
Invited TalksTopic 2 : Statistics education at school levelStatistics and hence statistics education are of vital importance to the whole of society and to all disciplines. The earliest introduction to statistical thinking occurs at the school level. Here a variety of techniques and concepts for data collection, handling and interpretation can be presented and explored, in harmony with concepts of chance. However the excitement of statistics and an appreciation of its utility can be sustained and enriched from perspectives beyond the school, and by an embracing of the challenges of engaging, empowering and intriguing young minds with the foundations of statistical thinking, and exhibiting its enormous power to harness and handle uncertainty.Session 2A: Early years statistics education: ages 4  82A1: Important ideas in statistics for children aged 48 yearsDenise Spangler University of Georgia, United StatesThis paper highlights important statistical ideas for children aged 48 years and the implications for teacher education and research. Drawing on both the Guidelines for Assessment and Instruction in Statistics produced by the American Statistical Association as well as standards/curriculum documents from other countries, I identify topics in statistics and probability that are appropriate for young children and to which they should be exposed in order to prepare them for later instruction. I also examine the knowledge that teachers of young children need to have in order to teach this content effectively. In addition, I suggest areas that are ripe for further research in this arena. Paper 2A2: Establishing statistical foundations early: data modeling with young learnersLyn English Queensland University of Technology, AustraliaThis paper addresses research from a threeyear longitudinal study that engaged children in data modeling experiences from the beginning school year through to third year (68 years). A data modeling approach to statistical development differs in several ways from what is typically done in early classroom experiences with data. In particular, data modeling immerses children in problems that evolve from their own questions and reasoning, with core statistical foundations established early. These foundations include a focus on posing and refining statistical questions within and across contexts, structuring and representing data, making informal inferences, and developing conceptual, representational, and metarepresentational competence. Examples are presented of how young learners developed and sustained informal inferential reasoning and metarepresentational competence across the study to become “sophisticated statisticians”. Paper 2A3: Exposing young children to activities that develop emergent inferential practices in statisticsDebra McPhee University of Queensland, AustraliaKatie Makar University of Queensland, Australia Informal statistical inference has gained increasing recognition as an effective approach to teaching statistics. Distinct from descriptive statistics, inference provides learners with access to the power of statistics by giving them tools to make predictions beyond their data. International research in this area has focused on students from primary school through university. A series of teaching experiments introduced informal statistical inference to very young children (aged 56). Although making predictions was familiar as an everyday task, initial attempts revealed challenges to teaching informal inferential reasoning to young learners. Prior to conducting a statistical inquiry involving inference, activities were designed to generate a need for recording and organising data, the language of uncertainty and using data as evidence. Results suggest that the activities prior to inquiry likely supported students in their emerging inferential practices. Paper Session 2B: Middle school statistics education: ages 8  132B1: Middle school (ages 10 – 13) students’ understanding of statisticsSteve Foti University of Florida, United StatesDouglas Whitaker University of Florida, United States Tim Jacobbe University of Florida, United States This paper will present results from the administration of the LOCUS assessments to measure students’ statistical understanding in grades 68 (ages 10 – 13). The development of these assessments utilized an Evidence Centered Design (ECD) (Mislevy & Riconscente, 2006) approach to establish their content validity. After an iterative development process, these assessments were administered to over 2,000 students in the United States. Student performance in each of the four areas of the statistical problem solving process — formulating questions, collecting data, analyzing data, and interpreting results — will be discussed, and examples of multiplechoice items will be provided. Paper 2B2: Where’s your evidence? Challenging young students’ equiprobability bias through argumentationJill FieldingWells University of Tasmania, AustraliaStudents come to formal schooling with prior probabilistic conceptions developed through informal experiential events. One such concept is that of chance outcomes being inherently equiprobable, even when not the case. In the designbased research described here, a class of 3rd Grade students was posed an inquiry problem embedded with nonequiprobable outcomes: What is the best addition bingo card? Argumentation was employed as a pedagogic approach to challenging students’ equiprobable beliefs, with students supported to develop an evidencebased argument in response. Students initially experienced conflict with the realisation of unequal frequencies, then developed representations to act as theoretical evidence. A shift from conceptualizing equiprobable outcomes towards responses reflecting theoretical distribution was observed. This exploratory research suggests potential for an evidentiary focus to challenge probabilistic conceptions. Paper 2B3: Student perspectives on being introduced to using Tinkerplots for investigationsTim Burgess Massey University, New ZealandThis paper reports on one aspect of the implementation of Tinkerplots into six primary school classrooms. Students’ perspectives are explored regarding their use of the software and their learning of statistics. These perspectives are compared with the “processes” and “products” of their statistics investigations, as evidenced by samples of Tinkerplots’ files, and video screen shots that captured the sequence of their computerbased work. The examples confirm some of the claims that students made about the ease of becoming acquainted with and using the software, and the ways in which the software helped them think more about the statistics. However there are also some implications for teachers with regard to using such software to enhance the teaching and learning of statistics. Paper Session 2C: Secondary school statistics education: ages 13 +2C1: On the delicate relation between informal statistical inference and formal statistical inferenceRolf Biehler University of Paderborn, GermanyInformal inferential reasoning has become a topic of concern in statistics education. Laying intuitive grounds for more formal procedures or shaping the reasoning of younger students for whom formal inferential procedures will be too difficult to understand can be found among the aims. However, formal inferential reasoning as such is controversial itself: Bayesian and nonBayesian ways of reasoning, NeymanPearsonian and Fisherian ways of hypothesis testing, confidence intervals. And: real applications of these procedures are context dependent. This raises questions with regard to which view of formal statistical inference we design preparatory informal inference activities for. The paper will critically discuss several approaches. Paper 2C2: High school (ages 14 – 18) students’ understanding of statisticsCatherine Case University of Florida, United StatesDouglas Whitaker University of Florida, United States Tim Jacobbe University of Florida, United States Steve Foti University of Florida, United States This paper will present results from the administration of the LOCUS assessments to measure students’ statistical understanding in grades 912 (ages 14 – 18). The development of these assessments utilized an Evidence Centered Design (ECD) (Mislevy & Riconscente, 2006) approach to establish their content validity. After an iterative development process, these assessments were administered to over 2,000 students in the United States. Student performance in each of the four areas of the statistical problem solving process  formulating questions, collecting data, analyzing data, and interpreting results  will be discussed, and examples of multiplechoice and constructedresponse items will be provided. Paper 2C3: Teaching statistics at secondary education in Italy: some issues on large scale standardized test resultsStefania Mignani University of Bologna, ItalyRobert Ricci National Institute for Educational Measurement, Italy M Gabriella Ottaviani Sapienza University of Rome, Italy This paper focuses on recent issues in statistics learning outcomes coming from the national largescale standardized tests administered to all students in 2013 by the National Institute for the Evaluation of the Educational System (INVALSI) at the end of 10th grade. Utilizing a representative sample of around 32000 units, we study several items assessing statistics skills that are part of the test concerning mathematics. The INVALSI data set allows replying issues as: which kind of contents do students learn with more difficulty? Do males perform better than females? Does the level of achievement differ among type of schools? Do the results show differences between mathematics and statistics outcomes according school types? The answers to these issues help to explain the quality of student learning with the view to improve taught curricula and suggest more effective pedagogical strategies for statistics. Paper 2C4: Analysis of teachers’ understanding of covariation in the Vitruvian Man contextIrene Cazorla State University of Santa Cruz, BrazilVerônica Yumi Kataoka State University of Santa Cruz, Brazil Cláudia Borim da Silva University of São Judas, Brazil This work aims to analyze how 24 high school inservice teachers understand covariation, within the context of Vitruvian Man. This concept was explored in an informal way during a teaching intervention mode, by applying tasks in which teachers observed the height and arm span of students. They had to describe both variables, and answer whether it was possible to say that the height and arm span measurements were equal. They were also asked to construct the scatter plot with and without drawing the linear function. Just after the construction of the scatter plot and the linear function, teachers considered that the measurements of both variables were close. We believe that at the end of the set of tasks, the understanding of covariation in this group of teachers was improved, leading us to think that proposals such as these can aid the training of math teachers for teaching this topic in schools. Paper Session 2D: Statistical education at the Secondary/Higher Education interface2D1: What did they learn? Statistics skills: from French secondary school to universityAlain BihanPoudec Catholic University of the West, Angers, FrancePhilippe Dutarte Academic Inspection Créteil, France The statistics curriculum in the French Secondary schools is evolving. According to the French Education Ministry’s Instructions, the statistics courses are cursory and essentially descriptive in middle school, while the high school curriculum is oriented towards inferential statistics. However, the limited amount of time available for teaching statistics and the teachers’ lack of training lead to the following question: what do students learn about statistics? Our own practice teaching statistics at university and our previous research lead us to put forward a few answers. Overall, undergraduate students in Humanities and Social Sciences associate statistics with mathematics, numbers, calculus, percentages, though interesting exceptions do exist. Regarding the basic notions themselves, it seems that: algorithmic conceptions prevail over the meaning of concepts; and difficulties in learning and understanding statistics are linked with the secondary school curriculum. Paper 2D2: Bridging the statistical gap: creating successful secondary/higher education partnershipsDaren Starnes Lawrenceville School, United StatesMany school mathematics curricula now include sophisticated