. Observing others and practicing in such set-tings help learners internalize the concepts thatthey are attempting to master or understand(Stevens, Slavin, and Farnish 1991).

One of the greatest benefits of cooperative learn-ing is that it increases students skills in communi-cating mathematics (Artzt 1999). This communica-tion helps them better understand the subjectmatter. In fact, Johnson and Johnson (1989, p. 235)state, If mathematics instruction is to help studentsthink mathematically, understand the connectionsamong various mathematical facts and procedures,and be able to apply formal mathematical knowledgeflexibly and meaningfully, cooperative learning mustbe employed in mathematics classes. Cooperativelearning promotes learning mathematics in anactive way, rather than in a passive way (Johnsonand Johnson 1989). Teachers encourage their stu-dents to explain their mathematical understandingbecause it forces them to evaluate, integrate, andelaborate their knowledge in new ways (Stevens,Slavin, and Farnish 1991). Students can learn bestfrom one another when they are required to providereasoning for their answers or explain how theyarrived at the answers. Teachers who require stu-dents to reflect on how they answered a problem andexplain or elaborate to the other students in thegroup help the entire group learn more and empha-size mathematical communication skills (NCTM2000). Cooperative learning allows students to bothgive and receive elaborate explanations. They thenlearn more than students who simply get the cor-rect answers (Stevens, Slavin, and Farnish 1991).Mathematical communication, which is sometimesdifficult, is important for a students developmentin mathematics. Leiken and Zaslavsky (1999) foundthat using cooperative learning encouraged stu-dents to be actively engaged in mathematical learn-

A benefit ofcooperativelearning isthe increaseof studentsskills in com-municatingmathematics

114 MATHEMATICS TEACHER

ing and to communicate with one another aboutmathematics. High achievement was a result.

Another benefit of cooperative learning is that itallows students to experience working with otherstoward a common goal. Students increased theirability to use mathematics in their social inter-actions. Some of the short-term outcomes includeincreased learning, retention, and critical thinking(Whicker, Bol, and Nunnery 1997). Compared withthe traditional individually competitive classroom,cooperative learning experiences promote higherlevels of self-esteem for the students (Johnson,Johnson, and Holubec 1984; Johnson and Johnson1989). Cooperative learning can reinforce a stu-dents feeling of self-acceptance, whereas competi-tiveness can negatively affect self-acceptance, andindividualistic attitudes tend to be related to basicself-rejection (Johnson, Johnson, and Holubec1984). Students working cooperatively often enjoythe experience and believe that their classmateslike them. This belief that they are accepted by oth-ers also allows the students to believe that they aremore successful academically. This perception ofsuccess increases students self-esteem.

The long-term outcomes of cooperative learninginclude greater employability and career success(Johnson and Johnson 1989). Many employersvalue an employee who has skills in verbal commu-nication, responsibility, initiative, interpersonalinteraction, and decision making. All these quali-ties can be developed by experiencing cooperativelearning. Thus, cooperative learning not only helpsstudents with mathematics but also prepares themfor life after they graduate.

OUR CONNECTION OF RESEARCH TOTEACHING: IMPLEMENTING COOPERATIVE LEARNING IN A HIGHSCHOOL MATHEMATICS CLASSROOMSince little research exists on cooperative learningin high school mathematics classrooms, we decidedto conduct an informal experiment. Our participantswere students in two geometry classes that one ofthe authors of this article taught. The classes hadsimilar academic ability. The study compared thegrades and attitudes of the students in these twoclasses. In both classes, the students worked mostlyindependently during the first quarter. Both classesworked on the same lesson every day and had thesame homework and tests. Cooperative learningwas used sparingly in both classes throughout thefirst quarter. However, we began to help studentsdevelop their abilities to work cooperatively. Thestudents learned how to work in groups; and fromthe individual work submitted for grading, wecould tell that they understood what it meant to beindividually accountable for their learning. Duringthe first quarter, we analyzed the achievement of

the two classes and compared them with each otherso that we could verify that the groups were indeedsimilar.

