This study investigates the effects of implementing mathematical problem solving instruction in a regular calculus course taught at the college level. Principles associated with this research are: i) mathematics is developed as a response to finding solutions to mathematical problems, ii) attention to the processes involved in solving mathematical problems helps students understand and develop mathematics, and iii) mathematics is learned in an active environment which involves the use of guesses, conjectures, examples, counterexamples, and cognitive and metacognitive strategies. Classroom activities included use of nonroutine problems, small group discussions, and cognitive and metacognitive strategies during instruction. Prior to the main study, in an extensive pilot study the means for gathering data were developed, including a student questionnaire, several assignments, two written tests, student task-based interviews, an interview with the instructor, and class observations. The analysis in the study utilized ideas from Schoenfeld (1985) in which categories, such as mathematical resources, cognitive and metacognitive strategies, and belief systems, are considered useful in analyzing the students' processes for solving problems. A model proposed by Perkins and Simmons (1988) involving four frames of knowledge (content, problem solving, epistemic, and inquiry) is used to analyze students' difficulties in learning mathematics. Results show that the students recognized the importance of reflecting on the processes involved while solving mathematical problems. There are indications suggesting that the students showed a disposition to participate in discussions that involve nonroutine mathematical problems. The students' work in the assignments reflected increasing awareness of the use of problem solving strategies as the course developed. Analysis of the students' task-based interviews suggests that the students' first attempts to solve a problem involved identifying familiar terms in the problem and making some calculations often without having a clear understanding of the problem. The lack of success led the students to reexamine the statement of the problem more carefully and seek more organized approaches. The students often spent much time exploring only one strategy and experienced difficulties in using alternatives. However, hints from the interviewer (including metacognitive questions) helped the students to consider other possibilities. Although the students recognized that it was important to check the solution of a problem, they mainly focused on whether there was an error in their calculations rather than reflecting on the sense of the solution. These results lead to the conclusion that it takes time for students to conceptualize problem solving strategies and use them on their own when asked to solve mathematical problems. The instructor planned to implement various learning activities in which the content could be introduced via problem solving. These activities required the students to participate and to spend significant time working on problems. Some students were initially reluctant to spend extra time reflecting on the problems and were more interested in receiving rules that they could use in examinations. Furthermore, student expectations, evaluation policies, and curriculum rigidity limited the implementation. Therefore, it is necessary to overcome some of the students' conceptualizations of what learning mathematics entails and to propose alternatives for the evaluation of their work that are more consistent with problem solving instruction. It is recommended that problem solving instruction include the participation or coordinated involvement of all course instructors, as the selection of problems for class discussions and for assignments is a task requiring time and discussion with colleagues. Periodic discussions of course directions are necessary to make and evaluate decisions that best fit the development of the course. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This project explored the Concrete to Representational to Abstract instruction (CRA instruction) as a strategy to teach abstract math concepts for secondary students with learning disabilities. Through the review of literature, multiple researchers suggested that students with learning disabilities need to be exposed to a variety of instructional strategies to develop problem solving skills in algebra concepts.

This study analyzed the mathematical problem solving behavior of mathematically gifted students. It focused on a specific fourth step of Polya's (1945) problem solving process, namely, looking back to find alternative approaches to solve the same problem. Specifically, this study explored problem solving using many different approaches. It examined the relationships between students' past mathematical experiences and the number of approaches and the kind of mathematics topics they used to solve three non-standard mathematics problems. It also analyzed the aesthetic of students' approaches from the perspective of expert mathematicians and the aesthetic of these experts' preferred approaches from the perspective of the students. Fifty-four students from a specialized high school were selected to participate in this study that began with the analysis of their past mathematical experiences by means of a preliminary survey. Nine of the 54 students took a test requiring them to solve three non-standard mathematics problems using many different approaches. A panel of three research mathematicians was consulted to evaluate the mathematical aesthetic of those approaches. Then, these nine students were interviewed. Also, all 54 students took a second survey to support inferences made while observing the problem solving behavior of the nine students. This study showed that students generally were not familiar with the practice of looking back. Indeed, students generally chose to supply only one workable, yet mechanistic approach as long as they obtained a correct answer to the problem. The findings of this study suggested that, to some extent, students' past mathematical experiences were connected with the number of approaches they used when solving non-standard mathematics problems. In particular, the findings revealed that students' most recent exposure of their then-AP Calculus course played an important role in their decisions on selecting approaches for solution. In addition, the findings showed that students' problem solving approaches were considered to be the least "beautiful" by the panel of experts and were often associated with standard approaches taught by secondary school mathematics teachers. The findings confirmed the results of previous studies that there is no direct connection between the experts' and students' views of "beauty" in mathematics.

