1.1 Theoretical accounts of PES

The Dual Mechanisms of Control (DMC) framework proposes that cognitive control comprises reactive and proactive components which can be used adaptively by adults depending on the demands of a task (Braver, 2012). With proactive control, individuals anticipate and prepare to respond to upcoming events. This approach can reduce detrimental interference during the events but is quite demanding on working memory capacity. In contrast, reactive control allows individuals to respond to unforeseen events, such as an error. The reactive mode is less demanding on working memory but also less efficient and takes time and may therefore lead to PES. Two main theoretical accounts of PES have been proposed. The functional account argues that PES serves the purpose of taking more time to plan an action to prevent future errors and increase performance. This is supported by the conflict monitoring theory (Botvinick et al., 2001), which states that slowing down reflects increased reactive control evoked by a conflict, in this case, an error. People monitor their performance and consequently adjust their response thresholds leading to slower but more accurate responses. This was also shown by Dutilh et al. (2012b)’s drift-diffusion model, where participants increased their boundary separation, the model parameter for response caution, after making an erroneous decision. A more cautious response leading to more accurate behaviour in post-error trials is thought to underly adaptive behaviour (Ridderinkhof et al., 2004), although a coupling of post-error changes in accuracy (PEA) and reaction time is not always found (for a review, see Ullsperger et al., 2014). An alternative explanation supporting the functional account is considering PES as the product of automatic inhibition of the next response after an error (Gupta et al., 2009; Marco-Pallarés et al., 2008), where inhibitory mechanisms increase response cautiousness after making an error.

Other studies have suggested a non-functional account of PES, where an error is thought to have a negative effect on subsequent post-error performance. The orienting account argues that PES occurs especially with infrequent errors since the attention orients towards this surprising event instead of the task (Houtman et al., 2012; Notebaert et al., 2009). This attention shift disrupts the information process, inducing slower responses and worse performance. Notably, Danielmeier and Ullsperger (2011) point out in their review that there is evidence for both the functional and non-functional account and that they are not necessarily mutually exclusive, as multiple mechanisms may contribute to PES.

Although PES effects have been presented as markers of adaptive behaviour, the majority of studies are done with relatively simple and rapid tasks such as the flanker task (e.g. Schroder et al., 2019), or the Go/No-go task (e.g. Jonker et al., 2013). These tasks are limited in terms of the adaptation strategies they allow and performance can only be increased by paying more attention to the stimuli, in line with Notebaert's (2009) conflict monitoring theory. However, there are studies that have investigated PES in more complex settings with adults, for example using a grasping task (Ceccarini & Castiello, 2018). In academic context, studies have also used mental arithmetic tasks with university students and found that multiple strategies were available, with larger response times after an error than after a correct response (Borght et al., 2016; Desmet et al., 2012; Lavro et al., 2018b; Núñez-peña et al., 2017). Interestingly, Borght et al. (2016) found that switching to a different strategy was associated with an increase in PEA and reduced PES. Recently, children aged 4 through 15 were tested outside the laboratory in their Montessori classrooms using a flanker task and showed post error slowing with increased self-monitoring performance later (Denervaud et al., 2020).

