Abstract
To support students’ understanding of quantity-dependent concepts (e.g., proportions) it is important for middle school mathematics teachers to be able to identify quantities and reason quantitatively. This study investigated how middle school mathematics teachers describe the concept of a quantity and how the concept of quantity appears in teachers’ quantitative work. The research was grounded in my understanding of the concept of quantity developed in concert with Professor Chandra Hawley Orrill (University of Massachusetts Dartmouth) and Associate Professor Rachael Eriksen Brown (The Pennsylvania State University Abington).We define a quantity as a numeric value (actual or hypothetical) that describes a measurable attribute of an object (or phenomenon, situation, etc.) based on a unit of measure. We operationalize this definition as how much, of what, measured how? (MWH). This study’s qualitative methodology used inductive coding and memoing to analyze one-on-one, task-based interviews with eight middle school mathematics teachers. Findings revealed a range of teacher views about quantity, with few holding an integrated MWH view. Teachers’ tended to associate a quantity with whole-number totals; discrete objects; and single, whole units. Teachers’ quantitative work varied markedly along the dimensions of flexibility and encapsulation. Some teachers applied the core elements of a quantity (e.g., attribute, unit) flexibly (e.g., to rates and new units) and encapsulated key aspects of the underlying situation in their written mathematical work. Other teachers showed both low flexibility and encapsulation, sometimes not quantifying relevant attributes and often reducing quantitative work to number manipulation. A third group of teachers’ flexibility was more moderate and encapsulation more variable. The potential importance of flexibility and encapsulation as well as potential interactions between flexibility and encapsulation are discussed in a proposed theory of how middle school mathematics teachers understand and use the concept of quantity.