Reciprocal and multiplicative relational reasoning with rational numbers

Abstract Developmental research has focused on the challenges that fractions pose to students in comparison to whole numbers. Usually the issues are blamed on children's failure to properly understand the magnitude of the fractional number because of its bipartite notation. However, recent research has shown that college-educated adults can capitalize on the structure of the fraction notation, performing more successfully with fractions than decimals in relational tasks, notably analogical reasoning. The present study examined whether this fraction advantage also holds in a more standard mathematical task, judging the veracity of multiplication problems. College students were asked to judge whether or not a multiplication problem involving either a fraction or decimal was correct. Some problems served as reciprocal primes for the problem that immediately followed it. Participants solved the fraction problems with higher accuracy than the decimals problems, and also showed significant relational priming with fractions. These findings indicate that adults can more easily identify relations between factors when rational numbers are expressed as fractions rather than decimals

Relational Reasoning with Rational Numbers: Developmental and Neuroimaging Approaches

The study of how children and adults learn mathematics has given rise to a rich set of psychological phenomena involving mental representation, conceptual understanding, working memory, relational reasoning and problem solving. The subfield of understanding rational number processing and reasoning focuses on mental representation and conceptual understanding of rational numbers, and in particular fractions. Fractions differ from other number types, such as whole numbers, both conceptually and in format. Previous research has highlighted the extent to which fractions and other rational numbers pose challenges for children and adults with respect to magnitude estimation and misconceptions. The goal of this dissertation is to highlight the distinct differences in reasoning with different types of rational numbers. First, a neuroimaging study provides evidence that fractions yield a distinct pattern of neural activation during magnitude estimation that differs from both decimals and integers (Chapter 2). Second, a set of behavioral studies with adults highlights the affordances of the bipartite format of fractions for relational reasoning tasks (Chapter 3). Finally, a developmental study with pre-algebra students provides evidence for a significant relationship between relational understanding of fractions and algebra performance, and specifically algebraic modeling. This work is presented in the context of viewing mathematical notation as a type of conceptual modeling. In particular, decimals have advantages in measurement and representing magnitude. Fractions, on the other hand, have advantages in relational contexts, due to the fact that fractions, with their bipartite (a/b) format, inherently specify a relation between the cardinalities of two sets. When mathematics is viewed as a type of relational modeling, rational expressions provide a gateway to more complex mathematical notations and concepts, such as those in algebra

How Adults Understand and Reason about Fractions

<p>We investigated what strategies adults use to compare magnitudes of fractions, and how the strategies vary with overall mathematical knowledge. Because little is known about how people think about fractions, prior research has attempted to assess the extent to which people think of fractions compared to how they think about whole numbers (Bonato, Fabbri, Umilta & Zorzi, 2007; Schneider & Siegler, 2010; Meert, Gregoire, Noel, 2009). In particular, Bonato et al. (2007) has examined the degree to which fractional magnitude comparisons can yield an understanding of the mental representation of fractions. Schneider & Siegler (2010) argue that these assessments cannot be made independently from understanding the strategies for comparison that a particular pair of fractions elicits. The current study extends prior research by comparing fraction magnitude comparison strategies of students at a selective university, with high math proficiency, with those of students at a community college, with lower math proficiency. The goals of this thesis were to identify, explain and define the strategies that are used in fraction magnitude comparisons by adults, investigate how these strategies vary with the math proficiency of the adults, and evaluate whether adults who do not consistently use desirable strategies recognize desirability of good alternative strategies. Our findings indicate that strategy use and consistency of using good and poor strategies vary with overall math knowledge and performance on our magnitude comparison task; that lower performing participants more frequently used strategies that would yield incorrect results and less often recognize when to switch to good alternative strategies compared to the high performing participants.</p