Brazilian math teacher’s magnitude representation and strategy use in fraction comparison: a mixed methods study

Abstract

Fractions have a fundamental importance in the design and construction of mathematical knowledge, being the basis for algebra and other advanced mathematical content (BAILEY et al., 2012; BOOTH; NEWTON, 2012; SIEGLER et al., 2012; TORBEYNS et al., 2015). However, historically this representational form of rational numbers has presented several obstacles in the teaching and learning process, both for students and teachers (PINTO, 2011; SERRAZINA; RODRIGUES, 2018; SIEGLER; THOMPSON; SCHNEIDER, 2011). One of the conceptual bases for the development of fractional thinking is understanding the magnitude of these numbers (SIEGLER et al. 2011, 2013; VAN HOOF et al., 2018), yet how teachers process fractions magnitudes remain unknown. Thus, this study aimed to use the lens of experimental math cognition research to investigate how postgraduate mathematics teachers process the magnitude of fractions. A convergent parallel mixed method was carried out: quantitative data were collected based on research in neuroscience and cognitive psychology, followed by a qualitative task where participants explained their answers on a subset of the comparisons. Teachers’ comparison performance indicated they were not thinking about fractions according to the fractions’ individual components. Instead, they used other strategies to compare symbolic fractions, mostly anchored in the part-whole perspective. The most commonly used strategy was the non-generalizable Gap strategy, where participants choose the fraction with a smaller difference between numerator and denominator rather than fully processing the magnitude of fractions. Thus, we can say that the investigated teachers were not influenced by a componential view, but we cannot affirm a holistic view either, since the most used strategy contains misconceptions and is based on an arithmetic procedure. Interestingly, during the qualitative task, which asked teachers textually to justify the choice of the largest fraction, some users of the Gap strategy recognized that it was not mathematically valid. Therefore, a direct implication of this research is the need for teaching strategies that illustrate failures of the Gap strategy. More broadly, instructional approaches that challenge intuitive thinking, allowing us to understand the magnitude of the fraction (SIEGLER et al., 2011) may prove useful for this population.

High School Students’ Fraction Comparison Strategy Profiles Relate to Rational Number Difficulties and Cognitive Outcomes

“…Although there was virtually no difference between their accuracy on compatible (94.9%) and misleading (91.1%) trials with common components, high-achieving students were worse at comparing compatible (69.7%) than misleading (89.7%) trials. This pattern of stronger performance on misleading than compatible has been reported in mathematicians (Obersteiner et al, 2013) and math teachers (Toledo et al, 2023). Such a pattern of performance may reflect the use of the gap strategy.…”

“…Consistent with these predictions, math teachers (Toledo et al, 2023) and even mathematicians (Obersteiner et al, 2013) show both componential and rational distance effects, but componential distances modulate performance as predicted by their relation with rational distance. Critically, when the effects of rational and componential distances are contrasted, rational distance has the strongest modulation (Toledo et al, 2023), suggesting that, at least among highly proficient individuals, the effects of componential distance may be partially explained by the intertwined relation between these distances and rational distance. In the Common Component block, the relations between rational and componential distance (for the metrics that varied) were strong and positive.…”

“…Participants engage in holistic processing depending on a variety of extrinsic (e.g., type of fractions, Bonato et al, 2007) and intrinsic (e.g., age) factors. Young adults typically show robust holistic processing when comparing fractions (Cui et al, 2020;Meert et al, 2009;Mock et al, 2018;Obersteiner et al, 2013;Schneider & Siegler, 2010;Toledo et al, 2023), while younger populations show mixed results. Some studies have found rational distance effects in middle-school-aged children Siegler et al, 2011b) and even before formal fraction instruction (Kalra et al, 2020;Szkudlarek & Brannon, 2021), while others have failed to find them (Liu et al, 2013).…”

“…Such a pattern of performance may reflect the use of the gap strategy. In fact, when math teachers were asked to describe the strategy they use to compare fractions, the gap strategy was the one they most frequently reported, 25.6% (Toledo et al, 2023). This flawed strategy is characterized by choosing the fraction with the smaller gap between denominator and numerator as the largest fraction (e.g., choosing 24/25 over 16/19 because the gap for the first fraction is 1 while for the second one is 3) (Clarke & Roche, 2009) such an approach works for all misleading stimuli, but only some compatible ones (Rosenberg- Lee, 2021).…”

Competing numerical magnitude codes in decimal comparison: Whole number and rational number distance both impact performance

“…Several studies have demonstrated that rational numbers display distance effects Bonato et al, 2007;DeWolf et al, 2014;Hurst & Cordes, 2016;Hurst & Cordes, 2018;Kalra, Binzak, Matthews, & Hubbard, 2020). More recent efforts have attempted to disentangling the effects of rational vs. whole distance (Ischebeck et al, 2009;Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013;Toledo, Abreu-Mendoza, & Rosenberg-Lee, 2022, August 10). These studies have focused on fractions, where the actual rational distance between fraction pairs has been contrasted with the distance between the components (numerators and denominators).…”