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Mathematics Problem-solving Challenges For Secondary School Students And Beyond (Problem Solving In Mathematics And Beyond Book 4) Kindle Edition

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• ASIN ‏ : ‎ B01D41D8QY
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• Publication date ‏ : ‎ February 25, 2016
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Mathematics Problem-Solving Challenges for Secondary School Students and Beyond

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Mathematics Problem-Solving Challenges For Secondary School Students And Beyond: 4 (Problem Solving in Mathematics and Beyond) Paperback – 25 Feb. 2016

• ISBN-10 9814730033
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• ISBN-10 ‏ : ‎ 9814730033
• ISBN-13 ‏ : ‎ 978-9814730037
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Mathematics Problem-solving Challenges for Secondary School Students and Beyond Mathematics Problem-solving Challenges for Secondary School Students and Beyond

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• Sultan, Alan, Author

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• Mathematics — Problems, exercises, etc.
• Mathematics — Competitions.
• Mathematics — Study and teaching (Secondary)
• New Jersey : World Scientific, [2016]

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内容简介  · · · · · ·

This book is a rare resource consisting of problems and solutions similar to those seen in mathematics contests from around the world. It is an excellent training resource for high school students who plan to participate in mathematics contests, and a wonderful collection of problems that can be used by teachers who wish to offer their advanced students some challenging nontrad...

This book is a rare resource consisting of problems and solutions similar to those seen in mathematics contests from around the world. It is an excellent training resource for high school students who plan to participate in mathematics contests, and a wonderful collection of problems that can be used by teachers who wish to offer their advanced students some challenging nontraditional problems to work on to build their problem solving skills. It is also an excellent source of problems for the mathematical hobbyist who enjoys solving problems on various levels. Problems are organized by topic and level of difficulty and are cross-referenced by type, making finding many problems of a similar genre easy. An appendix with the mathematical formulas needed to solve the problems has been included for the reader's convenience. We expect that this book will expand the mathematical knowledge and help sharpen the skills of students in high schools, universities and beyond.

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This book is a rare resource consisting of problems and solutions similar to those seen in mathematics contests from around the world. It is an excellent training resource for high school students who plan to participate in mathematics contests, and a wonderful collection of problems that can be used by teachers who wish to offer their advanced students some challenging nontraditional problems to work on to build their problem solving skills. It is also an excellent source of problems for the mathematical hobbyist who enjoys solving problems on various levels.Problems are organized by topic and level of difficulty and are cross-referenced by type, making finding many problems of a similar genre easy. An appendix with the mathematical formulas needed to solve the problems has been included for the reader's convenience. We expect that this book will expand the mathematical knowledge and help sharpen the skills of students in high schools, universities and beyond.

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This book is a rare resource consisting of problems and solutions similar to those seen in mathematics contests from around the world. It is an excellent training resource for high school students who plan to participate in mathematics contests, and a wonderful collection of problems that can be used by teachers who wish to offer their advanced students some challenging nontraditional problems to work on to build their problem solving skills. It is also an excellent source of problems for the mathematical hobbyist who enjoys solving problems on various levels. Problems are organized by topic and level of difficulty and are cross-referenced by type, making finding many problems of a similar genre easy. An appendix with the mathematical formulas needed to solve the problems has been included for the reader's convenience. We expect that this book will expand the mathematical knowledge and help sharpen the skills of students in high schools, universities and beyond.

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• Publisher Wspc
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• ISBN 10  9814730033
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Problem Solving In Mathematics and Beyond

Alan Sultan

World Scientific Publishing Company

25 February 2016

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Mathematical Problem Solving Beyond School: Digital Tools and Students’ Mathematical Representations

• First Online: 01 January 2015

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By looking at the global context of two inclusive mathematical problem solving competitions, the Problem@Web Project intends to study young students’ beyond-school problem solving activity. The theoretical framework is aiming to integrate a perspective on problem solving that emphasises understanding and expressing thinking with a view on the representational practices connected to students’ digital mathematical performance. Two contextual problems involving motion are the basis for the analysis of students’ digital answers and an opportunity to look at the ways in which their conceptualisations emerge from a blend of pictorial and schematic digital representations.

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This work is part of the research developed under the Problem@Web Project, Nº PTDC/CPE-CED/101635/2008, funded by Fundação para a Ciência e Tecnologia.

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Carreira, S. (2015). Mathematical Problem Solving Beyond School: Digital Tools and Students’ Mathematical Representations. In: Cho, S. (eds) Selected Regular Lectures from the 12th International Congress on Mathematical Education. Springer, Cham. https://doi.org/10.1007/978-3-319-17187-6_6

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Mathematics Problem-Solving Challenges for Secondary School Students and Beyond.

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ORIGINAL RESEARCH article

Mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

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

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

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

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

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

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

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

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

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

The Present Study

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

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

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

Participants

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

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

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

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

Intervention

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

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

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

Implementation of the Intervention

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

Control Group

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

Tests of Mathematical Problem-Solving

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

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

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

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

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

Measures of Peer Acceptance and Friendships

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

Statistical Analyses

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Limitations

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

Implications

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

Data Availability Statement

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

Ethics Statement

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

Author Contributions

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

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

Conflict of Interest

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

Publisher’s Note

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

Acknowledgments

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

Supplementary Material

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

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

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

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

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*Correspondence: Nina Klang, [email protected]

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• Published: 10 December 2022

Secondary school students’ attitude towards mathematics word problems

• Robert Wakhata   ORCID: orcid.org/0000-0001-9144-0420 1 ,
• Védaste Mutarutinya 2 &
• Sudi Balimuttajjo 3  

Humanities and Social Sciences Communications volume  9 , Article number:  444 ( 2022 ) Cite this article

