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Basic numerical abilities are generally assumed to influence more complex cognitive processes involving numbers, such as mathematics. Yet measuring non-symbolic number abilities remains challenging due to the intrinsic correlation between numerical and non-numerical dimensions of any visual scene. Several methods have been developed to generate non-symbolic stimuli controlling for the latter aspects but they tend to be difficult to replicate or implement. In this study, we describe the NASCO method, which is an extension to the method popularized by Dehaene, Izard, and Piazza (2005). Their procedure originally controlled for two visual dimensions that are mediated by Number: Total Area and Item Size (i.e., N = TA/IS). Here, we additionally propose to control for another twofold dimension related to the array extent, which is also mediated by Number: Convex Hull Area and Mean Occupancy (i.e., N = CH/MO). We illustrate the NASCO method with a MATLAB app—NASCO app—that allows easy generation of dot arrays for a visually controlled assessment of non-symbolic numerical abilities. Results from a numerical comparison task revealed that the introduction of this twofold dimension manipulation substantially affected young adults’ performance. In particular, we did not replicate the relation between non-symbolic number abilities and arithmetic skills. Our findings open the debate about the reliability of previous results that did not take into account visual features related to the array extent. We then discuss the strengths of NASCO method to assess numerical ability, as well as the benefits of its straightforward implementation in NASCO app for researchers.

^{i}. This document describes how the authors manipulated two critical perceptual dimensions that are intrinsically related to Number (

The main property of the relation between TA and IS is its proportionality: for a given

In the original document from ^{ii} (MO), considering

We name our dot generation method NASCO, as it controls for Number, Area, Size, Convex hull, and Occupancy. In the continuity of existing methods, NASCO aims at manipulating the unwanted visual dimensions in a more satisfying way than the original procedure suggested by

Another elegant procedure to disentangle numerosity from other non-numerical dimensions is the modeling approach of

In the previous section, we discussed the methods used to design non-symbolic number stimulus sets. In this section, we focus on software solutions allowing the creation of dot arrays (i.e., the generation and presentation of image files). We will not further develop how the script from

First of all, we need to mention ^{iii} dot arrays in a very similar way than Dehaene and colleague’s method: half of the trials have (on average) the same Item size, while the other half have (on average) the same TA. We specify “on average” as

The generation algorithm of

NASCO app aims at overcoming the latter issue by highlighting the intrinsic relation between the visual dimensions in a simple way, so that the user can easily create stimulus sets. It is important to note that NASCO app emphasizes the proportional relation between all dimensions: since

NASCO app has three functionalities: First, in the single array creation mode, and thanks to the straightforward relation between all considered numerical and non-numerical dimensions (^{iv}. The values of the other dimensions—in this case, TA and MO—will automatically be computed based on the introduced values. For instance, if the user wants to generate arrays with one hundred 10 px dots within an area of 100,000 px, they just need to introduce

In the current study, we illustrate the use of NASCO app by generating dot arrays with it as described in the previous section. We want to emphasize that the use of NASCO app is not limited to the generation of dots that follows the NASCO method. Since the user can set the four visual properties at its own discretion, it is possible to generate dot arrays following other recommendations. For instance, NASCO app can generate arrays following

We conducted an empirical study to assess the NASCO method on actual participants. The objectives of this study were twofold. Firstly, we aimed at providing a methodological evaluation of the non-symbolic stimuli designed by NASCO method and generated with NASCO app. We used these stimuli in a numerosity judgment task where participants were instructed to respond to the most numerous dot array. Since we aimed at assessing the approximate numerical ability of the participants, we expected to observe a numerical ratio effect (i.e

Secondly, we aimed at identifying the domain-general cognitive abilities related to numerical comparison tasks under investigation. Some authors indeed surmised that the different procedures to generate dot arrays in numerical comparison tasks involve different cognitive processes, such as inhibitory control (

We followed APA ethical standards to conduct the present study. The Ethic Review Panel from the Université Libre de Bruxelles approved the methodology and the implementation of the experiment before the start of data collection.

Seventy-two undergraduate students participated in exchange of course credits (58 women, mean age was 20.36 years). Participants did not report any uncorrected visual impairment or any math disability (or history of math learning disability). In our analyses, we had to exclude one participant who failed responding to the inhibition task due to severe misunderstanding of the instructions (she systematically responded to the

Participants were tested in a large room in groups of five to six people, for an approximate duration of 45 minutes. Each participant sat in front of a computer screen, isolated from the other ones with the help of separation panels. All tasks except the paper-and-pencil arithmetic test were displayed on a computer screen with MATLAB (The MathWorks), using the Psychophysics Toolbox extension (

We assessed arithmetic fluency with the Tempo-Test Rekenen (TTR,

We specifically generated non-symbolic stimulus pairs by using NASCO app (see Introduction). We generated 192 dot array pairs divided in four stimulus categories of 48 pairs each, see

We presented pairs of dot arrays and participants were instructed to determine as accurately as possible the array that contained the greater number of dots, by pressing the button on the side of the larger quantity. The onset of each trial was preceded by a fixation cross appearing 500 ms before the dots. Although speed was not emphasized, the dot arrays only remained on the screen for a maximal duration of 800 ms; they were then suppressed by an active mask displayed until participant’s response. The mask was followed by a blank screen for 400 ms, for an inter-stimulus interval of 900 ms (including the fixation cross). We analysed both accuracies and response times, but we only considered Correct Response rates (CR) for correlation analyses since they sufficiently depicted the performance at this task. We did not compute the Weber fractions, as recent evidence suggested they are not more informative than accuracies (

