Integrated Knowledge of Rational Number Notations Predicts Children's Math Achievement and Understanding of Numerical Magnitudes

Abstract

We propose that integrated number sense, the ability to fluidly translate and compare magnitudes within and across notations, is central to understanding of rational numbers. Consistent with this hypothesis, two studies of 6th through 8th grade students (N=264 and N=46) indicated that accuracy comparing magnitudes within and across notations predicted overall math achievement and fraction number line and arithmetic estimation accuracy. Cross-notation magnitude comparison accuracy (i.e., fraction vs. decimal, percentage vs. fraction, and percentage vs. decimal) accounted for variance in math outcomes beyond that explained by magnitude representations of individual notations. The findings also revealed a percentages-are-larger bias, in which percentages are perceived as larger than equivalent fractions and decimals. Theoretical and instructional implications are discussed.

Number lines can be more effective at facilitating adults’ performance on health-related ratio problems than risk ladders and icon arrays.

“…This work suggests that a well-chosen and well-designed visual display can highlight mathematical relations and promote mathematically accurate reasoning—the mere presentation of a number line without instructional intervention was sufficient to improve performance. Number lines may derive their advantage from alignment with the mental number line (Okamoto & Case, 1996; Dehaene, 2011; Siegler et al, 2011) and from implicitly drawing connections between the fractional components and percentages (Moss & Case, 1999; Schiller, 2020). This research bridges prior work in education contexts that has demonstrated a number line advantage over other types of visual displays for helping learners reason with ratios (Fazio et al, 2016; Fuchs et al, 2013, 2014; Gunderson et al, 2019; Hamdan & Gunderson, 2017; Moss & Case, 1999; Rittle-Johnson et al, 2001; Saxe et al, 2013; Schneider et al, 2009; Sidney, Thalluri, et al, 2019; Siegler et al, 2011).…”

Leveraging Math Cognition to Combat Health Innumeracy

“…As it turns out, many people have more negative attitudes about math involving fractions and percentages than about math involving natural numbers (Sidney et al, 2021) and prefer to avoid doing math. Furthermore, objective numeracy scales are often presented as word problems, which are notoriously difficult to parse (Koedinger & Nathan, 2004) and may result in impulsive calculation (Schiller, 2020). Impulsive calculation (i.e., “plug and chug”) occurs when people read a math problem and immediately begin solving it before trying to make sense of the underlying structure of the problem (Chi et al, 1981).…”

“…Alerting adults that the natural-number bias exists may prompt them to slow down and “make sense” of a fraction-related math problem (i.e., rely on relational reasoning). This may encourage people to use more effective strategies to translate rational-number information into an easier-to-interpret format (Alibali & Sidney, 2015; Fazio et al, 2017; Fitzsimmons et al, 2020b; Mielicki, Fitzsimmons, Schiller, et al, 2022; Moss & Case, 1999; Sidney, Thalluri, et al, 2019; Siegler et al, 2011; Siegler & Thompson, 2014; Thompson et al, 2021) rather than impulsively calculating (Schiller, 2020) and overapplying natural-number knowledge to fraction problems. Such interventions would help adults with low numeracy overcome their misunderstanding of rational numbers to engage strategically with rational-number magnitudes (see Alibali & Sidney, 2015; Fitzsimmons et al, 2020b; Schiller et al, 2022; see Fig.…”

“…Translating among fractions, decimals, and percentages can help people accurately interpret the magnitude of rational numbers. College students, who engaged with a playful educational intervention about monsters racing along a number line, demonstrated improvements in their ability to translate among fractions, percentages, and decimals (Schiller et al, 2022). Translation skills are helpful to adults as they attempt to reason about the magnitude of health risks or statistics in difficult-to-handle formats, such as the 1-in- x format.…”

“…4). Another promising solution is scaffolding learners’ understanding of rational numbers by helping them make connections across percentages, decimals, and fractions (Moss & Case, 1999; Schiller, 2020; Schiller et al, 2022; see Fig. 5) rather than introducing these concepts separately across several grade levels.…”