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.

By manipulating constraints and goals, this study had generated some ill-posed problems in "Fractions" which were packed into 2 mathematical tasks for teacher uses in an intended exploration of their perceived effectiveness of teaching mathematical problem-solving against their student responses through the lens of the theory of formative assessment. Each ill-posed problem was characterized by certain descriptive "instability" that users would have to define own sets of mathematical assumptions for problem-solving inquiries. 3 highly qualified, experienced, and trained mathematics teachers were purposefully recruited, and instructed to acquire and mark student responses without any prior teaching and intervention. Each of these teachers' perceptions of ill-posed problems was acquired through a semi-structured clinical case-interview. All teachers in common demonstrated only individual singular mathematical problem-solving inquiries as major instructional adjustments during evaluation, even though individuals had ample opportunities in manipulating the described intention of each problem. Although some could realize inquiries from students being alternative to own used, not all would intend to change initial instructional plans of each problem and could design dedicated tasks in extending given problem-solving contexts for subsequent teaching and maintaining the described problem-solving intentions merely because of evaluation purposes. The resulting thick teacher perceptions were then analyzed by the Mayring's (2015) Qualitative Content Analysis (QCA) method for exploring particularly those who could intend to influence and get influenced by students' used mathematical assumptions in interviews. Certain unanticipated uses of assumptions of student individuals and groups were evidently found to have influenced cognitively some teachers' further problem-solving inquiries at some interview instants and stimulated their perception changes. In the lack of subject implementation in mathematics education for the theory of "formative" assessment (Black & Wiliam, 2009), based on its definition, these instants should be put as their potential creations of and/or capitalizations upon certain asynchronous moments of contingency according to their planning of instructional adjustments for more comprehensive learning and definite growths of mathematical inquiries of students according to individuals' needs of problem-solving. Due to QCA, these perception changes might be characterized by four certain inductively formed categories of scenarios of perceptions, which were summarized as 1) Evaluation Perception, 2) Assumption Expansion Perception, 3) Assumption Collection Perception, and 4) Intention Indecision Perception. These scenarios of perceptions might be used to explore teachers' intentions, actions, and coherency in accounting for students' used assumptions in mathematical inquiries for given problem-solving contexts and extensions of given intentions of mathematical inquiries, particularly in their designs of mathematical tasks. Teacher uses of ill-posed problems were shown to have provided certain evidences in implementing formative assessments which should substantiate a subject implementation of its theory in the discipline of mathematics education. Methodologically, the current study also substantiate how theory-guided designs of ill-posed problems as well as generic plain text analysis through QCA have facilitate effectiveness comparisons of instructional adjustments within a teacher, across different teachers, decided prior knowledge, students of prior mathematical learning experiences, and students in different levels of schooling and class size.

The dissertation thesis deals with mathematical problem-solving at lower secon- dary level, as viewed from the perspective of heuristic strategies. The aim of the thesis is to comprehensibly summarize the results of research which began in 2012 and runs until now. The results concern both with theoretical and empirical parts of our research. This research study was conducted in fifteen lower secondary and upper secondary classes. Three dimensional classification of use of heuristic strategies and the structure of heuristic strategies' characteristics were developed by the author, and these constructs are presented in this work. The theory of mathematical problem and mathematical problem solving method is an integral part of this thesis too. Furthermore, the author presents a summary of all strategies used in the experiments; each strategy is fully described and illustrated by an appropriate example. The results of several short-term research studies (three months) and a longitudinal research study (sixteen months) are analysed in the empirical part of the thesis. This part also strives to find answers to several research questions, e.g.: " Could certain strategies be taught in a short-term period (three months)?", " Which strategies are suitable for an average pupil?" or " Are the pupils able to...

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)

