Slow drift of individuals' magnitude-to-number mapping

Slow drift of individuals’ magnitude-to-number mapping. Edward Vul, David Barner, & Jess Sullivan (evul, barner, jsulliva@ucsd.edu) Department of Psychology, 9500 Gilman Dr. # 109 La Jolla, CA 92093-109 USA Abstract When estimating the number of dots in a set, adults show bias and variability that scale with numerosity. Increasing variance in estimation is thought to reflect constant Weber noise on per- ceptual magnitude representations, while the increasing bias reflects miscalibrated mappings of number words onto mag- nitudes. Here we argue that response variability in numeri- cal estimation increases with numerosity in part due to uncer- tainty and slow drift in the mapping of numbers onto magni- tudes. We show that individuals’ number-to-magnitude map- ping functions drift slowly over the course of the experiment, with a shared-variance half-life of over 100 trials (∼ 10 min). We thus propose a model that treats the word-to-magnitude mapping function as a major source of estimation variability, and that accounts for cross-subject differences in estimation bias and variability, as well as changes to estimation perfor- mance within a given subject over time. In doing so, we rec- oncile the existing literature on the sources of estimation vari- ability, and provide evidence that uncertainty in the word-to- magnitude mapping function is a key limiting factor in estima- tion performance. Keywords: Approximate number, number words, numerical estimation function. First, are the mappings between verbal and nonver- bal numerical representations stable across individuals? Sec- ond, within individuals, are these mapping stable across time? As argued below, the answer to these questions suggests a novel model of numerical estimation, which explains signifi- cant aspects of error by appealing to a dynamically changing mapping function, rather than to the ANS. Introduction Human adults have access to at least two systems for repre- senting numerical quantity. The first is a noisy and evolution- arily ancient nonverbal number system, called the Approxi- mate Number System (ANS; see Dehaene, 1997; Feigenson, Dehaene, & Spelke, 2004 for review). The second is the ver- bal number word system (e.g., one, two, three, etc.). This system unique to humans allows for the precise represen- tation and manipulation of numerical content. When mak- ing estimates, adults draw on both of these systems: they use the ANS to represent the magnitude of the stimulus being es- timated, and they use the verbal number system to attach a linguistic label to this magnitude. The interface between these two systems has been an area of recent interest because it is crucial to characterizing how people can provide explicit verbal estimates of numerosity, and more generally how language relates to perception (e.g., Carey, 2009; Izard & Dehaene, 2008; Thompson & Opfer, 2011; Sullivan & Barner, 2012; Sullivan, Juhasz, Slattery, & Barth, 2011). The question is important for at least two rea- sons. First, estimation performance has been shown via in- tervention studies to be causally related to academic success, raising the question of why, and which aspects of training are most important to educational outcomes (Ramani & Siegler, 2011; Siegler & Ramani, 2009). Second, estimation tasks are often argued to elucidate properties of the ANS or their com- prehension of number word meanings. In the present study, we tested what contribution – if any – this mapping function makes to estimation error. Specifically, we asked two ques- tions about the nature of the number-to-magnitude mapping Figure 1: (Top) Our account of the bias and variability in human nu- merical estimation assumes two transformations between the phys- ical stimulus and a verbal numerical response (following Izard & Dehaene, 2008). First, the approximate number system maps the physical stimulus onto a logarithmic magnitude estimate with con- stant Weber noise. Second, the magnitude estimates are mapped onto the verbal number; we approximate this mapping as bilinear in log-log space with a variable slope. (Bottom) Two novel fea- tures of this account are necessary to capture the patterns of errors in human estimation data (one representative subject shown): (1) the mapping function must be non-linear in log-log space, otherwise the pattern of veridical calibration for small numbers, and systematic mis-estimation for large numbers will not hold, (2) the slope of the high end of the mapping function must be variable to capture the increasing variability of estimation for larger numbers. In the absence of training or feedback, adults are notori- ously inaccurate estimators (Kaufman, Lord, Reese, & Volk- mann, 1949; Izard & Dehaene, 2008; Minturn & Reese,