^{a}

^{a}

^{a}

When asked to estimate the outcome of arithmetic problems, participants overestimate for addition problems and underestimate for subtraction problems, both in symbolic and non-symbolic format. This bias is referred to as operational momentum effect (OM). The attentional shifts account holds that during computation of the outcome participants are propelled too far along a spatial number representation. OM was observed in non-symbolic multiplication and division while being absent in symbolic multiplication and division. Here, we investigate whether (a) the absence of the OM in symbolic multiplication and division was due to the presentation of the correct outcome amongst the response alternatives, putatively triggering verbally mediated fact retrieval, and whether (b) OM is correlated with attentional parameters, as stipulated by the attentional account. Participants were presented with symbolic and non-symbolic multiplication and division problems. Among seven incorrect response alternatives participants selected the most plausible result. Participants were also presented with a Posner task, with valid (70%), invalid (15%) and neutral (15%) cues pointing to the position at which a subsequent target would appear. While no OM was observed in symbolic format, non-symbolic problems were subject to OM. The non-symbolic OM was positively correlated with reorienting after invalid cues. These results provide further evidence for a functional association between spatial attention and approximate arithmetic, as stipulated by the attentional shifts account of OM. They also suggest that the cognitive processes underlying multiplication and division are less prone to spatial biases compared to addition and subtraction, further underlining the involvement of differential cognitive processes.

Numerical cognition has long been thought to represent a prime example of an abstract propositional symbolic system with no obvious reference to the outer world. For example, number names such as ‘one’, ‘two’ and so on do not bear any obvious association to the referenced cardinal meaning. Recent evidence, however, implies that numbers and mental arithmetic bear numerous associations with space and time (

This process is not bias-free. When approximating the outcome of simple addition or subtraction problems humans are likely to provide a biased response that deviates systematically from the correct outcome. The results of addition problems tend to be overestimated while the results for subtraction problems are underestimated. This cognitive bias is referred to as operational momentum (OM;

More evidence for spatial contributions to mental arithmetic comes from a recent study reporting systematic interference between arm or eye movements and mental arithmetic (

Despite the growing number of studies demonstrating OM in different settings, the underlying mechanisms of the OM effect are currently debated. Three major hypotheses vie with one another. First, it has been suggested that the OM effect reflects the outcome of a simple heuristic that would associate different arithmetic operations with expectations concerning the numerical relationship between the outcome and the operands (

Finally, spatial accounts have been proposed to account for OM. According to the spatial competition hypothesis (

The aim of the current study was two-fold. First, we aimed at testing the presence of OM in multiplication and division by eliminating the presence of the correct outcome amongst the response alternatives. By encouraging participants to approximate even in the symbolic notation we aimed at increasing sensitivity to detect any systematic biases during multiplication and division with Arabic digits. Second, engaging participants in both an OM task and a Posner paradigm allowed us to test whether potential OM biases actually correlate with attentional measures. According to the above theoretical accounts of the OM we can break this question down into four aspects. Do attentional parameters correlate with (a) a heuristic according to which multiplication leads to larger outcomes and division to smaller outcomes, (b) flawed decompression, (c) competing spatial biases by the operands, the results or the outcome, or (d) attentional shifts along the mental number line? According to the heuristics account and the compression-decompression approach, no correlation with attentional measures would be predicted. Among the spatial accounts, only the attentional explanation predicts a correlation between attentional parameters and OM effect. No such correlation is predicted by the competing spatial biases account.

Participants (

The study consisted of two experiments; a calculation task involving symbolic and non-symbolic multiplication and division problems, and a variant of the Posner task to test different aspects of visuo-spatial attention (orienting/selection and reorienting/executive attention).

The calculation task was created and presented using OpenSesame (

Calculation task. Symbolic and non-symbolic multiplication and division problems were presented in random order. Participants selected the responses with a mouse. (A) Non-symbolic multiplication. A set of 7 dot- arrays were created so that neither area subtended nor average dot-size correlated with quantity. (B) Symbolic multiplication. Response choices were jittered so that the exact answer was never shown.

