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COOPERATIVE LEARNING IN MATHEMATICSAuthor(s): Roza Leikin and Orit ZaslavskySource: The Mathematics Teacher, Vol. 92, No. 3 (MARCH 1999), pp. 240-246Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 22/10/2013 22:17Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at ms.jsp.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected] Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.http://www.jstor.orgThis content downloaded from on Tue, 22 Oct 2013 22:17:52 PMAll use subject to JSTOR Terms and Conditions

iResearchConnectingto TeachingRoza Leikin and Orit ZaslavskyCOOPERATIVELEARNINGIN MATHEMATIHowcanthe teacherbest organizeand managethe classroomcoopduringerative work so that discipline prone problems do not arise, interactionbetween students primarily involves task, and pupils still have sufficientfreedom to contribute and toparticipate in thegroup discussion?(Good,Mulryan, and McCaslinStudentsmutuallyandpositivelydepend onone anotherthis article we describe a method of implementinga cooperative-learning setting that we callexchangeof knowledge. The design meets the goals suggestedby Good, Mulryan, and McCaslin and gives students an opportunity to gain experience with somelearningmaterial and then to explain it to others.This method was developed on the basis ofguidelines for cooperative learning inmathematics classrooms (Arhipova and Sokolov 1988). Thissettingwas implemented and investigated fora variety ofmathematics topics in secondary school with students of differentage groups and ability levels inmathematics (Leikin 1993; Leikin and Zaslavsky1997). We hope that our discussion of the exchangeof-knowledgemethod will furnish specific suggestions forpromoting cooperative learning in yourclassroom, as well as a framework for consideringthe issues involved in evaluating cooperativelearningmethods in general.WHATIS COOPERATIVELEARNING?Davidson (1990a) notes that it is difficultto preciselydefine cooperative learning because of the largevariety of learning settings that are regarded asfacilitating cooperative learning and the differencesamong them.However, on the basis of informationinArtzt and Newman (1990) and Sutton (1992), wepropose fournecessary conditions that togetherconstitute a cooperative-learning setting:Students learn in small groups with two to sixmembers in a group.The learning tasks inwhich students areengaged require that the students mutually andpositively depend on one another and on thegroup's work as a whole.The learning environment offersall members ofthe group an equal opportunity to interactwith2401992,185)one another regarding the learning tasks andencourages them to communicate their ideas invarious ways, forexample, verbally.Each member of the group has a responsibility tocontribute to the group work and is accountablefor the learning progress of the group.To be cooperative, a learning setting should ensure the existence of all these conditions.Contraryto common belief, forminggroups in the classroomis not sufficientto create a genuine cooperativelearning setting. Of the four conditions,we considerthe third to be particularly significant (Bishop1985; Clement 1991; Jaworski 1992).THE EXCHANGE-OF-KNOWLEDGEMETHODWe turn to a detailed description, based on d byThomas [email protected] st. eduOregonState UniversityCorvallis,OR 97331-4605Penelope H. [email protected] CollegeAllentown,PA 18104RozaLeikin,is [email protected],teachingassociate at theTechnion?Israel InstituteofTechnology.She isinterestedmathematicalmathematicswhilein cooperative-learningprocesses,andthinking, and preserviceshe wasteachereducation.a ewaswrittenat LRDC?UniverissityofPittsburgh.OritZaslavsky, [email protected],a senior lecturerat theTechnion?Israel InstituteofTechnology.She isparticularly interestedinprofessionaldevelteachers and teacheropment of secondary mathematicsin learning processeseducatorsandthat motivateandenhance mathematicalthinking and reasoning.THEMATHEMATICSTEACHERThis content downloaded from on Tue, 22 Oct 2013 22:17:52 PMAll use subject to JSTOR Terms and Conditions

exchange-of-knowledgemethod. This learningmethod shares some characteristics with the jigsawmethod (Aronson et al. 1978) in that itgives students an opportunity to play the role of a teacherand to offerexplanations to their peers. However,the exchange-of-knowledgemethod also allows students towork individually when appropriate. Inaddition, the tasks are designed to have studentswork in pairs to ensure that every student has theopportunity to both study and teach each type oflearningmaterial.The learning setting presented in this articleresembles some features ofSlavin's (1987) teamassisted individualization program,which fostersstudents' individual work within larger groups andencourages them to check and help each otherwhen necessary by using