Numerical Magnitude Knowledge: Are All Numbers Perceived Alike?

Abstract

A robust knowledge of numbers, and their magnitudes, is thought to provide students a strong basis for later mathematics learning and achievement (see Siegler, 2016). The current study examined 7th grade students’ (N = 193) knowledge of numerical magnitudes, how this knowledge varied depending on the number’s type (integer or non-integer) and the number’s polarity (positive or negative), and the strategies that students use while estimating different types of numbers. The first experiment of the current study assessed students’ magnitude knowledge through a number line packet that used all-positive, all-negative, and bidirectional scales that spanned from negative to positive numbers; on these number line scales, students were asked to estimate whole numbers, fractions, and decimals. While prior literature has commonly assessed magnitude knowledge of positive integers (i.e., whole numbers) and non-integers (i.e., non-whole numbers), and the literature on negative numbers is growing, the current study is the first to directly explore students’ understanding of positive and negative magnitudes together with the use of all-negative and all-positive number line scales. Results from mixed linear models illustrated that a number’s polarity affects students’ estimates on the all-positive and all-negative scales, as estimates of negative and positive numbers differed in both accuracy and linearity. However, negative and positive estimates on the bidirectional scales were not significantly different from one another. Composite scores were created to reflect students’ performance on four types of number line scales, those that asked students to estimate positive integers, negative integers, positive non-integers, and negative non-integers. Analyses with these composite scores established that both polarity and number type separately affect students’ estimates—negative estimates had more error and were less linear than positive estimates, and non-integer estimates had more error and were less linear than integer estimates. The second experiment of this study used a think-aloud task to examine the strategies that students used while completing the number line task, and how these strategies differed depending on the number line’s overall scale, polarity, and the type of number being estimated (i.e., integers or non-integers). While some strategies were found to be prevalent across all types of number line scales, other strategy choices differed depending on the polarity of the scale, or the type of numbers being estimated. Findings from this study support the integrated theory of numerical development; mainly, that by the 7th grade students have integrated their knowledge of numbers into a unified system that houses both positive and negative numbers, and integers and non-integers. Educational implications are also discussed.