Cliff Konold

This study builds on research of Konold, Finzer, and Kreetong (2014), which demonstrated that students with little or no training in statistics or data analysis created nested tables or narratives to encode data that had an underlying hierarchical structure. The question remains whether students can understand and work with such structures when they are presented with them. In this study, we examine the reasoning of a seventh-grade student as she used the data-analysis environment CODAP to explore data about the growth of plants. The data were represented in a case table that structured them hierarchically at three levels. We attend in particular to her developing understanding of both the hierarchical case table and related graphs at the level Curcio (1987) refers to as “reading the data.” Critical to her coming to understand the data was 1) coming to see the graphs and table as dynamic rather than static representations, and 2) using dynamic linking among the objects to explore their relations. These findings have the potential for informing curricula, software development, and teachers’ knowledge for teaching.

A core component of informal statistical inference is the recognition that judgments based on sample data are inherently uncertain. This implies that instruction aimed at developing informal inference needs to foster basic probabilistic reasoning. In this article, we analyze and critique the now-common practice of introducing students to both “theoretical” and “experimental” probability, typically with the hope that students will come to see the latter as converging on the former as the number of observations grows. On the surface of it, this approach would seem to fit well with objectives in teaching informal inference. However, our in-depth analysis of one eighth-grader's reasoning about experimental and theoretical probabilities points to various pitfalls in this approach. We offer tentative recommendations about how some of these issues might be addressed.

For the past 15 years, pre-university students in many countries including the United States have encountered data analysis and probability as separate, mostly independent strands. Classroom-based research suggests, however, that some of the difficulties students have in learning basic skills in Exploratory Data Analysis stem from a lack of rudimentary ideas in probability. We describe a recent project that is developing materials to support middle-school students in coming to see the “data in chance” and the “chance in data.” Instruction focuses on four main ideas: model fit, distribution, signal-noise, and the Law of Large Numbers. Central to our approach is a new modeling and simulation capability that we are building into a future version of the data-analysis software TinkerPlots. We describe three classroom-tested probability investigations that employ an iterative model-fit process in which students evaluate successive theories by collecting and analyzing data. As distribution features become a focal point of students’ explorations, signal and noise components of data become visible as variation around an “expected” distribution in repeated samples. An important part of students’ learning experience, and one enhanced through visual aspects of TinkerPlots, is becoming able to see things in data they were previously unable to see.

Research has shown that adults have intuitions about probability and statistics that, in many cases, are at odds with accepted theory. The existence of these strongly-held ideas may explain, in part, why learning probability and statistics is especially problematic. One objective of introductory instruction ought to be to help students replace these informal conceptions with more normative ones. Based on this research, items are currently being developed to assess conceptual understanding before and after instruction.