Cross-Notation Knowledge of Rational Numbers Predicts Rational Number Arithmetic

• In Event: Connecting Fractions and Decimals

Abstract

Unlike whole numbers, rational numbers are frequently represented using multiple symbolic notations—fractions, decimals, and percentages. Understanding rational numbers requires not only understanding each notation alone, or "within-notation knowledge," but also understanding relations between notations, or "cross-notation knowledge." For example, to understand rational number magnitudes, students should know not only that 3/4>1/2 and 0.75>0.50, but also that 3/4=0.75, 3/4>0.50, and 0.75>1/2.

Acquiring cross-notation knowledge presents a challenge, but such knowledge could also confer benefits. First, knowledge of connections between notations could enable learners to leverage knowledge of one notation to help solve problems involving the other notation. Second, reflecting on connections between notations could highlight underlying properties shared by all of them. These observations suggest that cross-notation knowledge could facilitate learning about rational numbers.

To test this hypothesis, this study built on previous research that has found positive relations between individual differences in fraction or decimal magnitude knowledge and rational number arithmetic skill (Bailey et al., 2017; Rittle-Johnson & Koedinger, 2009; Siegler & Pyke, 2013; Siegler et al., 2011; Torbeyns et al., 2015). If the hypothesis is correct, then individual differences in cross-notation magnitude knowledge should predict rational number arithmetic skill even when controlling for within-notation magnitude knowledge.

This prediction was tested by re-analyzing data from three published studies (Study 1: N = 277 fourth to sixth graders, Authors, 2016; Study 2: N = 39 fourth to seventh graders, Authors, 2018; Study 3: N = 394 seventh and eighth graders, Authors, 2019). All studies included measures of within-notation magnitude knowledge (e.g., "1/2>3/4?" or "0.50>0.75?"), cross-notation magnitude knowledge (e.g., "1/2>0.75?" or "0.50>3/4?"), and arithmetic with fractions and/or decimals. Analyses assessed relations between both types of magnitude knowledge and arithmetic accuracy.

As predicted, in all three datasets, cross-notation magnitude knowledge predicted rational number arithmetic when controlling for within-notation magnitude knowledge. In most cases, effects of cross-notation magnitude knowledge were larger than those of within-notation magnitude knowledge. These results suggest that assessments of rational number magnitude knowledge that include only within-notation tasks fail to capture an important aspect of individual differences in rational number magnitude knowledge. The findings therefore argue for inclusion of cross-notation tasks in such assessments. Results are also consistent with the possibility that cross-notation knowledge facilitates learning rational number arithmetic, a possibility that should be explored in future experimental studies.

Another key finding was that relations between within-notation magnitude knowledge and arithmetic accuracy were not notation-specific. That is, fraction magnitude knowledge did not predict arithmetic with fractions more than arithmetic with decimals; similarly, decimal magnitude knowledge did not predict arithmetic with decimals more than arithmetic with fractions. Performance on fraction and decimal magnitude tasks may at least in part reflect a generalized, rather than fraction-specific or decimal-specific, understanding of rational number magnitudes that is related to arithmetic proficiency with both notations.

Authors

• David Braithwaite, Florida State University
  Presenting Author

• Jake McMullen, Turun yliopisto (UTU)
  Non-Presenting Author

• Michelle A Hurst, University of Chicago
  Non-Presenting Author