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Invited TalksTopic 6 : Innovation and reform in teaching probability within statisticsConcepts and models of uncertainty and variability are at the heart of statistical thinking and analysis. Hence probabilistic concepts and thinking underpin all of statistics. Chance and Data developments should be intertwined, in harmony and both driven by data, real contexts and problem-solving. As in all of statistics, students bring to formal education in probability concepts and intuitions from everyday real experiences, and the teaching of probability needs to build on these and link formal notions to the everyday, to real contexts and – constantly – to data. In order to enrich conceptual understanding and underpin formal developments with experiential insights, sustainable representations of probability, integrated with data and statistical thinking, will render concepts accessible and more easily exemplified. These invited sessions will address much-needed reforms in teaching probability in harmony with statistics and will explore current innovations and new directions.Session 6A: Bayesian inference (probability) goes to school: meanings, tasks and instructional challenges6A1: Will the real Bayesian probablity please stand up!?Egan Chernoff University of Saskatchewan, CanadaWhether examined from a mathematical perspective or especially from a philosophical perspective, as Bayesian inference (probability) enters school mathematics curricula worldwide it definitely brings with it “some baggage.” On its way to school, Bayesian inference (probability) will also pick up some baggage from the field of mathematics education (e.g., informal inference, “subjective” probability). Further adding to this baggage, mathematics and school mathematics are not one and the same. Through examining Bayesian inference (probability) from the perspectives of philosophy, mathematics, mathematics education and school mathematics (and popularization), this session will, ultimately, try to better establish what is meant by Bayesian inference (probability)...as it goes to school. In other words, “Will the real Bayesian probability please stand up!?” Paper 6A2: Proto-Bayesian reasoning of children in fourth classLaura Martignon Ludwigsburg University of Education, GermanyTim Erickson Epistemological Engineering Oakland, United States This paper is dedicated to the ecological rationality of certain information formats that facilitate proto-Bayesian reasoning of children. The formats of representation introduced are en-active, iconic and even symbolic and are based on so-called natural frequencies. Experiments are described that have been performed with fourth-graders recently at schools in Berlin and Ludwigsburg and which exhibit the success of these formats for fostering proto-Bayesian reasoning. Results on activities with a special website dedicated to conditional probabilities and Bayesian reasoning (www.eeps.com/riskicon ) are also presented and discussed. Paper 6A3: Exploring realistic Bayesian modeling situationsPer Nilsson Örebro University, SwedenPer Blomberg Linnaeus University, Sewden Jonas Bergman Ärlebäck Linköping University, Sweden The study reported in the present paper is part of a larger project, which aims to explore possibilities and challenges in developing a teaching practice that supports students’ ability to model random dependent situations by a Bayesian approach. A central premise is that modeling should be based on situations that appear realistic to the students. Given this premise, the specific purpose of the present study is to identify and characterize uncertain situations that are realistic and suitable for a Bayesian treatment. The study involves reviewing some of the literature related to Bayesian applications. Based on that review we distinguish detecting (test) situations and construction composition situations as two general types of Bayesian modeling situations. Paper Session 6B: Probability and p-values — probing the problems6B1: Impact of a simulation/randomization-based curriculum on student understanding of p-values and confidence intervalsBeth Chance California Polytechnic State University, United StatesKaren McGaughey Cal Poly - San Luis Obispo, United States This paper describes how changes in course sequencing and pedagogy have impacted students’ understanding of p-values and confidence intervals in introductory algebra-based and calculus-based tertiary-level courses. Using assessment data (for example CAOS, common exam questions, and a transfer problem) across several institutions in various stages of implementation, this paper focuses on how the use of simulation and randomization-based inference has developed, what have been found to be the main student gains, and potential cautions with the approach. Paper 6B2: Teaching probability: using levels of dialogue and proportional reasoningIan Hay University of Tasmania, AustraliaUnderstanding probability and conditional probability are important aspects of a statistical literacy program. Unfortunately many students find these difficult concepts to grasp. Enhancing students’ understandings of probability is explored from two perspectives. The first is based on the notion that students are better able to construct their understanding of probability when teachers use a hierarchy of dialogue questions from descriptive to the abstract, depending on the students’ understanding and responses. The second approach to enhance students’ understanding of probability is to extend their understanding of proportional reasoning and their ability to conceptualise how the same numerical value can be expressed in different forms. Paper 6B3: The interpretation of effect size in published articlesRink Hoekstra University of Groningen, The NetherlandsSignificance testing has been criticized, among others, for encouraging researchers to focus on whether or not an effect exists, rather than on the size of an effect. Confidence intervals (CIs), on the other hand, are expected to encourage researchers to focus more on effect size, since CIs combine inference and effect size. Although the importance of focusing on effect size seems undisputed, little is known about how often effect sizes are actually interpreted in published articles. The present paper will present a study on this issue. Interpretations of effect size, if they are presented in the first place, are categorized as unstandardized (content-related) or standardized (not content-related). Moreover, the interpretations of effect