Problemlösning genomsyrar hela läroplanen och är en viktig del av matematik-undervisningen i skolan (Skolverket, 2011a). Att lösa problem kommer naturligt för barn och det är lärarens uppgift att ta vara på den förmågan och hjälpa elever att bli effektiva problemlösare. Förmågan att lösa problem är en viktig kunskap som varje elev har fördel av att kunna (Lester, 1996). Studiens syfte är att undersöka variationen av problemlösnings-strategier som elever använder samt undersöka hur eleverna resonerar när de löser ett problem. 39 elever från två olika skolor i England och Sverige fick lösa ett matematiskt problem och sedan intervjuades 12 av dem med olika lösningsstrategier. Resultatet visade att eleverna använde sig av fyra olika kategorier av lösningsstrategier. De olika kategorierna var: lösningsstrategi genom addition, lösningsstrategi genom addition och subtraktion, lösningsstrategi genom att gissa och resonera, och lösningsstrategi genom att söka mönster. Det fanns även en grupp elever som inte hade någon utläsbar lösningsstrategi. Slutsatsen av studien är att elever behöver explicit undervisning i problemlösning för att till fullo kunna behärska den. / Problemlösning genomsyrar hela läroplanen och är en viktig del av matematik-undervisningen i skolan (Skolverket, 2011a). Att lösa problem kommer naturligt för barn och det är lärarens uppgift att ta vara på den förmågan och hjälpa elever att bli effektiva problemlösare. Förmågan att lösa problem är en viktig kunskap som varje elev har fördel av att kunna (Lester, 1996). Studiens syfte är att undersöka variationen av problemlösnings-strategier som elever använder samt undersöka hur eleverna resonerar när de löser ett problem. 39 elever från två olika skolor i England och Sverige fick lösa ett matematiskt problem och sedan intervjuades 12 av dem med olika lösningsstrategier. Resultatet visade att eleverna använde sig av fyra olika kategorier av lösningsstrategier. De olika kategorierna var: lösningsstrategi genom addition, lösningsstrategi genom addition och subtraktion, lösningsstrategi genom att gissa och resonera, och lösningsstrategi genom att söka mönster. Det fanns även en grupp elever som inte hade någon utläsbar lösningsstrategi. Slutsatsen av studien är att elever behöver explicit undervisning i problemlösning för att till fullo kunna behärska den. Problem solving permeates the Swedish national curriculum and it is an important part of mathematics education (Skolverket, 2011a). To solve problems comes naturally to children and it is the teacher’s task to harvest this ability and help pupils to be effective problem solvers. The ability to solve problems is an important knowledge and if known provides an advantage in life (Lester, 1996). The purpose of this study is to investigate the variation of problem solving strategies that pupil use and to investigate their mathematical reasoning while solving a mathematical problem. 39 pupils from two different schools in England and Sweden got to solve a mathematical problem and then 12 of them, which had different solution strategies, were selected for an interview. The result showed that the pupils used four categories or solving strategies. The categories were: finding a solution though addition, finding a solution though both addition and subtraction, finding a solution though guessing and reasoning and finding a solution though seeking patterns. There was also one group of pupils who did not have a distinguishable solution strategy. The conclusion of this study is that pupils need explicit teaching about problem solving to be able to fully master it.

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.

Pour résoudre un problème de mathématiques ou comprendre une démonstration, une figure bien choisie est parfois d’un grand secours. Ce fait souvent remarqué peut être vu comme un cas particulier d’un phénomène plus général. Utiliser une figure plutôt que des phrases, reformuler un problème sous la forme d’une équation, employer telles notations plutôt que telles autres : dans tous ces cas, en un sens, on ne fait que représenter sous une nouvelle forme ce qu’on sait déjà, et pourtant, cela peut permettre d’avancer. Comment est-ce possible ? Pour répondre à cette question, la première partie de cette thèse étudie ce qu’apporte un changement notationnel précis introduit par Leibniz à la fin du XVIIe siècle. La suite de ce travail analyse, et confronte à l’exemple précédent, plusieurs manières de penser les différences représentationnelles proposées dans la littérature philosophique récente. Herbert Simon, étudié dans la deuxième partie, s’appuie sur le modèle informatique des structures de données : deux représentations peuvent être « informationnellement » équivalentes, mais « computationnellement » différentes. Les logiciens Barwise et Etchemendy, étudiés dans la troisième partie, cherchent à élargir les concepts de la logique mathématique (en particulier ceux de syntaxe et de sémantique) aux diagrammes et figures. Enfin, certains philosophes des mathématiques contemporains, comme Kenneth Manders, remettent en cause la notion même de représentation, en soutenant qu’elle n’est pas éclairante pour comprendre l’usage de figures, formules ou autres supports externes en mathématiques. C’est à ces critiques qu’est consacrée la quatrième et dernière partie. / When solving a mathematical problem or reading a proof, drawing a well-chosen diagram may be very helpful. This well-known fact can be seen as an instance of a more general phenomenon. Using a diagram rather than sentences, reformulating a problem as an equation, choosing a particular notation rather than others : in all these cases, in a sense, we are only representing in a new form what we already knew; and yet, it can help us make progress. How is this possible? To address this question, the first part of this thesis explores the benefits afforded by a specific notational change introduced by Leibniz in the late seventeenth-century. The rest of this work analyses, and puts to the test of the preceding case study, several ways of understanding representational differences which have been put forward in the recent philosophical literature. Herbert Simon, studied in the second part, relies on a comparison with the notion of data structures in computer science: two representations, he writes, can be “informationally” equivalent yet “computationnally” different. The logicians Barwise and Etchemendy, studied in the third part, try to broaden the concepts of mathematical logic (in particular those of syntax and semantics) to cover diagrams and figures. Finally, some contemporary philosophers of mathematics, for instance Ken Manders, argue that the notion of representation itself is not helpful to understand the use of diagrams, formulas or other external reasoning tools in mathematics. Such arguments are the focus of the fourth (and last) part.

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.