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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

Barmby, P., Harries, T., Higgins, S., and Suggate, J. (2009). The array representation and primary children's understanding and reasoning in multiplication. Educ. Stud. Math. 70 (3), 217–241. doi:10.1007/s10649-008-914510.1007/s10649-008-9145-1

CrossRef Full Text | Google Scholar

Bates, D., Mächler, M., Bolker, B., and Walker, S. (2015). Fitting Linear Mixed-Effects Models Usinglme4. J. Stat. Soft. 67 (1), 1–48. doi:10.18637/jss.v067.i01

Capar, G., and Tarim, K. (2015). Efficacy of the cooperative learning method on mathematics achievement and attitude: A meta-analysis research. Educ. Sci-theor Pract. 15 (2), 553–559. doi:10.12738/estp.2015.2.2098

Child, S., and Nind, M. (2013). Sociometric methods and difference: A force for good - or yet more harm. Disabil. Soc. 28 (7), 1012–1023. doi:10.1080/09687599.2012.741517

Cillessen, A. H. N., and Marks, P. E. L. (2017). Methodological choices in peer nomination research. New Dir. Child Adolesc. Dev. 2017, 21–44. doi:10.1002/cad.20206

PubMed Abstract | CrossRef Full Text | Google Scholar

Clarke, B., Cheeseman, J., and Clarke, D. (2006). The mathematical knowledge and understanding young children bring to school. Math. Ed. Res. J. 18 (1), 78–102. doi:10.1007/bf03217430

Cohen, E. G. (1994). Restructuring the classroom: Conditions for productive small groups. Rev. Educ. Res. 64 (1), 1–35. doi:10.3102/00346543064001001

Davidson, N., and Major, C. H. (2014). Boundary crossings: Cooperative learning, collaborative learning, and problem-based learning. J. Excell. Coll. Teach. 25 (3-4), 7.

Google Scholar

Davydov, V. V. (2008). Problems of developmental instructions. A Theoretical and experimental psychological study . New York: Nova Science Publishers, Inc .

Deacon, D., and Edwards, J. (2012). Influences of friendship groupings on motivation for mathematics learning in secondary classrooms. Proc. Br. Soc. Res. into Learn. Math. 32 (2), 22–27.

Degrande, T., Verschaffel, L., and van Dooren, W. (2016). “Proportional word problem solving through a modeling lens: a half-empty or half-full glass?,” in Posing and Solving Mathematical Problems, Research in Mathematics Education . Editor P. Felmer.

Doerr, H. M., and Tripp, J. S. (1999). Understanding how students develop mathematical models. Math. Thinking Learn. 1 (3), 231–254. doi:10.1207/s15327833mtl0103_3

Fujita, T., Doney, J., and Wegerif, R. (2019). Students' collaborative decision-making processes in defining and classifying quadrilaterals: a semiotic/dialogic approach. Educ. Stud. Math. 101 (3), 341–356. doi:10.1007/s10649-019-09892-9

Gillies, R. (2016). Cooperative learning: Review of research and practice. Ajte 41 (3), 39–54. doi:10.14221/ajte.2016v41n3.3

Gravemeijer, K. (1999). How Emergent Models May Foster the Constitution of Formal Mathematics. Math. Thinking Learn. 1 (2), 155–177. doi:10.1207/s15327833mtl0102_4

Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., and Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? Int. J. Sci. Math. Educ. 15 (S1), 105–123. doi:10.1007/s10763-017-9814-6

Hamilton, E. (2007). “What changes are needed in the kind of problem-solving situations where mathematical thinking is needed beyond school?,” in Foundations for the Future in Mathematics Education . Editors R. Lesh, E. Hamilton, and Kaput (Mahwah, NJ: Lawrence Erlbaum ), 1–6.

Hannula, M. S. (2015). “Emotions in problem solving,” in Selected Regular Lectures from the 12 th International Congress on Mathematical Education . Editor S. J. Cho. doi:10.1007/978-3-319-17187-6_16

Hwang, W.-Y., and Hu, S.-S. (2013). Analysis of peer learning behaviors using multiple representations in virtual reality and their impacts on geometry problem solving. Comput. Edu. 62, 308–319. doi:10.1016/j.compedu.2012.10.005

Johnson, D. W., Johnson, R. T., and Johnson Holubec, E. (2009). Circle of Learning: Cooperation in the Classroom . Gurgaon: Interaction Book Company .

Johnson, D. W., Johnson, R. T., and Johnson Holubec, E. (1993). Cooperation in the Classroom . Gurgaon: Interaction Book Company .

Jordan, M. E., and McDaniel, R. R. (2014). Managing uncertainty during collaborative problem solving in elementary school teams: The role of peer influence in robotics engineering activity. J. Learn. Sci. 23 (4), 490–536. doi:10.1080/10508406.2014.896254

Karlsson, N., and Kilborn, W. (2018a). Inclusion through learning in group: tasks for problem-solving. [Inkludering genom lärande i grupp: uppgifter för problemlösning] . Uppsala: Uppsala University .

Karlsson, N., and Kilborn, W. (2018c). It's enough if they understand it. A study of teachers 'and students' perceptions of multiplication and the multiplication table [Det räcker om de förstår den. En studie av lärares och elevers uppfattningar om multiplikation och multiplikationstabellen]. Södertörn Stud. Higher Educ. , 175.

Karlsson, N., and Kilborn, W. (2018b). Tasks for problem-solving in mathematics. [Uppgifter för problemlösning i matematik] . Uppsala: Uppsala University .

Karlsson, N., and Kilborn, W. (2020). “Teacher’s and student’s perception of rational numbers,” in Interim Proceedings of the 44 th Conference of the International Group for the Psychology of Mathematics Education , Interim Vol., Research Reports . Editors M. Inprasitha, N. Changsri, and N. Boonsena (Khon Kaen, Thailand: PME ), 291–297.