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Students’ positive attitude towards mathematics leads to better performance and may influence their overall achievement and application of mathematics in real-life. In this article, we present the findings of an investigation on students’ attitude towards linear programming (LP) mathematics word problems (LPMWPs). An explanatory sequential quasi-experimental design involving a pre-intervention-intervention-post-intervention non-equivalent control group was adopted. A sample of 851 grade 11 Ugandan students (359 male and 492 female) from eight secondary schools (public and private) participated. Cluster random sampling was applied to select respondents from eight schools; four from central Uganda and four from eastern Uganda. The attitude towards mathematics inventory-short form (ATMI-SF) was adapted (with α  = 0.75) as a multidimensional measurement tool for measuring students’ attitude towards LPMWPs. The results revealed that students’ attitude towards LPMWPs was generally negative. Enjoyment, motivation, and confidence were weekly negatively correlated while usefulness was positively correlated. Additionally, the results found no significant statistical relationship between students’ attitudes towards LPMWPs and their age, gender, school location, school status, and school ownership. The discrepancy is perhaps explained by both theoretical and/or psychometric limitations, and related factors, for instance, students’ academic background, school characteristics, and transitional beliefs from primary to secondary education. This study acknowledges the influence of and supplements other empirical findings on students’ attitude towards learning mathematics word problems. The present study provides insight to different educational stakeholders in assessing students’ attitude towards LPMWPs and may provide remediation and interventional strategies aimed at creating students’ conceptual change. The study recommends that teachers should cultivate students’ interests in mathematics as early as possible. Varying classroom instructional practices could be a remedy to enhance students’ understanding, achievement, and, motivation in learning mathematics word problems. The teachers’ continuous professional development courses should be enacted to improve instruction, assessment, and students’ attitude. Overall, the study findings support the theoretical framework for enhancing the learning of mathematics word problems in general and LP in particular.

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Introduction.

The term attitude is not a new concept in mathematics education. It has been defined by different authors in different settings and contexts. For instance, Aiken ( 1970 ) defined attitude as “a learned disposition or tendency on the part of an individual to respond positively or negatively to some object, situation, concept or another person” (p. 551). According to Lin and Huang ( 2014 ), attitude towards mathematics can be referred to as positive, negative, or neutral feelings and dispositions. Attitude can be categorized as bi-dimensional (person’s emotions and beliefs) or multidimensional (affect, behavior, and cognition). Over the last decades, an extensive body of research from different settings and contexts have investigated variables that influence students’ attitude towards Science, Technology Engineering and, Mathematics (STEM) (e.g., Aiken, 1970 ; Gardner, 1975 ; Kempa and McGough, 1977 ). In this study, we are particularly concerned with students’ attitude towards mathematics word problems, and linear programming (LP) in particular due to the significant roles LP plays in constructing models for understanding the three (STE).

Numerous studies have been published on students’ attitude towards mathematics, which is always translated as liking and disliking of the subject (Arslan et al., 2014 ; Davadas and Lay, 2020 ; Pepin, 2011 ; Utsumi and Mendes, 2000 ). To some secondary school students, mathematics appears to be abstract, difficult to comprehend, sometimes boring, and viewed with limited relationship or relevance to everyday life experiences. At primary and secondary school levels, students start well but gradually start disliking mathematics feeling uncomfortable and nervous. Consequently, they may lack self-confidence and motivation during problem-solving. To some students, persevering and studying advanced mathematics has become a nightmare. Indeed, some students do not seem to know the significance of learning mathematics beyond the compulsory level. Students may (or may not) relate mathematical concepts beyond the classroom environment if they have a negative attitude towards the subject. This may lead to their failure to positively transfer mathematical knowledge and skills in solving societal problems.

Mathematicians have attempted to research and understand affective variables that significantly influence students’ attitude towards mathematics (e.g., Barmby et al., 2008 ; Davadas and Lay, 2020 ; Di Martino and Zan, 2011 ; Evans and Field, 2020 ; Grootenboer and Hemmings, 2007 ; Hannula, 2002 ; Maamin et al., 2022 ; Marchis, 2011 ; Pongsakdi et al., 2019 ; Yasar, 2016 ; Zan et al., 2006 ). Some researchers have gone ahead to ask fundamental questions on whether or not students’ attitude towards mathematics is a general phenomenon or dependent on some specific variables. To this effect, some empirical findings report students’ attitude towards specific units or topics in mathematics aimed at enhancing the learning of specific mathematical content and mathematics generally (e.g., Arslan et al., 2014 ; Estrada and Batanero, 2019 ; Gagatsis and Kyriakides, 2000 ; Julius et al., 2018 ; Mumcu and Aktaş, 2015 ; Selkirk, 1975 ; Townsend and Wilton, 2003 ).

Rather than investigating students’ general attitudes toward mathematics, recent research has also attempted to identify background factors that may provide a basis for understanding students’ attitude towards mathematics. Thus, students at different academic levels may have a negative or positive attitude towards mathematics due to fundamentally different reasons. Yet, other studies show the existence of a positive relationship between attitude and achievement in mathematics (e.g., Berger et al., 2020 ; Chen et al., 2018 ; Davadas and Lay, 2020 ; Grootenboer and Hemmings, 2007 ; Hwang and Son, 2021 ; Lipnevich et al., 2011 ; Ma, 1997 ; Maamin et al., 2022 ; Mazana et al., 2018 ; Mulhern and Rae, 1998 ; Opolot-okurut, 2010 ; Sandman, 1980 ; Tapia, 1996 ). From the above studies, it appears that multiple factors ranging from students’ to teachers’ classroom instructional practices may influence students’ attitudes towards, and achievement in mathematics.

Ugandan context

In Uganda, studies on predictors of students’ attitude towards science and mathematics are scanty. There are no recent empirical findings on secondary school students’ attitude towards Mathematics and mathematics word problems in particular. Solving LP tasks (by graphical method) is one of the topics taught to 11th-grade Ugandan lower secondary school students (NCDC, 2008 , 2018 ). Despite students’ general and specific learning challenges in mathematics and LP, the objectives of learning LP are embedded within the aims of the Ugandan lower secondary school mathematics curriculum (Supplementary Appendix 3 ). Some of the specific aims of learning mathematics in Ugandan secondary schools include …enabling individuals to apply acquired skills and knowledge in solving community problems, instilling a positive attitude towards productive work…” (NCDC, 2018 ). Generally, the learning of LP word problems aims to develop students’ problem-solving abilities, application of prior algebraic concepts, knowledge, and understanding of linear equations and inequalities in writing models from word problems, and real-life-world problems. Despite the learning challenges, the topic of LP is also aimed at equipping learners with adequate knowledge and skills for doing advanced mathematics courses beyond the 11th-grade (locally called senior four) minimum mathematical proficiency at Uganda Certificate of Education (UCE).

