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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training 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 of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study 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 small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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Development and differences in mathematical problem-solving skills: A cross-sectional study of differences in demographic backgrounds

Ijtihadi kamilia amalina.

a Doctoral School of Education, University of Szeged, Hungary

Tibor Vidákovich

b Institute of Education, University of Szeged, Hungary

Associated Data

Data will be made available on request.

Problem-solving skills are the most applicable cognitive tool in mathematics, and improving the problem-solving skills of students is a primary aim of education. However, teachers need to know the best period of development and the differences among students to determine the best teaching and learning methods. This study aims to investigate the development and differences in mathematical problem-solving skills of students based on their grades, gender, and school locations. A scenario-based mathematical essay test was administered to 1067 students in grades 7–9 from schools in east Java, Indonesia, and their scores were converted into a logit scale for statistical analysis. The results of a one-way analysis of variance and an independent sample t -test showed that the students had an average level of mathematical problem-solving skills. The number of students who failed increased with the problem-solving phase. The students showed development of problem-solving skills from grade 7 to grade 8 but not in grade 9. A similar pattern of development was observed in the subsample of urban students, both male and female. The demographic background had a significant effect, as students from urban schools outperformed students from rural schools, and female students outperformed male students. The development of problem-solving skills in each phase as well as the effects of the demographic background of the participants were thoroughly examined. Further studies are needed with participants of more varied backgrounds.

1. Introduction

Problem-solving skills are a complex set of cognitive, behavioral, and attitudinal components that are situational and dependent on thorough knowledge and experience [ 1 , 2 ]. Problem-solving skills are acquired over time and are the most widely applicable cognitive tool [ 3 ]. Problem-solving skills are particularly important in mathematics education [ 3 , 4 ]. The development of mathematical problem-solving skills can differ based on age, gender stereotypes, and school locations [ [5] , [6] , [7] , [8] , [9] , [10] ]. Fostering the development of mathematical problem-solving skills is a major goal of educational systems because they provide a tool for success [ 3 , 11 ]. Mathematical problem-solving skills are developed through explicit training and enriching materials [ 12 ]. Teachers must understand how student profiles influence the development of mathematical problem-solving skills to optimize their teaching methods.

Various studies on the development of mathematical problem-solving skills have yielded mixed results. Grissom [ 13 ] concluded that problem-solving skills were fixed and immutable. Meanwhile, other researchers argued that problem-solving skills developed over time and were modifiable, providing an opportunity for their enhancement through targeted educational intervention when problem-solving skills developed quickly [ 3 , 4 , 12 ]. Tracing the development of mathematical problem-solving skills is crucial. Further, the results of previous studies are debatable, necessitating a comprehensive study in the development of students’ mathematical problem-solving skills.

Differences in mathematical problem-solving skills have been identified based on gender and school location [ [6] , [7] , [8] , [9] , [10] ]. School location affects school segregation and school quality [ 9 , 14 ]. The socioeconomic and sociocultural characteristics of a residential area where a school is located are the factors affecting academic achievement [ 14 ]. Studies in several countries have shown that students in urban schools demonstrated better performance and problem-solving skills in mathematics [ 9 , 10 , 15 ]. However, contradictory results have been obtained for other countries [ 6 , 10 ].

Studies on gender differences have shown that male students outperform female students in mathematics, which has piqued the interest of psychologists, sociologists, and educators [ 7 , 16 , 17 ]. The differences appear to be because of brain structure; however, sociologists argue that gender equality can be achieved by providing equal educational opportunities [ 8 , 16 , 18 , 19 ]. Because the results are debatable and no studies on gender differences across grades in schools have been conducted, it would be interesting to investigate gender differences in mathematical problem-solving skills.

Based on the previous explanations, teachers need to understand the best time for students to develop mathematical problem-solving skills because problem-solving is an obligatory mathematics skill to be mastered. However, no relevant studies focused on Indonesia have been conducted regarding the mathematical problem-solving skill development of students in middle school that can provide the necessary information for teachers. Further, middle school is the important first phase of developing critical thinking skills; thus relevant studies are required in this case [ 3 , 4 ]. In addition, a municipal policy-making system can raise differences in problem-solving skills based on different demographic backgrounds [ 10 ]. Moreover, the results of previous studies regarding the development and differences in mathematical problem-solving skills are debatable. Thus, the present study has been conducted to meet these gaps. This study investigated the development of mathematical problem-solving skills in students and the differences owing demographic backgrounds. Three aspects were considered: (1) student profiles of mathematical problem-solving skills, (2) development of their mathematical problem-solving skills across grades, and (3) significant differences in mathematical problem-solving skills based on gender and school location. The results of the present study will provide detailed information regarding the subsample that contributes to the development of mathematical problem-solving skills in students based on their demographic backgrounds. In addition, the description of the score is in the form of a logit scale from large-scale data providing consistent meaning and confident generalization. This study can be used to determine appropriate teaching and learning in the best period of students’ development in mathematical problem-solving skills as well as policies to achieve educational equality.

2. Theoretical background

2.1. mathematical problem-solving skills and their development.

Solving mathematical problems is a complex cognitive ability that requires students to understand the problem as well as apply mathematical concepts to them [ 20 ]. Researchers have described the phases of solving a mathematical problem as understanding the problem, devising a plan, conducting out the plan, and looking back [ [20] , [24] , [21] , [22] , [23] ]. Because mathematical problems are complex, students may struggle with several phases, including applying mathematical knowledge, determining the concepts to use, and stating mathematical sentences (e.g., arithmetic) [ 20 ]. Studies have concluded that more students fail at later stages of the solution process [ 25 , 26 ]. In other words, fewer students fail in the phase of understanding a problem than during the plan implementation phase. Different studies have stated that students face difficulties in understanding the problem, determining what to assume, and investigating relevant information [ 27 ]. This makes them unable to translate the problem into a mathematical form.

Age or grade is viewed as one factor that influences mathematical problem-solving skills because the skills of the students improve over time as a result of the teaching and learning processes [ 28 ]. Neuroscience research has shown that older students have fewer problems with arithmetic than younger students; however, the hemispheric asymmetry is reduced [ 29 ]. In other words, older students are more proficient, but their flexibility to switch among different strategies is less. Ameer & Sigh [ 28 ] obtained similar results and found a considerable difference in mathematical achievement; specifically, older students performed better than younger students in number sense and computation using one-way analysis of variance (ANOVA) ( F ) of F (2,411) = 4.82, p  < 0.01. Molnár et al. [ 3 ] found that the student grade affects domain-specific and complex problem-solving skills. They observed that the development of problem-solving skills was noticeable across grades in elementary school but stopped in secondary school. The fastest development of domain-specific problem-solving occurred in grades 7 and 8 [ 3 ], but the fastest development of complex problem-solving occurred in grades 5–7 [ 3 ]. No development was detected between grades 4 and 5 as well as grades 6 and 7 for domain-specific and complex problem-solving skills, respectively. Similarly, Greiff et al. [ 4 ] concluded that students developed problem-solving skills across grades 5–11 with older students being more skilled. However, the grade 9 students deviated from the development pattern, and their problem-solving skills dropped. The theories from Molnár et al. [ 3 ] and Greiff et al. [ 4 ] are the benchmark cases herein.

The above studies showed that problem-solving skills mostly developed during compulsory schooling and developed most quickly in specific grades. This indicates that specific development times can be targeted to enhance the problem-solving skills [ 3 ]. However, Jabor et al. [ 30 ] observed contradictory results showing statistically significant differences with small effects in mathematical performance between age groups: those under the age of 19 outperformed those over the age of 19 years old. Grissom [ 13 ] observed a negative correlation between age and school achievement that remained constant over time.

2.2. Effects of school location and gender on mathematical problem-solving skills

School location has been shown to affect mathematical achievement [ 9 , 14 ]. In 15 countries, students in rural schools performed considerably worse than students in urban schools in mathematics [ 9 , 10 ], science and reading [ 9 ]. In addition, Nepal [ 15 ] discovered that urban students significantly outperformed rural students in mathematical problem-solving skills ( t  = −5.11, p  < 0.001) and achievement ( t  = −4.45, p  < 0.001) using the results of an independent sample t -test (t). However, other countries have found that rural students outperformed urban students in mathematics [ 6 , 10 ]. These variations may be attributed to a lack of instructional resources (e.g., facilities, materials, and programs), professional training (e.g., poorly trained teachers), and progressive instruction [ 6 ]. The results of Williams's study [ 10 ] serve as the basis for the current study.

