The purpose of this descriptive case study analysis was to provide portraits of the heuristics students used and difficulties they encountered solving conditional probability problems prior to and after two-week instruction on sample space, probability, and conditional probability. Further analysis consisted of evaluating the data in relation to a previously designed Conditional Probability Framework for assessing students levels of thinking developed by Tarr and Jones (1997). Five volunteer participants from a contemporary college mathematics course participated in pre-and post-interviews of a Probability Knowledge Inventory. The Inventory consisted of seven tasks on sample space, probability, and conditional probability. The semi-structured interviews provided participants' explanations on the development of their solutions to the seven tasks. Among the five participants, rationalizing, finding the odds, computing the percentages, and stating the ratio of a problem were the preferred heuristics used to solve the problems on the Probability Knowledge Inventory. After the two-week instruction, two of the four participants who did not previously use computation of probability to solve the problem changed their use of heuristics. The difficulties the students encountered prior to instruction included understanding the problem; recognizing the original sample space and when it changes; lacking probability vocabulary knowledge; comparing probability after the sample space changed; understanding the difference between probability and odds; and interchanging ratio, odds, and percentages-sometimes incorrectly-to justify their solution. After the two-week instruction, the students' difficulties diminished. Some improvements included a greater ability to understand the question of interest, to recognize the change in the sample space after a conditioning event, to use probability terminology consistently, and to compare probability after the sample space has changed. Comparisons to the Probability Framework revealed that four of the five participants exemplified Level 3 thinking-being aware of the role that quantities play in forming conditional probability judgements. One participant exemplified a Level 4 thinking-being aware of the composition of the sample space, recognizing its importance in determining conditional probability and assigning numerical probabilities spontaneously and with explanation. / Graduation date: 2001

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D.Ed. / Adolescents who achieve inadequately in mathematics often experience that the different aids available to assist them on their road to recovery are not sufficient. Most adolescents do not know how to bring about desired change. They are not aware what their problem areas are and how it should be addressed. Due to growing numbers in the classrooms in formal teaching, the teacher cannot assist the student individually on an ongoing basis. Factors within the student and his circumstances may also prevent him from benefiting from assistance. Furthermore, most adolescents in the senior secondary phase, who want to study further, are pressurized by educational bodies for higher marks. Therefore, limited time also becomes an issue. Although several studies have been carried out to investigate this matter and create models, the majority of practical programmes lacked depth and a holistic approach.

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In today's highly competitive landscape, new product development strategies are imperatives for companies to create and sustain competitive advantages. The objective of this research is to develop an integrated approach to automate, or aid, the design problem solving process. An Integrated Design System (IDS) is proposed focusing on the parametric and detail design. In this system, generation and evaluation of new design problems occur quickly and easily by changing the inputs for the design model. The IDS provides an integrated platform to incorporate available application programs such as CAD and FEM tools into a single system. Four major frameworks, namely information handling, problem decomposition, design space characterization, and problem solving, are proposed and embedded in it to implement the product development process. The information handling includes five aspects. A naming protocol is devised to organize the historical design cases. A search algorithm is proposed to retrieve a design case. A system-generated report is used to distribute the design information. A constraint definition frame is presented to define the relationships between design parameters. Two schemas, information matrix and constraint tree, are developed to represent information in the IDS. A diagonal-centered decomposition scheme is developed utilizing a Genetic Algorithm to decompose a complex design problem. In addition to the conventional genetic operators, two novel genetic operators, unequal position crossover and insertion mutation, are proposed. To characterize the design space, two methods, Incremental Response Method (IRM) and Artificial Neural Network (ANN), are presented. The IRM is derived from response surface method, while the back-propagated ANN is coded to be self-evaluated. The presented problem solving algorithm constitutes the solving mechanism of the IDS. Based on the assessment of the design objectives, all design parameters are given a priority index to facilitate the parameter selection. An independent recursive method is introduced in this algorithm to handle the design constraints. The case studies are performed on two design problems: a hard disk drive actuator arm and a shaft. The results show that the system can automatically align parameter values towards the objective values in a reasonable manner, and thus verify the feasibility of the embedded frameworks.

