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The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials.

Previous research has hinted at a functional association between the concepts of numbers and space (

Extending spatial-numerical associations to the realm of mental arithmetic,

Other accounts were put forward to explain the OM effect. According to the

Further evidence for the involvement of spatial representations during mental arithmetic comes from the

These attentional biases in mental arithmetic cannot be explained by all above theories. The heuristics account and the compression account alone do not predict spatial biases as they contain no spatial components. The AHAB model, on the other side, stipulates an association between addition signs and the right side of space and subtraction signs and the left side of space. This component which takes action in situations “when stimuli or responses are spatially distributed” can explain these spatial biases (

When it comes to the processes underlying mental arithmetic, a global and approximate evaluation is distinguished from an exact retrieval process. This dual-process assumption was based on the reaction time advantage for incorrect response alternatives with a large numerical distance from the correct outcome over correct response alternatives (

The question arises what process gives rise to the hypothesised attentional shifts during mental arithmetic. The OM effect which has originally been interpreted as a consequence of the movement on the spatial representation of magnitude (attentional shift hypothesis) has been observed both during exact and approximate arithmetic but was stronger in a non-symbolic (i.e. purely approximate) calculation context (

The present study intended to fill this gap by investigating whether the approximate solution process to non-symbolic addition and subtraction tasks induces attentional shifts to the right and left, respectively. The arithmetic task made use of dot arrays that contained a minimum of 8 dots to avoid exact processing and rapid counting strategies. We sequentially presented an operation cue, the first operand dot array followed by a second presentation of the operation cue, the second operand dot array, and the concurrent presentation of four response option (RO) arrays. The participant’s task was to indicate verbally which of these four ROs was the correct result to the arithmetic problem. Spatial attention during the approximate calculation phase was measured with a target detection task between the arithmetic problem presentation and the RO presentation. To investigate the time-course of potential attentional shifts, we varied the delay between the offset of the second operand and the onset of the target detection task (150, 300, 500 ms). Delays were chosen in accordance with previous studies (e.g.

We found no operation-dependent shifts in the target detection task but a spatial bias in the choice of RO locations of the arithmetic task (SOAR effect). To increase overall accuracy and overcome the overall underestimation bias that might have masked spatial effects in the target detection task, we conducted a second experiment that involved feedback on the arithmetic response during the practise phase. Again, no operation-dependent shifts were observed in the target detection task, but we replicated the SOAR effect in the arithmetic task. Exploratory analyses revealed that the OM effect is subject to sequence effects. The OM effect in trial

Eighteen German-speaking students from the Humboldt-Universität zu Berlin (_{age} = 22.67 years, _{age}

9 addition and 9 subtraction problems that were matched with regard to their operands from ^{(}^{r}^{+}^{i}^{/2)} (^{nd} RO (upper range)/ 3^{rd} RO (lower range) was correct.

The non-symbolic dot stimuli for the arithmetic task were created using MATLAB (R2016a) and the Psychtoolbox library (

The experiment was presented via MATLAB (R2016a) and the Psychtoolbox package (

The visual stimuli were presented on a light grey background. For fixation we used a white asterisk. All dot stimuli consisted of white dots on a grey circular background (radius 9.3°). Targets consisted of dark grey squares (.75° × .75°) that were presented with a distance of 5.12° from fixation (measured from the centre of the target).

The 396 trials were presented in 11 blocks of 36 trials each. Practise blocks contained 30 trials drawn randomly from the set of experimental trials. One practise block was mandatory. Another block was optional. Addition and subtraction trials were presented in interleaved, randomised order.

Every trial started with the fixation asterisk at the centre of the screen for 300 ms (

The factors operation (addition, subtraction), delay (bins around 150, 300, 500 ms) and target side (left, right) were varied within subjects. All condition combinations were repeated 27 times (i.e. 162 trials per operation). Additionally, 72 catch trials (no target) were included to ensure participants’ attention. Hence, the experiment consisted of a total of 396 trials.

Reaction time (RTs) measurement in the target detection task started with the presentation of the target. We eliminated RTs that deviated more than two standard deviations from the subject’s mean (4.25%) or that were shorter than 200 ms (0.1%). For inferential analysis, RTs were log_{10}-transformed because the distribution of non-transformed RTs is not symmetrical.

For the analysis of the arithmetic task, catch trials were included as they contained an arithmetic task response worth investigating. As the ROs were jittered, the correct RO did not contain the same amount of dots for all repetitions of that task. Therefore, we calculated the mean of these amounts over all repetitions in the experiment, so that the “correct value” in the result section constitutes the mean of the correct amounts. The correct value and the chosen value by the participant were log_{10}-transformed. Whenever Mauchley’s test of sphericity indicated a violation of the sphericity assumption, the Greenhouse-Geisser correction was used. Raw data are available as

In no-target (catch) trials the error rate (i.e. false alarm rate) was around 0.28%. For the further analysis of the target detection task, only left- and right-sided target trials (i.e. no catch trials) were considered. We predicted an interaction between target side and operation in the form of faster RTs for left-sided targets in subtraction trials and faster RTs for right-sided targets in addition trials. Mean RTs (and

Target side | Addition |
Subtraction |
||||
---|---|---|---|---|---|---|

150 | 300 | 500 | 150 | 300 | 500 | |

527 (170) | 477 (151) | 475 (160) | 508 (174) | 481 (153) | 471 (152) | |

511 (165) | 488 (168) | 475 (165) | 495 (161) | 470 (150) | 462 (160) |

We needed to examine whether the participants chose among the four ROs randomly. A non-random distribution (centred around the correct value) would indicate that they did indeed base their judgments on the arithmetic problem at hand.

In line with previous results, we observed that responses for addition problems were on average more accurate compared to subtraction. Two observations underline this. First, on average the correct response alternative was chosen significantly more often in addition (

For the analysis of the Operational Momentum (OM) effect, we calculated the response bias as the difference between the logarithm of the chosen value and the logarithm of the correct value.

Even though, we found no operation-dependent spatial bias in the target detection task, it is conceivable that the arithmetic operation had an impact on the locations of the ROs chosen (SOAR effect; cf.

In this exploratory analysis we analysed the response bias (defined as the difference between the logarithm of the chosen value and the logarithm of the correct value) and a CV_block variable (CV =

This study set out to investigate spatial biases and attentional shifts in the context of approximate addition and subtraction processing. We sequentially presented participants with arithmetic task components in the form of dot arrays. In the calculation phase, i.e. before the presentation of four ROs of which the correct solution had to be chosen, a target detection task was used to measure the locus of spatial attention.

The analysis of the target detection task showed no operation-dependent differences in target detection times for left-sided and right-sided targets. Hence, this experiment did not reveal spatial biases in approximate arithmetic in the target detection task. An increase in chosen numerosities and in response variability as a function of the correct value indicates that the ROs were not chosen randomly. We observed a response bias, defined as the difference between the chosen option and the correct result, that differed significantly between the operations (OM effect). Participants tended to select smaller ROs than the correct result in subtraction trials (underestimation). A block-wise analysis further revealed that this pattern of underestimation gradually develops over the course of the first blocks of the experiment and then remains nearly constant.

We also found that the arithmetic operation influenced the location of the ROs chosen: Participants preferentially selected left-sided ROs in subtraction trials, and right-sided ROs in addition trials. This SOAR effect implies some form of operation-dependent spatial bias. These findings are evocative of the observations made by

Although some spatial bias was observed in form of a SOAR effect, we hypothesised that providing feedback might increase overall accuracy and calibrate the responses on the correct outcome which would potentially help overcoming the overall underestimation bias. It is known from

Twenty-one participants took part in the experiment (_{age} = 25.67 years, _{age}

Experiment 2 was completely identical to Experiment 1 except for the fact that practise trials involved feedback and that the number of practise trials was increased. The feedback involved the presentation of a green frame for 2 seconds around the correct option once the participant had given a verbal response. Each practise block consisted of 25 trials drawn randomly from the set of experimental trials. Three practise blocks were mandatory and two further blocks were optional. Again, participants were instructed to press the space bar with their preferred hand once a target was detected in the target detection task (in one case the hand used did not match the self-reported handedness). Before the practise blocks, they were informed that after each response to the arithmetic task, a green frame around one of the four ROs would indicate the correct solution.

