M.Ed. (Subject Didactics of Mathematics) / The phenomenon commonly known as 'mathematics anxiety' has been discovered by researchers endeavouring to determine the cause of problems with mathematics, the antagonism against it, and the deliberate avoidance of the subject. The way in which we view and experience mathematics has a radical influence on our decision-making. Mathematics anxiety is a dominant factor in the decisionmaking process of pupils in a wide spectrum of situations, and research has confirmed that mathematics anxiety invariably has a negative, hampering effect. The development of mathematics anxiety is a gradual process which can commence at any stage of a pupil's school career. By means of a questionnaire it has been established that the problem regarding mathematics anxiety stems primarily from the very nature and structure of the subject. Secondary factors such as the language of mathematics, textbooks, personality, society, the curriculum, evaluation, time limitation and problem solving have been identified empirically as variables in the development of mathematics anxiety. Since mathematics anxiety is an emotional and not an intellectual problem, it can be overcome. Early intervention can prevent the establishment of a negative attitude. Affordable individual tuition and more confidence are according to the responses received, the most important means by which mathematics anxiety can be reduced. It is essential to develop the mathematical reading skills of mathematics anxious pupils. Once skilled, pupils can fill the gaps in their pre-knowledge by using study manuals which are carefully structured for self-study purposes...

The purpose of this study is to investigate various solution strategies employed by Grade 7 learners and their teachers when solving a given set of mathematical tasks. This study is oriented in an interpretive paradigm and is characterised by qualitative methods. The research, set in nine schools in the Eastern Cape, was carried out with nine learners and their mathematics teachers and was designed around two phases. The research tools consisted of a set of 12 tasks that were modelled after the Third International Mathematics and Science Study (TIMSS), and a process of clinical interviews that interrogated the solution strategies that were used in solving the 12 tasks. Aspects of grounded theory were used in the analysis of the data. The study reveals that in most tasks, learners relied heavily on procedural understanding at the expense of conceptual understanding. It also emphasises that the solution strategies adopted by learners, particularly whole number operations, were consistent with those strategies used by their teachers. Both learners and teachers favoured using the traditional, standard algorithm strategies and appeared to have learned these algorithms in isolation from concepts, failing to relate them to understanding. Another important finding was that there was evidence to suggest that some learners and teachers did employ their own constructed solution strategies. They were able to make sense of the problems and to 'mathematize' effectively and reason mathematically. An interesting outcome of the study shows that participants were more proficient in solving word problems than mathematical computations. This is in contrast to existing research on word problems, where it is shown that teachers find them difficult to teach and learners find them difficult to understand. The findings of this study also highlight issues for mathematics teachers to consider when dealing with computations and word problems involving number sense and other problem solving type problems.

The aim of the present study was to investigate the effect of metacognition-based teaching and teaching mathematics via problem solving on students' understanding of mathematics, and the ways in which the students' beliefs about themselves as doers and learners of mathematics and about mathematics and mathematical problem solving were influenced by the instruction. The 60 hours of instruction occurred in the context of a day-to-day mathematics course for undergraduate non-science students, and that gave mea chance to teach mathematics via problem solving. Metacognitive strategies that were included in the instruction contributed to the students' mathematical learning in various ways. The instruction used journal writing, small groups, and whole-class discussions as three different but interrelated strategies that focused on metacognition. Data for the study were collected through four different sources, namely quizzes and assignments (including the final exam), interviews, the instructor's and the students' autobiographies and journals, and class observations (field notes, audio and video tapes). Journal writing served as a communication channel between the students: and the-instructor, and as a result facilitated the individualization of instruction. Journal writing provided the opportunity for the students to clarify their thinking and become more reflective. Small groups proved to be an essential component of the instruction. The students learned to assess and monitor their work and to make appropriate decisions by working cooperatively and discussing the problems with each other. Whole-class discussions raised the students' awareness about their strengths and weaknesses. The discussions also helped students to a great extent become better decision makers. Three categories of students labeled traditionalists, incrementalists, and innovators, emerged from the study. Nine students, who rejected the new approach to teaching and learning mathematics were categorized as traditionalists. The traditionalists liked to be told what to do by the teacher. However, they liked working in small groups and using manipulative materials. The twelve incrementalists were characterized as those who propose to have balanced instruction in which journal writing was a worthwhile activity, group work was a requirement, and whole-class discussions were