Following international trends, statistics is a relatively new addition to the South African mathematics curriculum at school level and its implementation was fraught with problems. Since 2001 teaching statistics in the Further Education and Training Phase (Grades 10 to 12) has been optional due to lack of professional development of teachers. From 2014 teaching statistics will be compulsory. This study is therefore timely as it provides information about different discourses in discussions of an ill-structured problem in a data-rich context, as well as in discussions of the meaning of the statistical mean. A qualitative case study of informal statistical reasoning was conducted with a group of students that attended an introductory course in descriptive statistics as part of an honours degree in mathematics education at the University of the Witwatersrand. The researcher was the course lecturer. Transcripts of the discussions in four video recorded sessions at the start of the semester long course form the bulk of the data. The discussions in the first three sessions of the course were aimed at structuring the data-context, or grasping the system dynamics of the data-context, as is required at the start of a cycle of statistical investigation. The discussion in the fourth session was about the syntactical meaning of the mean algorithm. It provides guidelines for meaningful disobjectification of the well-known mean algorithm. This study provides insight into informal statistical reasoning that is currently described as idiosyncratic or verbal according to statistical reasoning models. Discourse analysis based on Sfard’s (2008) theory of Commognition was used to investigate and describe discursive patterns that constrain shifting from colloquial to informal statistical discourse. The main finding is that colloquial discourse that is aimed at decision making in a data-context is incommensurable with statistical discourse, since comparison of data in the two discourses are drawn on incommensurable scales – a qualitative evaluation scale and a quantitative descriptive scale. The problem of comparison on a qualitative scale also emerged in the discourse on the syntactical meaning of the mean algorithm, where average as a qualitative judgement conflicted with the mean as a quantitative measurement. Implications for teaching and teacher education are that the development of statistical discourse may be dependent on alienation from data-contexts and the abstraction of measurements as abstract numerical units. Word uses that confound measurements as properties of objects and measurements as abstract units are discussed. Attention to word use is vital in order to discern evaluation narratives as deed routines from exploration narratives and routines.

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The past two decades have seen a substantial increase in the availability of numerical data that individuals are faced with on a daily basis. In addition, research uncovering the multiple facets of statistical reasoning has become increasingly prominent. Both gender differences and the effect of experience or training have emerged as two key factors that influence performance in statistics. Surprisingly, though, the combined effects of these two variables have not been studied. This gap in understanding the joint effect of gender and experience on statistical reasoning is addressed in the present dissertation with six studies. In Study 1 (N = 201), participants with various levels of experience in statistics were asked to complete the Statistical Reasoning Assessment (SRA; Garfield, 2003). Although the performance of both genders improved with experience, the gender gap persisted across all experience levels. Multiple measures of individual differences were used in a confirmatory structural equation model. This model supported the idea that differences in statistical reasoning are not uniquely a matter of cognitive ability. In fact, gender was found to influence statistical reasoning directly, as well as indirectly through its influence on thinking dispositions. In Studies 2 (N = 67), 3 (N = 157), and 4 (N = 206), the role of stereotype threat was examined as a potential cause of the persisting gender gap in statistics, and value affirmation was tested as an intervention to overcome stereotype threat. Despite the fact that many women believed negative stereotypes about the ability of women in statistics, value affirmation had no significant impact on performance. To help explain this lack of effect, and in keeping with the results of the structural equation model suggesting a multi-pronged approach, efforts were turned towards a different (and potentially richer) cognitive factor. Specifically, mental representations were explored to help shed light on the root causes of those conceptual understanding differences in statistics. In Studies 5 and 6, gender differences in mental representations of statistical features were examined using a categorization paradigm. In Study 5 (N = 219), extending some of the key findings in Studies 1, 3 and 4, it was established that two courses in statistics are necessary to create a significant difference in the quality of mental representations of statistical concepts. More importantly, Study 6 (N = 208) demonstrated how constraining the task format particularly benefits women in that the quality of their reasoning significantly improved, where that of men was equal across tasks. Theoretical and practical implications of these findings are discussed.

