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The hand with which one starts to count has been shown repeatedly to influence numerical performance. However, methods vary greatly in how researchers determine starting hand. As such, it is impossible to say whether starting hand reflects one construct that is being differently measured, or if these methods reflect different constructs. To investigate these possibilities, we employed a binary magnitude comparison task known to elicit spatial-numerical biases and embodied number magnitude effects, as well as both cardinal and ordinal assessments of starting hand. In addition to this, we further examined whether being made aware of one’s finger-counting habits prior to the numerical task (through a finger-counting inventory) may alter performance during a spatial-numerical reaction-time task. Ordinal and cardinal starting hand classifications disagreed significantly in their classification of left vs. right-starters and predicted different aspects of numerical performance, which further interacted with procedure-order. The pattern of results suggest that 1) ordinal and cardinal aspects of finger-counting are dissociable and predict differing aspects of embodied numerosity, and 2) that assessing finger counting habits before performing a numerical task may affect performance on that task. Therefore, these methodological variations have important theoretical ramifications and need to be reported in greater detail in future work.

Finger-counting habits have become an important area of study for embodied numerosity. Finger counting is the most common form of bodily representation for numbers (

Furthermore, and of particular interest to this paper, people who start counting on their left hand (left-starters) differ from those who start counting on their right hand (right-starters) in both numerical performance (

Nevertheless, there are two reasons why it is difficult to explain the right-starter/left-starter difference. First, there are different methods to differentiate between left-starters and right-starters. For example, some studies have used a cardinal method, where participants are asked to show a set of numbers, one at a time, on their fingers and the hand that is used to show the numbers 5 or less is taken to be the starting hand (also called finger montring gestures,

The purpose of this study was to explicitly investigate whether different methods of assessing starting hand have differing relations with measures of numerical cognition. The objective is to ascertain if one of these methods better differentiates numerical ability, or if these different methods actually reflect different constructs. Our further goal was to discern what these results might mean theoretically for these previously observed differences between left- and right-starters.

One possible mechanism by which finger-counting may lead to embodiment of numeracy is through a SNARC-like spatial reference frame. Among populations that read from left to right, SNARC is characterized by faster association of smaller numbers with left hand space and larger numbers with right-hand space (

So far, research that has investigated individual differences between left-starters and right-starters has found several preliminary differences.

Left-starters and right-starters have also been found to exhibit performance differences in a variety of numerical tasks aside from those meant to measure SNARC. In an addition study,

As it turns out, there is more than one way to question someone about their finger counting habits. For example, cardinal finger-number gestures include finger numeral configurations which may be used to represent an individual number symbolically, similarly to written numerals, such as when showing a number to another person (

While cardinal and ordinal gestures are related, they are not always the same. Wasner and colleagues (

In addition to these differences in assessment, there is also some variability across studies about when the assessment is given. Some experimenters questioned participants about their counting habits prior to the main task (

The current study is a combination of a retrospective study using data from

For this reason, the present study separately evaluates the effect of starting hand differences, using both the ordinal and cardinal classifications, on two separate numerical cognition phenomena. The first of these was the embodiment of SNARC-like response compatibility effects. The second was the differences in response time for certain number comparisons whose numbers are represented on two hands, as reported by

The main difference between this sort of SNARC-like response compatibility effect and a more typical categorical SNARC task is in the explicitness of space in participants’ decision-making process. A typical SNARC magnitude comparison test present number-digit stimuli in the center of the screen, and participants rate these items as smaller, or larger, than some reference value, typically 5. Therefore, relatively faster responses for larger number-digits with the right hand, and vice versa, are argued to be a function of an implicit spatial association of number digits (

A number of investigations have demonstrated ordinal starting-hand as a moderator of SNARC (

Furthermore, by counterbalancing the order of task administration, there was an additional opportunity to test

