Affordances of Fractions and Decimals for Arithmetic

• In Event: The Importance of Mathematical Competencies: When and for Whom Do They Matter?

Abstract

1. Objectives
Rational numbers are represented using multiple notations—fractions, decimals, and percentages—which suggests that each notation is suitable for different tasks. This assumption has generated a productive line of research aimed at identifying affordances of different rational number notations (DeWolf et al., 2015; Liu & Braithwaite, 2022; Tian et al., 2020). Building on that research, the present study investigated whether fractions and decimals have different affordances for arithmetic calculation among children.
2. Perspectives
Affordances of rational number notations have been analyzed from two theoretical perspectives: semantic alignment (Bassok et al., 1998) and strategy choice (Siegler, 1996). Analyses from both perspectives suggested that fractions afford multiplication whereas decimals afford addition. Consistent with this hypothesis, when solving addition and multiplication problems involving fractions and decimals, university students spontaneously converted addition problems from fraction form into decimal form more often than vice versa, whereas the opposite was true for multiplication problems (Liu & Braithwaite, 2022). These findings suggested that university students prefer decimals for addition and fractions for multiplication.
3. Method
This study tested whether children display explicit notation preferences analogous to the revealed preferences displayed by university students (Liu & Braithwaite, 2022), and whether children’s accuracies when solving problems in different notations parallel their notation preferences. Fifth to eighth graders (Table 1) were presented 12 pairs of equivalent fraction and decimal problems involving either addition (e.g., 2/8+1/5, 0.25+0.2) or multiplication (e.g., 3/4×1/2, 0.75×0.5) and were asked which problem they would prefer to solve. After doing so for all problem pairs, children solved all of the problems.
4. Results
ANOVA on children’s preference for fraction problems over equivalent decimal problems revealed an effect of arithmetic operation (F(1, 83) = 50.6, p < .001). Decimal notation was preferred for addition problems and fraction notation was preferred for multiplication problems (Figure 1A). ANOVA on the difference in accuracy on fraction versus decimal problems similarly revealed an effect of arithmetic operation (F(1, 83) = 6.3, p = .01). Children were more accurate on decimal than fraction addition problems, whereas accuracy did not differ between fraction and decimal multiplication problems (Figure 1B).
5. Conclusions and Significance
Children preferred decimals for addition and fractions for multiplication. Accuracies paralleled notation preferences, implying that the preferences were adaptive. The findings converge with those of Liu and Braithwaite (2022) in suggesting that each rational number notation has distinct strengths and weaknesses that make them more suitable for some tasks than others. A potential implication for educational practice is that each arithmetic operation could be introduced first with the notation that best affords the operation. For example, when introducing procedures for adding rational numbers in fourth grade, decimal addition could be introduced before fraction addition. Then, when introducing procedures for multiplying rational numbers in fifth grade, fraction multiplication could be introduced before decimal multiplication. Doing so would enable children to use, for each arithmetic operation, the notation that better affords the operation as a foundation for learning how to perform the operation with the other notation.

Authors

• Qiushan Liu, Florida State University

• David William Braithwaite, Florida State University