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See discussions, stats, and author profiles for this publication at: Mathematical Problem-Solving Through Cooperative Learning-TheImportance of Peer Acceptance and FriendshipsArticle in Frontiers in Education · August 2021DOI: 10.3389/feduc.2021.710296CITATIONSREADS0535 authors, including:Nina KlangNatalia Lebedeva KarlssonMälardalen UniversitySödertörn University23 PUBLICATIONS 354 CITATIONS5 PUBLICATIONS 0 CITATIONSSEE PROFILEMartin KarlbergUppsala University14 PUBLICATIONS 128 CITATIONSSEE PROFILESome of the authors of this publication are also working on these related projects:Solidarity and Education for Sustainable Development. Doctoral thesis project View projectEARLI SIG 15 JURE - Special Educational Needs View projectAll content following this page was uploaded by Nina Klang on 24 August 2021.The user has requested enhancement of the downloaded file.SEE PROFILE

ORIGINAL RESEARCHpublished: 24 August 2021doi: 10.3389/feduc.2021.710296Mathematical Problem-SolvingThrough Cooperative Learning—TheImportance of Peer Acceptance andFriendshipsNina Klang 1,2*, Natalia Karlsson 3, Wiggo Kilborn 4, Pia Eriksson 1 and Martin Karlberg 11Department of Education, Uppsala University, Uppsala, Sweden, 2Department of Education, Culture and Communication,Malardalen University, Vasteras, Sweden, 3School of Natural Sciences, Technology and Environmental Studies, SodertornUniversity, Huddinge, Sweden, 4Faculty of Education, Gothenburg University, Gothenburg, SwedenEdited by:Dor Abrahamson,University of California, Berkeley,United StatesReviewed by:Kathryn Holmes,Western Sydney University, AustraliaTaro Fujita,University of Exeter, United Kingdom*Correspondence:Nina Klangnina.klang@edu.uu.seSpecialty section:This article was submitted toSTEM Education,a section of the journalFrontiers in EducationReceived: 15 May 2021Accepted: 09 August 2021Published: 24 August 2021Citation:Klang N, Karlsson N, Kilborn W,Eriksson P and Karlberg M (2021)Mathematical Problem-SolvingThrough Cooperative Learning—TheImportance of Peer Acceptanceand Friendships.Front. Educ. 6:710296.doi: 10.3389/feduc.2021.710296Frontiers in Education www.frontiersin.orgMathematical problem-solving constitutes an important area of mathematics instruction,and there is a need for research on instructional approaches supporting student learning inthis area. This study aims to contribute to previous research by studying the effects of aninstructional approach of cooperative learning on students’ mathematical problem-solvingin heterogeneous classrooms in grade five, in which students with special needs areeducated alongside with their peers. The intervention combined a cooperative learningapproach with instruction in problem-solving strategies including mathematical models ofmultiplication/division, proportionality, and geometry. The teachers in the experimentalgroup received training in cooperative learning and mathematical problem-solving, andimplemented the intervention for 15 weeks. The teachers in the control group receivedtraining in mathematical problem-solving and provided instruction as they would usually.Students (269 in the intervention and 312 in the control group) participated in tests ofmathematical problem-solving in the areas of multiplication/division, proportionality, andgeometry before and after the intervention. The results revealed significant effects of theintervention on student performance in overall problem-solving and problem-solving ingeometry. The students who received higher scores on social acceptance and friendshipsfor the pre-test also received higher scores on the selected tests of mathematical problemsolving. Thus, the cooperative learning approach may lead to gains in mathematicalproblem-solving in heterogeneous classrooms, but social acceptance and friendships mayalso greatly impact students’ results.Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms,hierarchical linear regression analysisINTRODUCTIONThe research on instruction in mathematical problem-solving has progressed considerably duringrecent decades. Yet, there is still a need to advance our knowledge on how teachers can support theirstudents in carrying out this complex activity (Lester and Cai, 2016). Results from the Program forInternational Student Assessment (PISA) show that only 53% of students from the participatingcountries could solve problems requiring more than direct inference and using representations fromdifferent information sources (OECD, 2019). In addition, OECD (2019) reported a large variation in1August 2021 Volume 6 Article 710296

