students' attitude towards problem solving

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Brief research report article, the influence of attitudes and beliefs on the problem-solving performance.

• 1 Department of Mathematics and Computer Science, University of Education of Ludwigsburg, Ludwigsburg, Germany
• 2 Hamburg Center for University Teaching and Learning, University of Hamburg, Hamburg, Germany

The problem-solving performance of primary school students depend on their attitudes and beliefs. As it is not easy to change attitudes, we aimed to change the relationship between problem-solving performance and attitudes with a training program. The training was based on the assumption that self-generated external representations support the problem-solving process. Furthermore, we assumed that students who are encouraged to generate representations will be successful, especially when they analyze and reflect on their products. A paper-pencil test of attitudes and beliefs was used to measure the constructs of willingness, perseverance, and self-confidence. We predicted that participation in the training program would attenuate the relationship between attitudes and problem-solving performance and that non-participation would not affect the relationship. The results indicate that students’ attitudes had a positive effect on their problem-solving performance only for students who did not participate in the training.

Introduction

Mathematical problem solving is considered to be one of the most difficult tasks primary students have to deal with ( Verschaffel et al., 1999 ) since it requires them to apply multiple skills ( De Corte et al., 2000 ). It is decisive in this respect that “difficulty should be an intellectual impasse rather than a computational one” ( Schoenfeld, 1985 , p. 74). When solving problems, it is not enough to retrieve procedural knowledge and reproduce a known solution approach. Rather, problem-solving tasks require students to come up with new ways of thinking ( Bransford and Stein, 1993 ). Problem-solvers must activate their existing knowledge network and adapt it to the respective problem situation ( van Dijk and Kintsch, 1983 ). They have to succeed in generating an adequate representation of the problem situation (e.g., Mayer and Hegarty, 1996 ). This requires conceptual knowledge, which novice problem-solvers have to acquire ( Bransford et al., 2000 ). As problem solving is the foundation for learning mathematics, an important goal of primary school mathematics teaching is to strengthen students’ problem-solving performance. One central problem is that problem-solving performance is highly influenced by students’ attitudes towards problem solving ( Reiss et al., 2002 ; Schoenfeld, 1985 ; Verschaffel et al., 2000 ).

Attitudes and beliefs are considered quite stable once they are developed ( Hannula, 2002 ; Goldin, 2003 ). However, students who are novices in a particular content area are still in the process of development, as are their attitudes and beliefs. It can therefore be assumed that their attitudes change over time ( Hannula, 2002 ). However, such a change does not take place quickly ( Higgins, 1997 ; Mason and Scrivani, 2004 ). Nevertheless, in a shorter period of time, it might be possible to reduce the influence of attitudes on problem-solving performance ( Hannula et al., 2019 ). In this paper, we present a training program for primary school students, which aims to do exactly that.

Problem-Solving Performance

Successful problem solving can be observed on two levels: problem-solving success and problem-solving skills. Many studies measure the problem-solving performance of students on the basis of correctly or incorrectly solved problem-solving tasks, that is, the product (e.g., Boonen et al., 2013 ; de Corte et al., 1992 ; Hegarty et al., 1992 ; Verschaffel et al., 1999 ). In this case, only problem-solving success, that is, specifically whether the numerically obtained result is correct or incorrect, is evaluated. This is a strict assessment measure, since the problem-solving process is not taken into account. As a result, the problem-solving performance is only considered from a single, product-oriented perspective. For instance students’ performance is assessed as unsuccessful when they apply an essentially correct procedure or strategy but achieve the wrong result, or it is considered successful when students achieve the right result even though they have misunderstood the problem ( Lester and Kroll, 1990 ). An advantage of this operationalization, however, is that student performance tends to be underestimated rather than overestimated.

A more differentiated view of successful problem solving includes the solver’s problem-solving process ( Lester and Kroll, 1990 ; cf. Adibnia and Putt, 1998 ). In this way, sub-skills such as understanding the problem, adequately representing the situation, applying strategies, or achieving partial solutions are taken into account. These are then incorporated into the evaluation of performance and, thus, of problem-solving skills ( Charles et al., 1987 ; cf. Sturm, 2019 ). The advantage of this operationalization option is that it also takes into account smaller advances by the solver, although they may not yet lead to the correct result. It is therefore less likely to underestimate students’ performance. In order to assess and evaluate the problem-solving skills of students in the best way and, thus, avoid over- and under-estimating their skills, direct observation and questioning should be implemented (e.g., Lester and Kroll, 1990 ). An analysis of written work should not be the only means of assessment ( Lester and Kroll, 1990 ).

Attitudes and Beliefs

Attitudes are dispositions to like or dislike objects, persons, institutions, or events ( Ajzen, 2005 ). They influence behavior (Ajzen, 1991). Therefore, it is not surprising that attitudes–which are sometimes also synonymously referred to as beliefs–are a central construct in psychology ( Ajzen, 2005 ).

Individual attitudes to word problems influence, albeit rather unconsciously, approaches to such problems and willingness to learn mathematics and solve problems ( Grigutsch et al., 1998 ; Awofala, 2014 ). Research on attitudes of primary students to word problems is scarce. Most research focuses on students with well-established attitudes. However, the importance of the attitudes of younger children is undisputed ( Di Martino, 2019 ). Di Martino (2019) conducted a study on kindergarten children as well as on first-, third-, and fifth-graders and found that, with increasing age, students’ perceived competence in problem solving decreases, and negative emotions towards mathematical problems increase. Whether a solver can overcome problem barriers when dealing with word problems depends not only on his or her previous knowledge, abilities, and skills, but also on his or her attitudes and beliefs ( Schoenfeld, 1985 ; Verschaffel et al., 2000 ; Reiss et al., 2002 ). It has been shown many times that attitudes towards problem solving are influencing factors on performance and learning success which should not be underestimated ( Charles et al., 1987 ; Lester et al., 1989 ; Lester & Kroll, 1990 ; De Corte et al., 2002 ; Goldin et al., 2009 ; Awofala, 2014 ). Learners associate a specific feeling with an object, in this case with a word problem, triggering a specific emotional state ( Grigutsch et al., 1998 ). The feelings and states generated are subjective and can therefore vary between individuals ( Goldin et al., 2009 ).

Attitudes towards problem solving can be divided into willingness, perseverance, and self-confidence ( Charles et al., 1987 ; Lester et al., 1989 ). This distinction comes from the Mathematical Problem-Solving Project, in which Webb, Moses, and Kerr (1977) found that willingness to solve problems, perseverance in attempting to find a solution, and self-confidence in the ability to solve problems are the most important influences on problem-solving performance. When students are willing to work on a variety of mathematics tasks and persevere with tasks until they find a solution, they are more task oriented and easier to motivate ( Reyes, 1984 ). Perseverance is defined as the willing pursuit of a goal-oriented behavior even if this involves overcoming obstacles, difficulties, and disappointments ( Peterson and Seligman, 2004 ). Confidence is an individual’s belief in his or her ability to succeed in solving even challenging problems as well as an individual’s belief in his or her own competence with respect to his or her peers ( Lester et al., 1989 ). Students’ lack of confidence in themselves as problem-solvers or their beliefs about mathematics can considerably undermine their ability to solve or even approach problems in a productive way ( Shaughnessy, 1985 ). The division of attitudes into these three sub-categories can also be found in current studies ( Zakaria and Yusoff, 2009 ; Zakaria and Ngah, 2011 ).