However, every academic year, the Uganda National Examinations Board (UNEB) highlights students’ strengths and weaknesses in previous examinations at UCE. The consistent reports (e.g., UNEB, 2016 , 2018 , 2019 , 2020 ) on previous examinations on the work of candidates show that students’ performance in mathematics is not satisfactory, especially at the distinction level. In particular, the above previous examiners’ reports show students’ poor performance in mathematics word problems. The examination reports have consistently revealed numerous students’ specific deficiencies in the topic of LP (please see Supplementary Appendix 1 ). Students’ challenges in LP mainly stem from comprehension of word problems to the formation of wrong linear equations and inequalities (in two dimensions) from the given word problems in real-life situations. Thus, wrong models derived from questions may result in incorrect graphical representations, and consequently wrong solutions and interpretations of optimal solutions. These challenges (and others) may consequently hinder and/or interfere with students’ construction of relevant models in science, mathematics, and technology. Moreover, learners have consistently demonstrated cognitive obstacles in answering questions on LP, while the majority elude these questions during national examinations by answering questions from presumably “simpler” topics. Noticeably absent in all the UNEB reports are factors that account for students’ weaknesses in learning LP and the specific interventions to overcome students’ challenges. Some students have, however, developed a negative attitude towards the topic. Yet, students’ attitudes may directly impact their learning outcomes (Code et al., 2016 ).

Although some empirical findings (e.g., Opolot-okurut, 2010 ) have reported on students’ attitude towards mathematics in the secondary school context, this paper presents results from a more specific investigation into students’ attitudes towards mathematics word problems. Specifically, the present study investigated secondary school students’ attitude towards solving linear programming mathematics word problems (LPMWPs). This is because studies concerning attitudes towards and achievement in mathematics have begun to drift from examining general attitudes to a more differentiated conceptualization of specific students’ attitude formations, and in different units (topics). Although different attitudinal scales (e.g., Code et al., 2016 ; Fennema and Sherman, 1976 ; Tapia, 1996 ) were developed to measure different variables influencing students’ attitudes towards mathematics, this study specifically investigated the influence of some of these constructs on students’ attitude towards learning LP. According to the above-stated authors (and other empirical findings), students’ attitude is a consequence of both general and specific latent factors.

Mathematics word problems

Verschaffel et al. ( 2010 ) define word problems as “verbal descriptions of problem situations wherein one or more questions have raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement.” The authors categorized word problems based on their inclusion in real-life world scenarios. Thus, mathematics word problems play significant roles in equipping learners with the basic knowledge, skills, and, understanding of problem-solving and mathematical modeling. Some empirical findings (e.g., Boonen et al., 2016 ) show that mathematics word problems link school mathematics to real-life world applications. However, the learning of mathematics word problems and related algebraic concepts is greatly affected by students’ cognitive and affective factors (Awofala, 2014 ; Jupri & Drijvers, 2016 ; Pongsakdi et al., 2019 ). Mathematics word problems are an area where the majority of students experience learning obstacles in secondary schools and beyond (Abdullah et al., 2014 ; Awofala, 2014 ; Dooren et al., 2018 ; Goulet-Lyle et al., 2020 ; Julius et al., 2018 ; Pearce et al., 2011 ; Sa’ad et al., 2014 ; Verschaffel et al., 2010 , 2020a , 2020b ). By contrast, comprehension of mathematics word problems explains relational difficulties. Consequently, this has undermined students’ competence, confidence, and achievement in word problems and mathematics in general.

Yet, mathematics word problems are intended to help learners to apply mathematics beyond the classroom in solving real-life-world problems. Verschaffel et al. ( 2020a , 2020b ) and Boonen et al. ( 2016 ) have argued that mathematics word problems are difficult, complex, and pause comprehension challenges to most learners. This is because word problems require learners to understand and apply previously learned basic algebraic mathematical principles, rules, and techniques. Indeed, most learners find it difficult to understand text in word problems before transformation into models. This is partly due to variations in their comprehension abilities and language (Strohmaier et al., 2020 ). Consequently, learners fail to write required mathematical algebraic symbolic operations and models. Yet, incorrect models lead to wrong algebraic manipulations and consequently wrong graphical representations and solutions.

Notably, research findings by Meara et al. ( 2019 ), and Evans and Field ( 2020 ) indicate that students’ mathematical inefficiency is due to their transitional epistemological and ontological challenges from primary to secondary education. Other studies (e.g., Georgiou et al., 2007 ; Grootenboer and Hemmings, 2007 ; Li et al., 2018 ; Norton, 1998 ; Sherman, 1979 ; Sherman, 1980 ) attribute students’ poor performance and achievement in mathematics to gender differences. Thus, students may start learning mathematics well from primary but gradually lose interest in some specific units and finally in mathematics generally. For the case of LP, and as indicated above, it is likely that students’ attitude towards mathematics and equations, inequalities, and LP in particular gradually drop in favor of other presumably simpler topics. However, to boost performance in mathematics word problems, Goulet-Lyle et al. ( 2020 ) proposed a step-by-step problem-solving strategy to enhance mastery and develop a positive attitude towards learning.

