^{*}

^{a}

^{a}

^{a}

^{b}

^{a}

^{c}

^{d}

^{a}

A small but growing body of evidence suggests a link between individual differences in processing the order of numerical symbols (e.g., deciding whether a set of digits is arranged in ascending/descending order or not) and arithmetic achievement. However, the reliability of behavioral correlates measuring symbolic and non-symbolic numerical order processing and their relationship to arithmetic abilities remain poorly understood. The present study aims to fill this knowledge gap by examining the behavioral correlates of numerical and non-numerical order processing and their unique associations with arithmetic fluency at two different time points within the same sample of individuals. Thirty-two right-handed adults performed three order judgment tasks consisting of symbolic numbers (i.e., digits), non-symbolic numbers (i.e., dots), and letters of the alphabet. Specifically, participants had to judge as accurately and as quickly as possible whether stimuli were ordered correctly (in ascending/descending order, e.g., 2-3-4; ●●●●-●●●-●●; B-C-D) or not (e.g., 4-5-3; ●●●●-●●●●●-●●●; D-E-C). Results of this study demonstrate that numerical order judgments are reliable measurements (i.e., high test-retest reliability), and that the observed relationship between symbolic number processing and arithmetic fluency accounts for a unique and reliable portion of variance over and above the non-symbolic number and the letter conditions. The differential association of symbolic and non-symbolic numbers with arithmetic support the view that processing the order of symbolic and non-symbolic numbers engages different cognitive mechanisms, and that the ability to process ordinal relationships of symbolic numbers is a reliable and unique predictor of arithmetic fluency.

Numerical abilities are an integral and important part of modern society. Partly because of this insight, the last decades have seen a remarkable growth in interest to better understand the neurocognitive mechanisms of number processing and their relationship with arithmetic abilities. While early research associated the acquisition of arithmetic and mathematical skills with the development of domain-general knowledge (e.g., stages of logical thinking;

Another important property of numbers is ordinality, which has been a largely overlooked topic of research in the past. Ordinality denotes the relative position or rank of a number, and it relates to the knowledge that one number comes before or after another number. For example, the ordinal position of numbers allows us to instantly infer that one-thousand-and-two comes right after one-thousand-and-one. Inferences like this would be difficult to make using a purely intuitive understanding of cardinality alone. In the last years, there has been a growing number of studies which have aimed to better understand the cognitive and neural mechanisms of numerical order processing (

While a large body of evidence has indicated a significant correlation between cardinality processing and arithmetic (e.g.,

Another useful metric that has been used to investigate ordinal processing of symbolic numbers is the numerical reverse distance effect. While a canonical distance effect—that is, greater response times for numbers that are close in numerical distance compared to numbers that are far in numerical distance—is typically found for cardinal judgments, distance effects have been shown to be reversed for ordinal judgments (

Although there is increasing evidence for an association between individual differences in processing the order of numbers and arithmetic abilities, the number of studies that have investigated this question is still limited. While results of the discussed studies indicate a certain degree of reliability, especially for mean reaction times, the test-retest reliability of the behavioral correlates frequently used to assess ordinal processing as well as their associations with arithmetic abilities have not yet been investigated within the same sample.

The evidence discussed so far has revealed a relationship between symbolic numerical order processing (i.e., Arabic numerals) and arithmetic abilities. In other words, the ability to order number symbols may play a crucial role in arithmetic. An interesting question that follows is: How specific is this relationship between number symbols and arithmetic? Similar to solving arithmetic problems, the processing of number symbols is culturally transmitted via formal education. This common background may explain the observed relationship between arithmetic and number symbol ordering (

A large number of studies have shown that preverbal infants (e.g.,

Given these findings, one would expect that individual differences in judging the order of non-symbolic numerical stimuli are also related to arithmetic abilities. To the best of our knowledge, only one study has thus far investigated individual differences in non-symbolic numerical ordering processing and its relationship to arithmetic abilities (

The ability of humans to process ordinal relationships is not restricted to numerical information, but rather extends to other non-numerical dimensions such as the letters of the alphabet (e.g.,

While the result above indicates common ordinal mechanisms between numerical and non-numerical sequences, some evidence also suggests differences between the two dimensions. For example,

