The purpose of this investigation was to explore how differences in an elementary mathematics teacher's subject matter knowledge structure impact classroom teaching and student learning. The study included two phases. Phase 1 focused on the selection of a single case. An open-ended questionnaire and interview were used to identify the subject matter knowledge structure for addition, subtraction, multiplication, and division of three elementary teachers. One teacher was selected who demonstrated clearly different levels of knowledge for multiplication and division. An additional interview provided information on the teacher's specific climate for teaching mathematics and details about the unit on multiplication and division to be observed. Phase 2 included daily classroom observations for approximately one hour each day of a seven-week unit on multiplication and division. Informal interviews were conducted with the teacher throughout the unit to better understand the lessons and allow the teacher an opportunity to clarify statements and actions. A final teacher interview occurred after the last classroom observation. At the conclusion of the observations, the students were assessed to determine their knowledge of multiplication and division based on the teacher's unit objectives. And six students, representing the range of class performance, were interviewed to provide additional insights into the students' learning. The teacher's subject matter knowledge of multiplication was strong but her knowledge of division was faulty and incomplete on several topics including the different meanings of division, the conceptual underpinnings of division procedures, the relationships between symbolic division and real life problems, and the idea of divisibility. Although the translation of the teacher's subject matter knowledge was complex, it seemed to be directly related to classroom teaching and students' learning. The teacher's narrow understandings were associated with an incomplete developing of the full range of division situations. Although the students had significantly more success on the post assessment problems involving multiplication than on those involving division (understandable since the teacher spent more time teaching multiplication than division), a more worrisome concern was that the students in this study exhibited serious misconceptions associated with the meanings of division, division computation, and notions of divisibility. / Graduation date: 2000

The purpose of the study was to investigate students' learning of the function concept and the role of the graphing calculator in a College Algebra course. Differences between students with high symbolic manipulation skills. and students with low symbolic manipulation skills were also examined. On the basis of an algebraic skills test administered by the instructor (high/low) and students' academic majors (math & science, business, and liberal arts), 25 students from one College Algebra class were placed into six categories. To gather data on students' understanding of functions, a pretest and posttest were administered. The Function Test consisted of four identification questions given in each of the representations, three questions asking for the definition, an example, and a nonexample of functions, and 15 questions consisting of three problem situations given in the numerical, graphical, and symbolic representations. To gather data on the role of the graphing calculator, daily classroom observations were conducted. To verify students' responses and classroom observations, formal interviews with students and informal interviews with the instructor were conducted. Students' personal definition progressed towards the formal definition of functions. Yet, students had difficulties with the univalence requirement in three areas: (a) order of domain and range, (b) preference for simple algorithms, and (c) the restriction that functions were one-to-one. Compared to students with low symbolic manipulation skills, students with high symbolic manipulation skills were more flexible working between representations of functions. Half of the interviewed students with low symbolic manipulation skills perceived a single function given in numerical, graphical, and symbolic representations as separate entities. The graphing calculator played a role in all phases of the solution process. During the initial phases, students used calculators to develop a symbolic approach. The prime motivation for using graphing calculators during the solution-execution phase was to avoid careless errors. The most common use of graphing calculators was to check answers during the solution-monitoring phase. However, graphing calculators created difficulties for students who accepted graphs at face value. Interpreting the truncated graph shown by the calculator, students determined that exponential functions possessed a bounded domain because they did not explore the graph. / Graduation date: 2001

Organizations leading education reform of the 1980's have challenged teacher education programs at colleges and universities across the nation to improve the subject matter content preparation of teachers. Past methods of program development and techniques to assess teacher's knowledge competence have been one-sided in their approach. New research studies on expert vs novice teachers show that expert teachers are more efficient in carrying out standard patterns of instruction. This nation's mathematics community has engaged in a revitalization of mathematics curricula. Traditional mathematics is being transformed to become a powerful science. Using the growing body of research, the National Council of Teachers of Mathematics have developed standards for improving the teaching and learning of mathematics. Oregon's Department of Education has also established standards to meet the needs of a changing mathematics curricula and the challenges of a changing society. This study identified the specific content knowledge taught in the mathematics curricula within colleges and universities which offer four, five or fifth year teacher education programs. It then compared these findings against teacher identified origins of elementary, middle and high school teachers' mathematics content knowledge relative to the Essential Learning Skills of Oregon. It was found that teachers' content knowledge of the Essential Learning Skills of Oregon was not directly related to their preparation as teachers; at the elementary and high school levels, there was no direct relationship found between teachers' degrees and their teaching assignment; there was no apparent relationship between teachers' knowledge of the Essential Learning Skills of Oregon and graduation from an Oregon college or university; there was no apparent relationship between teachers' lack of knowledge of the Essential Learning Skills of Oregon and graduation from a non-Oregon college or university. / Graduation date: 1990

