<p> Researchers have argued that gesture and speech, two elements of discourse, are neurologically related, and that language and mental imagery are intertwined. Because of this relationship between language, gesture and image, these discourse elements may allow a teacher to make inferences about the reasoning the student is using. In order for the teacher to make these inferences, students must engage in discourse, which I am initially defining here as written and spoken language and the accompanying gestures. This requires that students work on open ended, contextual problems that provide opportunities for discourse. An area that provides opportunities for discourse includes functions and the relationship between the covarying quantities that the function expresses. </p><p> By investigating discourse and covarying quantities, I will attempt to answer two, related research questions. What is the nature of students' use of metaphor and gesture when working collaboratively on tasks designed to provide opportunities for covariational reasoning? What information might the students' use of metaphor and gesture provide about the student's covariational reasoning? In order to answer these two questions, I analyzed data from four, ninth grade students during work on two task-based interviews in which the students completed a version of a widely-used bottle problem. The data analysis consisted of multiple passes coding for the quantitative operation, gesture and metaphor used by the students. </p><p> Gesture and metaphor helped make inferences about the quantitative operation the students were using and whether they were comparing or coordinating covarying quantities. The students' gesture allowed me to infer more about the underlying imagery they were using than did metaphor, however, the two were most powerful when considered together. Two of the four students were primarily comparing amounts of change in the two quantities and the other two students coordinated the two quantities. The results led me to a conjecture about the relationship of language, imagery and gesture, and how this relationship might be used in both educational and research settings. </p><p> I proposed a relationship between imagery, language and gesture that I referred to as the Language-Imagery-Gesture Triad with imagery and gesture forming the foundation supporting language. Linguistic structures such as metonymy and metaphor facilitate the relationship between imagery and language. </p>

Source: Dissertation Abstracts International, Volume: 40-09, Section: A, page: 4936. / Thesis (Ph.D.)--The Florida State University, 1979.

This study investigated, first students' use of rote and meaningful strategies to explain their computational procedures in division of decimals, and then their abilities to comprehend meaningful strategies in division of decimals and apply them to a meaningful explanation of their computational procedures. / Thirty-eight students randomly selected from a sample of 152 ninth and tenth grade students enrolled in Algebra One and Geometry classes were individually interviewed about their computational strategies on four division of decimal questions. After each computation, 28 of these students were guided through a discussion routine involving a problem circumstance that focused on one of these four computational strategies: measurement, money, fraction and estimation. Following each problem circumstance, they were given an opportunity to return to the prior computation question and apply the particular strategy to an explanation of their computational procedures. / The majority of the 28 students used predominantly rote explanations on the first computation question, before the introduction of problem circumstance one. Then the ten students who were not exposed to the problem circumstances used rote explanations on all four computation questions. All four problem circumstances were comprehended by the 28 students. Only in the case of estimation did a substantial majority of the students use the specific strategy being illustrated by the problem circumstance. However, after the first problem circumstance, there was a decided shift to more meaningful explanations of computational procedures. / This study should be replicated with improved sampling techniques to include high school students in all types of mathematics classes. Future research should also consider the alternative interview designs suggested in the text of the dissertation. / Source: Dissertation Abstracts International, Volume: 43-05, Section: A, page: 1456. / Thesis (Ph.D.)--The Florida State University, 1982.

This study was an initial assessment of mathematics instruction in Egyptian elementary schools. Models used were the International Study of Achievement in Mathematics (IEA) and research in the U.S.A. in basic division facts. Five hypotheses tested in the IEA study were revised and tested, covering teacher preservice training, teacher inservice training, students' parents' income, students' parents' occupation, students' sex. Student achievement was based on scores on a test of 80 basic division facts. Five additional IEA hypotheses were discussed, covering school enrollment, class size, teacher freedom, weekly hours of mathematics instruction, urban/rural differences. / The basic division facts test was administered to nine classes (310 students) in three elementary schools in Al-Minya City, Egypt--a third, fourth and fifth grade class in each school. Data were also collected about students, teachers and schools. / Multiple regression analysis was used to determine the relationship between students' performance on the division facts test and the five variables of the five hypotheses tested. Also processed were class frequency distributions, means, standard deviations, lists of item difficulty and item analysis. Comparisons were made between study results and corresponding results from England, Germany and the U.S.A. / Conclusions for Third Grade. (1) The mean test score of about 47 indicated lack of mastery of basic division facts. (2) Teacher preservice training, inservice training and parents' occupation level explained 20% of the variance and were the only variables which had a significant relationship with student performance on the division facts test. / Conclusions for Fourth Grade. (1) The mean score of about 73 indicated sufficient mastery of the basic division facts. (2) Preservice training for teachers explained 8% of the variance and was the only variable having a significant relationship with student performance. / Conclusions for Fifth Grade. (1) The mean score of about 68 indicated sufficient mastery of division facts. (2) Both preservice and inservice teacher training contributed 4% to an explanation of the variance, and were the only variables making a significant contribution or having a significant relationship to student performance. / Source: Dissertation Abstracts International, Volume: 43-02, Section: A, page: 0388. / Thesis (Ph.D.)--The Florida State University, 1982.