In light of the discussed literature, a number of important questions remain to be answered. First, it is currently unknown whether behavioral correlates of ordinality processing are reliable measures (i.e., test-retest reliability of ordinal processing). Furthermore, while a growing number of studies have reported an association between symbolic numerical ordinal processing and arithmetic, little is currently known about the association between non-symbolic numerical and symbolic non-numerical ordinal processing and arithmetic. In addition, it is not known whether symbolic numerical ordinal processing constitutes a reliable and unique predictor of arithmetic abilities over and above these other dimensions. To address these questions, we asked a group of healthy adults to perform a symbolic numerical order task (i.e., Arabic numerals), a non-symbolic numerical order task (i.e., dot arrays), and a symbolic non-numerical order task (i.e., letters of the alphabet), as well as a paper-pencil test assessing arithmetic performance. Individuals were asked to perform all conditions at two different time points to investigate the test-retest reliability of the discussed measures and their associations with arithmetic fluency. We formulated the following hypotheses:

We expect a reverse distance effect (i.e., faster reaction times for consecutive numbers) for in-order conditions of symbolic numbers as well as for letters of the alphabet. Furthermore, we expect a canonical distance effect for in-order trials of the non-symbolic number task. If these behavioral measures are reliable, we expect significant distance effects at both time points for all conditions.

If behavioral measures of ordinal processing are reliable, we expect them to correlate between the first time point (T1) and the second time point (T2).

Finally, we expect to find a reliable and unique association between symbolic numerical ordinal processing and arithmetic abilities. More specifically, symbolic numerical ordinal processing predicts arithmetic abilities and explains unique variance over and above possible associations between non-symbolic numbers and letters of the alphabet at T1 and at T2.

A total of 36 healthy right-handed adults (18 females; mean 23.5 years; range 20–33 years) were invited to participate in the present study. Four participants were excluded from the analyses due to missing data points. These participants had low accuracy rates (0-10% correct answers) in at least one of the distances in one of the conditions (i.e., symbolic numerical, non-symbolic numerical or letters), and therefore no canonical and/or reverse distance effects could be calculated. Thus, the final sample comprised 32 subjects (14 females; mean 23.5 years; range 20–33 years). Participants were recruited from undergraduate faculties of the University of Graz as well as from the surrounding community in Graz, Austria. All participants reported normal or corrected-to-normal vision and reported no history of neurological disorders. Participants gave informed consent prior to participating in the study. The study was approved by the local ethics board of the University of Graz.

The test sessions took place in a standardized setting within specific test rooms at the Institute of Psychology, University of Graz, Austria. Participants performed three computerized ordinality tasks as well as a paper-pencil test to measure arithmetic fluency. To assess the test-retest reliability of all conditions and their relationship with arithmetic fluency, participants performed all conditions at two test sessions six to nine days apart.

Participants completed a symbolic numerical ordinality (i.e., Arabic numerals), a non-symbolic numerical ordinality (i.e., dot arrays), and a symbolic non-numerical ordinality (i.e., letters of the alphabet) task (see

Examples of trial sequences for symbolic numbers, non-symbolic numbers and letters of the alphabet.

Stimuli were presented on a 14'' LCD (resolution 1366 x 768) using the presentation software

To assess individual differences in arithmetic fluency, a new paper-pencil test based on the French Kit test of arithmetic skills (

Prior to the test session, a short practice session with feedback was administered. This assured that participants understood the task, and that a correct stimulus-response assignment was established. For this, seven randomized trials were selected from the test set and presented to the participants. For each trial, feedback was given on whether the response was correct or incorrect. After this practice session, the test session started. The administration of the ordinality task was counterbalanced in such a way that one third of the participants started with the symbolic number condition (i.e., Arabic numerals), one third with the non-symbolic number condition (i.e., dot arrays), and one third with letters of the alphabet. Participants were allowed to make a short break (no longer than five minutes) between the computerized ordinality tasks. After participants finished the ordinality tasks, the arithmetic fluency test was administered. Participants performed all operations of the easy problems prior to the difficult problems. The entire test session lasted 70 minutes (individuals also participated in a second experiment—the data of this experiment are not reported in this study).

Data were analyzed using R (

The first analysis tested whether participants performed all conditions above chance and whether participants made less errors at T2 compared to T1 (i.e., training effect), a repeated measure analysis of variance (ANOVA) was calculated, using ^{2}_{G} = .070) and a significant main effect of ^{2}_{G} = .345). The interaction term ^{2}_{G} = .005). The main effect of _{FDR} < .001) and symbolic numbers (6% errors) (_{FDR} < .001). Participants also committed more errors with non-symbolic numbers than with symbolic numbers (_{FDR} < .001). The analysis above demonstrates a training effect between T1 and T2, and that participants performed above chance level (i.e., the highest error rate was 27% errors) in all conditions. Since previous work has found associations between the reaction time and arithmetic fluency (e.g.,

To investigate the reliability of mean reaction time patterns for all experimental conditions, three repeated ANOVAs were calculated, using