Klang et al.Mathematical Problem-Solving Through CLachievement with regard to students’ diverse backgrounds. Thus,there is a need for instructional approaches to promote students’problem-solving in mathematics, especially in heterogeneousclassrooms in which students with diverse backgrounds andneeds are educated together. Small group instructionalapproaches have been suggested as important to promotelearning of low-achieving students and students with specialneeds (Kunsch et al., 2007). One such approach is cooperativelearning (CL), which involves structured collaboration inheterogeneous groups, guided by five principles to enhancegroup cohesion (Johnson et al., 1993; Johnson et al., 2009;Gillies, 2016). While CL has been well-researched in wholeclassroom approaches (Capar and Tarim, 2015), few studies ofthe approach exist with regard to students with specialeducational needs (SEN; McMaster and Fuchs, 2002). Thisstudy contributes to previous research by studying the effectsof the CL approach on students’ mathematical problem-solvingin heterogeneous classrooms, in which students with specialneeds are educated alongside with their peers.Group collaboration through the CL approach is structured inaccordance with five principles of collaboration: positiveinterdependence, individual accountability, explicit instructionin social skills, promotive interaction, and group processing(Johnson et al., 1993). First, the group tasks need to bestructured so that all group members feel dependent on eachother in the completion of the task, thus promoting positiveinterdependence. Second, for individual accountability, theteacher needs to assure that each group member feelsresponsible for his or her share of work, by providingopportunities for individual reports or evaluations. Third, thestudents need explicit instruction in social skills that are necessaryfor collaboration. Fourth, the tasks and seat arrangements shouldbe designed to promote interaction among group members. Fifth,time needs to be allocated to group processing, through whichgroup members can evaluate their collaborative work to planfuture actions. Using these principles for cooperation leads togains in mathematics, according to Capar and Tarim (2015), whoconducted a meta-analysis on studies of cooperative learning andmathematics, and found an increase of .59 on students’mathematics achievement scores in general. However, thenumber of reviewed studies was limited, and researcherssuggested a need for more research. In the current study, wefocused on the effect of CL approach in a specific area ofmathematics: problem-solving.Mathematical problem-solving is a central area ofmathematics instruction, constituting an important part ofpreparing students to function in modern society (Gravemeijeret al., 2017). In fact, problem-solving instruction createsopportunities for students to apply their knowledge ofmathematical concepts, integrate and connect isolated piecesof mathematical knowledge, and attain a deeper conceptualunderstanding of mathematics as a subject (Lester and Cai,2016). Some researchers suggest that mathematics itself is ascience of problem-solving and of developing theories andmethods for problem-solving (Hamilton, 2007; Davydov, 2008).Problem-solving processes have been studied from differentperspectives (Lesh and Zawojewski, 2007). Problem-solvingFrontiers in Education www.frontiersin.orgheuristics Pólya, (1948) has largely influenced our perceptionsof problem-solving, including four principles: understanding theproblem, devising a plan, carrying out the plan, and looking backand reflecting upon the suggested solution. Schoenfield, (2016)suggested the use of specific problem-solving strategies fordifferent types of problems, which take into considerationmetacognitive processes and students’ beliefs about problemsolving. Further, models and modelling perspectives onmathematics (Lesh and Doerr, 2003; Lesh and Zawojewski,2007) emphasize the importance of engaging students inmodel-eliciting activities in which problem situations areinterpreted mathematically, as students make connectionsbetween problem information and knowledge of mathematicaloperations, patterns, and rules (Mousoulides et al., 2010;Stohlmann and Albarracín, 2016).Not all students, however, find it easy to solve complexmathematical problems. Students may experience difficulties inidentifying solution-relevant elements in a problem or visualizingappropriate solution to a problem situation. Furthermore,students may need help recognizing the underlying model inproblems. For example, in two studies by Degrande et al. (2016),students in grades four to six were presented with mathematicalproblems in the context of proportional reasoning. The authorsfound that the students, when presented with a word problem,could not identify an underlying model, but rather focused onsuperficial characteristics of the problem. Although the studentsin the study showed more success when presented with a problemformulated in symbols, the authors pointed out a need foractivities that help students distinguish between differentproportional problem types. Furthermore, students exhibitingspecific learning difficulties may need additional support in bothgeneral problem-solving strategies (Lein et al., 2020; Montagueet al., 2014) and