We assessed symbolic number processing with a number symbol comparison task similar to the one by

We evaluated general processing speed with a match-to-sample task (for a similar task see

We assessed visuo-spatial Working Memory (VSWM) because of their well-documented link to the acquisition of number skills (

To assess inhibitory control, we adapted the task of

Descriptive statistics for all control tasks are summarized in

Task / Measure | 95% CI | ||
---|---|---|---|

Symbolic comparison | |||

Accuracy | .965 | .183 | [.960, .970] |

Correct RT | 0.453 | 0.155 | [0.449, 0.458] |

IES | 0.468 | 0.064 | [0.452, 0.483] |

Arithmetic test | |||

Raw score (out of 200) | 128 | 23 | [122, 133] |

General processing speed | |||

Accuracy | .956 | .204 | [.945, .966] |

Correct RT | 0.527 | 0.137 | [0.520, 0.535] |

Visuo-spatial WM | |||

Accuracy | .763 | .425 | [.746, .779] |

Correct RT | 1.903 | 1.426 | [1.840, 1.966] |

1.675 | 0.944 | [1.451, 1.898] | |

Inhibition | |||

Accuracy | .960 | .194 | [.955, .965] |

Correct RT | 0.649 | 0.340 | [0.640, 0.658] |

Δ IES | 66.478 | 67.424 | [50.519, 82.437] |

Overall, participants correctly detected the more numerous array of dots in 89% of the cases, 95% CI [88.6, 89.6], with an average latency of 657 milliseconds, 95% CI [650, 664]. As expected, the numerical ratio affected performance; for the smallest ratio (i.e., 1.1), performances dropped to a mean accuracy of 77%, 95% CI [75.3, 78.7], and mean correct RT increased to 754 milliseconds, 95% CI [739, 770]. Conversely, the largest ratio (1.6) led to the best performance, with a mean accuracy of 97%, 95% CI [96.3, 97.7] and mean correct RT at 602 milliseconds, 95% CI [592, 613]. More relevant to the purpose of the current study, the stimulus properties significantly affected performance. The effects of the experimental manipulations are depicted on

First dimension confounded | Second dimension confounded | Accuracy | Correct RT |
---|---|---|---|

Total area | Convex Hull | .944 [.937, .952] | 618 [604, 632] |

Total area | Mean Occupancy | .894 [.884, .904] | 647 [637, 658] |

Dot size | Convex Hull | .913 [.903, .922] | 669 [648, 689] |

Dot size | Mean Occupancy | .813 [.800, .826] | 701 [689, 712] |

We analysed the statistical effects of both numerical ratio and stimulus properties with linear mixed effect models. We constructed two full models (i.e^{2}(5) = 758.06, ^{2}(5) = 208.06, ^{2}(5) = 374.07, ^{2}(9) = 103.26, ^{2}(4) = 31.92, ^{2}(4) = 4.132,

Overall, participants performed significantly better when TA and CH were confounded with number. We further looked at performance across all trials to disentangle the impact of these two cues. On one hand, we grouped all trials where TA varied with numerosity (irrespective of CH/MO, lower part of

We considered one measure per task to conduct correlation analyses: mean CR rates of the non-symbolic magnitude judgments, raw scores of the arithmetic test, and latencies of the general processing speed task. For the other tasks, we computed other measures that combined response times and accuracies (

Measure | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1. Non-symbolic number comparison (CR) | – | -.032 | .007 | -.081 | .247* | -.169* |

2. Symbolic digit comparison (IES) | – | -.148 | .289* | -.202* | .101 | |

3. Arithmetic (Raw Score) | – | -.198* | .177* | -.025 | ||

4. General processing speed (Correct RT) | – | -.207* | .064 | |||

5. Visuo-spatial working memory ( |
– | .001 | ||||

6. Inhibitory control (Δ IES) | – |

Correlational analyses revealed that performances of non-symbolic magnitude judgments did not correlate significantly with arithmetic performances, τ = .007,

In this study, we designed a new non-symbolic stimulus generation method—NASCO—extending the recommendations from

Moreover, manipulation of the IS and the TA influenced the numerical magnitude judgments. Participant performed very well when TA varied together with numerical magnitude (i.e

More critically for the purpose of the current study, which proposes to consider and control also the CH and the MO of dot sets, the manipulation of the latter two visual attributes was not negligible. It substantially influenced numerical comparison performances. Participants consistently had more difficulties in judging numerical magnitude when CH was kept constant (i.e

As we clearly observed, the manipulation of two additional visual dimensions to the classic method of

As a final reminder note, NASCO method does not aim at isolating the numerical dimension from every other visual dimension, or at suppressing the influence of the latter. NASCO method and app were designed for researchers or practitioners who want an easy and straightforward way to generate dot arrays. We suggest them to use NASCO app to create stimulus set that follows NASCO method. Researchers in need of sophisticated control method could still use a more elaborate method such as for instance the one from

Most published articles following the methodological document from

Mean Occupancy is actually the inverse of Density (i.e

Note that we consider here the default parameters, although the user is able to modify some properties (such as the difficulty level, the dot color, and the presentation time). Nevertheless, the user cannot precisely specify the extent of the Convex Hull or the Mean Occupancy.

It should be noted that NASCO app generates random positions during each iteration. In practice, it is thus not reliable to generate a given value of the size of CH with the precision of one pixel; an approximate value is rather provided (it is theoretically possible to get a CH with the wanted size but it may require a lot of iterations). By default, we tolerate a maximal error of only 1% for determining the size of CH (e.g

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The authors have declared that no competing interests exist.

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