The purpose of this study was to investigate the effect of teachers' behaviors and characteristics upon the development of student mathematical critical thinking skills. From a pool of 20 teachers, whose students had been pre- and post-tested for a measure of critical thinking skills, 10 middle level teachers were selected to complete extensive questionnaires on their backgrounds and experiences, submit videotaped records of classroom activity, and to maintain detailed data on their classroom actions. The teachers were ranked in accordance with their respective classes' mean gain scores on the assessment tool. From the pool of 20 teachers, the top-ranked 25% (five teachers) and the bottom-ranked 25% (five teachers) were selected for the study. Extremes of the ranking order were used to increase the probability of determining potential differences in teacher behaviors and characteristics between the two groups. The two extremes were thus placed in two groups to identify those variables which contributed to differences between the groups. Identified variables from pairwise comparisons of the teachers within each group were analyzed, following corroboration from a minimum of three data sources, to generate groups profiles. A 5 x 5 matrix was constructed for each potential group variable. Comparisons were conducted between all pairs of teachers within each group, and the differences between the two groups were compiled in the form of group profiles. The five top-ranked teachers, based upon student performances, were distinguished from the lowest-ranked five teachers by greater use of small group instruction, math manipulatives, and warmup activities; as well as by provision for teaching higher-order thinking skills, frequency of transitions between classroom activities, and the use of activities which required the application of concepts. The lowest-ranked teachers were characterized by the greater frequency of teacher-directed instruction, a higher amount of computer usage, assignment of individual student work, highly structured classes, and extensive reliance on textbooks as the primary source of instructional materials. / Graduation date: 1993

The purpose of this study was to investigate the knowledge base and instructional practices of two highly qualified experienced secondary mathematics teachers within the context of their classrooms during a unit in a geometry class. Data collected from interviews, classroom observations, pre and post-observation questionnaires, and detailed analyses of several lesson segments were used to create case studies for each teacher, which were compared to reveal any patterns in their instructional practices. The theoretical framework used for this study was Schoenfeld's (1998) model of teaching-in-context that included three factors that affected teachers' decisions during instruction: beliefs, goals, and their knowledge bases. The supporting questions that were investigated in this study dealt with teachers' conceptions of mathematics and teaching and learning mathematics, instructional goals, instructional strategies and curricular materials used during the unit, and any modifications made to instruction. Both teachers in this study used a more traditional lecture and discussion style of instruction that closely followed an explicit model of teaching instead of a more reform-based style of teaching. The teachers incorporated the processes of mathematics such as proof and reasoning and representation into their instruction through modeling instead of incorporating activities into instruction designed to engage students in the processes. Although both teachers were aware of and had used reform-based methods, they perceived that the traditional instructional methods were more efficient and effective. Contextual factors played a dominant role in the decisions the teachers made about their instruction. The contextual factor that had the greatest effect on instruction for these two teachers was the pressure to teach all of the topics in the required curriculum to prepare their students for the state standardized high stakes test. Other contextual factors were large class sizes, limited physical space, and limited access to technology. The results of this study indicated that although the teachers had strong content knowledge and knowledge of both traditional and reform-based pedagogy, they chose a more traditional instructional style and this decision was affected by contextual factors such as high stakes testing, a required curriculum, and the demands of their jobs. / Graduation date: 2006