The past two decades have seen a substantial increase in the availability of numerical data that individuals are faced with on a daily basis. In addition, research uncovering the multiple facets of statistical reasoning has become increasingly prominent. Both gender differences and the effect of experience or training have emerged as two key factors that influence performance in statistics. Surprisingly, though, the combined effects of these two variables have not been studied. This gap in understanding the joint effect of gender and experience on statistical reasoning is addressed in the present dissertation with six studies. In Study 1 (N = 201), participants with various levels of experience in statistics were asked to complete the Statistical Reasoning Assessment (SRA; Garfield, 2003). Although the performance of both genders improved with experience, the gender gap persisted across all experience levels. Multiple measures of individual differences were used in a confirmatory structural equation model. This model supported the idea that differences in statistical reasoning are not uniquely a matter of cognitive ability. In fact, gender was found to influence statistical reasoning directly, as well as indirectly through its influence on thinking dispositions. In Studies 2 (N = 67), 3 (N = 157), and 4 (N = 206), the role of stereotype threat was examined as a potential cause of the persisting gender gap in statistics, and value affirmation was tested as an intervention to overcome stereotype threat. Despite the fact that many women believed negative stereotypes about the ability of women in statistics, value affirmation had no significant impact on performance. To help explain this lack of effect, and in keeping with the results of the structural equation model suggesting a multi-pronged approach, efforts were turned towards a different (and potentially richer) cognitive factor. Specifically, mental representations were explored to help shed light on the root causes of those conceptual understanding differences in statistics. In Studies 5 and 6, gender differences in mental representations of statistical features were examined using a categorization paradigm. In Study 5 (N = 219), extending some of the key findings in Studies 1, 3 and 4, it was established that two courses in statistics are necessary to create a significant difference in the quality of mental representations of statistical concepts. More importantly, Study 6 (N = 208) demonstrated how constraining the task format particularly benefits women in that the quality of their reasoning significantly improved, where that of men was equal across tasks. Theoretical and practical implications of these findings are discussed.

Increased attention to statistical concepts has been a prevalent trend in revised mathematics curricula across grade levels. However, the preparation of secondary school mathematics educators has not received similar attention, and learning opportunities provided to these educators is oftentimes insufficient for teaching statistics well. The purpose of this study is to analyze pre-service teachers' conceptions about confidence intervals. This research inquired about statistical reasoning from the perspective of students majoring in mathematics education enrolled in an undergraduate statistics education course who have previously completed an introductory course in statistics. We found common misconceptions among pre-service teachers participating in this study. An unanticipated finding is that all the pre-service teachers believed that the construction of a confidence interval relies on a sampling distribution that does not contain every possible sample. Instead, they believed it is necessary to take multiple samples and build a distribution of their means. I called this distribution the Multi-Sample Distribution (MSD).

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1 SUMMARY In general this thesis deals with the question whether or to what extent human thinking is rational in terms of the optimality of the way people achieve their goals and in terms of the consistency between people's beliefs and the structure of the world. This question is quite difficult to answer unequivocally because the answer will always depend on the nature of the particular task and the exact way in which we define rationality. Among other things, that's the reason why we can meet two contradictory schools of thought within the so-called Great Rationality Debate, one of which is convinced of the systematic irrationality of human thinking (in the sense of the systematic deviation of human thinking from normative predictions stemming from the principles of rational thinking as they are captured by the statistical theory of probability, formal logic or decision theory), while the other one considers human thought to be more or less rational, and finds the source of its (alleged) failure elsewhere. In the case of the latter, however, the question is how to explain the apparent existence of irrational behavior and interindividual differences in such behavior. One possible answer to this question is illustrated by Griffiths and Tenenbaum's Bayesian model of causal reasoning based on perceived...