The primary objective of this investigation was to examine two separate numerical phenomena known to exhibit right/left-starter differences and ascertain whether cardinal and ordinal classifications of finger-counting habits would lead to different conclusions about the impact of starting hand. A secondary objective within this is to further examine whether being made aware of one’s finger-counting habits prior to the numerical task (through a finger-counting inventory) may alter performance during a spatial-numerical reaction-time task in ways consistent with prior investigations that have suggested that this is indeed possible when left/right hands are emphasized instead of left/right buttons (

The data used in this study were a combination of two different datasets, with all participants having participated in an identical number comparison task. Dataset one consisted of 98 participants specifically recruited for this investigation who all received both cardinal and ordinal finger counting inventories, although two participants were later excluded after committing more than 25% errors. Dataset two consists of published and unpublished data for 135 participants tested during

Overall, there were 96 participants included in Dataset one and 135 participants in Dataset two, with an average accuracy of 95.2% (

The stimuli and procedure for both datasets were the same as that used

A single numerical task was used for all participants. All number pairs were Arabic Digits in black Arial 60pt font, and presented on a white screen, using E-prime 2.0 on either a 15” or 18” lab computer monitor (

In Dataset one, participants were given both a cardinal and an ordinal finger-counting inventory to determine whether they were left-starters or right-starters. In Dataset two, 101 participants only received the cardinal inventory while 34 received a cardinal and an ordinal finger-counting inventory. The cardinal finger-counting inventory included a brief questionnaire about counting habits, where participants were asked to respond to different numbers from 1-10 by demonstrating the relevant number gesture. Participants were instructed to provide number gestures as quickly as possible and with the gesture that feels most natural. The hand used for each number gesture was recorded on a sheet picturing a variety of hands using different counting gestures. Right- and left-starters were classified by which hand was used to represent one through five. The ordinal finger-counting habits inventory instead asked participants to count from 1-10 on their fingers as they would normally, and their starting hand was recorded.

Each participant in the study answered demographic questions about ethnicity, first language, gender, language spoken in primary school, and nationality. Participants who went to school outside of Canada or who used a first language other than English or French were excluded, as previous work has shown that different cultures may impact patterns of numerical performance (

Participants were coded as being in the before-task or after-task condition; depending on if either the cardinal or ordinal finger counting inventory was given prior to the number comparison task. It has been shown recently in the literature that situated factors and experimental procedure can impact self-reported finger-counting habits (

Inventory / Starting hand | Dataset one |
Dataset two |
||||
---|---|---|---|---|---|---|

Cond. 1 | Cond. 2 | Cond. 3 | Cond. 4 | Cond. 3 | Cond. 4 | |

Cardinal | ||||||

Right-starter | 20 | 30 | 26 | 1 | 61 | 55 |

Left-starter | 9 | 5 | 5 | 0 | 7 | 12 |

Ordinal | ||||||

Right-starter | 16 | 21 | 24 | 1 | 2 | 21 |

Left-starter | 13 | 14 | 7 | 0 | 0 | 11 |

A preliminary chi-squared analysis was conducted to test for frequency differences between cardinal left- and right-starters between the two datasets. There was no significant difference in proportion of left- and right-starters between those in Dataset one (77 right-starters vs. 19 left-starters), and the additional 135 participants from Dataset two (116 right-starters vs. 19 left-starters), χ^{2}(1, ^{2}(1, ^{2}(1,

Because this study has more people who were classified on the cardinal inventory than were classified on the ordinal inventory, it could be argued that this study has more power to detect an effect with cardinal starting hand than with ordinal starting hand. As it turns out, the statistical power for comparing left-starters and right-starters is actually quite similar for cardinal and ordinal classifications, despite there being more cardinal data. Using a moderate effect size,

The proportion of left-starters and right-starters were examined as a function type of finger counting inventory. Because the off-diagonal frequencies were low (see ^{2}(1,

Starter hand | Ordinal Right-starter | Ordinal Left-starter |
---|---|---|

Cardinal Right-starter | 83 | 22 |

Cardinal Left-starter | 2 | 23 |

These analyses tested whether type of finger-counting inventory, as well as whether they were given this inventory before or after the reaction time task, affected participants’ SNARC-like performance. Median reaction time scores were utilized in all analyses, as medians are more robust to violations of normality that are typical of reaction-time data. Reaction time scores follow a positively-skewed distribution, due to participants’ inability to achieve a negative reaction time score. This is particularly important in a task like this where significant differences in errors and mean reaction time are expected between different items, independent of any variables of interest.

