Brief articleSampling from the mental number line: How are approximate number system representations formed?☆
Introduction
Fluency with mathematics is essential for day-to-day life. To successfully interact in a modern society it is frequently necessary to interpret, compare and calculate with numerical quantities. Along with a capacity to understand numerical ideas when represented symbolically, humans also have an Approximate Number System (ANS) which can be used to perform arithmetic operations on non-symbolic quantities such as arrays of dots or tones. The ANS is present in very young infants and some non-human animals (for a review see Dehaene, 1997), and recently some theorists have begun to speculate that it serves as the cognitive basis for symbolic mathematics (e.g. Halberda, Mazzocco, & Feigenson, 2008).
The ANS is widely believed to follow Weber’s law: the standard model proposes that when we encounter non-symbolic stimuli, a box of n oranges say, the distribution of possible ANS representations follows a normal distribution with mean n and standard deviation wn. Here w is the Weber fraction, a parameter which represents the acuity of an individual’s ANS (e.g., Barth, La Mont, Lipton, Dehaene, & Kanwisher, 2006). Several recent studies have shown that individuals’ ANS acuities are correlated with achievement in symbolic mathematics (e.g. Gilmore et al., 2010, Halberda et al., 2008, Libertus et al., 2011, Mazzocco et al., 2011a, Mazzocco et al., 2011b; but see Inglis et al., 2011, Price et al., 2012), lending credence to the suggestion that the ANS is implicated in the development of symbolic mathematics competence.
Although the capabilities of the ANS are now fairly well understood, the process by which the ANS forms representations from visual numerical stimuli is less clear. Several researchers have proposed that a mental ‘accumulator’ is central to this process (e.g. Dehaene and Changeux, 1993, Gallistel and Gelman, 2000, Piazza and Izard, 2009, Verguts and Fias, 2004). Gallistel and Gelman drew an analogy between filling up a beaker with cups of liquid, and filling up the accumulator with “accumulator units”. They suggested that when an array of objects is observed, the scene is first normalized to remove numerically-irrelevant between-object differences (color, shape, size etc.), then one cupful of ‘liquid’ is added to the accumulator per item. The contents of the accumulator are then emptied into memory which introduces noise proportionate to the accumulator’s contents (the “sloshing” of liquid in the memory beaker, in Gallistel and Gelman’s analogy). It is this noise, when the contents of the memory beaker are read off (converted into a numerical quantity), which causes the approximate nature of ANS representations.
It is notable that both Barth et al.’s (2006) computational model of the ANS, and Gallistel and Gelman’s (2000) accumulator beaker analogy assume that the duration for which a numerical stimulus is displayed is irrelevant to the ANS representation that an individual encodes from it. To date this assumption has not been tested. We see two reasons for questioning it. First, earlier researchers have reported different Weber fractions in studies which have used different display times. For example, a dot comparison task with a stimuli duration of 200 ms resulted in less precise ANS representations (mean w = 0.3, Halberda et al., 2008) than one with a display time of 750 ms (mean w = 0.1, Halberda & Feigenson, 2008). Second, it is well known that performance on many other visual tasks is dependent on stimuli durations (e.g., visual search: Guest and Lamberts, 2011, McElree and Carrasco, 1999; absolute identification: Guest, Kent, & Adelman, 2010).
Our goal in this paper is to explore whether the precision of an individual’s ANS representation varies with the length of time they spend studying the numerical stimulus. This question is important for at least two reasons. First because, as discussed above, it sheds light on the underlying mechanism that the ANS uses to form representations. Second, because numerical cognition researchers have to date adopted widely varying methods when conducting experimental studies. When presenting numerical comparison tasks (where participants are shown two dot arrays and asked to determine which is more numerous), some researchers have permitted participants to decide how long to study the stimuli before reaching a judgement (e.g., Inglis et al., 2011, Pica et al., 2004), whereas others have displayed the stimuli for a fixed period. Among those who have used fixed stimuli durations, some have displayed stimuli for as little as 200 ms (e.g., Halberda et al., 2008) whereas others have used up to 2500 ms (e.g., Halberda & Feigenson, 2008), and some researchers have used different stimuli durations for different participants within the same experiment (e.g., Halberda and Feigenson, 2008, Mazzocco et al., 2011b). All these authors have assumed that these methods investigate the same underlying process, but if the formation of ANS representations is time dependent then it is questionable whether results from these studies are comparable.
Here we report two experiments which directly investigated whether the acuity of ANS representations encoded from visual stimuli varies with stimuli duration. In Experiment 1 we demonstrate that individuals’ accuracies and Weber fractions are strongly dependent on stimuli duration, in Experiment 2 we show that this is not the result of differing onset-to-decision latencies, and in the general discussion we propose an adaptation of the standard model of the ANS which accounts for these data.
Section snippets
Participants
Participants were 12 staff or students of Loughborough University with normal or corrected-to-normal vision, who participated in exchange for a small inconvenience allowance. The study took place in a quiet laboratory using a 15” laptop.
Procedure
Each of 400 trials began with a fixation cross which was displayed for 1000 ms. This was followed by two dot arrays (a red array on the left of the screen and a blue array on the right) which were displayed for either 16 ms (the refresh rate of the monitor), 300
Experiment 2
The primary goal of Experiment 2 was to determine whether the effect found in Experiment 1 was due to stimuli duration, or the onset-to-decision latency. Here we held stimuli duration constant and varied the onset-to-decision latencies.
Summary of main findings
Our primary goal was to investigate how the duration of numerical stimuli influences the acuity of resultant ANS representations. Noting that earlier researchers have used dramatically different display times to estimate the acuity of individuals’ ANSs, in Experiment 1 we systematically varied stimuli display times on a dot comparison task. We found that participants were able to perform at above chance levels when stimuli were displayed for only 16 ms, but that, contrary to the assumption of
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Author Note: This research was supported by a British Academy Postdoctoral Fellowship (C.G.), and a Royal Society Worshipful Company of Actuaries Research Fellowship (M.I.). We are extremely grateful to two anonymous reviewers for their insightful comments on this work.