The purpose of this study was to develop a theoretical model for the use of counterintuitive examples in the introductory non-calculus-based statistics course at the college level. While intuition and misconceptions continue to be of great interest to mathematics and science educators, there has been little research, much less consensus or even internal consistency, in statistics curriculum development concerning the role of examples with counterintuitive results. Because the study intended to provide educators with useful connections to content, instructional methods (e.g., cooperative learning) and learning theory constructs that have been successfully used in mathematics or science education, the model that emerged was organized around a typical syllabus of topics. The study critiqued and the reconciled "Traditional" and "Alternative" perspectives. The Traditional Position attempts to minimize possible confusion and frustration by avoiding such examples, while the Alternative Position uses them to motivate and engage students in critical thinking, active learning, metacognition, communication of their ideas, real-world problem solving and exploration, reflection on the nature and process of statistics, and other types of activities encouraged by current reform movements. The study delineated specific criteria and conditions for selecting and using counterintuitive examples to achieve numerous cognitive and affective objectives. Examples explored include the Monty Hall problem, Simpson's Paradox, the birthday problem, de Mere's Paradox, The Classification Paradox, the Inspection Paradox, and required sample size. The study connected many of these examples (especially Simpson's Paradox) with other conterintuitive examples, with known probability or statistics misconceptions many students have, with representations from other branches of mathematics, and with the constructivist paradigm. Problematic issues addressed include difficulty in constructing assessment instruments and a multiplicity of terminologies and typologies. Additional directions for research were suggested, including several empirical investigations of various facets of the model. The connections, examples, and representations presented should be extremely useful for teachers of statistics, but should also enrich the pedagogy of teachers of other courses.
This descriptive ex post facto study analyzed conceptual understanding in descriptive statistics among 249 traditional age undergraduates at two northeastern universities after an instructional program which decreased the time spent on lecture and added activities, small group collaboration, and discussion that responded to students' varied learning styles.
This study examined students' development of reasoning about quantitative bivariate data during a one-semester university-level introductory statistics course. There were three research questions of interest: (1) What is the nature, or pattern of change in students' development in reasoning about vbivariate data?; (2) Is the sequencing of bivariate data within a course associated with changes in the pattern of change in students' reasoning about bivariate data?; and (3) Are changes in students' reasoning about the foundational concepts of distribution associated with changes in the pattern of change in students' reasoning about bivariate data?
Innovation and efficiency were examined for their effects on collaboration and learning in two experiments with university students. From the first experiment, the Innovation task promoted more knowledge-sharing behaviors than the Efficiency task. In the second experiment (built from the first experiment), participants learned about the Chi-square formula and their understanding of it was assessed with basic calculation questions, comprehension questions, and difficult transfer problems. As part of the transfer problems, a preparation for future learning (PFL) assessment was used to measure participants' ability to adapt their knowledge of the chi-square formula (Bransford & Schwartz, 1999). Participants in the Innovation condition scored significantly higher on the transfer problems, and Innovation dyads showed he greatest performance on the target PFL question.
This study sought to create a dispositional attribution model to describe differences in the development of statistical proficiency. To what extent can differences in psychological dispositions explain differences in the development of statistical proficiency and, in particular, students' understanding of hypothesis testing? A framework to describe statistical proficiency was created. Study subjects were undergraduates who have taken an algebra-based statistics course. The three emergent themes found in student discussions of hypothesis testing were, how students consider the experimental design factors of a hypothesis test situation, what types of evidence students find convincing, and what students understand about p-values.
This dissertation focused on the notion of distribution as a conceptual link between data and chance. The goal of this study was to characterize a conceptual corridor that contains possible conceptual trajectories taken by students based on their conceptions of probability and reasoning about distributions. A small-group teaching experiment was conducted with six fourth graders to investigate students' development of probability concepts and reasoning about distributions in various chance events over the course of seven weeks. The two major findings are as follows: First, students' qualitative reasoning about distributions involved the conceptions of groups and chunks, middle clump, spreadout-ness, density, symmetry and skewness in shapes, and "easy to get/ hard to get" outcomes. Second, students' quantitative reasoning arose from these quanlitative descriptions of distributions when they focused on different group patterns and compared them to each other.
The study sough to describe the statistical thinking of high school students. The two research questions guiding the study were: (i) What are the defining characteristics of different patterns of high school students' statistical thinking within the processes of describing, organizing and reducing, representing, analyzing, and collecting data? (ii) What levels of statistical thinking can be associated with each of the patterns? In order to answer the two research questions, high school students of various grade levels and mathematical backgrounds and recent high school graduates were asked to solve statistical thinking tasks during clinical interview sessions. The cognitive model described by Biggs and Collis (1982, 1991) was applied in differentiating among patterns of sophistication in the students' responses to the interview tasks. The study identified and characterized levels of thinking which provide the basis of a framework useful for advising instruction, curriculum development, and further research in the area of high school statistics.
Probability and statistical inference are important ideas with a remarkably wide range of applications. However, psychological and instructional studies conducted in the last two decades have consistently documented poor understanding of these ideas among different population across different settings. The purposes of this dissertation study are to understand teachers' understandings of probability and statistical inference; and to develop theoretical frameworks for understanding teachers' understandings. To this end, our research team conducted an eight-day seminar with eight high school statistics teachers in the summer of 2001. The data we collected include videotaped sessions and interviews, teachers' written work, and researchers' field notes. My analysis of the data revealed that: 1) There was a complex mix of conceptions and understandings of probability and statistical inference, both within individual teachers and among the group of teachers, that are often situationally triggered, which are often incoherent when the teachers try to reflect on them, and which do not support their attempts to develop coherent pedagogical strategies regarding probability and statistical inference; 2) teachers' conceptions of probability and statistical inference are highly compartmentalized: They did not understand probability and statistical inference as a scheme of interconnected ideas, but rather, ideas that are isolated from one another; 3) many teachers had a conception of learning as "knowing how to solve problems", and teaching as "displaying the expertise of problem solving". These conceptions of learning did not support their engagement in reflective conversations about the ideas in probability and statistical inference. The implications of these results include: 1) Understanding statistical inference and teaching effectively entails a substantial departure from teachers' prior experience and their tacit beliefs; and 2) the goal of teachers professional development should be helping the teachers develop understandings of probability and statistical inference as a scheme of interrelated ideas by exerting a great amount of coerced effort in helping teachers develop the capacity and orientation in thinking of a distribution of sample statistics.
The aim of this dissertation was to study the difficulties that some students of education, psychology and social science experience in their quantitative research courses at university. The problem is approached from the perspectives of anxiety studies, studies on conceptions and beliefs, orientations in learning situations and theories of conceptual change. Together, these five studies showed that students' difficulties experienced in quantitative methods, courses, research orientations and motivational factors, do constitute an interconnected web that may also have implications for content learning and to students' views of the importance of research skills for their future work.