For the analysis of the target detection task, again, we eliminated RTs that deviated more than two standard deviations from the subject’s mean (4.26%) or that were shorter than 200 ms (0.12%). In Experiment 2 (in contrast to Experiment 1) we also analysed the practise data because it contained feedback and was, therefore, deemed worth investigating (see “Arithmetic Task: Effect of feedback”). We failed to collect the practise trial data of one subject, so that for the analysis of the arithmetic performance within the practise (feedback) blocks, we could only use the data of 20 subjects. The experiment offered five practise blocks, but only three blocks were mandatory. For that reason, data for the 4th and 5th practise block was not available from all subjects. In fact, only three subjects made use of the 4th block and no subject used the 5th practise block. Therefore, we only analysed the arithmetic performance data of the mandatory three practise blocks. Raw data are available as

In no-target (catch) trials the error rate (i.e. false alarm rate) was around 0.37%. For the further analysis of the target detection task, only left- and right-sided target trials (i.e. no catch trials) were considered. As in Experiment 1, we predicted an interaction between target side and operation (see

Target side | Addition |
Subtraction |
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150 | 300 | 500 | 150 | 300 | 500 | |

468 (151) | 455 (151) | 436 (141) | 469 (144) | 448 (157) | 434 (154) | |

459 (140) | 440 (137) | 423 (133) | 467 (144) | 437 (143) | 426 (136) |

Similar to Experiment 1, in addition tasks the correct RO was chosen significantly more often (

To investigate whether feedback during practise blocks of Experiment 2^{1}

Note, that contrary to Experiment 2, in Experiment 1 no practice trial data was collected. Furthermore, it is important to point out that trial amounts slightly differed between practice and experimental blocks in both experiments: Experiment 1 included one mandatory and one optional practice block á 30 trials. Experiment 2 included three mandatory and two optional practice blocks á 25 trials. Both experiments consisted of 11 experimental blocks á 36 trials. Consequently, practice and experimental blocks are not fully comparable.

had an impact on arithmetic performance we focussed on the response bias (defined as the difference between the logarithm of the chosen value and the logarithm of the correct value) to quantify the OM effect and a CV_block variable (CV =To examine whether the results differed between experiments, we performed two mixed ANOVAs on the response bias variable (OM effect) and the response location variable (SOAR effect) data of both experiments. With regard to the OM effect, the ANOVA on the response bias variable revealed a significant effect of operation,

In order to examine the spatial distribution of responses and their operational origin, we pooled the data of both experiments and performed four separate, directional t-tests against zero. In additions, participants chose significantly less options from the left-hand side,

In an exploratory analysis we analysed the response bias (OM) depending on whether the previous trial (n-1) involved the same or different operation as the current trial (repeat trial: Add → Add, Sub → Sub; switch trial: Add → Sub, Sub → Add)^{2}

The switch trial property was not experimentally varied, but the data involved a nearly equal amount of switch (.52%) and repeat trials (.48%).

. We observed a significant interaction between the operation and the switch trial property,In Experiment 2, we provided feedback in the practise blocks to assure that participants performed the approximate calculation adequately. No operation-dependent effects were observed in the target detection task (as in Experiment 1). Additionally, the feedback did not lead to an improvement in the arithmetic performance. The remaining performance pattern matched the results from Experiment 1 extremely closely. Participants selected smaller numerosities as correct results in subtraction tasks compared to addition tasks (OM effect) and preferentially selected left-sided ROs in subtraction trials and right-sided ROs in addition trials (SOAR effect). A pooled analysis over both experiments revealed increased OM effects when the operation is repeated over trials.

Similar to Experiment 1, we found that performance in subtraction trials was initially centred on the mean outcome and only successively became biased towards underestimation.