1.2 Development of PES

Depending on the context, adults flexibly switch to the most adaptive mode of control, either reactive or proactive (Chiew & Braver, 2013). Cognitive control is an ability that children are still developing: younger children seem to mostly rely on the reactive control mode (even when the proactive control is more efficient), but begin to engage in proactive control from the age of 8 years (Chevalier et al., 2015; Niebaum et al., 2020). The development of cognitive control is not only associated with the improvement of these core cognitive processes, but also with improvements in the adaptive selection of a mode of control to engage in in a specific context or point in time. The development of error monitoring as measured by PES has mostly been examined using neuroscientific methods with children between 5 and 12 years of age (for reviews, see Ferdinand & Kray, 2014; Tamnes et al., 2013). Some behavioural studies such as Fairweather (1978) found that young children from the age of 5 slowed down after making errors during a two- to eight-choice RT task. Ever since, PES in children has been repeatedly examined and observed (Berwid et al., 2014; Fairweather, 1978; Gupta et al., 2009; Jones et al., 2003; Schachar et al., 2004; Smulders et al., 2016). Following the orienting account where errors lead to interference, children are thought to be more prone to interference than young adults and therefore also exhibit more PES (Smulders et al., 2016; van der Molen, 2000). Conflict arising from interference is also related to children's still-developing inhibitory control skills (Durston et al., 2002; Garon, Bryson, & Smith, 2008; Welsh, Friedman, & Spieker, 2006; B. R. Williams, Ponesse, Schachar, Logan, & Tannock, 1999). This indirect relation implies that improving inhibitory control skills would therefore lead to a reduction in PES magnitude. On the other hand, greater PES may also reflect a developmental increase in cognitive control, with greater performance monitoring and strategic adjustment of the balance between proactive and reactive control, according to the functional account (Jones et al., 2003; Tamnes et al., 2013).

The directionality of changes in PES during development described in the literature is heterogeneous. In recent research, Smulders et al. (2016) used standard two-choice RT tasks and found that post-error response slowing was present from 5 years to adulthood, i.e., the whole of their examined age range. They reported PES stability with age rather than developmental differences. Jones et al. (2003) reported increasing PES from 3- to 4-years of age using a Simon Says inhibitory control task. These changes were interpreted as reflecting a developmental increase in cognitive control. The results of Gupta et al. (2009) showed a non-linear development of PES in 6-11-year-old children performing two task-switching digit tasks, with a peak in magnitude of PES at age 7. Schachar et al. (2004) also found a decreasing PES magnitude from 7 to 16 years of age in a stop-signal task.

1.3 Factors influencing the presence or magnitude of PES

Prior research has shown that PES varies as a function of the type of task as well as the difficulty of the task. Although PES has been found in both easy tasks (e.g., flanker task, Schroder et al., 2020) and more complex tasks (e.g., mental arithmetic task, Desmet et al., 2012), the non-functional account noted that response slowing after an error occurs mostly under conditions when errors are rather unexpected and infrequent events (Danielmeier & Ullsperger, 2011; Lavro et al., 2018a; Notebaert et al., 2009).

Another factor influencing PES is the type of error that preceded it. Errors can be found to be systematically faster or slower than correct responses, with a range of underlying causes (Ratcliff & Rouder, 1998). Slow errors typically occur when accuracy is emphasized and the task is relatively difficult, whereas fast errors occur when responding is rushed. Damaso et al. (2020) categorised fast and slow responses as the 50% slowest and fastest responses of each participant in two simple recognition memory experiments. Results showed that PES mostly occurred after fast errors (Damaso et al., 2020).

1.4 The current study

In the present study, we investigated students’ post-error performance in a large scale online adaptive learning environment for primary school children aged 5–13 years old. The first aim of this study was to assess the presence of PES in the various learning activities relating to different mathematical and language skills. In addition, we examined whether PES associated with increased PEA, which was expected if PES functionality is adaptive of nature, but not if PES reflected interference, as proposed in the orienting account.

The second aim was to investigate predictors of the magnitude of PES and of the association between PES and PEA. At the task level, we investigated whether the type of skill practised (mathematics vs. language) was associated with the magnitude of PES. In the learning environment used for this study, children can choose not only between different types of games but also the difficulty level of the games (hard, medium and easy), which relate to the probability of solving the items correctly based on their ability (Brinkhuis et al., 2018; Jansen et al., 2013). This allowed us to test the prediction, based on previous research (Danielmeier & Ullsperger, 2011; Lavro et al., 2018a; Notebaert et al., 2009), that children who choose the easy level, where there is less chance of making errors, were more likely to show PES after an error than children playing the medium and hard level, as the errors were unexpected.

Finally, we investigated the influence of the speed of responding on the PES. We first assessed the effect of time pressure (i.e., how much time children were given to respond in each task) on PES magnitude. Secondly, we distinguished fast and slow errors by categorising whether RT was higher or lower than the median split of the correct trials before the error within a task (Damaso et al., 2020) – to control for, global fluctuations in skills and motivation of the participant. The prediction was that trials with fast errors would lead to greater PES.

We also assessed individual differences between participants as potential predictors of PES as well as of the PES-PEA association. We first examined associations with age. Given the development of cognitive control, it was expected that younger children would show more PES, reflecting greater reliance on reactive control, which is not necessarily beneficial for accuracy. Older children were expected to show a greater PES-PEA association, reflecting their improving proactive control skills. Second, we investigated how individual differences in PES may associate with children's ability in the task examined. This has not been investigated before, however, large differences in mathematical abilities occur within school grades (Straatemeier, 2014), so only considering the age of children may not be a good proxy for comparing children's developmental stage. Therefore, we predicted that ability on a particular task would be associated with PES and PEA above and beyond age.

2.1 Participants

The response data of 149,747 Dutch primary school children playing in the learning environment were collected. Their age was between 5 and 13 years old (M = 9.4 y, SD = 1.8 y, 48.6% female). Primary schools buy accounts for students to practise their language and mathematical skills in the learning environment, while their responses are logged for scientific purposes such as this study. Children (their parents or schools) can opt out of being part of the research done in the learning environment, in which case they were not included in this study. All anonymized data are available to researchers, and access to the data can be acquired by contacting the first author.

2.2 Materials and equipment

2.3 Procedure

Participants decide for themselves when they play, how long they play for and which learning activities they want to complete. Data from a total of 45 million trials were collected in 6 months (June 2019 until November 2019) from 23 different learning activities (13 mathematics-related and 10 language-related, see Appendix A.1 for more details about the learning activities). Since the students decide themselves which learning activity they participate in and these learning activities vary in which minimum age is required to practise, the number of participants and the average age is different for each activity (see Appendix A.1 for details). Trials with an RT faster than 200 ms were regarded as guess responses and were therefore excluded. In the learning environment, children are discouraged to guess due to a visible and direct penalty of fast and incorrect responses. Trials where no response was made within the time limit were labelled as missing.

To ensure a reliable measure of post-error behaviour, RTs were selected from a specific pattern of response. In a game session, a sequence of four problems was selected when the accuracy pattern was 1-1-0-1 (1 = correct; 0 = incorrect), such that a minimum of two correct responses precede an error and the trial after the error is correct. The additional correct pre-error response was added to the response sequences to ensure that a correct post-error response in a session could not at the same time be a correct pre-error response. In the 45 million trials, 8 million of such sequences were found. Similarly for the PEA measure, all occurrences of the pattern 1-1-0, i.e., a minimum of two correct responses followed by an error, were identified in the same dataset. Next, a proportion PEA was computed based on the accuracy of the first trial following this pattern for each learning activity and each participant.

2.4 Measures

2.4.1 Post-error slowing quantification

The majority of previous studies quantify the magnitude of PES as the difference in mean RT between trials following an error and trials following a correct response. As Dutilh et al. (2012b) pointed out, this method can be confounded by the global fluctuations in ability and motivation during the task, since post-error responses are more likely to originate from the second half of a task where responses are inevitably slower due to motivation and tiredness. Dutilh's solution is to quantify PES as the average, across the selected sequences, of the difference between the RT after the error (RT_E+1) and the RT before the error (RT_E-1) (PES_diff):

(1)

To ensure that our results do not fully depend on the choice of this absolute PES measure, we use two additional methods. We also report PES relative to the overall ability speed, calculated by dividing the RT difference with the average speed of the two trials (i.e., ). This ensures that individual differences in PES magnitude can be compared regardless of the overall speed/ability of the student:

(2)

The third method is a robust way of measuring post-error behaviour overall, by quantifying PES as the number of sequences where the RT was larger after the error (RT_{E + 1}) than before the error (RT_{E − 1}), relative to the number of sequences (N). This is a measure that is not affected by the PES effect size in certain sequences, which can vary considerably in such a big dataset. The disadvantage of this robust method is its low power.

(3)

2.5 Analyses

The main analysis focused on whether PES could be found for different learning activities. To do this, a linear mixed model was performed in R (Team, 2013) using the package lme4 (Bates et al., 2015) and lmerTest (Kuznetsova et al., 2017) to calculate p-values. Participant was treated as a random effect because the number of sequences collected per participant differed as participants can play whenever they want and as much as they want. Learning activity was treated as a fixed effect to investigate whether there were differences in PES effect per learning activity. When a general PES effect was found, the PEA measure was added along with a variety of predictors to a basic linear mixed model where both learning activity and participant were treated as random effects, to account for variance between the tasks and participants. A chi-square test was used to see if the added predictors explain PES better.

The first categorical predictor added, was the type of learning activity, distinguishing between mathematical and language skills. The second task predictor was time pressure since the tasks differed in the time limit given to participants to solve a problem, from 8 to 60 s (see Appendix A.1). The third categorical predictor, difficulty level (easy, medium, hard level), reflected the probability of answering correctly chosen by the child. At the participant level, we also included age (5-13 years old) and child's ability level (a continuous scale from −10 to 10) as predictors of PES magnitude. These last two predictors were analysed in a linear and quadratic fashion since the study of Gupta et al. (2009) also found a non-linear development of PES with age. The ability level of a child in each learning activity is provided by the learning system itself. It is estimated with an adaptive Elo algorithm and is updated after every item. Lastly, type of error was investigated at the trial level, where a comparison of the magnitude of PES was made between fast and slow errors. All errors were categorised as fast error when the RT was lower than the median RT of the correct items before the error or as slow error when the RT was higher than the median RT.

3.1 Overall PES effect

To measure whether there was PES in the learning activities, a linear mixed model was fitted to the data with learning activity as fixed effect and participant as random effects. Using the absolute PES_diff measure we found that in 20 of the 23 learning activities participants showed a significantly larger RT after an error than before, with a mean difference across learning activities of 225 ms (Figure 1 and Appendix A.2 for details on the statistics). The average age of participants differed considerably between the learning activities (8.2 – 10.8 years), but this did not account for the variation in PES magnitude between the learning activities, ̅r = -.08, p = .70. For the PES_rel measure we found the same pattern, where the learning activities letterchaos and spelling did not show PES but post-error speeding (Appendix A.2). The learning activity practising grammar was not significantly different from zero in PES_diff.

Since PES_robust is a proportion measure, we performed a separate proportion test for every learning activity, testing whether RTs were greater after an error than before on more than 50% of trials, corresponding to a mean proportion score greater than 0.5. The results were in line with the linear mixed models. The average proportion across learning activities was .52 [range .48 – .55], meaning that in 52% of the sequences the RT after an error was greater than before the error.

3.2 PES-PEA association

PES estimates per learning activity were expected to be positively associated with PEA, reflecting an adaptive response. This association was analysed with a one-sided Pearson correlation test. Only the learning activities with a PES_diff greater than zero were analysed since these showed PES. The test indicated a positive PES-PEA association, r = .437, t(19) = 1.72, p = .049, such that the learning activities showing greater PES also showed greater post-error accuracy (Figure 2A). A linear mixed model of PES data at the trial-level with PEA per learning activity and participant as random effect predictors also predicted PES magnitude, β = 96.90, t(121000) = 2.36, p = .018.

3.3 Type of learning activity and time pressure

Next, a variety of predictors were added and compared to a basic model with participant and learning activity as random effects to see whether these predictors could explain variance in the magnitude of PES. Learning activity PEA was also included as a fixed effect in all the analyses, to look at whether the predictors influenced the PES-PEA association. Since the different PES calculations showed comparable results, we only discuss PES_diff.

