Swedish students' knowledge of both mathematics and reading comprehension has deteriorated in recent years. Scientists are discussing whether there is a connection between these areas and that the pupils deteriorating math skills may have something to do with their increasingly lower results in terms of reading comprehension. To investigate this possible connection, I conducted a survey among students in ninth grade and have come to the conclusion that the scientists are right: this connection absolutely exist. Students who received a high score on tasks designed to test students' mathematical problem-solving skills, also received high results on the reading comprehension test. And students who received a poor performance on the problem-solving tasks, were also low performers in the reading comprehension test. The students who received low scores on the problem-solving tasks, wasn’t automatically scoring low on the mathematics test, as you might think. Therefore, I conclude that there is a greater connection between students' reading comprehension and ability to solve mathematical problem-solving tasks than between their abilities in problem-solving and pure mathematics. From this I conclude that reading has a major impact on students' problem-solving skills, which is why I believe that reading should have a greater role in mathematics education.

The purpose of this research is to study the mathematical problem solving processes, strategy use and success factors of middle grade gifted and talented (GT) elementary school students. This research is based on 9 mathematical problems edited by the author and divided into the following categories: ¡§numbers and quantity,¡¨ ¡§shape and space,¡¨ and ¡§logical thinking.¡¨ Seven GT students from Ta-Tung elementary school in Kaohsiung were selected as target students in the study. Besides, the seven students were translated into original cases using a thinking aloud method. Here are the conclusions: First of all, when facing non-traditional problems, GT students may use different problem solving steps to solve different problems and may not show all detailed steps for every single problem. The same types of problems may not have the same problem solving steps. Missing any single step would have no impact on the answers. Problem solving sequence may not fully follow the traditional 5-step sequence: study the problem, analyze, plan, execute, and verify, and, instead, may dynamically adjust the steps according to the thinking. Secondly, GT students¡¦ problem solving strategy includes more or less the following 19 methods: trial and error, tabling, looking for all possibilities, a combination of numbers, listing all possible answers, classifying the length of each side, classifying graphics, classifying points, adding extra numbers (the triangle problem), drawing, identifying rules and repetition, summarizing, forward solving, backward solving, remainder theory, polynomials, organizing data, direct solving, and making tallies. Finally, problem solving success factors are tightly coupled with problem solving knowledge, mathematical capability, and problem solving behavior. Problem solving knowledge includes knowledge of language, understanding, basic models, strategy use, and procedural knowledge. Instances of mathematical capability are capability of abstraction, generalization, calculation, logical thinking, express thinking, reverse thinking, dynamic thinking, memorizing, and space concept. Problem solving behavior includes the sense of understanding the problem and mathematical structure, keeping track of all possible pre-conditions, good understanding of the relationship between the problems and the objectives, applying related knowledge or formulas, verifying the accuracy of the answers, and resilience for problem solving. In addition to discussing the research results, future directions and recommendations for teaching mathematics for GT and regular students are highlighted.

Problem solving has been a core theme in education for several decades. Educators and policy makers agree on the importance of the role of problem solving skills for school and real life success. A primary purpose of this study was to investigate the influence of cognitive abilities on mathematical problem solving performance of students. The author investigated this relationship by separating performance in open-ended and closed situations. The second purpose of this study was to explore how these relationships were different or similar in boys and girls. No significant difference was found between girls and boys in cognitive abilities including general intelligence, general creativity, working memory, mathematical knowledge, reading ability, mathematical problem solving performance, verbal ability, quantitative ability, and spatial ability. After controlling for the influence of gender, the cognitive abilities explained 51.3% (ITBS) and 53.3% (CTBS) of the variance in MPSP in closed problems as a whole. Mathematical knowledge and general intelligence were found to be the only variables that contributed significant variance to MPSP in closed problems. Similarly, after controlling for the influence of gender, the cognitive abilities explained 51.3% (ITBS) and 46.3% (CTBS) of the variance in mathematical problem solving performance in open-ended problems. General creativity and verbal ability were found to be the only variables that contributed significant variance to MPSP in open problems. The author concluded that closed and open-ended problems require different cognitive abilities for reaching correct solutions. In addition, when combining all of these findings the author proposed that the relationship between cognitive abilities and problem solving performance may vary depending on the structure (type) and content of a problem. The author suggested that the content of problems that are used in instruments should be analyzed carefully before using them as a measure of problem solving performance.