Students’ attitudes should, therefore, be investigated as well as their influence on their conceptual changes. Several empirical studies have also investigated the relationship between attitude towards, and achievement in mathematics across all levels, and in different contexts (e.g., Bayaga and Wadesango, 2014 ; Camacho et al., 1998 ; Chun and Eric, 2011 ; Davadas and Lay, 2020 ; Karjanto, 2017 ; Khavenson et al., 2012 ; Ozdemir and Ovez, 2012 ; Quaye, 2015 ; Selkirk, 1975 ; Tahar et al., 2010 ; Utsumi and Mendes, 2000 ; Yáñez-Marquina and Villardón-Gallego, 2016 ). In particular, these studies generally focused on students’ attitude towards mathematics, and many of them were conducted from the western context (Kasimu and Imoro, 2017 ). Yet, students may have different perceptions and attitudes towards specific content (topics) in mathematics irrespective of their setting, context, and learning environment.

To enhance mathematical conceptual proficiency, educators should target and/or boost students’ cognitive and affective domains in specific mathematics content. In a related genre, students’ proficiency in LP word tasks may largely depend on their prior algebraic knowledge, skills, and experiences. Julius et al. ( 2018 ) noted that prior conceptual understanding coupled with students’ attitudes towards solving algebraic concepts impacted students’ inherent procedures in writing relational symbolic mathematical models (inequalities) from word problems, and provision of correct numerical solutions. Despite numerous difficulties encountered by students in algebraic inequalities as reported in Fernández and Molina ( 2017 ), Molina et al. ( 2017 ), Bazzini and Tsamir ( 2004 ), Tsamir and Almog ( 2001 ), Tsamir and Bazzini ( 2004 , 2006 ), and Tsamir and Tirosh ( 2006 ) have suggested a combination of approaches, methodologies, and strategies than applying one specific method. Adopting this instructional and assessment approach may help to overcome students’ learning and related algebraic challenges, which are all aimed at enhancing the learning of mathematics.

The theoretical framework

This study is situated on the theoretical framework according to constructivism, and Eccles, Wigfield, and colleagues’ expectancy-value model of achievement motivation (Wigfield, 1994 ; Wigfield and Eccles, 2000 ). The expectancy-value model is based on the expectancy-value theories of achievement. Thus, the theory is based on the premise that success on specific tasks and the values inherent in those tasks is positively correlated with achievement, and consequently students’ attitude towards specific mathematical tasks. In the context of the attitude towards mathematics inventory-short form (ATMI-SF), the theory combines motivation, enjoyment, confidence, value (usefulness), and related latent variables to explain students’ success in learning mathematics. Constructivism is a form of discovery learning that is based on the premise that teachers facilitate learning by actively involving learners so that they construct their world knowledge and understanding based on individual prior experiences and schema (Olusegun, 2015 ; Ültanır, 2012 ). Thus, previous knowledge, understanding, and reflection with new knowledge are inevitable for supporting subsequent learning and acquisition of both conceptual and procedural knowledge. These knowledge components may later arouse learners’ attitude towards specific mathematics content and mathematics achievement generally.

We are particularly concerned about students’ efforts, and persistence, their perceived difficulties and related challenges in learning LPMWPs and the experiences learners may encounter when solving LP word tasks. Empirical findings and our own experiences as mathematics educators show that students’ challenges in LP largely depend on their insufficient previous algebraic knowledge and experiences in applying the knowledge of equations and inequalities. In this article, we discuss students’ attitude towards LPMWPs using the expectancy-value model theory within the constructivism paradigm. Using this paradigm helped to explain the ATMI-SF constructs and their significance in enhancing the learning of mathematics in secondary schools. The expectancy-value theory and constructivism have been widely applied to enhance the learning of mathematics and science (Awofala, 2014 ; Fielding-Wells et al., 2017 ; Meyer et al., 2019 ; Wigfield and Eccles, 2000 ; Yurt, 2015 ). To foster a positive attitude, teachers (educators) should assign different tasks to students based on their academic level so that they apply previously acquired knowledge, understanding, and experiences in subsequent learning. Stein et al. ( 2000 ) reasoned that students’ proficiency and competency are determined by the mathematical tasks they are given. Tasks at the lower cognitive stage (memorization level), for example, must be different from those at the highest cognitive level (doing mathematics). In the context of learning LP, students should first understand and appropriately apply the basic knowledge of equations and inequalities to adequately and proficiently solve non-routine LPMWPs.

Attitude towards mathematics and the learning of linear programming word problems

Linear programming is one of the algebraic topics that require students’ understanding of basic mathematical principles and rules before the application of computer software for solving and optimizing more advanced and complex LP problems. Linear programming is a classical unit, “the cousin” of mathematics word problems, which has gained significant applications in mathematics, science, and technology (Aboelmagd, 2018 ; Colussi et al., 2013 ; Parlesak et al., 2016 ; Romeijn et al., 2006 ) because the topic is used for formulating models that link theoretical to practical mathematical applications. Thus, LP provides basic elementary modeling skills (Vanderbei, 2014 ).

Previous empirical studies have revealed that LP and/or related concepts are not only difficult for learners but also challenging to teach (Awofala, 2014 ; Goulet-Lyle et al., 2020 ; Kenney et al., 2020 ; Verschaffel et al., 2020a , 2020b ). Different factors account for learners’ challenges in mathematics word problems (e.g., Ahmad et al., 2010 ; Haghverdi et al., 2012 ; Heydari et al., 2015 ). The challenges range from students’ comprehension of word problem statements, and their attitude towards the topic, to their transformation from conceptual to procedural knowledge and understanding. Learners’ attitude towards solving algebraic word problems should, therefore, be investigated and integrated during classroom instruction to help educational stakeholders provide appropriate and/or specific instructional strategies and remedies.

