Browsing by Subject "Rational Numbers"
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Item Preservice Elementary Teachers’ Understandings of the Connections Among Decimals, Fractions, and the Set of Rational Numbers: A Descriptive Case Study(2015-12) Pettis, ChristyThe 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.Item The Role of Rational Numbers in Mathematical Achievement and Decision Making(2016-12) Houseworth, JamesUnderstanding rational numbers requires reorganizing our initial understanding of numbers as whole numbers. Coordinating the relationship between the different symbolic formats for expressing rational numbers (i.e., as fractions and as decimals) and their underlying non-symbolic magnitudes is an important component of mathematical development in children (Fazio, Bailey, Thompson, & Siegler, 2014; Siegler & Pyke, 2013; Mazzocco et al., 2013). It is also an important component of decision making in everyday life (Simon, Fagley, & Halleran, 2004; Peters et al., 2006). The goal of the present experiments was to investigate the relationship between rational numbers, expressed in various formats, on one hand and general mathematical achievement and decision-making on the other. Two experiments demonstrated that the format of rational numbers impacts processing: the fraction format hinders magnitude processing compared to the decimal format. Experiment 1 additionally demonstrated that the precision of rational number magnitudes is related to general mathematical achievement. This is evidence that a better understanding of rational numbers is important for more abstract mathematics in adults. Experiment 2 showed that individual differences in rational number ability are also associated with individual differences in bias in decision-making. These findings have practical implications. Educationally, these results suggest that using number lines and intermixing decimal and fraction formats might improve rational number ability and therefore better prepare children for later, more abstract mathematics. Pragmatically, the results of this study suggest numerical ability alone is not a sufficient guard against biased decision making when probabilities are involved. Instead it appears other, non-numerical task features cause bias and need to be identified to make decision making more normative.