Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People

Abstract

Fraction and decimal arithmetic pose large difficulties for many children and adults. This is a serious problem, because proficiency with these skills is crucial for learning more advanced mathematics and science and for success in many occupations. This review identifies two main classes of difficulties that underlie poor understanding of rational number arithmetic: inherent and culturally contingent. Inherent sources of difficulty are ones that are imposed by the task of learning rational number arithmetic, such as complex relations among fraction arithmetic operations, and that are present for all learners. Culturally contingent sources of difficulty are ones that vary among cultures, such as teacher understanding of rational numbers, and that lead to poorer learning among students in some places than others. We conclude by discussing interventions that can improve learning of rational number arithmetic.

Fraction Symbols and their Relation to Conceptual Knowledge

“…There are two theories that describe fraction learning as continuous. First, according to the integrated theory of numerical development proposed by Siegler and colleagues, fraction knowledge develops as students adapt their knowledge of whole number magnitude to accommodate fractions (Siegler et al, 2011;Siegler & Braithwaite, 2017;Siegler & Lortie-Forgues, 2017). On this view, learning that fractions, like whole numbers, "have magnitudes that can be ordered and assigned a specific location on a number line" is integral for students' developing fraction skills (Siegler, 2016;Siegler et al, 2011Siegler et al, , p. 274, 2013.…”

“…Many children and adults experience difficulty with fractions (Brizuela, 2006;Hecht et al, 2007). These difficulties start in elementary school and persist into high school and beyond (Siegler & Lortie-Forgues, 2017;Siegler & Pyke, 2013). Educators are concerned about students' difficulties with fractions because fraction knowledge is a foundation for later success in algebra and for more advanced math learning (Booth & Newton, 2012;Hurst & Cordes, 2018a;Siegler et al, 2012).…”

El conocimiento de las fracciones. Una revisión de su relación con factores cognitivos

“…El aprendizaje de ambas dimensiones ha sido relacionado con factores que pueden considerarse externos o internos al aprendiz. Entre los principales factores externos se han propuesto el currículo, las prácticas pedagógicas y el conocimiento del docente sobre la temática (e.g., Fuchs et al, 2013;Siegler y Lortie-Forgues, 2017). Por otro lado, dentro de los principales factores internos se encuentran el conocimiento matemático previo y las habilidades cognitivas que los estudiantes poseen para afrontar este aprendizaje (e.g., Hansen et al, 2015;Hecht y Vagi, 2010;Jordan et al, 2013).…”

Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers

“…Pupils' difficulties to learn rational numbers are believed to be caused by several factors, and one of them is insufficient teachers' knowledge (Siegler & Lortie-Forgues, 2017). Ma (1999), for instance, has showed that U.S teachers who merely mastered a procedural technique for the division task of fractions, were unable to construct real-life problems whose solution involves the same computations.…”