Made available in DSpace on 2016-04-29T14:23:36Z (GMT). No. of bitstreams: 1 Antonio Carlos Bento.pdf: 19317474 bytes, checksum: 3345ae7a48f0769e446ab9888361725d (MD5) Previous issue date: 2015-06-18 / The research focuses on the performance evaluation methodology within the models of multiple-choice questions from methodeutics criteria of Fuzzy Logic proposed by Lotfi Zadeh associated with statistical reasoning presented by Santos, Pigari and Tolentino present in the theories of uncertainty discussed by Russell and Norvig and item response theory presented by Fontanive. The study was based on an exploratory and experimental research used questionnaires that were submitted to academic groups. The collected data were compared with a bibliography on the field evaluation, education and logic, resulting in the organization of a methodology that enabled the construction of a prototype computer system to validate the realized and expected outcomes studies. Following the classification s heuristics Clancey it concludes by identifying a set of rules that have been identified as strong enough to guide work and software modeling, aimed at building models of multiple-choice questions based Fuzzy Logic rules. Finally, the results were gathered and presented by means of a system prototype able to characterize the main established rules, as well as studies, being able to be fully utilized in the implementation of performance assessments that may present evidence with expected results, supporting identifying random responses during evaluations / A pesquisa enfoca a metodologia da avaliação de desempenho, dentro dos modelos de questões do tipo múltipla escolha, a partir de critérios metodêuticos da Lógica Fuzzy (Lógica Difusa ou nebulosa) proposta por Zadeh Lotfi associados ao raciocínio estatístico apresentado por Santos, Pigari e Tolentino presente nas teorias da incerteza discutido por Russell e Norvig e teoria da resposta ao item apresentado por Fontanive. O estudo partiu de uma pesquisa exploratória e experimental que utilizou questionários que foram submetidos a grupos de acadêmicos. Os dados colhidos foram cotejados com uma bibliografia sobre o campo de avaliação-educação e lógica, resultando na organização de uma metodologia que possibilitou a construção de um sistema protótipo computacional para a validação dos estudos realizados e resultados esperados. Seguindo a Classificação Heurística de Clancey a pesquisa conclui pela identificação de um conjunto de regras que foram identificadas como suficientemente fortes para nortear trabalhos e de modelagem de softwares, visando a construção de modelos de questões do tipo múltipla escolha baseado em regras de Lógica Fuzzy. Finalmente, os resultados foram reunidos e apresentados por meio de um sistema protótipo capaz de caracterizar as principais regras estabelecidas, bem como os estudos realizados, sendo capaz de serem utilizados plenamente na aplicação de avaliações de desempenho que possam apresentar evidências com resultados esperados, oferecendo suporte a identificação de respostas ao acaso durante avaliações

Made available in DSpace on 2016-04-27T16:58:51Z (GMT). No. of bitstreams: 1 Roberto Canossa.pdf: 1455069 bytes, checksum: 4384e636c4f411f9c2c2a3d9efb4dbb2 (MD5) Previous issue date: 2009-03-02 / Secretaria da Educação do Estado de São Paulo / The study of Statistics became, in 1997, part of the curriculum of basic school, therefore, the preparation of the mathematics teachers became necessary in order to approach this theme, once many of the teachers either did not have this content in his/her initial formation or had it, but in a superficial way. The surveys performed by our group have also shown these results. These results and the great difficulties some mathematics teachers from public schools, especially in Diadema, have in developing, with the students, the Statistics contents and its interpretations is our motivation to the realization of this paper. In order to do so, we intended to answer the following question of research: What are the didactics characteristics to a continued formation to High School teachers, aiming at working with concepts of median and quartiles, so that students will be able to make decisions from the analysis of the realized variation, with the help of Dot-Plot and Box-Plot? To such verification, we have elaborated and applied a diagnose questionnaire (appendix 1), we have realized continued formation workshops from the results of these questionnaires, and, at last, we have observed a class from a volunteer teacher. We could notice that the majority of teachers do not work the concepts of median and quartiles: they limit themselves to the concept of mean, variance and standard deviation, inserted only as mathematics formulas, without giving sense to such concepts; besides, they do not have knowledge of the graphs Dot-Plot and Box-Plot. The workshop allowed an advancement concerning reasoning and statistical literacy to the volunteer teacher. However, we could realize that the two sections of workshop were not enough to get to level 5 (integrated process reasoning) of statistical reasoning proposed by Garfield (2002) / O estudo da Estatística passou, em 1997, a fazer parte do currículo da Escola Básica, sendo necessária então a preparação dos professores de matemática para a abordagem desse tema, uma vez que muitos deles não tiveram esse conteúdo em sua formação inicial ou tiveram de forma superficial e tecnicista. As pesquisas realizadas em nosso grupo apontam também para esses resultados. Nossa motivação para a realização deste trabalho deve-se a esses resultados e à grande dificuldade que alguns professores de matemática da rede pública do Estado de São Paulo, especificamente na região de Diadema, têm em desenvolver com seus alunos os conteúdos relativos a Estatística e suas interpretações. Para isso, pretendemos responder a seguinte questão de pesquisa: Quais as características didáticas de uma formação continuada para professores do Ensino Médio, visando o trabalho com conceitos de mediana e quartis, para que os alunos possam tomar decisões a partir da análise da variação percebida, com o auxílio do Dot-Plot e do Box-Plot? Para tal verificação, elaboramos e aplicamos um questionário diagnóstico (apêndice 1), realizamos oficinas de formação continuada a partir dos resultados desse questionário e, por fim, observamos uma aula com a professora colaboradora. O que pudemos notar é que a maioria dos professores não trabalha os conceitos de mediana e quartis: limitam-se aos conceitos de média, variância e desvio-padrão, inseridos apenas com fórmulas matemáticas, sem dar sentido para tais conceitos; além disso, não têm conhecimento dos gráficos Dot-Plot e Box-Plot. A oficina permitiu um avanço no nível de raciocínio e alfabetização estatística da professora colaboradora, mas podemos perceber também que as duas sessões de oficinas realizadas não foram suficientes para chegar ao nível 5 (processos de raciocínio integrados) de raciocínio estatístico proposto por Garfield (2002)