Gender differences in mathematics have received attention because studies show that male students outperform female students on higher-level cognitive tasks [ 31 ]. This is a shift from a meta-analysis study that found gender differences in mathematics to be insignificant and favored female students [ 32 ]. At the college level, female students slightly outperform male students in computation while male students outperform female students in problem solving. However, no gender differences have been observed among elementary and middle school students. This result was strengthened by other meta-analysis studies [ 7 , 8 ], which concluded that there was no gender difference in mathematical performance and problem-solving skills [ 15 , [33] , [35] , [34] ]. Gender similarity in mathematics is achieved when equal learning opportunities and educational choices are provided and the curriculum is expanded to include the needs and interests of the students [ 16 , 18 , 31 ].

From a sociological perspective, gender similarity in mathematics makes sense. If there is a gender difference in mathematics, this has been attributed to science, technology, engineering, and mathematics (STEM) being stereotyped as a male domain [ 8 ]. Stereotypes influence beliefs and self-efficacy of students and perceptions of their own abilities [ 8 , 19 ]. This is the reason for the low interest of female students in advanced mathematics courses [ 18 , 19 ]. However, Halpern et al. [ 16 ] found that more female students are entering many occupations that require a high level of mathematical knowledge. Moreover, Anjum [ 36 ] found that female students outperformed male students in mathematics. This may be because female students prepared better than the male students before the test and were more thorough [ 36 , 37 ]. The study of Anjum [ 36 ] is one of the basis cases of the current study.

Differences in brain structure support the argument that there are gender differences in mathematical performance [ 16 , 17 ]. Females have less brain lateralization (i.e., symmetric left and right hemispheres), which helps them perform better verbally. Meanwhile, males have more brain lateralization, which is important for spatial tasks [ 17 ]. In addition, the male hormone testosterone slows the development of the left hemisphere [ 16 ], which improves the performance of right brain-dominant mathematical reasoning and spatial tasks.

3.1. Instrumentation

In this study, a science-related mathematical problem-solving test was used. This is a mathematics essay test where the problems are in the form of scenarios related to environmental management. Problems are solved by using technology as a tool (e.g., calculator, grid paper). The test was developed in an interdisciplinary STEM framework, and it is targeted toward grades 7–9. There were six scenarios in total: some were given to multiple grades, and others were specific to a grade. They included ecofriendly packaging (grade 7), school park (grade 7), calorie vs. greenhouse gas emissions (grades 7–9), floodwater reservoir (grade 8), city park (grades 8–9), and infiltration well (grade 9). These scenarios cover topics such as number and measurement, ratio and proportion, geometry, and statistics. Every scenario had a challenge, and students were provided with eight metacognitive prompt items to help them explore their problem-solving skills.

The test was administered by using paper and pencils for a 3-h period with a break every hour. At the end of the test, students were asked to fill in their demographic information. Each prompt item had a maximum score of 5 points: a complete and correct answer (5 points), a complete answer with a minor error (4 points), an incomplete answer with a minor error (3 points), an incomplete answer with a major error (2 points), and a completely wrong and irrelevant answer (1 point). Each scenario had a maximum total score of 40 points.

The test was validated to determine whether it contained good and acceptable psychometric evidence. It had an acceptable content validity index (CVI >0.67), moderate intraclass correlation coefficient (ICC) (rxx = 0.63), and acceptable Cronbach's alpha (α = 0.84). The construct validity indicated all scenarios and prompt items were fit (0.77 ≤ weighted mean square ≤1.59) with an acceptable discrimination value (0.48 ≤ discrimination value ≤ 0.93), acceptable behavior of the rating score, and good reliability (scenario reliability = 0.86; prompt item reliability = 0.94).

3.2. Participants

The test was administered to grades 7–9 students in east Java, Indonesia (n = 1067). The students were selected from A-accreditation schools in urban and rural areas; random classes were selected for each grade. The majority of the students were Javanese (95.01%), with the remainder being Madurese (3.3%) and other ethnicities. Table 1 describes the demographics of the participants.

Demographic characteristics of participants.

3.3. Data analysis

Data were collected between July and September 2022. Prior to data collection, ethical approval was sought from the institutional review board (IRB) of the Doctoral School of Education, University of Szeged and was granted with the ethical approval number of 7/2022. In addition, permission letters were sent to several schools to request permission and confirm their participation. The test answers of the students were scored by two raters – the first author of this study and a rater with master's degree in mathematics education – to ensure that the rating scale was consistently implemented. The results showed good consistency with an ICC of 0.992 and Cronbach's alpha of 0.996.

The scores from one of the raters were converted to a logit scale by weighted likelihood estimation (WLE) using the ConQuest software. A logit scale provides a consistent value or meaning in the form of intervals. The logit scale represents the unit interval between locations on the person–item map. WLE was chosen rather than maximum likelihood estimation (MLE) because WLE is more central than MLE, which helps to correct for bias [ 38 ]. The WLE scale was represented by using descriptive statistics to profile the students' mathematical problem-solving skills in terms of the percentage, mean score ( M ) and standard deviation ( SD ) for each phase. The WLE scale was also used to describe common difficulties for each phase. The development of students’ mathematical problem-solving skills across grades was presented by a pirate plot, which is used in R to visualize the relationship between 1 and 3 categorical independent variables and 1 continuous dependent variable. It was chosen because it displays raw data, descriptive statistics, and inferential statistics at the same time. The data analysis was performed using R studio version 4.1.3 software with the YaRrr package. A one-way ANOVA was performed to find significant differences across grades. An independent sample t -test was used to analyze significant differences based on gender and school location. The descriptive statistics, one-way ANOVA test, and independent sample t -test were performed using the IBM SPSS Statistics 25 software.

4.1. Student profiles

The scores of students were converted to the WLE scale, where a score of zero represented a student with average ability, a positive score indicated above-average ability, and a negative score indicated below-average ability. A higher score indicated higher ability. The mean score represented a student with average mathematical problem-solving skills ( M  = 0.001, SD  = 0.39). Overall, 52.1% of students had a score below zero. The distribution of scores among students was predominantly in the interval between −1 and 0. When the problem-solving process was analyzed by phase, the results showed that exploring and understanding were the most mastered problem-solving skills ( M  = 0.24, SD  = 0.51). Only 27.9% of students had below-average scores for the exploring and understanding phases, which indicates that they mostly understood the given problem and recognized the important information. However, the problem-solving skills decreased with higher phases. The students had below-average abilities in the phases of representing and formulating ( M  = −0.01, SD  = 0.36), planning and executing ( M  = −0.15, SD  = 0.41), and monitoring and reflecting ( M  = −0.16, SD  = 0.36). About 57.9% of the students had below-average scores for the representing and formulating phase, which indicates that they had problems making hypotheses regarding science phenomena, representing problems in mathematical form, and designing a prototype. The obvious reason for their difficulty with making hypotheses was that they did not understand simple concepts of science (e.g., CO 2 vs. O 2 ). In the planning and executing phase, 66.8% of the students failed to achieve a score greater than zero. This happened because they failed to apply mathematical concepts and procedures. Because they were unable to plan and execute a strategy, this affected the next phase of the problem-solving process. In the monitoring and reflecting phase, 68.0% of the students had a below-average score.

4.2. Development of mathematical problem-solving skills across grades

The development of the mathematical problem-solving skills of the students across grades was observed based on the increase in the mean score. The problem-solving skills developed from grade 7 to grade 8. The students of grade 7 had a mean score of −0.04 while grade 8 students had the highest mean score of 0.03. The students in grades 7 and 8 also showed more varied problem-solving skills than the grade 9 students did. In contrast, the grade 9 students showed a different pattern of development, and their mean score dropped to 0.01. Although the difference was not large, further analysis was needed to determine its significance.

Fig. 1 displays the development of the mathematical problem-solving skills of the students. The dots represent raw data or WLE scores. The middle line shows the mean score. The beans represent a smoothed density curve showing the full data distribution. The scores of the students in grades 7 and 9 were concentrated in the interval between −0.5 and 0. However, the scores of the grade 8 students were concentrated in the interval between 0 and 0.5. The scores of the students in grades 7 and 8 showed a wider distribution than those of the grade 9 students. The bands which overlap with the line representing the mean score, define the inference around the mean (i.e., 95% of the data are in this interval). The inference of the WLE score was close to the mean.

Differences in students' mathematical problem-solving skills across grades.

Note : PS: Problem-Solving Skills of Students.

The one-way ANOVA results indicated a significant difference among the problem-solving skills of the students of grades 7–9 ( F (1,066) = 3.01, p  = 0.046). The students of grade 8 showed a significant difference in problem-solving skills and outperformed the other students. The students of grades 7 and 9 showed no significant difference in their mathematical problem-solving skills. Table 2 presents the one-way ANOVA results of the mathematical problem-solving skills across grades.

One-way ANOVA results of the mathematical problem-solving across grades.

Note. Post hoc test: Dunnett's T3. 7, 8, and 9: subsample grade. <: direction of significant difference ( p  < 0.05).

Fig. 2 shows the development of the mathematical problem-solving skills of the students across grades based on school location and gender. The problem-solving skills of the urban students increased from a mean score of 0.07 in grade 7 to 0.14 in grade 8. However, the mean score of urban students in grade 9 dropped. In contrast, the mean scores of the rural students increased continuously with grade. The improvements were significant for both the rural ( F (426) = 10.10, p  < 0.001) and urban ( F (639) = 6.10, p  < 0.01) students. For the rural students, grade 9 students showed a significant difference in problem-solving skills. In contrast, urban students in grades 8 and 9 showed significant differences in problem-solving skills but not in grade 7.

Differences in students' mathematical problem-solving skills across grades and different demographic backgrounds.

(a) Differences in students grade 7 of mathematical problem-solving skills across grades and different demographic backgrounds

(b) Differences in students grade 8 of mathematical problem-solving skills across grades and different demographic backgrounds

(c) Differences in students grade 9 of mathematical problem-solving skills across grades and different demographic backgrounds

Note: WLE_PS: The students' problem-solving skills in WLE scale; F: Female; M: Male; ScLoc: School location; R: Rural; U: Urban.

When divided by gender, both female and male students showed improvements in their problem-solving skills from grades 7 and 8. However, female students in grade 9 showed a stable score while the scores of male students in grade 9 declined. Only male students in grade 7 showed a significant difference in the mean score. In urban schools, the scores of male and female students increased and decreased, respectively, from grade 7 to grade 8. Male students in rural schools showed an increase in score from grade 7 to grade 9. However, the scores of female students in rural schools decreased from grade 7 to grade 8. Table 3 presents the one-way ANOVA results for the mathematical problem-solving skills of the students considering gender and school location.

One-way ANOVA results for mathematical problem-solving skills across grades and different demographic backgrounds.

Fig. 2 shows that the distributions of the male and female scores of students were similar for every grade except rural grade 9 students. The scores of the rural female students were concentrated in the interval between 0 and 0.5 while the scores of the rural male students were mostly below 0. The scores of rural students in grade 7 and urban students in grade 9 (both male and female) were concentrated in the interval between −0.5 and 0. The scores of urban students in grades 7 and 8 were concentrated in the interval between −0.5 and 0.5.

Fig. 3 shows a detailed analysis of the development of mathematical problem-solving skills across grades for each phase of the problem-solving process. Similar patterns were observed in the exploring and understanding and the representing and formulating phases: the mean score increased from grade 7 to grade 8 but decreased from grade 8 to grade 9. Grade 8 students had the highest mean score and differed significantly from the scores of students in other grades.

Differences in students' mathematical problem-solving skills in every phase across grades: (1) Exploring & understanding, (2) Representing & formulating, (3) Planning & executing, (4) Monitoring & reflecting.

(a) Differences in students' mathematical problem-solving skills in exploring and understanding phase

(b) Differences in students' mathematical problem-solving skills in representing and formulating phase

(c) Differences in students' mathematical problem-solving skills in planning and executing phase

(d) Differences in students' mathematical problem-solving skills in monitoring and reflecting phase

Note: WLE_Exp_Un: The WLE score in exploring and understanding; WLE_Rep_For: The WLE score in representing and formulating; WLE_Plan_Ex: The WLE score in planning and executing; WLE_Mon_Ref: The WLE score in monitoring and reflecting.

The scores of the students for the planning and executing phase increased with grade. However, the difference was only significant at grade 9. Grades 7 and 8 students showed an increase in score, but the improvement was not significant. There was no pattern detected in the monitoring and reflecting phase. The score was stable for grades 7 and 8 students but improved for grade 9 students. The mean score for each phase and the one-way ANOVA results are presented in Table 4 .

One-way ANOVA results for every phase of problem-solving across grades.

Fig. 3 shows that the distributions of the problem-solving skills of the students were similar across grades and phases. However, the distributions were different for grade 9 students in the representing and formulating, planning and executing, and monitoring and reflecting phases, where 95% of the data were in the interval between −0.5 and 0.5.

4.3. Effects of demographic background

4.3.1. school location.

The mathematical problem-solving skills of the students differed significantly based on school location. Urban students scored higher than rural students. The results of the t -test for mathematical problem-solving skills based on school location are presented in Table 5 .

T-test results for mathematical problem-solving skills based on school location.

The effects of the school's location on the performances of male and female students were analyzed. The results showed that the scores of the female students differed significantly based on school location ( t (613) = −6.09, p  < 0.001). Female students in urban schools ( M  = 0.18, SD  = 0.39) outperformed female students in rural schools ( M  = −0.08, SD  = 0.37). Similar results were observed for male students with urban students ( M  = −0.01, SD  = 0.35) outperforming rural students ( M  = −0.12, SD  = 0.39) by a significant margin ( t (382.764) = −3.25, p  < 0.01).

When analyzed by grade, grades 7 and 8 students contributed to the difference based on school location with t (377.952) = −6.34, p  < 0.001 and t (300.070) = −5.04, p  < 0.001, respectively. Urban students in grades 7 and 8 performed significantly better than their rural counterparts did. However, there was no significant difference between rural and urban students in grade 9 ( t (354) = 0.71, p  = 0.447).

4.3.2. Gender

Male and female students showed a significant difference in their mathematical problem-solving skills. Overall, female students outperformed male students. The detailed results of the independent sample t -test for mathematical problem-solving skills based on gender are presented in Table 6 .

T-test results for mathematical problem-solving skills based on gender.

The results were analyzed to determine whether the school location contributed to the gender difference. The gender difference was most significant among urban students ( t (596.796) = −4.36, p  < 0.001). Female students from urban schools ( M  = 0.12, SD  = 0.39) outperformed male students from urban schools ( M  = −0.01, SD  = 0.35). There was no significant difference between female and male students from rural schools ( t (425) = −1.31, p  = 0.191).

Grades 7 and 9 students contributed to the gender difference with t (372.996) = −3.90, p  < 0.001 and t (354) = −2.73, p  < 0.01, respectively. Female students in grades 7 and 9 outperformed their male counterparts. However, there was no significant gender difference among grade 8 students ( t (329) = −0.10, p  = 0.323).

5. Discussion

The mathematical problem-solving skills of the students were categorized as average. In addition, the difficulties of students increased in line with the problem-solving phase. Fewer students failed the exploring and understanding phase than the subsequent phases. This confirms the results of previous studies indicating that more students failed further along the problem-solving process [ 25 , 26 ]. Because the problem-solving process is sequential, students who have difficulty understanding a problem will fail the subsequent phases [ 27 ].

The development of mathematical problem-solving skills was evaluated according to the mean WLE score. The mathematical problem-solving skills of the students developed from grade 7 to grade 8 based on the increase in their mean scores. However, the development dropped in grade 9. This agrees with previous results that concluded that higher grades had the highest problem-solving skills, but the fastest skill development took place in grades 7–8 after which it dropped [ 3 , 4 ]. These results indicate that the mathematical problem-solving skills of the students should improve and be strengthened in grades 7–8, which will help them perform better in grade 9.

In this study, the effects of the demographic background of the students were analyzed in detail, which is an aspect missing from previous studies. The results showed that the mathematical problem-solving skills of urban students increased from grade 7 to grade 8 but decreased in grade 9. The same pattern was found among male and female students. However, a different pattern was observed for rural students, where the skills of grade 9 students continued to increase. The different patterns may be attributed to a structural reorganization of cognitive processes at a particular age [ 3 ]. However, more research is needed on the effects of the demographic backgrounds of students on mathematical problem-solving skills. These results were different from previous results because the previous studies only analyzed the development in general, without focusing on their demographic background. Hence, different patterns of development were observed when it was thoroughly examined.