Reducing the Influence of Attitudes and Beliefs

As it seems impossible to change attitudes within a short time frame, we developed a training program to reduce the influence of attitudes on problem solving, on the one hand, and to foster the problem-solving performance of primary-school students, on the other hand.

The training program was an integral part of regular math classes and focused on teaching students to generate and use external representations ( Sturm, 2019 ; Sturm et al., 2016 ; Sturm and Rasch, 2015 ; see also Supplementary Appendix A ). Such a program that concentrates on the strengths and weaknesses of novices and on their individually generated external representations can be a benefit for primary-school students in two ways. The class discusses how the structure described in the problem can be adequately represented so that the solution can be found, working out multiple approaches based on different student representations. The students are thus exposed to ideas about how a problem can be solved in different ways. Such a training program fulfils, albeit rather implicitly, another essential component. By respectfully considering their individual thoughts and difficulties, the students are made aware of their strengths and their creativity and of the fact that there is not a single correct approach or solution that everyone has to find ( Lester and Cai, 2016 ; Di Martino, 2019 ). This can counteract fears of failure and lack of self-confidence, and generate positive attitudes ( Lester and Cai, 2016 ; Di Martino, 2019 ). The teacher pays attention to the solution process rather than to the numerical result in order to reduce the influence of attitudes on performance ( Di Martino, 2019 ). In the same way, experiencing success and perceiving increasing flexibility and agility can reduce the influence of attitudes. As a result, we expected attitudes and beliefs to have a smaller effect on problem-solving performance.

Based on previous research, our goal was to reduce the influence of attitudes on the problem-solving performance of students (see Figure 1 ). To this end, the hypothesis was derived that participation in the training program would minimize the effect of attitudes and beliefs on problem-solving success, so that students would succeed at the end of the training despite initial negative attitudes and beliefs.

FIGURE 1 . The moderation model with the single moderator variable training influencing the effect of attitudes and beliefs on problem-solving success.

Participants

In total 335 students from 20 Grade 3 classes from eight different primary schools in the German state of Rhineland-Palatinate took part in the intervention study (172 boys and 163 girls). Nineteen students dropped out because of illness during the intervention. The age of the participants ranged between seven and ten years ( M = 8.10, SD = 0.47).

This investigation was part of a large interdisciplinary project 1 . A central focus of the project was to investigate whether representation training has a demonstrable effect on the performance of third-graders (cf. Sturm, 2019 ). For this reason, we implemented a pretest-posttest control group design. The intervention took place between Measurement Points 1 and 2. We measured the problem-solving performance of the students with a word-problem-solving test (WPST) at Measurement Points 1 and 2. All other variables were measured at Measurement Point 1 only (factors to establish comparable experimental conditions: intelligence, text comprehension, and mathematical abilities; co-variates for the mediation model: metacognitive skills, mathematical abilities).

In the intervention, third-grade students worked on challenging word problems for one regular mathematics lesson a week. The intervention was based on six task types with different structures ( Sturm and Rasch, 2015 ): 1) comparison tasks, 2) motion tasks, 3) tasks involving comparisons and balancing items or money, 4) tasks involving combinatorics, 5) tasks in which structure reflects the proportion of spaces and limitations, and 6) tasks with complex information. Two word problems were included for each task type and were presented to all classes in the same random sequence. Each task had to be completed in a maximum of one lesson.

The training was implemented for half of the classes and was conducted by the first author; the other half worked on the tasks with their regular mathematics teacher. They were not informed on the purpose of the intervention and not given any instructions on how to process the tasks. In the lessons for students doing the training, the students were explicitly cognitively stimulated to generate external representations and to use them to develop solutions. They were repeatedly encouraged to persevere and not to give up. The diverse external representations generated by the students were analyzed, discussed, and compared by the class during the training. They jointly identified the characteristics of representations that enabled them to specifically solve the tasks and identified different approaches (for more details about the study, see Sturm and Rasch, 2015 ). With the goal of reducing the influence of attitudes on performance, the class worked directly on the students’ own representations instead of on prefabricated representations. The aim was that students realized that it was worthwhile investing effort into creating representations and that they were able to solve problem tasks independently.

Thus, the study was composed of two experimental conditions: training program ( n = 176; 47% boys) (hereinafter abbreviated to T+) and no training program ( n = 159; 58% boys) (hereinafter abbreviated to T-). In order to control potential interindividual differences, the 20 classes were assigned to the experimental conditions by applying parallelization at class level ( Breaugh and Arnold, 2007 ; Myers and Hansen, 2012 ). The classes were grouped into homogeneous blocks using the R package blockTools Version 0.6-3 and then randomly assigned to the experimental conditions ( Greevy et al., 2004 ; Moore, 2012 ; see also Supplementary Appendix B for more information).

Word-Problem-Solving Test

Before the intervention and immediately after it, the students worked on a WPST, which we created. It consisted in each case of three challenging word problems with an open answer format. Each of the three tasks represented a different type of problem. The word problems from the WPST at Measurement Point 1 and the word problems from the WPST at Measurement Point 2 had the same structure. We implemented two parallel versions; only the context was changed by exchanging single words (see Supplementary Appendix C ). An example of an item from the test is a task with complex information ( Sturm, 2018 ): Classes 3a and 3b go to the computer room. Some students have to work at a computer in pairs. In total there are 25 computers, but 40 students. How many students work alone at a computer? How many students work at a computer in pairs? Direct observation and questioning could not be conducted due to the large number of participants in the project; only the students’ written work was available for analysis. The problem-solving process of the students could therefore only be assessed indirectly. For this reason, the performance of students in the two tests was evaluated based on problem-solving success, ruling out overestimation of performance.

Problem-Solving Success

The success of the solution was measured dichotomously in two forms: 1) correct solution and (0) incorrect solution. Only the correctness of the result achieved was evaluated. This dependent variable acted as a strict criterion that could be quantified with high observer agreement (κ = 0.97; κ min = 0.93, κ max = 1.00). A confirmatory factor analysis using the R package lavaan version 0.6-7 confirmed that the WPST measured the one-dimensional construct problem-solving success. The one-dimensional model exhibited a good model fit ( Nussbeck et al., 2006 ; Hair et al., 2009 ): χ 2 (27) = 36.613, p = 0.103; χ 2 /df = 1.356, CFI = 0.985, TLI = 0.981, SRMR = 0.032, RMSEA = 0.033 ( p = 0.854). The reliability coefficients at Measurement Point 1 were classified as low (Cronbach’s α = 0.39) because the test consisted of only three items ( Eid et al., 2011 ) and a homogeneous sample was required at this measurement point ( Lienert and Raatz, 1998 ). The Cronbach’s alpha for the second measurement point (α = 0.60) was considered to be sufficient ( Hair et al., 2009 ). The test score represented the mean value of all three task scores.

