Core systems of number and arithmetic

Feigenson et al. (2004) named two core systems based on the number sense theory of Dehaene (1997). Core system 1 allows the approximate representation of numerical magnitudes, while core system 2 is used for the precise representation of distinct individuals.

Core system 1 helps when solving comparison problems with two given sets. The distinction is rather imprecise and its precision is based on the ratio of the magnitudes of the two sets. This means that while a person might be able to distinguish two sets of 40 and 80 objects (ratio 4:8), there may be problems when

doing the same for two sets of 70 vs. 80 objects (ratio 7:8). Thus this just-noticeable- difference in ratio follows the Weber-Fechner law, which states that this difference is proportional to the magnitude of the stimuli. Furthermore, this ratio increases with age. Xu and Spelke (2000) found by means of habituation studies that six-month-old children are able to distinguish sets at a 1:2 ratio. Xu & Arrigia (2007) showed that ten-month-old children can already distinguish sets at a 2:3 ratio. Adults can even handle ratios up to 7:8, though performance still gets worse the closer the ratio approaches 1 (Barth, Kanwisher & Spelke, 2003).

Core system 2 helps people to quickly keep track of small amounts. The exact number of objects that can be tracked by this system is a controversial issue. Antell and Keating (1983) showed that neonates up to an age of one week can distinguish sets of two and three items, but were unable to do so with bigger sets like four and six. Similarly, Feigenson, Carey and Hauser (2002) showed that ten- and twelve- month-old children can do this for sets up to three items, but not for any larger sets. Balakrishnan and Ashby (1992) reanalyzed data from various studies and found no evidence for the claim that a subitizing limit exists for at least up to six items. They conclude that other findings claiming the finding of such a limit just measure limited attention, similar to the findings of Miller (1956) regarding the ‘magical number seven’.

Current research showed that during development children learn to use core system 1 based abilities not only to solve simple comparison problems, but to solve approximate basic arithmetic operation problems as well. Barth, LaMont, Lipton and Spelke (2005; Barth et al., 2006) showed that preschool children can solve addition problems in a number range up to 60 above chance, when those problems were embedded in comparison tasks. This means children solved the addition of two given sets, compared this sum to another given set and indicated which one was larger.

Gilmore et al. (2007) found that five- and six-year-old kindergarten children solved comparison, addition and subtraction problems in a number range up to 100, even when problems were presented with Arabic numerals (symbolic) instead of sets (non- symbolic), although these findings lack replication. Furthermore, Barth et al. (2009) showed that kindergarten children and first grade students can solve non-symbolic doubling (multiplication) and halving (division) problems with given sets. McCrink and Spelke (2010) expanded these results and showed that children solved problems like ‘multiply by 2.5’ or ‘multiply by 4’. In all these studies the ratio of the two sets was crucial for solution probabilities, which indicates involvement of core system 1.

There have been several studies dealing with abilities based on core system 1. Common names for phenomena based on this core system are based on the way they are presented to subjects like ‘non-symbolic arithmetic’ or ‘symbolic arithmetic without instruction’. As both presentation types rely on the approximate number system, we will use the term ‘approximate arithmetic’ in this paper to deal with problems that include (a) a basic arithmetic operation on two given sets and (b) the comparison of the result with another given set (c) without relying on counting but on the core system for the approximate representation of numerical magnitudes.

Many researchers showed that children have some understanding of the four basic arithmetic operations prior to schooling. The more important question for education is whether this understanding is helpful for the learning of formal arithmetic. Gilmore et al. (2007, supplemental information) showed that it is correlated with arithmetic achievement in kindergarten, assessed with counting, ordinal number knowledge, number lines, measurement, sets and graphs. Even more, it is also an important predictor of first grade mathematic success, even when

controlling for age, verbal intelligence and reading literacy (Gilmore et al., 2010). Halberda, Mazzocco and Feigenson (2008) found that ninth grade approximate

number system (ANS) acuity retrospectively correlated with math performance back in kindergarten. De Smedt, Verschaffel and Ghesquiere (2009b) found significant correlations between symbolic number comparison and school mathematics. On the other hand Holloway and Ansari (2009) found no significant correlation of either symbolic or non-symbolic distance effect or reaction time – when solving

comparison tasks – with school mathematics. Up to now it is not fully understood how core system 1 influences school mathematics, as the current findings

inconclusive. It could be that core system 1 is more important for early school arithmetic, than it is for later years.

There are only few studies regarding core system 2 and its influence on school success. Desoete and Grégoire (2006) conducted a longitudinal study which showed that weak subitizing one year before school predicts weak school arithmetic in first grade. Furthermore, Kroesbergen, van Luit, van Lieshout, van Loosbroek and van de Rijt (2009) showed that subitizing predicts preschool arithmetic achievement, even when controlling for language and executive functioning.