Because solving problems is a cognitive process, the development of problem-solving skills for particular phases and processes needed to be analyzed. The students showed the same pattern for knowledge acquisition (i.e., exploring and understanding, and representing and formulating phases), with an increase in skill from grade 7 to grade 8 but a decrease in grade 9. However, the students showed increasing skill in knowledge application (i.e., planning and executing, as well as monitoring and reflecting phases) across grades. This means that the difference between the mean scores in grade 9 was not significant across phases. Grade 9 students had lower scores than grade 8 students for the knowledge acquisition phase but higher scores for the knowledge application phase. In contrast, the gap between the mean scores of grades 7 and 8 was large across phases.

These results proved that there is a significant difference in the mathematical problem-solving skills of students based on their demographic backgrounds. The urban students outperformed rural students, which confirms the results of previous studies [ 9 , 10 , 15 ]. The difference can be attributed to the availability of facilities, teacher quality, and interactive teaching and learning instruction [ 6 ]. In Indonesia, the policies for the public educational system for middle schools are set at the municipal level. This means that each city has its own policies for teacher training, teacher recruitment, teaching and learning processes, facilities, etc. Urban schools mostly have stricter policies as well as various programs to help students improve their knowledge and skills. In addition, they have supportive facilities for teaching and learning. This unequal environment is the strongest reason for the difference in mathematical problem-solving skills.

The results were analyzed in detail to observe which groups in the rural and urban schools contributed to the difference. Both male and female students in urban schools performed better than their counterparts in rural schools did. In addition, urban students in grades 7 and 8 outperformed their rural counterparts. There was no significant difference between urban and rural students in grade 9. This may be because grade 9 is the last grade in middle school, so students have to prepare for high school entrance requirements, including exam and/or grade point average scores. Hence, both rural and urban schools focus much effort on the teaching and learning process in this grade.

In this study, the female students surprisingly had better mathematical problem-solving skills than the male students did. This confirmed the results of the meta-analysis by Hyde et al. [ 32 ] and study by Anjum [ 36 ], which found that female students slightly outperformed male students in mathematics. This difference may be because of motivation and attitude [ 39 , 40 ]. Female Indonesian students are typically more diligent, thorough, responsible, persistent, and serious with their tasks.

A detailed analysis was performed to evaluate which group of students contributed to the gender differences. The results showed that female students outperformed male students in urban schools. This may be because male students at urban schools typically display an unserious attitude toward low-stake tests. In addition, female students outperformed their male counterparts in grades 7 and 9. The reason for this difference requires further analysis.

6. Conclusion

Studying the problem-solving skills of students is crucial to facilitating their development. In this study, the conclusions are presented as follows:

• • The mathematical problem-solving skills of the students were categorized as average. More students failed at higher phases of the problem-solving process.
• • Students showed development of their mathematical problem-solving skills from grade 7 to grade 8 but a decline in grade 9. The same pattern was detected across grades for urban students, both female and male. However, the problem-solving skills of rural students increased with the grade.
• • A similar development was observed for the individual problem-solving phases. In the knowledge acquisition phase, the problem-solving skills of the students developed from grade 7 to grade 8 but decreased in grade 9. However, problem-solving skills increased across grades in the knowledge application phase.
• • The school location was shown to have a significant effect on the mathematical problem-solving skills of the students. Urban students generally outperform students in rural schools. However, gender and grade contributed to differences in mathematical problem-solving skills based on school location. Female and male urban students in grades 7 and 8 outperformed their rural counterparts.
• • In general, female students outperformed male students in mathematical problem-solving skills, particularly those from urban schools and in grades 7 and 9.

The sampling method and the number of demographic backgrounds limited the scope of this study. Only students from A-accreditation schools were selected because higher-order problem-solving skills were considered assets. Moreover, the study only included three demographic factors: grade, gender, and school location. More demographic information, such as school type, can be added (public or private schools). Hence, future studies will need to broaden the sample size and consider more demographic factors. Despite these limitations, this study can help teachers determine the best period for enhancing the development of mathematical problem-solving skills. Moreover, the differences in mathematical problem-solving skills due to demographic background can be used as a basis for educational policymakers and teachers to provide equal opportunity and equitable education to students.

Author contribution statement

Ijtihadi Kamilia Amalina: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Tibor Vidákovich: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data.

Funding statement

This work was supported by University of Szeged Open Access Fund with the grant number of 6020.

Data availability statement

Additional information.

No additional information is available for this paper.

Declaration of competing interest

No potential conflict of interest was reported by the authors.

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Mathematical Problem-Solving Style and Performance Of Students

• Luvie Jhun S. Gahi
• Ronald E. Almagro
• Richie Ryan C. Sudoy
• Dec 21, 2023

Mathematical Problem-Solving Style and Performance of Students

1 Luvie Jhun S. Gahi, 2 Ronald E. Almagro, 3 Richie Ryan C. Sudoy

1 Student, Master of Arts in Education Major in Mathematics, St. Mary’s College of Tagum,

2 Student, Master of Arts in Education Major in English, St. Mary’s College of Tagum,

3 Student, Doctor of Philosophy in Educational Management, Davao del Norte State College, Philippines

DOI: https://dx.doi.org/10.47772/IJRISS.2023.7011142

Received: 10 October 2023; Revised: 17 November 2023; Accepted: 20 November 2023; Published: 21 December 2023

This study aimed to determine whether the mathematical problem-solving style significantly affects the students’ performance in which a descriptive-correlational research design was used. Through stratified sampling, there were 291 first-year college respondents in the local college in Sto. Tomas, Davao del Norte who were chosen. This study used one adapted questionnaire and one researchers-made questionnaire with Mean, Pearson r, Standard Deviation, T-test, and Analysis of Variance used as the statistical tools. The students’ mathematics attained very good performance with the mathematical problem-solving style of students’ sensing, intuition, feeling, and thinking moderately observed. The findings revealed that the mathematical problem-solving style has a significant relationship with Students’ performance. However, there is a significant difference in the mathematical problem-solving style of students when grouped according to various programs. The result also revealed that there is no significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female).  Students, instructors, college administrators, and Commission of Higher Education officials (CHED) are encouraged to value the importance of mathematical problem-solving style in the performance of the students. Instructors, college administrators, and CHED officials must establish programs that will enhance the mathematical problem-solving styles and performance of students. College instructors and administrators must work collaboratively to achieve better performance in the mathematics. Therefore, these should ensure that the necessary materials, resources, activities, and differentiated instruction are provided and used to meet the students’ needs to learn and to encourage in the problem-solving style.

Keywords: Student’s Profile, Mathematical Problem-Solving Style, and Performance of Students

INTRODUCTION

Good problem-solving abilities are required for all issues emerging from daily activity or to progress through the developmental stages. Effective problem-solving style has been linked to beneficial psychological outcomes such as competence, productivity, and optimism (Carver & Scheier, 1999; Chang & D’Zurilla, 1996; Elliott, et al., 1994). Additionally, according to the National Council of Teachers of Mathematics (NCTM), problem-solving ability is an essential component of all mathematics learning. The ability to solve problems can provide significant benefits in everyday life and the workplace. However, problem-solving is not only a goal of learning mathematics, but also a major method of learning mathematical concepts.

Likewise, the process of problem-solving begins with the observation of a gap, the application, and the complete evaluation of a theory to close that loophole. Styles of problem-solving are viewed as contrasting individuals’ unique characteristics with the behaviors that people prefer to draw and concentrate on their efforts to arrive at some comprehension or awareness, generate ideas, and make plans for the work(Sutherland, 2002).In the local college of Sto. Tomas, Davao del Norte, the varying levels of student performance in mathematics courses was used in this study. Surprisingly, there seemed to be a gap in the published research addressing this issue (Almagro etal., 2023)

The ability and style to solve problems increase the students’ comfort level when solving mathematical problems and practical difficulties. In turn, having the ability to solve problems has several advantages. For instance, problem-solving style is a feature of mathematical activity and a key method for developing mathematical understanding (NCTM, 2000). This statement implies that problem-solving style is an essential component of mathematics education. Furthermore, students learn to apply their mathematical skills in various ways; they gain a deeper understanding of mathematical concepts and gain firsthand experience as a mathematician by solving problems (Badger et al., 2012). Consequently, instruction should be advanced to enable the students to recognize and address the issues they encountered in real-life situations (Phonapichat et al., 2014). Nevertheless, several research findings suggest that children struggle to solve problems because of (Herawatty et al., 2018). Thus, learning mathematics should encourage students to solve problems confidently using mathematics. Learning mathematics in school should assist the students in understanding and applying mathematics to their problems that occur in their daily lives and in the workplace. The learning program must enable students to develop new mathematical knowledge through problem-solving style, solve mathematics and other problems, implement and adjust various problem-solving strategies, and monitor and reflect on the problem-solving style (NCTM, 2000).