Non-symbolic stimuli were created using MATLAB and the Psychophysics Toolbox extension (

To catch random responding, symbolic and non-symbolic control trials (16 ÷ 1, 16 x 1) were intermixed with calculation trials. Participants whose performance deviated more than 3

To control for response choice magnitude effects, the same result values (with a random jitter in symbolic trials) or quantities (non-symbolic trials) were presented for multiplication and division; this resulted in different operands for multiplication versus division. We created a geometric series of 11 values ranging from 1/3 to three times the correct value (

Participants completed 576 calculation trials (144 per condition) and 120 control trials (16 ÷ 1, 16 x 1; 30 per condition). To prevent the correct result always being the median value (^{th} response value rank of 7 choices), we varied the range of response alternatives and presented only seven out of the eleven response alternatives for each problem. The seven response alternatives corresponded to the smallest (low), largest (high) or middle (middle) range of response alternatives (

Attention was assessed using an endogenous Posner cueing task (

Attentional cueing task. An arrow appeared that either pointed in the direction of the subsequent target (valid cue, shown), the opposite direction (invalid cue) or in both directions (double-headed arrow, neutral cue). Participants were instructed to press the space bar as quickly as possible when they saw the target (white disk) that appeared after a variable SOA (1200 ms – 1800 ms).

Symbolic catch trials were always 16 multiplied or divided by 1. Although responses were jittered, so that ‘16’ was never presented, if participants were paying attention and following the task directions, they should have been able to choose the value closest to 16 most of the time. Some degree of inaccuracy (i.e. < 100%) was expected since response choices were jittered. Therefore, we first eliminated subjects with accuracy less than 50% correct or greater than 3

Next, because multiplication and division problems were not presented in separate blocks (_{10} of the chosen value and the log_{10} of the correct value was more than 3 SD from the subject’s mean. When considering all conditions together, this excluded 40 calculation trials (0.4% of calculation responses, all non-symbolic). This was less than previous studies using this method (

In the Posner task, we first eliminated responses that were faster than 200ms, because these likely reflect premature responses. This eliminated 146 trials (7.3% of responses). The number of responses faster than 200ms per subject ranged from 0 to 37 responses (

We computed validity effect (RT invalid minus RT valid), benefit (RT neutral minus RT valid) and cost (RT invalid minus RT neutral) as indices for orienting/selection and re-orienting, respectively.

We first analyzed the distribution of responses to exclude the possibility that participants responded randomly. If participants responded non-randomly, then range and rank should have a significant effect on response choice (

Previous studies found that behavior was well described by Weber’s law, suggesting a logarithmic compression of the underlying representation. This also appeared to be true in the present data (see

Linear scale |
Log scale |
|||||
---|---|---|---|---|---|---|

Slope | 95% CI | Slope | 95% CI | |||

Multiplication | ||||||

Symbolic | 123.1** | 0.96 | 0.94, 0.97 | 169.0** | 0.98 | 0.97, 1.00 |

Non-symbolic | 40.7** | 1.24 | 1.18, 1.30 | 50.4** | 1.05 | 1.01, 1.09 |

Division | ||||||

Symbolic | 129.1** | 0.96 | 0.95, 0.97 | 148.0** | 0.98 | 0.97, 0.99 |

Non-symbolic | 37.0** | 0.95 | 0.90, 1.00 | 47.9** | 0.92 | 0.89, 0.96 |

**Bonferroni corrected for multiple comparisons,

To investigate operational momentum, we entered the response bias, defined as the difference between the log chosen and the log correct values, into an ANOVA comprising the factors notation (symbolic, non-symbolic) and operation (multiplication, division). There was a significant interaction between operation and notation on response bias (

Response bias in symbolic and non-symbolic calculation. F-values represent simple main effects of operation (upper half) and notation (lower half). T-values represent one sample t-tests against a test value of zero. For symbolic problems (left), operation did not have a significant effect on response bias. The response bias for symbolic division (dark grey) was not significantly different than multiplication (light grey). However, symbolic multiplication problems were significantly underestimated. For non-symbolic problems (right), there was a significant effect of operation on response bias. Non-symbolic division (dark grey) problems were underestimated relative to multiplication (light grey). Only division, which was underestimated, showed a response bias significantly different from zero.