A SNARC-like effect was defined as reacting more quickly to SNARC congruent responses, (e.g., choosing the larger number of a pair with the right hand or choosing the smaller digit with the left hand) than to SNARC incongruent responses, (e.g., choosing the larger digit with the left hand or choosing the smaller digit with the right hand). SNARC-like performance effects were treated as categorical, similar to

Errors have already been shown in

Two separate 2 (Finger counting timing) x 2 (Starting hand) x 2 (single-digit vs. other number pairs) between/within ANOVAs were conducted on these SNARC-like advantage scores, with finger counting timing and starting hand as between-subject factors and single-digit vs. other number pairs as a within-subject factor. One ANOVA used the cardinal finger-counting inventory in order to differentiate right vs. left-starters, while the second ANOVA used the ordinal inventory for this purpose. The within-subject factor splits up the analyses into number pairs with single-digit numbers, as compared to all other number pairs. The ANOVA with the cardinal inventory used both Datasets while the ANOVA with the ordinal inventory included only those participants who received the ordinal finger counting inventory.

For the ANOVA using the cardinal inventory, one participant, a right-starter in the before-test condition, was excluded due to a disproportionate number of errors for a particular number pair, leading to an empty cell. As expected, there was a robust overall effect of SNARC-like response compatibility effects, as shown by the intercept, (21.037ms), ^{2}_{p} = .110, and this was not significantly moderated by whether single-digit numbers, (27.122 ms), or other number pairs (14.952 ms) were included, ^{2}_{p} = .014. The within-subject factor did not interact with starting hand, ^{2}_{p} = .002, nor with inventory timing, ^{2}_{p} < .0005. There was also no evidence of a three way interaction among these factors, ^{2}_{p} = .003. The results for the two between-subjects factors are shown in ^{2}_{p} = .043. Inconsistent with ^{2}_{p} = .003. However, there was an interaction of the timing of the finger counting inventory with starting hand, ^{2}_{p} = .022. Bonferroni-corrected post-hoc comparisons indicated that the interaction was driven by a significant difference between left-starters in the before-test (2.848 ms) and after-test (46.210 ms) conditions,

Average reaction-time difference between SNARC-congruent and SNARC-incongruent trials. Error bars are 95% confidence intervals. BTRS indicates before-test right-starters (

The same repeated measures ANOVA model was tested again using the reported ordinal starting hand of these participants. As predicted, there was a large SNARC-like response compatibility effects (21.558 ms), as indicated by the intercept, ^{2}_{p} = .199, and this was not significantly moderated by whether single-digit numbers, (27.003 ms), or other number pairs (13.571 ms) were included, ^{2}_{p} = .022. The within-subject factor did not interact with starting hand, ^{2}_{p} = .007, nor with inventory timing, ^{2}_{p} = .012. There was also no evidence of a three way interaction among these factors, ^{2}_{p} < .0005. The results for the two between-subjects factors in this analysis are shown in ^{2}_{p} = .042. Ordinal left-starters also showed a non-significantly greater impact of SNARC-congruency of their responses (left: 27.929 ms, right: 15.186 ms), ^{2}_{p} = .021, consistent with the findings of ^{2}_{p} = .04. Bonferroni-corrected post-hoc comparisons indicated that the interaction was driven by a significant difference between left-starters in the before-test (10.037 ms) and after-test (45.822 ms) conditions,

