Based on the cognitive load theory (CLT) by Sweller et al. (1998), the worked example effect postulates that interspersing already worked-out solutions in a learning unit instead of solely presenting problem-solving tasks can be beneficial for knowledge acquisition (Sweller and Cooper, 1985). CLT understands learning as a process in which schemata (i.e., associative knowledge structures) are either constructed by linking new information elements with prior knowledge in working memory or they get automated through repeated recall. A central assumption of CLT is that the capacity of working memory, in contrast to long-term memory, is severely limited in terms of the amount that can be processed simultaneously. Another important assumption of CLT is the distinction between different types of cognitive load, which have an additive character concerning the capacity of working memory. Recent conceptions of the CLT distinguish between the intrinsic cognitive load and the extraneous cognitive load (Sweller, 2010; Kalyuga, 2011; Kalyuga and Singh, 2016; Sweller et al., 2019; Jiang and Kalyuga, 2020). The intrinsic load is caused by the complexity of the subject at hand for the learner and by that, it is considered to be essential or productive for understanding. The complexity is in turn caused by the element interactivity (i.e., the number of information elements that have to be processed simultaneously) and the domain-specific prior knowledge. High prior knowledge lowers the number of the to-be-processed elements in working memory as the learner is familiar with the topic and recognizes the meaning of the given combination of information elements automatically. By that, several subordinate information elements are structured into a few superordinate elements which reduces the total complexity. For the design of learning materials, CLT thus suggests that the intrinsic load should be sensibly controlled and successively adapted to the learner’s expertise (Sweller et al., 1998; Kalyuga, 2007). The extraneous load, on the other hand, is not caused by the learning subject itself, but by the design of the learning environment, and by that it is considered to be unessential or unproductive for understanding the subject. For example, unimportant additions or awkward wording can increase the extraneous load. The CLT thus suggests that the extraneous load should be reduced in the design of learning materials so that the cognitive resources are not wasted (Sweller et al., 1998; Kalyuga, 2007). Therefore, the CLT generally supports guided teaching methods through which learners are directed and supported in their learning process. In contrast, strategies such as discovery learning are rejected (Kirschner et al., 2006).

Regarding worked examples, the CLT (Sweller et al., 1998) predicts a learning-promoting effect compared to problem-solving tasks since the former exerts a lower extraneous load (Sweller and Cooper, 1985). Sweller and Cooper (1985) argue that pure problem-solving is associated with a heavy load because cognitive resources are expended on solving the task at hand. The learner has to perform a means-ends analysis to find appropriate operators for transforming an actual state into the desired state. This analysis or search requires the maintenance of many elements of information, is usually very unsystematic, and proceeds slowly, in particular for novices with a low domain-specific prior knowledge (Renkl, 2014). However, according to Sweller and Cooper, the goal of the learning process is not the solution of a concrete task, but the construction of generalized problem-solving schemata by the learner, which can be applied after the learning phase. The means-ends analysis during problem-solving thus can be seen as an additional extraneous load that interferes with the learning process. In general, worked examples have the advantage of eliminating the need for a means-ends analysis during learning. The individual solution steps and operators are already given so that the learner can concentrate solely on deriving general patterns for problem-solving from the example. In this respect, worked examples are generally regarded as a comparatively guided and structure-preserving instructional method (e.g., Sweller et al., 1998; Kapur and Bielaczyc, 2012).

Many studies have empirically confirmed that a mixture of worked examples and problem-solving tasks is better than problem-solving only regarding learning performance (e.g., Sweller and Cooper, 1985; Cooper and Sweller, 1987; Paas, 1992; Carroll, 1994; Paas and Van Merrienboer, 1994; Nievelstein et al., 2013), cognitive load (e.g., Sweller and Cooper, 1985) and learning efficiency when learning performance is related to time expenditure (McLaren et al., 2008, 2016). The effect was also replicated in studies in which the control group received additional assistance (Cooper and Sweller, 1987; Schwonke et al., 2009) and in studies in which the experimental group received worked examples only instead of a mixture of worked examples and problem-solving tasks (Paas, 1992). Overall, Crissman (2006) reported a meta-analytically determined mean effect size of d = 0.57 of worked examples versus problem-solving tasks concerning learning performance.

Traditional research on the worked example effect has mainly focused on completely correct examples (e.g., Paas, 1992; Sweller et al., 1998). Over time, however, a branch of research has emerged that investigates the learning effectiveness of erroneous worked examples (e.g., Joung et al., 2006; Große and Renkl, 2007). These can be described as worked examples that contain at least one incorrect solution step. In this respect, worked examples can be divided into correct examples (CE) and erroneous examples (EE). In the literature, various explanations are given for the learning-promoting effect of EE compared to CE (Bransford and Schwartz, 1999; VanLehn, 1999; Siegler, 2002; Joung et al., 2006; Barbieri and Booth, 2016). One widely cited explanation is offered by cascade theory (VanLehn, 1999) which describes the learning or problem-solving process as a continuous sequence of impasses, reflections, and repairments. An impasse is a kind of dead end, which means the learner finds themselves at a state during the problem-solving in which they lack the correct operators to transform the actual state into the target state. According to cascade theory and contrary to CLT (Sweller et al., 1998), such impasses are valuable for generating problem-solving schemes because they encourage reflection. Reflection expands previous knowledge. In the best case, the learner finds the correct operator and can free himself or herself from the impasse. EE may be more likely to provoke such impasse situations than CE (Große and Renkl, 2007; Heemsoth and Heinze, 2014). The assignment to comprehend why a correctly applied problem-solving operator is correct probably does not stimulate as much reflection on the valid rules and limits of this operator as the assignment to comprehend why an incorrectly applied operator is incorrect (Siegler and Chen, 2008). In the first case, it is obvious to simply rely on the given answer. In the second case, the person must delve deeper into the subject matter to generate an answer (Siegler and Chen, 2008).

Overall, empirical studies show that EE compared with CE may indeed have a positive impact on learning (Joung et al., 2006; Kopp et al., 2008; Booth et al., 2013; Heemsoth and Heinze, 2014). For example, Booth et al. (2013) showed that students presented with multiple EE on algebra developed better conceptual understanding after the intervention than those presented with multiple CE. However, some studies found either no learning advantage (Wang et al., 2015; McLaren et al., 2016; Heemsoth and Kleickmann, 2018) or, in line with the contrasting CLT framework, even a learning disadvantage compared to CE (Große, 2018). This suggests that, as with most design principles, there are factors that moderate the learning effect of EE compared to CE.

Furthermore, it was investigated how effective a mix of CE and EE is for learning. Some studies compared the effectiveness of such a condition with the exclusive presentation of EE (Booth et al., 2013; Heemsoth and Kleickmann, 2018). Heemsoth and Kleickmann (2018) found a significant difference in favor of the mixed condition, while Booth et al. (2013) found no difference. Other studies compared the effectiveness of a mixed condition with CE presentation only (Große and Renkl, 2007; Durkin and Rittle-Johnson, 2012; Booth et al., 2013; Zhao and Acosta-Tello, 2016; Huang, 2017; Heemsoth and Kleickmann, 2018; Loibl and Leuders, 2019). Here, only one of the studies found no significant differences (Huang, 2017). In the remaining studies, however, especially those which explicitly prompted the learner to compare both solution attempts, the mixed condition consistently proved to be more conducive to learning than the presentation of CE alone (Große and Renkl, 2007; Durkin and Rittle-Johnson, 2012; Booth et al., 2013; Zhao and Acosta-Tello, 2016; Heemsoth and Kleickmann, 2018; Loibl and Leuders, 2019). Overall, these studies indicate that implementing both types of examples in a learning process can be useful. However, the question remains unclear whether the order of presentation (i.e., the order of cognitive processing) is decisive for learning success and, if so, whether CE should be presented first or EE should be presented first.

In the following, two theoretical perspectives are presented which allow deriving assumptions on the question of whether more guided instructional methods such as CE or less guided instructional methods such as EE should take place first in a learning environment.

