This research study probes students’ understandings of various aspects of probability in a 3-week Probability unit in a Grade 7 classroom. Informing this study are the perspectives of constructivism and sociocultural theory which underpin the contemporary reform in mathematics education as codified in the NCTM standards and orient much of the teaching and learning of mathematics in today’s classrooms. Elements of culturally responsive pedagogy were also adopted within the research design. The study was carried out in an urban school where I collaborated with the teacher and students as co-teacher and researcher. As the population of this school was predominantly Aboriginal, the lessons included discussion of the tradition and significance of Aboriginal games of chance and an activity based on one of these games. Data sources included the responses in the pre- and post-tests, fleidnotes of the lessons, and audiotapes of student interviews. The key findings of the study are that the students had some understanding of formal probability theory with strongly-held persistent alternative thinking, some of which did not fit the informal conceptions of probability noted in the literature such as the outcome approach and the gambler’s fallacy. This has led to the proposal of a Personal Probability model in which the determination of a probability or a probability decision is a weighting of components such as experience, intuition and judgment, some normative thinicing, and personal choice, beliefs and attitudes. Though the alternative understandings were explored in interviews and resolved to some degree, the study finds that the probability understandings of students in this study are still fragile and inconsistent. Students demonstrated marked interest in tasks that combined mathematics, culture and community. This study presents evidence that the current prescribed learning outcomes in the elementary grades are too ambitious and best left to the higher grades. The difficulties in the teaching and learning of the subject induced by the nuances and challenges of the subject as well as the dearth of time that is needed for an adequate treatment further direct that instructional resources at this level be focused on deepening and strengthening the basic ideas.

This research study probes students’ understandings of various aspects of probability in a 3-week Probability unit in a Grade 7 classroom. Informing this study are the perspectives of constructivism and sociocultural theory which underpin the contemporary reform in mathematics education as codified in the NCTM standards and orient much of the teaching and learning of mathematics in today’s classrooms. Elements of culturally responsive pedagogy were also adopted within the research design. The study was carried out in an urban school where I collaborated with the teacher and students as co-teacher and researcher. As the population of this school was predominantly Aboriginal, the lessons included discussion of the tradition and significance of Aboriginal games of chance and an activity based on one of these games. Data sources included the responses in the pre- and post-tests, fleidnotes of the lessons, and audiotapes of student interviews. The key findings of the study are that the students had some understanding of formal probability theory with strongly-held persistent alternative thinking, some of which did not fit the informal conceptions of probability noted in the literature such as the outcome approach and the gambler’s fallacy. This has led to the proposal of a Personal Probability model in which the determination of a probability or a probability decision is a weighting of components such as experience, intuition and judgment, some normative thinicing, and personal choice, beliefs and attitudes. Though the alternative understandings were explored in interviews and resolved to some degree, the study finds that the probability understandings of students in this study are still fragile and inconsistent. Students demonstrated marked interest in tasks that combined mathematics, culture and community. This study presents evidence that the current prescribed learning outcomes in the elementary grades are too ambitious and best left to the higher grades. The difficulties in the teaching and learning of the subject induced by the nuances and challenges of the subject as well as the dearth of time that is needed for an adequate treatment further direct that instructional resources at this level be focused on deepening and strengthening the basic ideas.

This research study probes students’ understandings of various aspects of probability in a 3-week Probability unit in a Grade 7 classroom. Informing this study are the perspectives of constructivism and sociocultural theory which underpin the contemporary reform in mathematics education as codified in the NCTM standards and orient much of the teaching and learning of mathematics in today’s classrooms. Elements of culturally responsive pedagogy were also adopted within the research design. The study was carried out in an urban school where I collaborated with the teacher and students as co-teacher and researcher. As the population of this school was predominantly Aboriginal, the lessons included discussion of the tradition and significance of Aboriginal games of chance and an activity based on one of these games. Data sources included the responses in the pre- and post-tests, fleidnotes of the lessons, and audiotapes of student interviews. The key findings of the study are that the students had some understanding of formal probability theory with strongly-held persistent alternative thinking, some of which did not fit the informal conceptions of probability noted in the literature such as the outcome approach and the gambler’s fallacy. This has led to the proposal of a Personal Probability model in which the determination of a probability or a probability decision is a weighting of components such as experience, intuition and judgment, some normative thinicing, and personal choice, beliefs and attitudes. Though the alternative understandings were explored in interviews and resolved to some degree, the study finds that the probability understandings of students in this study are still fragile and inconsistent. Students demonstrated marked interest in tasks that combined mathematics, culture and community. This study presents evidence that the current prescribed learning outcomes in the elementary grades are too ambitious and best left to the higher grades. The difficulties in the teaching and learning of the subject induced by the nuances and challenges of the subject as well as the dearth of time that is needed for an adequate treatment further direct that instructional resources at this level be focused on deepening and strengthening the basic ideas. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

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.