Average reaction-time difference between SNARC-congruent and SNARC-incongruent trials. Error bars are 95% confidence intervals. BTRS indicates before-test right-starters (

On the surface, both the analysis involving the cardinal inventory and the analysis involving the ordinal inventory demonstrated similar patterns of results. SNARC-like response compatibility effects were observed, but differences between after-test left-starters exhibited these effects more strongly than other conditions. Assessed after the numerical task, left-starters showed a stronger SNARC-like response than right-starters for both the cardinal and ordinal classifications, although this difference was not statistically significant for after-test right and left-starters when using the cardinal classification. Given the extent to which these two classifications overlap, and given also the fact this overlap is not evenly distributed (i.e., out of the 130 participants who were coded on both, 2 were cardinal left-starters and ordinal right starters, while 22 demonstrated the opposite pattern), it is difficult to test whether the ordinal classification determined a reliably larger effect size for right and left-starters in the after-test condition using standard statistical methods. After collapsing SNARC-like difference scores evenly across all number-digit pairs, we used a bootstrapping procedure. With this procedure, it was possible to create a confidence interval around the difference between the effect found in the ordinal analysis and the effect found in the cardinal analysis. Resampling was done with 10 000 iterations while sampling with replacement. The resampling was set so that each sample were proportionally drawn from the 130 participants who had both classifications and the 101 participants who only had the cardinal classification. The 95% confidence interval generated by the procedure in milliseconds was [-1.73, 14.86]. This interval does include zero, so the effect for the ordinal classifications could not be said to be significantly larger than the effect for cardinal classifications.

Like the previous section, cardinal and ordinal finger counting inventories were tested separately.

We conducted a pair of independent samples t-tests, with Welch’s corrected degrees of freedom, with cardinal and ordinal starting hand as the predictors and participants’ log-fit slope as the dependent variable. Cardinal left-starters showed a slightly, but statistically significantly, steeper log-fit slope than right-starters, with 54.06 ms vs. 46.44 ms per log-digit magnitude increase respectively,

Using the cardinal inventory, a 2 (starting hand) x 2 (finger counting timing) between-subjects ANOVA was conducted on the mean log-residualized response latencies for comparisons of 6 vs. 8 and 7 vs. 9. These two comparisons were singled out in ^{2}_{p} = .040, ^{2}_{p} = .006, and no interaction, ^{2}_{p} = .0005.

Standardized residual scores across the 18 comparisons, when right-starters and left-starters are classified via cardinal finger-counting habits. Error bars are 95% confidence intervals.

Standardized residual scores across the 18 comparisons, when right-starters and left-starters are classified via ordinal finger-counting habits. Error bars are 95% confidence intervals.

The above analyses were repeated using the ordinal starting hand classifications. Unlike cardinal starting hand, ordinal starting hand did not predict any mean differences in log-residualized response latency between the 45 left-starters, ^{2}_{p} = .001, ^{2}_{p} =.008, and no interaction, ^{2}_{p} = .003.

As was case in the analyses of the SNARC-like response compatibility effects, it was necessary to directly compare whether ordinal classifications yielded different results than cardinal classifications. In this case, there was a significant effect of starting hand in the cardinal analyses but not one for the ordinal analyses. However, having one effect that is statistically significant and another one that is not does not necessarily mean that these two effects are different from each other. To test if the difference in these effects differed from zero, again a bootstrapping procedure was performed using the same parameters as above. The 95% confidence interval of the difference in the effect using the cardinal classifications and the effect using the ordinal classifications was [0.044, 0.592]. This confidence interval did not include zero, and so the effect found using the cardinal classifications was stronger than the effect found using the ordinal classifications.

