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

Abstract

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.

ACKNOWLEDGMENTS

The authors thank Jack Smith and two anonymous reviewers for their constructive comments. The present study was funded through the program EPEAEK II in the framework of the project “Pythagoras—Support of University Research Groups” with 75% contribution from European Social Funds and 25% contribution from National Funds.

Notes

We stress that the terms “discrete” and “dense” are used here with respect to the usual order relation. In this sense, natural numbers are discrete, whereas rational and real numbers are dense, the latter being not merely dense, but continuous (e.g., Lucas, 2000). This clarification is in order because the term “discreteness” is sometimes associated with the countability of the rational numbers (i.e., the fact that one-to-one correspondence can be established between natural and rational numbers). However, if rational numbers are “re-ordered” to match the natural numbers, then two successive numbers in this respect are not successive in terms of the usual ordering.

In the following, when we use the term “decimal” we will refer only to decimals that are elements of the rational numbers set, as opposed to decimals that are not rational because they do not repeat, such as π, or .0101101110….

For the purposes of this article, we use the following the term “fraction” to refer to numbers that are actually presented symbolically in the form a/b, and the term “rational number” to refer to any number that is or can be expressed in the form a/b, regardless of its symbolic representation. Thus, the term “rational numbers” is not used in the strict mathematical sense of the word.