OM bias (log-scale) as a function of attentional parameters (A) validity effect (invalid – valid), (B) cost (neutral – invalid) and (C) benefit (neutral – valid) from the Posner task. Each data point corresponds to one subject.

For non-symbolic problems, operation had a significant effect on response bias (

Notation did not have a significant effect on response bias for multiplication (

In sum, we replicated the results from

It has been put forward that the OM effect reflects the consequences of an attention-induced spatial displacement along the mental number line during the process of approximating the outcome of an arithmetic problem (

We first tested whether the relative OM bias, defined as the difference between operation-specific OM bias ((log correct minus log chosen Multiplication) minus (log correct minus log chosen Division)) correlated with the validity effect (valid cue minus invalid cue).

The validity effect was consistently larger than zero (i.e., faster response for valid than invalid) for all participants (^{2} = .269).

To further examine the attentional mechanisms potentially driving the OM effect in non-symbolic notation, we separately examined the effect of attentional orienting (benefit of valid cue compared to neutral cue; ^{2} value (^{2}) was used to determine effect sizes using the cutoffs: small = .01 or 1%, medium = .1 or 10%, and large = .25 or 25% (^{2} = .464. Valid cue benefit could not predict response bias difference in non-symbolic problems; ^{2} = -.044. Results are shown in

All correlations remained by-and-large unchanged after partialing out age as a potential confound (

These results suggest that in non-symbolic problems, attentional shifts, most likely re-orienting but not orienting, largely account for the difference in response bias between multiplication and division.

In this study we examined two questions. First, we explored whether the previously reported absence of an OM effect in symbolic multiplication and division (

The absence of a significant OM in symbolic multiplication and division in the current study is in line with previous results and implies that symbolic multiplication and division strongly rely on verbally mediated fact retrieval which is less prone to cognitive biases such as the OM. While a recent study described how the compression of the mental number line biases arithmetic fact retrieval (

We demonstrated a significant correlation between non-symbolic OM and measure of reorienting attention after invalid cues in the Posner paradigm. No correlation was observed between OM and orienting attention after valid cueing. This is somewhat unexpected since the attention account of OM holds that attentional shifts propel participants too far along the mental number line. Hence, the benefit from a valid cue should correlate positively with OM. Yet, we did not observe any correlation between OM and cue benefit, a measure of attentional orienting. This might be due to a reduced variability in participants’ performance in our Posner task which had slightly longer SOAs compared to the classical Posner paradigm. Variability may have been particularly reduced for the benefit measures as compared to cost measures, which, all else being equal, lowers correlation coefficients.

Reorienting is not a unitary process but has traditionally been subdivided into disengaging, shifting and reengaging attention to the new location (

Another question that arises from the current findings concerns the direction of the observed correlation. Why is the correlation between OM and reorienting positive, meaning that people with a large OM effect, i.e. larger deviations from the correct outcome exhibit larger costs for invalid cue both compared to neutral cues and valid cues? According to the above theoretical accounts of the OM we can break this question down into four aspects. Why would attentional reorienting correlate with (a) a heuristic according to which multiplication leads to larger outcomes and division to smaller outcomes, (b) flawed decompression, (c) competing spatial biases by the operands, the results or the outcome, or (d) attentional shifts along the mental number line?

According to the first account, no correlation with attentional measures would have been predicted. If any, there would have been a prediction that the expectation of larger or smaller outcomes creates attentional shifts to the left or right which might generate a coarse approximation of the result that can be used to check if a given outcome is plausible or not. However, what renders this hypothesis rather unlikely to account for the given results is that no gradation beyond a coarse “more or less” expectation is predicted.