specific strategies pertaining to underlyingmodels in problems. The CL intervention in the present studyfocused on supporting students in problem-solving, throughinstruction in problem-solving principles (Pólya, 1948),specifically applied to three models of mathematical problemsolving—multiplication/division, geometry, and proportionality.Students’ problem-solving may be enhanced throughparticipation in small group discussions. In a small groupsetting, all the students have the opportunity to explain theirsolutions, clarify their thinking, and enhance understanding of aproblem at hand (Yackel et al., 1991; Webb and Mastergeorge,2003). In fact, small group instruction promotes students’learning in mathematics by providing students withopportunities to use language for reasoning and conceptualunderstanding (Mercer and Sams, 2006), to exchange differentrepresentations of the problem at hand (Fujita et al., 2019), and tobecome aware of and understand groupmates’ perspectives inthinking (Kazak et al., 2015). These opportunities for learning arecreated through dialogic spaces characterized by openness to eachother’s perspectives and solutions to mathematical problems(Wegerif, 2011).However, group collaboration is not only associated withpositive experiences. In fact, studies show that some studentsmay not be given equal opportunities to voice their opinions, dueto academic status differences (Langer-Osuna, 2016). Indeed,2August 2021 Volume 6 Article 710296

Klang et al.Mathematical Problem-Solving Through CLstudents and 51 classes were required, with an expected effect sizeof 0.30 and power of 80%, provided that there are 20 students perclass and intraclass correlation is 0.10. An invitation to participatein the project was sent to teachers in five municipalities via e-mail.Furthermore, the information was posted on the website ofUppsala university and distributed via Facebook interestgroups. As shown in Figure 1, teachers of 1,165 studentsagreed to participate in the study, but informed consent wasobtained only for 958 students (463 in the intervention and 495 inthe control group). Further attrition occurred at pre- and postmeasurement, resulting in 581 students’ tests as a basis foranalyses (269 in the intervention and 312 in the controlgroup). Fewer students (n 493) were finally included in theanalyses of the association of students’ social acceptance andfriendships and the effect of CL on students’ mathematicalproblem-solving (219 in the intervention and 274 in thecontrol group). The reasons for attrition included teacher dropout due to sick leave or personal circumstances (two teachers inthe control group and five teachers in the intervention group).Furthermore, some students were sick on the day of datacollection and some teachers did not send the test results tothe researchers.As seen in Table 1, classes in both intervention and controlgroups included 27 students on average. For 75% of the classes,there were 33–36% of students with SEN. In Sweden, no formalmedical diagnosis is required for the identification of studentswith SEN. It is teachers and school welfare teams who decidestudents’ need for extra adaptations or special support (SwedishNational Educational Agency, 2014). The information onindividual students’ type of SEN could not be obtained due toregulations on the protection of information about individuals(SFS 2009). Therefore, the information on the number of studentswith SEN on class level was obtained through teacher reports.problem-solvers struggling with complex tasks may experiencenegative emotions, leading to uncertainty of not knowing thedefinite answer, which places demands on peer support (Jordanand McDaniel, 2014; Hannula, 2015). Thus, especially inheterogeneous groups, students may need additional supportto promote group interaction. Therefore, in this study, weused a cooperative learning approach, which, in contrast tocollaborative learning approaches, puts greater focus onsupporting group cohesion through instruction in social skillsand time for reflection on group work (Davidson and Major,2014).Although cooperative learning approach is intended topromote cohesion and peer acceptance in heterogeneousgroups (Rzoska and Ward, 1991), previous studies indicatethat challenges in group dynamics may lead to unequalparticipation (Mulryan, 1992; Cohen, 1994). Peer-learningbehaviours may impact students’ problem-solving (Hwang andHu, 2013) and working in groups with peers who are seen asfriends may enhance students’ motivation to learn mathematics(Deacon and Edwards, 2012). With the importance of peersupport in mind, this study set out to investigate whether theresults of the intervention using the CL approach are associatedwith students’ peer acceptance and friendships.The Present StudyIn previous research, the CL approach has shown to be apromising approach in teaching and learning mathematics(Capar and Tarim, 2015), but fewer studies have beenconducted in whole-class approaches in general and studentswith SEN in particular (McMaster and Fuchs, 2002). This studyaims to contribute to previous research by investigating the effectof CL intervention on students’ mathematical problem-solving ingrade 5. With regard to the complexity of mathematical problemsolving (Lesh and Zawojewski, 2007; Degrande et al., 2016;Stohlmann and Albarracín, 2016), the CL approach in thisstudy was combined with problem-solving principlespertaining to three underlying models of andproportionality. Furthermore, considering the importance ofpeer support in problem-solving in small groups (Mulryan,1992; Cohen, 1994; Hwang and Hu, 2013), the studyinvestigated how peer acceptance and friendships wereassociated with the effect of the CL approach on students’problem-solving abilities. The study aimed to find answers tothe following research questions:InterventionThe intervention using the CL approach lasted for 15 weeks andthe teachers worked with the CL approach three to four lessonsper week. First, the teachers participated in two-days training onthe CL approach, using an especially elaborated CL manual(Klang et al., 2018). The training focused on the five principlesof the CL approach (positive interdependence, individualaccountability, explicit instruction in social skills, promotiveinteraction, and group processing). Following the training, theteachers introduced the CL approach in their classes andfocused on group-building activities for 7 weeks. Then, 2 daysof training were provided to teachers, in which the CL approachwas embedded in activities in mathematical problem-solvingand reading comprehension. Educational materials containingmathematical problems in the areas of multiplication anddivision, geometry, and proportionality were distributed tothe teachers (Karlsson and Kilborn, 2018a). In addition tothe specific problems, adapted for the CL approach, theeducational materials contained guidance for the teachers, inwhich problem-solving principles (Pólya, 1948) were presentedas steps in problem-solving. Following the training, the teachersapplied the CL approach in mathematical problem-solvinglessons for 8 weeks.a) What is the effect of CL approach on students’ problemsolving in mathematics?b) Are social acceptance and friendship associated with the effectof CL on students’ problem-solving in mathematics?METHODSParticipantsThe participants were 958 students in grade 5 and their teachers.According to power analyses prior to the start of the study, 1,020Frontiers in Education www.frontiersin.org3August 2021 Volume 6 Article 710296

Klang et al.Mathematical Problem-Solving Through CLFIGURE 1 Flow chart for participants included in data collection and data analysis.TABLE 1 Background characteristics of classes and teachers in intervention and control groups.Number of classesMean number of students per classProportion of children with SEN per class1st quartile2nd quartile3rd quartileTeachers who reported implementing the CL approach at least three lessons a weekClasses, for which teachers reported using the CL approach in problem-solving at least one lesson per weekClasses in control group, for which teachers reported working with problem-solving at least one lesson per weekSolving a problem is a matter of goal-oriented reasoning,starting from the understanding of the problem to devising itssolution by using known mathematical models. This presupposesthat the current problem is chosen from a known context(Stillman et al., 2008; Zawojewski, 2010). This differs from theproblem-solving of the textbooks, which is based on an aim totrain already known formulas and procedures (Hamilton, 2007).Moreover, it is important that students learn modelling accordingto their current abilities and conditions (Russel, 1991).Frontiers in Education www.frontiersin.orgCL groupControl group23272527.17.27.3318 (20 responses)11 (14 responses).27.33.3610 (14 responses)In order to create similar conditions in the experiment groupand the control group, the teachers were supposed to use the sameeducational material (Karlsson and Kilborn, 2018a; Karlsson andKilborn, 2018b), written in light of the specified view of problemsolving. The educational material is divided into roportionality—and begins with a short teachers’ guide,where a view of problem solving is presented, which is basedon the work of Polya (1948) and Lester and Cai (2016). The tasks4August 2021 Volume 6 Article 710296