We first investigated task level predictors of PES, to understand why the learning activities differed in PES magnitude. We found that type of skill practised (mathematics vs language) significantly predicted PES magnitude. The Tukey comparison test showed that greater PES was shown in learning activities practising mathematical skills (M = 276 ms, SE = 12.4 ms) than language skills (M = 82.3 ms, SE = 11 ms), p = .02 (Figure 1). The type of skill also moderated the PEA-PES association, p < .001. A separate linear mixed model for learning activities with mathematical skills showed no PES-PEA association, but for the learning activities practising language skills a higher PES average associated to greater proportion post-error correct, β = 447.80, t(2011) = 5.59, p < .001.

As an addition to this first model, we found that how much time participants were given to respond on each trial (time pressure) predicted PES, such that greater PES was found in learning activities with more time to respond, β = 142.8, t(2942) = 8.27, p < .001. Moreover, time to respond moderated the PES-PEA association positively, β = 57.5, t(25027) = 13.36, p < .001, meaning that with more time to respond participants showed more PES along with greater post-error accuracy (Figure 2B). The time to respond did not differ between the learning activities practising language (M = 26.0 s) and mathematical skills (M = 29.2 s), p = .45. Time pressure also did not interact with the type of skill practised, p = .92.

While all learning activities showed on average slower RT on error trials than on correct trials, there was an association between time to respond and error RT, such that longer time to respond associated with slower error RTs, β = 0.46, t(1519124) = 422.8, p < .001. Furthermore, when there was more time to respond, participants also exhibited slower errors compared to the RTs of the correct trials, β = 0.15, t(1519124) = 189.1, p < .001.

In addition, at the trial level, a model including the type of error (fast vs slow errors) was found to be a better model to explain PES than the basic model, χ² =  380.96, p < .001. Here, participants showed greater PES effect after slow errors (M = 276 ms, SE = 11.6 ms) than after fast errors (M = 147 ms, SE = 12.4 ms), in line with what was found at the task level.

3.4 Difficulty level (chosen by the participant)

At the participant level, we examined whether including the difficulty level chosen by the student (easy, medium, hard), which affects the selection of items and hence the error rate, would change the basic model (participant and learning activity as random effects) fit. There were no significant differences in age between the children choosing the difficulty levels. We found that adding difficulty level as a predictor significantly improved the model, χ² =  205.08, p < .001. Figure 3 shows that participants choosing the most difficult level in learning activities also had the greatest PES effect (M = 270 ms, SE = 12.6 ms) compared to participants choosing the medium level (M = 208 ms, SE = 11.8 ms) and easy level (M = 129 ms, SE = 10.8 ms). Follow up comparisons using a Tukey test showed that participants choosing the hard level had a significantly greater PES than participants choosing the medium and the easy levels, p’s < .001, and children choosing the medium level also showed significantly greater PES than children choosing the easy level, p = .006. To examine the PES-PEA association in comparison to the difficulty level, the expected proportion correct given the difficulty level was subtracted from the post-error correct metric. There was no interaction between the difficulty level and the PES-PEA association, such that even though participants showed more PES in the hard level, these participants did not show significantly more improved post-error performance in comparison to the participants in the other levels. But the PES-PEA association remained significant when controlling for difficulty level, β = 131.4, t(122982) = 3.2, p = .001.

3.5 Age and ability

To investigate developmental differences in PES, we examined the association of PES magnitude with children's age and ability level at a participant level, above and beyond PEA. We found that age predicted PES linearly (β = −50.6, p < .001) and quadratically (β = −26.5, p < .001), where the magnitude of PES increased from 6 to 9 years old and decreased from 9 to 13 (Figure 4A). In the same model, the children's ability level was found to predict PES effect positively in a linear way (β = 156.6, p < .001), but not quadratically, p = .60 (Figure 4B). Neither the age nor the ability level of the child moderated the association between PES and PEA, age p = .26, ability level p = .07, but PEA remained a significant predictor of PES in the model, β = 130.1, t(358880) = 3.15, p = .002.