As an alternative to Cartesian dualism, recent work by some theorists has proposed a more integrated approach to imagery and formal thought. Researchers have argued that although visual reasoning is generally considered helpful in mathematics learning it has low status in the classroom. Mathematics educators for the most part either ignore visual reasoning completely or see it as some pre-formal stage to be replaced by more rigorous methods of reasoning. A growing body of research questions the level of mathematical sense making of students who just obtain high scores on school related tasks, suggesting links between the mathematical sense making activity of students and more image-based reasoning processes. / The purpose of this research was to investigate imaging in children's mathematical activity. Four fourth/fifth grade students participated in this study which lasted for one year. Bi-weekly individual problem solving sessions, together with field notes taken in the participants' classrooms constituted the data. / Tasks which had the potential to be problematic to the participants were presented in these sessions. Participants were encouraged to describe the ways in which they were constructing meaning for the tasks. A detailed description and analysis of the researcher's interpretation of their actions is included. / The researcher found that children's imaging activity was at the heart of their sense making and problem solving. The various types of imaging and the ways in which imaging contributed to mathematical sense making and the construction of viable solutions to tasks is described. / Source: Dissertation Abstracts International, Volume: 54-04, Section: A, page: 1274. / Major Professor: Grayson H. Wheatley. / Thesis (Ph.D.)--The Florida State University, 1993.

This study concerned limits of sequences. Since limits are such an important mathematical concept for students to "understand," the major purposes of this study were to: (1) Develop a meaning of "the understanding of the limit of a sequence" based upon students' behavior. (2) Construct an instrument for measuring the understanding described in 1. An additional purpose was to: (3) Investigate subskills related to understanding the limit concept. / A good test for measuring the understanding in "1" would prove useful in helping teachers at various levels to answer the question, "Do my students understand limits?" as opposed to just finding limits. / Naturally, such an endeavor would require some thought on what indeed it means to understand limits. Prior to this study such a definition of understanding limits appeared to be lacking. / Thus, behavioral objectives were established by identifying the main features of limits and gaining a consensus from well-qualified professionals whose work involves an intimate knowledge of limits. / Test development involved constructing an initial version of the limits instrument, and then performing many revisions so that certain standards of measurement theory were satisfied. The final version of the instrument was administered to 263 subjects who had studied limits. The results for this 53 item test were reliability, alpha = 0.817; mean, 35.9 (67.7%); and standard deviation, 6.99 (13.2%). Validity checks were made on the instrument by comparing performance on this instrument and other related measures. / This study also involved identifying specific subskills related to understanding limits. This is noteworthy in that a variety of illustrious professors shared their views with regard to these subskills. Linear relationships were found between scores received on the limits instrument and scores on five subskills test. / Finally, specific information gleaned from the analyses performed in this study would directly benefit classroom teachers. Students did poorly on absolute value, distance, inequality, and segments or intervals. They do not have a good formal level of understanding limits, although they did fine at seemingly lower levels of understanding. Repeating decimals caused students confusion. Also some specific misconceptions of which teachers should be aware, surfaced during this study. / Source: Dissertation Abstracts International, Volume: 44-12, Section: A, page: 3619. / Thesis (Ph.D.)--The Florida State University, 1983.

Recent work by researchers in the area of teaching practice proposes that assisting teachers to reconstruct their epistemologies and beliefs about the nature of the subject to be learned appears to be a powerful way of enhancing teaching and learning in mathematics classrooms. A growing body of research questions the level of mathematical sense making of students in classroom based activities, suggesting that current classroom practice conceptualizations need improvements. / The purpose of this research was to investigate the factors relating to the decisions that teachers make about their practice, more specifically, teacher and students beliefs about mathematics, teaching and learning mathematics, various interactions and its relationships with what happen during classroom practice, via a case study of a high school teacher, in the context of Costa Rica. The case study provides a detailed description and analysis of the researcher's interpretation of the teacher and some of her students. This study was conducted under a constructivist framework. / Data for this research were collected over a six month period. The primary data sources were field notes from class observations, and formal and informal recorded interviews/discussions. The investigation also involved participant observations in the classroom and planning sessions. / A narrative of Sofia's experiences during the research progress was developed using themes such as metaphors, beliefs (about mathematics, teaching mathematics, and learning mathematics), and actions, to describe Sofia's teaching style. Another theme was regarding teacher's and students' roles, and their views about each other. / The researcher found that teaching is very complex. Throughout Sofia's actions there were evidences that different components were woven together. Sofia held two contrasting sets of beliefs. Sofia's stated beliefs were that the teacher's main role is to provide students with opportunities to construct meanings for themselves, while her beliefs-in-practice suggested that direct instruction (teacher as dispenser of knowledge) is an effective way to teach. When planning, Sofia was more influenced by the syllabus and topics to be covered than student knowledge. / Source: Dissertation Abstracts International, Volume: 54-12, Section: A, page: 4342. / Major Professors: Elizabeth M. Jakubowski; Grayson H. Wheatley. / Thesis (Ph.D.)--The Florida State University, 1993.

