Incorrect Ways of Thinking About the Size of Fractions

Abstract

The literature has amply shown that primary and secondary school students have difficulties in understanding rational number size. Many of these difficulties are explained by the natural number bias or the use of other incorrect reasoning such as gap thinking. However, in many studies, these types of reasoning have been inferred from comparing students’ accuracies in multiple-choice items. Evidence that supports that these incorrect ways of reasoning are indeed underlying is scarce. In the present work, we carried out interviews with 52 seventh grade students. The objective was to validate the existence of students’ incorrect ways of thinking about fraction size that were previously inferred from patterns of correct and incorrect answers to multiple-choice items, by looking at students’ verbalizations, and examine whether these ways of thinking are resistant to change. Students’ verbalizations support the existence of the different incorrect ways of thinking inferred from previous studies in fraction size. Furthermore, the high levels of confidence in their incorrect reasoning and the fact that they were reluctant to change their answer when they were confronted with other reasoning suggest that these ways of thinking may be resistant to change.

Testing the whole number interference hypothesis: Contributions of inhibitory control and whole number knowledge to fraction understanding.

“…In the present study, we chose to control for magnitude difference between the fraction pairs and partial distance. Although, we were not able to exactly match between the conditions on factors like gap distance, half benchmarking, simplified forms, and familiarity, which have been shown to impact performance (Clarke & Roche, 2009; González-Forte et al, 2018, 2020), there does not seem to be a definitive impact of these factors on students’ performance in the current sample (). For example, applying the gap strategy (selecting the fraction with the smallest distance between numerator and denominator) would only lead to an error on one congruent pair (2/7 vs. 3/9), yet performance was no worse on this pair (see ).…”

“…Contrary to our expectations, participants were more accurate on the incongruent distinct component problems than on the congruent problems. However, a number of previous studies have documented stronger performance for incongruent relative to congruent problems with distinct components (Gómez & Dartnell, 2019; González-Forte et al, 2018, 2020; Obersteiner et al, 2013; Rinne et al, 2017; Viol Ferreira Toledo et al, 2023), which may reflect the strategy that participants engaged in when solving these problems. In these cases, participants (or at least a subset of participants), may have used a “select the smaller number strategy,” which would lead to the correct answer for incongruent but not congruent problems.…”

Not realizing that you don't know: Fraction state anxiety is reduced by natural number bias

“…Merenluoto and Lehtinen (2004) stated that even students in upper secondary school can be unaware of their misconception and show overconfidence in their natural number-based answers. Similarly, González-Forte et al (2023) showed that seventh-grade students with a clear NNB profile had high confidence levels, when answering a fraction task incorrectly with a natural number-based answer. Moreover, these students were reluctant to adapt their reasoning, when confronted with a student's answer who had reasoned correctly.…”

“…It aims to extend the findings by Halme et al (2022) by assessing whether fraction state anxiety responses differ between low‐performing students with qualitatively different fraction understanding. Previous studies have shown that students with a strong NNB are unaware of their incorrect reasoning and show overconfidence in their incorrect answers (González‐Forte et al, 2023; Merenluoto & Lehtinen, 2004). Therefore, students with a clear NNB may also have low fraction state anxiety despite their poor fraction understanding.…”

La competencia mirar profesionalmente de futuros profesores de matemáticas: uso de representaciones de la práctica

“…En particular, el documento teórico está formado por información procedente de investigaciones sobre la comprensión de los números racionales en estudiantes de secundaria (González-Forte et al, 2020;González-Forte et al, 2022). Esta información integra diferentes formas de razonar (correctas e incorrectas) de los estudiantes en i) actividades de comparar y ordenar fracciones y números decimales, ii) actividades de suma, resta, multiplicación y división con fracciones y números decimales, y iii) actividades sobre el concepto de densidad.…”