cally, these spaces do not trouble a student who holds this “boundary-filling” conception of measurement (Lehrer, 2003; McClain et al., 1999).

Data

Researchers agree that scientific thinking entails the coordination of theory with evidence (Klahr and Dunbar, 1988; Kuhn, Amsel, and O’Loughlin, 1988), but there are many ways in which evidence may vary in both form and complexity. Achieving this coordination therefore requires tools for structuring and interpreting data and error. Otherwise, students’ interpretation of evidence cannot be accountable. There have been many studies of students’ reasoning about data, variation, and uncertainty, conducted both by psychologists (Kahneman, Solvic, and Tversky, 1982; Konold, 1989; Nisbett et al., 1983) and by educators (Mokros and Russell, 1995; Pollatsek, Lima, and Well, 1981; Strauss and Bichler, 1988). Particularly pertinent here are studies that focus on data modeling (Lehrer and Romberg, 1996), that is, how reasoning with data is recruited as a way of investigating genuine questions about the world.

Data modeling is, in fact, what professionals do when they reason with data and statistics. It is central to a variety of enterprises, including engineering, medicine, and natural science. Scientific models are generated with acute awareness of their entailments for data, and data are recorded and structured as a way of making progress in articulating a scientific model or adjudicating among rival models. The tight relationship between model and data holds generally in domains in which inquiry is conducted by inscribing, representing, and mathematizing key aspects of the world (Goodwin, 2000; Kline, 1980; Latour, 1990).

Understanding the qualities and meaning of data may be enhanced if students spend as much attention on its generation as on its analysis. First and foremost, students need to grasp the notion that data are constructed to answer questions (Lehrer, Giles, and Schauble, 2002). The National Council of Teachers of Mathematics (2000) emphasizes that the study of data should be firmly anchored in students’ inquiry, so that they “address what is involved in gathering and using the data wisely” (p. 48). Questions motivate the collection of certain types of information and not others, and many aspects of data coding and structuring also depend on the question that motivated their collection. Defining the variables involved in addressing a research question, considering the methods and timing to collect data, and finding efficient ways to record it are all involved in the initial phases of data modeling. Debates about the meaning of an attribute often provoke questions that are more precise.

For example, a group of first graders who wanted to learn which student’s pumpkin was the largest eventually understood that they needed to agree



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