The rate at which individuals acquire new cognitive skills may vary quite substantially, some acquiring a new skill more rapidly and efficiently than others. It has been shown through the analysis of think aloud protocols that learning performance on a map learning task, for instance, is associated with the use of certain learning procedures. In the domain of mathematical problem solving, it has also been shown that performance is associated with strategic as opposed to tactical decision making. Previous research on learning and problem solving has tended to focus on tactical processes, ignoring the role of strategic processes in learning and problem solving. There is clearly a need to examine the role of strategic processes in learning and to determine whether they might be an important source of individual differences in learning performance. A related question concerns teaching thinking skills. If it is possible to determine those learning procedures that differentiate good from poor learners, is it then possible to teach the effective procedures to a group of novice students in order to enhance the rate of skill acquisition? Results from the experiments reported here show that novices differ, and that learning performance is related to the use of certain learning procedures, as revealed by subjects' think aloud protocols. A follow-up study showed that novices taught to use the procedures differentiating good from poor learners performed at a higher level than two control groups of novices. A coding scheme was developed to explicitly examine learning at macroscopic and microscopic levels, and to contrast tactical with strategic processes. Discriminant function analysis was used to examine differences between good and poor learners. It was shown that good learners more frequently use executive processes in learning episodes. A study of the same subjects learning to use statistical packages on a microcomputer corroborate these findings. Thus, results extend those obtained from the first study. A study of the knowledge structures possessed by novices was complicated by differences in levels of statistical knowledge. Multidimensional scaling techniques revealed differences between novices with three statistical courses behind them, but not among those with only two statistical courses behind them. Among those novices with three statistical courses behind them, faster learners' knowledge structures more closely resembled those of experienced users of statistical packages than did those of slower learners.

The aim of this dissertation is to describe the role of educational strategy especially in field of the teaching of mathematics and to compare the mathematical problem solving strategies of primary school pupils which are taught by using different educational strategies. In the theoretical part, the main focus is on divergent educational strategies and their characteristics, next on factors affected teaching/learning process and finally on solving the problems. The empirical part of the dissertation explores the effect of educational strategy on problem solving strategies of third graders of primary school by using the method of experiment that includes a didactic test and interviews with selected pupils. The result of the dissertation is providing the evidence of influence of educational strategy on mathematical problem solving strategies. KEYWORDS Educational strategy, teacher, pupil, mathematical problem solving, experiment.

Student Number : 8710172X - MSc research report - School of Education - Faculty of Science / This study investigates mathematics for teaching, specifically in the case of functions at the grade 10 level. One teacher was studied to gain insights into the mathematical problem-solving a teacher does as he/she goes about his/her work. The analysis of data shows that the mathematical problems that this particular teacher confronts as he goes about his work of teaching can be classified as defining, explaining, representing and questioning. The resources that he draws on to sustain and drive this practice can be described as coming from aspects of mathematics, his own teaching experience and the curriculum with which he works. Of interest in this study are those features of mathematical problemsolving in teaching as intimated by other studies, particularly restructuring tasks and working with learners’ ideas; which are largely absent in this practice. This report argues that these latter aspects of mathematical problem-solving in teaching are aligned to a practice informed by the wider notion of mathematical proficiency. The report concludes with a discussion of why and how external intervention is needed to assist with shifting practices if mathematical proficiency is a desired outcome, as well as with reflections on the study and its methodology.

This thesis describes changes in the spatial thinking of Year 2 and Year 4 students who participated in a six-week long spatio-mathematical program. The main investigation, which contained quantitative and qualitative components, was designed to answer questions which were identified in a comprehensive review of pertinent literatures dealing with (a) young children's development of spatial concepts and skills, (b) how students solve problems and learn in different types of classrooms, and (c) the special roles of visual imagery, equipment, and classroom discourse in spatial problem solving. The quantitative investigation into the effects of a two-dimensional spatial program used a matched-group experimental design. Parallel forms of a specially developed spatio-mathematical group test were administered on three occasions—before, immediately after, and six to eight weeks after the spatial program. The test contained items requiring spatial thinking about two-dimensional space and other items requiring transfer to thinking about three-dimensional space. The results of the experimental group were compared with those of a ‘control’ group who were involved in number problem-solving activities. The investigation took into account gender and year at school. In addition, the effects of different classroom organisations on spatial thinking were investigated~one group worked mainly individually and the other group in small cooperative groups. The study found that improvements in scores on the delayed posttest of two-dimensional spatial thinking by students who were engaged in the spatial learning experiences were statistically significantly greater than those of the control group when pretest scores were used as covariates. Gender was the only variable to show an effect on the three-dimensional delayed posttest. The study also attempted to explain how improvements in, spatial thinking occurred. The qualitative component of the study involved students in different contexts. Students were video-taped as they worked, and much observational and interview data were obtained and analysed to develop categories which were described and inter-related in a model of children's responsiveness to spatial problem-solving experiences. The model and the details of children's thinking were related to literatures on visual imagery, selective attention, representation, and concept construction.

The purpose of the present study is (1) to investigate factors that influence mathematical problem solving competence for elementary students, (2) to understand the current studies regarding the development of mathematical problem solving competence, and (3) to probe background factors that affect the development of mathematical problem solving competence. The subjects of the study included 710 fifth-graders in Tainan city. The surveys of Thinking Style Inventory, Mathematical Learning Perception Check List as well as Mathematical Problem Solving Competence Test were used as instruments for data collection. A total of 710 questionnaires were delivered and 587valid questionnaires were collected, with fairly high 82.60% return rate. The collected data was tested with descriptive analysis, independent t test¡BANOVA¡Bproduct-moment correlation coefficient,multiple correlation and multiple regression. Based on the data analysis, the six findings of this study are summarized as follows: 1. The low satisfaction with mathematics class was revealed from the analysis of students¡¦ Mathematical Learning Perception Check List. It is suggested boosting subjects¡¦ satisfaction with the mathematics class will enhance the development of mathematical problem solving competence. 2. The positive correlation between administration style and mathematical problem solving competence was shown eminently among all types of thinking styles. The result indicated different function of the thinking styles influenced the development of mathematical problem solving competence in a varied degree. 3. From the analysis of students¡¦ background factor and mathematics problem solving competence, the statistic indicated the length of extra curriculum students devoted to does not affect their mathematical problem solving competence. The factors that influence students¡¦ mathematical problem solving competence the most were shown in the following order: administration district, the social status of father, the social status of mother, gender and the size of school. 4. The comparative variance of the mathematics learning achievement and mathematics problem solving competence was 24.3%. It implied the two influences each other. Students with low mathematical learning achievement show low mathematical problem solving competence and vice versa. 5. When predicting students¡¦ development of mathematical problem solving competence via the data of parents¡¦ social status and mathematical learning perception check list, the result showed the prediction via parents¡¦ social status is less significant. Yet the prediction via mathematical learning perception check list gained the highest variance ratio in this case. 6. In terms of the distribution of parents¡¦ social status, East, North and Middle East were of eminent as compared to South, An-Ping and An-Nan district in Tainan city. The finding implied parents¡¦ social status was a major factor that influence students¡¦ mathematics problem solving ability in administration district, as the £b2 ¡]Eta Squared¡^¡×25.3¢H shown in this study.

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.

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.

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.