Edited by: Carmelo Mario Vicario, University of Queensland, Italy
Reviewed by: Christian Agrillo, University of Padova, Italy; Zaira Cattaneo, University of Milano-Bicocca, Italy
*Correspondence: Valérie Dormal, Centre de Neuroscience Système et Cognition, Institut de Recherche en Sciences Psychologiques, Université Catholique de Louvain, Place Cardinal Mercier, 10, B-1348 Louvain-la-Neuve, Belgium e-mail:
This article was submitted to Frontiers in Cognition, a specialty of Frontiers in Psychology.
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Numerosity, length, and duration processing may share a common functional mechanism situated within the parietal cortex. A strong parallelism between the processing of these three magnitudes has been revealed by similar behavioral signatures (e.g., Weber–Fechner's law, the distance effect) and reciprocal interference effects. Here, we extend the behavioral evidence for a common magnitude processing mechanism by exploring whether the under- and overestimation patterns observed during numerical perception and production tasks are also present in length and duration perception and production. In a first experiment, participants had to perform two estimation tasks (i.e., perception and production) on three magnitudes (i.e., numerosities, lengths, and durations). The results demonstrate similar patterns for the three magnitudes: underestimation was observed in all perception tasks, whereas overestimation was found in all production tasks. A second experiment ensured that this pattern of under- and over-estimation was not solely generated by the mere process of perceiving or producing something. Participants were required to estimate the alphabetical position of a letter (i.e., perception task) or to produce the letter corresponding to a given position (i.e., production task). No under- or overestimation were observed in this experiment, which suggests that the process of perceiving or producing something alone cannot explain the systematic pattern of estimation observed on magnitudes. Together, these findings strengthen the idea that magnitude estimations share a common metric system, requiring similar mechanisms and/or representations.
Number, space and time are fundamental properties of the environment constantly used by humans and animals to adapt and regulate their behavior to the external world. The idea of a unique functional mechanism supporting magnitude processing was primary developed for numerosity and duration in the Accumulator model (Meck and Church,
At the neurofunctional level, brain areas located along the right intraparietal sulcus (IPS) are involved in numerosity, length, and duration discrimination. The involvement of these areas has been highlighted in neuroimaging (e.g., Pinel et al.,
At the developmental level, several studies have shown that discriminating numerosities, surface areas and durations leads to similar patterns of performance in babies (see respectively, Xu and Spelke,
Finally, at the behavioral level, various similarities have been reported between the discrimination of numerosities, lengths, and durations. First, discriminating all three magnitudes obey Weber–Fechner's law (Stevens and Greenbaum,
Together, these behavioral similarity and interference results, and the common activation areas support Walsh's (
To explore the characteristics of the accuracy estimation profile observed during numerosity, length, and duration, perception and production tasks were used jointly for the first time in order to highlight similarities and differences in estimation processes. In perception tasks, non-symbolic stimuli, such as collections of dots, are presented to participants who have to estimate their numerosity by providing symbolic outputs, such as verbal or Arabic numerals. Conversely, in production tasks, participants produce non-symbolic numerosities (e.g., collection of dots or sequences of sounds) corresponding to symbolic stimuli (e.g., Arabic digits or verbal numerals). A specific pattern of behavioral results has been revealed in the literature on numerical cognition. While numerosities are systematically underestimated in perception tasks (Kaufman et al.,
The present study was conducted to assess whether other magnitudes, such as length or duration, share or not a common metric system with numerosity, by determining whether under- and overestimation are also observed during length and duration perception and production tasks. In order to directly compare the performance patterns, Experiment 1 required each participant to perform both perception and production judgments on numerosity, length, and duration. If numerical, spatial and temporal estimations rely on a common mechanism and/or involve a common representation, the participants should underestimate these three magnitudes in the perception tasks, while a general overestimation should be observed in the production tasks. Dissimilarities in the performance patterns during the estimation of the three magnitudes would imply several core mechanisms using different metrics. To control whether those under- or overestimation patterns could be due to specific magnitude mechanisms and not to non-specific task requirement (perceiving or producing a stimulus), Experiment 2 was conducted with the same tasks but using unquantifiable material (i.e., letters).
Sixteen volunteers (5 males, 1 left-handed, mean age: 19.50 ± 0.97) took part in this experiment. They all had normal or corrected-to-normal vision and were unaware of the purpose of the study. All the procedures were non-invasive and were performed in accordance with the ethical standards laid down in the 1964 Helsinki Declaration.