Similarly, according to the compression-decompression approach OM results from flawed compression-decompression mechanisms. No attentional mechanism is involved in this process. According to this account, OM would scale with the size of both operands and results. However, previous research found that OM increases with increasing outcome (

With respect to the spatial accounts of the OM, a clear prediction comes from the attentional shift hypothesis which predicts a clear association between attentional parameters and OM. The current results partially confirmed this by the correlation between OM and reorienting, providing further evidence for a role of spatial attention during approximate arithmetic and, more specifically, for the idea that the OM results from attentional mechanisms. Larger reorienting costs may reflect highly efficient orienting mechanisms that need to be overruled after invalid cueing. These results are also in line with the finding that symbolic addition problems are solved faster when the second operand is presented on the right compared to left-sided presentation (

The spatial competition account, in contrast, does not predict any correlation between attentional shifts and OM. According to this account OM is largest when competition between spatial positions of operands and the results on the MNL are minimal. The strong OM in zero problems where only the first operand and the operational sign induce a spatial bias is explained by the absence of a spatial bias induced by zero (

As this study provides only correlational evidence, no causal inference can be drawn. Further experimental work is required to elucidate the neurocognitive mechanisms underlying the OM effect. The absence of significant correlations between operational momentum bias and attentional measures may in part be due to the long SOAs in our version of the Posner task. Future studies could increase variability by using shorter SOAs, which might favor finding higher correlations. The fact that we related accuracy-based operational momentum bias with speed–based measures of the attentional capacities might further have reduced our statistical power. The limited sample size may raise concerns about (a) stability and reliability of the data and (b) to what extent the observed correlations were due to the increased variability in our sample. We checked whether our results were driven by some outliers by separately excluding all possible combinations of 1, 2 or 3 participants from the sample. We found single participants to have only a minor impact on the correlation pattern. Even when excluding the two participants with the most extreme value pairings, correlations by and large remained significant or marginally significant. To protect against the possibility that age was a confounding variable that drives our results, we recalculated the main findings after partialing out age and found the major findings of the study unchanged. Finally, while one may be concerned about the role of counting in computing the approximate results, we would reason that counting does play a major role in explaining our results. These additional analyses are can be found in

To sum up, we failed to observe an OM effect in symbolic multiplication and division. This is in line with previous findings (

The problems we used were identical to the ones in

Operands |
Response alternatives |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1^{st} |
2^{nd} |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Multiplication | ||||||||||||

