Academic journal article The Mathematics Enthusiast

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Decimals

Academic journal article The Mathematics Enthusiast

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Decimals

Article excerpt

Introduction

The historical evolution of decimals as a representation of quantity rests largely on the development of place value and the use of zero in the numeration system. Far more difficult than using the notational system is understanding the quantities represented with the system (Irwin, 2001} in context. Of particular difficulty are decimal fractions (decimals}, rational numbers "which originate by subdivision of each unit interval into 10, then 100,1000, etc., equal segments" (Courant & Robbins, 1996, p. 61}. Research on children's conceptions of decimals illustrates a series of conceptual hurdles involved in interpreting and using the notational system (Resnick et al., 1989; Sackur-Grisvald & Leonard, 1985}. Because children build their understandings of decimals from their existing or coemergent understandings of multidigit whole numbers and fractions, they tend to over-apply concepts for these more familiar objects when the numerals being discussed are decimals. Findings from studies of children's understandings encouraged researchers to begin to explore prospective teachers' (PTs'} understandings of decimal notations (Putt, 1995; Thipkong & Davis, 1991}. Such studies unearthed parallels between categories of reasoning used by children and reasoning used by PTs, encouraging researchers to identify teachers' misconceptions as a source of children's faulty reasoning.

Research on PTs' knowledge of decimal fractions has focused on exploring how decimals are interpreted and used in computation, and how mathematics educators might challenge existing beliefs about the use of decimal fractions. In this report, we focus primarily on terminating decimals that are included in primary school curriculum. A very small collection of reports focused on PTs' knowledge of decimals has been published over the last 25 years, but findings point to the importance of place value in PTs' understanding and application of decimals.

Approaches and Orientations

In the sections that follow, we have summarized historical influences in the study of PTs' knowledge of decimals, findings of published peer-reviewed papers from 1998 to 2011, and additional insights drawn from more recent work. Our approach to identification of articles was consistent with the method described in the introductory article of this Special Issue. In addition, our perspective on decimal understanding influenced our interpretations of the articles. We share this perspective to enable readers to gain insight into our interpretations.

Our view of decimal is informed by explorations of PTs' understandings (D'Ambrosio & Kastberg, 2012; Kastberg & D'Ambrosio, 2011) of decimals using a framework including units, relationships between units, and additivity. As Courant and Robbins (1996) suggest, decimal units in the place value system involve repeatedly "subdividing" an individual unit into 10 parts. So if we begin with 1, then subdividing this unit into 10 parts produces 10 subunits 0.1. This action creates the opportunity for the development of relationships between 1 and 0.1, namely, that 1 is 10 times 0.1 and 0.1 is one tenth of 1. While this example involves adjacent units in the set of place value units {..., 10,1, 0.1, 0.01,...}, any two units in the set can be thought of as related multiplicatively. Finally, sums of multiples of the units can be used to create new decimals, an idea that is represented in expanded notation. For example, if we compare 0.606 and 0.66 using the additive structure, we can see that 0.606 = 0.6 + 0.006 and 0.66 = 0.6 + 0.06. This understanding and understanding of multiples of the units 0.001 and 0.01 allow us to quickly determine that 0.606 is less than 0.66. Understanding decimals as linear combinations of place value units allows us to compose and decompose decimals to quickly compare them. While there are certainly other views of decimals, it was this view that we held and used to make sense of the findings reported in the research. …

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