preferred for clarifying concepts and problems more than for generating and developing new ideas. The nineteen other students were categorized as innovators, those who welcomed the new approach and utilized it and preferred it. For them, journal writing played a major role in enhancing and communicating the ideas. Working in small groups seemed inevitable, and whole-class discussions were a necessity to help them with the meaning-making processes. The incrementalists and the innovators gradually changed their beliefs about mathematics from viewing it as objective, boring, lifeless, and unrelated to their real-lives, to seeing it as subjective, fun, meaningful, and connected to their day-to-day living. The findings of the study further indicated that most of the incrementalists and the innovators changed their views about mathematical problem solving from seeing it as the application of certain rules and formulas to viewing it as a meaning-making process of creation and construction of knowledge. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This study investigated whether critical thinking can be fostered in home economics through teaching a problem solving approach in Family Management. Secondarily, it investigated teacher behaviours which may foster critical thinking abilities, the moral and ethical issues which the teaching of critical thinking addresses, and whether the students were able to use problem solving in real life situations. The research involved the students and teacher in a Family Management eleven class in rural British Columbia. All students in the class chose to participate in the study. The study was conducted during twenty-six classroom hours. The study used action research as the research methodology. The research included action/research cycles with time between for analysis and reflection. The phase of data analysis and reflection was called the reconnaissance. Data was collected through audio tapes of the classes, entries in the teacher's journal, a checklist, and collected student work. The data collected in the first reconnaissance phase established a description which served as a point of reference for comparing and analyzing later observations. Two cycles of action/research followed. Observations were made and data collected as the critical thinking concepts were introduced. The introduction of the macro-thinking skill of problem solving was combined with the micro- thinking skills of avoiding fallacies, observing, reporting and summarizing. The research found that there was an increase in critical thinking activities at the end of the study. Factors that were found to have effected this change were: the teaching of a problem solving process, the teaching of micro-thinking skills, certain teacher behaviours, and the classroom atmosphere. Home economics was found to play a unique role in providing practice in real life problem solving. Further research is needed to determine if the skills the students learned while problem solving in Family Management will carry over to everyday life. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

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The paper documents the need for an effective technology to teach problem-solving. It asserts that a behaviorological analysis of problem-solving can speed the development of an effective technology to teach problem-solving behavior. A behaviorological definition of problem-solving is proposed. The history of behaviorological approaches to problem-solving is traced and suggestions are offered that may facilitate further empirical and theoretical work. One application of a behaviorological analysis to the teaching of problem-solving is illustrated by some preliminary data on the effectiveness of a technique for teaching a type of problem-solving behavior. Suggestions for further research are provided.

This dissertation examines the response of a group of educators to a state mandate to integrate computational thinking (CT) into all levels of the curriculum. It explores the historical development of CT and its significance within the broader context of Science, Technology, Engineering, and Mathematics education, emphasizing the rapid growth and evolving nature of this interdisciplinary field. By examining the challenges and potential strategies for incorporating CT into K-12 curricula, the research highlights the critical role of school leadership in navigating the complexities associated with this integration. Utilizing Everett Rogers’s Diffusion of Innovation theory, the dissertation explores how new knowledge is integrated into schools and examines the pivotal role of educational leaders in steering this endeavor. A mixed-methods research design was used to gather the attitudes and perceptions of school leaders toward CT, identifying key factors that influence the adoption and implementation of CT in schools. The study reveals that leadership awareness, involvement, and support are pivotal in overcoming obstacles to CT integration. It also underscores the importance of developing a shared understanding of CT among educators and administrators, aligning CT initiatives with school priorities, and providing adequate resources and professional development opportunities to ensure effective implementation. The findings of the dissertation offer valuable insights for policymakers, educators, and educational leaders, suggesting that a comprehensive approach to integrating CT into K-12 education requires strategic planning, collaboration, and sustained support. By addressing the gaps in current research and practice, this dissertation contributes to the discourse on effective strategies for embedding CT within the educational curriculum, with the goal of enhancing students’ preparedness for an increasingly computational world. This research sheds light on the challenges and opportunities of CT integration and contributes to the development of a roadmap for future efforts to integrate new bodies of knowledge into the K-12 curriculum.