given answer sheets.However, the proposed method develops more complex problem-solving and explaining activities. Allstudents have to explain to one another mathematical ideas and principles, figure out for themselveshow to solve problems, and decide on the acceptableor correctanswers.Card 1Relationship between a Function and Its Derivative FunctionPartdistancetraveledby a car as a functionof time.Draw a graph of speed of thecar as a function of time.Most of the time, students learn in pairs within alarger group of four students.Each student is required to explain to his or herpartner how to solve theworked-out example inwhich the student has gained expertise on theprevious card and to listen to the explanationsgiven by the partner on how to deal with theworked-outexampleon a newSpeed is the rate ofchange of thedistance over an intervalAn averagespeedof time.Solution:i; (km/min)is va .ASAtInstantaneous speed is v? lim4 S'.*- 0AtAfter completing thework on a pair of cards, students change partners within the group. Thismove gives each group member an opportunity toact in the role of both a student and a teacher.Guidelinesforpreparinga setofstudycardsEach set of cards constitutes a learning unit. Eachset consists of two, four,or six study cards. The orderinwhich the cards can be applied is not important.Figures 1 and 2 show examples of learning cards.Each card consists of two or three parts. Part 1consists of a worked-out example. The extent of theexplanations on the card depends on the students'level and on their learning experience in the topic.Part 2 includes a problem similar to theworked-out10 20Instantaneousspeed (rateofchange ofdistance) is a slope ofa graph ofdistance.50t (min)On thefirstsegment:AS 10.At10' 0On thesecondsegment: ?-t zu 1uOn the thirdsegment: ? ou?zu 1.5Part II?Solve a problemProblem: The givengraph representsthedistancetraveledby a car as a functionS (km)of time.Of the followinggraphs offunctions,which could representthe speed of thecar in this trip?40t (min)t (min)(km/min) 40t (min)(km/min)(km/min?card.Each student is required to solve a problemsimilar to the previous worked-out example thatthe student's partner explained to the student?and is entitled, ifneeded, to ask the partner?who already tackled the problem?for help insolving it.t (min)Explanation:Descriptionof thelearningsettingThe method is based on study cards and is carriedout as follows:I?ExampleProblem: The givengraph representsthe(km/min)80Examplet (min)Fig. 1of a working cardt (min)(card1)example on the first part of the card, for students'individual solutions. Part 3, ifappropriate, includesan additional problem to be solved bymoreadvanced students. For each study card a corresponding homework card is available.Arrangementof learningwithina classroomThe learning setting is divided into twomainstages, as shown infigure 3: groups of experts andgroups for exchange ofknowledge.Vol. 92,No. 3 ?March 1999241This content downloaded from on Tue, 22 Oct 2013 22:17:52 PMAll use subject to JSTOR Terms and Conditions

Card 2Relationship between a Function and Its Derivative FunctionPartI?ExampleProblem: The givengraph representsthespeedStage 1 :Groups ofexpertsirdfl &Card (km/min)of a car as a function of time.Draw a graph ofdistance traveledbythe car as a functionof time.Explanation: The distance traveledbythe car in each segmentof time is thearea bounded between thegraph of thespeed and the jc-axison this segmentofCard'41?j?/ (min)CardSolution:S (km)time.1040Stage 2: Groups ofpartnersexchange-of-knowledgeS(2) 1? 10? S(5) 10 10 (5 2) 40S(5) 40 (10 30)-(55-5)t (min).?4 40 1000 1040o4Part II?Solve a problemProblem: The givengraph representsthespeed(km/min)High Achieversof a car as a function of time.Of the followinggraphs offunctions,which could representdistance traveledby the car in this trip?25S (km)4545t (min)2545t (min)S (km)2545Examplet (min)Fig. 2of a working25cardLow AchieversFig. 3Two stages of workS (km)45-1 (min)(card 2)Groups of experts.All students learn within thegroups of experts. No more than six students are ineach group. Each student within a particular groupgets the same card. The numbers of studentsreceiving differentcards are equal. All these groupscontain students of varying achievement levels. Theteacher makes sure that one student in each groupis at the highest achievement level so that studentcan help the teacher check the pace and correctnessof the group's work. The teacher monitors theworkof this student, who is responsible for reviewing thework of all the group members. Students shouldunderstand theworked-out example presented inthe first part of the card and are required to solveindividually the problems given in the second part.Each studentmay ask forany needed help. Students compare their solutions within their groups242Middle-Low Achieverst (min)S (km)25IMiddle-High AchieversMiddle Achieversand revise their solutions accordingly. The work ina group of experts is completed when the studentsagree on the solutions of the problems frompart 2of the card. The students then continue toworkwithin new groups of exchange ofknowledge.Groups of exchange of knowledge. The number ofstudents within a group should equal the numberof cards within the learning unit. Each student hasgained expertise in his or her own card, which isdifferentfrom cards of the other students withinthe exchange-of-knowledge group, as in the jigsawmethod. For example, iffour cards are in the set ofcards, the group contains four students, each ofwhom has a differentcard. High achievers learnwithin homogeneous groups, and students ofmiddle and low levels work in heterogeneous groups topace thework according to students' needs. Thiskind of arrangement enables the low achievers tofeelmore comfortable and believe that they cansucceed inmathematics. The high achievers canlearn additional material, as given on part 3 of thecards, for example. Middle-level students candevelop confidence in theirmathematical ability byhelping other students.Within the exchange-ofknowledge groups, students work in pairs all thetime.Suppose that studentMike, who gained expertise in the learningmaterial presented in card 1, cbTHEMATHEMATICSTEACHERThis content downloaded from on Tue, 22 Oct 2013 22:17:52 PMAll use subject to JSTOR Terms and Conditions

is learning in the pair with student Nick, whogained expertise in card 2, c2.1.Mike explains toNick part 1 of card 1 because heisworking as a teacher explaining theworked-outexample, asks Nick questions regarding the solution,verifies thatNick's understanding of the solution is acceptable, and answers Nick's questions.2. Nick explains toMike part 1 of card 2 in thesameway.3.When Mike and Nick finish their explanations,theyhave to solve part 2 of a new card simultaneously. They can ask each other questions andhelp each other ifneeded.4.When they finish solving the problems frompart2 of the cards,Mike and Nick check each other'ssolutions and revise them.5. Ifboth of them accept their partners' solutions ascorrect, thework in the pair is completed.froma worked-out example of one of the problems,received from their partners explanations regarding the other three tasks, explained worked-outexamples of three of the four cards in the learningunit, and solved all types of problems individually.This cooperative setting requires that studentsknow the underlying principles and have factualknowledge relevant to solving the unit problems.If the number of students is not evenly divisibleinto groups, a teacher can let a student who is at alow achievement level work with a middle-levelachiever all the time. This pair then works as onestudent in the learning arrangement. They solveproblems individually and explain differentcards totheir partners in turn.LEARNING OUTCOMESIn Leikin and Zaslavsky's study (1997), students'learning in traditional settingswas compared withtheir learning by the exchange-of-knowledgemethod. Four middle-level ninth-grade classes wereincluded in the study. This study investigated threemain questions with respect to the experimentalexchange-of-knowledge learning setting:1.What is the effectof this cooperative small-grouplearning setting on students' activeness?2.What kinds of students' interactions take place,and in particular, what kinds ofhelp do students receive in this learning setting?3.What are students' attitudes toward the experimental method?Fig. 4Work within a groupexchange-of-knowledgeofpartnersPairs' work within a group for exchange of knowledge. (See fig. 4.)1. Students Mike and Nick work as described inthe previous section.At the same time,Kathy andLora work with cards 3 and 4 (c3,c4).When the twopairs complete theirwork and each student hasacquired expertise in two cards, they begin theirnext stage and move towork within a new pair.2.Mike works with Lora, and Nick works withKathy, using the cards that they received at theprevious stage.When the students complete theirwork in these pairs, they return to their previouspeer with the new card.3.Mike works with Nick, and Lora works withKathy, using the cards that they received at theprevious stage.When students complete theirworkin these pairs, the unit has been completed.In thisway, by the end of the last stage, studentshave worked with all the learning cards, learnedThe findings of this study show that the experimental small-group cooperative-learningsettingfacilitates a higher level of learning activities. Classroomobservations indicated an increase in students' activeness. Altogether, students spentmuch more timeactively involved in the experimental cooperativesetting.We attributed this change to the increase inmathematical communications, which were definedin general as student-student and student-teacherinteractions