The purpose of this qualitative study was to investigate the thought processes of secondary level novice programmers engaged in computer programming for the purpose of generating hypotheses for consideration in future research on the relationship between computer programming and problem solving. A high school BASIC programming course with 14 students from a single school in the tenth through the twelfth grades was selected for the sample. Data describing students' thought processes while programming were collected during double periods in the 11th and 16th weeks of the fall semester. Students worked in role-assigned partnerships, wherein one student was the problem solver and the other was the recorder. The problem solver's task was to solve the problem using a "think aloud" strategy, while the recorder took notes describing the problem solver's actions to assure that audiotape recordings of the problem solver's voice were maintained. Following the solution of one problem, these roles were switched. Analysis of novice programmers' thought processes revealed two categories of student problem solution strategies: coded thinking and debugging. In the coded thinking strategy, students approached the problems primarily from the perspective of BASIC codes. This strategy was similar in nature to activities involved in verbal association learning, a low level thinking strategy identified by Gagne (1970). Students relied on two techniques for debugging syntax and logic errors. They applied a guess-and-check technique to correct syntax errors or asked the teacher for assistance. Similarly, when logic errors were revealed, the subjects typically asked the teacher for assistance and then used the guess-and-check technique to correct the errors. Both techniques utilized lower level thought processes than that required for problem solving learning. Analysis of the subject programming processes revealed that problem solving processes, as identified by Polya (1988), were not involved. Future research should examine students thought processes when working with a compiled language such as Pascal. In addition, future research should investigate the thought processes of students who have had more experience than a single term of programming. A case study of from two to three students explored over a longer period of time may provide a clearer description of student thought processes. / Graduation date: 1993

published_or_final_version / Education / Master / Master of Education

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.

This study explores how Lesotho primary school teachers understand and interpret problem-solving (PS) and how they teach and support it. Observation schedules and semi-structured interviews were used to collect data from classrooms, teachers and learners. The findings revealed that teachers have considerable understanding of (PS) and value it but are not teaching it. Teachers attribute this to their lack of knowledge, the difficult conditions in their schools, policy constraints (such as assessment) and their own habits and behaviours. However, the data also indicated that teachers, with support, can successfully design and teach appropriate lessons in their schools, raising issues about their knowledge, beliefs, identity and structures. / Theses (Ph.D.)-University of KwaZulu-Natal, Durban, 2005.

Are there differences among freshman, sophomore, junior and senior baccalaureate nursing students in levels of critical thinking skills? What demographic variables are related to critical thinking?Critical thinking was equated with the problem-solving process in the nursing process and defined as the ability to collect and interpret facts, develop problem statements, identify interventions and evaluate the outcomes.A stratified random sample of nursing students (N = 114) from freshman, sophomore, junior and senior classes completed the Watson-Glaser Critical Thinking Appraisal and a demographic questionnaire.No significant differences were found among the four grade levels in critical thinking skills (F = 2.506, p = .0628). Scholastic Aptitude Test (SAT) verbal and quantitative scores and grade point averages (r = .55, .30, .41 respectively) were positively correlated to critical thinking (p = .05). Age, total number of credit hours completed, credit hours completed In physical/earth/life sciences, behavior and social sciences, humanities and fine arts, professional nursing and general electives were also statistically significantly related (r < .30, indicates little practical significance) to critical thinking. SAT verbal, grade point average, humanities and fine arts entered a regression equation to collectively account for 41% of the variance in critical thinking (p = < .001).Either (a) critical thinking skills are not increasing, (b) the WGCTA does not measure the critical thinking skills used by nurses or (c) nursing curricula may not develop critical thinking skills in nursing students. using an analysis of the uniqueness of the nursing process, nurse educators should develop an instrument that measures the critical thinking "process" component of the nursing process as well as the "logic" component with items specific to the nursing knowledge base.If critical thinking skills are important to nursing practice, then curriculum content, teaching methodologies and learning experiences should increase critical thinking skills in nursing students. Longitudinal studies need to be conducted to determine what curriculum content, teaching methodologies and learning experiences are most effective in increasing the critical thinking skills in nursing students. / Center for Lifelong Education