Implementation of Oregon’s Educational Reform Act (HB 3565 and HB 2991) provides the context for this inquiry as its emphasis on problem solving has impacted mathematics teaching and learning throughout the state. Even though all Oregon teachers are responding to the same policy, their goals in teaching problem solving vary. These goals and these practices are influenced by the way teachers view the role of problem solving in the curriculum. Further, their practice is influenced by their knowledge and beliefs about mathematics content, teaching, learning, and the reform policy. The questions addressed in this study are: (1) What do exemplary middle school math teachers do to engage students in mathematical problem solving? and (2) On what bases do these teachers make decisions about what to emphasize when teaching problem solving? how to teach problem solving?, and when to teach problem solving? This qualitative study provides a fuller description of Standards-based classroom practice than presently represented in the literature by offering both examples of problem solving practice and the related influences on that practice. It considers the influences of policy, curriculum, professional development, administrators, and colleagues on teachers’ developing practice. The study also grounds the implementation of the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) in the work of practicing middle school teachers. Finally, the study shows how, for these teachers, their curriculum has played a significant role in developing their perspectives on learning, teaching, and the nature of math, which has in turn, influenced their knowledge, beliefs, and instructional practice. This study demonstrates that teachers are able to teach in ways consistent with the NCTM Standards when their knowledge and beliefs about practice align with the recommendations. Further, they teach in this manner when professional development experiences are geared toward understanding and developing Standards-based instructional practice, curriculum is consistent with this vision of practice, and administrators and school cultures are supportive of such practice. When these internal and external conditions exist within and for teachers, their students have the opportunity to learn to become “problem solvers,” not just “problem performers.”

It is common for students to make mistakes while solving mathematical problems. Some of these mistakes might be caused by the false ideas, or misconceptions, that students developed during their learning or from their practice. Calculus courses at the undergraduate level are mandatory for several majors. The introductory course of calculus—Calculus I—requires fundamental skills. Such skills can prepare a student for higher-level calculus courses, additional higher-division mathematics courses, and/or related disciplines that require comprehensive understanding of calculus concepts. Nevertheless, conceptual misunderstandings of undergraduate students exist universally in learning calculus. Understanding the nature of and reasons for how and why students developed their conceptual misunderstandings—misconceptions—can assist a calculus educator in implementing effective strategies to help students recognize or correct their misconceptions. For this purpose, the current study was designed to examine students’ misconceptions in order to explore the nature of and reasons for how and why they developed their misconceptions through their thought process. The study instrument—Calculus Problem-Solving Tasks (CPSTs)—was originally created for understanding the issues that students had in learning calculus concepts; it features a set of 17 open-ended, non-routine calculus problem-solving tasks that check students’ conceptual understanding. The content focus of these tasks was pertinent to the issues undergraduate students encounter in learning the function concept and the concepts of limit, tangent, and differentiation that scholars have subsequently addressed. Semi-structured interviews with 13 mathematics college faculty were conducted to verify content validity of CPSTs and to identify misconceptions a student might exhibit when solving these tasks. The interview results were analyzed using a standard qualitative coding methodology. The instrument was finalized and developed based on faculty’s perspectives about misconceptions for each problem presented in the CPSTs. The researcher used a qualitative methodology to design the research and a purposive sampling technique to select participants for the study. The qualitative means were helpful in collecting three sets of data: one from the semi-structured college faculty interviews; one from students’ explanations to