Effects | η^{2}_{G} |
|||
---|---|---|---|---|

Digits | ||||

Time point | 20.424 | (1, 31) | .001** | .397 |

Order | 51.933 | (1, 31) | .001** | .626 |

Distance | 33.529 | (1.663, 51.539) | .001** | .520 |

Time point x Order | 0.300 | (1, 31) | .588 | .010 |

Time point x Distance | 4.240 | (1.891, 58.611) | .021* | .120 |

Order x Distance | 35.797 | (1.956, 60.625) | .001** | .536 |

Time point x Order x Distance | 3.404 | (1.802, 55.873) | .045* | .099 |

Dots | ||||

Time point | 23.361 | (1, 31) | .001** | .430 |

Order | 16.075 | (1, 31) | .001** | .341 |

Distance | 30.053 | (1.763, 54.665) | .001** | .492 |

Time point x Order | 0.000 | (1, 31) | .986 | .000 |

Time point x Distance | 0.948 | (1.971, 61.093) | .393 | .030 |

Order x Distance | 1.343 | (1.789, 55.462) | .269 | .042 |

Time point x Order x Distance | 0.060 | (1.881, 58.317) | .933 | .002 |

Time point | 1.822 | (1, 31) | .187 | .056 |

Order | 11.602 | (1, 31) | .002* | .272 |

Distance | 15.532 | (1.957, 60.671) | .001** | .334 |

Time point x Order | 0.003 | (1, 31) | .959 | .000 |

Time point x Distance | 4.663 | (1.791, 55.528) | .016* | .131 |

Order x Distance | 44.745 | (1.919, 59.494) | .001** | .591 |

Time point x Order x Distance | 0.221 | (1.848, 57.296) | .785 | .007 |

^{2}_{G} denotes generalized eta squared. Values are reported with Greenhouse-Geisser correction where necessary.

*

Bar graphs depicting reaction times of a) symbolic numbers, b) non-symbolic numbers, and c) letters of the alphabet as a function of

Results for symbolic numbers showed a significant main effect _{FDR} = .038), but not compared to Distance 3 (952ms; _{FDR} = .258). There was no reaction time difference between Distance 2 and Distance 3 (_{FDR} = .255). For in-order trials at T2, no significant differences in reaction time between Distance 1 (915ms) compared to Distance 2 (912ms; _{FDR} = .861) and compared to Distance 3 (881ms; _{FDR} = .074) were found. Participants were, however, significantly slower to judge the order for Distance 2 compared to Distance 3 (_{FDR} = .016). For mixed-order trials at T1, significantly slower reaction times were found for Distance 1 (1221ms) compared to Distance 2 (1150ms; _{FDR} = .010) and compared to Distance 3 (1032ms; _{FDR} < .001). Participants were also significantly slower to judge the order of Distance 2 compared to Distance 3 (_{FDR} < .001). The same pattern was found for mixed-order trials at T2. Reaction times were significantly slower for Distance 1 (1116ms) compared to Distance 2 (1007ms; _{FDR} < .001) and compared to Distance 3 (938ms; _{FDR} < .001). Participants were also significantly slower to judge the order of Distance 2 compared to Distance 3 (_{FDR} < .001). The results of the ANOVA are in line with a reversal of the distance effect (i.e., faster reaction times for Distance 1 compared to Distance 2) in the in-order condition at T1, and with a canonical distance effect in mixed-order trials at T1 and T2. The absence of a significant distance effect for in-order trials at T2 indicated that reaction time in the in-order condition slightly differed as a function of whether participants performed the task at T1 or T2.

The ANOVA on reaction times of the non-symbolic number condition (see _{FDR} < .001) and Distance 3 (998ms; _{FDR} < .001). Participants were also slower to judge the order of Distance 2 compared to Distance 3 (_{FDR} < .001). This pattern is consistent with a canonical distance effect (i.e., decreasing reaction times with an increase in distance). The absence of an

The final ANOVA performed on the reaction times of the letter condition showed a significant main effect of _{FDR} < .001) and compared to Distance 3 (1570ms; _{FDR} < .001). Participants were, however, slower to judge the order of Distance 2 compared to distance 3 (_{FDR} = .010). For the mixed-order condition, results showed significantly slower reaction times for Distance 1 (1663ms) compared to Distance 2 (1608ms; _{FDR} = .045) and Distance 3 (1507ms; _{FDR} < .001). Participants were also slower to judge the order of Distance 2 than Distance 3 (_{FDR} = .003). These results are in line with a reversal of the distance effect in the in-order condition, and with a canonical distance effect in the mixed-order condition. The absence of a significant