size for articles that include a CI will be contrasted with articles in which significance testing is the only used inferential measure. Implications for the current research practice are discussed. Paper Session 6C: Interdisciplinarity and innovation6C1: Sampling in the wildMichelle Cotterman Vanderbilt University, United StatesLeona Schauble Vanderbilt University, United States Richard Lehrer Vanderbilt University, United States In professional practice, scientific claims are made in light of how samples are constructed and how sampling variability is considered. Yet students are rarely invited to grapple with the complexities of these practices. This presentation describes a design research study in which, over the course of a three-week investigation of a local ecosystem, six classes of sixth-grade students constructed or appropriated measures, grappled with how variability in time and space affected samples of these measures, and made informal inferences about ecosystem functioning in light of the uncertainties involved in measure and sampling variability. Our methods include analyses of student’s sampling plans, interviews with students about how their own investigations, and video recordings of students’ planning sessions, fieldwork, and class “research meetings” wherein they contested claims about ecosystem relationships. We conclude by considering how practices of sampling in the wild influenced students’ conceptions of data collection, samples and sampling variability, and ecosystem functioning. Paper 6C2: Using re-sampling and sampling variability in an applied context as a basis for making statistical inferences with confidenceLuis Saldanha University of Quebec at Montreal, CanadaMichael McAllister Arizona State University, United States We describe an instructional sequence that engaged a class of 9th grade students in making statistical inferences on the basis of distributions of a sample statistic. The sequence involved a scenario and tasks that entailed comparing samples of two types of organisms on a common attribute. Students engaged in: 1) making sense of the scenario and a TinkerPlots™ simulation that produced distributions of a sample statistic, 2) examining and interpreting a sequence of such distributions in relation to increases in sample size, and 3) drawing a conclusion about the attribute in the sampled population and assessing their confidence of the conclusion. We highlight aspects of students’ understandings of what an empirical sampling distribution represented in terms of the scenario’s context, and their abilities to track the multi-tiered re-sampling process that began with a population and culminated with distributions of the sample statistic. Paper 6C3: A case study of an elementary school student’s understanding of stochastic prognosesJudith Stanja University of Duisburg-Essen, GermanyStochastic prognoses are assumed to be a key concept for elementary school stochastics. They may be characterized by the following structural components: focus, evaluation and justification. A qualitative research project with third graders (age 8-9) was conducted in the frame of epistemological interaction research (Steinbring, 2009). The aim of the study was to learn about elementary school students understanding of stochastic prognoses. The study encompassed pre- interviews, a series of lessons and post- interviews. The contribution aims at analyzing and comparing one student’s understanding in the pre- and post-interview. The data used for analysis are transcribed episodes from the pre- and post-interview. The analysis and comparison gives insights in the students developing understanding of stochastic prognoses. Paper Session 6D: Teaching probability to future teachers of mathematics and statistics6D1: Step-by-step activities in the classroom preparing to teach the frequentist definition of probabilityLisbeth Cordani University of São Paulo, BrazilDue to the explicit insertion of Statistics and Probability into the National Mathematics Curriculum in Brazil, since the end of the twentieth century, increasing numbers of local teachers at the basic school (pre-university) level have shown an interest in learning statistics. This paper describes the details of an activity for explaining the frequentist definition of probability often offered by the author as a workshop in pre-service training courses. Hidden concepts used to facilitate comprehension by teachers in training are presented, and some of the main difficulties encountered by students during the activity are described. These difficulties include the practice of transforming qualitative results (e.g., Heads or Tails in a coin throw) into quantitative ones (i.e., 1 or 0, respectively); the concept of cumulative frequencies; and, finally, the alternation style of Heads and Tails to consider the experiment a random phenomenon. Paper 6D2: Learning and teaching probability in the 21st centuryMaurizio Manuguerra Macquarie University, AustraliaPeter Petocz Macquarie University, Australia Contemporary technology affords considerable enhancements to the pedagogy of probability. Traditional approaches focused on mathematical presentation of theory, supported by theoretical and numerical exercises. Utilising current electronic media in the form of e-books, available to students on mobile devices such as the iPad, allows probability to be presented and learned in a more active mode, with seamless integration of text, images, video, animations, presentations and self-assessment exercises. Such a format promotes deep learning of the underlying theory and an essential connection to practical application of the ideas. While this is beneficial for all students (and teachers), it is especially so for those who will be future teachers of mathematics and statistics. They not only get access to contemporary learning materials in probability, but they are also exposed to models of teaching from which they can develop their own pedagogical repertoire. Paper 6D3: Transforming media items into classroom tasks in the context of a study groupDionysia Bakogianni University of Athens, GreeceDespina Potari University of Athens, Greece Efi Paparistodemou Ministry of Education, Cyprus Interpretation of statistical information which is included in media extracts is considered as a critical aspect of statistical literacy. In the context of a study group, seven secondary teachers were given a media item and were asked to analyze it and transform it into a classroom task. In this paper we focus on the process of this