This study set out to investigate spatial biases and attentional shifts in the context of approximate addition and subtraction. In two experiments, participants were presented with non-symbolic addition and subtraction problems and had to choose the correct solution amongst four ROs. Spatial attention was measured via a target detection task that was presented after the arithmetic task and before the RO presentation. In the second experiment we introduced a feedback during initial training trials to increase the reliance on approximate calculation and its accuracy.

We replicated the OM effect in both experiments (

How can this pattern of results be linked to the models explaining the OM effect? The models differ regarding their predictions of spatial biases: The compression account and the heuristics account do not predict spatial biases, while the AHAB model contains a spatial component (sign-space association). Note however, that the present study did not involve operation signs but the letters “A” and “S”, respectively. Nevertheless, if this component would be generalised to an “operation-space association” it would still predict a spatial bias. The heuristics and compression accounts could make spatial predictions under the additional assumption of magnitude-space associations. Hence, all models could potentially predict operation-dependent spatial biases, i.e. effects in the target detection task and the arithmetic choice locations (SOAR). But this is not what we found.

This prompts the question of why spatial biases have been observed in one task (arithmetic task: locations) but not the other (target detection task). We would like to discuss two possible explanations. First, the way we realised the target detection task might not have been sensitive enough to measure spatial biases. Previous studies using this paradigm differ largely with respect to the timings (target onset and durations), use of no-target trials and response (to target or target side). In the present study, the choice to leave the target on screen until the response might have impeded the detection of RT effects: Participants had no incentive to react as fast as possible because even if they didn’t detect the target at first, they would at some point. However, previous experiments (

Another possible explanation might be that the time window of the target detection task did not capture the approximate calculation process. Attentional modulations of activity in posterior parietal cortex and of reaction times have been observed in response to the presentation of an arithmetic operator. For example, Mathieu and colleagues (

The time window might also differ as a function of the type of response, depending on whether a free response (e.g. verbal input or dot production) must be given or a forced-choice decision has to be made on one or multiple ROs. The present study involved an approximate arithmetic task in combination with a forced-choice response (choice among four ROs). In that case, it is plausible that actual processing - i.e. operation upon the numerical representation - occurred only when the ROs were presented and the correct solution had to be selected. Consequently, spatial biases should have been observed in the RO presentation part of the arithmetic task but not before. This is exactly what we observed: Spatial biases were not observed before the ROs via the target detection task but during the RO presentation phase via the arithmetic task response. Future studies need to systematically measure spatial attention during the arithmetic response, i.e. during the RO selection phase in non-symbolic approximate arithmetic.

Another major finding of the present study is that both experiments did not differ in their result patterns even though the second experiment involved feedback during the practise blocks. In contrast with the hypotheses, however, participants’ performance (accuracy and CV) did not improve via feedback, and the OM and SOAR effect remained the same. This means that the observed effects are relatively invariant to feedback. Two potential explanations for the lack of improvement in arithmetic performance after feedback can be discussed.

Firstly, it is plausible that due to its shortness, the feedback used in the present study was not efficient enough to improve arithmetic performance. Other studies involving some form of training often used multiple training sessions (e.g.

Secondly, simply relying on a feedback might not have been sufficient to induce improvements of performance. Several studies investigated the effects of correct/incorrect feedback on approximate numerical processing or approximate arithmetic - with mixed results: Two studies described improvement in symbolic arithmetic after approximate arithmetic training (

Furthermore, earlier studies investigating the processing of non-symbolic stimuli were in fact able to observe reduced variability after providing feedback of the real numerosity of stimuli (

