Rational numbers and the common core state standards : a descriptive case study

Abstract

Research suggests that some students and teachers have a problematic sense of number. While previous standards documents have emphasized facility with representations of rational numbers, the Common Core State Standards for Mathematics (CCSS) places comparatively more emphasis on rational numbers in general and the repeating decimal representation of rational numbers in particular. The limited body of literature on repeating decimal concepts suggests that teachers are ill-equipped to teach rational numbers in the way described in CCSS. The purpose of this study was to describe the ways that inservice middle grades teachers, selected from a sample, understand rational numbers, how they interpreted a statement that rational numbers can be written as a repeating decimal, and how these understandings manifested during instruction. The setting is a mid-size town in the Rocky Mountain region. The study was conducted in two phases. The interview phase consisted of 40-60 minute interviews with ten different inservice middle school teachers regarding their rational number sense. The observation phase involved observing four of the ten interview participants delivering a lesson on repeating decimals. Data were analyzed both between and within cases using a standardized open-ended interview protocol designed by the researcher as a framework for comparisons. Results indicated that participants primarily understood rational numbers as a collection of mutually exclusive sets. Rational numbers were defined primarily in terms of their decimal representation, with a majority of participants describing rational numbers as "not irrational." The statement "A number is rational if it can be written as a repeating decimal" was interpreted as incomplete by many participants due to a sharp distinction drawn between terminating and repeating decimals. Each observed teacher displayed at least one understanding of rational number in classroom instruction that manifested during the interview, with many participants repeating their understandings verbatim. Recommendations for inservice teacher professional development are offered, as well as suggestions for future research.