Elementary School Children’s Fraction Knowledge and Its Association With Algebraic Reasoning

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

Abstract

1. Objectives
Fractions play an important role in mathematics, and fraction knowledge in middle school is a predictor of later mathematics achievement (Bailey et al., 2012). It is imperative to consider when and how we measure fraction knowledge—including its emergence in elementary school and different tasks used to assess it. We conducted an empirical study with second- and third-graders to evaluate their fraction knowledge and its association with algebraic skills.

2. Theoretical Framework
The integrated theory of numerical development (Siegler et al., 2011) highlights the role of fractions in building a comprehensive understanding of number—including the realization that all numbers (fractions and whole numbers) have magnitudes that can be placed on a number line. Unfortunately, fraction tasks tend to be difficult, and children have negative attitudes toward fractions (Sidney et al., 2021). However, fraction knowledge is multi-faceted and can depend on how it is assessed. Our goal was to measure children’s fraction knowledge using two tasks and its relation to early algebra.

3. Methods
Fifty-eight children in Grades 2 and 3 from a midwestern town in the U.S. (M age = 7.6 years, SD = 0.6, range = 7-9) completed an early algebra task, a fraction mapping task, and a fraction equivalence task in a one-on-one setting with a researcher.

4. Data Sources
See Figure 1 for example items. The algebra task included 9 items: four equations to solve, four equations to reconstruct from memory, and defining the equal sign. The fraction mapping task included 8 items; children were shown a symbolic fraction (e.g., 1/3) and selected the shaded shape that represented the same magnitude. The fraction equivalence task included 8 items; children saw two symbolic fractions on either side of an equal sign and had to fill in one missing numerator or denominator. On the fraction tasks, we coded children’s accuracy and their whole-number errors (see Figure 2 for examples of these errors).

5. Results
Children’s accuracy on the fraction mapping task (M=75%, SD=37%) was significantly higher than their accuracy on the fraction equivalence task (M=29%, SD=32%), p<.001. However, even though the sheer number of mistakes was lower on the mapping task, the proportion of whole-number misconceptions was similar across tasks. For fraction mapping, children made 117 mistakes (across all 464 trials) and 48% of those mistakes were whole-number bias errors. For fraction equivalence, children made 330 mistakes (across all 464 trials), and 41% of those mistakes were whole-number bias errors. The tasks shared other similarities; accuracy on fraction mapping and fraction equivalence was significantly (all ps<.05) and positively correlated with age (mapping r=.41, equivalence r=.32) and algebra knowledge (mapping r=.44, equivalence r=.51).

6. Scholarly Significance
Elementary school children exhibited varying amounts of fraction knowledge depending on the task. However, when they were incorrect, they consistently showed a high proportion of errors that were based on mis-applying their knowledge of whole numbers. Importantly, performance on both fraction tasks positively predicted children’s algebra knowledge, corroborating previous evidence on the importance of fractions for algebra (Booth & Newton, 2012).

Authors

• Emily R. Fyfe, Indiana University

• Amanda Grenell, Indiana University