Kazak, S., Wegerif, R., and Fujita, T. (2015). Combining scaffolding for content and scaffolding for dialogue to support conceptual breakthroughs in understanding probability. ZDM Math. Edu. 47 (7), 1269–1283. doi:10.1007/s11858-015-0720-5

Klang, N., Olsson, I., Wilder, J., Lindqvist, G., Fohlin, N., and Nilholm, C. (2020). A cooperative learning intervention to promote social inclusion in heterogeneous classrooms. Front. Psychol. 11, 586489. doi:10.3389/fpsyg.2020.586489

Klang, N., Fohlin, N., and Stoddard, M. (2018). Inclusion through learning in group: cooperative learning [Inkludering genom lärande i grupp: kooperativt lärande] . Uppsala: Uppsala University .

Kunsch, C. A., Jitendra, A. K., and Sood, S. (2007). The effects of peer-mediated instruction in mathematics for students with learning problems: A research synthesis. Learn. Disabil Res Pract 22 (1), 1–12. doi:10.1111/j.1540-5826.2007.00226.x

Langer-Osuna, J. M. (2016). The social construction of authority among peers and its implications for collaborative mathematics problem solving. Math. Thinking Learn. 18 (2), 107–124. doi:10.1080/10986065.2016.1148529

Lein, A. E., Jitendra, A. K., and Harwell, M. R. (2020). Effectiveness of mathematical word problem solving interventions for students with learning disabilities and/or mathematics difficulties: A meta-analysis. J. Educ. Psychol. 112 (7), 1388–1408. doi:10.1037/edu0000453

Lesh, R., and Doerr, H. (2003). Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning and Teaching . Mahwah, NJ: Erlbaum .

Lesh, R., Post, T., and Behr, M. (1988). “Proportional reasoning,” in Number Concepts and Operations in the Middle Grades . Editors J. Hiebert, and M. Behr (Hillsdale, N.J.: Lawrence Erlbaum Associates ), 93–118.

Lesh, R., and Zawojewski, (2007). “Problem solving and modeling,” in Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics . Editor L. F. K. Lester (Charlotte, NC: Information Age Pub ), vol. 2.

Lester, F. K., and Cai, J. (2016). “Can mathematical problem solving be taught? Preliminary answers from 30 years of research,” in Posing and Solving Mathematical Problems. Research in Mathematics Education .

Lybeck, L. (1981). “Archimedes in the classroom. [Arkimedes i klassen],” in Göteborg Studies in Educational Sciences (Göteborg: Acta Universitatis Gotoburgensis ), 37.

McMaster, K. N., and Fuchs, D. (2002). Effects of Cooperative Learning on the Academic Achievement of Students with Learning Disabilities: An Update of Tateyama-Sniezek's Review. Learn. Disabil Res Pract 17 (2), 107–117. doi:10.1111/1540-5826.00037

Mercer, N., and Sams, C. (2006). Teaching children how to use language to solve maths problems. Lang. Edu. 20 (6), 507–528. doi:10.2167/le678.0

Montague, M., Krawec, J., Enders, C., and Dietz, S. (2014). The effects of cognitive strategy instruction on math problem solving of middle-school students of varying ability. J. Educ. Psychol. 106 (2), 469–481. doi:10.1037/a0035176

Mousoulides, N., Pittalis, M., Christou, C., and Stiraman, B. (2010). “Tracing students’ modeling processes in school,” in Modeling Students’ Mathematical Modeling Competencies . Editor R. Lesh (Berlin, Germany: Springer Science+Business Media ). doi:10.1007/978-1-4419-0561-1_10

Mulryan, C. M. (1992). Student passivity during cooperative small groups in mathematics. J. Educ. Res. 85 (5), 261–273. doi:10.1080/00220671.1992.9941126

OECD (2019). PISA 2018 Results (Volume I): What Students Know and Can Do . Paris: OECD Publishing . doi:10.1787/5f07c754-en

CrossRef Full Text

Pólya, G. (1948). How to Solve it: A New Aspect of Mathematical Method . Princeton, N.J.: Princeton University Press .

Russel, S. J. (1991). “Counting noses and scary things: Children construct their ideas about data,” in Proceedings of the Third International Conference on the Teaching of Statistics . Editor I. D. Vere-Jones (Dunedin, NZ: University of Otago ), 141–164., s.

Rzoska, K. M., and Ward, C. (1991). The effects of cooperative and competitive learning methods on the mathematics achievement, attitudes toward school, self-concepts and friendship choices of Maori, Pakeha and Samoan Children. New Zealand J. Psychol. 20 (1), 17–24.

Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (reprint). J. Edu. 196 (2), 1–38. doi:10.1177/002205741619600202

SFS 2009:400. Offentlighets- och sekretesslag. [Law on Publicity and confidentiality] . Retrieved from https://www.riksdagen.se/sv/dokument-lagar/dokument/svensk-forfattningssamling/offentlighets--och-sekretesslag-2009400_sfs-2009-400 on the 14th of October .

Snijders, T. A. B., and Bosker, R. J. (2012). Multilevel Analysis. An Introduction to Basic and Advanced Multilevel Modeling . 2nd Ed. London: SAGE .

Stillman, G., Brown, J., and Galbraith, P. (2008). Research into the teaching and learning of applications and modelling in Australasia. In H. Forgasz, A. Barkatas, A. Bishop, B. Clarke, S. Keast, W. Seah, and P. Sullivan (red.), Research in Mathematics Education in Australasiae , 2004-2007 , p.141–164. Rotterdam: Sense Publishers .doi:10.1163/9789087905019_009

Stohlmann, M. S., and Albarracín, L. (2016). What is known about elementary grades mathematical modelling. Edu. Res. Int. 2016, 1–9. doi:10.1155/2016/5240683

Swedish National Educational Agency (2014). Support measures in education – on leadership and incentives, extra adaptations and special support [Stödinsatser I utbildningen – om ledning och stimulans, extra anpassningar och särskilt stöd] . Stockholm: Swedish National Agency of Education .

Swedish National Educational Agency (2018). Syllabus for the subject of mathematics in compulsory school . Retrieved from https://www.skolverket.se/undervisning/grundskolan/laroplan-och-kursplaner-for-grundskolan/laroplan-lgr11-for-grundskolan-samt-for-forskoleklassen-och-fritidshemmet?url=-996270488%2Fcompulsorycw%2Fjsp%2Fsubject.htm%3FsubjectCode%3DGRGRMAT01%26tos%3Dgr&sv.url=12.5dfee44715d35a5cdfa219f ( on the 32nd of July, 2021).

van Hiele, P. (1986). Structure and Insight. A Theory of Mathematics Education . London: Academic Press .

Velásquez, A. M., Bukowski, W. M., and Saldarriaga, L. M. (2013). Adjusting for Group Size Effects in Peer Nomination Data. Soc. Dev. 22 (4), a–n. doi:10.1111/sode.12029

Verschaffel, L., Greer, B., and De Corte, E. (2007). “Whole number concepts and operations,” in Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics . Editor F. K. Lester (Charlotte, NC: Information Age Pub ), 557–628.

Webb, N. M., and Mastergeorge, A. (2003). Promoting effective helping behavior in peer-directed groups. Int. J. Educ. Res. 39 (1), 73–97. doi:10.1016/S0883-0355(03)00074-0

Wegerif, R. (2011). “Theories of Learning and Studies of Instructional Practice,” in Theories of learning and studies of instructional Practice. Explorations in the learning sciences, instructional systems and Performance technologies . Editor T. Koschmann (Berlin, Germany: Springer ). doi:10.1007/978-1-4419-7582-9

Yackel, E., Cobb, P., and Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. J. Res. Math. Edu. 22 (5), 390–408. doi:10.2307/749187

Zawojewski, J. (2010). Problem Solving versus Modeling. In R. Lesch, P. Galbraith, C. R. Haines, and A. Hurford (red.), Modelling student’s mathematical modelling competencies: ICTMA , p. 237–243. New York, NY: Springer .doi:10.1007/978-1-4419-0561-1_20

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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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

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At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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Usable Math: Math Learning through Problem Solving and Design

Usable Math is a free and accessible online interactive math problem solving platform for elementary school-age children and their teachers, tutors, caregivers, and families. It is being developed by a team of UMass Amherst researchers led by Sharon Edwards in the College of Education. Our workshop will highlight the site's design and functions, including:

• 17 problem solving modules featuring essential math concepts taught in elementary schools.
• Use of virtual coaches to support students' math thinking and learning.
• Motivation strategies to engage and sustain student learning.
• Storytelling modules connecting math with science, history and English/language arts.
• Students writing their own math problems and solving strategies
• GenAI-enhancements for teachers.
• Sharon A. Edwards, Clinical Faculty
•  Robert W. Maloy, Senior Lecturer
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• Aubrey Coyne, Undergraduate Project Assistant

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Some St. Patrick’s Day math

At today’s Gathering 4 Gardner social, Colm Mulcahy presented on two Irish figures in recreational mathematics with whom Martin Gardner corresponded: Victor Meally and Owen O’Shea. Owen O’Shea is the natural successor to Gardner’s “Professor I.J. Matrix” as a prolific generator of numerological coincidences — see for example The Magic Numbers of the Professor (2007). Victor Meally shows up occasionally in Gardner’s Mathematical Games columns, and also in “Problem 3.14” (appropriate for Pi Day!) in The Colossal Book of Short Puzzles and Problems (2006):

One of the satisfactions of recreational mathematics comes from finding better solutions for problems thought to have been already solved in the best possible way. Consider the following digital problem that appears as Number 81 in Henry Ernest Dudeney’s Amusements in Mathematics . (There is a Dover reprint of this 1917 book.) Nine digits (0 is excluded) are arranged in two groups. On the left a three-digit number is to be multiplied by a two-digit number. On the right both numbers have two digits each: \[158\times 23 = 79\times 46\] In each case the product is the same: 3,634. How, Dudeney asked, can the same nine digits be arranged in the same pattern to produce as large a product as possible, and a product that is identical in both cases? Dudeney’s answer, which he said “is not to be found without the exercise of some judgment and patience,” was 5,568: \[174\times 32 = 96\times 58\] Victor Meally of Dublin County in Ireland later greatly improved on Dudeney’s answer with 7,008: \[584\times 12 = 96\times 73\] This remained the record until a Japanese reader found an even better solution. It is believed, although it has not yet been proved, to give the highest possible product. Can you find it without the aid of a computer?

With the aid of a computer ( code ), it’s easy to confirm that that Japanese reader’s solution is indeed the best of the 11 basic solutions (and Meally’s the runner-up):

But now consider the ruminations of another Irishman, James Joyce’s Leopold Bloom, who in Chapter 17 of Ulysses recounts how he became aware of

the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that […] 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers […] the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers.

There’s some confusion as to what number Joyce was really talking about (if any); but the mathematical community has apparently settled on \(9^{9^9}\), as seen in e.g. the “ Ulysses sequence” OEIS A054382 : \(\lceil\log_{10} 1^{1^1}\rceil, \lceil\log_{10} 2^{2^2}\rceil, \lceil\log_{10} 3^{3^3}\rceil,\ldots\)

The ninth element of this sequence — the number of decimal digits in \(9^{9^9}\) — is \(369\,693\,100\). If “closely printed” at the resolution of A Million Random Digits with 100,000 Normal Deviates , printed by the RAND Corporation (no relation) in 1955, the recording of the entire expansion of \(9^{9^9}\) would take up 148 thousand-page volumes. (Or, if you postulate a thousand physical pages, each printed on both sides: 74 volumes.)

Suppose we allow solutions of Dudeney’s problem to contain powers: not merely \(abc\times de = gh\times ij\), but for example \(ab^c\times d^e = g^h\times i^j\). Then there are five more basic solutions possible, the largest of which is

The solutions are:

Clement Wood — compiler of The Best Irish Jokes (1926) — asserts, in the same Book of Mathematical Oddities (1927) which we previously mined in “Mathematical Golf” (2023-03-23), that there are only two solutions to the double equality

Wood is correct, and the administration of powers produces no further solutions to that puzzle.

What to watch on Super Tuesday: Why the delegate math shows Haley has little room to stop Trump

Former President Donald Trump enters Super Tuesday with a big delegate lead over former U.N. Ambassador Nikki Haley in the GOP presidential race — and by the end of the night, he will most likely find himself closing in on the magic number he needs to officially end the contest: 1,215 delegates. 

Haley has won just one contest — the one in Washington, D.C. — and her reliance on a coalition that leans heavily on affluent, college-educated suburbanites (including non-Republicans who may simply see her as a vehicle to register disgust with Trump) faces two mighty Super Tuesday headwinds.

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First, numerous contests are in states with demographic profiles decidedly unfavorable to Haley but right in Trump’s white, working-class wheelhouse. And second, even where the demographics are Haley-friendly, party rules in many cases limit non-Republicans’ participation and all but require outright majorities to collect delegates.

While a strong showing in New Hampshire netted her a significant share of the state’s delegates, more states coming up this month have delegate rules like those in South Carolina, where 40% of the vote netted Haley just 6% of the state’s delegates. In other words, even relatively strong showings aren’t likely to translate into meaningful delegate hauls on Super Tuesday if she can’t win states outright. And if Haley can’t either broaden her coalition or supercharge friendly turnout, she’ll finish Tuesday buried beneath a Trump delegate avalanche. 

To illustrate the scale of Haley’s challenge, we have looked closely at the demographics, participation rules and delegate allocation formulas for every GOP caucus and primary taking place Tuesday and in the next wave of contests March 12 and March 19. For each one, we have generated our best estimate of each candidate’s delegate ceiling and floor. These estimates are more art than science — but we will lay out our best explanation, reporting and analysis in each state. 

There may be areas of disagreement in one state or another, but the broad conclusion is unavoidable: Unless Haley somehow finds a way to win outright majorities in multiple states, her campaign will be overwhelmed by the cold, hard realities of delegate math. 

Keep reading for our state-by-state analysis of what’s on tap Super Tuesday — and see how yours compares with ours.

Super Tuesday

Alaska — 29 delegates.

Trump best case: Trump 29, Haley 0

Haley best case: Trump 23, Haley 6

A caucus limited to registered Republicans is the worst set of conditions Haley could face, with a small universe of highly engaged core GOP voters holding sway. In 2016, the last contested Republican race, Sen. Ted Cruz of Texas won Alaska’s caucuses narrowly over Trump, with their combined share of the vote reaching 70%. Sen. Marco Rubio of Florida, the 2016 candidate whose coalition most resembles Haley’s, took only 15%.

GOP Sen. Lisa Murkowski did endorse Haley, but Haley’s real glimmer of hope here is that the 29 delegates are awarded proportionally — but only for candidates who clear a 13% threshold. That’s no guarantee given that Haley won’t have any independents or Democrats to lean on. But she’ll walk away with at least a handful of delegates if she clears that bar.

Alabama — 50 delegates

Trump best case: Trump 50, Haley 0

Haley best case: N/A 

The state’s open primary allows Haley to tap into non-Republicans, including Trump-hostile voters who plan to vote for President Joe Biden in the fall. That should boost her share of the vote somewhat, but that kind of voter is in short supply in Alabama, and it’s not likely to matter in the delegate math. 

Twenty-nine delegates are dished out based on the statewide result, and Trump can claim all of them with at least 50% of the vote. The rest are parceled out by congressional district, three apiece in each of the seven districts. Again, finishing with 50% of the vote in a district gives a candidate all three district delegates. While Haley can leverage the same rules that allowed her to carry one district and win three delegates in South Carolina, none of Alabama’s districts come close to fitting the unique demographic makeup that made that one South Carolina district fertile pro-Haley turf.

Arkansas — 40 delegates

Trump best case: Trump 39, Haley 1

The rules in this GOP primary are similar to those in many other Super Tuesday states, and they are likely to be an impenetrable barrier to Haley’s collecting more than a single delegate. Twenty-seven delegates are awarded based on the statewide result, with just a simple majority needed to gobble up all of them. A dozen more are divided equally among four congressional districts, but a candidate who wins a simple majority in one district wins all three delegates there. 

Demographically, neither the state as a whole nor any individual district features anything approaching the mix that has powered Haley elsewhere, even in an open primary. She has leaned heavily on white, college-educated voters, while Trump has racked up massive margins in areas where white voters without four-year degrees predominate. And in Arkansas, that pro-Trump demographic reigns supreme. 

However, any candidate who breaks 20% of the statewide vote wins a single delegate. Haley should be able to do that, but not much else. If you want to see a close race in Arkansas, you’re probably better off heading to Hot Springs to take in the action at Oaklawn Park.

California — 169 delegates

Trump best case: Trump 169, Haley 0

A gigantic, deep-blue state with large pockets of economically upscale, college-educated voters is, at first glance, a ripe target for Haley. But the state GOP’s rules make it all but impossible for her to win delegates. 

First, it’s a closed primary — the huge chunk of independent voters who fit right into Haley’s demographic base can’t participate. Instead, the vastly smaller universe of registered Republicans, a group Trump has been winning by around 50 percentage points in primaries so far, will cast all of the ballots. 

What’s more, the California GOP changed its allocation rules last year , scrapping a format that doled out most delegates by congressional district in favor of what effectively amounts to a winner-take-all primary based solely on the statewide result. As a result, all Trump has to do to claim all 169 delegates is win a simple majority of the vote. In a closed primary, that’s a pretty low bar for him to clear. 

Colorado — 37 delegates

Trump best case: Trump 27, Haley 10

Haley best case: Trump 21, Haley 16 

The variety of delegate outcomes here is limited, because all 39 delegates are awarded proportionally to candidates winning at least 20% of the vote. With independents participating (but Democrats not allowed to cross over) and the suburbs of Denver rich with college-educated, white-collar professionals, Haley can easily cross that threshold. 

But just how high can she climb? Working in Haley’s favor is the state’s vote-by-mail system, under which independent voters were all sent Republican primary ballots. Will the ease of filling out ballots at home and dropping them in mailboxes mean more Haley-friendly/Trump-hostile independents participate? That’s Haley’s best hope. 

Still, Haley capped out at 43% in New Hampshire under even more favorable demographic conditions. If she can hit 40%, it would probably be enough to grab 16 delegates. And if the mass independent participation she’s banking on fizzles and she just barely crosses the 20% line, her delegate haul would fall to about 10. Colorado’s delegate rules mean the practical difference between a 50-point Trump win and a 20-point Trump win just isn’t that much. 

Maine — 20 delegates

Trump best case: Trump 20, Haley 0

Haley best case: N/A

Here’s another state where a candidate who breaks 50% wins every delegate. And Maine is far friendlier to Trump, who carried one of its two congressional districts in the 2016 and 2020 general elections, than any other New England state. Anchored by the old mill towns of Bangor and Lewiston and taking in a wide rural swath that contains one of the highest concentrations of white voters without college degrees in the country, Maine’s 2nd District is ideally suited to Trump, who should rack up massive margins there.

The rest of the state does include some well-to-do ZIP codes, especially along the southern coast, where Haley figures to score. But there are also plenty more blue-collar pockets for Trump to tap. Three times since 2010, the Republican electorate here has chosen Paul LePage, an inflammatory, Trump-style populist, as its nominee for governor. For Trump, attaining a simple majority should be no problem.

Massachusetts — 40 delegates

Trump best case: Trump 40, Haley 0

Haley best case: Haley 40, Trump 0

This is essentially a winner-take-all contest, with all 40 of the state’s delegates going to any candidate breaking 50% of the vote. And while it’s a stretch to float the possibility of Haley’s actually carrying Massachusetts, it should at least be mentioned, if only because of the extraordinary imbalance between the number of registered Republicans in the state (a mere 9% of the electorate) and the number of independents (60%) who can participate if they want. 

Trump figures to clean up among that core group of registered Republicans — voters who turned strongly against the state’s Republican former governor, Charlie Baker, due in no small part to his outspoken criticism of Trump. Facing the prospect of defeat in his own party’s primary, Baker declined to seek a third term in 2022. Trump also easily won the state’s primary in 2016, claiming 49% of the vote, 31 points ahead of his nearest foe. 

Still, there are those independents. With such a vast pool of them, including many Trump-phobic and college-educated suburbanites, there’s a slim chance that just enough of them see the GOP primary as an enticing opportunity to vote against Trump and come out for Haley. It would take a lot of them to offset Trump’s core Republican strength — but again, there are a lot of them. 

That having been said, the scant GOP primary polling conducted in Massachusetts has shown Trump consistently over 50%. And according to the secretary of state’s office , the vast majority of the (many) vote-by-mail ballots received so far have been for the Democratic primary. To borrow a phrase from the state’s political past, it would take a “Massachusetts Miracle” for Haley to pull this off. 

Minnesota — 39 delegates

Trump best case: Trump 39, Haley 0

Haley best case: Trump 17, Haley 14, 8 unbound

Complicated rules mean the Land of 10,000 Lakes has about 10,000 delegate scenarios. The 15 at-large and 24 congressional district delegates are awarded proportionally, but solely by the statewide results. A candidate can win all the state’s delegates with a statewide result above 80%. But short of that, things get tricky. 

Finishing above 60% nets a candidate two delegates from each congressional district, and the second-place finisher gets the third delegate from each district, provided he or she wins at least 20% of the statewide vote. 

If the winner falls somewhere below 60% statewide (or the second-place finisher falls below 20% with a winner below 80%), the state would send some unbound delegates to the Republican convention.

This is an open primary in which Rubio won and Trump placed third in 2016. So Haley’s expected to pick up some delegates here, even if a win is unlikely.  

North Carolina — 74 delegates

Trump best case: Trump 60, Haley 14

Haley best case: Trump 55, Haley 19  

Haley has a few things going for her in North Carolina. First, there’s no statewide winner-take-all threshold here, so Haley will net some delegates if she can eclipse 20%. Congressional delegates are awarded proportionally, too, as long as one candidate doesn’t break two-thirds of the vote. And independents can participate in the primary, as well.

A few demographically Haley-friendly metro areas — Raleigh-Durham-Cary and Charlotte — could allow her to pick up some delegates here. But the proportional rules mean there’s a ceiling, too, as North Carolina Republicans are largely on board with a Trump-friendly brand of politics. Trump’s preferred candidate for governor, Mark Robinson, has stoked controversy throughout his political career, and he is likely to lock up his nomination Tuesday, as well, in one of the most watched governor’s races this fall.

Oklahoma — 43 delegates

Trump best case: Trump 43, Haley 0

With the exception of that of Washington, D.C., closed primaries are Haley’s bane, with no independents or Democrats available for her to offset Trump’s overwhelming advantage among registered Republicans.

