Probability is an area of mathematics that remains a mystery to many people and is problematic for others who engage in its study in secondary school or its use in fields such as science and business. Yet, ordinary people frequently encounter the informal application of aspects of probability in daily-life situations that require decision-making under uncertainty, understanding of random behaviour and consideration of likelihood. With the purpose of discovering more about children’s ‘natural’ probability strategies, in a particular part of the world, task-based interviews were conducted with 74 children aged four to twelve years from three schools. These children had not received any formal instruction in probability, as it was not part of their school curriculum. The children’s interaction with games involving random generators prompted a range of intuitive strategies for making probabilistic judgments. These invented strategies are related to the development of proportional reasoning, but are also interlaced with the development of understanding of randomness. Examination of the strategies revealed an age-related hierarchy of sophistication of reasoning and mathematical precision. The study confirmed the presence of three developmental stages, but also revealed two distinct transitional stages not reported in previous research. The characteristics of children’s thinking at each stage of development provide a further research on probabilistic thinking and contribute to planning for teaching. / Doctor of Philosophy (PhD)

The purpose of this descriptive case study analysis was to provide portraits of the methods college students used to solve probability problems and the factors that supported or impeded their success prior to and after two-week instruction on probability. Fourteen-question Pre- and Post-Instructional Task-Based Questionnaires provided verbal data of nine participants enrolled in a college finite mathematics course while solving problems containing simple, compound, independent, and dependent probabilistic events. Overall, the general method modeled by the more successful students consisted of the student reading the entire problem, including the question; breaking down the problem into sections, analyzing each section separately; using the context of the question to reason a solution; and checking the final answer. However, this ideal method was not always successful. While some less successful students tried to use this approach when solving their problems, their inability to work with percents and fractions, to organize and analyze data within their own representation (Venn diagram, tree diagram, table, or formula), and to relate the process of solving word problems to the context of the problem hindered their success solving the problem. In addition, the more successful student exhibited the discipline to attend the class, to try their homework problems throughout the section on probability, and to seek outside help when they did not understand a problem. However, students did try alternate unsuccessful methods when attempting to solve probability problems. While one student provided answers to the problems based on his personal experience with the situation, other students sought key words within the problem to prompt them to use a correct representation or formula, without evidence of the student trying to interpret the problem. While most students recognized dependent events, they encountered difficulty stating the probability of a dependent event due to their weakness in basic counting principles to find the size of the sample space. For those students who had not encountered probability problems before the first questionnaire, some students were able to make connections between probability and percent. Finally, other inexperienced students encountered difficulty interpreting the terminology associated with the problems, solving the problem based on their own interpretations. / Graduation date: 2003

Lawrence S. Frishkopf. / "March 1, 1956." "This work was presented as a thesis submitted to the Department of Physics, M.I.T., 1956, in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Bibliography: p. 71-74. / Army Signal Corps Contract DA36-039 sc-42607 Project 102B Dept. of the Army Project 3-99-10-022

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Thesis (M.Sc.)--Memorial University of Newfoundland, 2001. / Bibliography: leaves 59-62.

Thesis (M.S.)--University of Alabama at Birmingham, 2009. / Title from PDF title page (viewed Jan. 21, 2010). Additional advisors: Yulia Karpeshina, Ian Knowles, Rudi Weikard. Includes bibliographical references (p. 19).

Thesis (Ph.D.)--York University, 2008. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 81-83). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR39029

Thesis (Ph. D.)--Florida State University, 2003. / Advisor: Ladislav Kohout, Florida State University, College of Arts and Sciences, Dept. of Computer Science. Title and description from dissertation home page (viewed Mar. 9, 2004). Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Department of Statistics, August 2003. / Includes bibliographical references. Also available on the Internet.