their solutions; and the other one from semi-structured student interviews. In addition, the researcher administered two surveys (Faculty Demographic Survey for college faculty participants and Student Demographic Survey for student participants) to learn about participants’ background information and used that as evidence of the qualitative data’s reliability. The semantic analysis techniques allowed the researcher to analyze descriptions of faculty’s and students’ explanations for their solutions. Bar graphs and frequency distribution tables were presented to identify students who incorrectly solved each problem in the CPSTs. Seventeen undergraduate students from one northeastern university who had taken the first course of calculus at the undergraduate level solved the CPSTs. Students’ solutions were labeled according to three categories: CA (correct answer), ICA (incorrect answer), and NA (no answer); the researcher organized these categories using bar graphs and frequency distribution tables. The explanations students provided in their solutions were analyzed to isolate misconceptions from mistakes; then the analysis results were used to develop student interview questions and to justify selection of students for interviews. All participants exhibited some misconceptions and substantial mistakes other than misconceptions in their solutions and were invited to be interviewed. Five out of the 17 participants who majored in mathematics participated in individual semi-structured interviews. The analysis of the interview data served to confirm their misconceptions and identify their thought process in problem solving. Coding analysis was used to develop theories associated with the results from both college faculty and student interviews as well as the explanations students gave in solving problems. The coding was done in three stages: the first, or initial coding, identified the mistakes; the second, or focused coding, separated misconceptions from mistakes; and the third elucidated students’ thought processes to trace their cognitive obstacles in problem solving. Regarding analysis of student interviews, common patterns from students’ cognitive conflicts in problem solving were derived semantically from their thought process to explain how and why students developed the misconceptions that underlay their mistakes. The nature of how students solved problems and the reasons for their misconceptions were self-directed and controlled by their memories of concept images and algorithmic procedures. Students seemed to lack conceptual understanding of the calculus concepts discussed in the current study in that they solved conceptual problems as they would solve procedural problems by relying on fallacious memorization and familiarity. Meanwhile, students have not mastered the basic capacity to generalize and abstract; a majority of them failed to translate the semantics and transliterate mathematical notations within the problem context and were unable to synthesize the information appropriately to solve problems.

Thesis (PhD)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: The globalisation of business is probably the most important topic of discussion in business around the world. Globalisation impacts on business processes, and changes in the processes of economic and political liberalisation signify a radical shift in thinking about how the world works and how it should be organised. The process of change consequently means having to deal with increasing complexity. It is the complexity of change that prompts the urgency to improve the thinking of management in an organisation. As adult learners, managers need different competencies to operate; in addition their thinking skills need to be developed, as they are the key drivers in an organisation going through transformation. In a needs' assessment in a financial services organisation operating in the global environment, a need for training of middle managers to help them improve their thinking skills in order to become more effective thinkers was determined. The need to train adults to develop cognitive skills prompted a specific thinking skills intervention and the facilitation thereof. A Thinking Skills Programme aimed at teaching thinking dispositions and thinking skills was implemented as a programme to teach the managers to become effective thinkers. The study was approached from a programme evaluative perspective. The purpose of the programme was an improvement-orientated evaluation and was designed to evaluate the outcomes of the programme. Qualitative data was collected through a pre- assessment and postassessment process. Open-ended questionnaires as well as other qualitative methods were used in the