The purpose of this study was to build a descriptive model of social-interaction processes of natural and artificially imposed student partnerships engaged in mathematical problem-solving activity. The theoretical perspective of this study was based on the socio-cognitive model of learning which hypothesizes that cognitively effective social interactions will generate perturbations or disequilibrations in subject's existing knowledge schemes. / To enable the development of the model, questions relating to partnerships roles, differences in problem-solving strategies between partners, and evidence of coordinated problem-solving activity were of particular interest. Through the use of nonroutine mathematics tasks that had the potential of being problematic, an environment for discrepant points of view was provided. / The study was conducted in two phases. First a fourth grade class was observed biweekly for a period of eight weeks to document and analyze interaction patterns. Based on the initial observation, two natural dyads and three natural triads were selected for the second phase of the study. In the second phase of the study the selected natural partnerships and researcher imposed artificial partnerships were videotaped in problem-solving sessions where nonroutine mathematics tasks were given to the partnerships. Artificial partnerships were determined through researcher imposed changes in partnership participants based on the observation phase of the study and an initial analysis of the natural partnership videotapes. / Major themes that emerged in the qualitative analysis of the data were: gender differences, levels of collaboration, partnership roles, methods of resolving conflict, and effects of setting changes. A synthesis of major themes revealed a descriptive model in which three factors contributed to the level and quality of task-focused interactions. The three factors were: the type of mathematics task posed, the presence of a socially dominant partner, and the degree of cognitive difference between partners. / Source: Dissertation Abstracts International, Volume: 53-09, Section: A, page: 3092. / Major Professor: Janice Flake. / Thesis (Ph.D.)--The Florida State University, 1992.

The main objective of this study was to investigate the hierarchical nature of the van Hiele levels in the learning of transformation geometry. Secondary school students in Singapore completed tasks using the concepts of reflection, rotation, translation and enlargement. In addition, the van Hiele levels of two current Singapore textbooks were analyzed for transformation geometry. / A level characterization for transformation geometry was written after interpreting related research reports. Test items were then developed, and critiqued by a nationally-based panel of mathematics educators. The items, revised for the first four levels were used in interviews with twenty secondary four students (ages 15-16) from a school. The audiotaped and videotaped interviews took two sessions of one and a half hours each. In the analysis, two persons independently assigned levels based on students' responses. These responses were analyzed for existence of level hierarchy using a Guttman Scalogram and for patterns of thinking. Textbooks were analyzed to identify levels for the content and the sequencing of the levels in the material. / Results indicated the levels form a possible hierarchy. The percentage of responses at each level of thinking was: 42.5%, Basic; 36.25%, Level 1; 6.25%, Level 2; 12.5%, Level 3. Analysis of responses revealed students: (1) had misconceptions with enlargement which is the least achieved concept; (2) perceived transformations in terms of motion before attending to the properties associated with the transformation; (3) lacked precise vocabulary to describe transformations; (4) had difficulties in relating a matrix to a transformational picture; (5) continually referenced teachers and text as reasons for their solutions; (6) did proofs using particular examples. The textbook analysis showed expository lessons with many worked examples and exercises characterized at Level 1 and Level 2 pertaining to coordinate and matrix system. Also there was an absence of hands-on activities and applications to real life situations with little opportunity for students to explore, reflect and conjecture. / The study has implications for teacher-educator in preparing the teachers to provide appropriate learning environments. The implication for the curriculum developer and textbook writer is in restructuring curricula. / Source: Dissertation Abstracts International, Volume: 50-03, Section: A, page: 0619. / Major Professor: Janice Louise Flake. / Thesis (Ph.D.)--The Florida State University, 1989.

This study compared two different approaches to the teaching of elementary probability to 196 community college students. These two approaches were identified as the single concept approach and the multi-formula approach. In the single concept approach the students solved probability problems by relying solely upon the definition of 'probability'. Students in the multi-formula approach solved probability problems by the traditional approach of using several formulas. / The multi-formula group and the single concept group were compared on achievement, retention, and transfer. An analysis of variance was used to analyze the achievement scores. The single concept group scored significantly higher (p-value = 0.0001). An analysis of covariance was used to analyze the retention scores. The single concept group scored significantly higher (p-value = 0.025). An analysis of variance was performed on the transfer items. Again, the single concept group scored significantly higher than the multi-formula group on the transfer items both on achievement and retention. The p-value was equal to 0.0001 for both analyses. / A depth of understanding may account for these results. Whereas the multi-formula group divided their time and effort among several concepts associated with their formulas, the single concept group concentrated their efforts and attention on the single definitional concept. One might conjecture that students versed in a single concept would outperform those spreading the same amount of time over many concepts (formulas). / In addition to investigating the learning of probability, this study relates to two types of understanding identified by Richard Skemp. Instrumental understanding is identified with the multi-formula group and Relational understanding with the single concept group. The results of this study suggest that the single concept approach may be better for learning other mathematical concepts. For example, the idea of perimeter as the distance around a figure contrasted with a collection of formulas for finding the perimeters of various figures. Another example is the definitional meaning of integral exponents contrasted with a variety of formulas addressing operations with exponents. / In view of the success with the single concept approach used in this study, additional research would tell if similar success may be realized with other mathematical topics. / Source: Dissertation Abstracts International, Volume: 51-12, Section: A, page: 4052. / Major Professor: Herbert Wills, III. / Thesis (Ph.D.)--The Florida State University, 1990.

