The consideration of deliberate versus automatic processing of numeric representations is important to both math education and decision-making. Numeric information is ubiquitous in the decisions that we make, and thus numeric representation may play an important role. Basic number skills have only recently received attention in the decision literature. Numeracy, defined as the ability to process mathematical and probabilistic concepts, for example, has been shown to reduce susceptibility to framing effects and improve judgment accuracy (Peters et al. Reference Peters, Västfjäll, Slovic, Mertz, Mazzocco and Dickert2006). However, the effects of numeric representations in decisions have been largely ignored with two recent exceptions (Furlong & Opfer Reference Furlong and Opfer2009; Peters et al. Reference Peters, Slovic, Västfjäll and Mertz2008). These lines of research raise important questions concerning relations between numeric representations, math skills, and the use of numeric information in decisions. This commentary adds to the Cohen Kadosh & Walsh (CK&W) viewpoint by addressing the possible roles of automatic and deliberate processing of numeric representations in higher-level cognitive processing.
Numeric representations and math skills
Peters et al. (Reference Peters, Slovic, Västfjäll and Mertz2008) tested a large sample of healthy younger and older adults (mean age=20 and 70 years, respectively) using a distance-effect reaction-time task with Arabic integers and dots (“Is the quantity shown bigger or smaller than 5?”), among other notations. In a reanalysis of the younger-adult data including only these two notations, the size of the distance effect varied by notation, with more precise representations (smaller distance effects) for Arabic integers than dots, supporting CK&W's point that numeric representations are not necessarily abstract. CK&W claim further support from findings that the distance effect with integers relates to math achievement, but the distance effect with dots does not. However, in the reanalysis of Peters et al.'s data, more precise numeric representations for both Arabic integers and dots were associated with higher numeracy scores. Halberda et al. (Reference Halberda, Mazzocco and Feigenson2008) demonstrated a similar positive relationship between 14-year-old children's performance on a task that used only numerosities (dots) and their math ability back to kindergarten.
The question remains whether this association is based on automatic or deliberate processing of the representations. An age comparison may be illuminating because of the shift that occurs in reliance on more deliberative to more automatic processes from younger to older adulthood across a variety of domains. If the relation between numeracy and numeric representations is based on more automatic processes, then this correlation should increase, as older adults tend to rely relatively more on automatic processes (see Peters et al. Reference Peters, Hess, Västfjäll and Auman2007 for a review); if it is based on more deliberative processes, one might expect a decreased correlation. In Peters et al.'s (Reference Peters, Slovic, Västfjäll and Mertz2008) original data, the distance effect was less associated with numeracy for older than younger adults (r = .06, p = .64 and r = .41, p < .001, respectively). These data best support deliberate processes underlying numeracy's association with numeric representations. In addition, Castel (Reference Castel2007) found that older adults who were accountants and bookkeepers demonstrated memory ability remarkably similar to younger adults for arbitrary numbers, but not for arbitrary non-numeric information (where the usual age declines in memory were found). In another study, older adults were exceptionally good at remembering grocery-store prices, but not arbitrary prices (Castel Reference Castel2005). Such data could support a more precise abstract representation for these individuals (who have more experience with numbers in general or grocery prices in particular) that is carried into later life. However, in combination with Peters et al.'s data, these findings are also consistent with motivated selectivity in deliberative processing (Hess Reference Hess, Hecker, Dutke and Sedek2000), with some numeric information (or all numbers for some individuals) being particularly important and valued.
Numeric representations and decision-making
In accord with CK&W's notion of a deliberate level of numeric representations, decision-makers are thought to use numeric information intentionally and deliberatively (e.g., stock-market indicators, mortgage rates, and grocery bills). Because human decision-making likely derives in part from the same mechanisms evolved by other animals in response to risky natural environments, intuitive representations of symbolic numbers should relate to how individuals respond to numeric information in decision-making. Peters et al. (Reference Peters, Slovic, Västfjäll and Mertz2008) developed and tested hypotheses relating an individual-difference measure of the size of the distance effect to decision-making. They hypothesized that individuals with more precise representations would weight proportional differences between numeric attributes in choice more than individuals with less precise representations. In addition, a larger proportional difference should result in a bigger difference between individuals than a smaller proportional difference.
Results of two decision studies in Peters et al. (Reference Peters, Slovic, Västfjäll and Mertz2008) supported these hypotheses. In the first study, individuals with more precise representations (compared to those with less precise representations) were more likely to choose larger prizes received later than smaller, immediate prizes, particularly with a larger proportional difference between the two monetary outcomes. In a second study, they were more likely to choose a normatively worse option that saved a greater proportion of lives at risk (but a smaller number of lives) compared to those with less precise representations. Importantly, these findings were more consistent with the abstract-representation view, because the results of both studies held after controlling for numeracy and various measures of intelligence associated with prefrontal activity. The precision of number representations appears to underlie: (a) perceived differences between numbers, (b) the extent to which proportional differences are weighed in decisions, and, ultimately, (c) the valuation of decision options. Human decision processes involving numbers important to health and financial matters may be rooted in elementary, biological processes shared with other species, and which depend on an automatic representation of numerical information across notations.
It is critical to better understand number representation in the context of how individuals use numbers. The CK&W article forces us to consider whether the role of these representations in higher-order cognitive processing emerges from a shared representation across notations resulting more from prefrontal activity or whether their role results from an abstract representation. The current results are most consistent with numeracy being related to the former shared representation and decision-making being associated with the latter automatic processing of the representations. Human decision-making, however, often involves prefrontal activity (Rangel et al. Reference Rangel, Camerer and Montague2008), and further consideration of numeric representations being some combination of abstract and deliberate may converge with and ultimately explain some findings in the neuroanatomy of decision-making. It remains plausible that a shared deliberate representation (that is separate from what is associated with numeracy) could explain Peters et al.'s findings. A better understanding of the automatic versus deliberate nature of these representations' influences on decision-making will illuminate the important contribution of numeric representations on everyday decisions and should ultimately lead to improvements in decision aids.