A simple majority is all that’s needed to claim all 28 statewide delegates, and the state’s five congressional districts also award three delegates apiece to the majority winners there. There is some real estate around Oklahoma City with Haley-friendly demographics, and under different rules, perhaps she’d be competitive in one district. But not in a closed primary system.

Tennessee: 58 delegates

Trump best case: Trump 58, Haley 0

Haley best case: Trump 44, Haley 14

Tennessee holds an open primary with a twist — voters don’t register by party, so they can technically vote in whatever primary they want, but a state law enacted last year requires polling places to inform voters it’s illegal to vote in a primary without being a “bona fide member” of that party . The law could dissuade non-Republican voters from crossing over, so the state’s results may not look like those of other states without party registration. 

The statewide margin will be very important, as Tennessee will award all of its 31 at-large delegates to a candidate who eclipses two-thirds of the statewide vote. Otherwise, it will award those delegates proportionally. If Haley can perform here like she did in South Carolina, winning 40% of the vote, she’ll win a handful of delegates. But if her Tennessee margin is more like Michigan’s (27%), she risks getting blown out in terms of delegates. She also has to keep an eye on her lower threshold, as the state requires hitting the 20% mark to receive delegates. 

Haley’s best-case scenario involves her keeping Trump under two-thirds and then snagging a delegate or two in the Nashville-area 5th District.

Texas: 161 delegates

Trump best case: Trump 155, Haley 6

Haley best case: Trump 146, Haley 15

While Texas has an open primary, the rightward shift of the state GOP electorate, as reflected in the intraparty civil war that has been brewing in recent years, suggests a majority of the statewide vote — and 36 of the state’s 47 at-large delegates — will be a lock for Trump. The remaining 11 will be awarded based on a vote at the party's convention, so those delegates may not be allocated yet (and could be subtracted from Trump's totals above). But if Trump wins a majority of the statewide vote, he'll be the heavy favorite to win those remaining delegates at the convention.

Then things get interesting in the 38 congressional districts, because each will award three delegates — two to a plurality winner, three to a majority winner — based on the results in each individual district. Texas is the largest state whose primary isn’t expected to function as a de facto winner-take-all contest. (See California and its rules above.)

The Texas rules give Haley a few opportunities around the major metropolitan areas, particularly in the highly educated 37th District in Austin and the 7th in the Houston area. But in a one-on-one race, she doesn’t have much room to grow past that.    

Utah — 40 delegates

Caucuses haven’t been kind to Haley, putting her at an immediate disadvantage here. Independents do have the opportunity to participate if they show up and change their party registrations on site. But that requirement, along with the fact that voting in the caucuses will require attending a meeting at a fixed time, will surely cut into the number who actually take part.

It’s also true that Utah hasn’t been kind to Trump. Utah’s large Mormon population, typically a stalwart Republican group, proved remarkably resistant to him in 2016, when he finished third in the caucuses, 55 points behind Cruz in first place. That fall, Trump earned just 45% of the vote in Utah — still enough to win, but also the third-lowest share ever recorded by a GOP presidential candidate in the state’s history.

But that was then. While some have remained hostile to Trump, many other Republicans have made their way to his bandwagon. As president in 2020, Trump took 87% of the GOP primary vote even with a protest candidate — former Massachusetts Gov. Bill Weld — on the ballot. Trump’s most prominent Republican critic in the state, Sen. Mitt Romney, has seen his own standing with Republican voters erode .  

As in many other Super Tuesday states, a simple majority is all Trump will need to gobble up all 40 delegates. And Republican Gov. Spencer Cox, who has refused to endorse Trump and urged his party to select a different candidate, says there’s “ no doubt ” Trump will achieve that. 

Vermont — 17 delegates

Trump best case: Trump 17, Haley 0

Haley best case: Haley 17, Trump 0

Vermont may be Haley’s best shot at a Super Tuesday victory. The state has no party registration, meaning anyone can vote in the GOP primary, and it’s a deep-blue state with a unique, highly engaged political culture. At least in theory, there are hordes of anti-Trump voters here who don’t think of themselves as Republicans but could be motivated to vote in the GOP primary to express their profound distaste for him.

Haley’s chances rest on that. And she’s helped by the lack of drama on the Democratic side and the endorsement of the state’s moderate Republican governor, Phil Scott.

Scott is serving his fourth two-year term and has survived primary challenges thanks in no small part to the kind of crossover voters Haley needs. There also remains a strain of old school, middle-of-the-road Yankee Republicanism in quirky Vermont: In the 2016 GOP primary, John Kasich finished just 2 points behind Trump (who won with only 32%), and the combined vote share for Kasich and Rubio reached 50% — higher than in almost any other contest that year. Both candidates attracted the type of upscale, college-educated support that Haley has this year.

But while the ingredients for a potential Haley win are largely present, Trump does have his own strengths. For one, there’s a sizable population of working-class white voters without college degrees, his core demographic, particularly in the state’s Northeast Kingdom. Trump carried Essex County, the heart of that region, in both the 2016 and the 2020 elections — after it had twice voted for Barack Obama. Overall, 41% of white adults in the state have college degrees; that’s 6 points lower than in South Carolina’s 1st District, which Haley narrowly carried over Trump on Feb. 24. There’s also a University of New Hampshire poll that just last week put Trump ahead of Haley by 30 points in Vermont. 

The rules here award all 17 delegates to anyone who can attain a simple majority. It’s very likely that either Trump or Haley will, though other candidates are on the ballot. In a tight race, it’s conceivable they could both fall short of the threshold, in which case delegates would be divided proportionally. 