4 | 3 | 6 | 7 | 8 | 9 | 10 | 12 | 14 | 16 | 18 | 21 | 24 |

6 | 3 | 9 | 10 | 12 | 14 | 16 | 18 | 21 | 24 | 27 | 31 | 36 |

6 | 4 | 12 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 36 | 42 | 48 |

7 | 3 | 11 | 13 | 14 | 16 | 19 | 21 | 25 | 28 | 32 | 37 | 42 |

7 | 4 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 37 | 42 | 49 | 56 |

7 | 6 | 21 | 24 | 28 | 32 | 37 | 42 | 48 | 55 | 64 | 73 | 84 |

8 | 3 | 12 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 36 | 42 | 48 |

8 | 4 | 16 | 18 | 21 | 24 | 28 | 32 | 37 | 42 | 49 | 56 | 64 |

8 | 6 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

9 | 3 | 14 | 16 | 18 | 21 | 24 | 27 | 31 | 36 | 41 | 47 | 54 |

9 | 4 | 18 | 21 | 24 | 27 | 31 | 36 | 41 | 48 | 55 | 63 | 72 |

9 | 6 | 27 | 31 | 36 | 41 | 47 | 54 | 62 | 71 | 82 | 94 | 108 |

12 | 3 | 18 | 21 | 24 | 27 | 31 | 36 | 41 | 48 | 55 | 63 | 72 |

12 | 4 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

13 | 4 | 26 | 30 | 34 | 39 | 45 | 52 | 60 | 69 | 79 | 91 | 104 |

13 | 6 | 39 | 45 | 51 | 59 | 68 | 78 | 90 | 103 | 118 | 136 | 156 |

14 | 3 | 21 | 24 | 28 | 32 | 37 | 42 | 48 | 55 | 64 | 73 | 84 |

14 | 6 | 42 | 48 | 55 | 64 | 73 | 84 | 96 | 111 | 127 | 146 | 168 |

16 | 3 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

16 | 4 | 32 | 37 | 42 | 49 | 56 | 64 | 74 | 84 | 97 | 111 | 128 |

17 | 3 | 26 | 30 | 34 | 39 | 45 | 51 | 59 | 68 | 78 | 89 | 102 |

17 | 4 | 34 | 39 | 45 | 52 | 59 | 68 | 78 | 90 | 103 | 118 | 136 |

19 | 3 | 29 | 33 | 38 | 44 | 50 | 57 | 66 | 76 | 87 | 99 | 114 |

19 | 4 | 38 | 44 | 50 | 58 | 66 | 76 | 87 | 100 | 115 | 132 | 152 |

Division | ||||||||||||

36 | 2 | 9 | 10 | 12 | 14 | 16 | 18 | 21 | 24 | 27 | 31 | 36 |

48 | 2 | 12 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 36 | 42 | 48 |

48 | 4 | 6 | 7 | 8 | 9 | 10 | 12 | 14 | 16 | 18 | 21 | 24 |

54 | 2 | 14 | 16 | 18 | 20 | 24 | 27 | 31 | 36 | 41 | 47 | 54 |

63 | 3 | 11 | 12 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 37 | 42 |

96 | 2 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

96 | 4 | 12 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 36 | 42 | 48 |

108 | 3 | 18 | 21 | 24 | 27 | 31 | 36 | 41 | 48 | 55 | 63 | 72 |

112 | 4 | 14 | 16 | 18 | 21 | 24 | 28 | 32 | 37 | 42 | 49 | 56 |

126 | 3 | 21 | 24 | 28 | 32 | 37 | 42 | 48 | 55 | 64 | 73 | 84 |

128 | 2 | 32 | 37 | 42 | 49 | 56 | 64 | 74 | 84 | 97 | 111 | 128 |

128 | 4 | 16 | 18 | 21 | 24 | 28 | 32 | 37 | 42 | 49 | 56 | 64 |

136 | 2 | 34 | 39 | 45 | 52 | 59 | 68 | 78 | 90 | 103 | 118 | 136 |

144 | 3 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

144 | 4 | 18 | 21 | 24 | 27 | 31 | 36 | 41 | 48 | 55 | 63 | 72 |

152 | 2 | 38 | 44 | 50 | 58 | 66 | 76 | 87 | 100 | 115 | 132 | 152 |

153 | 3 | 26 | 29 | 34 | 39 | 44 | 51 | 59 | 67 | 77 | 89 | 102 |

156 | 2 | 26 | 30 | 34 | 39 | 45 | 52 | 60 | 69 | 79 | 91 | 104 |

156 | 3 | 26 | 30 | 34 | 39 | 45 | 52 | 60 | 69 | 79 | 91 | 104 |

162 | 3 | 27 | 31 | 36 | 41 | 47 | 54 | 62 | 71 | 82 | 94 | 108 |

168 | 2 | 42 | 48 | 55 | 64 | 73 | 84 | 96 | 111 | 127 | 146 | 168 |

168 | 4 | 21 | 24 | 28 | 32 | 37 | 42 | 48 | 55 | 64 | 73 | 84 |

171 | 3 | 29 | 33 | 38 | 43 | 50 | 57 | 65 | 75 | 86 | 99 | 114 |

192 | 4 | 24 | 28 | 32 | 36 | 42 | 48 | 55 | 63 | 73 | 84 | 96 |

Exemplary depiction of how we varied the ordinal rank of the correct outcome (gray column) within the low (blue), middle (brown), and high (red) range of response alternatives (RA 1 to RA 11). Example depicts the outcomes for the problem 3 × 4 in the symbolic notation.