During the last week of the first quarter, bothclasses completed a survey, given in figure 1, thatmeasured the students attitudes about cooperativelearning and mathematics. The results of this Likert-style questionnaire showed that both the experi-mental and control groups agreed that cooperativelearning was a positive learning strategy to use inmathematics class. The students had similar opin-ions about working in groups, and not much dispar-ity occurred in their answers to the questionaire.The responses of the group that was to become theexperimental group indicated that those students,as a group, liked mathematics slightly more thantheir counterparts.

During the second nine weeks of the first semes-ter, one of the geometry classes became an experi-mental group and the other class was a controlgroup. Students in the control group were basicallytaught the same way that they had been taughtduring the first quarter; however, they did not workin cooperative learning groups during the secondquarter. We introduced a significant instructionalchange for the experimental group. This class wastaught using cooperative instruction exclusively,and the students desks were placed in groups offour. Other factors were the same in both classes.The experimental group was taught the samelessons as the control group, both classes weretaught at the same pace, and they took the sametests and quizzes. The only difference was that theexperimental group worked cooperatively.

As stated previously, two very important aspectsof implementing cooperative learning are to providegroup rewards and to reinforce individual account-ability. The group rewards were explained to theexperimental group on the first day of the secondnine weeks. The rewards were based on individualperformance on each chapter test. Since the stu-dents had similar ability levels, we first designatedan acceptable minimum score of 90 percent forevery test. If each student in the group scored 90percent or higher on a chapter test, each student inthe group received four extra-credit points on thetest. This type of reward system encouraged stu-dents in each group to make sure that everyone inthe group understood the material before a test. Itreinforced the value of individual accountabilityand at the same time created the possibility ofearning extra-credit points if everyone in the groupdid well on the test.

At the end of the second quarter, we comparedthe grades of the experimental class to those of thecontrol class. The average grade for each class hadincreased from the first quarter to the second quar-ter. When comparing the first-quarter grades to the

The only difference

was that theexperimental

groupworked

cooperatively

Vol. 96, No. 2 February 2003 115

second-quarter grades, the overall average second-quarter grade of the experimental group was 9 per-cent higher than that groups overall average first-quarter grade, and the overall average second-quarter grade of the control group was 4 percenthigher than that groups overall average first-quartergrade. The experimental group also had a highersecond-quarter average than the control group: 97percent, as compared to 95 percent in the controlgroup. Table 1 and table 2 show quarter gradesfor each student in both classes.

Within the experimental group, ten students(more than half the class) increased their overallgrades by more then 9 percent. One student in par-ticular had an increase of 20 percent from the firstquarter to the second quarterfrom 82 percent to102 percent. Cooperative learning certainly seemedto have a positive effect on the experimental group.

At the end of the second semester, students inthe experimental group completed the same atti-tude survey that both groups had completed at thebeginning of the year. We compared the scores ofthe experimental groups postsurvey with theirscores from the presurvey to determine whetherimmersion in a cooperative group setting for theentire semester had changed their attitudes aboutcooperative learning. The students in the experi-mental classroom appeared to have an even morepositive attitude toward cooperative learning by theend of the second semester. Overall, the survey

Please circle one of the following numbers in each of the questions below.

1. strongly disagree 2. somewhat disagree 3. somewhat agree 4. strongly agree

11. I learn more when I work in groups in math class.

1 2 3 4

12. I like math.

1 2 3 4

13. I like working on math homework alone.

1 2 3 4

14. I like studying for math alone.

1 2 3 4

15. I would rather work in groups when doing math homework awayfrom school.

1 2 3 4

16. Learning math can be fun and enjoyable.

1 2 3 4

17. I dislike studying for math in groups.

1 2 3 4

18. I feel more comfortable asking a student in my group for help ratherthan asking my teacher.

1 2 3 4

19. I learn less when working in groups in math.

1 2 3 4

10. I dislike learning math.

1 2 3 4

Fig. 1Attitude survey

TABLE 1Quarter Grades (%) for Control Group

Student Quarter 1 Quarter 2

11 87 19612 88 19513 81 19314 98 10215 88 19616 85 19317 94 19618 88 18219 87 19310 90 18711 91 19912 90 19213 98 19914 97 10115 89 19616 91 10217 97 10118 89 19219 96 197

