Numerical magnitude understanding of natural and rational numbers in secondary-school students: a number line estimation study

Abstract

Rational numbers, such as fractions and decimals, are harder to understand than natural numbers. Moreover, individuals struggle with fractions more than with decimals. The present study sought to disentangle the extent to which two potential sources of difficulty affect secondary-school students’ numerical magnitude understanding: number type (natural vs. rational) and structure of the notation system (place-value-based vs. non-place-value-based). To do so, a 2 (number type) × 2 (structure of the notation system) within-subjects design was created in which 61 secondary-school students estimated the position of four notations on a number line: natural numbers (e.g., 214 on a 0–1000 number line), decimals (e.g., 0.214 on a 0–1 number line), fractions (e.g., 3/14 on a 0–1 number line), and separated fractions (3 on a 0–14 number line). In addition to response times and error rates, eye tracking captured students’ on-line solution process. Students had slower response times and higher error rates for fractions than the other notations. Eye tracking revealed that participants encoded fractions longer than the other notations. Also, the structure of the notation system influenced participants’ eye movement behavior in the endpoint of the number line more than number type. Overall, our findings suggest that when a notation contains both sources of difficulty (i.e., rational and non-place-value-based, like fractions), this contributes to a worse understanding of its numerical magnitude than when it contains only one (i.e., natural but non-place-value-based, like separated fractions, or place-value-based but rational, like decimals) or neither (i.e., natural and place-value-based, like natural numbers) of these sources of difficulty.