Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

Abstract

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people's underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals' performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that numberline estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.

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

“…To summarize, we have shown that when mapping number to space subjects adjust responses to take into account recent history, fitting well with suggestions that the spatial representation of numbers is not an inert map but is a highly dynamic process (8). If this notion is correct, we are in a position to make predictions for a wide range of research.…”

“…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.…”

Categories of Large Numbers in Line Estimation

“…The number line estimation task has recently experienced increased interest and scrutiny due to both novel interpretations of the mechanisms subserving linear and logarithmic behavior in children (Chesney & Matthews, ; Moeller et al., ) and its clear importance in predicting future mathematical success (Siegler, ; Thompson & Opfer, ). The data presented here do not speak directly to the mechanisms involved in line estimation in young children over relatively small ranges, but the theoretical perspective relates to positions taken in that domain.…”

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

“…Recently, an alternate view has emerged that focuses on the role that strategies play in the bounded number-line task (see e.g., Barth, Slusser, Cohen, & Paladino, 2011;Chesney & Matthews, 2013;Cohen & Blanc-Goldhammer, 2011;Cohen & Sarnecka, 2014; Friso-van den Bos, Kroesbergen, Van Luit, Xenidou-Dervou, Jonkman, Van der Schoot, et al, 2015;Hurst & Cordes, 2015;Hurst, Relander, & Cordes, 2016;Slusser, Santiago, & Barth, 2013). The counter argument is that (i) that successful completion of the bounded number-line task depends on the deployment of an appropriate cognitive strategy, and, (ii) only when these strategies are modelled, can a meaningful estimate of the participantÕs underlying quantity representation be obtained.…”