# Theory

• ### Teaching and assessing statistical reasoning

Statistical reasoning may be defined as the way people reason with statistical ideas and make sense of statistical information. This involves making interpretations based on sets of data, representations of data, or statistical summaries of data. Much of statistical reasoning combines ideas about data and chance, which leads to making inferences and interpreting statistical results. Underlying this reasoning is a conceptual understanding of important ideas, such as distribution, center, spread, association, uncertainty, randomness, and sampling. This chapter begins by distinguishing reasoning from mathematical reasoning, and then outlines goals for students studying statistics. Challenges in assessing statistical reasoning are described and information is provided on a unique paper and pencil instrument, the Statistical Reasoning Assessment. The final section suggests ways teachers may help students develop sound statistical reasoning skills.

• ### Modeling aspects of students' attitudes and achievement in introductory statistics courses

This report presents a path model of students' attitudes and achievement in statistics; it also presents correlations among attitudinal measures and correlations between attitudinal measures and test performance for both graduate students and undergraduate students.

• ### Correspondence analysis as a tool to analyze the relational structure of students' intuitive strategies in judging statistical association

The theoretical basis of this paper is the modeling of students' knowledge about a specific subject as a qualitative and systemic construct. Following therefrom, a discussion about the role of multivariate analysis for studying the structure of this knowledge and for building explanatory models relating its structure to task, cognitive and instructional variables. Correspondence analysis in an empirical study referring to statistical association is used as an example.

• ### Emergent ideas of chance and probability in primary grade children

The National Council of Teachers of Mathematics has recommended that we incorporate chance and probability as a thematic strand throughout children's formal schooling, beginning at the kindergarten level. This recommendation comes at a time when there is little agreement concerning the psychology of chance and probability. This is particularly disconcerting, given the importance now given to building on children's intuitions in the instructional process. The chapter addresses these issues as they involve the successful implementation of the K-4 statistics and probability standard, through an analysis of releveant research literatures. These literatures indicate that primary grade children have a number of intuitions that are directly relevant to statistical instruction. These include intuitions about relative magnitude and part/whole relations, incertitude and indeterminancy, the likelihood of a given event and, much more limited, intuitions about expected distrubutions of outcomes. The chapter also considers challenges to primary grade children's reasoning in this sphere that instruction needs to address, including conceptual gaps such as the idea of patterns in outcomes over the long haul, the restructuration and elaboration of children's intuitions, raising the meta-level of children's knowledge, and the issue of appropriate application.

• ### Will our students be statistically literate?

One key goal of adult literacy education is to empower students and enable them to become more informed citizens. This numeracy column focuses on a critical but often neglected aspect of what becoming an informed citizen entails. That aspect involves developing students' statistical literacy skills.

• ### The objective-format question in statistics: Dead horse, old bath water, or overlooked baby?

Cobb attempts to make the claim that we have been too quick to dismiss the objective-format question means of assessment in statistics courses in favor of authentic assessment (e.g., projects, oral presentations, writing assignments).

• ### Some emerging influences underpinning assessment in statistics

A number of influences impinge on statistics education. This chapter focuses on three of these that are especially noticeable at the K-12 level but also operate to some extent at the college level: a) the changing place of statistics in the curriculum, b) the emphasis on processes in the mathematics curriculum, and c) new ideas about learning and the ways that assessment is viewed in education in general. These, together with the purposes and principles of assessment, are explored because they underpin aspects of assessment in statistics education.

• ### Authentic assessment models for statistics education

Authentic assessment is an emerging field within assessment models. It claims to measure by direct means the student performance on tasks that are relevant to the student outside of the school setting. Most educators will agree with the need to assess learning within the context of applications. This chapter will address the following issues: (1) a vision of an effective assessment system must be articulated--what are the standards (visions) for an effective assessment system?, (2) a well-thought-out plan for designing an effective program must be constructed--what are the components of a process for designing an effective authentic assessment program?, (3) classroom teachers readiness to change assessment plans is crucial to any program of assessment--how do you determine the degree of readiness of classroom teachers for a new assessment plan?, and (4) promises abound in the assessment field but limitations can strangle an assessment program at conception--what are the promises and limitations of recent assessment reforms? A crucial aspect of teaching and learning is knowing what and how much is learned. Assessment should be the source of this information. This chapter will give a glimpse of how to design an authentic assessment plan in statistics education.

• ### Using "real-life" problems to prompt students to construct conceptual models for statistical reasoning

The purpose of this chapter is to examine a "model-eliciting activity", based on a "real-life" problem situation, in which students were provided with an opportunity to construct powerful ideas relating to data analysis and statistics, without explicitly being taught. Student results of this activity will be examined that reveal the somewhat surprising fact that children, even those who traditionally do not perform well in mathematics, can invent more powerful ideas relating to trends, averages, and graphical representations of data than their teacers ever anticipated. The student results shared in this chapter are not unique. In classrooms where we have piloted and refined problems (including the ones presented), one common observation is that many of the children who emerge as "most productive" are often those whose mathematical abilities had not been recognized or rewarded by their teachers in the past.

• ### Assessing students' connected understanding of statistical relationships

We believe that connected understanding among concepts is necessary for successful statistical reasoning and problem solving. Two of our major instructional goals in teaching statistics at any level are to assist students in gaining connected understanding and to assess their understanding. In this chapter, we will explore the following questions: (1) Why is connected understanding important in statistics education?, (2) What models of connected understanding are useful in thinking about statistics education?, (3) How can connected understanding be represented visually?, and (4) What approaches exist for assessing connected understanding?