Analogical Reasoning with Rational Numbers: Semantic Alignment Based on Discrete Versus Continuous Quantities

Analogical Reasoning with Rational Numbers: Semantic Alignment Based on Discrete Versus Continuous Quantities Melissa DeWolf (mdewolf@ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA, USA Miriam Bassok (mbassok@u.washington.edu) Department of Psychology, University of Washington Seattle, WA, USA Keith J. Holyoak (holyoak@lifesci.ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA, USA Abstract without having grasped how fractions relate to decimals, or how either number type relates to integers. This conceptual disconnection in turn contributes to a compartmentalization of mathematical operations (e.g., multiplication of fractions is treated as unrelated to multiplication of integers; Siegler et al., 2011; Siegler & Pyke, 2012). Although mathematical relations are typically construed as internal to the formal system of mathematics, the application of mathematics to real-world problems also depends on grasping relations between mathematical concepts and the basic ontological distinctions among the concepts to which mathematics must be applied. Rather than treating mathematical concepts as purely formal, both children and adults are naturally guided by a process of semantic alignment, which favors mapping certain mathematical concepts (and their associated operations) onto certain conceptual types. Bassok, Chase and Martin (1998) demonstrated that the basic mathematical operations of addition, subtraction, multiplication, and division are typically conceptualized within a system of relations between mathematical values and objects in the real world. Specific mathematical operators are semantically aligned with particular relationships among real-world objects. For example, addition is aligned with categorical object relations (e.g., people find it natural to add two apples plus three oranges, because both are subtypes of a common category, fruit), whereas division is aligned with functional object relations (e.g., a natural problem would be to divide ten apples between two baskets). Semantic alignment has been demonstrated with both children and adults (e.g., Martin & Bassok, 2005), and for many adults the process is highly automatic (Bassok, Pedigo, & Oskarsson, 2008). Although natural semantic alignments are implicitly acknowledged in the construction of textbook word problems (Bassok et al., 1998), teachers seldom discuss these alignments with their students. This gap may contribute to the difficulty of conveying how and why mathematical formalisms “matter” in dealing with real- world problems. Non-integer rational numbers, such as fractions and decimals, pose challenges for learners, both in conceptual understanding and in performing mathematical operations. Previous studies have focused on tasks involving access and comparison of integrated magnitude representations, showing that adults have less precise magnitude representations for fractions than decimals. Here we show the relative effectiveness of fractions over decimals in reasoning about relations between quantities. We constructed analogical reasoning problems that required mapping rational numbers (fractions or decimals) onto pictures depicting either part- whole or ratio relations between two quantities. We also varied the ontological nature of the depicted quantities, which could be discrete, continuous, or continuous but parsed into discrete components. Fractions were more effective than decimals for reasoning about discrete and continuous-parsed (i.e., discretized) quantities, whereas neither number type was particularly effective in reasoning about continuous quantities. Our findings show that, when numbers serve as models of quantitative relations, the ease of relational mapping depends on the analogical correspondence between the format of rational numbers and the quantity it models. Keywords: analogy; relational reasoning; number concepts; fractions; decimals; semantic alignment; math education Introduction Mathematical Understanding as Relational Reasoning Mathematics is in essence a system of relations among concepts based on quantities. A core problem with math education, particularly in the United States, is that greater focus is placed on execution of mathematical procedures than on understanding of quantitative relations (Richland, Stigler & Holyoak, 2012; Stigler & Hiebert, 1999; Rittle- Johnson & Star, 2007). An early manifestation of this problem involves teaching of non-integer rational numbers in the standard curriculum—typically, first fractions and subsequently decimals. Students often leave middle-school (and often enter community college: see Stigler, Givvin & Thompson, 2010; Givvin, Stigler & Thompson, 2011)