Virginia — 48 delegates

Trump best case: Trump 42, Haley 6

Haley best case: Haley 27, Trump 21

It would be easy to see how Haley’s victory in the Washington, D.C., primary would carry over to momentum in Northern Virginia, with significant blocs of Trump-resistant members of the professional working class. But there’s a lot more to Virginia beyond the Washington suburbs. Overall, the person who gets 50% or more, statewide and in each congressional district, will win the delegates in Virginia. 

Haley was polling at 43% in the state in the most recent Roanoke College poll . In her best-case scenario, she can win the state. On the other hand, in 2016, Rubio and Kasich combined to pull in 42% of the state vote. And in 2024, it’s difficult to see Trump falling under the 50% threshold.

Beyond Super Tuesday

That’s the situation through Super Tuesday. But we have a bonus for loyal readers: Here’s our outlook through the next two weeks of the contest, when Trump could hit the magic number to become the presumptive nominee.

American Samoa: 9 delegates

Trump best case: Trump 9, Haley 0

Haley best case: Trump 0, Haley 0, unbound 9

This territory is holding a territorial caucus, which is typically good news for Trump. 

But there’s an important wrinkle: The delegates aren’t always bound. Party rules say the delegates will be “instructed” by a caucus resolution as to the “disposition of their vote.” In 2016, the territorial delegates started off as unbound but ended up announcing they’d all back Trump by May . 

The best case for Trump is that the nine delegates decide to back him like they did in 2016. Haley’s best bet is likely to be that they stay unbound — and that the caucus comes and goes without Trump’s getting any closer to the nomination. 

Georgia: 59 delegates

Trump best case: Trump 52, Haley 7

Haley best case: Trump 43, Haley 16

Trump has had a rocky time in Georgia. He and his allies face charges there related to their attempts to overturn the 2020 presidential election, and his preferred candidates lost primaries for governor, attorney general and secretary of state in 2022, as well as the general election for the U.S. Senate months later. The state’s Republican governor, Brian Kemp, remains popular despite having distanced himself from Trump. 

An open primary in a state where Trump has that much baggage could give Haley an opening. But it would be tough to see her doing much better than her performance in South Carolina, where she won about 40%. 

While 17 at-large delegates will be doled out proportionally to every candidate who hits 20%, the remaining 42 will be split up across congressional districts. And the candidate who wins 50% in a district will win all three delegates.

The demographics look best for Haley in the 4th, 5th and 6th districts in suburban Atlanta — Rubio and Kasich combined for over 50% of the vote in DeKalb and Fulton counties in 2016. But again, the best-case scenario for Haley is limiting damage, not winning the state. 

Hawaii: 19 delegates

Trump best case: Trump 16, Haley 3

Haley best case: Trump 13, Haley 6

Another closed caucus gives the advantage to Trump. He won 42% of the caucus vote here in 2016, with Cruz winning 33%, while the more establishment Rubio and Kasich combined for 24%.

Expect Trump to get vote share about similar to what he and Cruz combined to get that year, as he continues to enjoy sky-high polling among Republicans, the only voters who get to participate here. 

Mississippi: 40 delegates

On paper, an open primary could seem like good news for Haley. But the delegate rules leave her little room. A candidate who hits 50% statewide in Mississippi wins every delegate in the state, meaning the primary could be about as competitive as an early-season Ole Miss out-of-conference football game. 

Washington: 43 delegates

Trump best case: Trump 40, Haley 3

In another closed primary, Trump is the heavy favorite to win the majority here, which would give him all 13 at-large delegates (assuming he eclipses 50% in the two-way race). The state awards its 30 congressional district delegates by similar rules, giving them out proportionally in each district unless a candidate eclipses 50% and wins all three from that district. 

Washington’s 7th District, in Seattle, is the only one with significantly favorable demographics for Haley. A win here would net her three delegates, and that seems likely unless the bottom falls out. 

Northern Mariana Islands: 9 delegates

Another closed caucus and another contest in which Haley’s underwater favorability with Republicans may put her out of contention. 

Guam: 9 delegates

Trump and Haley best case: N/A

Guam is one of the few states sending unbound delegates to the convention, so they won’t be allocated to one candidate and will be free to vote as they wish on the convention floor. Individual delegates may have personal loyalties, but they won’t be locked in for one candidate. 

Arizona: 43 delegates

There’s just no evidence that Haley can win a closed primary in a state whose Republican Party has shifted rightward, even as the state grows increasingly competitive in general elections. Primary voters who chose Republicans Kari Lake, Mark Finchem and Blake Masters in key races last cycle aren’t likely to back Haley in this winner-take-all race. 

Florida: 125 delegates

Trump best case: Trump 125, Haley 0

It’s the same story in Florida, Trump’s adopted home. It’s hard to cobble together a path for anything else to happen except Trump’s winning it — just like he did in 2016 over Rubio, the home-state candidate. 

Illinois: 64 delegates

Trump best case: Trump 61, Haley 3

Haley best case: Trump 51, Haley 13

This is effectively an open primary, but one Trump won in 2016, combining with Cruz for 69% of the vote. The winner here takes home all 13 at-large delegates, and then it gets complicated.

That’s because congressional district delegates get elected individually on the state’s primary ballot — the ballot includes individual names alongside the candidates they’ve said they’ll support. With so many permutations, an exact split is hard to predict. But Trump’s 39% finish here in 2016 netted him 78% of the delegates, so he doesn’t need to put up a big statewide number to win the vast majority of the delegates. 

Kansas: 39 delegates

This is another (effectively) closed contest, as non-Republicans had to register with the party by Feb. 20 to participate in the primary. That spells bad news for Haley. 

Ohio: 79 delegates

Trump best case: Trump 79, Haley 0

Ohio voters can participate in whichever party’s presidential primary they want — the only stipulation being that by doing so, they are registering with the party through that election cycle.

Kasich won the state’s primary in 2016 with 47% of the vote. But Haley will be hard-pressed to eclipse that in a one-on-one contest in a winner-take-all state — and especially in a state where she’s not the current governor, as Kasich was.