The existing literature acknowledges challenges in mathematical problem-solving, but there is a significant gap in understanding the specific difficulties students face and the effectiveness of different problem-solving strategies. GanzonandEdig (2022), recognize the challenges, there is a need for in-depth investigations into the categories of difficulties encountered during the problem-solving process, low academic performance during in the pandemic. Additionally, the literature notes the importance of problem-solving models (Foshay & Kirkley, 2003; Almagro & Edig,2023), but a gap exists in understanding the comparative effectiveness of these motivated learning strategies. Moreover, within Realistic Mathematics Education (RME), recognized for its real-world emphasis, there is a need to examine the specific contextual factors that contribute to or hinder students’ success in mathematical problem-solving. Addressing these gaps will provide valuable insights for supporting students in developing effective problem-solving skills in mathematics.

The objective of this study is to investigate the relationship between mathematical problem-solving styles and the performance of students. Specifically, it aims to identify the various problem-solving styles employed by students, explore the challenges they face in mathematical problem-solving, and assess the impact of different problem-solving strategies on overall performance. The study seeks to contribute valuable insights that can inform educational practices and enhance students’ proficiency in mathematical problem-solving.

Statement of the Problem

The purpose of this study was to determine the relationship between mathematical problem-solving style and performance of first-year college students in Sto. Tomas College of Agriculture, Science and Technology (STCAST)in the academic year 2022-2023.

Specifically, these research questions sought to answer the following:

1. What is the measurable level of students’ performance in solving math problems? 2. What is the quantifiable level of students’ mathematical problem-solving style, considering the dimensions of sensing, intuition, feeling, and thinking? 3. Is there a significant and measurable relationship between students’ mathematical problem-solving style and their performance? 4. Can the mathematical problem-solving style of students be significantly differentiated when grouped according to sex and program, making it a specific and measurable analysis? 5. What specific and measurable instructional interventions can be proposed based on the study’s results, ensuring relevance and time-bound applicability?

The following hypotheses was tested at a 0.05 level of significance. Specifically, this was drawn to determine whether mathematical problem-solving style of students differ in terms of their sex and program.

• There is no significant relationship between the mathematical problem-solving style and performance of students.
• There is no significant difference on the mathematical problem-solving style of the students when classified according to sex and programs.

Theoretical Framework

The study is grounded in the original problem-solving style model, rooted in the concept of psychological functions as proposed by Jung (1923) and further developed by Moon (2008) and Taylor & Mackenny (2008). This model encompasses thinking, feeling, sensation, and intuition as the four psychological functions (Ghodrati et al., 2014). Building upon this foundation, the research draws attention to gender-specific problem-solving tendencies, with Burkey and Miller (2005) finding that women often employ intuition in work settings, contrasting the emphasis on rational problem-solving linked to masculinity by Wang, Heppner, and Berry (2007). Additionally, Conner (2000) identifies women as more intuitive global thinkers, emphasizing simultaneous, interconnected processing of information. The study aligns with the belief that students’ problem-solving methods significantly influence their academic achievement and success (Poshtiban, 2007; Morton, 2001).

Conceptual Framework

The study’s conceptual paradigm, which is shown in Figure 1, summarizes the variables which composed of mathematical problem-solving style and performance of students. On the one hand, the independent variable consists of mathematical problem-solving style which includes the indicators of sensing, intuitive, feeling, and thinking. On the other hand, the dependent variable consists of the performance of students in mathematics which composed of the moderating variable, i.e., the student respondent’s profile which are classified into sex and program. The researchers’ interpretation explains what the study wants to achieve as emphasized in the figure below. As such, the study aims to assess the Mathematical Problem-Solving Style of College students, and its relationship to their performance, and produce Students’ Instructional Intervention Plan that will improve their styles in solving mathematical problems.

 Figure 1. The Conceptual Paradigm of the Study

METHODOLOGY

This section covers the study’s numerous methodologies, which include the research design, respondents, research instrument, data gathering procedures, statistical treatment of data, and ethical considerations.

Research Design

This study employed descriptive and correlational study design. Descriptive research entails gatherings of quantitative data that may be tabulated along a scale in numerical forms, such as test scores. It entails collecting data by describing occurrences and then arranging, tabulating, displaying, and summarizing the data (Glass & Hopkins, 1984). The researcher will utilize this design to determine and describe the variables employed in this study. It utilized the mean test since this aimed to measure the level of performance of students.

Correlational study, meanwhile, tried to establish correlations between two or more variables. It looked to see if a rise or drop in one variable corresponded to an increase or decrease in another (Tan, 2014). This design will be utilized by the researcher to examine and determine the existing correlations between the variables in this research.

This study was concerned with data collection utilizing adopted research instrument and a pilot-tested researchers’ made examination to evaluate the hypotheses whether the mathematical-problem solving style influences the student performance. It will test the data using the proper statistical tools. Furthermore, the study’s major objective is to distinguish between the mathematical problem-solving styles of students when they are classified by sex and program. Thus, the study intends to look into the relationship between mathematical problem-solving style and performance of freshmen students in various programs at Sto. Tomas College of Agriculture, Sciences, and Technology.

Participants of the Study

The respondents of this research were the first-year college students enrolled in bachelor of Technical and Vocational Teacher Education (BTVTED), Bachelor of Science in Agricultural Business (BSAB), Bachelor of Science in Office Administration (BSOA), and Bachelor of Public Administration (BPA)programs for the school year 2022-2023. The respondents’ total population size of this study comprises of 1,188 students coming from four (4) programs in Sto. Tomas College of Agriculture, Sciences and Technology (STCAST). Specifically, BSOA department consists of 426 first-year students, BSAB department consists of 381 first-year students, BPA department consists of 217 first-year students, and BTVTED department consists of 164 first-year students. By using Qualtrics online sample size calculator, given the identified collective population size of 1,188 students, the ideal sample size of this quantitative study will consist of 291 students in total.

Moreover, this study utilized a stratified sampling technique to determine the sample size and determine the final total number of respondents. As a result, the BSOA program has an ideal sample of 176 students, BSAB program with 94 students, BPA with 53 students, and BTVTED with 40 students.

Materials/Research Instrument

One adapted research instrument and one researcher-made examination were used in this study. This was selected and modified to match the overall objectives of the study. These research instruments were validated by a panel of experts.

Problem-Solving Style Questionnaire (PSSQ). This instrument contains a 20-item survey questionnaire comprising the six (4) components problem-solving in mathematics such as Sensing (5 items), Intuitive (5 items), Feeling (5 items), and Thinking (5 items). This questionnaire was anchored on a 5-point Likert scale ranging from 5 as strongly agree to 1 as strongly disagree.

The following parameter limits, with its corresponding descriptions, were applied for the level of students’ mathematical problem-solving style.

The instrument for performance of students in Mathematics was a pilot-tested researcher-made questionnaire worth 40-item questionnaire. This instrument had been determined to possess good psychometric validity and reliability. The value of Cronbach’s a for the total scale is 0.747. All items of problem-solving skills are acceptable.

The percentage of the test score was computed by dividing the number of correct responses over the total highest possible score by multiplying it by 70 add 30. The highest probable score to achieve will be 40.

For the level of the mathematical problem-solving skills, the following parameter was used.

Data Gathering Procedure

The necessary data was gathered in a systematic procedure, which will involve the following.

Seeking permission to conduct the study. The researcher sought approval to conduct the research project. Primarily, the researcher will acquire a letter of recommendation from the College President of Santo Tomas College of Agriculture, Sciences, and Technology (STCAST). After acceptance, the researcher submitted a copy of the recommendation to the respective Department Heads of the following four (4) Degree Programs such as Bachelor of Science in Agriculture and Business (BSAB), Bachelor of Technical Vocational Teacher Education (BTVTED), Bachelor of Science in Office Administration (BSOA), and Bachelor of Public Administration (BPA) to finalize the approval to conduct the entire study.

General orientation and seeking of consent from research respondents . The study’s conduct was to regulate by ethical values, i.e., respect for individuals, beneficence, and justice, particularly in terms of data privacy and protection. Prior to data collection, the researcher will generate informed consent or assent forms and request them from respondents by e-mail. As evidence of their voluntary involvement in the full study, all forms will be delivered and signed electronically by those research respondents through e-mail message.

In addition, the researcher gave a brief 30-minute virtual presentation about the findings to the respondents. In accordance to this procedure, respondents who confirmed their voluntary involvement in the study were given a unique connection to Google Classroom developed by the researcher where the participants can partake in a brief virtual presentation. This was done specifically before conducting the survey. However, those respondents who were unable to attend the orientation due to unforeseen circumstances or personal reasons were educated about the research by phone call or chat through Facebook Messenger by the researcher. In addition, all respondents received a recorded video from the virtual presentation, which can be observed within the Google Classroom built by the researcher for the research.

Administration and retrieval of the questionnaire . The study took place in February of the school year 2022-2023. In order to carry out the research, the researcher will first develop Google Forms that will be utilized to collect responses from respondents based on the survey questions from the questionnaires. The quantitative data for this study was collected online using Google Forms. The researcher managed all direct contact and administration of surveys to respondents.

All surveys were allotted within one 90-minute session commencing with the Mathematical Problem-Style Questionnaire (MPSQ) and researcher-made examination. To protect the data, respondents was required to take the survey at a location and on a technological device (e.g., laptop, cellphone, or tablet) where only they have access to those offered online survey surveys via Google Forms. This was explicitly stated in their Informed Consent Form (ICF). In addition, the data questionnaire was returned to the researcher on time. Furthermore, the researcher handled personal communication and questionnaire administration. Finally, questionnaires were administered when the respondents’ 90-minute session has expired.

Checking, collating, and processing of data. The researcher gathered, validated, and quantified the respondents’ scores collected in an Excel file throughout this step. Following the tabulation, the data were submitted to an expert or certified statistician for data analysis. The researcher analyzed the results based on the data analysis for specific discoveries, discussions, and conclusions. This was accomplished mostly through data table and graphical presentations. Furthermore, descriptive statements were used to further explain and easily grasp the findings in relation to the study’s variables.

Statistical Tool for Data Analysis

The study’s findings were examined and comprehended properly using statistical methods such as Mean, Standard Deviation, Pearson r, T-test, and Analysis of Variance (ANOVA).

Mean . This method of analysis was used to measure the level of performance of students and their mathematical problem-solving style. Specifically, this was addressed in the first and second research questions.

Standard Deviation . A standard deviation is a statistical measure of the dispersion of a dataset in reference to its mean. This kind of analysis was used to determine how widely scattered the data is or how close the scores are to the mean. This was specifically answer the first and second research questions.

Pearson r. This statistical analysis was utilized to establish the existence of a significant relationship between mathematical problem-solving style and the performance of students. This will be utilized to specifically address the third research question.

T-test. This statistical analysis was utilized to determine if there was significant difference in the mathematical problem-solving style of students when classified into sex. This will specifically answer the fourth research question.

Analysis of Variance . This statistical analysis was utilized to determine if there was a significant difference in the mathematical problem-solving style of students when classified into four (4) different programs.

RESULTS AND DISCUSSION

In this chapter, the researchers present the results and discussions from the data gathered. In particular, this shows the data in tables and its corresponding descriptive interpretations.

Level of Mathematical Problem-Solving Style of Students in terms of Sensing

Table 1 presents the level of Mathematical Problem-Solving Style of Students in terms of Sensing. The item “As a student, I like to solve math problems and I am comfortable to trying to learn new skills.” has the highest mean of 3.69 with a descriptive equivalent of high. This is followed by the item “Before I put energy into solving math problems, I want to know first the benefits I can get from it.”, with a mean of 3.62 and high descriptive equivalent. On the contrary, the item “I tend to focus on immediate problems and let others worry about the distant future.” with the lowest mean of 3.20 and descriptive equivalent of moderate.

Table 1 Level of Mathematical Problem-Solving Style of Students in terms of Sensing

Furthermore, it has a category mean of 3.42 with descriptive equivalent of high. This indicates that the mathematical problem-solving style of students in terms of sensing is observed. Moreover, it has an Standard Deviation (SD) of 0.97.

The dispersion of the mathematical problem-solving style of students in terms of sensing based on the answers of the students revealed that the SD is 0.97. This indicates that the measures of variability of sensing as a mathematical problem-solving style of students are near the mean.

The result shows that students are interested in solving math problems and they are comfortable in trying new skills. It is also much observed that before solving math problems, students want to identify first the benefits they can get from it. Furthermore, students focus on immediate problems. Moreover, Vicente et al. (2002) observed that individuals with a sensation-type problem-solving style tend to focus on details and gather specific, factual data from their environment using their five senses. This approach involves a preference for concrete, practical, and tangible information, rather than abstract or theoretical concepts. These individuals tend to use a step-by-step approach to problem-solving, relying on established rules and procedures, and often prefer to work with real-world problems that have clear and immediate applications. Similarly, the study of Hsieh and Lin (2006) investigated the connection between mathematical problem-solving style and sensory preference among high school students. The study established that students with a sensing preference tended to use a more practical, sequential, and concrete problem-solving approach.

Level of Mathematical Problem-Solving Style of Students in terms of Intuitive

Table 2 presents the level of Mathematical Problem-Solving Style of Students in terms of Intuitive. The item “ As a student, I solve math problems accurately by knowing all the details of the problem.” has the highest mean of 3.68 with a descriptive equivalent of high. This is followed by the item “As a student, I enjoy solving mathematical problems.” with a mean of 3.36 and a descriptive equivalent of moderate. On the contrary, the item “ As a student, I solve mathematical problems quickly without wasting a lot of time on details.” has a mean of 3.05 with a descriptive equivalent of moderate.

Table 2 Level of Mathematical Problem-Solving Style of Students in terms of Intuitive

Furthermore, it has a category mean of 3.26 with a descriptive equivalent of moderate. It means that the mathematical problem-solving style of students in terms of intuitive is moderately observed. Moreover, the standard deviation of 1.01 in the category mean indicates that the measures of the variability of the mathematical problem-solving style of students in terms of intuition are near the mean.

It is observed that the students solve math problems accurately by knowing all the details of the problem. Additionally, it is moderately observed that students enjoy solving mathematical problems quickly and without wasting a lot of time on details. Hafriani (2018) suggests that students rely heavily on their intuition when it comes to solving mathematical problems. Students who use intuitive thinking to solve mathematical problems exhibit several characteristics: directness, self-evidence, intrinsic certainty, perseverance, coercion, extrapolation, globality, and implicitness.

Similarly, in the study conducted by Wuryanieet al., (2020) they found that students tend to rely on intuition when solving problems, exhibiting traits such as directness, self-evidence, extrapolation, intrinsic certainty, coercion, and decisiveness.

Level of Mathematical Problem-Solving Style of Students in terms of Feeling

Table 3 presents the level of Mathematical Problem-Solving Style of students in terms of feeling. The item “I want to solve math problems within a group and not individually.” has the highest mean of 3.73 with a descriptive equivalent of high. This is followed by the item “As a student, I can tell how others feel about solving math problems.” with a mean of 3.60 and a descriptive equivalent of high. On the other hand, the item “I try to please others and need occasional praise for myself.” has the lowest mean of 3.05 with a descriptive equivalent of moderate.

Table 3 Level of Mathematical Problem-Solving Style of Students in terms of Feeling

Moreover, it has a category mean of 3.48 with a descriptive equivalent of high. It implies that the mathematical problem-solving style of students in terms of feeling is observed. Consequently, the standard deviation of 0.98 in the category mean indicates that the measures of variability of the mathematical problem-solving style of students in terms of feeling are close to the mean.

Based on the results, it is observed that students want to solve math problems within a group and not individually. In addition, it is also observed that students can tell how others feel about solving math problems. It is moderately observed that in solving math problems, students try to please others and need occasional praise for their selves. A study by Goez et al. (2005) found that students must gain information and abilities related to feelings. Moreover, Altun (2003) examined the students who tend to rely on their feelings when solving problems prioritize their emotional and personal approaches in the problem-solving process. Along with this, Ahmed et al. (2014) stated that the mathematical problem-solving style has a favorable effect on the student’s attention, motivation to learn, choice of learning tools, self-regulation of learning, and academic performance.

Level of Mathematical Problem-Solving Style of students in terms of Thinking

Table 4 presents the level of mathematical problem-solving style of students in terms of thinking. The item “ As a student, I don’t let mathematics word problems discourage me, no matter how difficult they are.” has the highest mean of 3.65 with a descriptive equivalent of high. This is followed by the item “I solve math problems by analyzing all the facts and putting them in systematic order.” with a mean of 3.56 with a descriptive equivalent of high. On other hand, the item “ When I have a math problem to be solved, I solve it, even if others’ feelings might get hurt in the process.” has the lowest mean of 3.08 with a descriptive equivalent of moderate.

Furthermore, it has a category mean of 3.36 with a descriptive equivalent of moderate. This implies that the mathematical problem-solving style of students in terms of intuitive is moderately observed. Consequently, the standard deviation of 1.01 in the category means indicates that the measures of variability of the mathematical problem-solving style of students in terms of intuition are close to the mean.

Table 4 Level of Mathematical Problem-Solving Style of Students in terms of Thinking

Based on the results, it is observed that students approach mathematics word problems in a thoughtful and analytical manner. They are not easily discouraged by the difficulty of the problems and instead persevere until they find a solution. Students employ a systematic approach to problem-solving, carefully considering all the relevant facts and putting them in a logical order. This approach aligns with the findings of Khan et al. (2016), who emphasize the importance of analysis and research in effective problem-solving. Additionally, students demonstrate a commitment to objectivity and impartiality, seeking solutions that are grounded in evidence and reason. This aligns with the notion that mathematical problem-solving requires a high level of reflective thinking, as highlighted by Kneeland (2001) and Macaso and Dagohoy (2022). Moreover, the results of this study suggest that students are generally well-equipped to tackle mathematics word problems. They possess the necessary skills and mindset to approach these problems in a thoughtful, analytical, and objective manner. Thinking based on a mental process is an essential component of problem-solving solving, and problem-solving abilities are dependent on the correct application of thinking and solution processes. Furthermore, high-level thinking skills are involved in the intricate process of problem-solving (Gürsan & Yazgan, 2020).

Summary of the Level of Mathematical-Problem Solving Style of Students

Table summarizes the level of mathematical problem-solving style of students. Among the four indicators, “feeling” acquire the highest mean of 3.48 with descriptive equivalent of high. “Sensing” developed a mean of 3.42 with a descriptive equivalent of high. They have an SD of 0.98 and 0.97, respectively. It is followed by “thinking” with a mean of 3.36 with a descriptive equivalent of moderate and an SD of 1.01. On other hand, “intuitive” got the lowest mean of 3.26 with a descriptive equivalent of moderate and an SD of 1.01.

Table 5 Summary of the Level of Mathematical-Problem Solving Style of Students

Furthermore, it has an overall mean of 3.38 with a descriptive equivalent of moderate. This means that the mathematical problem-solving style of students is moderately observed. The findings suggest that there is a high degree of homogeneity in the mathematical problem-solving styles of the students, as evidenced by the small standard deviation of 0.99 in the overall mean. This indicates that the measures of variability in the students’ responses are clustered closely around the mean. Such a narrow range of variability in the problem-solving styles implies that the students have similar levels of proficiency in this variable.

Particularly, the results suggest that “feeling and sensing” as students’ mathematical problem-solving style is observed. This means that students want to solve math problems within a group and they like to solve math problems and are comfortable trying to learn new things. Moreover, “thinking and intuitive” as students’ mathematical problem-solving style is moderately observed. This indicates that students solve math problems by analyzing all the facts and knowing all the details of the problem.

This finding is supported by a recent study by TIMSS and PISA, which found that students can use their mathematical understanding and knowledge to solve problems (IEA, 2016). PISA assesses students’ ability to use their knowledge and skills in recognizing, analyzing, and solving problems in a variety of situations(OECD, 2019). Moreover, the result confirms the findings of Schoenfeld (2013), who mentioned that the mental state of students is an essential aspect of learning mathematics. The belief system of the student regarding himself, mathematics, and problem-solving determines student progress in problem-solving. This is also agreed by Hendriana et al., (2017) who mentioned that one of the fundamental mathematical abilities that students who study mathematics must develop is the ability to solve problems.

Level of Performance of Students in Solving Math Problems

Table 6 depicts the level of performance of students in solving math problems. The level of performance of students in terms of answering the 40-item Mathematics test has a mean of 60.84 with a descriptive equivalent of above average. This indicates that the performance of students in mathematics is very good.  Furthermore, it has an SD of 10.48. This demonstrates whether one’s scores on mathematical problem-solving style were extremely high or extremely low. This suggests that students’ capacity to solve mathematical problems is more likely to deviate from the mean.

The result shows that the students were able to select and correctly identify the appropriate answer to the given questions. They could choose the correct equation from the problem and accurately determine the answer in the problem scenario. Additionally, they were able to verify their responses by selecting the correct answer to the given problem.

Table 6 Level of Performance of Students in Solving Math Problems

As cited by Heris and Sumarmo (2014), problem-solving style are basic mathematical skills students need to learn. Fernandez et al. (2017) also added that problem-solving is a significant part of learning, making it of particular importance for the study of mathematics. Furthermore, another component of learning mathematics is problem-solving. This means that students need to become proficient in a variety of problem-solving style in mathematics in order to improve their creativity, reasoning, critique, and systematic thinking (NCTM, 2000). Therefore, completing mathematical problems is a crucial component of the learning objectives that must be encountered (Surya et al., 2017).

Significance of the Relationship Between the Variables

Table 7 presents the relationship between Mathematical Problem-Solving Style and the Performance of Students in Mathematics.

The correlation of Mathematical Problem-Solving Style has a significant relationship with the Performance of Students in Mathematics (p<0.05) with a coefficient determination of 0.733. Specifically, there is a strong positive correlation between the variables, and the p-value of the two variables is less than the 0.05 level of significance, which indicates that there is a significant relationship between the mathematical problem-solving style and the performance of students in mathematics (r=0.733,p=0.000. Thus, the null hypothesis is rejected.

Since the result confirmed that the mathematical problem-solving style and performance of students have a very high relationship, this means that the mathematical problem-solving style of the students significantly affects their performance in math. It can be seen from the aforementioned discussion that the mathematical problem-solving style of students is an important factor that affects their performance.

Table 7 Significance of the Relationship Between the Variables

It can be deduced that if students’ mathematical problem-solving styles will be developed, then it enhances their performance in school. This is supported by the study conducted by Suratno et al., (2020) which found that there is a significant positive correlation between students’ problem-solving styles and academic performance of students. Additionally, Mustafić et al. (2017) established that a student would perform exceptionally well in any science subject if they propose a high level of self-concept toward problem-solving skills.

The significant difference in the Mathematical Problem-solving Style of students when grouped according to sex

To determine if there was a significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female), a t-test was used. Table 8 shows the result.

Using the t-test, the obtained t-value is -2.211 and the resulting p-value is 0.9860. This result indicates that the difference in mathematical problem-solving style between males and females is not statistically significant at the 0.05 level. Therefore, the researchers fail to reject the null hypothesis that there is no significant difference in mathematical problem-solving style between male and female students.

Table 8 The significant difference in mathematical-problem solving style of students when grouped according to sex

This means that students’ mathematical problem-solving style when grouped according to sex (male and female) does not vary. This finding is consistent with previous research conducted by Rusdiet al., (2020) who mentioned that men and women have similar mathematical problem-solving styles. Male and female students may understand the information, describe their knowledge, and ask relevant questions. They can use notations, symbols, and mathematical models to describe the problem and solution properly.  Moreover, Goos et al. (2017) studies examined the impact of gender differences on students’ mathematical learning outcomes have yielded inconsistent results. While some studies have demonstrated differences between genders, indicating that either men or women perform better, other studies have found no significant gender differences.

The significant difference in mathematical problem-solving style of students when grouped according to Program

Table 9 presents the significant difference in the mathematical problem-solving style of students when grouped according to four different programs: BTVTED, BSAB, BSOA, and BPA.

The result shows that there is a positive significant difference in the mathematical problem-solving style of students when grouped according to different programs since the f-value is 12.46 and the p-value is 0.000 which is lesser than 0.05 alpha level of significance. It indicates the mathematical problem-solving style of students from BTVTED, BSAB, BSOA, and BPA is significantly different. Therefore, the null hypothesis is rejected.