Klang et al.Mathematical Problem-Solving Through CLare constructed in such a way that conceptual knowledge was infocus, not formulas and procedural knowledge.symmetry, transformation, and patterns. van Hiele (1986)prepared a new taxonomy for geometry in five steps, from avisual to a logical level. Therefore, in the tests there was a focus onproperties of quadrangles and triangles, and how to determineareas by reorganising figures into new patterns. This means thatstructure was more important than formulas.The construction of tests of proportionality (M3) was morecomplicated. Firstly, tasks on proportionality can be found inmany different contexts, such as prescriptions, scales, speeds,discounts, interest, etc. Secondly, the mathematical model iscomplex and requires good knowledge of rational numbersand ratios (Lesh et al., 1988). It also requires a developed viewof multiplication, useful in operations with real numbers, not onlyas repeated addition, an operation limited to natural numbers(Lybeck, 1981; Degrande et al., 2016). A linear structure ofmultiplication as repeated addition leads to limitations interms of generalization and development of the concept ofmultiplication. This became evident in a study carried out in aSwedish context (Karlsson and Kilborn, 2018c). Proportionalitycan be expressed as a/b c/d or as a/b k. The latter can also beexpressed as a b·k, where k is a constant that determines therelationship between a and b. Common examples of k are speed(km/h), scale, and interest (%). An important pre-knowledge inorder to deal with proportions is to master fractions asequivalence classes like 1/3 2/6 3/9 4/12 5/15 6/18 7/21 8/24 . . . (Karlsson and Kilborn, 2020). It was importantto take all these aspects into account when constructing andassessing the solutions of the tasks.The tests were graded by an experienced teacher ofmathematics (4th author) and two students in their final yearof teacher training. Prior to grading, acceptable levels of interrater reliability were achieved by independent rating of students’solutions and discussions in which differences between thegraders were resolved. Each student response was to beassigned one point when it contained a correct answer andtwo points when the student provided argumentation for thecorrect answer and elaborated on explanation of his or hersolution. The assessment was thus based on quality aspectswith a focus on conceptual knowledge. As each subtestcontained three questions, it generated three student solutions.So, scores for each subtest ranged from 0 to 6 points and for thetotal scores from 0 to 18 points. To ascertain that pre- and posttests were equivalent in degree of difficulty, the tests wereadministered to an additional sample of 169 students in grade5. Test for each model was conducted separately, as studentsparticipated in pre- and post-test for each model during the samelesson. The order of tests was switched for half of the students inorder to avoid the effect of the order in which the pre- and posttests were presented. Correlation between students’ performanceon pre- and post-test was .39 (p 0.000) for tests ofmultiplication/division; .48 (p 0.000) for tests of geometry;and .56 (p 0.000) for tests of proportionality. Thus, the degree ofdifficulty may have differed between pre- and post-test.Implementation of the InterventionTo ensure the implementation of the intervention, the researchersvisited each teachers’ classroom twice during the two phases ofthe intervention period, as described above. During each visit, theresearchers observed the lesson, using a checklist comprisingthe five principles of the CL approach. After the lesson, theresearchers gave written and oral feedback to each teacher. Asseen in Table 1, in 18 of the 23 classes, the teachers implementedthe intervention in accordance with the principles of CL. Inaddition, the teachers were asked to report on the use of the CLapproach in their teaching and the use of problem-solvingactivities embedding CL during the intervention period. Asshown in Table 1, teachers in only 11 of 23 classes reportedusing the CL approach and problem-solving activities embeddedin the CL approach at least once a week.Control GroupThe teachers in the control group received 2 days of instruction prehension. The teachers were also supported witheducational materials including mathematical problemsKarlsson and Kilborn (2018b) and problem-solving principles(Pólya, 1948). However, none of the activities during training orin educational materials included the CL approach. As seen inTable 1, only 10 of 25 teachers reported devoting at least onelesson per week to mathematical problem-solving.MeasuresTests of Mathematical Problem-SolvingTests of mathematical problem-solving were administered beforeand after the intervention, which lasted for 15 weeks. The testswere focused on the models of multiplication/division, geometry,and proportionality. The three models were chosen based on thesyllabus of the subject of mathematics in grades 4 to 6 in theSwedish National Curriculum (Swedish National EducationalAgency, 2018). In addition, the intention was to create avariation of types of problems to solve. For each of these threemodels, there were two tests, a pre-test and a post-test. Each testcontained three tasks with increasing difficulty (SupplementaryAppendix SA).The tests of multiplication and division (Ma1) were chosenfrom different contexts and began with a one-step problem, whilethe following two tasks