transformation and we explore factors that frame the integration of media in the teaching of statistics. Results show that among the main aspects that affect the transformation process are: teachers’ familiarity with the context in media reports, difficulties in defining learning goals and teachers’ low self-confidence regarding classroom management issues. Paper Session 6E: Modeling distributions to connect chance processes, data production, and data distributions6E1: Model-based informal inferenceRichard Lehrer Vanderbilt University, United StatesMin-Joung Kim Vanderbilt University, United States Ryan Seth Jones Vanderbilt University, United States Informal inference refers to making decisions informed by conceptions and representations of sampling variability without recourse to theorems governing formal approaches. To support informal inference, grade 6 middle-school students invented and revised models of chance processes they experienced first-hand by generating repeated measures. These processes provided an accessible interpretation of variability as composed of fixed (signal) and random (error) components. Conducting an analysis of variance and then creating random device analogs of these sources informed model building. Students then generated empirical sampling distributions of model parameters to assist model test and revision. Students used these sampling distributions to guide informal inference. We report on the intelligibility of this form of model-based inference as indicated by results of a flexible interview. Paper 6E2: Visual representations of empirical probability distributions when using the granular density metaphorJ Todd Lee Elon University, United StatesHollylynne Stohl Lee North Carolina State University, United States This paper is an expository on searching for intuitive visualizations of empirical probability distributions. The visualizations use the granular density metaphor for probability density functions (Lee & Lee, 2009) to provide a way for students to build from equiprobable intuitions of probability to standard discrete and continuous distributions. The visualizations dynamically link the sampling process with the formation of an empirical distribution, while remaining true to the granular density metaphor for both the empirical and theoretical density functions. We present and discuss several examples along with conceptual and mathematical ideas that support use of these visualizations. Paper 6E3: Multidirectional modelling for fostering students’ connections between real contexts and data, and probability distributionsTheodosia Prodromou University of New England, AustraliaThe article investigates how 14- to 15-year-olds build informal conceptions about data distributions, and theoretical probability distributions as they engaged in a multidirectional modelling process using computer-based simulations. The students of this study are engaged in modelling. First, students examined data from an unknown stochastic process and built a model of the processes that might explain the outputs. Second, the students constructed representations that generated data whose distributions were well predictive of real world samples. This study shows shifts in the conceptual structures across the two directions and points to the potential of specific aspects of multidirectional modelling for fostering the development of students’ robust knowledge of the logic of inference when using computer-based simulations to model and investigate connections between real contexts and data, and probability distributions. Paper Session 6F: Teachers’ awareness of conceptual connections between probability and statistics6F1: Teachers and students: from an intuitive approach to a rational evaluation of probabilityMaria Pia Perelli D’Argenzio Trieste University, ItalySilio Rigatti-Luchini Padua University, Italy The paper describes a research conducted in Italian schools and connected to the m@t.abel project for the improvement of teaching and learning mathematics. A group of selected teachers produced 56 didactic units on mathematical subjects. Other teachers involved in the experimentation, received a short in-person presentation and then worked with the units in a virtual environment available on line under the supervision of a tutor. The in-class experimentation was on 4 units and at least one had to be a statistical one. Teachers had to write a log book for each unit including a report with the reasons for their unit choice, a close-ended questionnaire on the experimentation and a final report with the results. By analyzing the context of the unit “Stem and leaf plots“, one of those implemented in the classroom, we observed the teachers and students transition from a purely intuitive probability approach to a rational estimate. Paper 6F2: Challenges for learning about distributions in courses for future Mathematics teachersMarcos Magalhães University of Sao Paulo, BrazilKnowledge about distribution is important and necessary for the process of learning other statistical concepts, in particular, formal or informal inference. The differences between empirical and theoretical distributions are sometimes hard to understand for students of all levels. At the University of Sao Paulo, Brazil, future mathematics teachers attend two basic statistics courses as part of their curriculum structure. In Brazilian middle and high schools, these are the professionals responsible for teaching the statistical concepts included in the contents of the mathematics discipline. Therefore, good knowledge of the concept of distribution will improve their teaching. In this paper, we focus on the misconceptions observed in basic Statistics courses attended by future mathematics teachers. We also present suggestions for activities that include the use of graphical and computer tools to improve the learning process. Paper 6F3: Odds that don’t add upMichael Fletcher Canterbury Christ Church University, United KingdomThis paper examines a common probability misconception that students have. Although students are generally able to correctly calculate the probability of two independent events occurring they are often not able to use this calculation in a meaningful way. This paper will contextualise this probability misconception by using examples from television game shows. Extracts from the game shows will demonstrate how contestants routinely overestimate the probability of combined events. The decisions made by the contestants reduce their probability of success by a significant margin. Paper |