Several assessment studies document African-American children's achievement in mathematics learning at various grade levels. However, little research exists which systematically examines the role of visual imagery in mathematics learning within this population. This study examined how African-American fifth-graders used visual-imagery in constructing geometric tiling patterns and (indirect) area measurement concepts. It was conducted within the constructivist theoretical framework and made explicit basic processes of knowledge acquisition. / This investigation consisted of clinical interviews of eight African-American fifth-grade students, exhibiting high or low spatial-thinking ability according to the Space Thinking (FLAGS) Test. Each participant engaged in three imagery-building tasks designed to facilitate mathematical thinking and develop spatial reasoning in the content area. All interviews were videotaped; they provided data for use in the development of a cognitive model of the participants' spatial and related mathematical constructions. / Major themes which emerged from the data refer to the participants' Construction of Visual Units, Acknowledgement of Tiling Patterns, and Interpretation Levels of Size and Measurement. Careful analysis of the themes revealed important answers to the major research questions. High-spatial thinkers, particularly, made more use of dynamic imagery to recall repeating aspects of geometric tiling patterns. Low-spatial thinkers' images were more concrete and static. High-spatial learners used cognitive reorganization to formulate equivalences of area measures of plane regions. High- and low-spatial learners interpreted "size" on diverse levels--ranging from a concrete level to an intuition level. / Research themes and answers supported the development of abstract constructs comprising a model of the African-American fifth-graders' constructed activity (i.e., their mental actions and operations). Major components of the model along with some associated elements include: Nonverbal/Verbal Cues--motor activity, verbal discourse; Anticipatory Images--using images dynamically, forming dynamic images, mental transformations; Reflective Abstraction--mental restructuring, reversibility of thought; and Cognitive reorganization--chunking, decomposing/recombining images. The model suggests instructional and curricular implications to educators whose goal is to enhance children's mathematics learning. / Source: Dissertation Abstracts International, Volume: 52-02, Section: A, page: 0454. / Major Professor: Janice L. Flake. / Thesis (Ph.D.)--The Florida State University, 1991.

The concept of absolute value is central to several important mathematical topics. Among these are distance, limits, continuity, metric spaces and the square root function. Some studies reveal that students perform surprisingly poorly on problems related to absolute value. / Several authors of articles in professional journals have identified a variety of errors, difficulties, and misconceptions arising from attempts to solve sentences involving absolute value. / This study compared the results of two groups of college algebra students studying the solution of absolute value sentences. One group was instructed using the Distance approach, and the other group used the Enhanced Traditional approach. The comparison was made on achievement, retention, and transfer. This research used four intact college algebra classes from four different colleges as the population sample. / Four different junior colleges were involved in this study. Two classes of college algebra students at two different junior colleges were assigned to the Distance approach. Two classes from two other junior colleges were assigned to the Enhanced Traditional approach. Each of the four classes took four tests on solving absolute value sentences. Subjects were given a fifty minute pretest on the day the experiment started. Then the instructional materials dealing with the absolute value concept were presented during four fifty-minute lessons. The subjects were given the fifty minute achievement test at the next scheduled class meeting. Two weeks later the retention test was given to each of the four groups. / Analysis of covariance was used to test whether there were any significant differences between the two groups on the three tests with the pretest as the covariate. Three null hypotheses of no differences between means scores were rejected at the 0.05 level for the achievement, the retention, the transfer test. In addition, no gender differences were found in this study. We may conclude that students in the Distance approach perform better than the students in the Enhanced Traditional approach in each of the three areas investigated. / We found from interviews that most teachers view absolute value as an important part of mathematics. Those contacted supported this view with reference to specific areas of mathematics that draws upon absolute value. The interviews were also valuable in identifying specific diverse difficulties experienced by students studying or using absolute value. / Students interviewed from the distance group demonstrated greater competency in solving problems involving absolute value. Not only did they get more correct answers but did so in less time. Several students from the Traditional group found problems too long and boring. / Source: Dissertation Abstracts International, Volume: 52-11, Section: A, page: 3820. / Major Professor: Herbert Wills, III. / Thesis (Ph.D.)--The Florida State University, 1991.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

This study investigated the relationship among three high school mathematics teachers definitions and beliefs about mathematical problem solving, their problem solving practices and how they teach mathematical problem solving. Each teacher was interviewed three times and observed once during a problem solving lesson. Data comprised of transcriptions of audio tapes, field notes, and completed problem solving checklists were used to prepare the case studies. While the definitions, practices and teaching of the teachers varied, the findings were consistent within each case. The results suggest that how teachers are taught and what they learn as students are related to how they teach mathematical problem solving.