Paper 1. The purpose of this analysis is to critically evaluate the Lower Primary Mathematics programme within the context of the Namibian educational reform, against the backdrop of learner-centred education from within the Constructivist epistemology. Through the analysis of a small-scale survey, I will try to determine the extent to which learners, teachers and other educationalists, involved in the Lower Primary school phase, understand the new approach to Mathematics and their pedagogical and theoretical insight into the new programme. I will also analyse the syllabus documents in view of the educational policies and further discuss the social, historical and economic background to the reform. I will then analyse the progress or development of the reform process through discussing the learning environment as well as the learners who are the central participants in education. Paper 2.This critical discussion aims to explore the prospects and underlying principles of the epistemologies of two opposing paradigms of education: Behaviourism and Constructivism. I have critically examined and compared the theoretical aspects that shape and inform the model of instruction and the systemic implications of the learning process. The purpose was to compare both learning theories and to draw a conclusion of which the better epistemology is for the teaching and learning process. Paper 3. This is an investigation into the theories that underpin and inform mathematics teaching in the Lower Primary phase of the primary school in Namibia. The Namibian society requires the development of knowledge and understanding, skills and competencies, attitudes and values, which everyone must have to be able to function adequately in society on a social, economic and political level (MBESC 1996). We need to seek ways of enhancing mathematics teaching in Namibia if we want to live up to the expectations of our society. Paper 4. Mathematicians and researchers across the globe have theorised and speculated about education reform movements, which aim for more than just structural knowledge where it concerns Mathematics. This action research study is based on the findings of a pilot study about 'Problem-based Learning', using this as a basis to investigate the relationship between the intentions of the Lower Primary Mathematics curriculum and its implementation at classroom level. This paper reports on a case study of two grade one teachers' perceptions of problem solving as a key component of Mathematics learning and how the implementation of the problem-based approach could be supported through intervention. Data was gathered from a series of cycles of planning, acting, observing and reflecting. Analysis of the data indicates that teachers' professional development lies within a willingness to change and in reflexive practice. Given focused support to teachers could result in the application of contemporary approaches to Mathematics teaching, with an overall improvement of constructivist-oriented learner-centred education.

Thesis (MTech (Further Education and Training))--Cape Peninsula University of Technology, 2012. / This research explored the utilization of problem solving techniques by Computer Applications Technology (CAT) teachers in grade 11. The requirements and criteria as set out in the National Curriculum Statement (NCS) and the National Curriculum and Assessment Policy Statement (CAPS), were applied in this study. CAT is only introduced in grade 10 in the Further Education and Training (FET) phase. The researcher thus focussed on the surrounding schools that offer CAT as a subject. The research focussed on the teacher and the teaching of problem solving techniques in CAT. The researcher used a case study design and employed a constructivism as the theoretical framework. Classroom observations, interviews and both teacher generated and formal documents served as data sources. A thematic analysis approach was adopted to make sense of the data. The analysis of the different data sources indicated that the respondents did not fully comply with the requirements set by the NCS and CAPA with reference to the teaching of problem solving techniques in CAT. The researcher recommends that courses are presented to address the gaps in the knowledge and skills of teachers in CAT referring to problem solving techniques. This creates both the opportunity and challenge to tertiary institutions to align their curricula and training programmes to address the gaps identified by this research.