The purpose of this study was to investigate certain aspects of the task of helping prospective elementary school teachers improve their problem-solving abilities. The subjects were 29 prospective elementary school teachers enrolled in a mathematics education methods course at the Florida State University. A computational prerequisite skills test was administered to the class 12 days before instruction began and one week before the problem-solving pretest was administered. The class then received five hours of instruction on the use of the table or the model strategies to solve unfamiliar problems. A problem solving posttest was given on the day immediately following the instructional period. / The class was divided into 14 matched pairs based on the results obtained from these three tests, with one student from each pair randomly assigned to the experimental group, and the other to the control group. The experimental group taught one problem-solving strategy to a small group of elementary school children for two days. Concurrently, the control group practiced solving problems by using both strategies. Six weeks later the retention test was administered to both groups. / Results indicate that students do have the necessary computational skills to solve problems appropriate for grades 1-6. On the problem-solving pretest most students were not able to use table or model strategies to solve nonroutine problems. Only 10% scored 70% or above on the pretest (mean = 27.1%). The results indicated that students can learn to use tables and models to solve nonroutine problems. On the problem-solving posttest 69% scored 70% or above (mean = 77.5%). The teaching experience (experimental group) seemed to be valuable in helping students retain the use of table and model strategies. The experimental group mean was 6.5% higher than the control group mean; the experimental group continued using the instructional strategies more often than the control group, and students' comments indicated a positive attitude toward teaching problem solving strategies to elementary school children. / Source: Dissertation Abstracts International, Volume: 44-11, Section: A, page: 3312. / Thesis (Ph.D.)--The Florida State University, 1983.

The purpose of this study was to determine whether the achievement of the specified entering prerequisites facilitated the achievement of the major learning outcomes in selected mathematics courses by determining, for each course, whether the achievement of the major learning outcomes by a group of students who achieved the entering prerequisities was significantly higher than the achievement of the major learning outcomes by a group of students who did not achieve these prerequisites. / The subjects for the study were the students enrolled in each of the selected learning components (59 in Interim Algebra, 165 in College Algebra, and 52 in Plane Trigonometry) at Pensacola Junior College who took both the entering prerequisite and major learning outcome assessment tests for that learning component. These students, in each learning component, were divided into two groups; one group achieved at least 70% of the entering prerequisites and the other group achieved less than 70% of these prerequisites. The following null hypothesis, therefore, were established for each course. / Null hypothesis. The achievement mean on the posttest of major learning outcomes for a course will not be significantly higher (.05 level) for the group who had achieved the entering prerequisites for that course than it will be for the group who had not achieved these prerequisites. / Four learning outcome-based assessment tests were developed and used to assess the achievement of the entering prerequisites and the major learning outcomes for the selected learning components at the beginning and end of the term respectively. In addition, the procedures for the study included an academic program mapping approach. The validity of the program map depends on the rejection of the null hypotheses. Based on the t-test, the null hypothesis for each learning component was rejected. / Conclusions. It was concluded that: (1) The specified entering prerequisites facilitated the achievement of the major learning outcomes in the mathematics courses selected for the study. (2) The academic program contains a valid program map. / Source: Dissertation Abstracts International, Volume: 45-08, Section: A, page: 2428. / Thesis (Educat.D.)--The Florida State University, 1984.