related to the learningmaterial.Observations pointing to these communicativeinteractions took the formof giving an explanationand posing a question or requesting help. These twotypes of communicative interactions,which we callmathematical communication, fall intowhat Webb(1991) calls students' verbal interactions. These twocategories ofmathematical communication are considered very active and desirable. The importanceofmathematical communication is also manifestedin theNCTM's Curriculum and Evaluation Standards forSchool Mathematics (1989). We foundthat student-teacher learning interactions dominate whole-class settings,whereas student-studentlearning interactions tend to dominate the cooperative-learning setting.We suggest, therefore,thatVol. 92,No. 3 ?March 1999This content downloaded from on Tue, 22 Oct 2013 22:17:52 PMAll use subject to JSTOR Terms and Conditions

the increase of students' activeness in the experimental classroom situations occurredmainlybecause of the opportunity for increased studentstudent learning interactions.Students were asked to report about the types ofhelp they received while solving individually aproblem frompart 2 of a learning card. The students'self-reportsdealt with the part of the studentstudent learning interactions related to help. Thehigh percent of instances inwhich students statedthat they had requested help from their peers indicates that the supportive atmosphere created bythe experimental-learning environment encouragesstudents to ask forhelp despite their normal reluctance to do so (Newman and Goldin 1990). In fact,students received more help than they requested.This supportive atmosphere could be attributed tothe special arrangement of the small groups.We also examined students' attitudes withrespect to the cooperative-learning setting.Anextremely high percent of students, 90 percent,expressed positive attitudes toward the opportunities to pose questions and to explain the learningmaterial to their peers. Students' overall attitudestoward the learningmethod were highly positive.What types ofhelp did students offereach one another in the experimental small-group cooperativelearning setting? Explanations were the predominant type ofhelp. According toWebb, this type ofhelp is themost powerful. "The content-relatedhelpthat students give each other in small groups mightbe considered to lie on the continuum according toamount of elaboration. Detailed explanations wouldbe at the high end of such an elaboration scale,merely stating the answer to a problem or ld be at the low end, and providing otherkinds ofinformationwould fall in between the two extremes"(Webb 1991, 367). For most of the situations inwhich help was requested, the help offered includedexplanations. This finding indicates that the experimental cooperative-learning method allows students to construct explanations regarding underlying principles for solvingmathematical problems.GUIDELINES FOR FACILITATINGCOOPERATIVE LEARNING INMATHEMATICSTeachers can use many methods to facilitate cooperative learning. In designing a cooperative-learningsetting inmathematics, special attention is usuallygiven to the following issues (Hertz-Lazarowitz andFuks 1987;Kroll,Masingila,andMau 1992):(1) thestructure of the cooperative groups, (2) students'interactions in each group, (3) interactions amongthe differentgroups, (4) learning tasks and theteacher's role in the classroom, and (5) assessmentand evaluation of the learning process. These fivecriteria influence the type of cooperative-learning244setting that takes place in the classroom and itssuccess.The structure of the cooperative groupThe structure of a cooperative group is defined bythe number of students within a group and by thedegree ofheterogeneity of a group.Number of students within a cooperative group.The majority of the authors discussing cooperativelearning refer to this issue (Artztand Newman1990; Davidson 1990a; Davidson 1990b; Slavin1985;Webb 1985; Hertz-Lazarowitz and Fuks1987). The number of the students in a groupdepends on the type of themathematical activitythat is intended to take place in the classroom. Ingeneral, four is the optimal number ofmembers ina cooperative group. Some researchers recommendthat students work in pairs and emphasize thatworking in pairs facilitates active learning. Otherssuggest that a group of six students is the bestgroup size fora cooperative-learning setting.However, all the researchers agree that the number ofstudents in a group should not exceed seven. Theexchange-of-knowledge learning setting in this article gi

opment of secondary mathematics teachers and teacher educators and in learning processes that motivate enhance mathematical thinking and reasoning. 240 THE MATHEMATICS TEACHER This content downloaded from on Tue, 22 Oct 2013 22:17:52 PM All use subject to JSTOR Terms and Conditions