Theory

• Assessing students' statistical problem-solving behaviors in a small group setting

The ability to interpret and predict from data presented in graphical form is a higher-order thinking skill that is a necessity in our highly technological society. Recent recommendations for the mathematics and science education communities have therefore stressed the importance of engaging learners in real life statistical tasks given in a setting that will promote effective problem solving. Since the small-group setting has been shown to be a fertile environment in which problem solving can occur, we have used that setting for engaging students in data analysis tasks. However, there is a dearth of ideas related to how to assess students' behavior, thinking, and performance in such a setting. The purpose of this chapter is to describe a framework for assessing students' problem solving behaviors on a graph task as they work within a small-group setting.

• Dimensions in the assessment of students' understanding and application of chance

Randomness and chance variation are key ideas that can function as goals in young students' understanding and application of chance. In this chapter, I examine how these key ideas involve construction of new concepts, as well as beliefs about the place of chance in the world. These ideas are considered from the perspective of the mathematics or statistics classroom culture; i.e., how the classroom culture reflects and fosters beliefs about the place of uncertainty and chance in the world.

• Combinatorial reasoning and its assessment

In this chapter we provide some answers to the following questions: (1) What is combinatorics and what role does it play in teaching and learning probability?, (2) What components of combinatorial reasoning should we develop and assess in our students?, (3) Are there any task variables that influence students' reasoning and provoke mistakes when solving combinatorial problems?, and (4) What are the most common difficulties in the problem-solving process? How should we consider these variables in the teaching and assessment of the subject? We illustrate these points by presenting some examples and test items taken from different research work about combinatorial reasoning and samples of students' responses to these tasks.

• Probability distributions, assessment and instructional software: Lessons learned from an evaluation of curricular software

While mathematics education guidelines have encouraged substantial change in the introductory probability and statistics curriculum, probability distributions still remain an important topic in a first course. In fact, just as software has made data analysis more accessible to students in introductory courses, it also offers new ways to teach probability distributions. However, these new teaching technologies, which emphasize active experimentation and interpretation of displays, also raise new questions. Just what do students see when they exmaine a display of a probability distribution? Do the displays really help students acquire a clear conceptual understanding? Can interactive exercises for related concepts like sampling distributions make good use of displays? Finally, can good assessment practices help us learn when displays are effective and when they might be confusing? This chapter will discuss some interactive, computer-based exercises that use and teach probability distributions, and consider how assessment can help address some of the important questions these new teaching technologies raise.

• Reflections on statistics: Learning, teaching, and assessment in grades K-12

This book represents an interdisciplinary effort to construct an understanding of how to enhance statistics education and assessment for students in elementary and secondary school.

• A theoretical approach to understanding learners of statistics

This paper provides examples of students' reflections on learning statistics. The Mathematics Learning Centre, where I teach, offers help to students experiencing difficulty with basic mathematics and statistics courses at the university. The excerpts are drawn from surveys or interviews of these and other students studying statistics at the University of Sydney. Activity theory, which is based on the work of Vygotsky, provides a helpful conceptual model for investigating learning at the university level. From the perspective of activity theory, learning is viewed as a mediated activity in a sociohistorical context. In particular, the way a student monitors and controls the ongoing cognitive activity depends on how that individual reflects on his or her efforts and evaluates success. In Semenov's words, " Thought must be seen as a cognitive activity that involves the whole person" (1978, p. 5). Students' interpretations of their learning tasks and the educational goals for their self-development are discussed within this theoretical framework.

• New pedagogy and new content: The case of statistics

Statistical education now takes place in a new social context. It is influenced by a movement to reform the teaching of the mathematical sciences in general. At the same time, the changing nature of our discipline demands revised content for introductory instruction, and tehcnology strongly influences both what we teach and how we teach. The case for substantial change in statistics instruction is build on strong synergies between content, pedagogy, and technology. Statisticians who teach beginners should become more familiar with research on teaching and learning and with changes in educational technology. The spirit of contemporary introductions to statistics should be very different from the traditional emphasis on lectures and on probability and inference.

• Experience with authentic assessment techniques in an introductory statistics course

In an effort to align evaluation with new instructional goals, authentic assessment techniques (see, e.g., Archbald and Newman, 1988, Crowley, 1993, and Garfield, 1994) have recently been introduced in introductory statistics courses at the University of the Pacific. Such techniques include computer lab exercises, term projects with presentations and peer reviews, take-home final exam questions, and student journals. In this article, I discuss the University of the Pacific's goals and experiences with these techniques, along with strategies for more effective implementation.

• Mathematics, statistics, and teaching

How does statistical thinking differ from mathematical thinking? What is the role of mathematics in statistics? If you purge statistics of its mathematical content, what intellectual substance remains? In what follows, we offer some answers to these questions and relate them to a sequence of examples that provide an overview of current statistical practice. Along the way, and especially toward the end, we point to some implications for the teaching of statistics.

• A model for assessing higher order thinking in statistics

As in other areas of the school curriculum, the teaching, learning and assessment of higher order thinking in statistics has become an issue for educators following the appearance of recent curriculum documents in many countries. These documents have included probability and statistics across all years of schooling and have stressed the importance of higher order thinking across all areas of the mathematics curriculum. This paper reports on a pilot project which applied the theoretical framework for cognitive development devised by Biggs and Collis to a higher order task in data handling in order to provide a model of student levels of response. The model will assist teachers, curriculum planners and other researchers interested in increasing levels of performance on more complex tasks. An interview protocol based on a set of 16 data cards was developed, trialed with Grade 6 and 9 students, and adapted for group work with two classes of Grade 6 students. The levels and types of cognitive functioning associated with the outcomes achieved by students completing the task in the two contexts will be discussed, as will the implications for classroom teaching and for further research.