Several attitudinal scales (with both cognitive and behavioral components) have been developed (Lim and Chapman, 2013 ; Yáñez-Marquina and Villardón-Gallego, 2016 ) adopted or adapted (Lin and Huang, 2014 ) to assess students’ attitude towards mathematics and in specific mathematics content. For instance, Geometry Attitude Scales (Avcu and Avcu, 2015 ), Statistics Attitude Scales (Ayebo et al., 2019 ; Khavenson et al., 2012 ), Attitudes toward Mathematics Word Problem Inventory (Awofala, 2014 ), the Attitude towards Geometry Inventory (ATGI) instrument (Utley, 2007 ), and others. In this study, we adapted the ATMI-SF instrument (Lin and Huang, 2014 ) to investigate the 11th-grade students’ attitude towards learning LP word problems (see Supplementary Appendix 1 ). Taken together, research shows that a high percentage of educational stakeholders around the world are concerned about attitude towards mathematics and word tasks in particular. However, to fully understand students’ attitude towards mathematics, it is necessary to investigate beyond general mathematics attitudes and examine specific underlying aspects of these attitudes. Thus, the present study examines students’ attitude towards solving LP mathematics word problems.

Methodology

This study investigated students’ attitude towards linear programming mathematics word problems (LPMWPs). To achieve this purpose, a quantitative survey research design was used (Creswell and Plano Clark, 2018 ). The authors contend that the quantitative approach provides a more general understanding of the views of participants in an entire population. Thus, this approach was applied to collect, analyze, and describe the secondary school students’ ATLPWPs, their experiences, and latent behavior.

Research design

The present study was part of a large study that investigated the effect of active learning heuristic problem-solving approach on students’ achievement and attitude towards learning LP word problems. The present study adopted a quantitative approach to gain a deeper and broader understanding of students’ ATLPWPs (Creswell, 2014 ; Creswell and Plano Clark, 2018 ; Djamba and Neuman, 2002 ). A quasi-experimental pre-test, post-test, and non-equivalent control group study design was adopted. By using the stated approach and design, researchers ably compared and contrasted students’ ALPMWPs. Learners from the experimental group, and in their intact classes participated. The main reason for adopting intact classes was to avoid interference with the internal school-set timetables and already set operational schedules.

The analysis reported in this study comprised a research study of 851 grade 11 students from eight randomly selected private or public secondary schools (both rural and urban), four from Mbale district, eastern Uganda, and the remaining four from Mukono district, central Uganda. Cluster random sampling was used to select regions and schools. The sampled schools were allocated to the experimental and comparison groups by a toss of a coin. Four hundred thirty-two (50.8%) students were assigned to the comparison group while four hundred nineteen (49.2%), were assigned to the treatment group. Two schools from both regions were assigned to the experimental group. The selection of students from the two distant schools within/outside the regions and assigning them to treatment groups was to avoid spurious results. In a situation where a particular school had more than one class (“stream”), at the time of data collection, at least one hundred students were randomly picked from different classes in that specific school to respond to the attitudinal questionnaires. The main reason for selecting the 11th-grade students as research participants are based on curriculum materials in which LP is taught to the 11th-grade students (see NCDC, 2018 ). Indeed, at the time of data collection, students were preparing for UCE national examinations for the 2019/2020 academic year. The school heads revealed that the mathematics syllabus containing LP word problems (Supplementary Appendix 1 ) had been completed. The students were selected to provide their experiences and attitudes toward learning LP word problems. Of the 851 students who participated, 359 (42.2%) were males and 492 (57.8%) were females with a mean age of 18.32 (S.D. = 0.94) years. We predicted that the participants had adequate knowledge and understanding of solving LP word problems by graphical method. Identification numbers were allotted to participants before they anonymously and voluntarily completed adapted ATMI-SF questionnaire items.

Research instruments and procedure for administration

In addition to demographic questions, the ATMI-SF (Lin and Huang, 2014 ), a 14-item instrument questionnaire consisting of four subscales (enjoyment, motivation, value/usefulness, and self-confidence) was adapted to measure students’ attitude towards learning LP mathematics word problems. The ATMI-SF is a 5-point Likert-type scale with response options ranging from “Strongly Disagree (1)” to “Strongly Agree (5).” The ATMI-SF items were developed by Lim and Chapman ( 2013 ), which were also developed and validated from several mathematics attitudinal questionnaire items (Fennema and Sherman, 1976 ; Kasimu and Imoro, 2017 ; Mulhern and Rae, 1998 ; Primi et al., 2020 ; Tapia, 1996 ). The ATMI-SF was adapted because it directly correlates with the learning of LP, “the cousin of mathematics word problems.” English is the language of instruction in Ugandan secondary schools’ curricula, and translation of questionnaire items was not required. The content validity of the questionnaire was assessed by three experts (one senior teacher for mathematics, one senior lecturer for mathematics education, and one tutor at a teacher training institution). The experts were selected based on their vast experience in teaching mathematics at various academic levels. The experts further evaluated the appropriateness and relevance of the adapted questionnaire items. Based on their recommendations, suggestions, and comments, some questionnaire items were adjusted to suit students’ academic level and language to adequately measure students’ ATLPMWPs.

To adequately implement active learning heuristic problem-solving strategies, teachers from the treatment group were trained. First, students’ basic prior conceptual knowledge of equations and inequalities plus the basic algebraic principles and understanding were reviewed to link previous concepts to the learning of LP. Second, several learning materials were applied to help students adequately master the concepts. The materials included the use of graphs, grid boards, excel, and GeoGebra software. These strategies were further integrated with problem-solving strategies (Polya, 2004 ) by ensuring that students understand the LP word problem, devise a plan, adequately carry out the plan and finally look back to verify solution sketches and procedures. To ensure that students minimize errors and misconceptions, the learning of LP was further integrated with Newman Error Analysis (NEA) model prompts. The teachers emphasized question reading and decoding, comprehension, transformation, process skills, and encoding to cultivate students’ positive attitude towards LPMWPs.

