From a curriculum perspective, number sense has been described as an “intuition about numbers that is drawn from all varied meanings of number” (National Council for Teachers of Mathematics, 1989, p. 39). From the perspective of young children’s learning “we all know number sense when we see it but, if asked to define what it is and what it consists of, most of us, including the teachers among us, would have a much more difficult time” (Griffin, 2004, p. 173); while from the perspective of learning difficulties, “no two researchers have defined number sense in precisely the same fashion” (Gersten et al., 2005, p. 296).
This definitional uncertainty is not new. For example, and writing from the perspective of learning difficulties, Schmitt (1921, p.538) reports a teacher who described an eight-year old girl struggling to master arithmetic as, “absolutely lacking in number sense”. Later, Hildreth (1935, p.606) alludes to a mathematical phobia in her social psychological account of Peter, a six year-old with learning difficulties, who “sensed the anxiety that the family felt about his evident lack of number sense and detested nothing more than to be asked to count, to identify or to write numbers”. Others working in a cognitive psychological tradition construed number sense as an innate human characteristic, distinguishing between the innate estimation of small quantities based on, effectively, subitising and an acquired estimation of large quantities based on experience (Riess, 1943). A year later the same author notes that a child can “estimate the relative size of small groups, although this ‘number sense’ does not carry him far” (Riess (1944, p.26). However, Judd (1924, p.180), also writing from a psychological perspective, argues the opposite, asserting that since there “is no such thing as a number sense or an instinct for exactness”, all insights with respect to arithmetic and how numbers function must be taught. In other words, during the first decades of the twentieth century, number sense was not only construed differently in different education-related fields but carried with it very different expectations with respect to definitional clarity and the whether number sense was innate.
Also, there exists further confusion with respect to number sense and numeracy. McIntosh et al. (1992) construe numeracy “to mean only an ability to cope with the basic demands of everyday life” (p. 2), while number sense, somewhat self-referentially, is construed as “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” (p.3). The distinction seems subtle in that both emphasise basic competences, although the former alludes to everyday demands and the latter professional. This is interesting as Atweh et al’s (2014, p.5) summary of the literature with respect to the teaching of numeracy in low-income countries, argue that
"numeracy education is understood to consist of a wide variety of ‘contents’, including knowledge, skills and dispositions, needed to function in different ‘contexts’, including inside and outside the school, and for achieving different ‘aims’, including developing the learners’ identity and active participation in the world".
That is, Atweh et al.’s perspective on numeracy seems closer to McIntosh et al.’s definition of number sense than their definition of numeracy, a confusion exacerbated by Askew (2015, p.1), who writes that
"While parts of the education community cling to the ‘number sense’ and ‘basic arithmetic’ view of numeracy, there is a substantial body of work that argues that numeracy education needs to go beyond the basic skills model and to engage learners in thinking critically about how mathematics is used, the power that mathematics has (or how it is used by those in power), and to question the status quo".
Here, Askew seems to equate number sense more closely to McIntosh et al.’s numeracy and numeracy closer to McIntosh et al.’s number sense. That being said, the purpose here is not to reconcile different perspective on what some might construe as the same thing, but to clarify a particular form of number sense in ways that teachers, researchers and policy makers may find useful. Our aim is to characterise those essential number competences that require instruction and which year one children need to acquire if they are to become successful learners of mathematics. In so doing we
“accept that an innate numerical competence cannot account for the rapid expansion in mathematical ability seen in school-aged children, suggesting that socially mediated numerical activities also play a role in the development of mathematical skills” (Bull et al., 2011, p.453).