Attitudes and Beliefs About Problem Solving

The attitudes and beliefs of the learners were recorded with the Attitudes Inventory Items ( Webb et al., 1977 ; Charles et al., 1987 ). The original questionnaire comprises 20 items, which are measured dichotomously (“I agree” and “I disagree”). The Attitudes Inventory measures the three categories of attitudes and beliefs related to problem solving: a) willingness (six items), b) perseverance (six items), and c) self-confidence (eight items). An example of an item for willingness is: “I will try to solve almost any problem.” An example of an item for perseverance is: “When I do not get the right answer right away, I give up.” An example of an item for self-confidence is: “I am sure I can solve most problems.”

Because the reported reliabilities were only satisfactory to some extent (α = 0.79, mean = 0.64) ( Webb et al., 1977 ), the Attitudes Inventory was initially tested on a smaller sample ( n = 74; M = 8.6 years old; 59% girls). A satisfactory Cronbach’s α = 0.86 was achieved (mean α = 0.73). The number of items was reduced to 13 (four items for willingness, four items for perseverance, five items for self-confidence), which had only a minor influence on reliability (α = 0.83). For economic reasons, the shortened questionnaire was used in the study. The three-factor structure of the questionnaire was confirmed with a confirmatory factor analysis using the R package lavaan version 0.6–7. As the fit indices show, the three-factor model had a good model fit: χ 2 (62) = 134.856, p < 0.001; χ 2 / df = 2.175, CFI = 0.948, TLI = 0.935, RMSEA = 0.062 ( p = 0.086) ( Hair et al., 2009 ; Brown, 2015 ). The three-factor model had a better fit than the single-factor model ( p = 0.0014): χ 2 (65) = 152.121, p < 0.001; χ 2 / df = 2.340, CFI = 0.938, TLI = 0.926, SRMR = 0.061, RMSEA = 0.066 ( p = 0.028). The students were grouped into three groups ( M –1 SD ; M ; M +1 SD ). The responses were coded in such a way that high scores ( M +1 SD ) indicated positive attitudes and beliefs, and low scores ( M –1 SD ) indicated negative attitudes and beliefs.

Additional Influencing Factors

In order to ensure the internal validity of the investigation, we collected student-related factors that influence the solution of word problems from a theoretical and empirical point of view. It has been shown that the mathematical abilities and metacognitive skills of students significantly influence their performance ( Sturm et al., 2015 ).

Mathematical Abilities

The basic mathematical abilities were determined using a standardized German-language test as a group test (Heidelberger Rechentest HRT 1–4, Haffner et al., 2005 ). The test consists of eleven subtests, from which three scale values were determined: calculation operations, numerical-logical and spatial-visual skills as well as the overall performance for all eleven subtests. The reliability was only satisfactory (Cronbach’s α = 0.74). Total performance was included in the study.

Metacognitive Skills

The metacognitive skills of the students were measured using a paper-pencil version of EPA2000, a test to measure metacognitive skills before and/or after the solving of tasks ( Clercq et al., 2000 ). The prediction skills and evaluation skills of the students were collected for all three word problems of the WPST using a 4-point rating scale: 1) “absolutely sure, it’s wrong,” 2) “sure, it’s wrong,” 3) “sure, it’s right,” and 4) “absolutely sure, it’s right” ( Clercq et al., 2000 ). If the students’ assessments of “absolutely sure” matched their solution, they were awarded 2 points. If they agreed with “sure,” they received 1 point. No match was scored with 0 points ( Desoete et al., 2003 ). The reliabilities were only satisfactory (Cronbach’s α total =0.74, α prediction =0.56, α evaluation = 0.73). A confirmatory factor analysis revealed that prediction skills and evaluation skills represent a single factor (χ 2 (9) = 16.652, p < 0.001; χ 2 / df = 1.850, CFI = 0.952, TLI = 0.919, RMSEA = 0.053 ( p = 0.396)). The aggregated factor was used as a control variable in the moderator analysis.

In addition to the variables considered in this paper, text comprehension and intelligence were also surveyed in the project. However, they are not the focus of this paper; additional information can be found in Sturm et al. (2015) .

Descriptive Statistics and Correlations Between the Measures

The descriptive statistics and correlations of all scales are presented in Table 1 (see Supplementary Appendix D for a separate overview for each of the experimental conditions). The signs for all correlations were as expected. The variable training program is not listed because it is the dichotomous moderator variable (T+ and T−).

TABLE 1 . Descriptive statistics and correlations of all variables for both experimental conditions.

Moderated Regression Analyses

The hypothesis was tested with a moderated regression analysis using product terms from mean-centered predictor variables ( Hayes, 2018 ). This model imposed the constraint that any effect of attitudes and beliefs was independent of all other variables in the model. This was achieved by controlling for mathematical abilities, metacognitive skills, and problem-solving performance at Measurement Point 1. The estimated main effects and interaction terms are presented in Table 2 .

TABLE 2 . Results from the regression analysis examining the moderation of the effect of attitudes and beliefs on problem-solving success (t 2 ) by participation in the training program, controlling for mathematical abilities, metacognitive skills, and problem-solving success from the pretest.

When testing the hypothesis, we found a significant main effect of attitudes and beliefs, a significant main effect of the training program, and a significant moderator effect of the training on attitudes and beliefs as a predictor of problem-solving success. The main effect of the training program indicated that students who participated in the training performed better in the second WPST. The main effect of attitudes and beliefs showed that students with more positive attitudes and beliefs were more successful than students with negative attitudes and beliefs.

To further explore the interaction between attitudes and beliefs and the training program, we analyzed simple slopes at values of 1 SD above and 1SD below the means of attitudes and beliefs ( Hayes, 2018 ). As can be seen from the conditional expectations in Figure 2 , attitudes and beliefs did not affect the problem-solving success of students who participated in the training program. Attitudes and beliefs only had a positive effect on the problem-solving success of students who did not participate in the training.

FIGURE 2 . Moderator effect of the training program on problem-solving success at Measurement Point 2.

Our results confirm previous findings that the attitudes and beliefs of students correlate with their problem-solving performance. They indicate that this correlation can be moderated by student participation in a training program. Negative attitudes and beliefs did not affect the performance of students who participated in a problem-solving training program over several weeks. Whether the training program also causes a change in the attitudes and beliefs of the students over time has to be investigated in a follow-up study, which is planned with a longer intervention period with at least two measurements of attitudes and beliefs. A longer intervention period would have the advantage that attitudes develop depending on the individual experiences of a person ( Hannula, 2002 ; Lim and Chapman, 2015 ), for instance, when new experience is gathered or new knowledge is acquired (e.g., Ajzen, 2005 ).

Some limitations need to be considered when interpreting the results of the study. For example, the mitigating processes need to be investigated further. It is also unclear as to which components of the training are ultimately responsible for counteracting the effect of attitudes and beliefs. Although the study did not provide results in this regard, we assume that the following factors might have an effect: generating external representations, reflecting on the representations together as a group, and fostering an appreciative and constructive approach to mistakes. Further studies are needed to show whether and to what extent these factors actually attenuate the effect of attitudes and beliefs.

Furthermore, the measurement instruments for the control variables mathematical abilities and metacognitive skills were rather limited. If researchers are interested in understanding further effects of metacognitive skills, more aspects should be included. Furthermore, according to Lester et al. (1987), investigating attitudes and beliefs using a questionnaire is associated with disadvantages. How accurately students answer the questions depends on how objectively and accurately they can reflect on and assess their own attitudes. Misinterpretations and errors cannot be ruled out. The most serious disadvantage, however, is that data collection using an inventory can easily be assumed to have unjustified validity and reliability. For a deeper insight into the attitudes and beliefs of primary school students, qualitative interviews have to be implemented.

However, for the purpose of this study, it seems sufficient to consider the two control variables mathematical abilities and metacognitive abilities. We were able to ensure that the correlation between attitudes and beliefs and the mathematical performance of students was not influenced by these factors.

Regardless of the limitations, our study has some practical implications. Participation in the training program, independently of the mathematical abilities and text comprehension of students, reduced the influence of attitudes and beliefs on their performance. Thus, for teaching practice, it can be concluded that it is important not only to implement regular problem-solving activities in mathematics lessons, but also to encourage students to externalize and find their own solutions. The aim is to establish a teaching culture that promotes a variety of approaches and procedures, allows mistakes to be made, and makes mistakes a subject for learning. Reflecting on different possible solutions and also on mistakes helps students to progress. Thus, students develop a repertoire of external representations from which they can profit in the long term when solving problems.

Data Availability Statement

The original contributions presented in the study are included in the article/ Supplementary Material , further inquiries can be directed to the corresponding author.

Ethics Statement

The studies involving human participants were reviewed and approved by the Ethics Committee of the Department of Psychology, University of Koblenz and Landau, Germany. Written informed consent to participate in this study was provided by the participants' legal guardian. This study was also carried out in accordance with the guidelines for scientific studies in schools in the German state Rhineland-Palatinate (Wissenschaftliche Untersuchungen an Schulen in Rheinland-Pfalz), Aufsichts- und Dienstleistungsdirektion Trier. The protocol was approved by the Aufsichts- und Dienstleistungsdirektion Trier.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The project was funded by grants from the Deutsche Forschungsgemeinschaft (DFG, grant number GK1561/1).

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.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.525923/full#supplementary-material

1 This project was part of the first author’s PhD thesis

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Keywords: attitudes and beliefs, word problem, training program design, problem-solving, problem-solving success, primary school, moderation effect analysis

Citation: Sturm N and Bohndick C (2021) The Influence of Attitudes and Beliefs on the Problem-Solving Performance. Front. Educ. 6:525923. doi: 10.3389/feduc.2021.525923

Received: 21 May 2020; Accepted: 18 January 2021; Published: 17 February 2021.

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

*Correspondence: Nina Sturm, [email protected]

This article is part of the Research Topic

Psychology and Mathematics Education

Broadening the Scope of Research on Mathematical Problem Solving pp 401–434 Cite as

Students’ Attitudes in a Mathematical Problem-Solving Competition

• Nélia Amado 6 &
• Susana Carreira 6  
• First Online: 01 December 2018

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Part of the book series: Research in Mathematics Education ((RME))

The mathematical competitions SUB12 and SUB14 are web-based inclusive competitions where multiple affective dimensions come to the fore in relation to mathematics and problem solving. The moderate mathematical challenges proposed to participants, independent of the official curriculum and intended to be mathematically rich problems, allow young students to find their own strategies and approaches and incite them to come up with ways of expressing their thinking. As part of our research focusing on the competitions SUB12 and SUB14, one goal is to obtain an outline of students’ attitudes in the context of their participation. For this, we adopted a three-dimensional model that addresses students’ views, students’ emotional dispositions, and students’ perception of their competence, to study the participants’ attitudes towards mathematical problem solving and mathematics. In this chapter, we present and analyse data from a questionnaire, which are complemented by the voice of the participants themselves, collected through interviews and from a pool of students’ messages arising from their online participation. Our results indicate that the three affective dimensions are closely interconnected. The global positive attitude towards the participation in the competition reveals emotional dispositions such as fun and enjoyment in solving problems, beliefs such as the view that they become more confident and learn more, and perceived self-competence, when they realize that they are capable of success with different levels of performance and knowledge. The mathematical competitions SUB12 and SUB14 provide fulfilling experiences to the young participants (in each of the three dimensions of the model adopted), which are linked to the will to solve and express mathematical problems, to a favourable view about mathematics, and to a perception of self-improvement.

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Amado, N., Carreira, S. (2018). Students’ Attitudes in a Mathematical Problem-Solving Competition. In: Amado, N., Carreira, S., Jones, K. (eds) Broadening the Scope of Research on Mathematical Problem Solving. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99861-9_18

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Student attitudes towards learning mathematics through challenging, problem solving tasks: “It’s so hard–in a good way”

Research output : Contribution to journal › Article › Research › peer-review

• mathematics education
• student attitudes
• problem solving
• self-determination theory

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• 330313090-oa Final published version, 241 KB Licence: CC BY-NC-ND

T1 - Student attitudes towards learning mathematics through challenging, problem solving tasks

T2 - “It’s so hard–in a good way”

AU - Russo, James

AU - Minas, Michael

PY - 2020/12

Y1 - 2020/12

N2 - Despite a focus on teaching mathematics through challenging, problem solving tasks, there has been limited research into student attitudes towards these learning experiences. To address this gap in the literature, we asked 52 Australian primary students who had recently experienced mathematics taught in this manner to convey their feelings about learning through problem solving. Adopting a qualitative, exploratory, research design, student participants completed a brief questionnaire, and a sub-set also contributed to follow-up focus groups. Thematic analysis of the questionnaire data revealed that three-quarters of students reported unambiguously positive attitudes towards problem solving, most others were ambivalent, and no students expressed negative attitudes. Younger students (Year 3/4) were more likely to express positive attitudes than older students (Year 5/6) and boys more likely to express positive attitudes than girls. Positive attitudes arose from students enjoying learning through problem solving, the perception that it supported their learning, and students thriving on challenge. Follow-up focus groups also reinforced the power of working collaboratively, particularly the importance of learning through discussions with peers, and opportunities to explore authentic and purposeful tasks. The findings help explain why students frequently have positive reactions to learning mathematics through problem solving.

AB - Despite a focus on teaching mathematics through challenging, problem solving tasks, there has been limited research into student attitudes towards these learning experiences. To address this gap in the literature, we asked 52 Australian primary students who had recently experienced mathematics taught in this manner to convey their feelings about learning through problem solving. Adopting a qualitative, exploratory, research design, student participants completed a brief questionnaire, and a sub-set also contributed to follow-up focus groups. Thematic analysis of the questionnaire data revealed that three-quarters of students reported unambiguously positive attitudes towards problem solving, most others were ambivalent, and no students expressed negative attitudes. Younger students (Year 3/4) were more likely to express positive attitudes than older students (Year 5/6) and boys more likely to express positive attitudes than girls. Positive attitudes arose from students enjoying learning through problem solving, the perception that it supported their learning, and students thriving on challenge. Follow-up focus groups also reinforced the power of working collaboratively, particularly the importance of learning through discussions with peers, and opportunities to explore authentic and purposeful tasks. The findings help explain why students frequently have positive reactions to learning mathematics through problem solving.

