Literature Index

In response to the critical role that information and data play in our technological society, there have been national calls for reform in statistical education al all grade levels (Lajoie Romberg, 1998; National Council of Teachers of Mathematics, 1998; School Curriculum and Assessment Authority & Curriculum and Assessment Authority for Wales, 1996; Australian Education Council, 1994). These calls for reform have advocated a more pervasive approach to the study of statistics, one that includes describing, organizing and reducing, representing, and interpreting data. This broadened perspective has created the need for further research on the learning and teaching of statistics, especially at the elementary grades, where there has been a tendency to focus narrowly on some aspects of graphing rather than the broader topics of data handling and data analysis (Shaughness, Garfield, & Greer, 1996).

People make use of quantitative information on a daily basis. Professional education organizations for mathematics, science, social studies, and geography recommend that students, as early as middle school, have experience collecting, organizing, representing, and interpreting data. However, research on middle school students' statistical thinking is sparse. A cohesive picture of middle school students' statistical thinking is needed to better inform curriculum developers and classroom teachers. The purpose of this study was to develop and validate a framework for characterizing middle school students' thinking across 4 processes: describing data, organizing and reducing data, representing data, and analyzing and interpreting data. The validation process involved interviewing, individually, 12 students across Grades 6 through 8. Results of the study indicate that students progress through 4 levels of thinking within each statistical process. These levels of thinking were consistent with the cognitive levels postulated in a general developmental model by Biggs and Collis (1991).

Over the past several years, a community of researchers has been using and refining a particular framework for research and curriculum development in undergraduate mathematics education. The purpose of this paper is to share the results of this work with the mathematics education community at large by describing the current version of the framework and giving some examples of its application.<br>Our framework utilizes qualitative methods for research and is based on a very specific theoretical perspective that is being developed through attempts to understand the ideas of Piaget concerning reflective abstraction and reconstruct them in the context of college level mathematics. Our approach has three components. It begins with an initial theoretical analysis of what it means to understand a concept and how that understanding can be constructed by the learner. This leads to the design of an instructional treatment that focuses directly on trying to get students to make the constructions called for by the analysis. Implementation of instruction leads to the gathering of data, which is then analyzed in the context of the theoretical perspective. The researchers cycle through the three components and refine both the theory and the instructional treatments as needed.<br>In this report the authors present detailed descriptions of each of these components. In our discussion of theoretical analyses, we describe certain mental constructions for learning mathematics, including actions, processes, objects, and schemas, and the relationships among these constructions. Under instructional treatment, we describe the components of the ACE teaching style (activities, class discussion, and exercises), cooperative learning and the use of a mathematical programming language. Finally, we describe the methodology used in data collectoin and analysis. The paper concludes with a discussion of issues raised in the use of this framework, followed by an extensive bibliography.

This article is a discussion of and reaction to two collections of papers on research on Reasoning about Variation: Five papers appeared in November 2004 in a Special Issue 3(2) of the Statistics Education Research Journal (by Hammerman and Rubin, Ben-Zvi, Bakker, Reading, and Gould), and three papers appear in a Special Section on the same topic in the present issue (by Makar and Confrey, delMas and Liu, and Pfannkuch). These papers show that understanding of variability is much more complex and difficult to achieve than prior literature has led us to believe. Based on these papers and other pertinent literature, the present paper, written by the Guest Editors, outlines seven components that are part of a comprehensive epistemological model of the ideas that comprise a deep understanding of variability: Developing intuitive ideas of variability, describing and representing variability, using variability to make comparisons, recognizing variability in special types of distributions, identifying patterns of variability in fitting models, using variability to predict random samples or outcomes, and considering variability as part of statistical thinking. With regard to each component, possible instructional goals as well as types of assessment tasks that can be used in research and teaching contexts are illustrated. The conceptual model presented can inform the design and alignment of teaching and assessment, as well as help in planning research and in organizing results from prior and future research on reasoning about variability.