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

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.

The present study investigated imagery and visualisation characteristics of undergraduate students’ thinking processes in learning selected concepts of mathematical analysis. The aim was to discover the nature of images evoked by these undergraduate students and the role of imagery and visualisations when students were solving some selected problems related to mathematical analysis. The study was guided by the theory of registers of semiotic representations. Psychological notions on imagery were also fused to cater for a cognitive approach to the study. A sample of 50 undergraduate mathematics students participated in the study. The researcher employed both quantitative and qualitative methods. Before the main study, a pilot study was conducted to account for the reliability and validity of the research instruments. The data were collected through use of a cognitive test that was composed of 12 tasks with items selected from mathematical analysis. These tasks were specially designed to capture the variables of imagery and visualisations. A structured interview was also conducted as a follow-up to the results of the cognitive test. The study found that visual images were noticeable in the thinking processes of undergraduate students in solving problems related to mathematical analysis. The nature of the visual images evoked by the students varied from person to person. The nature of these images was also determined by the nature of the task. The most common types of imagery were diagrams, prototypes and symbols. On rare occasions the students also evoked metaphoric images. It was also observed that these images were used for illustrative purposes and to spark the idea for a proof. It was also interesting to note that some images were used to discover the limit of a converging series. The results confirmed the need to use visualisation with caution, especially when treating concepts which involve infinity. The study recommends that instructors of mathematics should encourage visual thinking in the learning and teaching of mathematical analysis. Knowledge of the students’ concept images helped the researcher to understand the nature of the learning difficulties of the students. Further research should focus on the strengths and weaknesses of visual-mediated learning and also on the relationship between creativity and visual thinking. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))