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

This study aimed to investigate students' models of probability in a modern physics context. The study was divided into three phases. The first phase explored student pre-knowledge about probability before modem physics instruction. The second phase investigated student understanding of concepts related to probability such as wave-particle behavior, the uncertainty principle, and localization. The third phase probed how students used the wave function to interpret probability in potential energy problems. The participants were students taking modem physics at Oregon State University. In the first phase, we developed a diagnostic test to probe mathematical probability misconceptions and probability in a classical physics content. For the mathematical probability misconceptions part, we found that students often used a randomly distributed expectancy resource to predict an outcome of a random event. For classical probability, we found that students often employed an object's speed to predict the probability of locating it in a certain region, which we call a classical probability reasoning resource. In the second and the third phases, we interviewed students in order to get more in-depth data. We also report the findings from Fall 03 preliminary interviews which indicated the need for a more detail theoretical framework to analyze student reasoning. Therefore, we employed the framework proposed by Redish (2003) to analyze the interview data into two perspectives - reasoning resources and epistemic resources. We found that most students used a classical probability resource to interpret the probability from the wave function. Additionally, we identified two associated patterns that students used to describe the traveling wave function in the potential step and barrier. Finally, we discuss some teaching implications and future research that the findings suggested. / Graduation date: 2005

As a result of the calls for reform in mathematics education and the ever-changing nature of mathematics, today's teachers face the challenge of teaching unfamiliar content in ways that are equally unfamiliar. In view of this challenge, the purpose of this study was to investigate middle school teachers' subject matter and pedagogical content knowledge of probability and its relationship to the teaching of probability. The study also explored the nature of the instructional tasks and classroom discourse during probability instruction. Case study methodology was used to examine the knowledge and practice of 4 middle school teachers. A pre-observation interview assessed the teachers' subject matter knowledge of probability. The teachers were then observed as they taught probability. Post-observation interviews further explored teacher knowledge and its relationship to teaching practice. Data sources included interview transcripts, observational field notes, video and audiotapes of classroom instruction, and written instructional documents. Individual case studies were written describing the teachers' background and probability instruction. Cross-case analyses compared and contrasted the cases in response to the research questions. The results of this study indicate the teachers generally (a) lacked an explicit and connected knowledge of probability content, (b) held traditional views about mathematics and the learning and teaching of mathematics, (c) lacked an understanding of the "big ideas" to be emphasized in probability instruction, (d) lacked knowledge of students' possible conceptions and misconceptions, (e) lacked the knowledge and skills needed to orchestrate discourse in ways that promoted students' higher level learning, and (f) lacked an integrated understanding of the nature of the reform. One teacher captured the essence of the reform effort in her probability instruction; the other 3 teachers generally fell short of the goal despite their efforts to implement aspects of the reform. Although students were actively involved in exploring probability content through the use of games, simulations, and other hands-on instructional tasks, the cognitive level of the tasks and discourse was limited by the nature of instruction. The findings of this study have implications for mathematics education reform, preservice teacher preparation, staff development, and curriculum development. / Graduation date: 1998

This study considered spontaneous versus feedback induced changes in probability strategies using grouped trials of two-choice problems. Third and sixth grade Anglo and Apache children were the focus of computer assisted instruction and diagnostics designed to maximize performance and measure understanding of probability concepts. Feedback, using indeterminate problems directed at specific strategies, in combination with a large problem set permitted examination of response latency and hypothesis alternation. Explicit training, in the form of computer based tutorials administered feedback as: (a) correctness and frequency information, (b) mathematical solutions, or (c) in a graphical format, targeted by weaknesses in the prevailing strategy. The tutorials encouraged an optimal proportional strategy and sought to affect the memorial accessibility or availability of information through the vividness of presentation. As the subject's response selection was based on the query to select for the best chance of winning, each bucket of the two-choice bucket problems was coded as containing target or winner (W) balls and distractor or loser (L) balls. Third and sixth grade subjects came to the task with position oriented strategies focusing on the winner or target elements. The strategies' sophistication was related to age with older children displaying less confusion and using proportional reasoning to a greater extent than the third grade children. Following the tutorial, the subjects displayed a marked decrease in winners strategies deferring instead to strategies focusing on both the winners and losers; however, there was a general tendency to return to the simpler strategies over the course of the posttest. These simpler strategies provided the fastest response latencies within this study. Posttest results indicated that both third and sixth grade subjects had made comparable gains in the use of strategies addressing both winners and losers. Based on the results of a long-term written test, sixth grade subjects appeared better able to retain or apply the knowledge that both winners and losers must be considered when addressing the two-choice bucket problems. Yet, for younger children, knowledge of these sophisticated strategies did not necessarily support generalization to other mathematical skills such as fraction understanding.

A Research Report submitted to the Faculty of Education, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Education by course-work and research report. Johannesburg, 1992 / This research report investigates the psychological dimensions in the learning and teaching of probability theory. It begins by outlining some problems arising from the author's own experience in the learning and teaching of probability theory, and develops a theoretical position using the Theory of Activity. This theory places education within the broad social context and recognises the centrality of affective aspects of cognition. [Abbreviated abstract. Open document to view full version]

A research report submitted to the WITS SCHOOL OF EDUCATION, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science (Science Education). Johannesburg 2015 / Using qualitative methods, this study reports on the selection and use examples in Probability by a teacher in a multilingual mathematics classroom where learners learn in a language which is not their first or home language. The study involved one teacher together with his Grade 11 multilingual class in a township school in Ekurhuleni South Johannesburg. Data was collected through audio-visual recording of four lessons. In addition two one-on-one semi-structured interviews were conducted with the teacher. Data was analysed using Rowland‘s (2008) categories of exemplification alongside Staples' (2007) conceptual model of collaborative inquiry mathematics practices. In the study it emerged that it is important for teachers to select examples by considering the context, ability of the example to be generalised, consistency in the use of symbols, syllabus requirements and accessibility. It also emerged that the selection of examples together with the accompanying mathematical practices has the potential to support or impede the learning of mathematics. In particular the findings revealed that the practice of ‗guiding the learners with the map‘ declines the cognitive level of examples and hence impedes learning. Code- switching and re-voicing were most frequently used practices seen in the findings with the use of code-switching encouraging full participation of the learners. The study recommends that methodology courses offered at tertiary institutions to pre-service teachers should include the selection, how to select or design and use examples in multilingual classrooms e.g. what constitutes a good example and how to maintain the cognitive level of an example. The study also recommends that more research needs to be done on effective mathematical practices that may be used to implement worked-out examples in multilingual classrooms.

Includes bibliographies and index.

Includes bibliographies and index. Electronic publication; Full text available in PDF format; abstract in HTML format. The teaching of probability in schools provides a good opportunity for examining how a new topic is integrated into a school curriculum. Furthermore, because probabilistic thinking is quite different from the deterministic thinking traditionally found in mathematics classrooms, such an examination is particularly able to highlight significant forces operating within educational practice. After six chapters which describe relevant aspects of the philosophical, cultural, and intellectual environment within which probability has been taught, a 'Broad-Spectrum Ecological Model' is developed to examine the forces which operate on a school system. Electronic reproduction.[Australia] :Australian Digital Theses Program,2001. 2 v. (xxxi, 1023 p.) : ill. ; 30 cm.

The teaching of probability in schools provides a good opportunity for examining how a new topic is integrated into a school curriculum. Furthermore, because probabilistic thinking is quite different from the deterministic thinking traditionally found in mathematics classrooms, such an examination is particularly able to highlight significant forces operating within educational practice. After six chapters which describe relevant aspects of the philosophical, cultural, and intellectual environment within which probability has been taught, a 'Broad-Spectrum Ecological Model' is developed to examine the forces which operate on a school system. The Model sees school systems and their various participants as operating according to general ecological principles, where and interprets actions as responses to situations in ways which minimise energy expenditure and maximise chances of survival. The Model posits three principal forces-Physical, Social and Intellectual-as providing an adequate structure. The value of the Model as an interpretative framework is then assessed by examining three separate aspects of the teaching of probability. The first is a general survey of the history of the teaching of the topic from 1959 to 1994, paying particular attention to South Australia, but making some comparisons with other countries and other states of Australia. The second examines in detail attempts which have been made throughout the world to assess the understanding of probabilistic ideas. The third addresses the influence on classroom practice of research into the teaching and learning of probabilistic ideas. In all three situations the Model is shown to be a helpful way of interpreting the data, but to need some refinements. This involves the uniting of the Social and Physical forces, the division of the Intellectual force into Mathematics and Mathematics Education forces, and the addition of Pedagogical and Charismatic forces. A diagrammatic form of the Model is constructed which provides a way of indicating the relative strengths of these forces. The initial form is used throughout the thesis for interpreting the events described. The revised form is then defined and assessed, particularly against alternative explanations of the events described, and also used for drawing some comparisons with medical education. The Model appears to be effective in highlighting uneven forces and in predicting outcomes which are likely to arise from such asymmetries, and this potential predictive power is assessed for one small case study. All Models have limitations, but this one seems to explain far more than the other models used for mathematics curriculum development in Australia which have tended to see our practice as an imitation of that in other countries. / Thesis (Ph.D.)--Graduate School of Education and Department of Pure Mathematics, 2001.

In the last three decades, there has been a significant growth in the number of undergraduate students taking introductory statistics. As a result, there is a need by universities and community colleges to find well-qualified instructors and graduate teaching assistants to teach the growing number of statistics courses. Unfortunately, research has shown that even teachers of introductory statistics struggle with concepts they are employed to teach. The data presented in this research sheds light on the statistical knowledge of graduate teaching assistants (GTAs) and community college instructors (CCIs) in the realm of probability by analyzing their work on surveys and task-based interviews on the p-value. This research could be useful for informing professional development programs to better support present and future teachers of statistics.