Preverbal number sense
Human beings are born with an innate ability, independently of subsequent counting skills, to estimate, compare and mentally manipulate numerical quantities (Butterworth, 2005; Feigenson, Dehaene, & Spelke, 2004; Lipton & Spelke, 2005). While some have referred to this intuitive and non-verbal competence as the approximate number system (Inglis & Gilmore, 2014; Mazzocco et al., 2011; Smets et al., 2012), other have described it as preverbal number sense (Feigenson et al., 2004 Gallistel & Gelman, 1992; Ivrendi, 2011; Tibber et al, 2013). In broad terms, preverbal number sense, reflects those number insights that are innate to all humans and comprises an understanding of small quantities in ways that allow for comparison (Butterworth, 2005; Ivrendi, 2011; Lipton & Spelke, 2005). For example, “6-month-olds can discriminate numerosities with a 1:2 but not a 2:3 ratio, whereas 10-month-old infants also succeed with the latter” (Feigenson et al., 2004, p. 307). Also, children at ages 3 and 4 can estimate accurately the numerosity of sets containing up to five items (Gelman & Tucker, 1975). Thus, numerical discrimination, which “becomes more precise during infancy” (Lipton & Spelke, 2005, p. 978), underpins verbal counting skills (Gallistel & Gelman, 2000) and arithmetic (Zur & Gelman, 2004). This preverbal number sense is independent of formal instruction, developing as an innate consequence of human, and other species’ evolution (Dehaene, 2001; Feigenson et al., 2004).
Of course, there are grey areas where the distinction between those elements of number sense that are innate and those that are not are unclear. For example, it has been argued that by age four or five children have normally acquired initial counting skills and an awareness of quantity that allows them to respond to questions about more or less, while by the time they start school children have typically acquired a sense of a mental number line (Aunio et al., 2006; Griffin, 2004). However, such number-related understandings are frequently dependent on individual family circumstances (Zur & Gelman, 2004), indicating that instruction, whether implicit or explicit, may be necessary. Moreover, as young children from high-socioeconomic status (SES) backgrounds are five times as successful as children from low SES backgrounds on tasks like, which number is bigger, 5 or 4? (Griffin et al. 1994), the case for intervention seems clear, particularly as “aspects of number sense development may be linked to the amount of informal instruction that students receive at home on number concepts” (Gersten et al. 2005, p. 297).
Relational number sense
Most curricular expect students to become functionally numerate and typically discuss the need for the teaching of numeracy or some form of generic number sense. Our interpretation is that these expected or desired competences, which permeate all mathematical learning (Faulkner 2009; National Council of Teachers of Mathematics 1989), can be described relational number sense. For some this refers to the “basic number sense which is required by all adults regardless of their occupation and whose acquisition by all students should be a major goal of compulsory education” (McIntosh et al. 1992, p. 3). For others it reflects understandings and skills that enable a person to
"look at a problem holistically before confronting details, look for relationships among numbers and operations and will consider the context in which a question is posed; choose or invent a method that takes advantage of his or her own understanding of the relationships between numbers or between numbers and operations and will seek the most efficient representation for the given task; use benchmarks to judge number magnitude; and recognize unreasonable results for calculations in the normal process of reflecting on answers" (Reys 1994, p. 115).
Either way, relational number sense is a long-term goal of school but is clearly dependent of a key set of number understandings and competences.
Foundational number sense
Connecting preverbal number sense and relational number sense is what we have come to call foundational number sense (FoNS). FoNS refers to a core set of number-related understandings and competences on which much later mathematical learning depends. Unlike preverbal number sense, FoNS is not innate but requires instruction. Unlike relational number sense, FoNS is not the outcome of learning but the root of that learning. In this sense, FoNS reflects the key number-related understandings that occur during the first year of school (Ivrendi 2011; Jordan and Levine 2009). It is a “construct that children acquire or attain, rather than simply possess” (Robinson et al. 2002, p. 85) and reflects, for example, elementary conceptions of number as a representation of quantity or a fixed point in the counting sequence (Griffin 2004). FoNS is to the development of mathematical competence what phonic awareness is to reading (Gersten and Chard 1999), in that early deficits tend to lead to later difficulties (Jordan et al. 2007; Mazzocco and Thompson 2005). Significantly, FoNS-related competences have been shown to be more robust predictors of later mathematical success than almost any other factor (Aunio and Niemivirta 2010; Byrnes and Wasik 2009).
Our reading of the literature has identified the following eight broad categories of FoNS: