Preservice Elementary Teachers’ Understandings of the Connections Among Decimals, Fractions, and the Set of Rational Numbers: A Descriptive Case Study

Abstract

The mathematical knowledge needed for teaching is a specialized form of mathematical knowledge, (Ball, Thames, & Phelps, 2008). One important area of mathematical understanding for elementary teachers is the area of number and numeration. Mathematically, the sets of whole and rational numbers and their corresponding notational systems are deeply interconnected. Ensuring that preservice elementary teachers understand the ways these sets of numbers and notations are connected, both mathematically and developmentally, is a critical component of teacher education coursework. This study is a descriptive case study (Yin, 2014) documenting preservice elementary teachers’ ways of understanding the relationships among fractions, decimals, and the sets of rational and irrational numbers. The unit of analysis was a single class of preservice elementary teachers participating in an eight-week instructional unit designed to support them in making explicit connections between concepts related to number and numeration. The broad agenda for this study is to support the development of curricula that may productively and efficiently develop preservice teachers’ understandings of the connections among fractions, decimals, and the sets of rational and irrational numbers. This study extends prior work on bridging tools (Abrahamson & Wilensky, 2007) by documenting how two bridging tools were used to promote understanding of the connections between fraction and decimal notation. Results from early in the unit indicate that preservice elementary teachers’ initial understandings of the connections among fractions, decimals, and the set of rational numbers were limited and often inaccurate. Limited understandings of decimal notation were also documented. Finally, the preservice teachers primarily used symbolic representations to explain the connection between fractions and decimals. After the unit, the preservice teachers showed a more connected understanding of the relationships among fractions, decimals, and the set of rational numbers. The majority of preservice teachers demonstrated the ability to use multiple, non-symbolic representations in order to find and explain connections between fractions and decimals. Widespread understandings of decimal notation were documented, but these understandings were applied inconsistently. Together, the results suggest that a connected approach to curriculum design shows promise as a way to address multiple areas of preservice teachers’ content understandings simultaneously.