The purpose of this research is to use Schoenfeld¡¦s mathematics problem-solving model to analyze processes, strategies, and affective characteristics of children in a gifted primary program, and then, to propose concrete suggestions for gifted class and general class teachers. Participants were six third-grade gifted children who were great in articulation, and enrolled in one primary school in Kaohsiung. The investigator analyzed think-aloud protocols of them who solved four non-routine problems selected by several expert teachers. The findings of this study were three. First, all six gifted students' thought processes mostly conformed to Schoenfeld¡¦s problem-solving model, though with various differences by individuals, and by problems. One of them provided two correct answers, having no verification stage in all problems. And one only provided one correct answer, had less analysis, exploration, design, and verification stage in solving all problems. Second, children exhibited diversified and flexible strategies. They used representing, drawing figures, working backward, introducing auxiliary element, and attempting mistakes to solve four non-routine mathematical problems. Last, the affective characteristics of students were positive. They were patient and perseverant and showed personal mathematics curiosity, excitement, and confidence, which were given as creative characteristics by Sternberg, and as mathematical talent or characteristics by Krutetskii. The investigator concluded that not all gifted students possessed meta-cognition ability: including exploration, design, and verification. The gifted class teachers could use non-routine mathematics problems to discipline students' meta-cognitive ability, including exploration, design, and verification, and encourage them to generate more solving strategies by group discussion in class. Finally, the general class teachers could adopt problem-solving characteristics of gifted students as materials for gifted students and general students to learn together in class.

The purpose of this study was to investigate the prospective elementary mathematics teachers&rsquo / use of strategies and their achievement levels in solving mathematical problems with respect to year level. The data were collected from 250 prospective elementary mathematics teachers enrolled in an elementary mathematics education program from a state university in Central Anatolian Region. Problem Solving Test (PST) was used to accomplish the purpose of the study. The data collection tool adapted by the researcher included nine open ended problems. In this study, item based in-depth analysis was employed to determine a variety of problem solving strategies used by prospective teachers.The frequencies and percentages of categories were gathered for each item and for each year level. The results of this study revealed that prospective elementary mathematics teachers&rsquo / problem solving achievement was moderately high. Prospective elementary mathematics teachers in each year level were able to use various problem solving strategies to a certain extent. More specifically, the results indicated that &lsquo / making a drawing&rsquo / and &lsquo / intelligent guessing and testing&rsquo / strategies were among the most prominent strategies frequently used by prospective teachers. Setting up an equation and using a formula was other strategies used by prospective teachers. On the other hand, finding a pattern strategy was the least frequent strategy used by prospective teachers.

My thesis deals with various methods of solving the correlation of~the~diametres of four mutually tangent circles. It also deals with the history of~the~derivation of mathematical property. The didactic part contains a book of solved tasks. It involves the topic of Apollony fractals, gothic vaults, examples of~mutually tangent circles and set of exercises for practising different solutions of the various cases.

This thesis provides some new insight into children’s strategies and behaviours relating to problem solving. Problem solving is one of the main aims in the renewed mathematics National Curriculum 2014 and has appeared in the Using and Applying strands of previous National Curriculums. A review of the literature provided some analysis of the types of published problem solving activities and attempted to construct a definition of problem solving activities. The literature review also demonstrated this study’s relevance. It is embedded in the fact that at the time of this study there was very little current research on problem solving and in particular practitioner research. This research was conducted through practitioner research in a focus institution. The motivation for this research was, centred round the curiosity as to whether the children (Year Six, aged 10 -11 years old) in the focus institution could apply their mathematics to problem solving activities. There was some concern that these children were learning mathematics in such a way as to pass examinations and were not appreciating the subject. A case study approach was adopted using in-depth observations in one focus institution. The observations of a sample of Year Six children engaged in mathematical problem solving activities generated rich data in the form of audio, video recordings, field notes and work samples. The data was analysed using the method of thematic analysis utilising Nvivo 10 to code the data. These codes were further condensed to final overarching themes. Further discussion of the data shows both mathematical and non-mathematical overarching themes. These themes are discussed in more depth within this study. It is hoped that this study provides some new insights into children’s strategies and behaviours relating to problem solving in mathematics.

The purpose of this study was to investigate the effect of teachers attending Cognitively Guided Instruction (CGI) professional development on students' problem solving strategies and the effect of students' use of strategies on their mathematics achievement as measured by a standardized test. First, the study analyzed the differences in students' use of strategies between treatment and control groups. The treatment was CGI professional development, and the teachers in the treatment group attended CGI workshops whereas the teachers in the control group did not. Next, the study analyzed the differences in the mathematics achievement of students between different strategy groups. A student posttest, which was ITBS (Math Problems and Math Computation), was used to compare students' mathematics achievement. A student pretest was used as a covariate. The results of this study showed that there were statistically significant differences in the students' use of strategies between the treatment and control groups at the second grade level. A greater percentage of treatment students used derived facts / recall strategies (the most advanced strategy for single-digit addition and subtraction) than control students did. The results related to the effect of students' use of strategies on their mathematics achievement showed that the students who used derived facst/recall strategies for single-digit problems had significantly higher mathematics achievement than students who used counting or concrete modeling strategies. Furthermore, the students who used invented algorithms for multi-digit problems had significantly higher mathematics achievement than the students who used standard algorithms.