Nevertheless, it is important to point out that studies by

The two experiments of the present study further revealed that the underestimation in subtraction trials gradually emerged over the first blocks of the experiment. In Experiment 1, where no practise trial data was collected, participants performed fairly accurate in the first experimental block and then tended to progressively underestimate subtraction results. In Experiment 2, this pattern was already detectable in the practise trials – despite them involving feedback. We did not expect this surprising finding. The current study is the first to describe this type of time-resolved performance in the context of a non-symbolic arithmetic task. Hence, we cannot know whether this is something seen frequently or not. We can only speculate what might have caused participants in subtraction trials to choose values that become smaller over time. The OM effect might require a contextual build-up phase in which participants create an internal distribution of the stimuli that are used in the experiment and elaborate their strategy. In our study, the operands were identical for addition and subtraction. Hence, participants may have learned over time that the average outcome for subtraction is smaller compared to addition which may have biased their decision. Note, that the CV remains constant throughout the entire experiment for both addition and subtraction, meaning that participants did not become increasingly inaccurate. Rather, the overall mean of the preferred outcome with respect to the correct outcome stabilizes only after ~75 trials during which the OM effect increases. This suggests that the OM effect builds up over time by biasing the decisional processes without affecting the precision of the perceptual basis. This is a new and exciting discovery that requires a systematic investigation in future experiments. It is unclear, for example, why this was observed only for subtraction. One potential reason might be that the approximate addition of visual stimuli is easier than approximate subtraction because such stimuli might be visually superimposed in visual working memory. Approximate subtraction, on the other hand, involves a more complex mechanism of mentally erasing stimuli from the mental representation of the minuend. What is more, real-life subtraction usually involves the “disappearance” of a certain amount of objects – whereas the operationalisation of approximate subtraction in the current study involved the additional presentation of a second visual stimulus as a subtrahend. As this mechanism deviates from the real-life visual subtraction, it might be more difficult for participants to achieve. Consequently, subjects might fall back to a simple heuristic of choosing the smallest option leading to the underestimation bias. Of course, this remains highly speculative, and it does not explain why participants start off fairly accurate in approximate subtraction. Future studies are needed to systematically investigate the differences between approximate addition and subtraction.

Finally, the joint analysis of both experiments revealed sequential dependencies in the form of increased OM biases in cases where the operation was the same as in the previous trial compared to when it was different (switch). This finding might indicate that more mechanisms than only attentional shifts are involved in the formation of the OM. This is an exploratory finding which needs to be treated with caution as the switch property was not experimentally manipulated. Nevertheless, it provides an interesting new field of research. In a variety of tasks, the perception of a current stimulus is modulated by previous perceptual history. Bayesian theories of perception propose that the perceptual history serves to predict current perception via changes of perceptual priors. One possible perceptual mechanism that could explain the observed increase in the OM bias on repeated operation trials and that does not require additional theoretical assumptions for explaining the OM effect operates on the perceived numerosity (

We corroborated these results by comparing the relative importance of both variables (by partitioning of the total explained variance (^{2}) of the model into individual ^{2} contributions; see ^{2}) for 29% and 34% of the variance in switch and repeat trials, respectively. Of these, the choice in the previous trial explained more variance (i.e., was a better predictor) compared to the mean numerical size of the response alternatives (81% vs. 79% for repeat trials and 80% vs. 20% for switch trials).

Based on these (exploratory) results, we tentatively conclude that the locus of the amplification of the OM bias is at a central cognitive level, rather than on the perceptual side of the process. It may be interpreted as the consequence of an inhibition of the irrelevant task set (here: the alternative arithmetic operation;

The present study investigated spatial attention in the context of approximate addition and subtraction. While no shifts could be observed via the target detection task, participants preferentially selected right ROs after addition and left ROs after subtraction processing (SOAR effect) implying a spatial bias in the context of approximate calculation during the RO selection stage. The feedback introduced in the second experiment in the form of highlighting the correct arithmetic answer after the participant had given their response during the practise trials, did not improve arithmetic performance. Consequently, the pattern of results was identical to the first experiment. Put positively, the observed effects and biases are robust and not easily malleable by feedback. The newly described serial dependency effect and the built-up of the OM effect over trials point to additional cognitive factors that require a more systematic exploration in future studies.

André Knops is the Editor-in-Chief of the

For this article, a data set is freely available (

The Supplementary Materials contain the following items (for access see

Raw data and codebook

Arithmetic stimuli used in both experiments. The stimuli were based on the stimuli of

The authors have no funding to report.

The authors have no additional (i.e., non-financial) support to report.