There was a significant interaction between range and rank (

Range | Partial η^{2} |
Sphericity^{a} |
|||
---|---|---|---|---|---|

χ^{2} |
ε | ||||

Symbolic multiplication | |||||

Low, 5^{th} correct** |
168.847 | <.001 | .913 | 157.0 | .230 |

Med., 4^{th} correct** |
131.614 | <.001 | .892 | 132.2 | .243 |

High, 3^{rd} correct** |
207.360 | <.001 | .928 | 183.4 | .275 |

Non-symbolic multiplication | |||||

Low, 5^{th} correct** |
7.209 | .001 | .311 | 39.5 | .474 |

Med., 4^{th} correct |
2.628 | .058 | .141 | 35.1 | .522 |

High, 3^{rd} correct* |
3.084 | .038 | .162 | 47.3 | .484 |

Symbolic division | |||||

Low, 5^{th} correct** |
109.958 | <.001 | .873 | 118.6 | .266 |

Med., 4^{th} correct |
142.816 | <.001 | .889 | 118.6 | .283 |

High, 3^{rd} correct |
116.792 | <.001 | .880 | 127.7 | .288 |

Non-symbolic division | |||||

Low, 5^{th} correct** |
3.043 | .046 | .160 | 59.4 | .433 |

Med., 4^{th} correct* |
4.916 | .012 | .235 | 75.8 | .350 |

High, 3^{rd} correct** |
14.402 | < .001 | .474 | 83.0 | .355 |

^{a}Greenhouse-Geisser corrected for violations of sphericity as measured by Mauchley’s Test of Sphericity.

*Significant simple main effect only.

**At least 1 significant post-hoc pairwise comparison (Bonferroni-corrected

There was a significant interaction between range and rank (

1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

Symbolic multiplication | |||||||

L | 0.1^{3,4,5,6,7} |
0.1^{3,4,5,6,7} |
0.3^{1,2,4,5} |
1.2^{1,2,3,5,7} |
5.6^{1,2,3,4,6,7} |
1.0^{1,2,5} |
0.4^{1,2,4,5} |

M | 0.2^{2,3,4} |
0.4^{3,4,5,7} |
1.2^{1,2,4,6,7} |
5.2^{1,2,3,5,6,7} |
1.1^{1,2,4,6,7} |
0.2^{3,4,5} |
0.1^{2,3,4,5} |

H | 0.4^{2,3,4,6,7} |
1.3^{1,3,5,6,7} |
5.4^{1,2,4,5,6,7} |
1.0^{1,3,5,6,7} |
0.1^{2,3,4} |
0.0^{1,2,3,4} |
0.0^{1,2,3,4} |

Symbolic division | |||||||

L | 0.1^{4,5,6} |
0.2^{4,5,6} |
0.3^{4,5,6} |
1.2^{1,2,3,5,7} |
5.0^{1,2,3,4,6,7} |
1.1^{1,2,3,5,7} |
0.4 ^{4,5,6} |

M | 0.3^{3,4,5} |
0.2^{3,4,5} |
1.2^{1,2,4,6,7} |
5.0^{1,2,3,5,6,7} |
1.0^{1,2,4,6,7} |
0.3^{3,4,5,7} |
0.1^{3,4,5,6} |

H | 0.4^{2,3} |
1.4^{1,3,5,6,7} |
5.0^{1,2,4,5,6,7} |
1.0^{3,5,6,7} |
0.3^{2,3,4} |
0.1^{2,3,4} |
0.0^{2,3,4} |

Non-symbolic multiplication | |||||||

L | 1.0^{5} |
1.0^{4,5,6} |
1.0^{5,6} |
1.2^{2} |
1.5^{1,2,3} |
1.8^{2,3} |
1.6 |

M | 1.0 | 1.0 | 1.1 | 1.6 | 1.4 | 1.5 | 1.0 |

H | 1.2 | 1.3 | 1.6 | 1.3 | 1.4 | 1.1 | 1.0 |

Non-symbolic division | |||||||

L | 1.0 | 1.4 | 1.5^{6} |
1.4^{6} |
1.4^{6} |
0.8^{3,4,5} |
1.0 |

M | 1.6 | 1.7 | 1.5 | 1.3 | 1.0 | 0.8 | 0.6 |

H | 2.4^{6,7} |
2.1^{3,4,5,6,7} |
1.3^{2,6,7} |
1.0^{2,6} |
0.8^{2,6} |
0.4^{1,2,3,4,5} |
0.5^{1,2,3} |

^{th} correct. M = Medium-4^{th} correct. H = High-3^{rd} correct.

^{1,2,3,4,5,6,7} Significantly different (Bonferroni-corrected) from rank ^{n}

There was a significant interaction between range and rank (^{th} choice (too large) was selected significantly more than the 2^{nd} or 3^{rd} choice, the 5^{th} choice (correct choice) was selected significantly more often that the 1^{st}, 2^{nd} or 3^{rd} and the 4^{th} choice significantly more than the 2^{nd} choice (^{th}, 5^{th} and 6^{th} choice were not significantly different for low range trials.