Two different measures of ordinality, which have been used in previous studies to investigate ordinal processing (e.g.,

To quantify the reliability of the distance effects, we first calculated the individual size of the distance effects for each subject by adopting the formula used in

_{Distance2,Distance3}– meanRT

_{Distance1})/meanRT

_{Distance1,Distance2,Distance3}

For the dot condition, the canonical distance effect was calculated by:

_{Distance1}– meanRT

_{Distance3})/meanRT

_{Distance1,Distance3}

Importantly, the distance effect was only calculated for in-order trials in which a reverse distance effect was found for numbers and letters. Results demonstrated significant correlations for the reverse distance effect of numbers (

The final test-retest analysis evaluated the reliability of the arithmetic fluency paper-pencil test used in the present study. Using a test-retest correlation analysis (Pearson’s correlation), results demonstrated a significant correlation between test session one and test session two (

Measure | Minimum | Maximum | Mean | Std. Deviation |
---|---|---|---|---|

AF_{T1} |
120 | 364 | 209.03 | 60.17 |

AF_{T2} |
128 | 365 | 224.13 | 58.29 |

The next analysis aimed to investigate whether a reliable and unique relationship between the mean reaction times of the ordinal conditions and arithmetic fluency exists. To this end, two multiple regression analyses (method: ENTER), in which all mean reaction time variables were entered into one model, were calculated; one for T1 and one for T2. Arithmetic fluency was entered as the dependent variable and mean reaction times as independent variables. Results of the regression analyses (see

Predictor | β | ||||
---|---|---|---|---|---|

T1 | |||||

Numbers | -0.119 | 0.048 | -2.492* | .019 | -.540** |

Letter | -0.033 | 0.056 | -0.593 | .558 | -.159 |

Dots | 0.012 | 0.053 | 0.228 | .821 | .377* |

Constant | 376.410 | 76.799 | |||

T2 | |||||

Numbers | -0.133 | 0.060 | -2.194* | .037 | -.491** |

Letters | -0.057 | 0.052 | -1.098 | .282 | -.149 |

Dots | 0.069 | 0.063 | 1.094 | .283 | .395* |

Constant | 370.922 | 63.628 |

^{2} for T1 is 0.225, and for T2 0.202.

^{a}The rightmost column

*

The final analysis investigated whether a reliable relationship between the distance effects and arithmetic fluency exists. For this, Pearson correlations between the distance effects of all conditions and arithmetic fluency were calculated for T1 and T2. The correlation analyses revealed no significant correlations between the reverse distance effect of digits and arithmetic fluency at T1 (

Numerical and arithmetical abilities are an important part of modern society. Recent research has demonstrated that the behavioral correlates of symbolic numerical order processing—the knowledge that one number comes before or after another number— are associated with arithmetic abilities in children and adults (

Results of the analyses revealed a significant reverse distance effect for letters (i.e., faster reaction times for consecutive items compared to non-consecutive items) of the alphabet and for number symbols (i.e., digits). This observation is consistent with an increasing body of literature that has reported a reversal of the distance effect in tasks in which participants are asked to indicate the order of numerical and non-numerical symbols (

For symbolic numbers, performed test-retest correlation analyses demonstrated that the reverse distance effect and mean reaction times are reliable measures. Both, reaction times of the reverse distance effect and mean reaction times, showed a significant correlation between the measures collected at time point one and the measures collected at time point two. Thus, demonstrating that individual differences in the behavioral correlates of symbolic numerical order processing are preserved across different time points. This result is consistent with a recent study (

In contrast to number symbols, a different test-retest pattern was found for letters of the alphabet. While a significant correlation was found for mean reaction times, no significant correlation was found for the reverse distance effect. In other words, while individual differences in mean reaction times are stable between the two time points, individual differences of the reverse distance effect of letters are not preserved across both time points. As such the test-retest reliability of the reverse distance effect in letters of the alphabet is significantly lower compared to the test-retest reliability of the reverse distance effect of number symbols. One potentiation reason for such inconsistency in the reverse distance effect of the letter condition may be that individuals are less familiar with processing the order of letters compared to processing the order of number symbols. The ordinal processing of letter sequences is less frequent in everyday life compared to the ordinal processing of numbers and, as a consequence, participants may use inconsistent strategies/cognitive mechanisms to judge the order of letters—resulting in a low reliability of the reverse distance effect in letters of the alphabet. For future work, it would be important to better understand how individuals solve numerical and non-numerical ordinal tasks. While recent research has found commonalities and differences between non-numerical and numerical ordinal processing (e.g.,