KW - mathematics education

KW - student attitudes

KW - problem solving

KW - self-determination theory

U2 - 10.26822/iejee.2021.185

DO - 10.26822/iejee.2021.185

M3 - Article

SN - 1307-9298

JO - International Electronic Journal of Elementary Education

JF - International Electronic Journal of Elementary Education

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Assessing the attitude and problem-based learning in mathematics through PLS-SEM modeling

Samina zamir.

School of Education, Shaanxi Normal University, Xi’an, P.R. China

Uzma Sarwar

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All relevant data are within the manuscript and its Supporting Information files.

Mathematics plays a leading part in day-to-day life and has enhanced a necessary component for human accomplishments. Students from many countries do not reach the expected level in mathematics. Therefore, it is essential to pay close consideration to the causes related to ability in mathematics. Mathematics attitude is considered as one of the critical variables in the process of mathematics learning. This study aimed to determine students’ attitudes and achievements through problem-based learning in mathematics. The selected study group contained 600 students and 35 teachers from rural public secondary schools in District Rawalpindi, Pakistan. The data collection was done using questionnaires from students and teachers and collected data analyzed by SPSS 23 and Amos 23. This study’s result was carried out using Partial Least square structural equation Model (PLS-SEM), descriptive analysis, and hypotheses testing. The outcomes in this study indicated that the mean fluctuated between 1 to 4.5, 3.71 to 4.20, and Std. Deviation fluctuated between 0.6 to 2.0 and 0.75 to 1.55 in the students and teacher models, respectively. The results of the PLS-SEM students’ model show a negative attitude towards mathematics. The teachers’ PLS-SEM model showed the Effects of using problem-based learning (PBL) on students’ achievements. According to the hypotheses testing, the acceptance of hypotheses by stating that the Confidence in Learning Mathematics Scale (C), Value of Mathematics Scale (V), and Student Mathematics Motivation Scale (M) are significant effects for the Students’ Attitude Toward Problem-Based Learning (ATPBL). But the Attitude Toward Enjoyment in Mathematics Scale (AE) was rejected, and it did not significantly affect the ATPBL. As well as, the Problem-solving learning and students’ achievement (PLA), Advantages of problem-solving learning (APL) and Difficulties in using problem-solving learning (DPL) have a significant positive effect on the ATPBL. Finally, this study suggested that teachers also adopt new teaching methods corresponding to mathematics, and there is a need to explore particular mathematics skills to enhance students’ learning abilities.

1. Introduction

The most imperative things about societies are their learning ability, and learning is a necessary behaviour in life through genetic intelligence and the environment. There has been a lot of advances in educational technology in the last few decades [ 1 ]. Learning ability constantly impact the human lifestyle. As well as it is, in both developed and developing economies, entrepreneurship is considered vibrant to the nation’s competitive ability and a high-powered resource for decreasing regional inequities thereby allowing the development of the country [ 2 , 3 ]. According to that, the knowledge capabilities of an individual affect the student’s way of life continually [ 4 ]. Scribbling, painting and drawing perform ana significant part in the growing up of children [ 5 , 6 ] and it is helping to build their knowledge, personality like that. Consequently, human societies attempt to increase the method of learning in education unceasingly. As well, the ability is the core dimension of personality between learning preferences and cognitive forms. It can be defined as the preferred personal method to collect and process information, decisions, interests, ideas, and attitudes [ 3 ].

Mathematics plays a dominant role in human lives, and it is broadly applied as an essential element in personal achievements and economics [ 7 , 8 ]. In the 21st century, mathematics plays a significant skill in individual satisfaction and involvement in society, school, and the labour market. It appears to be a key academic filter for students’ educational trajectories [ 9 ]. Mathematics plays an essential role in supporting people to grow to reason, problem-solving skills, and thinking, and the importance of mathematics in the education system has gradually increased. Not only that, the impact of students’ mathematics achievements, including students’ ability, family socioeconomic status (SES), curriculum, many factors, peer influence, parental participation, school environment, and teachers’ quality [ 10 , 11 ]. By taking a positive attitude towards mathematics, students will think that mathematics is fundamental, so they try to enhance their performance in mathematics [ 7 ]. However, learning mathematics has grown into a challenge for most students today. Lack of learning despair motivates many students to say, "I am not good at mathematics", even before trying to solve mathematical problems [ 11 ]. Hence, teachers have a significant part in enhancing students’ mathematics achievement [ 10 ]. Emotional understanding, belief, and attitude are three major categories in the effective field of mathematics education [ 12 ].

However, recent worldwide determinations showed that students from many countries do not accomplish as anticipated in mathematics [ 13 ]. Therefore, one must pay close attention to the factors related to mastering mathematics and attitudes are one of the variables that can play a crucial role in learning mathematics [ 7 ].

The recent regeneration of mathematics education has brought new requirements. These provide students with meaningful activities that allow them to share their information in society. Various learning approaches focus on actions are used mainly in primary schools. One method is "problem-based learning" (PBL), which is a skill-based learning technique used to investigate and solve complicated real-life difficulties [ 14 ]. Most of the newest studies on PBL accentuate that it is a technique to enable students to enthusiastically play the part of learners. Most of the studies on PBL focus on teaching in different fields of education. These studies focus on mathematics education, science, engineering, and medicine [ 15 ]. Kaptan [ 16 ] documented that the PBL method is very important for students to improve the skills and knowledge learned in mathematics class to their daily issues and daily life. Theoretically, PBL is based on constructivism, and its instructional design method is based on problem-solving and "contextual learning" [ 14 ]. In general, PBL is considered to contribute to increasing and maintaining academic success [ 17 ] increase performance abilities [ 18 ] have a confident impact on attitudes towards classes [ 15 ], self-learning abilities and improve communication, as well as independent working abilities and motivation, and produce more reasonable explanations to problems [ 19 ]. Therefore, students’ attitudes toward mathematics have been researching worldwide for several decades [ 20 ].

2. Theoretical reviews

Attitude is "a learned inclination on the part of an individual to respond positively or negatively to the concept, situation, some object, or another person" [ 21 ]. Thus, the attitude towards mathematics can be an aggregation of mathematical emotions and beliefs. Allport [ 22 ] defines an attitude as "a mental or neural state of readiness, prepared over practice, applying a directive or dynamic effect upon the individuals’ feedback to all objects and circumstances with which it is associated". Adediwura [ 23 ] describes attitude as a persons’ positive, neutral or negative thinking about mathematics. A positive attitude is very instructive because research shows that there is a link between student performance and their attitude toward mathematics [ 24 ]. Students who have a positive attitude toward mathematics have better problem-solving abilities and are better able to resolve unusual difficulties [ 25 ]. They capitalize more energy in solved problems and give up when the problem cannot be solved. Attitude is also be interchanged with personality and is recognized as a multidimensional structure, including self-confidence or anxiety, such as enjoyment or not, commitment or avoidance, beliefs about whether mathematics is difficult or easy, unimportant or important, uninteresting, interesting, and useless [ 12 ]. Köğce [ 26 ] showed that the mathematics attitude is subjective in some factors, and it can be considered as several groups: firstly, reasons connected with the student, secondly, reasons associated to the teacher and school, and finally reasons related to the society and environment. Reasons related to the students’ mathematical results, their past practices [ 27 ], and social image of the mathematics. Not only that, but the reasons also related with the teachers and their content of knowledge, resources used in the classroom, the teaching methods, personality, teaching topics with real-life enriched examples [ 28 ], and the teachers’ attitude towards mathematics. Therefore, their teachers’ attitudes influence students’ attitudes [ 29 ]; teachers’ wrong beliefs about mathematics powerfully affect their teaching practices [ 30 ]. As well as it is vital to improving a positive attitude towards mathematics between students and teachers.