assessment to capture the data. The data analysis was done through content analysis. In the process of determining the patterns and processes in the preassessment, observations, semi-structured interviews, field notes and post-assessment, the researcher looked for themes or interconnections that emerged in the units, sub-categories and categories. The sub-categories that emerged were derived from the frequency of appearance in the answers of the participants in the questionnaires. The post-assessment feedback categories indicated that a shift in effective thinking had taken place in the participants. The most significant difference in their thinking was the awareness of their thinking. They reflected on their thinking while solving problems, which is evidence of effective thinking. They used the thinking tools to help them solve problems. The intervention influenced the participants to be more creative in solving problems, which was not evident in the pre-assessment. The evaluation of the outcome of the programme through the application of a Thinking Skills Programme was successful. This was demonstrated by the fact that in the measurement of the outcome of the programme it was evident that the middle managers had become more aware of their thinking and that they applied the thinking dispositions and thinking tools in their daily managerial activities. They had become more effective thinkers. / AFRIKAANSE OPSOMMING: Die globalisering van die sakewêreld is waarskynlik die belangrikste besprekingspunt in dié sektor regoor die wêreld. Globalisering het 'n impak op sakeprosesse, en veranderings in die prosesse van ekonomiese en politieke liberalisering dui op 'n radikale verandering in denke oor hoe die wêreld werk en hoe dit georganiseer behoort te word. Die proses van verandering dui gevolglik daarop dat daar rekening gehou moet word met toenemende kompleksiteit. Dit is die kompleksiteit van verandering wat aanleiding gee tot die belangrikheid daarvan om die denkvaardighede van die bestuur in 'n organisasie te verbeter. As volwasse leerders benodig bestuurders verskillende vaardighede om hulle taak te verrig, en hulle denkvaardighede moet ontwikkel word aangesien hulle die dryfkrag is van 'n organisasie wat transformasie ondergaan. In 'n behoeftebepaling in 'n finansiële dienste organisasie wat op die wêreldmark meeding, is die behoefte geïdentifiseer aan opleiding vir middelvlak bestuurders om hulle denkvaardighede te help verbeter, ten einde van hulle meer effektiewe denkers te maak. Die behoefte om volwassenes op te lei in die ontwikkeling van kognitiewe vaardighede het aanleiding gegee tot In spesifieke denkvaardigheidsintervensie en die fasilitering daarvan. 'n Denkvaardigheidsprogram gemik op die onderrig van denkdisposisies en -vaardighede is as 'n program ingestelom bestuurders op te lei om effektiewe denkers te word. Die studie is benader vanuit 'n programevaluasie-perspektief. Die doel van die program was 'n verbeteringsgeoriënteerde evaluasie en dit is ontwerp om die uitkomstes van die program te evalueer. Kwalitatiewe data is versamel deur middel van 'n pre- en postassesseringsproses. Oopeindevraelyste asook ander kwalitatiewe metodes is in die assessering gebruik om data te versamel. Die data-analise is gedoen deur inhoudsanalise. Vir die bepaling van die patrone en prosesse in die pre-assessering, is observasies, semigestruktureerde onderhoude en veldaantekeninge gebruik. Vir die postassessering het die navorser gesoek na temas of interkonneksies wat na vore gekom het in die eenhede, subkategorieë en kategorieë. Die subkategorieë wat na vore gekom het, is geïdentifiseer op grond van die frekwensie van voorkoms in die antwoorde van die deelnemers wat die vraelyste voltooi het. Die terugvoeringskategorieë in die postassessering het aangetoon dat 'n verandering in effektiewe denke by die deelnemers plaasgevind het. Die mees beduidende verandering in hulle denke was die bewustheid van hulle denke. Deelnemers het gereflekteer op hulle denke tydens probleemoplossing, wat as bewys dien van effektiewe denke. Hulle het die denkinstrumente gebruik om hulle te help om probleme op te los. Die intervensie het tot gevolg gehad dat die deelnemers meer kreatief tydens probleemoplossing was, wat nie tydens die pre-assessering geblyk het nie. Die evaluering van die uitkoms van die program deur die toepassing van 'n Denkvaardigheidsprogram was suksesvol. Dit is gedemonstreer deur die feit dat in die meting van die uitkoms van die program dit duidelik geblyk het dat die middelvlak bestuurders meer bewus geraak het van hulle denke en dat hulle die denkdisposisies en denkinstrumente in hulle daaglikse bestuursaktiwiteite toegepas het. Hulle het meer effektiewe denkers geword.