The participants had to perform two tasks (i.e., perception and production) on three magnitudes (i.e., numerosity, length, and duration), giving a total of six different conditions (Figure
The presentation procedure and the number of trials were identical in the six conditions. Each condition was composed of 3 blocks of 24 experimental trials, with every target values presented twice. Ten training trials were presented before the experimental trials in order to familiarize the participants with the tasks, and were not analyzed. The stimuli were projected on a screen measuring 1.65 m wide and 1.20 m high. This methodological choice was made to ensure that the participants could not use their knowledge of computer screen size to infer their answer and to allow a similar potential variability in the answers of the participants across the different dimensions. The participants sat 95 cm from the screen in a dark room. Stimulus presentation and data collection were controlled by a PC computer connected to a data projector and using a customized E-prime 2 program (Schneider et al.,
The same range of values was used in the different conditions (21, 24, 28, 32, 37, 42, 49, 56, 64, 74, 85, 98 dots or cm for numerosity and length, respectively; 210, 240, 280, 320, 370, 420, 490, 560, 640, 740, 850, 980 ms for duration). These figures were chosen by applying a ratio between two consecutive values ranging from 0.85 to 0.93, as this ratio corresponds to the discrimination threshold in adults (Halberda and Feigenson,
Each trial began with the presentation of the sign “*” for 1000 ms. Then, an array of dots was flashed on the screen for a duration of 250 ms followed by the sign “=” (500 ms). After the sign “=,” the Arabic numeral “1” was presented, indicating to the participants that they had to give their estimation. To give their answers, participants had to turn a potentiometer on a response box (Mejias et al.,
For each target value and in every condition, any response which fell 2 or more standard deviations (
Firstly, to determine whether performance obeyed Weber–Fechner's law, linear mixed models (LMM) with magnitude (i.e., numerosity in the numerical estimation, length in the length estimation, and duration in the temporal estimation) as a fixed effect and participants as a random effect were conducted for perception and production conditions on log(mean) and log(
Secondly, a mean error rate (ER) for each condition was calculated as follows: ER = [(participant's response—target value)/target value]*100. An ER of zero indicates accurate estimation, a negative ER indicates underestimation, and a positive ER indicates overestimation. The mean ER for each task and magnitude was then submitted to
In the perception conditions, a significant underestimation was observed for each magnitude as confirmed by the results of
The ANOVA performed on the mean ER with condition and magnitude as within subject-variables demonstrated a significant main effect of magnitude,
For all magnitudes and tasks, both the means and the standard deviations of the magnitude judgments increased with target value, and the variability in the participants responses was always proportional to the mean for a given target (i.e., the average magnitude of the error increased in proportion to the target), reflecting the scalar property (i.e., the signature of Weber–Fechner's law) already described in various magnitudes estimation tasks (e.g., Stevens,
Under- and over-estimation were observed during numerosity perception and production tasks, respectively, in line with the error pattern observed in previous studies on numerosity processing (e.g., Whalen et al.,
A total of 16 volunteers (8 males, mean age: 29 ± 7 years) participated after they gave their informed consent. They all had normal or corrected-to-normal vision, were unaware of the purpose of the study and had not participated in Experiment 1. All the procedures were non-invasive and were performed in accordance with the ethical standards laid down in the 1964 Helsinki Declaration.
As in Experiment 1, participants had to perform two tasks (i.e., perception and production), but with letters of the alphabet. The same letters were used in the two tasks; the first four and last four letters of the alphabet were not used as their positions could be overlearned, hence too easy to perceive/produce. Seven consonants and 4 vowels were chosen among the 18 remaining letters (i.e., E, F, H, I, K, N, O, R, T, U, V). Each task was composed of 1 block of 44 experimental trials with every letter presented four times. Stimulus presentation and data collection were controlled by a Dell laptop using a customized E-prime 2 program (Schneider et al.,
In the perception task, participants were required to give the alphabetical position of a letter (e.g., “
In the production condition, participants were asked to produce the letter corresponding to a given number (e.g., “6” is “
The same data transformation and analyses as in Experiment 1 were applied.
For the perception task, the results of the LMM showed that the log of the mean estimates increased with the alphabetical position of the letter,
The mean ER did not significantly differ from zero in the perception [
In contrast to Experiment 1, perceiving and producing an unquantifiable ordered sequence does not lead to systematic under- and over-estimation, respectively. These data therefore suggest that the mere process of perceiving or producing is not sufficient to explain the systematic under- and overestimation patterns observed in Experiment 1.
Previous research on numerical cognition has demonstrated that the estimation of numerosity induces either under- or overestimation as a function of task (i.e., perception or production; Castronovo and Seron,
In Experiment 1, the pattern of under- and overestimation previously observed during perception and production of numerosity (e.g., Whalen et al.,
The scalar variability principle predicts imprecise performances, with the average magnitude of the error increasing in proportion to the target (e.g., Meck and Church,
Although processing numerosity, length, and duration present some similarities (e.g., global under- and overestimation), the mean overestimation error was significantly smaller in the length production task than in the production of the other two magnitudes, suggesting that participants were more precise in their estimations of length. This observation of better length performance might be accounted for by the fact that humans are confronted to the spatial magnitude earlier and more frequently than to the other magnitudes. Indeed, babies are able to move and use objects in their peripersonal space at a very early stage in their understanding of distance (Piaget and Inhelder,
To conclude, in Experiment 1, under- and overestimation were observed in perception and production tasks, respectively, for numerosity, length, and duration. As the results of Experiment 2 demonstrate that perceiving and producing
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This study was supported by grant FSR 2011 ADi/DB/1058.2011 from the Fonds Spéciaux de Recherche of the Université catholique de Louvain (Belgium). Stéphane Grade is a research fellow, Virginie Crollen and Valérie Dormal are post-doctoral researchers and Mauro Pesenti is a research associate at the National Fund for Scientific Research (Belgium). The authors are grateful to Xavier De Fays and Sandrine Mejias for their help in material construction and data collection.
1Weber–Fechner's law states that Δ
2Note that this analysis was also carried out separately for extensive and intensive sets and a significant difference from 0 was found in each condition (extensive:
3Similar differences from 0 were observed for the extensive (