The research reported here was prompted by recent work indicating that finger montring /cardinal finger-number associations may differ from spontaneous or ordinal finger-number gestures (

The first analysis examined how left- vs. right-starter would differ on a SNARC-like task. We also tested whether giving an inventory before the main task (which should draw participants’ attention to their hands) lessens the SNARC-like response biases, similar to

In addition, drawing attention to participants’ hands appears to be a more persistent manipulation of SNARC-like response biases than may have been implied by

There were, however, important differences between the current investigation and that of both

The second analysis in this study examined individual differences in number representation effects, such as those investigated in

Ordinal finger-counting habits did not differentiate right and left-starters on log-residualized reaction time scores, and this was not a function of differences in statistical power. This observation is also important in order to ensure that this finding can be replicated. The methods of

While research staff did not systematically record all aspects of bodily behaviour associated with producing finger counting gestures, it was also observed that cardinal gestures were not typically produced with participants looking at their hands, except for the production of number gestures for six through eight, which were not used in determining if participants were right or left-starters. Ordinal counting gestures were more often, but not universally, produced with a participant looking at their hands. Likewise, the highest rate of left-starters observed in the literature, appears to be for the written inventory used by

The findings discussed here raise several important points for research that involves finger counting habits and SNARC. The first point is methodological. It is very important that finger counting assessments be reported in greater detail, as different assessments result in different participants being classified as left or right-starters, as well as different overall rates in left and right-starters. While there was substantial discordance in starting hand for ordinal and cardinal finger counting habits, there are other inventories in use and so this does not capture the full range of how differences in finger counting inventories may alter the classification of right and left-starters. This suggests that methodological inconsistencies in the assessment of finger counting habits may be more pernicious in how they compromise the ability to replicate or directly compare research in this area. If studies are going to be reporting results for samples of ~30 participants, then the likelihood of one study group containing substantially more ordinal left-starters than another group is high. Combine this with unreported timing of finger-counting habits, or the emphasis of buttons versus hands in participant instructions (

There has been a recent discussion in the literature about a lack of successful replication in psychological research (

The second major point raised here is that of evidence for multiple independent finger-counting-based reference frames. Currently, this is the first study to demonstrate how inconsistencies in finger counting inventories can have consequences for the prediction of actual individual differences in numerical performance. If there were only one global left-right reference frame interacting with a hand-based reference frame, then these inventories should be predicting the same types of phenomena, but with varying degrees of success or clarity. This latter scenario appeared to be correct when describing SNARC-like response compatibility effects, with both ordinal and cardinal inventories predicting similar differences, with somewhat stronger effects detected with the ordinal inventory. However, what was particularly interesting about this analysis was not only that classifications of left and right-starters differed between the ordinal and cardinal inventories, but that this disagreement was almost entirely limited to the classification of ordinal left-starters. Despite this relative disagreement, both inventories predicted substantial individual differences with regard to SNARC-like response compatibility effects, while only the cardinal inventory found differences between right and left-starters when evaluating number representation effects. This fits with recent evidence suggesting that decoding ordinal and cardinal aspects of number symbols may be separable and each may predict unique variance in mathematical competence (

All in all, these results suggest that finger-based numerical representation may be more complex than originally thought. While we continue to find evidence that some participants respond more slowly than others when comparing numbers typically counted on two hands, it appears that only cardinal, and not ordinal, starting-hand is a useful predictor of this individual difference. While the use of a categorical SNARC-like task does soften our conclusions somewhat for spatial associations, these results do raise some concerns for interpreting other findings in the literature. Fortunately, the main recommendation that we make in order to avoid most of the issues raised here would be to simply transparently report timing and type of finger counting inventory in all studies using these variables. A possible side benefit for such improvements in reporting would be that differences between inventories, and between procedure-orders, may provide researchers with richer information as to what influences underlay certain aspects of numerical cognition, as we can see not only how certain manipulations impact performance, but how different types of participants may respond differently to these manipulations. This increased methodological exactness, and the further testing of the influence of finger counting habits, should help us to better understand what appears to be the increasingly complex connection between finger and numerical representations.

The authors have no support to report.