Comparing Symbolic and Nonsymbolic Number Lines: Consistent Effects of Notation Across Output Measures

Abstract

The mental number line (MNL) is a popular metaphor for magnitude representation in numerical cognition. Its shape has frequently been reported as being nonlinear, based on nonlinear response functions in magnitude estimation. We investigated whether this shape reflects a phenomenon of the mapping from stimulus to internal magnitude representation or of the mapping from internal representation to response. In five experiments, participants (total N = 66) viewed stimuli that represented numerical magnitude either in a symbolic notation (i.e., Arabic digits) or in a nonsymbolic notation (i.e., clouds of dots). Participants estimated these magnitudes by either adjusting the position of a mark on a ruler-like response bar (nonsymbolic response) or by typing the corresponding number on a keyboard (symbolic response). Responses to symbolic stimuli were markedly different from responses to nonsymbolic stimuli, in that they were mostly powershaped. We investigated whether the nonlinearity could be explained by effects of previous trials, but such effects were (a) not strong enough to explain the nonlinear responses and (b) existed only between trials of the same input notation, suggesting that the nonlinearity is due to input mappings. Introducing veridical feedback improved the accuracy of responses, thereby showing a calibration based on the feedback. However, this calibration persisted only temporarily, and responses to nonsymbolic stimuli remained nonlinear. Overall, we conclude that the nonlinearity is a phenomenon of the mapping from nonsymbolic input format to internal magnitude representation and that the phenomenon is surprisingly robust to calibration.

Keywords: calibration; nonsymbolic magnitude; number line; numerical cognition.

Table 1

Tasks and Stimuli Used in Experiments

Figure 1

Screenshots from each of our experiments. Top row, left = Experiments 1A-4A, ruler-based response, symbolic stimuli, symbolic endpoints; Middle = Experiment 2B, ruler-based response, nonsymbolic stimuli, symbolic endpoints; Right = Experiment 4B, rulerbased response, nonsymbolic stimuli, nonsymbolic (and sometimes flipped) endpoints; Right column, middle = Experiment 5A, ruler-based response, nonsymbolic stimuli, nonsymbolic endpoints; Bottom = Experiment 5B, typed response to nonsymbolic stimuli. The line asking for and displaying the response is magnified for visibility in this figure. The word Anzahl is German for number or numerousness; Bottom left, big panel = Experiments 3B and 5A, ruler-based response, nonsymbolic stimuli, nonsymbolic endpoints. See also Table 1 for a summary of the conditions.

Figure 2

Data from Experiments 1-4, with fitted linear, logarithmic, and power models. Left to right = Experiments 1-4; Top row = symbolic stimuli. Bottom row = nonsymbolic stimuli. For details on the experiments, see Table 1.

Table 2

Linear, Logarithmic, and Power Models Fit to Data from All Experiments

Figure 3

Exploring variability in responses to nonsymbolic stimuli. Left = responses in a log-log plot. A power function would be linear in such a plot; Right = SD by number, for all experiments.

Figure 4

Responses in Experiment 5, by feedback. Prefeedback panels show data from all groups. Dashed lines depict veridical performance. Top row = ruler-based task. Bottom row = typed number response; Left column = before feedback was introduced; Middle column = feedback blocks; Right column = post-feedback blocks without feedback.