Reduction procedures in fraction arithmetic

In elementary school, students learn about fraction concepts and procedures. Concepts are knowledge about fraction facts and principles, for example, that two different fractions (i.e., 4/8 and 5/10) can have the same value. Procedures are algorithms that, when applied correctly, allow students to produce a correct solution (e.g., addition, subtraction, multiplication, and division). Knowledge of fraction procedures and understanding how those procedures apply is central to mathematical knowledge that students acquire in school and is used frequently in everyday life. The goal of the present research was to explore adults' use of a fraction procedure called reduction, a procedure that is taught to students but is sometimes an optional step in fraction arithmetic. Fraction reduction involves dividing the numerator and denominator of a fraction by a common factor. However, there is no existing research on fraction reduction, despite considerable interest in how students learn and apply fraction procedures. In three studies, I assessed the factors that determined the selection and implementation of fraction reduction. I hypothesized that characteristics of the solvers and of the problems would both contribute to people's use of reduction in fraction arithmetic. In Study , 720 participants were assessed on their conceptual and procedural understanding of fractions and profiled as either conceptually- or procedurally-strong solvers. I found that solvers' strategy selection was related to their understanding of fraction concepts and procedures, the required operation, and whether fractions within a problem shared a common denominator (i.e., denominator relation). In two subsequent studies, a smaller group of 59 participants from the larger sample solved fraction arithmetic problems and provided written protocols. In Study 2, I found that a combination of denominator-relation, and operation predicted use of reduction. In Study 3 (n = 27), I found that, when solvers multiplied fractions, conceptually-strong solvers were more likely to reduce fractions than procedurally-strong solvers. These findings are discussed in relation to the potential impact of fraction concepts and procedures on selection of the reduction procedure.