While letters demonstrated significant reverse distance effects (albeit uncorrelated) at both time points, the pooled mean reverse distance effect for number symbols was only significant at T1 but no at T2. One possible explanation for this pattern is that participants showed a training/familiarity effect between the time points, resulting in a reduction of the reverse distance effect from time point one to time point two. In other words, individuals may become more fluent in judging symbol-to-symbol associations between the two test sessions. Supporting evidence that familiarity may alter the size of the reverse distance effect comes from a study with children (

In contrast to the symbolic conditions discussed above, a canonical distance effect (i.e., slower reaction times for distances that are small compared to distances that are large) was observed in the non-symbolic number condition. The canonical distance effect is in sharp contrast to the reverse distance effect observed for symbolic numbers and letters, indicating different mechanisms with which the order of non-symbolic numbers is evaluated. One interpretation of this canonical distance effect in ordinal processing is that participants engage in a multi-stage cardinal comparison (cardinal judgments typically reveal a canonical distance effect). At this, participants may judge whether the second dot array is larger compared to the first dot array and whether the third dot array is larger compared to the second dot array (i.e., ●<●● and ●●<●●● = ●<●●<●●●). The relative difficulty in discriminating dot arrays with a small numerical distance results in longer reaction times. As such the present data provide evidence that the ordinal representation of symbolic numbers is fundamentally different from the ordinal representation of non-symbolic number, potentially highlighting a key difference between these two numerical representations. While the underlying mechanisms between symbolic and non-symbolic ordinal judgments may differ, both numerical tasks yielded comparable test-retest reliability. Specifically, the performed test-retest analyses revealed significant correlations for the canonical distance effect as well as for mean reaction times—measures at time point one correlated significantly with the same measures at time point two. This indicates that individual differences in non-symbolic numerical order processing were preserved across time points and that individuals engaged similar mental operations.

The next analyses of the present work investigated the associations between the behavioral correlates of ordinality processing and arithmetic fluency as well as the test-retest reliability of these associations. Results of the performed multiple regression analyses revealed a significant and reliable association between mean reaction times of the number symbol condition and the arithmetic fluency test, which explained a unique portion of variance over and above non-symbolic numerical and symbolic non-numerical order processing. Since multiple regression analyses control for shared variance across the included conditions, such as domain-general ordinal processes or domain-general cognitive operations, significant results can be interpreted as task specific effects that do not relate to shared processing mechanisms. This result is in line with previous findings that have found comparable results in children and adults (

In contrast to our expectations, the results of the present work did not reveal a significant association between non-symbolic numbers and arithmetic. Evaluating the order of non-symbolic numbers was neither a significant predictor of arithmetic achievement in the multiple regression analyses, nor did the behavioral measures of the task demonstrate a zero-order correlation with arithmetic fluency. This is an interesting finding, given that a significant body of literature has reported an association between non-symbolic number processing and arithmetic in children and adults (e.g.,

One explanation for these differences and for the significant and reliable association between symbolic numerical ordinal processing and arithmetic is that symbolic numerical order processing and arithmetic are culturally transmitted via formal education. More specifically, formal education provides a common framework in which knowledge about symbolic relations and their manipulations are taught. Since both operations share a similar basic understanding of these ordinal symbol-to-symbol associations, formal education may be the bridge that links the mental operations that are afforded in symbolic numerical order processing and arithmetic (see

Contrary to our expectations, the results of the present study did not show a significant relationship between the distance effects (neither the reverse distance effect nor the canonical distance effect) and arithmetic fluency. This finding is in contrast to a recent study that reported a significant association between the reverse distance effect and arithmetic abilities in adults (

The present work showed that the behavioral correlates of numerical order (symbolic and non-symbolic) processing are reliable measurements. Consistent with previous findings, the results also demonstrated that processing the order of number symbols is a reliable and significant predictor of arithmetic fluency, explaining a unique portion of variance over and above the mean reaction time of non-symbolic numerical and symbolic non-numerical order processing. In contrast to number symbols, non-symbolic numerical order processing was not related to arithmetic fluency whatsoever. The differential association between symbolic and non-symbolic numbers with arithmetic fluency supports the view that symbolic and non-symbolic numerical order processing engage different cognitive mechanisms, and that the ability to process ordinal relationships of number symbols is a crucial ingredient to better understand the development of numerical and arithmetic abilities.

The authors have no funding to report.

The authors have declared that no competing interests exist.

Trent Haigh, Gerrit Sommerauer, and Melanie Spindler contributed equally to the work.

The authors have no support to report.