|Clifford Konold (USA)||email@example.com|
|Amy Robinson (USA)||firstname.lastname@example.org|
|Khalimahtul Khalil (USA)||email@example.com|
|Alexander Pollatsek (USA)||firstname.lastname@example.org|
|Arnold Well (USA)||email@example.com|
|Rachel Wing (USA)||firstname.lastname@example.org|
|Susanne Mayr (Germany)||email@example.com|
A number of researchers have observed people using various informal averages to reason about data (Cobb, 1999; Konold and Higgins, in press; Mokros & Russell, 1995; Noss, Pozzi, & Hoyles, 1999). Konold and Higgins (in press) suggest that students' ideal average would be a) an actual value in the data set, b) the most frequently occurring value (the mode), c) located midway between the two extremes both in terms of value (the midrange) and order (the median), and d) relatively close to the other values. In this article, we explore how young students use "modal clumps" - a cluster of values in the heart of a distribution - to characterize what is average in a set of data. We interviewed groups of students at two different schools (grades 5, 9) near the end of a six-week collaborative research project. During the project, students analyzed data they had collected that showed the types and frequencies of various animals killed on town roads. During the interviews, students worked with data similar to those they had collected to answer questions we posed about conditions that might affect the number of animals struck by cars. Analysis of these interviews reveals that the range of values students select as modal clumps manage to meet most of the ideal criteria listed above. Modal clumps may serve the additional purpose for students of allowing them to express at the same time both what is average and how variable the data are. We also examine how students used data to answer such questions as whether more animals were killed on warm or cool days. Despite the fact that they spontaneously used modal clumps to express what was average in a data set, they did not use these modal clumps to compare one group of data to another. This suggests that these students did not regard their modal clumps as group characteristics, or "central tendencies" as described by Konold and Pollatsek (in review). We offer ideas about activities that might help students come to see modal clumps, and other types of averages, as signals or signatures of processes that can be used to judge whether two sets of data were generated from the same process.
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