statistical content, like randomizationbased inference methods and experimental design. Secondary mathematics educators often lack the formal training, and hence the confidence, to help students develop sound statistical reasoning. By networking with college and university statistics educators, secondary school teachers can deepen their statistical understanding and improve their instructional practice. Students benefit directly from these professional partnerships, especially when they participate in statisticsrelated events sponsored by local colleges and universities. This paper describes several successful statistical partnerships started in the United States that have enhanced student learning and eased students’ transition from secondary school to higher education. Opportunities and challenges for launching similar initiatives in other countries are discussed. Paper 2D3: Preparing future teachers to teach statisticsGail Burrill Michigan State University, United StatesAs statistics and data analysis become increasingly important so does the need to prepare teachers to teach these topics. The paper addresses three aspects related to this work with prospective teachers. First, a theoretical framework related to fundamental statistical ideas in the school curriculum (Burrill & Biehler, 2011) might provide guidance for the preparation of teachers across the grade levels. Second is a description of a course for prospective elementary teachers at a large university, which is designed to introduce key statistical concepts using real data and relevant contexts. Third is a brief discussion of the preparation of prospective mathematics teachers at the secondary level and how the statistics courses they take prepare them, or not, to teach statistical content and reasoning processes, particularly the content suggested in the U.S. Common Core State Standards, which have a strong emphasis on informal inference at the upper secondary level. Paper Session 2E: Using technology at school level to enhance statistical understanding2E1: From hat plots to box plots in Tinkerplots: supporting students to write conclusions which account for variability in dataSue Allmond Jindalee State School, AustraliaKatie Makar University of Queensland, Australia A statistical question acknowledges that there is variability in data that needs to be accounted for in the conclusion. Accounting for variability is problematic if students do not have an understanding that a distribution shows patterns and can be described by the centre, spread and overall shape. TinkerPlots provides opportunities to build understandings of spread and measures of centre as students work with distributions, adding and manipulating dividers and hat plots. In this exploratory study, students in a middle school inquiry classroom used hat plots to compare distributions and write justified conclusions. Results suggest that the necessity to account for variability in data within their conclusions presented students with a purpose to transition from hat plots to box plots to provide evidence to answer the question. Paper 2E2: Constructing, refining and validating a task for developing reasoning on stabilized frequency distributions in the context of informal inferencesAna SerradóBayés La SalleBuen Consejo, SpainIn this paper we present the construction, refining and validation of a task to reason about sampling and stabilized relative frequencies distribution to be included in a possible learning trajectory in Secondary School that facilitates the link between Informal and Formal Inference. The task consists of an informal modeling process of the relative frequency of appearance of each vowel in strings of characters, with a pseudoconcrete model developed through a statistical process of investigation, statistical modeling and validation through the analysis of animations. Animations of the relative frequency distributions when varying the sample size were created using the dynamic opensource software, Geogebra, and were incorporated in an online quiz, constructed within the virtual learning environment, Moodle. The task, its learning goals and assessment activities were validated with n=49 Spanish students of Grade 9 in order to draw conclusions about the conjectured learning process. Paper 2E3: Games of chance: tools that help enhance teachers’ notions of statistics and probabilityNirmala Naresh Miami University, United StatesResearch reported in this paper is part of a larger study that focused on curriculum development and on prospective teachers’ content and pedagogical knowledge of probability. To address the first research goal, during the preliminary phase of the project, a probability teaching module was developed. To address the second research goal, during the subsequent phase, this module was taught to a group of prospective middle school teachers. In this report, the focus is on the first research goal  I will describe key components of the probability teaching module. In particular, I will share two activities from the teaching module and discuss related findings. Paper Session 2F: Innovative approaches to improve pedagogical content knowledge at the school level2F1: Improving the perceived value and affect of statistics in elementary and middle school teachers through the development of pedagogical content knowledgeTamara Pearson Clayton State University, United StatesAs Arthur Benjamin discusses in his TED talk (www.ted.com), statistics is often undervalued because we live in a society where mathematics curriculum follows a path from arithmetic, to algebra, and finally to calculus. Many elementary and secondary educators do not deem it worthy of substantial focus in their classrooms, and therefore thousands of college students struggle through statistics courses as they try to build on an unstable foundation. Research from an intensive professional development program for 4th8th grade teachers in which increases in perceived value and affect were achieved through a focus on pedagogical content knowledge is presented. Included are teacher journal entries, discussion of program development, and implications for future research and practice. Paper 2F2: Statistical knowledge for teaching: elementary preservice teachersChristine Browning Western Michigan