Numbers derive meaning from their positions relative to one another. It is for this reason that number sense is best measured by indexing a person’s accuracy in comparing sets. The Approximate Number System (ANS) evolved to do just that, but in an approximate fashion. For example, Neider and Miller (2003) isolated numerosity-selective neurons in primates and measured neural activity as the primates were presented with dot arrays. Primates were able to make number matches, but with approximate or “noisy”^{[1]} brain activations (Neider, 2004). They found that the neural activation associated with a given number clusters around that number as it’s mean, with decreasing activation on either side of the target number. For example, if a target array has five items, brain activation is highest at a numerosity^{[2]} of five, less intense at four and six, and even less intense at three and seven (Halberda, in press; Neider & Miller, 2003).

With a bigger number, such as 52, the estimates still cluster around the mean (52), but the range of potential activations by neighboring numerosities increases. The standard deviations, or spread of the distributions for big numbers are larger. This phenomenon has been observed repeatedly in the behavioral domain. For example, when human and primate subjects were required to produce a number of key presses without counting, the number of presses was normally distributed around the target with the spread of the distributions increasing linearly with the size of the target number (Cordes, Gallistel, Gelman, & Whalen, 2001; Neider & Miller, 2003; Whalen, Gallistel, & Gelman, 1999).

If we placed these distributions along a mental number line, there would be overlap between adjacent distributions (Figure 2.). Moving up the scale distributions become wider and the resulting overlap between numerosities is greater (Halberda, in press; Neider, 2004). Numbers that are far apart are easier to discriminate because there is less overlap between the distributions of possible guesses. Smaller numbers are easier to discriminate because of the limited range of possible activations.

These size and distance rules are not absolute. It is the ratio between sets, rather than number position, that drives discriminability. For example, the numbers 4 and 5 have the same discriminability as the numbers 16 and 20, because they differ by equal ratios. The distributions for 4 and 5 have less spread, but 16 and 20 are far enough apart that their larger standard deviations result in the same amount of overlap (Dehaene, 1997; Halberda, in press). Ratio-dependent performance operates in several areas of perception, including brightness, weight, and temperature. We call this phenomenon Weber’s Law. A person requires a certain magnitude of difference between two numbers to discriminate them, and this difference is a constant proportion of those numerosities (Whalen, et al., 1999). Discriminability increases as the ratio between two numbers increases.

Because the Weber fraction is a constant, we can use it to predict performance. It tells us the amount of variability in a person’s ANS representations (Halberda, in press). The Weber fraction could also be defined as the constant increase in standard deviations of the distributions for each numerosity along a person’s mental number line. Although there is a range of typical performance, some people have a smaller Weber fraction with more precision and less variability in their number distributions, and some people have a larger Weber fraction with less precision and more variability in their number distributions. The amount of variability in a person’s number representations equals each target number multiplied by the Weber fraction for that person. It is possible to calculate a person’s Weber fraction for quantity, or their ANS precision, by using a quantity comparison task to isolate the smallest ratio at which they can reliably discriminate one set from another. Higher ANS precision is associated with math achievement over time; while low ANS precision predicts dyscalculia^{[3]} (Geary, et al., 2008; Mazzocco, Feigenson, & Halberda, 2011).

**The Weber Fraction and Math Achievement**

** **In a groundbreaking longitudinal study, Halberda and colleagues (2008) found that a child’s Weber fraction, or quantity comparison ability, consistently correlates with math achievement. The researchers found a consistent association (r = .52) between ANS precision and math achievement in a group of 64 subjects. The association was robust from Kindergarten to grade 9; even when the authors controlled for IQ, working memory, spatial reasoning, and reading ability (Halberda, Mazzocco, & Feigenson, 2008).

These findings provided evidence that number sense is essential to mathematics, but the question of directionality remained. What part of the variation in student performance was attributable to formal mathematics instruction? To answer this question, Halberda, et al. measured the ANS precision of 200 three to five year old children. They found again that ANS acuity correlated with math ability, in children with limited exposure to formal instruction. Two years later, the association between ANS precision and math achievement persisted. ANS precision predicted math achievement, but not other cognitive abilities commonly associated with math skill such as rapid naming^{[4]} (Libertus, Feigenson, & Halberda, 2011). Taken together, these studies provide compelling evidence for the importance of core number ability. ANS precision is foundational to and not an artifact of symbolic mathematics instruction. This leads to questions about the role of language in the development of mathematical competence.

[1] The firing activity of neurons has an inherent variability, called “noise” (Gallistel & Gelman, 2000).

[2] Numerosity refers to the exact number of items in a set.

[3] Dyscalculia is a learning disability characterized by an inability to represent and manipulate exact quantities. People with dyscalculia have significant difficulty with counting, estimating, and calculation even in the small number range.

[4] Rapid Naming is the ability to automatically name common objects and symbols such as numbers and letters.