for the learning of mathematics

an international journal of mathematics education

Margaret Rathouz - Vol. 30 Num. 1 (2010)
 Ambiguity in units and the referents: two cases in rational number operations


I explore the impact of ambiguous referral to the unit on understanding of decimal and fraction operations during episodes in two different mathematics courses for pre-service teachers (PSTs). In one classroom, the instructor introduces a rectangular area diagram to help the PSTs visualize decimal multiplication. A transcript from this classroom exposes ambiguities in language that may hinder the ability of PSTs to track the elusive unit in this multiplicative situation. Placing the computation in a context is suggested as a remedy to help students resolve these ambiguities. The instructor for a second course intentionally introduces a task with some built-in ambiguities as a mechanism to catalyze student reasoning through discussion and debate. In a segment presented from this classroom, PSTs struggle with the changing nature of the unit as they create story problems for fraction operations. The impact of students' own language and pictorial representations is discussed as it relates to ability to track the unit during fraction operations.