Prospective Teachers’ Conceptions of Area

Abstract

This study explored 10 prospective teachers’ (PTs’) understanding of the area of a rectangular region using square and non-square rectangular area-units. In an hour-long interview, each PT was first asked to explain how they would find the area of a given rectangular region in terms of a non-square notecard. For several PTs, this task prompted discussion of a square unit defined by the edge of the notecard. In a second task, PTs were presented with a rectangular array of squares and were asked to explain to a fictional child why multiplying length times width does not count the top left corner square twice. An analysis of transcripts of the interview suggested three conceptual components to understanding the area of a rectangular region—area-units, multiplication, and their connection to the length times width formula. Based on the PTs’ responses, we propose multi-tiered levels of reasoning. Each new level of understanding was built on the prior level in a hierarchical fashion. Juxtaposing tasks involving different area-units revealed PTs’ thinking about non-square and square area-units and how these related to the formula “L × W”.