Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Juli K. Dixon. Includes bibliographical references (p. 251-259).

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains v, 30 p. Includes abstract. Includes bibliographical references (p. 29-30).

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains xviii, 339 p.; also includes graphics (some col.). Includes bibliographical references (p. 296-306). Available online via OhioLINK's ETD Center

Thesis (Ph. D.)--Illinois State University, 1996. / Title from title page screen, viewed May 24, 2006. Dissertation Committee: Albert D. Otto (chair), Barbara S. Heyl, Cheryl A. Lubinski, Nancy K. Mack, Jane O. Swafford, Carol A. Thornton. Includes bibliographical references (leaves 188-198) and abstract. Also available in print.

The purpose of this study was to determine whether fourth-, sixth-, and eighth-grade students used rational number reasoning to solve comparison of area tasks, and whether the tendency to use such reasoning increased with grade level. The areas to be compared were not similar and therefore, could not directly be compared in a straightforward manner. The most viable solution involved comparing the part-whole relationships inherent in the tasks. Rational numbers in the form of fractional terms could be used to express the part-whole relationships. The use of fractional terms provided a means for students to express the areas to be compared in an abstract manner and thus free themselves from the perceptual aspects of the tasks. The study examined how students solve unique problems in a familiar context where rational number knowledge could be applied. It also noted the effect of introducing fraction symbols into the tasks after students had indicated how they would solve the problems without any reference to fractions. Data were gathered through individual task-based interviews which consisted of 21 tasks, conducted with 36 elementary and junior high school students (12 students each in the fourth, sixth, and eighth grades). Each interview was video and audio taped to provide a record of the students' behavioral and verbal responses. The student responses were analyzed to determine the strategies the students used to solve the comparison of area tasks. The student responses were classified into 11 categories of strategies. There were four Part-Whole Categories, one Part-Whole/Direct Comparison Combination category and six Direct Comparison categories. The results of the study indicate that the development of rational number instruction should include: learning sequences which take students beyond the learning of a set of fraction concepts and skills, attention to the interaction of learning and the visual aspects of instructional models, and the careful inclusion of different types of fractions and other rational number task variables. This study supports the current national developments in curriculum and evaluation standards for mathematics instruction which stress the ability of students to problem solve, communicate, and reason.

A thesis submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / This study explores the potential for Web-based diagnostic assessments in the classroom, with specific focus on certain common challenges experienced by learners in the development of their rational number knowledge. Two schools were used in this study, both having adequate facilities for this study, comprising a well-equipped computer room with one-computer-per-learner and a fast, reliable broadband connection. Prior research on misconceptions in the rational numbers has been surveyed to identify a small set of problem types with proven effectiveness in eliciting evidence of misconceptions in learners. In addition to the problem types found from prior studies, other problem types have been included to examine how the approach can be extended. For each problem type a small item bank was created and these items were presented to the learners in test batteries of between four and ten questions. A multiple-choice format was used, with distractor choices included to elicit misconceptions, including those previously reported in prior research. The test batteries were presented in dedicated lessons to learners over four consecutive weeks to Grade 7 (school one) and Grade 8 (school two) classes from the participating schools. A number of test batteries were presented in each weekly session and, following the learners’ completion of each battery, feedback was provided to the learner with notes to help them reflect on their performance. The focus of this study has been on diagnosis alone, rather than remediation, with the intention of building a base for producing valid evidence of the fine-grained thinking of learners. This evidence can serve a variety of purposes, most significantly to inform the teacher on each learners’ stage of development in the specific micro-domains. Each micro-domain is a fine-grained area of knowledge that is the basis for lesson-sized teaching and learning, and which is highly suited to diagnostic assessment. A fine-grained theory of constructivist learning is introduced for positioning learners at a development stage in each micro-domain. This theory of development stages is the foundation I have used to explore the role of diagnostic assessment as it may be used in future classroom activity. To achieve successful implementation into time-constrained mathematics classrooms requires that diagnostic assessments are conducted as effectively and efficiently as possible. To meet this requirement, the following elements of diagnostic assessments were investigated: (1) Why are some questions better than others for diagnostic purposes? (2) How many questions need to be asked to produce valid conclusions? (3) To what extent is learner self-knowledge of item difficulty useful to identify learner thinking? A Rasch modeling approach was used for analyzing the data, and this was applied in a novel way by measuring the construct of the learners’ propensity to select a distractor for a misconception, as distinct from the common application of Rasch to measure learner ability. To accommodate multiple possible misconceptions used by a learner, parallel Rasch analyses were performed to determine the likely causes of learner mistakes. These analyses were used to then identify which questions appeared to be better for diagnosis. The results produced clear evidence that some questions are far better diagnostic discriminators than others for specific misconceptions, but failed to identify the detailed rules which govern this behavior, with the conclusion that to determine these would require a far larger research population. The results also determined that the number of such good diagnostic questions needed is often surprisingly low, and in some cases a single question and response is sufficient to infer learner thinking. The results show promise for a future in which Web-based diagnostic assessments are a daily part of classroom practice. However, there appears to be no additional benefit in gathering subjective self-knowledge from the learners, over using the objective test item results alone. Keywords: diagnostic assessment; rational numbers; common fractions; decimal numbers; decimal fractions; misconceptions; Rasch models; World-Wide Web; Web-based assessment; computer-based assessments; formative assessment; development stages; learning trajectories.

No description available.

Thesis (Ph.D.)--Australian National University, 2000.

The system of rational numbers can be extended to the real number system by several methods. In this paper, we shall extend the rational number system by means of rational nests of intervals, and develop the elementary properties of the real numbers obtained by this extension.