In the following, experimental studies that exclusively manipulated the order of more (e.g., explicit instruction or CE) and less guided (e.g., problem-solving) instructional methods will be discussed. The findings of these studies are quite inconsistent. Many findings support the expertise reversal effect as well as CLT and thus suggest that more guided instructional methods should be implemented first (Renkl et al., 2000, 2004; Atkinson et al., 2003; Renkl and Atkinson, 2003; Salden et al., 2010; Van Gog et al., 2011; Fyfe et al., 2014; Leppink et al., 2014; Hsu et al., 2015). For example, several studies investigated the guidance fading principle (Renkl et al., 2000, 2004; Atkinson et al., 2003; Renkl and Atkinson, 2003; Salden et al., 2010). In fading guidance conditions, learners receive only CE at the beginning of the learning phase, then they proceed with so-called completion tasks in which individual solution steps are missing, and at the end, they receive problem-solving tasks without any assistance. In all studies, this method proved to be more conducive to learning than a control group, which received multiple CE – problem pairs. Further studies indicate that CE should be presented before problem-solving tasks in the learning process rather than the other way around (Van Gog et al., 2011; Leppink et al., 2014). This order was found to lead to both better learning performance and lower mental effort. In addition, the study by Fyfe et al. (2014) indicates that conceptual instruction should be presented before problem-solving tasks as this order led to better problem-solving performance on a post-test.

However, there are also contradictory findings that support the productive failure approach and suggest that less guided methods should be implemented first (DeCaro and Rittle-Johnson, 2012; Kapur, 2014; Rittle‐Johnson et al., 2016; Lai et al., 2017; Weaver et al., 2018). Kapur (2014) investigated the effect of sequencing an instructional phase in which a teacher presented CE and problem-solving tasks on the topic of standard deviation and provided feedback on the latter and a pure problem-solving phase in which students were individually asked to collect as many solutions as possible. In the subsequent learning test, there were no differences between the groups in terms of problem-solving performance. However, students from the problem-solving – instruction condition performed better in terms of conceptual knowledge. Further studies confirm this finding concerning the expansion of conceptual knowledge (DeCaro and Rittle-Johnson, 2012; Lai et al., 2017; Weaver et al., 2018) but also partly concerning problem-solving performance (Lai et al., 2017), even though the two learning phases were sometimes presented differently in these studies. Sometimes, additional guiding aids were implemented in the problem-solving phase, such as the opportunity for collaborative exchange with classmates (Weaver et al., 2018) or corrective feedback (DeCaro and Rittle-Johnson, 2012). In another study, the direct instruction phase was made less guiding and more challenging by embedding several test questions (Lai et al., 2017).

Furthermore, several studies have been conducted examining the order of more and less guided methods in interaction with prior knowledge or topic complexity (Reisslein et al., 2006; Hsu et al., 2015; Ashman et al., 2020). Ashman et al. (2020) manipulated the sequence of explicit instruction and problem-solving. During instruction, a teacher explained to students how to calculate the energy efficiency of light bulbs. The students were also allowed to perform the calculations themselves and then participate in a discussion in which the teacher addressed, among other things, common erroneous solutions (comparable to EE). The first experiment showed that students from the explicit instruction – problem-solving order performed better on near transfer tasks in a delayed learning test a few days later. In a second experiment, the same design and learning topic was used again, but with more complex problem-solving. Here, the positive learning effect was confirmed with regard to the near transfer tasks and the same effect also occurred concerning the far transfer tasks. In the study by Hsu et al. (2015), a similar pattern emerged. They manipulated the order of CE and problem-solving and found an overall learning-promoting effect of the CE – problem-solving order. Subjects in this condition also reported lower mental effort. Furthermore, there was an interaction effect between order and expertise. The positive effect was only found for subjects who were classified as novices. In contrast, there was no difference for subjects who were classified as experts. Reisslein et al. (2006) found that expertise even reversed the direction of the order effect. Globally, there was no effect of order. However, subjects who were classified as novices benefited from the CE problem-solving order in a transfer test, in line with the previously discussed results. Subjects who were classified as experts, on the contrary, benefited from the problem-solving – CE order. These studies indicate that the contradictory findings on the effect of order described above may be explained by different levels of prior knowledge or element interactivity.