There was a significant interaction between range and rank (^{rd} (too small), 4^{th} (too small) and 5^{th} (correct) choices were selected significantly more than the 6^{th} choice (too large), but there was no significant difference between the 1^{st} through 5^{th} choice (^{st}, 2^{nd} & correct, 3^{rd}) were selected significantly more often than the two largest choices (6^{th} & 7^{th}) and the 6^{th} was selected less often than all of the smaller choices (1^{st}-5^{th}). The 2^{nd} choice was selected more often than all of the larger choices (3^{rd}-7^{th}), including the correct choice (^{nd} choice) than the correct choice when the 3^{rd} choice was correct (high range), a possibly random response pattern when the middle choice (4^{th}) was correct (medium range) and a non-random response pattern driven by a decreased likelihood of choosing the 6^{th} choice than the 3^{rd}, 4^{th} and 5^{th} when the 5^{th} was correct (low range).

Response value as a function of correct value on linear and log-scale non-aggregated data. (A) The non-aggregated response value was plotted as a function of correct value on the linear (left) and log (right) scale. Each point represents one case and number of cases is indicated by dot density (_{10} of the response and correct value was used, response value increased as a function of correct value, but variability remained constant. (C) The dispersion of the response choices, measured by the mean-centered coefficient of variation (CV), was more constant when the log-scale data was used (right).

In the following we report the results of additional analyses (1) to verify the stability and reliability of the data, as well as the (2) impact of age, a (3) Bayesian analysis of the reported null effects, and (4) why we think counting does not play a major role in the current experiment.

We checked the effect of separately excluding all possible combinations of 1, 2 or 3 participants from the sample on our main findings, that is the correlation between

Excluding 1 participant: When excluding the two participants (#13 and #16) marked in red in the

Scatter plot of the validity effect against the OM bias. Each dot represents one participant.

When excluding more than one participant, 32 out of 136 combinations of two participants (23.5%), or 229 out of 680 combinations of three participants were not significant (33.7%).

Excluding one participant did not change the fundamental pattern of results. All correlations remained significant (p > .05).

When excluding more than one participant, 0 out of 136 combinations of two participants (0%), or 11 out of 680 combinations of three participants were not significant (1.62%).

Only when excluding one participant (#17) the correlation (

When excluding more than one participant, 20 out of 136 combinations of two participants (14.7%), or 183 out of 680 combinations of three participants were not significant (26.9%).

Partialing out age did not significantly impact the correlation between OM bias and validity effect (_{age}(bias, validity effect) = .613, (_{age}(bias, cost) = .70, _{age}(bias, benefit) = -.09,

Together with the above results we think that these results imply rather stable data even given the small sample size. Age was not a major factor driving our results. On the contrary, our convenience sample increases the generalizability of the data to the population since we do not – as the majority of studies in the field of experimental psychology – artificially restrict our sample to university students in their early twenties. Further, it should be mentioned that we chose a rather conservative procedure when excluding one participant before engaging in our analyses based on the participant’s low performance.

For the non-significant correlation between benefit and OM bias, a Bayes factor BF10 = 0.214 indicates evidence in favor of the null hypothesis. It is 1/0.214 = 4.673 more likely that the data occurred under the H0 than the H1 (

For the following reasons we consider counting as an unlikely strategy to account for the results. The majority of operands cannot be counted during two seconds. Assuming a counting rate of ~250 ms per item, participants may have counted up to ~8 items. This allowed counting only those problems where the sum of the operands would be around 8. This was the case for four problems only. What makes it even more unlikely that participants counted is that the results screen contained seven dot patterns, clearly exceeding the time limit to count all dots within only 6 seconds. Mean reaction time for non-symbolic problems was 2.595 second (

This work was funded by a grant (KN 959/2) from Deutsche Forschungsgemeinschaft (DFG, German Research Council) to André Knops in support of Curren Katz and Hannes Hoesterey.

The authors have declared that no competing interests exist.

The authors have no support to report.