were multi-step problems. Concerningmultiplication, many students in grade 5 still understandmultiplication as repeated addition, causing significantproblems, as this conception is not applicable to multiplicationbeyond natural numbers (Verschaffel et al., 2007). This might bea hindrance in developing multiplicative reasoning (Barmbyet al., 2009). The multi-step problems in this study wereconstructed to support the students in multiplicative reasoning.Concerning the geometry tests (Ma2), it was important toconsider a paradigm shift concerning geometry in education thatoccurred in the mid-20th century, when strict Euclideangeometry gave way to other aspects of geometry likeFrontiers in Education www.frontiersin.orgMeasures of Peer Acceptance and FriendshipsTo investigate students’ peer acceptance and friendships, peernominations rated pre- and post-intervention were used.5August 2021 Volume 6 Article 710296

Klang et al.Mathematical Problem-Solving Through CLTABLE 2 Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.CL group (269)Ma totProblem-solvingMa 1Multiplication/DivisionMa 2GeometryMa 3ProportionalityControlgroup (312)Effect of CLBelow median at pre-testAbove median at pre-testPrePostPrePostb1 (95% CI)b1 (95% CI)b1 (95% (2.02)4.90(1.47)1.30***(.57; 2.03) .03( .41; .36)1.07***(.66; 1.48).26( .13; .64) .21( 1.10; .67) .50*( .97; .03) .90 3.87; 2.08)-.25 .74; .24).74( .09; 1.56) 09( .45; .27).55(.001; 1.11)-.27( .72; .19)Note. *p 0.05, **p 0.01, ***p 0.001.Students were asked to nominate peers who they preferred towork in groups with and who they preferred to be friends with.Negative peer nominations were avoided due to ethicalconsiderations raised by teachers and parents (Child andNind, 2013). Unlimited nominations were used, as these areconsidered to have high ecological validity (Cillessen andMarks, 2017). Peer nominations were used as a measure ofsocial acceptance, and reciprocated nominations were used asa measure of friendship. The number of nominations for eachstudent were aggregated and divided by the number ofnominators to create a proportion of nominations for eachstudent (Velásquez et al., 2013).multicollinearity was met, as the variance inflation factors(VIF) did not exceed a value of 10. Before the analyses, thecases with missing data were deleted listwise.RESULTSWhat Is the Effect of the CL Approach onStudents’ Problem-Solving inMathematics?As seen in the regression coefficients in Table 2, the CLintervention had a significant effect on students’ mathematicalproblem-solving total scores and students’ scores in problemsolving in geometry (Ma2). Judging by mean values, students inthe intervention group appeared to have low scores on problemsolving in geometry but reached the levels of problem-solving ofthe control group by the end of the intervention. The interventiondid not have a significant effect on students’ performance inproblem-solving related to models of multiplication/division andproportionality.The question is, however, whether CL intervention affectedstudents with different pre-test scores differently. Table 2includes the regression coefficients for subgroups of studentswho performed below and above median at pre-test. As seen inthe table, the CL approach did not have a significant effect onstudents’ problem-solving, when the sample was divided intothese subgroups. A small negative effect was found forintervention group in comparison to control group, butconfidence intervals (CI) for the effect indicate that it was notsignificant.Statistical AnalysesMultilevel regression analyses were conducted in R, lme4 packageBates et al. (2015) to account for nestedness in the data. Students’classroom belonging was considered as a level 2 variable. First, weused a model in which students’ results on tests of problemsolving were studied as a function of time (pre- and post) andgroup belonging (intervention and control group). Second, thesame model was applied to subgroups of students who performedabove and below median at pre-test, to explore whether the CLintervention had a differential effect on student performance. Inthis second model, the results for subgroups of students could notbe obtained for geometry tests for subgroup below median and fortests of proportionality for subgroup above median. A possiblereason for this must have been the skewed distribution of thestudents in these subgroups. Therefore, another model wasapplied that investigated students’ performances in math atboth pre- and post-test as a function of group belonging.Third, the students’ scores on social acceptance andfriendships were added as an interaction term to the firs

focused on supporting students in problem-solving, through instruction in problem-solving principles (Pólya, 1948), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality. Students' problem-solving may be enhanced through participation in small group discussions. In a .

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