The purpose of this study was to develop and evaluate a classroom model for diagnosing the problem-solving skills of elementary school students. The resultant model includes (a) traditionally administered tests that assess the students's ability to solve typical textbook problems, and (b) a test using the Logo computer lanuage that assesses the degree of procedural thought used by the student when solving process problems. The model was field tested with 12 students in Grades 4.9 through 6.9. A case study was written for each subject in the form of a diagnostic report. The model was evaluated by comparing the predicted and actual behaviors demonstrated by each subject while solving three textbook and two process problems. It was decided that the model would be considered a viable method for diagnosing problem-solving skills if the predictions matched the performance for 80% of the subjects. Results indicated that (a) the model can provide information for accurately diagnosing the problem-solving skills of selected elementary school students, (b) Logo can be used to assess the degree of procedural thinking students use when solving process problems, and (c) students seem to solve dissimilar problems using a consistent degree of procedural thinking. / Source: Dissertation Abstracts International, Volume: 45-09, Section: A, page: 2790. / Thesis (Ph.D.)--The Florida State University, 1984.

Purpose. This exploratory case study investigated the problem-solving strategies used by secondary mathematics teachers to solve and teach word problems conducive to proportional solutions. / Method. In individual interviews 20 randomly selected teachers described their thinking as they (1) solved four problems, two depicting direct relationships and two inverse; (2) explained how they would teach two of the problems; and (3) solved four similar problems after being asked to use proportional approaches. Responses were categorized using an eight-category classification scheme. Three categories are proportional--proportion formula (a/b = c/d form), proportion strategy (correct strategies which research indicates secondary students prefer to the formula) and proportion attempt (an incorrect use of proportion). The remaining categories are no answer, intuitive, additive (a focus on difference rather than ratio which is a common error of secondary students), algebraic (a correct equation other than the proportion formula), and Other. / Findings. On the first task, 74% of the responses were correct. Performance was almost perfect on the direct items; only half the responses to the inverse items were correct. There was no significant change in performance when teachers attempted to use proportional strategies. Middle school teachers were less successful than high school teachers, chiefly due to difficulties with the inverse items. When simply asked to solve the problems, the teachers used a variety of strategies which tended to vary with the problem; algebra accounted for one-fourth of the solutions and proportional approaches for one-half. The proportion formula was used far more often than other proportional strategies. The additive strategy was not used. When asked to use proportional approaches, the teachers increased their use of the proportion formula. The majority indicated that they would use the same strategies to solve and teach the selected items. Few teachers used informal proportional strategies to solve or teach these problems. The most unique and ingenious solutions tended to be devised on the inverse items for which many teachers had no ready solutions. / Source: Dissertation Abstracts International, Volume: 44-06, Section: A, page: 1715. / Thesis (Ph.D.)--The Florida State University, 1983.

Much research has been conducted by P. C. Wason and his followers that studies subjects' responses to a selection task. Most of this research excludes from consideration any subjects who had been exposed to a formal study of logic. It had been observed, by the author of this study, that those who had taken a course in logic were also subject to the same errors. Therefore this study included some instruction in symbolic logic. / The subjects for this experiment were a group of college students taking a course in Finite Mathematics. The instructional part of the experiment consisted of six class sessions in symbolic logic. Two posttests were given after the instruction. The first posttest contained question on symbolic logic and a question on a selection task. The success rate for the symbolic logic was very high. The selection task had a low success rate and very similar to the earlier results reported by Wason and his followers. / Because it was anticipated that errors would be made on the selection task, a second posttest was given to attempt to locate the source of error. The purpose of the second posttest was to find out if the subjects were able to identify, as equivalent, verbalizations in a variety of structural forms. The four structural forms selected for this posttest were: causally related, unrelated, perverse, and abstract. The overall performance of the subjects on this task indicated that they had a great deal of difficulty in identifying the equivalent forms. An open question as a result of the study is whether the low success rate was caused by lack of ability to translate the equivalent forms into symbolic form or by the individuals attempting to answer the question of equivalence without reference to the logic that they had learned. / The study did reveal that subjects, with previous experience in logic, do not have any apparent superior performance on the selection task. The study also reveals that these subjects had a great deal of difficulty in recognizing logically equivalent statements when stated verbally. It is left to further investigation to determine the cause of the high error rate. The subjects had not been given any practice in translating the verbal statements into symbolic form and thus it could have been lack of ability. Experience in such translation could greatly improve the results. / Source: Dissertation Abstracts International, Volume: 42-06, Section: A, page: 2547. / Thesis (Ph.D.)--The Florida State University, 1981.

Source: Dissertation Abstracts International, Volume: 40-10, Section: A, page: 5351. / Thesis (Ph.D.)--The Florida State University, 1979.