Mean 91 195

TABLE 2Quarter Grades (%) for Experimental Group

Student Quarter 1 Quarter 2

11 93 19512 96 10213 78 18814 97 10215 93 10316 83 19317 90 10218 90 19919 97 10710 94 19211 80 19712 84 18613 81 19814 88 19415 83 10316 93 19517 82 18818 87 19819 90 100

Mean 88 197

116 MATHEMATICS TEACHER

indicated that the students in the experimentalgroup liked working on mathematics with one an-other and thought that cooperative learning couldhelp them understand and learn mathematicalideas.

Our experience with trying cooperative learningin the classroom convinced us that cooperativelearning has a positive effect on students achieve-ment and attitudes toward mathematics. Webelieve that our focus on providing for individualaccountability and rewarding groups for realizingtheir group goals contributed to the success of thisproject. Each student in the class had a dualresponsibilityto himself or herself and to theother members of the group. Rewards were givenfor both efforts. Our initial success has encouragedus to continue using cooperative groups in ourclasses, and we continue to see it as a successfulmethod for teaching mathematics.

CONCLUSIONCooperative learning has many positive effects inthe mathematics classroom if it is properly imple-

mented. Studies have found that if teachers havesome kind of group reward system with provisionsfor individual accountability, cooperative learningcan be successful. Teachers must also prepare stu-dents to work cooperatively by emphasizing theneed for good listening skills and an openness tothe ideas of others. An effective use of cooperativelearning in the classroom can positively affect stu-dents social skills, self-esteem, and intergrouprelationships. Today, teachers have the resourcesto see that cooperative learning can work; but expe-rience is the best way to truly understand it (Artzt1999).

REFERENCESArtzt, Alice. Cooperative Learning in Mathematics

Teacher Education. Mathematics Teacher 92 (January 1999): 1117.

Johnson, David W., and Roger T. Johnson. Coopera-tive Learning in Mathematics Education. In NewDirections for Elementary School Mathematics, 1989Yearbook of the National Council of Teachers ofMathematics (NCTM), edited by Paul R. Trafton,pp. 23445. Reston, Va.: NCTM, 1989.

. Social Skills for Successful Group Work.Educational Leadership 47 (January 1990): 2932.

. The Cooperative Learning Center at the Uni-versity of Minnesota. University of Minnesota,2000. Available at www.clcrc.com. World Wide Web.

Johnson, David W., Roger T. Johnson, and E. J. Holubec.Circles of Learning: Cooperation in the Classroom.Edina, Minn.: Interaction Book Co., 1984.

Leiken, Roza, and Orit Zaslavsky. Cooperative Learn-ing in Mathematics. Mathematics Teacher 92(March 1999): 24046.

Lindauer, Patricia, and Garth Petrie. A Review ofCooperative Learning: An Alternative to EverydayInstructional Strategies. Journal of InstructionalPsychology 24 (September 1997): 18388.

National Council of Teachers of Mathematics. Princi-ples and Standards for School Mathematics. Reston,Va.: NCTM, 2000.

Posamentier, Alfred S., and Jay Stepelman. TeachingSecondary School Mathematics. Upper Saddle River,N.J.: Prentice-Hall, 1999.

Slavin, Robert E. Here to Stayor Gone Tomorrow?Educational Leadership 47 (January 1990): 250.

. Synthesis of Research on Cooperative Learn-ing. Educational Leadership 48 (February 1991):7182.

Stevens, Robert, Robert E. Slavin, and Anna Farnish.The Effects of Cooperative Learning and DirectInstruction in Reading Comprehension Strategies onMain Idea Identification. Educational Leadership83 (March 1991): 815.

Stuart, Vanessa. Math Curse or Math Anxiety.Teaching Children Mathematics 6 (January 2000):33035.

Whicker, Kristina, Linda Bol, and John A. Nunnery.Cooperative Learning in the Secondary Mathemat-ics Classroom. Journal of Educational Research 91(SeptemberOctober 1997): 4248.

Cooperativelearning has

a positiveeffect

on studentsachievements