Compressive mapping of number to space reflects dynamic encoding mechanisms, not static logarithmic transform

Abstract

The mapping of number onto space is fundamental to measurement and mathematics. However, the mapping of young children, unschooled adults, and adults under attentional load shows strong compressive nonlinearities, thought to reflect intrinsic logarithmic encoding mechanisms, which are later "linearized" by education. Here we advance and test an alternative explanation: that the nonlinearity results from adaptive mechanisms incorporating the statistics of recent stimuli. This theory predicts that the response to the current trial should depend on the magnitude of the previous trial, whereas a static logarithmic nonlinearity predicts trialwise independence. We found a strong and highly significant relationship between numberline mapping of the current trial and the magnitude of the previous trial, in both adults and school children, with the current response influenced by up to 15% of the previous trial value. The dependency is sufficient to account for the shape of the numberline, without requiring logarithmic transform. We show that this dynamic strategy results in a reduction of reproduction error, and hence improvement in accuracy.numerical cognition | predictive coding | approximate number system | Weber-Fechner law | serial dependency H umans have a strong intuition of the spatial nature of numbers, usually (but not always) a horizontal "mental numberline," with numbers increasing from left to right (1-4). However, the nature of number mapping is not identical for all, but changes during development, starting from a nonlinear representation, well characterized as logarithmic (placing, for example, the number 10 near the midpoint of a 1-100 scale), then becoming more linear over the first years of schooling (3,5,6). Similarly, logarithmic-like numberlines have been demonstrated in indigenous Amazonian populations without formal mathematical schooling (4).Several recent studies have shown that under certain circumstances even the math-educated tend to reproduce numbers logarithmically. For example, we showed that depriving attentional resources leads to logarithmic-like numberline responses (7), consistent with the possibility that the native logarithmic encoding emerges when attention is deprived. Other studies have shown that the use of unfamiliar numerical format (such as exponential) can induce a switch from a linear to a logarithmic-like response, even in math-educated adults (7,8). Most recently, Dotan and Dehaene (9) have devised a clever technique to record the whole trajectory of the pointing response (across the face of a touchscreen), rather than just the endpoint: The response begins quite logarithmically, then corrects toward linear mapping by the time contact is made. All these studies have led many to interpret the logarithmic map as the direct reflection of the internal native number representation (4, 10-12) that becomes corrected over time by education but can emerge under special circumstances.Whereas the nonlinear numberline is consistent with intrinsic logarithmic processes, other explanat...

The log–linear response function of the bounded number-line task is unrelated to the psychological representation of quantity

“…al, (2012; see also Cicchini, Anobile, & Burr, 2014). The log-linear function has two free parameters: (i) p(log) which identifies the proportion of log fit (vs. linear fit) and, (ii) a a scaling parameter.…”

Serial dependence promotes the stability of perceived emotional expression depending on face similarity

“…Participants then saw a test screen containing a random adjustment face, which they modified by scrolling through the continuous expression wheel to match the target expression. After picking a match expression, participants saw a 1000-ms noise mask followed by a 1000-ms fixation cross before the next trial began Cicchini et al, 2014;Cicchini, Mikellidou, & Burr, 2017;Corbett et al, 2011;Liberman et al, 2014;Manassi et al, 2017). In Experiment 3, serial dependence was not ethnicity selective.…”

“…Previous studies have shown that the perception of orientation, numerosity, and other low-level stimulus features is serially dependent-systematically biased (i.e., pulled) towards similar visual input from the recent past (Cicchini, Anobile, & Burr, 2014;Corbett, Fischer, &Taubert, Alais, & Burr, 2016;Xia, Leib, & Whitney, 2016), as well as in feature space (object similarity; Fritsche, Mostert, & de Lange, 2017;Liberman, Fischer, & Whitney, 2014;Manassi, Liberman, Chaney, & Whitney, 2017). The spatio-temporal region over which current object features, such as orientation, are pulled by previously seen features is known as the Continuity Field (CF).…”

What is in a reach? Domain-general spatial modulation of motor responses by number representations

“…Recent research on the mental number line reported that logarithmic compression can be explained by adaptive decision-making (Cicchini et al, 2014). Cicchini et al showed that each subject's spatial response during a number line task is predicted by that subject's previous exposures to numerical values, and greater logarithmic scaling (as opposed to linear scaling) is observed under high cognitive load than low cognitive load.…”

“…Number representations are organized along a mental number line wherein small and large numbers are placed to the left and right side of space, respectively (Dehaene, Bossini, & Giraux, 1993;de Hevia & Spelke, 2009;Rugani, Vallortigara, Priftis, & Regolin, 2015; but see Núñez & Fias, 2015). Numerical positions in the spatial representation conform to a logarithmic scale: as numbers get larger their distances in the mental line get compressed (Dehaene, Izard, Spelke, & Pica, 2008;Nieder & Dehaene, 2009; but see Cantlon, Cordes, Libertus, & Brannon, 2009;Cicchini, Anobile, & Burr, 2014). Under this hypothesis, the mental number line, its left-to-right organization, and logarithmic scale directly maps to the spatial positions of reach trajectories -and some data support this hypothesis (Dotan & Dehaene, 2013;Song & Nakayama, 2008).…”