Journal Article

  • In the May 2009 issue of The American Statistician, Brown and Kass (BK) offered thought-provoking answers to the question "What is Statistics?" which have direct implications for statistics education. For five years, St. Olaf College's Center for Interdisciplinary Research's (CIR) activities have aligned with BK in both philosophy and practice. We describe the program's motivation and design, how we recruit students and find faculty collaborators with suitable projects, and how the teams of faculty and students work together. A research skills seminar series parallels the research process and prepares students for working on teams. Inevitably, administrative issues arose which we identify and address. Landes (2009) identified significant issues related to recruiting. Our model of undergraduate education has proved to be fruitful on this front. Sending nearly 50 students to graduate school in five years from a college of fewer than 3000 speaks to the program's efficacy. Here we present a program based on authentic interdisciplinary research with undergraduates which embodies many of BK's ideas and addresses recruiting issues. Although this experience underscores the potential for new and exciting approaches to statistics education in the liberal arts environment, the model itself can be adapted by a variety of undergraduate programs. Supplemental materials are available online.

  • Although literature on challenges to students' learning of data analysis and probability has steadily accumulated over the past few decades, research on challenges encountered in teaching the content area is in its beginning stages. The present study aims to help build this area of research by identifying some knowledge elements necessary for teaching conditional probability and independence. Artifacts of classroom practice, including written plans and lesson video, were used to identify challenges encountered by teachers in establishing productive learning environments for students first learning the concepts. It is proposed that enhanced common and specialized content knowledge may help teachers address the challenges identified. Some salient aspects include knowledge of: distinctions among major concepts, data displays with pedagogical value, and the roles of fractions and combinatorial ideas in the psychology of learning conditional probability and independence. The discussion of these and other relevant knowledge aspects is drawn upon to propose potentially productive directions for teacher education efforts and future research.

  • In the topic study group on probability at ICME 11 a variety of ideas on probability education<br>were presented. Some of the papers have been developed further by the driving ideas of interactivity and use<br>of the potential of electronic publishing. As often happens, the medium of research influences the results and<br>thus - not surprisingly - the research change its character during this process. This paper provides a summary<br>of the main threads of research in probability education across the world and the result of an experiment in<br>electronic communication. For convenience of international readers, abstracts in Spanish and German have<br>been supplied, as well as hints for navigation to linked electronic materials.

  • The research question in this study was assessing possible relationships between formal<br>knowledge of conditional probability as well as biases related to conditional probability reasoning: fallacy of<br>the transposed conditional; fallacy of the time axis; base rate fallacy; synchronic and diachronic situations;<br>conjunction fallacy; and confusing independence and mutually exclusiveness. Two samples of university<br>students majoring in psychology and following the same introductory statistics course were given the CPR<br>test before (n = 177) and after (n = 206) formal teaching of conditional probability. Results indicate a<br>systematic improvement in formal understanding of conditional probability and in problem solving capacity<br>but little change in those items related to psychological biases

  • In this paper we summarize the research we have recently carried out on classifying problems<br>of conditional probability. We investigate a particular world of school word problems we call ternary<br>problems of conditional probability. With the help of a mathematical object, the trinomial graph, and the<br>analysis and synthesis method, we propose a framework for a structural, didactical and phenomenological<br>analysis of the ternary problems of conditional probability. Consequently, we have organized this world into<br>several types of problems. With respect to students' behaviour, we identify four types of thinking processes<br>related to data format and the use of data. We also illustrate our approach by use of the diagnostic test<br>situation, and in the particular context of health.<br>The main purpose of our work is to improve secondary school students' understanding of conditional<br>probability and their probability literacy by proposing a teaching approach based on problem solving within<br>appropriate contexts. We believe that the framework we present in this paper could help teachers and<br>researchers in this purpose.

  • What instructional materials and practices will help students make sense of probability<br>notions? Li (11 years) participated in an interview-based implementation of a design for the binomial. The<br>design was centered around an innovative urn-like random generator, creating opportunities to reconcile two<br>mental constructions of anticipated outcome distributions: (a) holistic perceptual judgments based in tacit<br>knowledge of population-to-sample relations and implicitly couched in terms of the aggregate events with no<br>attention to permutations on these combinations; and (b) classicist-probability analytic treatment of ratios<br>between the subset of favorable to all elemental events with attention to the permutations. We argue that<br>constructivist and sociocultural perspectives on mathematics learning can be reconciled by revealing<br>interactions of intuitive and formal resources in individual development of deep conceptual understanding.<br>Learning is the guided process of blending two constructions of problematized situations: the<br>phenomenologically immediate and the semiotically mediated.

  • The intention of this work is to exhibit how children can be provided with a kit of elementary<br>tools for judgment under uncertainty, for good decision making and for reckoning with risk. Children, we<br>claim, can acquire this tool kit through a mosaic of simple, play-based activities which are devised to make<br>them aware of the characteristics of uncertainty. We present a sequence of tasks that build upon each other,<br>beginning with the Wason selection task, moving on to probabilistic tasks, tasks in elementary Bayesian<br>reasoning comparing proportions and, finally, to comparing risks. This research is guided and inspired by<br>empirical results on human decision making in the medical and financial domain.

  • We investigate the evolution of probabilistic reasoning with age and some related biases, such<br>as the negative/positive recency effects. Primary school children and college students were presented with<br>probability tasks in which they were asked to estimate the likelihood of the next occurring event after a<br>sequence of independent outcomes. Results indicate that older children perform better than younger children<br>and college students. Concerning biases, the positive recency effect decreases with age whereas no age-<br>related differences are found for the negative recency effect. Theoretical and educational implications of<br>results are discussed.

  • In a society that generates information rapidly, schools have to fulfil their programmes<br>imaginatively. Thus, extra-curricular activities may be helpful for the students to acquire wider knowledge<br>than that they may get within the classrooms. On the other hand, randomness is present in almost all everyday<br>decisions, mainly based on prior information so it is important to have at least a rough idea on how specific<br>events may affect the chances of other events. We explore both ideas here in the context of a science fair, in<br>which two high-school senior students conducted an investigation about conditional probability using a game<br>called "Shut the box". We also want to pose, as a research question, if, after their participation in the science<br>fair, these students have reached higher levels in probabilistic reasoning compared to their classmates or have<br>acquired knowledge about concepts far beyond the official curriculum.

  • The purpose of this paper is to report on the conception and some results of a long-term<br>university research project in Budapest. The study is based on an innovative idea of teaching the basic notions<br>of classical and Bayesian inferential statistics parallel to each other to teacher students. Our research is driven<br>by questions like: Do students understand probability and statistical methods better by focussing on<br>subjective and objective interpretations of probability throughout the course? Do they understand classical<br>inferential statistics better if they study Bayesian ways, too? While the course on probability and statistics has<br>been avoided for years, the students are starting to accept the "parallel" design. There is evidence that they<br>understand the concepts better in this way. The results also support the thesis that students' views and beliefs<br>on mathematics decisively influence work in their later profession. Finally, the design of the course integrates<br>reflections on philosophical problems as well, which enhances a wider picture about modern mathematics and<br>its applications.

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