The procedure and data analysis

The ATLPMWP questionnaires were completed by individual students at their respective schools in their natural classroom settings. The 11th-grade students completed this study in at most 20 min on average. The survey contained a ‘filter statement’, as a Social Desirability Response (SDR) to verify and discard respondents’ questionnaires, especially those who did not read (see item 15 in Supplementary Appendix 1 ) or finish answering questionnaire items (Bäckström and Björklund, 2013 ; Latkin et al., 2017 ). Written consent was received from all participants and participation in this study was completely voluntary and confidential. Participants who felt uncomfortable completing the questionnaire were not penalized. Data were collected with the help of mathematics heads of the department who were selected from sampled schools as experts. Participants were explained, the purpose of the study before administering and/or filling in questionnaire items. In the presence of the principal researcher, research assistants, and some selected school administrators, participants completed and returned all the questionnaires. In addition to the administration of questionnaire items, 12 heads of department and 24 students (a boy and a girl from each sampled school) were interviewed to correlate the data collected in trying to adequately assess the learning of LP word problems. Descriptive and inferential statistics were used to analyze the collected data about the background characteristics. Data were analyzed using the Statistical Package for Social Sciences (SPSS) version 26. In addition, and where necessary, excerpts were used to make a judgment about students’ ATMWPs, and how this affects the learning and achievement in mathematics and LP in particular.

Preliminary results and interpretation

Psychometric properties of the atlpmwp scale.

IBM SPSS (version 26) software package was used for analysis. Preliminary statistical analysis revealed no evidence of missing data due to a few cases, which were ignored because they did not exceed 5% of sample cases (Barbara and Tabachnick, 2001 ; Kline Rex, 1998 ; Lim and Chapman, 2013 ). However, out of 885 questionnaires distributed, 31 questionnaires were removed because the participants did not either conform to SDR (Bäckström and Björklund, 2013 ; Latkin et al., 2017 ) or had incomplete data. Univariate analysis was run to examine the degree of normality (Hair et al., 2010 ; Pallant, 2011 ). The indices for skewness and kurtosis were within the acceptable ranges (±2 and ±7 respectively) (Byrne, 2010 ; Curran et al., 1996 ; Hair et al., 2010 ). Thus, data were fairly normally distributed (Table 1 ). Exploratory factor analysis was run using initial pilot data collected from 215 students outside the study sample to check the correlation between the items. Most of the ATMI-SF scale inter-item means were below 3.0; suggesting that students generally had negative attitude ALPMWPs. However, browsing through the data, psychometric average scores for items still confirmed and indicated that most students (both male and female) irrespective of the school type and location had a negative attitude towards learning LP word problems (albeit their agreement and consideration that LP is useful).

Factor analysis was performed to confirm the factor structure. Principal component (with varimax) analysis to was used to show interrelationships (Tabachnick, 2001 ; Pallant, 2011 ; Pituch, 2016 ). Four constructs with eigenvalues greater than 1 accounted for 55.89% of the total variance. All items loaded significantly on four factors (enjoyment: 0.91, motivation: 0.89, value/usefulness: 0.94, and self-confidence: 0.95 with p  < 0.05, respectively). The values obtained were consistent with previous empirical findings (see Lin and Huang, 2014 , Awofala, 2014 ). The Kaiser-Meyer-Olkin measure of sampling adequacy test (KMO) and Bartlett’s test of sphericity were conducted. The value of KMO in our analysis was 0.71 > 0.60, and that of Bartlett’s Test was significant ( X 2 (760) = 13792.55, p  < 0.005) indicating a substantial correlation in the data and an acceptable fit (Nunnally and Berstain, 1994 , Pallant, 2011 ). Following the above recommendations, all items were found to be acceptable with adequate construct validity, internal consistency, and homogeneity. Overall, these items were deemed fit to measure students’ ATLPWPs in secondary schools.

Tables 1 and 2 show descriptive statistics (mean, standard deviation, skewness, and kurtosis). Important to note are students’ scores on ATMI-SF questionnaires during the pre-test and post-test. The results show no significant differences between the two groups in the pre-test and for the four scales (enjoyment, motivation, usefulness, and self-confidence). Indeed, both experimental and comparison groups were similar during the pre-test. There was however a slight change in students’ ATLPWPs due to the intervention administered to students from the experimental group (Table 3 ). The findings, however, show that students generally had a negative attitude towards learning LP word problems. These findings are consistent with other research studies (e.g., see Awofala, 2014 ). Thus, the learning of LP word problems and related mathematics concepts should be structured using multiple problem-solving techniques to boost students’ understanding and attitude.

From the correlation matrix in Table 4 above, it is evident that most of the inter-item correlations are low. This suggests that the data collected shows students’ negative attitude towards LP word problems. Students’ responses may have revealed intrinsic traits as far as the learning of LP is concerned. These findings are not in any way different from UNEB annual reports on previous students’ performance in the topic of LP. The additional qualitative data collected from senior teachers on why students elude questions on LP during internal and national examinations confirmed our investigations.

The results found no significant statistical difference between students’ ATLPMWPs, and their age (Table 5 ), gender (Table 6 ), school location (Table 7 ), school status (Table 8 ), and school ownership (Table 9 ).

Discussions, conclusions, and recommendations

This study sought to investigate the 11th-grade Ugandan students’ attitude towards LPMWPs. The psychometric properties of the adapted ATMI-SF instrument were found acceptable. We were fundamentally interested in students’ motivation, confidence, usefulness, and enjoyment in learning LP, and related mathematics word problems. These were the four main reliable latent dimensions identified through principal component factor analysis to explain the underlying students’ attitude towards LPMWPs. At first, students’ attitude towards LPMWPs for both groups (comparison and experimental groups) were not significantly different irrespective of the student’s age, gender, school status, or school location. These findings show that students generally had negative attitude towards LPMWPs. Yet, Arslan et al. ( 2014 ) show that there exists a positive significant relationship between attitude and problem-solving.

Although students’ ratings were below the neutral attitude (please see Table 2 ), they indicated the usefulness of LP in daily life. The experimental group showed a slightly favorable attitude towards LP word problems (Table 3 ) after an intervention because the active learning heuristic problem-solving instruction was applied compared to students in the comparison group who learned LP conventionally. Face-to-face interviews with some students and teachers have not been provided in this quantitative study. However, a section of students whom we interacted with revealed that LP concepts are more stimulating, require prior conceptual knowledge and understanding of equations and inequalities and that these questions are not interesting to learn in comparison to other topics in mathematics. Our findings concord with Chen et al. ( 2018 ) who postulated that positive attitude influences early career performance.

