| The purpose of this study was to investigate and characterize conceptions of ratio underlying preservice and practicing teachers' approaches to (ratio) tasks and to explore and identify how those conceptions of ratio are related developmentally. I conducted a retrospective analysis of a 20-week whole-class teaching experiment in which the goal was to promote students' conceptions of ratio. I analyzed the videotapes and full transcripts of those classes, students' solutions to a written pretest, and students' journals.;I formulated a construct-the identical groups conception-as a way to characterize the conception of ratio that seemed to be underlying the students' work. Students with an identical groups conception understand that (a) in some situations the quality of interest is created by associating (grouping) two quantities, (b) the quality of interest as determined by the particular amounts of those two quantities remains invariant when that original group is partitioned into identical groups or when groups identical to that original group are accumulated, and (c) ratio is the mathematical model that labels and organizes that system of identical groups. The identical groups conception is a limited conception because not all ratio situations conform to the identical groups structure and hence the identical groups scheme is not triggered by all ratio tasks. Therefore, at this point in development, students still use the incorrect addition strategy to solve (ratio) tasks that do not trigger one of the schemes (e.g., partitive division, part-whole, identical groups) that students do have available.;I contrasted the identical groups conception with a more advanced conception---the conception of ratio as a quantity. The development of a conception of ratio as a quantity supersedes the identical groups conception. The defining characteristic of a conception of ratio as a quantity is the understanding that a ratio is an intensive quantity---a single quantity that measures the multiplicative relationship between two quantities rather than an amount. That is, a ratio is a measure of relative size. This conception of ratio as a measure of relative size enables students to appropriately differentiate between additive and multiplicative situations. |