The nature and function of Task Complexity, in the context of senior secondary mathematics, has been identified through: a search of the research literature; interviews with experts that focused on the nature of task complexity; expert use of the Williams/Clarke Framework of Complexity (1997) as a tool to categorise the complexity of a task, and observation and analysis of the responses of senior secondary mathematics students as they worked in collaborative groups to solve an unfamiliar challenging problem. Although frequently used in the literature to describe tasks, ‘complexity’ has often lacked definition. Expert opinion about the nature of mathematical complexity was ascertained by seeking the opinions of experts in the areas of mathematics, mathematics education, and gifted education. Expert opinion about task complexity was stimulated by questions about the relative complexity of two tasks. The experts then categorised the complexities within each of these tasks using the Williams/Clarke Framework of Complexity. This framework identifies the dimensions of task complexity and was found by experts to be both useful and adequate for this purpose. A theoretical framework was developed to assess student ability to solve challenging problems. This theoretical framework was used to design a test to assess student ability to solve challenging problems. The information this test provided about the problem solving ability of the students in this study informed my analysis of student response to complexity.

The purpose of this study was to investigate the effect of the use of an Alternative-Solution Worksheet (ASW) on American ninth-grade students' problem solving performance, and to determine the extent to which instruction in alternative solutions promotes "look back" strategies. "Look back" strategies are based on Polya's (1973) problem solving steps, and they are an examination of what was done or learned previously. The ASW was designed to encourage students to utilize "look back" strategies by generating alternative solutions to the problems. This mixed-methods study was conducted with two existing groups of ninth-grade Algebra I students. An experimental group of 18 students received instruction in utilizing the ASW for two 55-minute class periods a week for a period of four weeks. A comparison group of 14 students did not receive any instruction. Data for this study were collected by pre- and post-testing, ASWs, focus groups, and one student's "think aloud" process. For the quantitative analysis, a one-way ANCOVA was conducted to determine if there was a significant difference in the mean post-test scores between the experimental group and the comparison group. The students' pre-test score was the covariate. The findings indicated that the experimental group scored slightly better on the post-test, and R2=.345, a medium effect size. There were no significant correlations between the ASW scores and the pre- and post-test scores, but the ASW scores were significantly correlated with the students' EXPLORE9 math and reading percentiles. The qualitative findings indicated that "look back" occurred at all six levels of Bloom's Revised Taxonomy, but it is the "look back" that occurs at the upper three levels, in the context of higher order thinking skills, that results in better mathematical problem solving abilities. In addition, positive affective changes were evident despite little improvement in students' mathematical problem solving abilities. The results of this study indicated that higher order thinking skills need to be practiced regularly so students can use them effectively.

The purpose of this study was to determine the relationship between High School learners‟ mathematics-related belief systems and their approaches to mathematics non-routine problem-solving. A mixed methods approach was employed in the study. Survey questionnaires, mathematics problem solving test and interview schedules were the basic instruments used for data collection. The data was presented in form of tables, diagrams, figures, direct and indirect quotes of participants‟ responses and descriptions of learners‟ mathematics related belief systems and their approaches to mathematics problem solving. The basic methods used to analyze the data were thematic analysis (coding, organizing data into descriptive themes, and noting relations between variables), cluster analysis, factor analysis, regression analysis and methodological triangulation. Learners‟ mathematics-related beliefs were grouped into three Learners‟ mathematics-related beliefs were grouped into three categories, according to Daskalogianni and Simpson (2001a)‟s macro-belief systems: utilitarian, systematic and exploratory. A number of learners‟ problem solving strategies were identified, that include unsystematic guess, check and revise; systematic guess, check and revise; trial-and-error; logical reasoning; non-logical reasoning; systematic listing; looking for a pattern; making a model; considering a simple case; using a formula; numeric approach; piece-wise and holistic approaches. A weak positive linear relationship between learners‟ mathematics-related belief systems and their approaches to non-routine problem solving was discovered. It was, also, discovered that learners‟ mathematics-related belief systems could explain their approach to non-routine mathematics problem solving (and vice versa). / Mathematics Education / D.Phil. (Mathematics Education)

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-06-14T17:55:05Z No. of bitstreams: 1 DissGABF.pdf: 7214889 bytes, checksum: 6de3fc0b60e1ff02be332fb45551e261 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-22T13:08:18Z (GMT) No. of bitstreams: 1 DissGABF.pdf: 7214889 bytes, checksum: 6de3fc0b60e1ff02be332fb45551e261 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-22T13:08:28Z (GMT) No. of bitstreams: 1 DissGABF.pdf: 7214889 bytes, checksum: 6de3fc0b60e1ff02be332fb45551e261 (MD5) / Made available in DSpace on 2017-06-22T13:18:47Z (GMT). No. of bitstreams: 1 DissGABF.pdf: 7214889 bytes, checksum: 6de3fc0b60e1ff02be332fb45551e261 (MD5) Previous issue date: 2017-02-17 / Não recebi financiamento / This research project refers to an experience applied to study groups for high school students in order to provide them with an improvement of Mathematics contents, as well as an improvement in the practice of their studies, using the approach proposed and systematized by George Pólya, which is the method of solving problems in Mathematics. Some topics involving functions were explored, through problem solving, mathematical language, collective discussions and applications. The actions implemented in this project also aim to improve the teaching practice, provided by the methodology, to be developed at other times in the classroom, as an important teaching tool. / Este projeto de pesquisa se refere a uma experiência aplicada a grupos de estudos para estudantes do ensino médio com o objetivo de proporcionar-lhes um aprimoramento dos conteúdos de Matemática, bem como um aperfeiçoamento na prática de seus estudos, utilizando-se para isso a abordagem proposta e sistematizada por George Pólya, que é o método de resolução de problemas em Matemática. Foram selecionados alguns tópicos envolvendo funções e explorados, através da resolução de problemas, a linguagem matemática, discussões coletivas e aplicações. As ações implantadas neste projeto visam também aprimorar a prática docente, proporcionada pela metodologia, a ser desenvolvida em outros momentos na sala de aula, como uma importante ferramenta de ensino.

Syftet med denna studie var att kartlägga 13 gymnasieelevers uppfattningar om problemlösningsuppgifter i matematik. I studien användes individuella intervjuer som datainsamlingsmetod. Frågorna rörde elevernas syn på matematiska problem i stort samt deras syn på om uppgifterna i deras läromedel var problem eller ej. Detta ledde till användandet av två analysmetoder, den ena för att studera vilka uppgifter i läromedlen som var problemlösning, den andra för att kartlägga och kategorisera elevernas uppfattningar. Resultaten visar att eleverna hade vissa uppfattningar som överensstämmer med hur forskningslitteraturen definierar problemlösningsuppgifter, men det framkom även uppfattningar som inte tas upp av litteraturen. Studien visar därmed på vilka som kan vara elevernas missuppfattningar kring problemlösningsuppgifter. Detta kan vara till nytta för både lärare, elever och forskare, då denna studie har kartlagt eleverna missuppfattningar kring problemlösnings uppgifter. Där läraren kan rätta elevernas missuppfattningar kring problemlösningsuppgifter, detta innebär att eleverna får en mer korrekt uppfattning kring problemlösningsuppgifter som är mer nära forskning.

Problemformulering är en arbetsmetod som är tätt kopplad till problemlösning, men trots att matematiker och forskare under lång tid betonat dess betydelse, har det inte fått tillräckligt med uppmärksamhet inom forskning. Användningen av problemformulering kan resultera i förbättrad förståelse av matematik, ökat självförmåga, främjande av matematisk kreativitet och en positiv inställning till ämnet, samt ökad motivation. Denna studie har granskat användningen av problemformulering i undervisningen för elever i årskurs 4-6 genom att besvara följande frågor: "Hur motiveras användningen av problemformulering i matematikundervisning för elever i årskurs 4-6?" och "Vilka aspekter av problemformulering kan enligt tidigare forskning vara utmanande för genomförandet?". Det är en systematisk litteraturstudie, vilket innebär att den bygger på tidigare forskning som samlats in metodiskt och väldokumenterat, för att sedan analyserats och diskuterats. Resultaten av studien visar att vetenskapligt stöd för användningen av problemformulering i årskurs 4-6 finns och stödjer tidigare forskning som visar att problemformulering förbättrar elevers förståelse av matematik, färdigheter, kreativitet, självförmåga, motivation och attityder gentemot ämnet. Resultaten pekar också på vissa utmaningar som lärare och elever kan ställas inför vid genomförandet, främst på grund av att arbetsmetoden avviker från den mer traditionella undervisningen samt brist på erfarenhet inom detta område / Mathematical problem posing is a method closely linked to problem-solving, but despite mathematicians and researchers emphasizing its importance for a long time, it has not received sufficient attention in research. The use of mathematical problem-posing can result in an improved understanding of mathematics, increased self-efficacy, improvement of mathematical creativity, a positive attitude towards the subject, and increased motivation. This study has examined the use of mathematical problem-posing in the education of students in grades 4-6 by addressing the following questions: "How is the use of mathematical problem-posing in mathematics education for students in grades 4-6 motivated?" and "What aspects of mathematical problem-posing can, according to previous research, be challenging for implementation?". It is a systematic literature study, meaning that it is based on previous research that has been systematically and well-documented in its collection, analysis, and discussion. The results of the study show that there is scientific support for the use of mathematical problem-posing in grades 4-6 and support previous research indicating that mathematical problem-posing improves students' understanding of mathematics, skills, creativity, self-efficacy, motivation, and attitudes towards the subject. The results also point to certain challenges that teachers and students may face in implementation, primarily due to its deviation from more traditional teaching methods and the lack of experience in this area.

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