The subject of this thesis is the identification of strategies, which are used by lower-secondary school students when solving problem tasks in chemistry. The strategies were identified during talks with ninth-grade students. The talks conducted by using the Think-aloud method were connected with solution of selected problem tasks. The ascertained strategies were classified as expansive strategies (such strategy can be used to solve more types of problems) and limiting strategies (such strategy can be used to solve easy task, but they can fail when solving more difficult tasks). Furthermore, reader's strategies, which help students to understand the tasks were separately identified. Also, other problems that students had to face when solving the tasks were identified. To identify problematic elements a collection of problem tasks named Metodické komentáře a úlohy ke standardům pro základní vzdělávání - chemie (Methodical comments and tasks for educational standards for elementary education - chemistry) were used. The results showed that when students solved the tasks, which were larger and more difficult, they often used reading strategies, which consist in multiple reading and reading aloud. Some of the expansive strategies the students used consisted in analogous deducing and logical reasoning....

The purpose of this study was to investigate the problem solving strategies, problem solving episodes, and metacognition of five Turkish students just graduated from elementary school and explore the interplay of these factors on their problem solving success in mathematics. The research data had been collected by clinical interviews and a self monitoring questionnaire followed by the interviews. Ten mathematical problems that participant students had worked on were selected among the released mathematical literacy items used in Programme for International Student Assessment (PISA) 2003. The problem solving strategies used by participants were coded according to the descriptions given by Posamentier &amp / Krulik (1999). The cognitive-metacognitive problem solving framework developed by Artzt and Armour-Thomas (1992) has been used to observe the problem solving episodes of the participants. The coding system developed by Pappas et al. (2003) has been utilized to examine the major components of metacognition (mistake recognition, adaptability, awareness and expression of thought) of the participants. The self-monitoring questionnaire responses were analyzed to crosscheck the results obtained from the clinical interviews. The problem solving behaviors of the participants observed in the study confirmed their academic success levels. The study confirmed that the problem solving success is too complex to be clarified by a unique property or a behavior of the problem solver. The problem solving requires overcoming various obstacles to reach a successful result. Hence, not only the students should have the required mathematical knowledge and a good repertoire of different problem solving strategies, but also they should know when and how to use those strategies, and also they could monitor and regulate their problem solving processes using their metacognitive skills. So mathematics teachers should provide problems that require different problem solving strategies and encourage the students to explore new strategies, to take risks in trying and to discuss failures and successes with peers and teacher.

This work contains reasearch of the procedures and strategies used for mathematical problems of motion by high school students, as well as analysis of student´s problem solving. This part if followed by a compilation of problems in order to practice solving of the motion problems. This compilation is divided into several parts, while each of the parts focuses on different types of problems.

Syftet med studien är att undersöka huruvida elever använder problemlösningsstrategier under programmering i programmet ScratchJr. Forskningsfrågorna utgår ifrån hur eleverna använder sig av samt uttrycker sig om sina egna problemlösningsstrategier. Metoderna som använts för att besvara forskningsfrågorna har varit en triangulering mellan deltagande observation och intervju. Den deltagande observationen har bestått av ett lektionsupplägg. 40 elever från tre olika skolor har deltagit i studien genom observation och därefter intervjuer med samma elever om den lösning som framkommit. Det resultat som framkommit har varit att samtliga elever använt sig av problemlösning genom olika strategier som att prova sig fram, samarbeta eller komma ihåg hur de tidigare gjort. De allra flesta elever kunde uttrycka sig på vilket sätt de löst uppgiften under intervjun. Studiens resultat visar att programmering är en viktig del av undervisningen. Elever använder sin individuella problemlösningsförmåga och kan då bli medvetna om den. Programmering ska också ses som en del av det lustfyllda lärandet. / The purpose of the study is to investigate whether pupils use problem-solving strategies during programming in the ScratchJr program. The research questions are based on how the pupils make use of and express themselves about their own problem-solving strategies. The methods used to answer the research questions have been a triangulation between participant observation and interview. The participatory observation consisted of a lesson plan. 40 students from three different schools participated in the study through observation and then interviews with the same pupils about the solution that has emerged. The result that has emerged has been that all students have used problem solving through different strategies such as trying to develop, collaborate or remember how they did before. Most pupils were able to express how they solved the task during the interview. The study's results show that programming is an important part of teaching. Pupils use their individual problem-solving ability and can then become aware of it. Programming should also be part of the fun-filled learning.

Studien undersöker hur lärare för årskurs F-3 undervisar och ger återkoppling inom matematikundervisningen. För att undersöka detta användes semistrukturerade intervjuer med 8 lärare. Lärarna fick i början av intervjun ge återkoppling på två elevlösningar på ett matematiskt problem, en korrekt och en inkorrekt. När lärarna givit återkoppling på de båda elevlösningarna ställdes frågor för att komplettera materialet. Resultatet visar att några av lärarna väljer att tilldela strategier, medan majoriteten väljer att inte tilldela strategier eftersom de anser att syftet med problemlösning försvinner. Vidare visar resultatet att lärarna riktade sin återkoppling mot den felaktiga lösningen och gav därför förslag på framåtsyftande återkoppling. Återkopplingen till den korrekta elevlösningen handlade istället om att eleven löst uppgiften. Slutligen pratade majoriteten av lärarna om undervisning för problemlösning, vilket innebär att de lär ut procedurer och begrepp innan eleverna blir tilldelade problemlösningsuppgifter. / The study examines how teachers in years F-3 instruct and give feedback when teaching mathematics. To examine this, semi structured interviews was conducted with 8 teachers. In the beginning of the interview the teachers were asked to give feedback on two student-solutions to a mathematical problem, one correct and one incorrect. When the teachers had given their feedback on both of the student solutions, they were asked questions to complement the material. The result shows that some of the techers choose to assign strategies, while the majority chose not to since they believe the purpose of problem-solving would disappear. The result further shows that the teachers aimed their feedback towards the incorrect solution and suggested ways to improve in the future. The feedback aimed at the correct solution instead focused on the fact that it was correct. Finally, the majority of the teachers spoke about education for problem-solving, which means they teach procedures and concepts before giving the students problem-solving assignments. / <p>Matematikdidaktik</p>