THE RELATION BETWEEN LEARNERS’ SPONTANEOUS FOCUSING ON QUANTITATIVE RELATIONS AND THEIR RATIONAL NUMBER KNOWLEDGE

Abstract

IntroductionRational Number KnowledgeThere is a broad agreement in the literature a good understanding of rational numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction number line estimation, 0-5 fraction number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded fifth graders' rational number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, number knowledge, family income, and family education.Despite the critical importance of a good rational number knowledge, a large body of literature reported children and even adults have a lot of difficulties dealing with various aspects of rational numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for rational numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with rational number tasks are often - at least in part - attributed to the number (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of whole number bias), which is the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to rational numbers in the classroom, they have already formed an idea of what a number is. This idea is based on their experiences (both in daily life and in school) with natural numbers. Once the learners are then instructed about rational numbers, the properties of natural numbers are not always applicable anymore, leading to problems and misconceptions with rational numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in rational number tasks where reasoning purely in terms of natural numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in rational number tasks where reasoning in terms of natural numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural number bias reports three main aspects elicit such systematic errors. The first aspect relates to the density of the set of rational numbers. While natural numbers are characterized by a discrete structure (one can always indicate which number follows a given number; for example after 13 comes 14), rational numbers are characterized by a dense structure (you cannot say which number comes next, because between any two given rational numbers are always infinitely many other rational numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of rational numbers. Research indicates errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural numbers, longer decimals are larger, shorter decimals are smaller, or that a fraction's numerical value always increases when its denominator, numerator, or both increase (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …