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This study investigated whether sequential difficulty effects emerge during processing of a mixed set of small, easy and large, more difficult arithmetic problems. Furthermore, we assessed if these sequential difficulty effects are reflected in event-related (de-)synchronization (ERS/ERD) patterns. To this end, we analyzed data of 65 participants, who solved two separate blocks (additions and subtractions) of arithmetic problems while their EEG was recorded. In each block, half of the problems were difficult problems (two-digit/two-digit with carry/borrow), and the other half were easy problems (one-digit/one-digit). Half of the problems were preceded by a problem of the same difficulty (repeat trials), and half were preceded by problems of the other difficulty (switch trials). In subtractions a sequential difficulty effects pattern emerged. Participants solved easy repeat trials faster than easy switch trials, while difficult repeat trials were solved slower and less accurately than difficult switch trials. In the EEG, we found the strongest effects in left hemispheric beta band (13–30 Hz) ERD. Specifically, participants showed a stronger beta band ERD in easy switch trials than in easy repeat trials. Furthermore, beta band ERD was stronger in difficult problems than in easy problems within repeat trials, but stronger in easy problems than in difficult problems within switch trials. In summary, our results are in line with the presence of sequential difficulty effects, as processing of easy and difficult problems was impaired if they were preceded by a difficult problem. Furthermore, these sequential difficulty effects are reflected in ERD patterns.

Sequential difficulty effects in arithmetic have been first described by

An open question however is, whether these patterns of sequential difficulty effects also emerge in a common difficulty switching process in mental arithmetic that occurs in everyday life, namely the switch between fact retrieval and procedural calculation. In general, arithmetic problems can be solved by application of one of these strategies. Fact retrieval, being the mainly applied strategy in small, easy problems (e.g. 3 + 5 = 8 or 2 * 4 = 8), means solving a problem by directly retrieving the solution from long-term memory – a fast, effortless, and highly accurate process. Procedural calculation, on the other hand, occurs primarily in larger, difficult problems (e.g. 35 + 17). For instance, persons break the problem down in a series of smaller steps to calculate the correct solution (35 + 17 could be broken down into 35 + 10 = 45; 45 + 5 = 50; 7 – 5 = 2; 50 + 2 = 52). This requires recalling the correct procedure, maintaining the interim results in working memory, and monitoring progress across the single steps. Thus, these two types of strategies strongly differ in complexity and difficulty, and the cognitive load put on executive control functions and working memory is a discriminating feature between them (e.g.

However, whether these ERS/ERD patterns related to fact retrieval and procedural calculation are also modulated by difficulty switches and consequently by sequential difficulty effects has not been investigated. A previous study on the neurophysiological correlates of sequential difficulty effects in mental arithmetic focused on event-related potentials (ERP;

The present study aims to extend the current understanding of sequential difficulty effects in mental arithmetic in two ways: First, by investigating sequential difficulty effects in switching between fact retrieval and procedural calculation, the study tests whether and to what extent previously observed sequential difficulty effects can be generalized to everyday arithmetic demands. And, second, by investigating their oscillatory EEG correlates, the potential impact of these effects on ERS/ERD patterns related to fact retrieval and procedural calculations can be uncovered. Notably, in previous oscillatory EEG studies on mental arithmetic, switching between these trial types is common but sequential difficulty effects on the ERS/ERD patterns have not been investigated so far.

To this end, participants solved two separate sets of arithmetic problems. One consisting of additions and one consisting of subtractions. In each set, half of the problems where easy, fact retrieval problems, and the other half were difficult, procedural calculation problems. Based on prior literature we expected better performance in easy repeat trials than in easy switch trials, but a better performance in difficult switch trials than in difficult repeat trials. In other words, the effects of difficulty switching should be asymmetric, consisting of performance impairments due to possible switching costs inflicted by the difference in difficulty plus sequential difficulty effects in the easy (fact retrieval) switch trials and performance impairments due to sequential difficulty effects in the difficult (procedural) repeat trials. At the electrophysiological level, we expected effects of sequential difficulty effects in the alpha and beta bands as these are related to attentional processes and executive functions (e.g.

The sample consisted of 72 adult students. Seven participants were excluded because they showed an accuracy more than two standard deviations below the mean in at least one type of arithmetic problems. Hence, the final sample consisted of 65 participants with a mean age of 25.98 years (

The arithmetic task (as well as the working memory task) was presented on a 24 inch screen (LG Electronis Inc., Seoul, South Korea) and controlled by PsychoPy software (

At the end of each arithmetic block, participants were asked separately for easy and difficult problems whether they used more fact retrieval or procedural calculations. Thereby, participants could indicate their strategy use by positioning a cursor on a bar ranging from

Accuracy and calculation times were used as performance markers. Accuracy was calculated as the percentage of correctly and timely solved trials. Calculation time was assessed as the mean time between start of problem presentation and the first button press, indicating that the participant had come to a solution. Calculation times above or below three standard deviations of a participant’s mean Calculation time in the given tasks were excluded from further analyses. On average, this led to an exclusion of 2.97% (

To assess participants’ WM, a 2-back letter task was used. The task had a duration of 180 seconds and each letter was presented for 0.5 seconds, followed by a blank screen for 1.5 seconds before the next letter appeared. Hence, the WM task consisted of 90 trials. The letters appeared in a pseudorandomized order to achieve a ratio of 30 target trials to 60 non-target trials. Participants had to press a button if the current letter matched the letter seen two trials ago (target trial) and to refrain from pressing any button if not. Hence, a correct reaction (CR) was present when a target item was displayed and the participant pressed the button and a correct rejection (CRJ) was present if a non-target item was presented and the participant refrained from pressing a button. A miss (M), on the other hand, was present if a target item was presented and the participant failed to press the button, while a false alarm (FA) occurred if a non-target item was presented and the participant pressed the button. Reaction time (WM-RT) was assessed for all correct trials and overall WM accuracy (WM-ACC) was calculated as the percentage of correct reactions and correct rejections over all trials by the equation WM-ACC = (1 - ((FA + M) / 90)) * 100. As for the arithmetic task, written instructions were given at the beginning and after ten practice trials, participants were given the opportunity to ask final questions.

Basic arithmetic abilities were assessed by the IST-2000R (

EEG was recorded using a BioSemi ActiveTwo EEG system (BioSemi, Amsterdam, Netherlands) with 64 channels and hardware low-pass filter of -3 dB at 1/5 of the sampling rate of 256 Hz. Electrodes were mounted in a BioSemi headcap and contact was established using Signagel (Parker Laboratories Inc., Fairfield, USA). Channels F3, P3, AF8, and F8 were not connected because at these positions transcranial electrical stimulation electrodes were mounted beneath the head cap. Stimulation was not used during the assessments for this study, but followed immediately after as part of another study examining the same participants. EEG data was analyzed using MNE (

Examination and EEG measurement took place in normally lit, quiet rooms, in a single person setting. Tests and questionnaires conducted before the EEG measurement consisted of a short demographic questionnaire, followed by a test of hand dominance (HDT;

In the arithmetic tasks, switching costs (and with them sequential difficulty effects) were calculated separately for easy and difficult problems as well as additions and subtractions by assessing the difference between switch and repeat trials. For both calculation times and accuracy, the amount of switching costs/sequential difficulty effects was calculated as the difference between performance in switch trials and performance in repeat trials (switch – repeat). Hence, for calculation time positive values indicate better performance in repeat trials and negative values indicate better performance in switch trials. For accuracy, on the other hand, positive values indicate better performance in switch trials, while negative values indicate better performance in repeat trials.

All statistical analyses were carried out using SPSS 25 (IBM Corporation, Armonk, USA). Behavioral data (calculation times and accuracy) were analyzed separately for additions and subtractions, using repeated measurement ANOVAs with difficulty (easy vs. difficult) and order (switch vs. repeat trials) as within-subject factors. ERS/ERD data was analyzed similarly with separate analyses for each frequency band using repeated measurement ANOVAs with difficulty (easy vs. difficult), order (switch vs. repeat trials), and location cluster as within-subject factors. Furthermore, analyses were conducted separately for left and right hemispheres as well as midline areas. Hence, the location cluster factor consisted of AF, F, FC, C, CP, P, and PO for the analyses regarding the left and right hemispheres and of AM and PM for the analysis of midline areas. Bivariate correlations were calculated to assess the associations between behavioral switching costs/sequential difficulty effects, WM, and IST arithmetic scores. For analyses including WM we had to remove one additional participant as response recording had not worked properly and accuracy and reaction times could not be assessed. Hence, sample size for analyses including WM is 64 instead of 65. The main research question was whether there are switching costs or sequential difficulty effects on a behavioral level when switching between fact retrieval (easy) and procedural calculation (difficult) and, if any, if they are reflected on a neurophysiological level. Hence, for the sake of brevity, ERS/ERD patterns and correlations were only analyzed for those arithmetic operations in which behavioral switching costs/sequential difficulty effects appeared. In all analyses applicable, Greenhouse-Geisser correction was applied if the assumption of sphericity was violated.

Means and standard deviations for accuracy and calculation times are given in

Type | Difficulty | Order | ||
---|---|---|---|---|

Accuracy | ||||

Additions | Easy | Repeat | 96.43 | 4.85 |

Switch | 95.67 | 4.81 | ||

Difficult | Repeat | 86.15 | 12.18 | |

Switch | 84.58 | 9.77 | ||

Subtractions | Easy | Repeat | 96.25 | 4.91 |

Switch | 95.87 | 5.21 | ||

Difficult | Repeat | 73.65 | 16.18 | |

Switch | 78.15 | 13.95 | ||

Calculation Times | ||||

Additions | Easy | Repeat | 0.74 | 0.14 |

Switch | 0.74 | 0.13 | ||

Difficult | Repeat | 2.49 | 0.64 | |

Switch | 2.50 | 0.70 | ||

Subtractions | Easy | Repeat | 0.82 | 0.19 |

Switch | 0.85 | 0.20 | ||

Difficult | Repeat | 2.94 | 0.77 | |

Switch | 2.83 | 0.68 |

For accuracy, the repeated measurements ANOVA showed a significant main effect of difficulty,

Similarly, for calculation times the ANOVA only showed a significant main effect of difficulty,

For accuracy, the repeated measurements ANOVA showed significant main effects of difficulty,

Analysis of calculation times in subtractions showed significant main effects of difficulty,

Results for the beta band are depicted in

In the left hemisphere, there was a significant interaction difficulty * order,

In the right hemisphere, there was also a significant interaction difficulty * order,

In midline areas, there was a significant main effect of order,

Results for upper alpha band are depicted in

In the right hemisphere only an effect of location,

Regarding midline areas, there were no significant effects of difficulty, order, location, or any interactions.

In the left hemisphere, only effects of location,

In the right hemisphere, the only significant effect was location,

Regarding midline areas, results were similar to the left hemisphere, with the only significant effects emerging from location,

In the left hemisphere, the repeated measurements ANOVA only revealed a significant effect of difficulty,

Similarly in the right hemisphere, there only was an effect of difficulty,

For midline areas, only difficulty showed an effect,

A larger amount of sequential difficulty effects in accuracy in difficult subtractions (higher accuracy in switch as compared to repeat trials) was significantly related to lower IST arithmetic scores (

Participants indicated that in additions they solved 87.8% (

There was no difference between the average durations of the inter-trial intervals after easy (

The main objective of this study was to investigate whether sequential difficulty effect patterns, which appear when switching between procedural calculation problems of different difficulty or applied strategy (

On the behavioral level, the main findings were sequential difficulty effect patterns in calculation times and, to a lesser extent, accuracy in subtractions. In line with our predictions, both easy and difficult subtractions were solved more slowly if they were preceded by a difficult problem. A partially similar pattern emerged in accuracy. In easy problems, however, there were no differences in accuracy between switch and repeat trials. Hence, for calculation time in subtractions and to a lesser extent also for accuracy, the results are well in line with prior literature on sequential difficulty effects (e.g.

In additions, however, we did not find any differences between repeat and switch trials. The absence in the more difficult addition problems is thereby partly in line with prior literature, as asymmetries are assumed to arise from less switch costs in the more difficult or less automatized task or strategy as compared to the switch costs in the easy or more automatized one (e.g.

An additional interesting aspect regarding sequential difficulty effects is whether these effects are related to arithmetic performance and WM. In case of basic arithmetic abilities, only the magnitude of the sequential difficulty effects in accuracy (less accuracy in repeat than in switch trials) in difficult subtractions was related to performance. However, basic arithmetic abilities were assessed by a single test consisting of a series of more complex problems. This might be a reason why only the sequential difficulty effects in difficult problems appear to be related. Nevertheless, this is an interesting new aspect because it indicates that the magnitude of sequential difficulty effects might relate to arithmetic performance in general (although it has to be mentioned that the correlation was rather low and has to be interpreted cautiously). Furthermore, there were no correlations between sequential difficulty effects or switching costs and WM performance. This is somewhat surprising, as

At the electrophysiological level, the sequential difficulty effects in subtractions were reflected in EEG, albeit only partially as expected. Based on prior research (

The most interesting results regarding sequential difficulty effects were found in the beta band, with the strongest effects in the left hemisphere. Specifically, ERD was stronger in easy switch than in easy repeat trials (with a trend to but not significantly more ERD in difficult repeat than in difficult switch trials). Similar to the upper alpha band, ERD was even stronger in easy switch trials than in difficult switch trials, but lower in easy repeat than in complex repeat trials. A comparable pattern emerged over the right hemisphere and midline areas, although only the differences between easy repeat and switch trials proved to be significant. Hence, regardless of difficulty of the trial at hand, ERD in beta band (especially in the left hemisphere) seems stronger in the trials preceded by a difficult one. A stronger decrease of beta band power in right frontal, parietal, and temporal areas during procedural calculations (difficult) as compared to fact retrieval (easy) has been assumed to reflect higher demands on an executive control network (

Additionally, we could replicate earlier findings of stronger theta band ERS in easy, fact retrieval problems as compared to difficult, procedural calculation problems (

There are some limitations in this study. While the differences in average accuracy and calculation times between fact retrieval (easy) and procedural calculation (difficult) problems were as expected (lower accuracy and longer calculation times in the difficult procedural calculation problems), we did not assess the applied strategy trial by trial, but as a general self-rating at the end of each block. Thereby, participants indicated that they solved easy problems primarily by fact retrieval, and difficult problems primarily by procedural calculation, but not all of them were solved by application of the assumed strategy. Hence, it is possible that not every switch of problem difficulty was accompanied by a switch of strategy and vice versa. Applying a trial by trial assessment of strategy use and only analyzing those trials in which the reported strategy fits to the given problem difficulty might lead to even clearer sequential difficulty effect patterns in future studies. Furthermore, in the current study participants had to calculate the results and were then presented with three possible answers to choose from. This is different from a production task, where participants have to produce the answer directly, used in some other studies (

Overall, these results extend prior knowledge by showing that switching between easy and difficult subtractions induces sequential difficulty effects and, more importantly, that these effects are reflected in differences in ERS/ERD patterns and relate to basic arithmetic performance. Especially the effects on ERS/ERD patterns are valuable information for future studies, as they show that arithmetic strategy effects found earlier (e.g.

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The authors have declared that no competing interests exist.

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