The big picture

Adding our best-case scenarios for Haley and Trump together, here’s our estimate of the rough range of possible outcomes in the delegate race through next week:

Trump’s best scenario has him just sneaking past the finish line only a week after Super Tuesday, with Haley adding only a few dozen stray delegates to her tally. But even under Haley’s best-case results, she would still fall miles behind Trump and delay his win by only a week or two. The Republican nominating system is designed to produce a quick and decisive winner, with few rewards for finishing second in most states — and it’s likely to work as designed over the next few weeks.

Steve Kornacki, author of "The Red and the Blue: 1990s and the Birth of Political Tribalism," is a national political correspondent for NBC News and MSNBC.

Ben Kamisar is a deputy political editor in NBC's Political Unit. 

Producer at MSNBC with Steve Kornacki.

Former EY partner shares how he climbed the corporate ladder to walk away from it all and become a math teacher

• Deepak Swaroop was a partner at EY in London for 10 years before quitting in 2019.
• Swaroop left his position, with a reported average pay of $1 million in 2022, to change careers.
• He qualified to be a math teacher in his 50s and said teaching is more valuable than money. 

Deepak Swaroop spent nearly 20 years working at EY in London, 10 of which as a partner— a position known to attract an average salary of £803,000, about $1 million. Swaroop has an MBA and has completed courses in executive management at Harvard and AI at MIT.

In 2020, he left the high-flying world of finance behind to become a high school math teacher, taking a considerable pay cut in the process.

Swaroop started his career consulting for Arthur Andersen & Co. in India

Swaroop grew up in Delhi and Mumbai during the 1970s and 80s. He came from a high-achieving family of engineers, but his aptitude for finance helped him climb the ranks at global financial consultancy Arthur Andersen & Co.

"I built a valuation practice supporting foreign investments into India in the 90s. I made my way up from a senior consultant to manager and then a director," Swaroop told Business Insider.

In 2000, Swaroop leaped at the chance to be promoted to chief of staff to the company's EMEIA —  Europe, the Middle East, India, and Africa— managing partner. This opportunity included company benefits like relocating from Delhi to a four-bedroom apartment in London's expensive St John's Wood area and covering London private school fees for his two young children.

Soon after Swaroop was made a partner himself in 2001, Arthur Andersen & Co. collapsed. The accounting firm was part of a massive scandal involving the energy company Enron in the US. The energy company went bankrupt, and Arthur Andersen & Co. was dissolved.

Swaroop became a partner at EY

Swaroop was one of 85,000 employees left without a job. He was faced with the decision to return to India, or to stay in the UK without the corporate benefits. "My wife and I said we would not let the kids' education get impacted, so if we wanted to stay on in London, that would be the one constant thing," said Swaroop. "We said we'd give ourselves a year to see if it would work out."

He and his wife decided to cut costs by moving to a smaller apartment in a less expensive borough of London. They relied on their savings to keep their children in private education.

During this time, Swaroop began his career at EY. Swaroop, alongside many of his fellow Arthur Andersen & Co employees, moved to EY after it absorbed some of the Arthur Andersen & Co operations.

'I was at the right place at the right time'

At EY, Swaroop shone. He pioneered EY's adoption of automation and AI, building the company's Automation Central division in 2015.

Related stories

"I felt I was at the right place at the right time, with the right sort of knowledge of the organisation, which allowed me to rapidly build the team," Swaroop told BI. "People tell me that still remains the largest such operation in the corporate world. So we did something really good," he added.

EY restructured, and Swaroop no longer felt excited about his role

After a company restructuring in 2019, Swaroop told BI he faced another big decision: accept a significant change to his role or leave the business. He chose the latter.

"I felt that working there was no longer going to be as exciting for me," said Swaroop. "So I took early retirement, partly driven by my desire to start a startup."

He spent the next year co-founding an enterprise AI platform, from which he quickly parted ways. He told BI it wasn't the change he sought from corporate life.

Retired life was unfulfilling, but teaching was a way out

Swaroop was 56. He had paid off his mortgage and supported his now adult children through university. The next step for the ex-partner was early retirement. But he found his new schedule of golf, walks, and trips to the library unfulfilling.

Swaroop told BI he read about Lucy Kellaway, the Financial Times editor who left journalism to become a trainee teacher. Kellaway founded Now Teach in 2017, an organisation to support people to change careers into teaching. After sitting in on lessons at five different schools, his mind was made up. Swaroop wanted to retrain as a high school math teacher.

Three years on, Swaroop told BI he feels energized and inspired in his new career — even if he now earns a fraction of what he used to. The maximum salary of a qualified teacher in London is around £50,000, according to the UK government.

The pay cut didn't matter as long as Swaroop was making a difference

Swaroop said his salary wasn't a deciding factor in this career change as he knew it wouldn't come close to his old pay.

"I was keen to do something which had a purpose, so I could contribute back to society in a way," said Swaroop.

Swaroop said coming into teaching as a novice in his 50s was challenging. He was surrounded by more experienced peers who were much younger. He also found standing up in front of teenagers all day more grueling than working in an office. But he has been buoyed by his students' energy, humor, and affection and how he motivates them to learn.

"I just want to get into the classroom and teach. I don't have an objective of becoming a head of school. Previously, I would actively try to move up the ladder. That is being replaced by my desire to be more committed to my teaching," said Swaroop.

"I have had students write to me that I have helped them realize their potential and what path to take in the future. That is more valuable to me than money."

Correction: March 14, 2024 — An earlier version of this article misstated how long Deepak Swaroop was a partner at EY. Swaroop worked at EY for 20 years, 10 of which he held a partner position.

Watch: JIM GLASSMAN: It's not a wage problem, it's a skills problem

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