Table 9 The significant difference in mathematical problem-solving style of students when grouped according to Program.

The result means that students coming from four programs have different problem-solving styles in mathematics. This finding is concurrent to the first model of problem-solving styles based on the concept of psychological functions (Jung, 1923; Moon, 2008; Taylor & Mackenny, 2008). This model consists of four psychological functions as thinking, feeling, sensation, and intuition (Ghodrati et al., 2014). As such, Hacısalihlioğlu et al. (2003)stated that achieving success in problem-solving is linked to possessing abilities such as critical thinking, decision-making, reflective thinking, inquiring, analyzing, and synthesizing. This is further confirmed by Wen-Chun et al. (2015) that students use different problem-solving styles in solving math problems.

SUMMARY, CONCLUSION, AND RECOMMENDATIONS

This chapter presents the summary of the major findings of the study, the conclusion, and the proposed recommendations for possible implementations.

Summary of Findings

The major findings of the study are the following:

• For the level of performance of students in solving math problem, the level of performance of students in terms of answering the 40-item Mathematics test has a mean of 60.84 with a descriptive equivalent of high. This indicates that the performance of students in mathematics is very good. Furthermore, it has an SD of 10.48. This demonstrates whether one’s scores on mathematical problem-solving style were extremely high or extremely low.
• For the level of mathematical problem-solving style, “feeling” got the highest mean of 3.48 with an SD of 0.98. This is followed by “sensing” with a mean of 3.42 and an SD of 0.97. Both indicators got a similar descriptive equivalent of high. On other hand, “intuitive” got the lowest mean of 3.26 and an SD of 1.01 with a descriptive equivalent of moderate. Furthermore, it has an overall mean of 3.38 and an SD of 0.99 with a descriptive equivalent of moderate.
• The statistical analysis shows that there is a strong positive correlation (r=0.733) between the mathematical problem-solving style and students’ performance. The p-value of the two variables is less than 0.05, indicating that the correlation is statistically significant. This means that there is a significant relationship between the students’ mathematical problem-solving style and their academic performance. As a result, the null hypothesis is rejected.
• The statistical analysis shows that there is no significant difference (t=211, p=0.9860) in the mathematical problem-solving style of students when grouped according to sex (male and female). The result indicates that the difference in mathematical problem-solving style between males and females is not statistically significant at the 0.05 level. Therefore, the researchers fail to reject the null hypothesis that there is no significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female). Meanwhile, there is a significant difference (f=12.46, p=0.000) in the mathematical problem-solving style of students when grouped in accordance to different programs since the f-value is 12.46 and the p-value is 0.000 which is lesser than 0.05 alpha level of significance. This indicates the mathematical problem-solving style of students from BTVTED, BSAB, BSOA, and BPA is significantly different. Therefore, the null hypothesis is rejected.
• Based on the results of the study, here are the instructional intervention plan that can be proposed:
• For thinking and intuitive problem solvers – Provide explicit instruction on how to justify mathematical solutions using evidence and logical reasoning. Encourage intuitive problem-solvers to verbalize their thought process and explain how they arrived at their solution. Moreover, provide opportunities for students to develop their critical thinking skills through activities such as puzzles, brain teasers, and logic games. Encourage students to explain their reasoning and thought processes when solving problems.
• For sensing and feeling problem solvers – Provide opportunities for collaborative problem-solvers to work in pairs or small groups to solve problems. Encourage the students to take turns explaining their thought process and to ask questions to deepen their understanding. Provide opportunities for independent problem-solving and self-directed learning while allowing for collaborative work. Furthermore, provide support and guidance as needed to help students build their problem-solving skills. Gradually remove support as students become more confident and independent.

The findings from the study led the researcher to draw the following conclusions:

• Performance of students in solving math problems is very good.
• Mathematical problem-solving style of students is moderately observed.
• There is a significant relationship between mathematical problem-solving style and the performance of students.
• There is no significant difference in the mathematical problem-solving style of students when grouped according to sex. However, students’ mathematical problem-solving style is significantly different when grouped according to different programs.
• Understanding students’ different mathematical problem-solving styles can help instructors tailor their instruction to meet the needs of different learners and create a more inclusive classroom environment.

Recommendations

Based on the findings, analysis, and conclusion drawn in this study, the following recommendations were summarized:

• Students are encouraged to learn effectively and independently in solving mathematical tasks. They may discover that strengthening their different mathematical problem-solving style would boost their performance in math. This can be achieved by helping them discover their different problem-solving styles and identifying strategies that can assist them in solving mathematical problems. By doing so, they can improve their performance in mathematics and unleash their full potential in this subject area.
• Instructors, college administrators, and local college officials are urged to develop enrichment activities to help their students in developing their mathematical problem-solving styles. This is especially true when it comes to encouraging students to take the initiative, establish their own mathematical problem-solving strategy, set learning goals, and assess their abilities to specify the sources they need to learn, particularly in mathematics. Teachers may create engaging instructional intervention programs to deepen students’ interests in problem-solving in Mathematics. Furthermore, they can engage in and be creative with technology, mentoring, and coaching students who are having difficulty completing mathematical problems.
• The current study’s findings emphasize the relevance of problem-solving style in mathematics and provide ways to apply and improve it. The establishment of a curriculum and instructional strategy for applying it to students is required so that teachers and students can continue and maximize their mathematical problem-solving style. To ensure that these tactics are embedded in students, ongoing efforts will be required. To maximize students’ problem-solving style in Mathematics, STCAST instructors and administrators should work collaboratively to achieve the objective. They should ensure that the necessary materials, resources, activities, and differentiated instruction are available and used to meet students’ needs to be motivated in learning.
• Future research for developing other intervention programs needed to identify the factors that might improve the mathematical problem-solving style to enhance students’ performance.

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Integrals of groups. II

• Open access
• Published: 24 April 2024

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• João Araújo 1 ,
• Peter J. Cameron 2 , 3 ,
• Carlo Casolo 4 ,
• Francesco Matucci 4 &
• Claudio Quadrelli 5  

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G . This paper continues the investigation on integrals of groups started in the work [1]. We study:

A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).

The existence of integrals that are p -groups for abelian p -groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).

Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).

The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).

Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).

Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).

We end the paper with a number of open problems.

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J. Araújo, P. J. Cameron, C. Casolo and F. Matucci, Integrals of groups , Israel Journal of Mathematics 234 (2019), 149–178.

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Acknowledgements

The authors would like to thank the referee for a careful reading which led to several improvements of the exposition and proofs.

The first author was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 ( https://doi.org/10.54499/UIDB/00297/2020 ) and UIDP/00297/2020 ( https://doi.org/10.54499/UIDP/00297/2020 )(Center for Mathematics and Applications).

The first, second and fourth authors gratefully acknowledge the support of the Fundação para a Ciência e a Tecnologia (CEMAT-Ciências FCT projects UIDB/04621/2020 and UIDP/04621/2020).

The fourth and fifth authors are members of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM), and the fourth author gratefully acknowledges the support of the Universit’a degli Studi di Milano—Bicocca (FA project ATE-2017-0035 “Strutture Algebriche”).

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Center for Mathematics and Applications (NOVA Math), Department of Mathematics, Faculdade de Ciências e Tecnologia, NOVA University Lisbon, 2829-516, Caparica, Portugal

João Araújo

School of Mathematics and Statistics, University of St. Andrews, St. Andrews, KY16 9AJ, UK

Peter J. Cameron

CEMAT-Ciências, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal

Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, 20125, Milano, Italy

Carlo Casolo & Francesco Matucci

Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, 22100, Como, Italy

Claudio Quadrelli

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Correspondence to Peter J. Cameron .

Additional information

In April 2018, Carlo Casolo sent the other authors detailed answers to some of the questions in the first version of the paper [1], and we immediately invited him to join us. He was very dedicated and curious about integrals and inverse group theory problems. In fact, the current paper is in large part Carlo’s work, together with the fruits of a meeting in Florence in February 2020. Carlo passed away not long after. He was very generous and kind to all of us and is sorely missed. We dedicate this paper to his memory.

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Araújo, J., Cameron, P.J., Casolo, C. et al. Integrals of groups. II. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2610-4

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Received : 04 September 2020

Revised : 28 April 2022

Published : 24 April 2024

DOI : https://doi.org/10.1007/s11856-024-2610-4

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