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Adults, infants and some non-human animals share an approximate number system (ANS) to estimate numerical quantities, and are supposed to share a second, ‘object-tracking,’ system (OTS) that supports the precise representation of a small number of items (up to 3 or 4). In relative numerosity judgments, accuracy depends on the ratio of the two numerosities (Weber’s Law) for numerosities >4 (the typical ANS range), while for numerosities ≤4 (OTS range) there is usually no ratio effect. However, recent studies have found evidence for ratio effects for small numerosities, challenging the idea that the OTS might be involved for small number discrimination. Here we tested the hypothesis that the lack of ratio effect in the numbers 1–4 is largely dependent on the type of stimulus presentation. We investigated relative numerosity judgments in college students using three different procedures: a simultaneous presentation of intermingled and separate groups of dots in separate experiments, and a further experiment with sequential presentation. As predicted, in the large number range, ratio dependence was observed in all tasks. By contrast, in the small number range, ratio insensitivity was found in one task (sequential presentation). In a fourth experiment, we showed that the presence of intermingled distractors elicited a ratio effect, while easily distinguishable distractors did not. As the different ratio sensitivity for small and large numbers has been often interpreted in terms of the activation of the OTS and ANS, our results suggest that numbers 1–4 may be represented by both numerical systems and that the experimental context, such as the presence/absence of task-irrelevant items in the visual field, would determine which system is activated.
A large body of experimental evidence collected in cultural, developmental, comparative and cognitive psychology supports the existence of numerical systems that are not related to language and culture. For example, newborns are able to discriminate between 4 and 12 objects (
A common finding of experiments with adults and infants is that performance in relative numerosity judgments decreases as the numerical ratio between the smaller and the larger group increases. For instance, adult humans and infants are better at discriminating 8 vs. 16 items (0.50 ratio) than 8 vs. 12 items (0.67 ratio) (e.g.,
The behavioral difference between small (≤4) and large (>4) numbers has suggested the existence of two different pre-verbal numerical systems: one precise but operating only on small numbers and one approximate, with virtually no upper limit but whose discriminability is constrained by a ratio distance (
To date, the exact cognitive system underlying subitizing is unknown. Several authors believe that this phenomenon is based on a process linked to a system for representing and tracking individual objects (
Subitizing and numerical estimation of large numbers appear to differ in many respects, such as speed, accuracy and cognitive load (
There is also dissociation in human ontogeny. Newborns can discriminate 2 vs. 3 but not 4 vs. 6 items in a habituation task, showing a different ratio-dependence in the two numerical ranges (
Other studies, however, report ratio sensitivity in the small number range, challenging the idea of different mechanisms for small and large numbers. For instance, participants required to order two numerical values presented simultaneously showed ratio dependence across the range of values from 2 through 30 in accuracy and reaction time (
To summarize, while the existence of the ANS is generally accepted and supported by the universal ratio dependence reported in large number discrimination, researchers tend to disagree as to whether a distinct precise system operates for 3–4 objects. This debate has been recently enlarged to encompass non-human species. Some authors found evidence of a different ratio dependence of the performance within and outside the subitizing range in mammals, birds and fish (
It has proved difficult to resolve this issue with neuroimaging studies. If observers always
The question therefore remains as to what determines different ratio sensitivity in the small number range and, strictly related to this issue, what triggers the supposed subitizing response. Part of the problem is due to the fact that different studies use very different experimental paradigms that are likely to have a different impact on memory and cognitive load. For instance, in some tasks stimuli are simultaneously presented (e.g.,
To explain the inconsistency reported in the literature, it has been recently hypothesized that the ANS may sometimes be recruited to represent numbers in the subitizing range and the context in which the representation is elicited may determine which of them is employed in the subitizing range (
Here we tested the hypothesis that the task context – specifically, the type of stimulus presentation – could play a decisive role in the different ratio effect observed in literature in the small number range. To this end, we adopted three of the most common procedures for investigating numerosity discrimination: simultaneous presentation of intermingled elements (e.g.,
Twenty undergraduate students with a mean age of 23.3 years (range = 20 years to 27, five males) participated. The experiment was carried out at the Department of General Psychology, University of Padova. All participants were selected with normal or corrected vision.