(Uncorrected OCR) Abstract This study is designed to examine the effectiveness of Knowledge Forum (KF) on foresting mathematical concept understanding and problem solving; and on enhancing attitude towards and beliefs of mathematic word problems. Thirty-eight sixth graders from a Hong Kong pnmary school were introduced to KF in mathematics proportional reasoning learning for five weeks. The students spent about one hour a week at school to solve ratio and proportion problems through KF and another hour to reflect on their learning by whole class discussion. Before and after the experimental program, instruments like traditional mathematics tests, mathematics word problems tests, beliefs and attitude questionnaires were administered in both the control and the experimental groups. An ANCOVA analysis was conducted to the data collected by the four quantitative instruments. During the program, the experimental group's KF notes and learning diaries were collected and analysed qualitatively to evaluate the knowledge building process. The results suggested that the KF not only had a significant positive effect on mathematics performance and problem-solving competency, it also yielded a positive influence on beliefs of mathematic word problem. However, the KF 11 had no impact on attitude towards mathematic word problem. There were evidences that some subjects were applying the five-step metacognitive strategy and appropriate heuristics during the problem solving process. Furthermore, there were mathematical discourses on the concept of ratio and proportion. The information obtained from this study will be helpful in gaining a better understanding of the computer-supported collaborative learning environment, in particular, the Knowledge Forum and mathematics problem solving. Despite of the constraints, the findings of this study may have implications for developing instructional programs and improving mathematical teaching methods for teachers in elementary schools. 111 / abstract / toc / Education / Master / Master of Education

Presented here is a case study of the problem-solving behaviors of upper-division undergraduate physics majors. This study explores the role of visual representations in students' problem solving and provides a foundation for investigating how students' use of visualization changes in the upper-division physics major. Three independent studies were conducted on similar samples of students. At the time of these studies, all of the subjects were junior physics majors participating in the Paradigms in Physics curriculum at Oregon State University. In the first study, we found that while all students had high scores on the Purdue Spatial Visualization Test, the correlation between test scores and their grades in physics was not statistically significant. In the second study (N=5) and the third study (N=15), we conducted think-aloud interviews in which students solved electrostatics problems. Based on the interviews in the third study, we develop a model that describes the process by which students construct knowledge while solving the interview problems. We then use this model as a framework to propose hypotheses about students' problem-solving behavior. In addition, we identify several difficulties students have with the concepts of electric field and flux. In particular, we describe student difficulties that arise from confusing the vector and field line representations of electric field. Finally, we suggest some teaching strategies that may help to assuage the student difficulties we observed. / Graduation date: 2002

published_or_final_version / Education / Master / Master of Education

In this multiple-case study, I developed cases describing three students‘ (Mary, Ryan and David) solution methods for algebra word problems and investigated the effect of technology on their solution methods by making inferences about their preferences for visual or nonvisual solutions. Furthermore, I examined the students‘ solution methods when presented with virtual physical representations of the situations described in the problems and attempted to explain the effect of those representations on students‘ thinking preferences. In this study, the use of technology referred to the use of the dynamic software program Geogebra. Suwarsono‘s (1982) Mathematical Processing Instrument (MPI) was administered to determine their preferences for visual and nonvisual thinking. During the interviews, students were presented with paper-andpencil-based tasks (PBTs), Geogebra-based tasks (GBTs) and Geogebra-based tasks with virtual physical representations (GBT-VPRs). Each category included 10 algebra word problems, with similar problems across categories. (i.e., PBT 9, GBT 9 and GBT-VPR 9 were similar). By investigating students‘ methods of solution and their use of representations in solving those tasks, I compared and contrasted their preferences for visual and nonvisual methods when solving problems with and without technology. The comparison between their solutions of PBTs and GBTs revealed how dynamic software influenced their method of solution. Regardless of students‘ preferences for visual and nonvisual solutions, with the use of dynamic software students employed more visual methods when presented with GBTs. When visual methods were as accessible and easy to use as nonvisual methods, students preferred to use them, thus demonstrating that they possessed a more complete knowledge of problem-solving with dynamic software than their work on the PBTs. iii Nowadays, we can construct virtual physical representations of the problems in technology environments that will help students explore the relationships and look for patterns that can be used to solve the problem. Unlike GBTs, GBT-VPRs did not influence students‘ preferences for visual or nonvisual methods. Students continued to rely on methods that they preferred since their preferences for visual or nonvisual solutions regarding GBT-PRs were similar to their solution preferences for the problems on MPI that was administered to them to determine their preferences for visual or nonvisual methods. Mary, whose MPI score suggested that she preferred to solve mathematics problems using nonvisual methods, solved GBT-VPRs with nonvisual methods. Ryan, whose MPI score suggested that he preferred to solve mathematics problems using visual methods, solved GBT-VPRs with visual methods. David, whose MPI score suggested that he preferred to solve mathematics problems using both visual and nonvisual methods, solved GBT-VPRs with both visual and nonvisual methods.