Animals, human babies, and monolingual tribes without count words can all represent number via two systems that underlie numerical cognition (Butterworth, 2005; Carey, 1998; Halberda & Feigneson, 2008). One system apprehends small sets, and the other represents number approximately by relative magnitude, akin to a “mental number line” (Butterworth, 2005). Relative magnitudes are compared by the same mechanism that is used to compare length, duration, volume, and other continuous quantities (Carey, 1998). One of the central characteristics of this system at work is our ability to compare large sets of numbers. The precision of this system diminishes as the ratio of one set to the other approaches one, consistent with Weber’s law (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Butterworth, 2005).
Even two day old infants can represent number via this mechanism, but their precision is inferior to that of adults and the system gradually refines itself through development until early adolescence (Halberda, & Feigenson, 2008). Rhesus monkeys can distinguish between sets of elements on the basis of number, and they can also represent ordinality from 1 through 9 (Judge, Evans, & Vyas, 2005). The hallmark measurement of the approximate number system is the quantity comparison task. In a standard quantity comparison task, subjects are shown arrays of items side by side. These items are displayed tachistoscopically, so as to preclude counting. Subjects are told to choose the greater of the two quantities with a key press. When Halberda and Feigenson (2008) used this task with 3, 4, 5, and 6 year olds and adults, they found that discriminability changes with development. Refining the mental number line is a lengthy process. Halberda and Feigenson conclude: “ Given the central role this system plays in supporting mathematical intuitions, this protracted period of development highlights the importance of coming to understand the effects of changes in ANS acuity on math learning and achievement”. Brian Butterworth has developed a dyscalculia screener based on various quantity comparison tasks, and the instrument does appear to have good predictive validity for mathematics difficulty. Click the image below to go to a demo version:
That brings us to this post’s topic study: Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665-668. Halberda and colleagues from Johns Hopkins decided to find out whether there is individual variation in ANS (approximate number sense) acutity, and whether individual differences in ANS acuity are related to or predictive of math achievement. This is an important study because to date, it is unknown how the ANS comes to be linked to symbolic math ability, and whether dyscalculia is the result of functional anomalies in the ANS, problems with symbolic number reasoning, or a problem with some type of “linking” mechanism between the two.
To that end, Halberda, et al. (2008) used a standard quantity comparison task with 14 year old subjects and compared their scores. They did indeed find significant individual variability. Then, they compared individual ANS acuity with their subjects’ math achievement scores from Kindergarten through grade nine. Even when they controlled for overall cognitive ability and other cognitive processes that have been linked to mathematical ability (such as spatial reasoning), they found a significant relationship between ANS acuity and math achievement. Individual ANS acuity may be a determining factor in math ability. However, given the long developmental process of “tuning” the ANS as well as the long process required for symbolic number acquisition, it is possible that high-quality activities of the right type may mediate this process.
To test yourself on the quantity comparison task used in this study, follow the link below to a simulation.
Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3-18.
Carey, S. (1998). Knowledge of number: Its evolution and ontogeny. Science, 282, 641-642.
Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain imaging evidence. Science, 284(5416),970, retrieved, March 29, 2009 from Education Research Complete Database.
Halberda, J. & Feigenson, L. (2008). Developmental change in the acuity of the “number sense”: The approximate number system in 3-, 4-, 5-, and 6- year olds and adults. Developmental Psychology, 44(5), 1457-1465.
Judge, P. G., Evans, T. A., & Vyas, D. K. (2005). Ordinal representation of numeric quantities by brown capuchin monkeys (cebus paella). Journal of Experimental Psychology, 31(1), 79-94.