There have been a lot of improvements in educational technology in the last few decades [ 1 ] like online education. Students can use this technology any subject areas (especially mathematics) to improve their knowledge. But empirical studies have found that students feel that they learn better in physical classrooms than through online education [ 31 ]. Hence, Educational technology is affecting the students and teachers’ attitude toward problem-based learning mathematics.

In recent times, many researchers have pointed out the student attitude and teachers’ attitude towards problem-based learning in mathematics in several cities /countries around the world. The attitude towards mathematics has been considered for past years and shows a high relationship between attitude (including motivation, enjoyment, and self-confidence) and mathematical performance. Mezirow [ 32 ] defines learning as a cycle that starts from experience, continues to reflect, and leads to action, which becomes the experience of reflection. Valkenburg [ 33 ] found that children give their attention very rapidly to media content that was only moderately various from their existing capabilities and knowledge and teachers should give their attention for that [ 34 ]. Attitude towards mathematics is the students’ and teachers’ prepared preference to behave, perceive, feel, and think towards mathematics. Many studies have been established to assess the effect in mathematics [ 35 ].

Yılmaz [ 28 ] presented a positive and vital association between students’ attitudes towards mathematics use and mathematics accomplishment. Secondly [ 36 ], proposed a progressive connection between mathematics accomplishment and mathematics attitudes. They revealed that scholars improve attitudes, ideas, and feelings about school subjects from different sources. Thirdly, Colomeischi [ 37 ] analyzed a correlation between learning style and gender, attitude towards mathematics, and mathematical achievement. Thus, Bayaga [ 38 ] explained the students’ attitudes toward mathematics achievement using a variety of factors (attitude, mathematics self-concept, school condition, family background, teaching, and parent’s educational level) and approaches. We’ve looked into the relationship between math attitude and mathematics performance. A positive relationship between attitudes toward mathematics and academic achievement has been established in the majority of studies conducted across a range of age groups. According to some of the findings, having a negative attitude toward mathematics is associated with minor academic consequences in college students [ 39 , 40 ] and children [ 7 ]. However, in addition to doing so, Zsoldos-Marchis [ 24 ] investigated the problem-solving potential of various primary preschool teachers’ attitudes toward mathematics.

Moreover, Russo [ 41 ] documented the association between math teachers’ enjoyment and attitudes toward student struggle and the number of times teachers spent teaching math. There are more methods developed around the world to analyze attitudes towards mathematics. Among them, one of the most well-known analysis methods is the Partial Least Structural Equation Model (PLS-SEM), and it is a flexible modeling method without data distribution assumptions. It is also essential and suitable for various education analyses. The main aim of this study was to estimate the student attitude and teachers’ attitudes towards problem-based learning in mathematics.

3. Methodology

3.1. participants and data collection.

The study population comprised 3,300 secondary mathematics students and 35 mathematics teachers in District Rawalpindi’s 35 rural public secondary schools. The population is the mathematics students and teachers in North Punjab District Rawalpindi Government Areas as of the 2020/2021 academic session. This study selected the North Punjab district because the schools and education system are better than other areas. Moreover, belonging to the Rawalpindi district so it will be convenient to access the schools. This study selected rural schools because, in mathematics, 10 th class students score low compared to urban schools.

First, purposive sampling will be used in identifying and selecting schools that meet the following criteria:

• Evidence of continuous presentation of candidates for external examination in mathematics.
• Availability of qualified mathematics teachers who used the problem-based learning method in their class.
• Availability of teacher’s students and schools who agree to this study. Due to religious, cultural, regional, and local barriers.

The current study was approved by the Educational Research Ethics committee from the School of Education, Shaanxi Normal University. All procedures performed in the study involving human participants were in accordance with the ethical standards of the institutional research committee and consent was obtained from each respondent. Additional information regarding the ethical, cultural, and scientific considerations specific to inclusivity in global research is included in the Supporting Information ( S1 Appendix ).

By the above criteria, 35 schools will be purposively selected. In these schools, 600 mathematics (female, male) students and 35 mathematics teachers applied problem-based learning methods in their classes. The general overview of the students in the study is given in Tables ​ Tables1 1 and ​ and2 2 showed that the number of teachers in gender-wise and their qualifications.

3.2. Data analysis

A structured questionnaire with a Likert scale was used to investigate the mathematics attitudes toward students (see S2 Appendix ) and teachers (see S3 Appendix ). Descriptive statistics, hypothesis testing is used for analysis. Analysis was performed by examining the correlations, covariance patterns between the observed measures and hypotheses testing were used for this study. There are seven (7) proposed hypotheses (H 1 to H 7 ) used for analysis (see Fig 1 ).

• H 1 : Confidence in Learning Mathematics Scale is positively influenced to Student’s Attitude Toward Problem-Based Learning.
• H 2 : Value of Mathematics Scale is positively influence by Student’s Attitude Toward Problem-Based Learning.
• H 3 : Attitude Toward Enjoyment in Mathematics Scale is positively influence to Student’s Attitude Toward Problem-Based Learning.
• H 4 : Student Mathematics Motivation Scale is positively influencing to Student’s Attitude Toward Problem-Based Learning.
• H 5 : Problem-solving learning and students’ achievements positively influence Student’s Attitude Toward Problem-Based Learning.
• H 6 : Difficulties in using problem-solving learning is positively influenced Student’s Attitude Toward Problem-Based Learning.
• H 7 : Advantages of problem-solving learning is positively influenced to Student’s Attitude Toward Problem-Based Learning.

Descriptive statistics are shown that provide a general overview of the data of the respondents. The collected data was analyzed by SPSS 23 version and Amos 23. For data analysis, Partial Least square structural equation Model (PLS-SEM) was used, interpreted in two stages. The first was to evaluate the student model, and the second was to assess the teachers’ model. The first (Student’s Attitude Toward Problem-Based Learning -ATPBL) model consisted of four constructs with 46 indicators—Confidence in Learning Mathematics Scale (C) = 12 indicators; Value of Mathematics Scale (V) = 12 indicators; Attitude Toward Enjoyment in Mathematics Scale (AE) = 10 indicators; and Student Mathematics Motivation Scale (M) = 12 indicators. The second model consisted of three constructs with 22 indicators—Problem-solving learning and students’ achievement (PLA) = 8 indicators, Advantages of problem-solving learning (APL) = 7 indicators, and Difficulties in using problem-solving learning (DPL) = 7 indicators can be seen in S4 Appendix .

4. Results and discussion

4.1. student attitude towards problem-based learning in mathematics.

According to Table 3 , the mean fluctuated between 1 to 4.5 and Std. Deviation fluctuated between 0.6 to 2.0 and highly Std. Deviation reported from C4 ( I am always confused in my mathematics class .) in the Confidence in Learning Mathematics Scale group. But the low value of Std. Deviation value reported from AE5 ( I really like mathematics ) in attitude toward enjoyment in mathematics scale group. Table 3 shows the results of descriptive statistics in the SEM model’s exogenous variables.

According to Fig 2 , the Confidence in Learning Mathematics Scale group had a high regression weight from C11 (In terms of my adult life, it is not important for me to do well in mathematics in high school). It recorded 0.664. But in the C1 (I have a lot of self-confidence when it comes to mathematics) showed that a low regression weight. It is recorded -22. As well as Confidence in Learning Mathematics Scale and Student’s Attitude toward Problem-Based Learning presented the 0.11 Standardized Regression Weight. Value of Mathematics Scale (V) showed the high regression weight with V10 (Taking mathematics is a waste of time.), and it recorded 1.01. But in the V1 (Mathematics is a very worthwhile and necessary subject) showed a low regression weight. It is recorded -.11. As well as Value of Mathematics Scale and Student’s Attitude toward Problem-Based Learning presented the 0.12 of Standardized Regression Weight. Hence, the Attitude toward Enjoyment in Mathematics Scale (AE) had a high regression weight from AE7 (Winning a prize in mathematics would make me feel unpleasantly conspicuous), and it recorded 0.88. Hence in the AE4 (I really like mathematics.) showed a low regression weight. It is recorded -.12.

As well as Attitude toward Enjoyment in Mathematics Scale and Student’s Attitude toward Problem-Based Learning presented the 0.005 of Standardized Regression Weight. Furthermore, Student Mathematics Motivation Scale (M) had a high regression weight from M8 (The challenge of math problems does not appeal to me), and it recorded 0.90. Hence in the AE4 (I really like mathematics.) showed a low regression weight. It is recorded -.014. As well as Student Mathematics Motivation Scale and Student’s Attitude toward Problem-Based Learning presented 0.086 of Standardized Regression Weight. According to the SEM, the standardized estimation can be identified the most student have a negative attitude about mathematics. As well as Fig 2 showed that the squared multiple correlations (R 2 ). A strong positive correlation was reported in V10 (Taking mathematics is a waste of time) with a 1.0 value. V2 (I want to develop my mathematical skills), V12 (I expect to have little use for mathematics when I get out of school), M1 (I like math puzzles), M2 (Mathematics is enjoyable and stimulating to me), M4 (Once I start trying to work on a math puzzle, I find it hard to stop), M12 (I do as a little work in math as possible), C5 (I learn mathematics easily.), C12 (When I hear the word mathematics, I have a feeling of dislike.), AE1 (I have usually enjoyed studying mathematics in school.) showed that, the no correlation. Not only that, but there was also no negative correlation reported in this model.

4.2. Hypotheses testing for Student attitude towards problem-based learning in mathematics

The proposed hypotheses of this study were tested through the standardized coefficient values and p-values in AMOS 23.0. The students’ model’s dependent variable was the Student’s Attitude Toward Problem-Based Learning (ATPBL). The Confidence in Learning Mathematics Scale (C), Value of Mathematics Scale (V), Attitude Toward Enjoyment in Mathematics Scale (AE), and Student Mathematics Motivation Scale (M) were independent variables.

Table 4 showed the acceptance of hypothesizes states that the Confidence in Learning Mathematics Scale (C), Value of Mathematics Scale (V), and Student Mathematics Motivation Scale (M) and these states are significant effects on the Student’s Attitude Toward Problem-Based Learning. But hypotheses state that the Attitude Toward Enjoyment in Mathematics Scale (AE) was rejected, and it did not significantly impact the Student’s Attitude Toward Problem-Based Learning (ATPBL).

4.3. Effects of using Problem-based Learning (PBL) on student’s achievements

According to Table 5 , the mean fluctuated between 3.71 to 4.20 and Std. Deviation fluctuated between 0.75 to 1.55 and high std. Deviation reported from PLA3 ( When I use this method , student achievement is high .) in Problem solving learning and students’ achievement group. But the low value of std. Deviation value reported from APL7 ( Problem-solving reduces the need to revise prior to examinations .) in Advantages of the problem-solving learning group.

According to Fig 3 , the problem-solving learning and students’ achievement group had the high regression weight from PLA6 ( The mathematics curriculum is designed to use the problem-solving method frequently .), and it recorded 0.97. However, the PLA1 ( You always get a good response from students who are motivated actively to solve the problems by themselves . ) showed a low regression weight. It is recorded -.269. As well as problem-solving learning and students’ achievement (PLA) and Student’s Attitude toward Problem-Based Learning presented the 0.106 Standardized Regression Weight. Advantages of problem-solving learning (APL) showed the high regression weight with APL7 ( Textbooks are structured to support problem-solving strategies ) and recorded 0.314. But in the APL1 ( Problem-solving helps students to use mathematics in their daily life .) showed a low regression weight. It is recorded -.0.974. As well as Advantages of problem-solving learning (APL) and Student’s Attitude toward Problem-Based Learning presented the 0.031 of Standardized Regression Weight. Hence, Difficulties in using problem-solving learning (DPL) had a high regression weight from DPL2 ( This method is not suitable when the time span is short for teaching . ) , and it recorded 0.94. Moreover, the DPL4 (You need enough space, resources, and feasible environment in the class.) showed the low regression weight. It is recorded -.153. As well as Difficulties in using problem-solving learning (DPL) and Student’s Attitude toward Problem-Based Learning presented the 0.11 of Standardized Regression Weight.

As well as Fig 3 showed the squared multiple correlations (R 2 ) and strong positive correlation reported in APL1 ( You always get a good response from students are motivated actively to solve the problems by themselves .), DPL2 ( This method is not suitable when time span is short for teaching .), APL2 ( You find the problem-solving method supportive for learners of all abilities in the class .), DPL6 ( It is more difficult to satisfy slow and weak learners through problem solving .), APL4 ( Students learn to draw diagram and pictures themselves to solve problems .), with .949, .883, .883, .859, .844 .824, respectively. ATPBL ( Student’s Attitude toward Problem-Based Learning ), APL3 ( When I use this method , student achievement is high .) PLA5 ( Problem-solving is helpful to make a learner more skilled and confident . ) DPL1 ( This method is difficult when students are larger in number in the classroom .) showed the very week but positive correlation. Not only that, but there was also no negative correlation reported in this model.

4.4. Hypotheses testing for effects of using Problem-based Learning (PBL) on student’s achievements

The proposed hypotheses of this study were tested through the standardized coefficient values, and p-values in AMOS 23.0 for the teachers’ model. In the teachers’ model dependent variable was the Student’s Attitude Toward Problem-Based Learning (ATPBL). The Problem-solving learning and students’ achievement (PLA) Advantages of problem-solving learning (APL) and Difficulties in using problem-solving learning (DPL) were independent variables in this study.

Table 6 showed the acceptance of hypothesizes by stating that the Problem-solving learning and students’ achievement (PLA), Advantages of problem-solving learning (APL), and Difficulties in using problem-solving learning (DPL). These states have a significant positive impact on the Student’s Attitude Toward Problem-Based Learning (ATPBL).