Computer-Assisted Instruction (CAI) offers the potential to deliver cognitive and meta-cognitive strategies in mathematical word problem-solving for students with mathematics difficulties. However, there is a lack of commercially available CAI programs with cognitive and meta-cognitive strategies for mathematical word problemsolving that pay particular attention to the critical design features for students with mathematics difficulties. Therefore, empirical evidence regarding the effects of CAI program with cognitive and meta-cognitive strategies on the word problem-solving of students with mathematics difficulties has not been found. Considering the imperative need for a CAI program with cognitive and metacognitive strategies for students with mathematics difficulties, an interactive multimedia software, ‘Math Explorer,’ was designed, developed, and implemented to teach one-step addition and subtraction word problem-solving skills to students with mathematics difficulties. Math Explorer incorporates: (a) four-step cognitive strategies and corresponding three-step meta-cognitive strategies adapted from the research on cognitive and meta-cognitive strategies, and (b) instruction, interface, and interaction design features of CAI identified as crucial for successful delivery of cognitive and metacognitive strategies for students with mathematics difficulties. The purpose of this study was to investigate the effectiveness of Math Explorer, which was designed to be a potential tool to deliver cognitive and meta-cognitive strategy instruction in one-step addition and subtraction word problem-solving. Three research questions guided this study: (a) To what extent does the use of Math Explorer affect the accuracy performance of students with mathematics difficulties in grades 2-3 on computer-based tasks with one-step addition and subtraction word problem-solving?, (b) To what extent does the use of Math Explorer generalize to the accuracy performance of students with mathematics difficulties in grades 2-3 on paper/pencil-based tasks with one-step addition and subtraction word problem-solving?, and (c) To what extent does the use of Math Explorer maintain the accuracy performance of students with mathematics difficulties in grades 2-3 on computer- and paper/pencilbased tasks with one-step addition and subtraction word problem-solving? A multiple probe across subjects design was used for the study. Four students with mathematics difficulties participated in the pre-experimental (i.e., introduction; screening test; and computer training I) and experimental (i.e., baseline, computer training II, intervention, and follow-up) sessions over an 18-week period. Each week of the intervention phase, the students received an individual 20- to 30-minute Math Explorer intervention, at most, five days. After each intervention, they took the 10-minute computer- or paper/pencil-based tests developed by the researcher. The intervention phase for each student lasted five to seven weeks. Two weeks after termination of the intervention phase, their accuracy performance on the computer- and paper/pencil-based tests were examined during the follow-up phases. The findings of the study revealed that all four of the students were able to use the cognitive and meta-cognitive strategies to solve the addition and subtraction word problems and improved their accuracy performance on the computer-based tests. Their improved accuracy performance found on the computer-based tests was successfully transferred to the paper/pencil-based tests. About two weeks after termination of the intervention phase, except for one student who had many absences and behavioral problems during the extended intervention phase, the three students successfully maintained their improved accuracy performance during the follow-up phase. Taken together, the findings of the study clearly provide evidence that Math Explorer is an effective method for teaching one-step addition and subtraction word problem-solving skills to students with mathematics difficulties and suggest that the instruction, interface, and interaction design features of CAI program is carefully designed to produce successful mathematical performance of students with mathematics difficulties. Limitations of the research and implications for practice and future research were discussed. / text

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Doctoral / Doctor of Philosophy

published_or_final_version / Education / Doctoral / Doctor of Education

The reformed Hong Kong mathematics curriculum for the 21st century consists of three components, namely generic skills, values and attitudes and, lastly, traditional cognitive development. The first two are newly emphasized and expanded. Theoretically, these components correspond closely with communication, socioculture and constructivism respectively, which are the central concepts of dialogic learning (DL). In DL, students are autonomously engaged in egalitarian dialogue, in which they share, reflect and form a learning community. Through DL, a student is expected to develop into an all-rounded and life-long learner. Contrary to the reform, dialogue is still deficient in mathematics classrooms. The role of this study is to present examples of students’ experiences in DL, found in the mathematics club of a secondary girls’ school. This study aims to explore and investigate: (1) the existence of DL in the club, (2) what the members learnt and (3) how they did it. This is an ethnographic research, which emphasizes first hand understanding, grounded theories and thorough intricacies. Therefore, I observed the students’ activities as a participant, interviewed them, and then described, analyzed and interpreted my findings accordingly. Based on my synthesis of relevant literature and the insight I gained from decades of teaching and otherwise interacting with students, I constructed a pentahedral framework to help investigate DL in a more comprehensive and intensive way. It involves the development of various generic skills and the cultivation of values and attitudes, which are usually unrecognized in examination syllabuses and the old curriculum. It consists of five facets, concerning cognitive knowledge, sharing and negotiation, learning skills, metacognition and values and attitudes. And here are the findings. All significant elements of DL from literature have been identified to exist in the club. As for what the students learnt, they recalled fruitful experiences in all five facets of the DL pentahedron. These findings were then combined with the learning histories of three subjects to yield four representative learning patterns, namely those of a ‘cognitive developer’, a ‘communicative daily life explorer’, a ‘eureka torchbearer’ and a ‘frustrated sharer-explorer’. These 4 learning patterns were further combined with (i) the purposes for mathematics study from pure examination results to ‘liberation’ and (ii) the understanding of mathematics learning from pure cognitive knowledge to inclusion of generic skills and values and attitude, to form a conceptual model of learning styles. The styles of the ‘eureka torchbearer’ and the ‘communicative daily life explorer’ were found to be exemplars of the ideals of people who favour the most liberal implementation of the curriculum reform. The ‘frustrated sharer-explorer’ was stuck with the style favoured by conservatives who are against hasty reforms. The ‘cognitive developer’ was somewhere in between. These findings may contribute to the framework of policy debate on mathematics education. In the school and classroom level, they may help teachers overcome learning disaffection and other difficulties, in both theory and practice. Organizers of extracurricular activities may also be inspired by the students’ rich experiences of dialogic learning. / published_or_final_version / Education / Doctoral / Doctor of Education