Math teachers' implicit and explicit fraction knowledge: a mixed-methods approach
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,
... 1). 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.