5. Conclusion

Information about students’ attitudes towards problem-based learning in mathematics is influential to both the students and the teachers [ 30 ]. The current study Partial Least Structural Equation Model (PLS-SEM) approach investigates the student attitude and teachers’ attitude towards problem-based learning in mathematics. The demographic data of this study have also exposed those 600 students and 36 teachers are competent in handling mathematics.

In firstly, this study estimated the student attitude towards problem-based learning in mathematics. The PLS-SEM model showed that the mean fluctuated between 1 to 4.5 and Std. Deviation fluctuated between 0.6 to 2.0. Among the 46 indicators, the C4 (I am always confused in my mathematics class) showed a high Std. Deviation and, but the low value of Std. Deviation value reported from AE5 (I really like mathematics) indicator.

According to the regression weight, in the students’ model, the high weight record in C11 (In terms of my adult life it is not important for me to do well in mathematics in high school.) and it recorded 0.664, V10 (Taking mathematics is a waste of time.). It recorded 1.01, AE7 (Winning a prize in mathematics would make me feel unpleasantly conspicuous) and it recorded 0.88, M8 (The challenge of math problems does not appeal to me), and it recorded 0.90.

According to the regression weight, in the teachers’ model, the high weight record PLA6 (The mathematics curriculum is designed to use the problem-solving method frequently) and is recorded 0.97. APL7 (Textbooks are structured to support problem-solving strategies.), and it recorded 0.314. DPL2 (This method is not suitable when time span is short for teaching.), and it recorded 0.94.

According to the hypothesizes testing, the acceptance of hypothesizes by stating that the Confidence in Learning Mathematics Scale (C), Value of Mathematics Scale (V), and Student Mathematics Motivation Scale (M) and these states are significant effects on the Students’ Attitude Toward Problem-Based Learning. But it hypothesizes by stating that the Attitude Toward Enjoyment in Mathematics Scale (AE) was rejected, and it did not significantly affect the Students’ Attitude Toward Problem-Based Learning (ATPBL). As well as the acceptance of hypothesizes by stating that the Problem-solving learning and students’ achievement (PLA), Advantages of problem-solving learning (APL) and Difficulties in using problem-solving learning (DPL) has a significant positive impact on the Students’ Attitude Toward Problem-Based Learning (ATPBL).

This study significantly revealed students’ attitudes towards mathematics and the attitudes of teachers who use it to teach mathematics. Finally, this study suggested that teachers should also adopt new teaching methods corresponding to mathematics. There is a need to explore particular mathematics skills to enhance students’ learning abilities.

Supporting information

S1 appendix, s2 appendix, s3 appendix, s4 appendix, acknowledgments.

The first author would like to thank his parents who support him in this work. We thank those anonymous reviewers whose comments/suggestions helped to improve and clarify this manuscript.

Institutional review board statement

This study is approved by the Educational Research Ethics committee from the School of Education, Shaanxi Normal University. All procedures performed in the study involving human participants were in accordance with the ethical standards of the institutional research committee.

Informed consent statement

Informed consent was obtained from all subjects involved in the study.

Funding Statement

ZY received the grant for the Social Science Fund Project of Shaanxi Province "Legitimacy Analysis of Education and Training Market" (2019Q005). This study was also supported by the National Social Science Foundation of China Project (BAA170014).

Data Availability

‘Our goal is to get students to see problems as…

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Post-Tribune

‘our goal is to get students to see problems as opportunities’, teens pitch start-up ideas at innovate within regional competition.

Ten teams of Lake County high school students did their best to pitch their problem-solving entrepreneurial ideas to a panel of judges Friday at Purdue Northwest as part of the Innovate WithIN regional pitch competition.

Don Wettrick, CEO of STARTedUP Foundation, who organizes the event in cooperation with Purdue Northwest and the Purdue Society of Innovators, said this is the seventh year for the competition. Six regional events are held and the winners from those rounds go on to compete for the state title – along with $25,000 in company seed money and scholarship funds – on June 21 at Butler University.

Wettrick said the local anchor school is Hobart High School, where the competition started, and as word of the competition spread, more schools have become involved. This year there were three teams from Munster, two from Lowell, two from Hobart, and one each from Hanover Central, Hammond and Lake Central high schools.

More than 3,000 Indiana high schoolers sent in applications for consideration. That number was whittled down to 10 in each region. The winner of each region plus four wild cards will compete for the state title.

“Our whole goal is to get students to see problems as opportunities,” Wettrick said.

Whether students go into entrepreneurship after their experience or choose a different path, the skills they learn help change the way they look at the world, he said.

Jason Williams, manager director of the Society of Innovators at Purdue Northwest, said there were 600 applicants in Northwest Indiana alone.

“The quality of the applicants and the number of applicants continues to grow,” Williams said. Along with the growth of the competition is the growth of an ecosystem of support that is spreading around it, providing mentorship and assistance after the contest ends.

Dr. Rachel Clapp-Smith, dean of Purdue University Northwest College of Business at PNW, said it is inspiring to see the innovation brought out in the competition by the region’s youth. She described the university as fertile ground for entrepreneurship.

She said the solutions to tomorrow’s problems can be found in the minds of today’s high school students.

“I know I am probably going to see a CEO, a future change maker, a future business leader here today,” Clapp-Smith said.

One by one the teams took the stage to talk about the problem their business would fix.

Ashton Verbish from Hanover Central High School explained what brought him to his idea of “SafetyGlow,” an emergency lighting system for disabled semi-trucks.

As a newer driver who lives in Cedar Lake and often uses local highways and interstates, he said he noticed how difficult it is to see semi trucks that have pulled over on the shoulder. The three triangles they are required by law to put behind their trucks are difficult to spot in the few seconds a vehicle moving 40-plus miles an hour has to see something approaching at night.

His solution is a band of flashing lights that can be attached from corner to corner on the back of the truck to form a large “X.”

Noah Kaiser and Mikel Ivy from Hobart High School pitched their product “Safe Trips,” a product designed to keep youth accounted for when they are on classroom trips or with other organizations or businesses. The team said their idea stemmed from an incident where Kaiser’s younger brother was hiding and the family could not find him.

They have devised a bracelet and are working on an app that allows teachers or event organizers, for example, to provide each child with a GPS bracelet. The bracelet is registered to the child and the teacher and parent can track the child’s location. After the event, the bracelet is returned and can be reassigned for future use.

The Munster High School team behind “Wringo” walked away with first place. Ameen Musleh, Vasili Papageorge and Conner Gomez wowed the judges with their “perfect solution” designed to help instill confidence and reduce anxiety among people who have sweaty hands.

Wringo is a product the user holds in their hand for 15 seconds and it leaves hands dry and natural for 30 minutes.

The Wringo team will head to Butler University in June to make their pitch at the state competition. There also are four at-large positions in the six districts that will be filled with the top four scoring teams district-wide after the top six winners are named. Williams said the 10 teams heading to state will be announced Friday.

There will be a pep rally for the state finalists on June 7 at the Lake County Corn Dogs game in Crown Point.

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