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# Conference Paper

• ### Requirements for an ideal software tool in order to support learning and doing statistics

Features of software tools that can support learning and doing statistics in introductory courses are the focus of this paper. We will start with describing assumptions concerning the direction in which introductory statistics education should move and improve, which are discussed in more detail elsewhere (e.g. Biehler (1992); Biehler (1993); Gordon &amp; Gordon (1992); Thisted &amp; Vellemann (1992)). Our notion of introductory statistics education comprises the teaching of "stochastics" within mathematics education in schools and data analysis in other school subjects. After that, we will discuss requirements of software in more detail, and how we integrated requirements in the paper-and-pencil prototype of the MEDASS-project in Biehler &amp; Rach (1992).

• ### What counts as statistical understanding?: An ethnographic study in an AP statistics class

This study investigates students' statistical learning in a high school classroom setting. Using a theoretical framework derived from interpretive and sociolinguistic ethnography, the study explores the meaning of "statistical understanding" that developed in a statistics class over the course of a school year. Seventeen students enrolled in an AP statistics course participated in the study. Data were collected through participant observation, videotaping of classroom sessions, field notes, and interviews. Initial analysis identified a series of tensions that highlight the shared beliefs of "what counts as statistical understanding". In the paper, I will present and discuss the emergence of these tensions in relation to the classroom interaction and students' learning.

• ### A review of the literature on learning and understanding probability

One of the most challenging aspects of teaching statistics is helping students to understand concepts of randomness and probability. Research reveals that sutdents bring many misconceptions to their study of probability that are difficult to eradicate. Moreover, experience indicates that students often find the study of probability to be very frustrating and often disconnected from the study of statistics.

• ### Constructing Data, Modeling Chance in the Middle School

We describe the design and iterative implementation of a learning progession for supporting statistical reasoning as students construct data and model chance. From a disciplinary perspective, the learning trajectory is informed by the history of statistics, in which concepts of distribution and variation first arose as accounts of the structure inherent in the variability of measurements. Hence, students were introduced to variability as they repeatedly measured and attribute (most often, length), and then developed statistics as ways of describing "true" measure and precision. Both of these developments have historic parallels, and the intricate relation of measure and data are also key components of ongoing professional practice (see Hall et al., this symposium). From a learning perspective, the learning trajectory reflects a commitment to several related principles: (a) constituting a learning progession as encounters with a series of problematics; (b) representational fluency and meta-repesentational competence as constituents of conceptual development in a discipline; (c) invented measures as grounding students' understanding of statistics and (d) an agentive perspective for orienting student activity, according to which distribution of measures emerges as a result of the collective activity of measurer-agents. Instructional design and assessment design (see Wilson et al., this symposium) were developed in tandem, so that what we took as evidence for the instructional design was subjected to test a a model of assessment, resulting in revision to each.

• ### Model Competition: A Strategy Based on Model Based Teaching and Learning Theory

The purpose of this study is to develop a theoretical framework for describing different teaching strategies that can foster student model construction in large group discussions. Such a framework is necessary for developing new instructional principles about how to build mental models in large classroom settings. This particular paper focuses on a mode of interaction call model competition as one possible strategy. The teacher has an opportunity to promote model competition when the students contribute to a discussion with ideas that are contradictory to each other. The presence of these different kinds of ideas fosters dissatisfaction in the students' minds that can be productive. We follow the strategies a teacher uses to support this and other important modes of learning, such as model evolution, in a case study of classroom learning in the area of respiration.<br>We believe that the teacher played a key role during the teacher/student co-construction process described in the present study. The teacher particpated by constantly diagnosing the students' ideas and attemping to introduce dissatisfaction by suggesting constraints that led the students to evaluate and modify their ideas, producing cycles of model construction and criticism. In this way she was able to guide students toward targeted content goals. The learning model we develop includes nested teacher-student interaction organization patterns that the teacher used in order to encourage the students to disconfirm, recombine, restructure, or tune their ideas and to generate successive intermediate mental models. These patterns have been analyzed from the perspective of a theoretical framework of model construction theory. We believe that this framework can provide a set of lenses that complements other cognitive and sociological frameworks for analyzing classroom discussions.

• ### Applying an Action Research Model to Assess Student Understanding of the Central Limit Theorem in Post-Calculus Probability and Statistics Courses

The focus of this paper is our examination of students' graphical understanding of the CLT and related concepts as presented at the Joint Statistics Meeting in August, 2005. For a more detailed analysis that includes students' numerical understanding, results from the second course of two-semester post-calculus sequence, and comparison to results from previous studies, please see our paper and website (Lunsford, Rowell, Goodson-Espy 2005).

• ### Utilizing Distance Education to Offer Web-based Professional Development in Statistics Education to Teachers Across Europe

The paper provides an overview of a new program recently funded by the European Union that aims to enhance the teaching and learning of early statistical reasoning in European schools by utilizing distance education to offer high-quality professional development experiences to geographically-dispersed teachers across Europe. Acknowledging the fact that teachers are at the heart of any educational reform, the project will facilitate intercultural collaboration of European teachers using contemporary technological and educational tools and exemplary web-based materials and resources. Long-term sustainability will be assured through support of multilingual interfaces and online servies for the accumulation of collective knowledge from teachers and teacher educators. An online knowledge base will offer access to usable and validated pedagogical models, didactic approaches, and innovative instructional materials, resulting in a complete and flexible teacher professional professional development program.

• ### An Analysis of Students' Statistical Understandings

It is important to develop instructional sequences that build on students' current understandings and support shifts in their current ways of thinking. As part of the pilot work for a project on mathematics teaching, classroom performance assessments were conducted to obtain baseline data on students' current statistical understandings. The assessments were conducted in three sessions of a seventh-grade class. The assessment task was designed to provide information about students' current understandings of the mean and graphical representations of data because these ideas were the focus of a statistics chapter students previously studied. Students worked in small groups on the three performance tasks, each of which is described in detail. The analysis shows that students typically viewed the mean as a procedure that was to be used to summarize a group of numbers regardless of the task situation. Data analysis for these students meant "doing something with the numbers," an idea grounded in their previous mathematics experiences. Students' conversations about graphical representations highlight the procedures for constructing graphs with no attention to what the graphs signify and how that relates to the task situation. To help students develop a sense of data analysis as more than just "doing something with numbers," it is necessary to create tasks that are relevant to middle school students. An appendix contains a list of 69 sources for additional information. (Contains 8 figures and 11 references.) (SLD)

• ### From Data to Graphs to Words - But Where are the Models?

The pioneers of statistics focused on parametric estimation and summary to communicate statistical findings. The tradition of basing inference on parametric fits is a central mode in statistics education, but in statistics applications, computer-based graphical summary is playing an increasingly important role. A parallel development has been the spread of statistics education to almost all disciplines, and thus the need to communicate statistical results to non-specialists has become more acute. These influences of more graphics and a wider distribution require adaptation in our statistics courses. This paper provides examples of, and arguments for, the use of simulation and graphical display, and the role of these techniques in enhancing the verbalization of analytical results. The immediate goal of the paper is to persuade those who design curricula for early statistics courses to provide a serious introduction to grpahical data analysis, at the expense of some traditional parametric inference. The goals is to enable more students to communicate statistical findings effectively.

• ### Why Statistical Inference is Hard to Understand

The purpose of this study was to investigate the development of students' thinking in relation to their engagement in classroom instruction designed to support their conceiving sampling and inference as a part of a scheme of interrelated ideas including repeated random sampling, variability among sampling outcomes, distribution, and representativeness.