All the experiments were approved by the ethics committee of the Department of General Psychology of University of Padova. Subjects gave written informed consent in accordance with the Declaration of Helsinki.
Stimuli consisted on 240 arrays of intermixed yellow and blue dots and appeared on a gray background. Half of the trials presented numerical contrasts with numerosities in the range 1–4 and half with numerosities in the range 6–24. In half of the trials the larger set was composed by the yellow dots, in the other half it was composed by the blue ones (diameter: 0.3–0.7 cm). We presented the same five numerical ratios, 0.25, 0.33, 0.50, 0.67, 0.75, in range 1–4 and in range 6–24. In the former the numerical contrasts were respectively 1 vs. 4, 1 vs. 3, 1 vs. 2 or 2 vs. 4, 2 vs. 3, 3 vs. 4; in the latter the numerical contrasts were 6 vs. 24, 6 vs. 18 or 7 vs. 21, 6 vs. 12 or 10 vs. 20, 6 vs. 9 or 12 vs. 18, 6 vs. 8 or 12 vs. 16. Yellow and blue dots were randomly placed in the display, therefore continuous variables (see
Stimuli were displayed on a 17-inch monitor at a comfortable viewing distance (60 cm), using E-Prime software. The experiment was conducted in a quiet and dimly illuminated room. Participants were presented with three blocks of 80 trials each. All stimuli were randomly presented within each block. The test was preceded by a short familiarization block of 20 trials with feedback.
To prevent verbal processing of the stimuli, participants were asked to select, as quickly as possible, whether there were more blue or more yellow dots (
Proportion of correct choices was used as a dependent variable to assess participants’ accuracy (
Trials with reaction times greater than 2000 ms (0.49% of participants’ responses) were discarded. A repeated measure ANOVA with Numerical Ratio (0.25/0.33/0.50/0.67/0.75) as the within-factor was performed, separately for small and large numbers.
In the small number range, participants’ accuracy was influenced by Numerical Ratio [
To assess whether the control of continuous variables is a potential factor that may affect the slopes of small and large number discrimination, we analyzed, in this and in the following experiments, whether there was a significant interaction between Continuous variable control (number + continuous variables vs. number only) and Numerical Ratio in a 2 × 5 ANOVA. No interaction was found, either in the small [
To verify whether the slopes for small and large numbers differ statistically, we observed whether there was a significant interaction between Range (Small vs. Large) and Numerical Ratio in a 2 × 5 ANOVA. No interaction between the two factors was found [
Lastly, in order to assess whether the salience of the stimuli was the same for yellow and blue dots, we contrasted the overall accuracy when the larger group was composed by yellow dots with that observed when the larger group was composed by the blue dots: no significant difference was found, both in the small [yellow dots: 0.913 ± 0.045; blue dots: 0.909 ± 0.043,
Participants’ accuracy in this task was affected by the numerical ratio, both for small and large numbers. As ratio-dependence did not vary as a function of the numerical range, these data are in line with the idea of a single system of numerical representation below and beyond four units. To assess whether the same results could be obtained if the arrays were spatially separated we carried out Experiment 2.
Twenty undergraduate students with a mean age of 21.9 years (range = 19 years to 32, 4 males) participated.
We used the same numerical contrasts of Experiment 1, both for small and large numbers. Stimuli were 240 pairs of arrays of black dots of the same sizes presented as in Experiment 1 on a white background. The two arrays were displayed simultaneously, one set on the right side of the screen and the other on the left side (average distance between the two groups: 8 cm). The right–left position of the larger set was randomized. Half of the trials were controlled for continuous variables, namely cumulative surface area (summed area of dots), overall space (space encompassed by the most eccentric dots) and density of the elements (average inter-dot distance). Since density and overall space are inversely correlated, these two variables cannot be controlled simultaneously. Therefore in half of these controlled trials we equalized the overall space occupied by the arrays; in the other half of controlled trials, we equalized the density of the items. In the second half of the 240 trials stimuli were controlled for dot sizes.
As in Experiment 1, the test phase was preceded by a training phase with feedback. Each trial started with fixation cross in the center of the computer screen (1000 ms). Subsequently two arrays of dots appeared on the two sides of the screen and remained visible for 150 ms (
Trials with reaction times greater than 2000 ms (0.37% of participants’ responses) were discarded. As in Experiment 1, accuracy was analyzed with a repeated measure ANOVA, separately for small and large numbers. In the small number range, participants’ accuracy showed a significant main effect of Numerical Ratio [
No interaction Continuous variable control × Numerical Ratio was found, either in the small [
No interaction Numerical Ratio × Range was found in the 2 × 5 (Range × Numerical Ratio) ANOVA [
Thus, Experiment 2 confirmed the main finding reported in the previous experiment: participants’ accuracy is set by numerical ratio both for small and large numbers. In this sense, the fact that stimuli are presented either intermingled (Experiment 1) or separately (Experiment 2) does not appear to have significant effect in terms of ratio dependence of small and large number discrimination. In both experiments, however, stimuli were simultaneously presented. In Experiment 3 we investigated whether performance can be affected by the sequential presentation of the arrays.
Twenty undergraduate students with a mean age of 19.6 years (range = 19–22 years, two males) participated.
Stimuli were identical to those used in Experiment 2, with the exception that, in this test, the arrays were presented sequentially. The test phase was preceded by a training phase with feedback. Each trial started with a fixation cross for 1000 ms, followed by the presentation of an array of dots at the screen center for 150 ms. After a delay of 500 ms, another group of dots appeared for 150 ms (
Trials with reaction times greater than 2000 ms (0.90% of participants’ responses) were discarded. Accuracy was analyzed with a repeated measure ANOVA, separately for small and large numbers. In the small number range, participants’ accuracy was not influenced by Numerical Ratio [
In contrast, in the large number range, participants’ accuracy was affected by Numerical Ratio [
No interaction Continuous variable control × Numerical Ratio was found either in the small [
A significant interaction Range × Numerical Ratio in the 2 × 5 ANOVA (Range × Numerical Ratio) was found [
One may argue that a lack of significance in accuracy within the small number range was a consequence of a lack of power due to the small sample size. To test for this hypothesis we increased the sample size by adding 18 subjects tested in another study (
Thus, participants’ accuracy is differently affected by numerical ratio for small and large numbers. As suggested by some authors (
Accuracy was analyzed with a 3 (Task: simultaneous presentation of intermingled arrays/simultaneous presentation of separate arrays/sequential presentation of separate arrays) × 5 (Numerical Ratio: 0.25/0.33/0.50/0.67/0.75) ANOVA, separately for small and large numbers (
In the small number range, the accuracy was influenced by Numerical Ratio and Task [
In the large number range, the performance was affected by the Numerical Ratio [
To assess whether the different ratio dependence could be due to task difficulty, we performed a 2 × 3 (Numerical Ranges × Experiments) repeated measures ANOVA on the overall accuracy. Results showed a marginally significant effect of Experiments [
One may argue that participants seeing a single dot in the array might have used a non-numerical strategy (1 dot is always the smaller number and participants could have easily selected the second array as larger). In addition, the control of density in numerical contrasts with a single object is intrinsically problematic. Hence, we analyzed the performance of the three experiments without the two numerical ratios (0.25 and 0.33) involving the presentation of a single dot. We also excluded 1 vs. 2 contrast from the 0.50 ratio. Repeated measure ANOVA showed the same pattern of results (see
Results of the three experiments in the absence of arrays composed by a single dot [only 0.50 (2 vs. 4) / 0.67 and 0.75 numerical ratios].
Experiment | Range | Ratio effect (ANOVA) | Linear trend |
---|---|---|---|
1 | Small | ||
Large | |||
2 | Small | ||
Large | |||
3 | Small | ||
Large | |||
The results from Experiments 1 and 2 show that the simultaneous presentation of the arrays (either intermingled or separate) determines a ratio effect in the small number range. This may imply that both experimental procedures automatically activate the ANS. In particular we hypothesized that a number of items exceeding the subitizing range might automatically activate the ANS even if participants were required to discriminate between small numerosities. In both experiments participants were required to segregate two sets of dots (blue from yellow dots, or right set from left set) under time constraints, which may have induced an estimation strategy. By contrast, in Experiment 3 participants saw only one group at a time, and here no ratio effect on accuracy was reported in the small number range.
As a further test of this hypothesis, we set up Experiment 4. Separate groups of dots in the range 1–4 were presented in sequence as in Experiment 3, but a group of distractors were also introduced into each array. We wanted to assess whether the total number of objects or just the task-relevant objects trigger the ANS response. Also, we investigated whether the ease of visually segregating task-relevant from task-irrelevant elements modulated the potential triggering effect of additional visual elements. If the ANS is particularly activated when the task is beyond the limit to encode items as individual objects, a ratio effect is expected in the condition that requires higher attentional resources.
Twenty undergraduate students with a mean age of 21.81 years (range = 21 years to 26, four males) participated.
Target stimuli were identical to that used in Experiment 3, with the exception that, in this test, six distractors were also included within each array. Distractors were represented by dots (diameter: 0.3–0.7 cm) presented in two different ways: in the intermingled version (
Trials with reaction times greater than 2000 ms (0.46% of participants’ responses) were discarded. Accuracy was analyzed with a repeated measure ANOVA (Numerical Ratio: 0.25/0.33/0.50/0.67/0.75), separately for the type of presentation (control test with no distractors, separated arrays and arrays with intermingled distractors).
In the control set, participants’ accuracy was not influenced by Numerical Ratio [
Similarly with the separated arrays participants’ accuracy was not influenced by Numerical Ratio [
On the contrary, in the presence of the intermingled distractors, the participants’ accuracy was significantly affected by Numerical Ratio [
On the whole, results of our experiments suggest that sequential vs. simultaneous presentation of target items significantly affects ratio dependence in the small number range. However, in the sequential task (Experiment 3) stimuli were presented for 150 ms, followed by a 500 ms break, and followed by 150 ms for the second array, for a total of 800 ms processing time of the first array. The longer amount of time – compared to 150 ms used in Experiments 1 and 2 – represents a potential confound in our study. To dissociate the role of the type of presentation (simultaneous vs. sequential) or time processing of the stimuli, we set up a control experiment in which both arrays were presented simultaneously but for an extended period of time, 800 ms. If the numerical ratio continues to hold in the small number range for these extended presentations, it suggests that the format of the task, and not the presentation time, was driving the different pattern of results observed in Experiments 1–3.
Twenty undergraduate students with a mean age of 23.5 years (range = 21 years to 26, six males) participated.
The same stimuli used in Experiment 2 were presented, both for small and large numbers. The two arrays were displayed simultaneously, one set on the right side of the screen and the other on the left side.
As in Experiment 2, the test phase was preceded by a training phase with feedback. Each trial started with fixation cross in the center of the computer screen (1000 ms). Subsequently two arrays of dots appeared on the two sides of the screen and remained visible for 800 ms. Participants (tested in a verbal suppression condition) were required to estimate whether there were more dots in the right array or in the left one, by pressing spatially congruent keys on the keyboard.
Trials with reaction times greater than 2000 ms (0.15% of participants’ responses) were discarded. As previously, accuracy was analyzed with a repeated measure ANOVA, separately for small and large numbers.
In the small number range, participants’ accuracy showed a significant main effect of Numerical Ratio [
No interaction Numerical Ratio × Range was found in the 2 × 5 (Range × Numerical Ratio) ANOVA [
Thus, participants’ performance is close to ceiling in both numerical ranges, due to the extended presentation time. However, even in this experimental condition, numerical ratio similarly affects the accuracy of small and large numbers. It is worth noting that the two groups were simultaneously presented in comparable time conditions of Experiment 3. In this sense, we believe that the ratio insensitivity reported in the small number range of Experiment 3 cannot be due to longer presentation time, but instead is likely to be due to the format of the task (sequential stimulus presentation).
Although studies generally agree that numerical ratio affects discriminating large numerosities, they often disagree as to whether the same ratio dependence exists in the 1–4 numerical range. The exact nature of this different ratio dependence reported in literature in the small number range is currently unknown. In the present study we tested the hypothesis that the presence/absence of ratio effect in the range 1–4 might be due to the type of stimulus presentation. As the different ratio sensitivity between small and large numbers have been commonly interpreted as evidence of two separate numerical systems (OTS and ANS,
As predicted by the literature, the performance in the large number range showed ratio dependence with all the types of presentation. Participants became less accurate when the numerical ratio was increased in Experiments 1, 2, and 3, a condition that has been widely accepted as a clear signature of the activation of the ANS (
Experiment 4 was devised to test this hypothesis. In two conditions a group of distractors were introduced, either separated from the target stimuli or intermingled with them. Ratio dependence was observed only when target stimuli and distractors were intermingled. Even though our data are not fully conclusive on this issue, they suggest that it is not only the total number of objects (specifically, more than four items in the visual field) that would activate the ANS, but probably a combination of different factors, such as total number of items and the ease of visually segregating task-relevant from task-irrelevant items. Indeed when target stimuli and distractors were intermingled, participants firstly had to dissociate relevant from irrelevant stimuli; on the contrary when distractors were spatially separated, focusing attention on task-relevant stimuli is supposed to be easier and presumably the task required a lower visuo-spatial working memory load.
In accordance with this interpretation,
The fact that the ANS could be activated when the task is beyond the limit to encode items as individuals objects may imply that subitizing would be the default; if the number of items (or the attentional resources) exceed a threshold, ANS would be then activated. Of course it is theoretically possible also the opposite: ANS would be the default and, if the number of items falls into the small number range, subitizing mechanisms would be recruited. However, while the one-to-one correspondence between items and object-files represents in our view a potential explanation on how to assess whether the number of items exceeds the limit of 4, it appears difficult to theorize a precise mechanism that allows the ANS to assess whether or not this limit was crossed. With respect to this topic,
Some authors have raised the question whether it is necessary to postulate the existence of two different systems on the basis simply of ratio dependence (
Also, some authors hypothesized that pattern recognition may be the main reason about why we often improve the performance in the small number range, reaching a ceiling effect (
Within the theoretical framework of a single system, one may argue that the results of our study may be due to a variation in task difficulty only. The harder the task, the less optimal is the performance in the small number range. We cannot entirely exclude this hypothesis, as we found a marginally significant effect of Experiments (
With respect to the ‘one vs. two-system’ debate, it is worth noting that data in favor of two distinct systems do not come exclusively from different ratio dependence. Indeed a potential prediction of the single-system hypothesis is that manipulation of physical properties of the stimuli should never have opposed effects on small and large number estimation, while the two-system hypothesis would allow for this possibility.
In conclusion we are aware that other factors should be investigated in details before drawing firm conclusions. The absence of masking of Experiment 3, for example, was necessary to have a precise comparison with Experiments 1 and 2 in similar conditions, but in Experiment 3, we cannot exclude the possibility that an afterimage of the first array seen might have introduced a confound when comparing the three experiments. The focus of researchers should be also enlarged to encompass the effects of task paradigm. As studies on animals, infants and adults used very different stimuli – i.e., food, social companions, tones, dots (
The differences in ratio dependence reported here in the small number range are due to the differences in the experimental paradigms, but the debate about how many systems account our pre-verbal numerical estimation extends far beyond the aims of our work with human adults into studies of infants and the numerical abilities of non-human species.
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.
The authors would like to thank Chiara Avancini for her help in testing the participants, and the reviewers for their useful comments. This study was supported by FIRB grant 2013 (RBFR13KHFS) from ‘