University, United StatesDustin Smith Western Michigan University, United States Joshua Goss Western Michigan University, United States A major component of statistical thinking deals with the omnipresence of variability in data. Advances in technology allow for the development of tasks that can engage students more readily in data analysis so that they come to see this variability as early as the elementary grades. Yet how do we help prepare elementary preservice teachers (PSTs) to understand variability in data for themselves and to consider the statistical thinking of children? This paper will share tasks that were designed for a statistics course for elementary PSTs. These tasks make use of several forms of technology such as Tinkerplots® and Interactive Whiteboards (IWBs), and have the intent to develop PSTs’ statistical knowledge for teaching. Preliminary data analysis reveals that these tasks provided PSTs with a conceptual way of appropriately attending to measures of variability in a manner that the knowledge of procedures could not. Paper 2F3: High school teachers’ pedagogical content knowledge of variabilitySylvain Vermette University of Québec at TroisRivières, CanadaLinda Gattuso University of Quebec at Montreal, Canada This research sought to explore teachers’ pedagogical content knowledge of the concept of variability. Twelve mathematics high school teachers were tested on their knowledge of the concept of variability. Subjects were then asked to react when presented with scenarios describing students’ strategies, solutions and misconceptions when faced with a task based on the concept of variability. Outcomes of this study uncovered interesting teaching interventions that could prove useful to teachers faced with such scenarios. Results of both teachers’ tests and interviews revealed that teachers had difficulties and misconceptions related to the concept of variability. Furthermore, teachers’ reactions to some scenarios highlighted the influence of content knowledge of the concept of variability on the pedagogical content knowledge related to this concept. Paper Session 2G: Linking research and practice in teaching and learning statistics at the school level2G1: Developing statistics teachers’ identity: a look at communities of practiceDifariney González University of Antioquia, ColombiaLucía ZapataCardona University of Antioquia, Colombia In this work we studied the development of statistics teachers’ identity. The study took place within a teacher education program with weekly meetings. The program promoted the development of statistics teaching materials at the school level and reflection on several tensions that emerged during teachers’ practice in the classroom. Statistics teachers together with researchers participated as members of a community of practice that was characterized by their mutual commitment to the group. Data was collected from multiple sources: classroom observations, interactions within the meetings with teachers, autobiographies and interviews. We focused on the episodes that revealed pieces of teachers’ identity, those episodes that exposed teachers’ experience, background, and life story that helps us to understand how statistics teachers learn. Paper 2G2: Communities of practice: a theoretical framework to design for teachers’ statistical learningAna Luisa GómezBlancarte Center for Research and Advanced Studies of IPN, MexicoIsaias Miranda Viramontes National Polytechnic Institute, Mexico Drawing from Communities of Practice theory (Wenger, 1998), in this paper we propose a design for statistical learning for teachers enrolled in professional development programs. This design has to do with activities based on participation and reification: two processes that define learning as a negotiation of meaning. According to this theory, design for learning is to establish conditions to create experiences of meaning. In teachers training, such experiences involve interconnections between training practices and teaching practices (e.g. planning, teaching and analyzing classroom lessons). Taking into account the necessity to improve statistics teaching this article gives some elements to support teachers’ statistical learning. Paper 2G3: From observing and evaluating variation to measuring and comparing variationErna Lampen University of the Witwatersrand, South AfricaTeachers are tasked with engaging students’ everyday reasoning in order to develop statistical reasoning, but there is little guidance for statistics teachers to understand relevant patterns in everyday reasoning about data contexts. In this paper I report a discursive analysis of a discussion of the data context of prices of used cars by a group of teachers in an introductory statistics course. Everyday reasoning and informal statistical reasoning are proposed as points on a continuum along the development of statistical reasoning, and defined in terms of discursive patterns. I describe patterns in everyday discourse about data contexts and illustrate how such patterns influence reasoning about context in a classroom discussion. Analysis of the participants’ word use indicates that everyday terms like “value” and “price” should not be treated as synonyms in discussions of data contexts, even if they refer to the same numerical data. Paper 2G4: Teachers as key stakeholders in research in statistics educationMichael Shaughnessy Portland State University, United StatesTeachers and practitioners of statistics are an underutilized resource in research in statistics education. Historically practitioners have tended to be viewed as consumers of research by the educational research community rather than active participants in the generation of new knowledge about the teaching and learning of statistics. Recently researchers in mathematics education have begun to involve teachers more as key stakeholders in research in an effort to forge closer links between research and practice. A discussion of what it means for teachers to be key stakeholders in research is provided, along with some examples from recent research that involve teachers as key stakeholders. Research in statistics education can enrich the links between research and practice in teaching statistics by involving statistics teachers as key stakeholders. Paper 