1 3 DOI 10.1007/s11858-015-0669-4 ZDM Mathematics Education ORIGINAL ARTICLE Preschool spontaneous focusing on numerosity predicts rational number conceptual knowledge 6 years later Jake McMullen · Minna M. Hannula-Sormunen · Erno Lehtinen Accepted: 8 January 2015 © FIZ Karlsruhe 2015 of deep and extensive study within mathematics education and educational psychology research (Confrey et al. 2009; Jordan et al. 2013; Mazzocco and Devlin 2008; McMullen et al. 2014b; Merenluoto and Lehtinen 2004; Siegler et al. 2013; Vamvakoussi et al. 2012; Vamvakoussi and Vosnia- dou 2004, 2010; Van Hoof et al. 2015; Vosniadou 2014). One particularly difficult hurdle that must be overcome in understanding rational numbers is the inappropriate use of rules based on the natural number concept when reason- ing about rational numbers (e.g. Vamvakoussi and Vosnia- dou 2004). This overuse of natural number concepts may be explained by a lack of inhibition of intuitive concep- tions about numbers (Vamvakoussi et al. 2012; Van Hoof et al. 2013a, 2015; Obersteiner et al. 2013). In this way, it is argued that reasoning about rational numbers is often negatively influenced by this so-called natural number bias, which has its roots in the privileged role natural numbers play in everyday situations. Understanding rational numbers and the ability to use them in problem solving is not only a key contribu- tor to later mathematical knowledge (Siegler et al. 2012), but also is key for the understanding of a wide range of aspects of everyday life (Reyna and Brainerd 2007). Despite this, few studies have investigated early predictors of rational number conceptual knowledge (Bailey et al. 2014; McMullen et al. 2014a; Siegler et al. 2011; Vukovic et al. 2014).While rational number conceptual knowledge includes a wide range of dimensions (Hallett et al. 2010; Van Hoof et al. 2015), in the frames of this study we refer to rational number conceptual knowledge as covering par- ticularly those aspectsof the size and density of rational numbers which are incongruent with natural numbers. There is sufficient evidence that early spontaneous quan- titative focusing tendencies play a role in the development of mathematical skills (e.g. Hannula and Lehtinen 2005). Abstract Recent evidence suggests that early natural number knowledge is a predictor of later rational number conceptual knowledge, even though students’ difficul- ties with rational numbers have also been explained by the overuse of natural number concepts—often referred to as the natural number bias. Hannula and Lehtinen (Learn Instr 15:237–256, 2005) have shown that children’s ten- dency to spontaneously focus on numerosity (SFON) pre- dicts the development of natural number and arithmetic skills. The present study follows 36 children from the age of 6 years to the age of 12 years in order to determine how preschool SFON tendency and number sequence skills are related to rational number conceptual knowledge at the age of 12 years. The results show that children’s SFON tendency before school age is a strong predictor of later rational numbers conceptual knowledge, even after con- trolling for preschool number sequence skills. This finding has implications for the understanding of how the transition from reasoning about natural number concepts to reason- ing about rational numbers may be influenced by children’s self-initiated practice with numbers in everyday situations. 1 Introduction The difficulties students and many educated adults face with understanding rational numbers has been a subject J. McMullen (*) · M. M. Hannula-Sormunen · E. Lehtinen Centre for Learning Research and Department of Teacher Education, University of Turku, 20014 Turku, Finland e-mail: jake.mcmullen@utu.fi; jamcmu@utu.fi M. M. Hannula-Sormunen Turku Institute for Advanced Studies, University of Turku, Turku, Finland