The explanation provided indicated that some teachers either teach this topic hurriedly towards national examinations or some of them avoid teaching it completely. This means teachers have not adequately applied instructional techniques and suitable learning materials to fully explain the concepts of LP to the students. However, it was observed that teachers encouraged students to constantly practice model formation from word problem statements to demystify the negative belief that LP word problems are hard for students to conceptualize. Negative beliefs limit students’ understanding, thereby making them fear the topic and consequently develop a negative attitude towards learning LP. However, students’ attitudes towards LPMWPs from the experimental group slightly improved compared to their counterparts from the comparison group who almost had a similar attitude towards LP before and after an intervention.

Participants from the experimental group and the comparison groups acknowledged the fact that LP is a challenging topic, although they highly recognized its significance in constructing models, and in developing models for optimization in real-life scenarios. The importance of LP rests in its application and thus teachers were tasked to help learners to develop a positive attitude towards, and their conceptual understanding so that they can reason insightfully, think logically, critically and, coherently. The teachers’ competence in applying instructional strategies helped learners from the experimental group to gain deeper and broader insight, conceptual and procedural understanding, reasoning, and positive attitude towards LPMWPs. As Mazana et al. ( 2018 ) noted, aspects of attitude (motivation, confidence, value, increased anxiety and enjoyment) enhance students’ learning and hence performance. The control group, however, in their conventional instruction still perceived LP as one of the hardest topics. A negative attitude was observed in this particular group of students as indicated in the results of most learners’ ATMI-SF questionnaires.

Thus, teachers recognized that hard work and application of prior conceptual knowledge and understanding may favorably help students to develop a positive attitude and perform better. Generally, students seemed not to have adequately developed the knowledge of logical thinking and reasoning of basic and prior LP concepts to aid in learning LP. They did not view the learning of LP from a broader perspective beyond passing national examinations at UCE. The results of this study are likely to inform educational stakeholders in assessing students’ ATLPWPs and provide remediation and interventional strategies aimed at creating a conceptual change in students’ attitudes towards learning LP and related topics. This will further act as a lens in examining the relationships between students’ achievement and their attitude toward learning specific mathematics concepts, as indicators of students’ confidence, motivation, usefulness, and enjoyment in learning LP word problems and mathematics generally.

The study findings also point to important issues and may provide insight to the educational stakeholders in cultivating an early positive attitude in mathematics, aimed at investigating students’ challenges in specific topics from primary to secondary school mathematics. This may be a potential strategy for applying different active learning heuristic problem-solving approaches and methods to significantly improve students’ attitude and performance. The active learning heuristic problem-solving approach is likely to support collaboration and discussions between teachers and amongst students themselves during the learning process. The findings show that most students from the experimental group worked collaboratively in their small groups and individually hence the conceptual and attitudinal change. The students helped and guided each other during peer teaching, hence boosting their attitude. As noted by Asempapa ( 2022 ), suitable teachers’ instructional strategies that emphasize individual students’ academic differences may change students’ attitude towards LPMWPs, thereby providing both academic and social support.

Consequently, the low performers gained conceptual understanding, morale, and problem-solving strategies, hence positive attitude towards learning. This further enhanced students’ learning and attitude towards mathematics and LP in particular. Besides, the active learning heuristic problem-solving approach applied to the experimental group boosted students’ confidence in answering both routine and non-routine LP problems. Students’ fear of comprehending LP word problems and attempting to answer LP questions decreased. Moreover, the heuristic problem-solving approach boosted students’ attitude towards LPMWPs. Students were actively involved in problem-solving. This gradually built their motivation, competence, and confidence in learning LP and related concepts. This generally and significantly fostered students’ positive attitude towards LPMWPs.

Limitations of the study and future research directions

The purpose of this research was to explore students’ attitude towards LPMWPs. The findings provide preliminary insights into the fundamental concepts of the introduction of LP for supporting the learning of advanced mathematics. Our key observation is that the present study involved schools from two regions (Eastern Uganda and Central Uganda), and the study was specifically conducted in two districts (Mukono and Mbale). Yet, there are at least 120 districts in Uganda. Hence, the sample may not adequately represent all the 11th grade Ugandan students. Future studies should consider the inclusion of sampled students from all districts. While the quantitative study is important and valuable for yielding robust and comprehensive data in social sciences research, its limitations must be acknowledged. Triangulation of data collection and analysis methods might have yielded additional results. We, therefore, recommend future studies in different or similar settings and contexts, and in different mathematics topics (content) with a diversity of methods to compare and contrast our findings and to gain deeper and broader insights into students’ attitude towards LPMWPs.

Students’ attitudes point to issues related to demographic variables and latent constructs for learning mathematics. Specifically, to gain more insight, this research recommends that future researchers should use qualitative methods such as interviews and observation to provide more evidence on students’ experiences in learning LP. The teachers ‘attitude towards LPMWPs is also a potential area for further investigation aimed at improving the instructional strategies, pedagogical content knowledge, and mathematical knowledge for teaching. To achieve this, the teachers’ professional development programs should be enacted to emphasize content knowledge and pedagogical content knowledge of learning LPMWPs. Teachers coming together to share learning experiences and strategies, may improve students’ attitude towards learning LP, and other related but challenging topics. Indeed, teachers need continuous routine professional development support to successfully implement the learning activities. Despite some limitations, this study supplements other empirical shreds of evidence in support of enhancing students’ attitude towards learning mathematics word problems, and LP in particular.

Data availability

All the data analyzed and reported in this study is available and may be accessed on request.

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Acknowledgements

This research is part of the Ph.D. Thesis that investigated the effect of active learning heuristic problem-solving approach on students’ achievement and attitude towards mathematics word problems (LP) from sampled secondary schools in Uganda. The research was funded by the African Centre of Excellence for Innovative Teaching and Learning Mathematics and Science (ACEITLMS), [ACEII (P151847)]. We appreciate the useful information provided by the students and teachers in the study sample, which helped us write this research article. The views expressed herein are those of the authors and not necessarily those of ACEITLMS. This is because the ACEITLMS was not involved in identifying the suitable study design, methods of data collection and analysis, publication decision, or manuscript preparation.

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Wakhata, R., Mutarutinya, V. & Balimuttajjo, S. Secondary school students’ attitude towards mathematics word problems. Humanit Soc Sci Commun 9 , 444 (2022). https://doi.org/10.1057/s41599-022-01449-1

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© Bachudo Science Co. Ltd. This work is licensed under the creative commons Attribution 4.0 International license.

Ekok Edim Odor

Department of Science Education, Ebonyi State University, Abakaliki, Nigeria

Valentine Joseph Owan

Department of Educational Foundations, University of Calabar, Nigeria

Victor Ubugha Agama

Ultimate Research Network (URN), Calabar, Cross River State, Nigeria

Main Article Content

Process errors committed by senior secondary school students in solving problems in geometry in cross river state, nigeria.

This study aimed to identify the process errors made by senior secondary school students in geometry in Cross River State, Nigeria. A sample of 300 students, selected using a proportionate sampling technique, participated in the study. The Geometry Diagnostic Test (GDT) was used to collect data upon validation by mathematics education and psychometrics experts. The instrument was tested for reliability using the Kendal coefficient of concordance (W), with a coefficient of 0.89 providing sufficient evidence of good inter-rater reliability. The data collected were analysed using frequency counts, percentages, and the Chi-square test. The results showed that most students make errors in transformation, process skills, and encoding when solving geometry problems in mathematics. In contrast, the number of students who made reading and comprehension errors was relatively small. The study also revealed that the process errors made by students did not significantly depend on their gender and school location. The findings of this study have implications for teaching and learning mathematics, particularly geometry, in secondary schools. It underscores the need for teachers to focus on the process of arriving at the correct answer rather than just obtaining the right answer, which is a critical component of problem-solving in mathematics. The results also provide a basis for curriculum developers and designers to design appropriate instructional strategies and learning materials to help students overcome the identified process errors.

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DU Professor Helps Solve Famous 70-Year-Old Math Problem

Jordyn reiland.

[email protected]

Assistant Professor Mandi Schaeffer Fry is the first faculty member to be published in the Annals of Mathematics since the 1880s.

University of Kaiserslautern Professor Gunter Malle, University of Denver Assistant Professor Mandi Schaeffer Fry and University of Valencia Professor Gabriel Navarro pose for a photo after announcing their theorem in Oberwolfach, Germany.

Whether it be flying trapeze, participating in competitive weightlifting or solving math problems that have confounded academics for decades, Mandi Schaeffer Fry enjoys chasing the next adventure.

Schaeffer Fry, who joined the University of Denver’s Department of Mathematics in the fall of 2023, will be the first faculty member since the 1880s to be published in the Annals of Mathematics , widely seen as the industry’s most prestigious journal.

In 2022, Schaeffer Fry helped complete a problem that dates to 1955—mathematician Richard Brauer’s Height Zero Conjecture.

“Maybe one of the most challenging parts, other than the math itself, was the knowledge of the weight that this would have on the field,” Schaeffer Fry says. “If you’re going to make an announcement like this, you have to be darn sure that it’s absolutely correct.”

Over the years, number crunchers have worked on the problem at universities across the globe, and some found partial solutions; however, the problem was not completed until now.

“Mandi’s accomplishment is exciting. Solving Brauer's Height Zero Conjecture is remarkable,” Mathematics Department Chair Alvaro Arias says. 

The work is also a testament to DU’s achievement as a Research 1 (R1) institution.

Fry and her collaborators—University of Kaiserslautern Professor Gunter Malle, University of Valencia Professor Gabriel Navarro and Rutgers University Professor Pham Huu Tiep—worked around the clock over the course of three months in eight-hour shifts during the summer of 2022 to find a solution.

In April, that work was accepted for publication in the Annals of Mathematics.

'Brauer's Height Zero Conjecture (BHZ) was the first conjecture leading to the part of my field studying 'local-global' problems in the representation theory of finite groups, which seek to relate properties of groups with properties of certain nice smaller subgroups, letting us 'zoom in' on the group using just a specific prime number and simplify things," Schaeffer Fry says. 

"The BHZ gives us a way to tell from the character table of a group (a table of data that encodes lots, but not all, information about the group) whether or not certain of these subgroups, called defect groups, have the commutativity property," she adds.

This paper was especially meaningful to Schaeffer Fry as she had always wanted to work with Malle, Tiep and Navarro as they have been her primary mentors. Tiep was her PhD advisor and this was the first time they had worked together since then.

Fry believes she has solidified her place in the field and knows she’ll likely never top this accomplishment, but she’s always looking for the next adventure—whether that’s in or out of the classroom.

Flying high and pumping iron

When Schaeffer Fry isn’t on DU’s campus working with students or conducting research, you can find her flying trapeze and competitive weightlifting.

Schaeffer Fry became involved in competitive weightlifting during graduate school, and, in the last year of her PhD at the University of Arizona, she defended her dissertation one day and got on a plane and competed at the national level for “university-aged” athletes—which included Olympians.

While she now lifts weights more casually, Schaeffer Fry competed last September in an over-35 competition and qualified for the USA Weightlifting Masters National Championships.

It was a “field trip” during a conference in Berkeley, California, in 2018 that led Fry to become enamored with flying trapeze.

In fact, she enjoyed it so much she signed up to be a member of Imperial Flyers, an amateur flying trapeze cooperative located in Westminster. Once she found out about the sport, her previous experience as a gymnast made it a natural fit.

Not only is she working on her own intermediate tricks, she’s also a “teaching assistant” at Fly Mile High, the state’s only flying trapeze and aerial fitness school.

“It’s exhilarating; it’s gotten me a bit over my fear of heights,” she says.

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