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' under-standing 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.