Xenia Vamvakoussi

How Many Decimals Are There Between Two Fractions? Aspects of Secondary School Students’ Understanding of Rational Numbers and Their Notation

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students’ understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The results showed that the idea of discreteness (i.e., that fractions and decimals had “successors” like natural numbers) was robust in all age groups; that students tended to believe that the intermediate numbers must be of the same type as the interval endpoints (i.e., only decimals between decimals and fractions between fractions); and that the type of interval endpoints (natural numbers, decimals, or fractions) influenced students’ judgments of the number of intermediate numbers in those intervals. We interpret these findings within the framework theory approach to conceptual change.

The Transition from Natural to Rational Number Knowledge

In this chapter, we elaborate on learners’ difficulties with rational numbers. We begin with the importance of robust rational number understanding for learners’ general mathematics achievement and then move to the challenges that many learners face with respect to various aspects of rational numbers. Third, we elaborate on previous research that focused on a major source of learners’ difficulty, namely the natural number bias. Fourth, we introduce two complementary theoretical perspectives that have been employed in the past decade to study the natural number bias, namely the framework theory approach to conceptual change and the dual process perspective of reasoning, and then we review some studies that we conducted from these perspectives. We close with discussion of the theoretical and educational implications of these studies and provide suggestions for further research.

Bridging Psychological and Educational Research on Rational Number Knowledge

In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual change. This program focuses on the interference of prior natural number knowledge in the development of rational number learning. The other two are the research program by Moss and colleagues that uses Case’s theory of cognitive development to develop and test a curriculum for learning fractions, and the research program by Siegler and colleagues, who attempt to formulate an integrated theory of numerical development. We will discuss the similarities and differences between these approaches as a means of identifying potential meeting points between psychological and educational research on numerical cognition and in an effort to bridge research between the two fields for the benefit of rational number instruction.