# Proceedings

• ### CONCEPTIONS OF DISTRIBUTION HELD BY<br>MIDDLE SCHOOL STUDENTS AND PRESERVICE TEACHERS

This paper examines how preservice teachers and middle school students reason about distributions as they consider graphs of two data sets having identical means but different spreads. Results show that while both subject groups reasoned about the task using the aspects of average and variation, relatively more preservice teachers than middle school students combined both aspects to constitute an emerging form of distributional reasoning in their responses. Moreover, these emergent distributional reasoners were more likely to see the data sets as fundamentally different despite the identical means used in the task.

• ### REASONING ABOUT VARIATION OF A UNIVARIATE DISTRIBUTION: A STUDY WITH SECONDARY MATHEMATICS TEACHERS

Variation is a fundamental concept in statistics literacy; standard deviation is part of compulsory school curriculum in Brazil. The objective of this study is to explore reasoning about variability by teachers, using the model proposed by Garfield (2002). The sample was composed of nine in-service mathematics teachers who took part in a teacher-training course on statistics. An experimental focus made it possible for them to experience all the steps of a statistics research project in which the course content was designed to expose the reasoning about variability employed by these teachers. We identified an oscillation between idiosyncratic and procedural levels, but no teacher showed complete reasoning about variation. The most prevalent reasoning employed was verbal, when teachers interpreted standard deviation as a measure of variation among observations.

• ### PREPARING TO TEACH DATA ANALYSIS AND PROBABILITY<br>WITH TECHNOLOGY

Developing the pedagogical expertise needed to effectively engage students in learning data analysis and probability can be facilitated by engaging teachers in statistical thinking with technology tools. In this paper we present a framework and examples from teacher education materials designed to develop a specialized knowledge we call technological pedagogical statistical knowledge (TPSK).

• ### TEACHERS TRAINING IN A REALISTIC CONTEXT

There is an important need to prepare preservice teachers for the teaching of statistics. We will describe an experiment set up to achieve effective teacher training in statistics in the setting of an Italian university. Student teachers had to prepare lessons using a real data set collected from the Italian mail services. Not only did they look into all the usual basic concepts of statistics, but they also questioned and dealt with doubts and errors their pupils put forth. They discovered the richness of the concepts, the content of descriptive statistics and the basic analyses of observed data. This experiment showed that the data, although very simple, is rich and productive, and that effective teacher training can be set up even with modest resources when there is determination and motivation.

• ### ASSESSING AND DEVELOPING PEDAGOGICAL CONTENT AND STATISTICAL KNOWLEDGE OF PRIMARY SCHOOL TEACHERS THROUGH PROJECT WORK

In this paper we describe a model of pedagogical content knowledge with a formative cycle directed to simultaneously increase the teachers' statistical and pedagogical knowledge. In this cycle, teachers are first given a statistical project to work with and then carry out a didactical analysis of the project. An analysis guide, based on the notion of didactical suitability, helps increase the teachers' competence related to the different components of pedagogical content knowledge and their ability to carry out didactical analyses. At the same time it provides the teacher educator with information regarding the future teachers' previous knowledge and learning. Results of experimenting with this formative cycle for a particular project in a group of 55 prospective teachers indicated a need for better statistics preparation of these teachers and illustrated the usefulness of the formative cycle and analysis guide proposed.

• ### ACQUISITION OF NOTIONS OF STATISTICAL VARIATION BY IN-SERVICE TEACHERS

A study with six middle school teachers on the notion of variation in the task of prediction is hereby presented. The SOLO hierarchy in which notions like randomness, structure and variation are included is applied. Variation related activities used in this study arise from the questions used in research on statistical variation but were also adapted to include computer simulation. The notion of no singular event emerged as a key for evaluation of the transition from multi-structural to relational thinking.

• ### STATISTICS TEACHING FOR PROSPECTIVE TEACHERS OF MATHEMATICS: REFLECTIONS AND PERSPECTIVES

Basic knowledge of statistics has become necessary so that people do not become hostages of data interpretation. Since 1998, in Brazil, the National Curricula Parameters suggest the incorporation of statistics content into mathematics, although a main concern is the academic background of elementary school teachers. The aim of this work is to evaluate the basic statistical background of students in the first semester of mathematics at the Federal University of Lavras, by means of interviews and workshops. Results suggest that it is possible and necessary to plan, in the short and medium term, pedagogical workshops on statistics teaching for prospective teachers of mathematics in order to motivate and qualify them.

• ### TEACHING STATISTICS IN ELEMENTARY AND HIGH SCHOOL<br>AND TEACHER TRAINING

The focus of the reflections presented here is the teacher's point of view on the teaching and learning of statistical concepts. This paper reports the research efforts carried out recently by our group with converging results. We identified that teachers have difficulty in teaching topics related to statistics, particularly when some analysis of the data is required. Teacher discourse shows they favor the philosophy of Exploratory Data Analysis (EDA), but in practice, they restrict their work, according to the results of our study, to a more technical approach that emphasizes the use of algorithms. This research suggests the need for initial and continued training in statistics for mathematics teachers.

• ### THE TRAINING OF PRIMARY SCHOOL TEACHERS IN STOCHASTICS AND IN STOCHASTIC EDUCATION IN EUROPE

In this paper we analyse the available resources, potential difficulties and principles in the training of primary school teachers in stochastics and stochastic education and then present a proposal for a syllabus to train primary school teachers in this field that takes into account the restrictions and aims of the European Higher Education Area.

• ### VARIABILITY IN A SAMPLING CONTEXT: ENHANCING ELEMENTARY PRESERVICE TEACHERS' CONCEPTIONS

While recent and ongoing research has begun to reveal ways that precollege students think about variation, more research has been needed to understand the conceptions of variation held by elementary preservice teachers and also how to shape the university courses where those preservice teachers learn. This paper, sharing an excerpt from an exploratory study aimed at preservice teachers, describes changes in class responses to a sampling task where variation is a key component. Overall, going from before to after a series of instructional interventions, responses reflected a more appropriate sensitivity to the presence of variation.