Each year thousands of students enrol in introductory statistics courses at universities throughout Australia, bringing with them formal and informal statistical knowledge and reasoning, as well as a wide range of basic numeracy skills, mathematical inclinations and attitudes towards statistics, which have the potential to impact on their ability to develop statistically. This research develops and investigates measures of each of these components for students at the interface of secondary and tertiary education, and investigates the relationships that exist between them, and a range of background variables. The focus of the research is on measuring and analysing levels and abilities in statistical reasoning for a range of students at the tertiary interface, with particular interest also in investigating their basic numeracy skills and how these may or may not link with statistical reasoning allowing for other variables and factors. Information from three cohorts in an introductory data analysis course, whose focus is real data investigations, provides basis for the research. This course is compulsory for all students in degree programs associated with all sciences or mathematics. The research discusses and reports on the development of questionnaires to measure numeracy and statistical reasoning and the students' attitudes and reflections on their prior school experiences with statistics. Students' attitudes are found to be generally positive, particularly with regard to their self-efficacy. They are also in no doubt as to the links that exist between mathematics and statistics. The Numeracy Questionnaire, developed to measure pre-calculus skills relevant to an introductory data analysis course which emphasises real data investigations, demonstrates that many students who have completed a basic algebra and calculus senior school subject struggle with skills which are in the pre-senior curricula. Direct examination of the responses helps to understand where and why difficulties tend to occur. Rasch analysis is used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrate that a student's numeracy score depends on the result obtained in senior mathematics, whether or not the student is a mathematics student, gender, whether or not higher level mathematics has been studied, self-efficacy and year. The research indicates that either the pre-senior curricula need strengthening or that exposure to mathematics beyond the core senior course is required to establish confidence with basic skills particularly when applied to new contexts and multi- step situations. The Statistical Reasoning Questionnaire (SRQ) is developed for use in the Australian context at the secondary/tertiary interface. As with the Numeracy Questionnaire, detailed examination of the responses provides much insight into the range and features of statistical reasoning at this level. Rasch analyses, both dichotomous and polychotomous, are used to establish the appropriateness of this instrument as a measuring tool at this level. The polychotomous, Rasch partial credit model is also used to define a new approach to scoring a statistical reasoning instrument and enables development and application of a hierarchical model and measures levels of statistical reasoning appropriate at the school/tertiary interface. General linear models indicate that numeracy is a highly significant predictor of statistical reasoning allowing for all other variables including tertiary entrance score and students' backgrounds and self-efficacy. Further investigation demonstrates that this relationship is not limited to more difficult or overtly mathematical items on the SRQ. Performance on the end of semester component of assessment in the course is shown to depend on statistical reasoning at the beginning of semester as measured by the partial credit model, allowing for all other variables. Because of the dominance of the relationship between statistical reasoning (as measured by the SRQ) and numeracy on entry, some further analysis of the end of semester assessment is carried out. This includes noting the higher attrition rates for students with less mathematical backgrounds and lower numeracy.