However, three research gaps emerge from the overall review of the literature on the order effects of more and less guided instructional methods. Firstly, while the order effects of explicit instruction and problem solving as well as of CE and problem-solving have been empirically investigated several times, there is no investigation of an order effect of CE and EE on learning performance and cognitive load. Secondly, apart from the variables of prior knowledge or element interactivity, no other moderating variables have yet been investigated that may change the influence of the order of more and less guided methods. Thirdly, no study has yet investigated whether the order of more and less guided instructional methods affects metacognitive variables such as the JOL. JOL refers to one’s self-assessment of the level of knowledge related to a particular learning topic and is thought to be a relevant factor in long-term learning success because it controls the future allocation of resources during self-regulated learning activities (Nelson and Dunlosky, 1991; Dunlosky and Hertzog, 1998; Zimmerman, 2008; Metcalfe, 2009). For example, the discrepancy-reduction model proposes that more learning resources are used for topics that are perceived as more difficult or for which a learner’s understanding is estimated to be lower (Dunlosky and Hertzog, 1998; Metcalfe, 2009). In this sense, JOL accuracy is considered decisive for self-regulated learning (Thiede et al., 2003; Zimmerman, 2008; Baars et al., 2017). If the JOL controls the distribution of resources, then it is most conducive if learner neither underestimate nor overestimate in their JOL, but correctly allocate the required resources for each subtopic (Thiede et al., 2003; Metcalfe, 2009). The present study aimed at these three research gaps described above.

Since CE can generally be classified as a guided instructional method and EE as a comparatively less guided instructional method, two theoretical perspectives were presented, which allow for the derivation of (contrasting) hypotheses regarding the question of whether more or less guided methods should be used first in the learning process. First of all, both, the expertise reversal effect and the productive failure approach assume that presenting less guided instructional methods first and more guided methods second exerts higher cognitive load than vice versa (Kalyuga, 2007; Kapur, 2014). This is empirically supported by studies that manipulated the order of explicit instruction and problem-solving tasks as well as those that manipulated the order of CE and problem-solving tasks (Van Gog et al., 2011; Kapur, 2014). Therefore, the following hypothesis was formulated:

However, the two theoretical frameworks have contrasting suggestions for the effect of order on learning achievement. On the one hand, the expertise reversal effect (Kalyuga, 2007), following CLT (Sweller et al., 1998), claims that it is more conducive to learning if more guided instructional methods take place before less guided instructional methods. This assumption is supported by various findings (e.g., Van Gog et al., 2011). Moreover, studies show that expertise moderates the EE effect, and CE is more preferable when prior knowledge is low and EE is more preferable when prior knowledge is high (e.g., Große and Renkl, 2007). Therefore, according to the expertise reversal effect, it can be inferred that learners benefit from a CE-EE sequence. On the other hand, the productive failure approach (Kapur, 2008) predicts that it is more conducive to learning when less guided instructional methods precede more guided instructional methods. This assumption is also supported by several findings (e.g., Kapur, 2014). Therefore, according to the productive failure approach, it can be inferred that learners benefit from an EE-CE sequence. Due to the conflicting theoretical approaches and findings, the following research question was formulated:

Furthermore, the two theories differ in their assumption about an indirect effect of order via cognitive load on learning achievement. On the one hand, the expertise reversal effect assumes that cognitive load caused by the design (e.g., order of methods) has a negative effect because the capacities of the working memory are strongly limited and thus can lead to an overload for some learners (Kalyuga, 2007). The productive failure approach, on the other hand, assumes that higher load after the entire learning phase should mainly be attributed to the high demands in the initial exploration and generation phase. Although this presents a difficulty for the learner, it can also be helpful because it supports schema assembly and helps the learner make better use of the subsequent instruction phase (Kapur, 2014). Due to the conflicting theoretical assumptions, following research question was formulated:

Few studies examined the question of whether the influence of order is moderated by different variables (Reisslein et al., 2006; Hsu et al., 2015; Ashman et al., 2020). So far, these studies only identified prior knowledge and element interactivity as such variables. The present study aimed to investigate whether congruency between subsequent exemplified problems also interacts with the effect of order. According to cascade theory (VanLehn, 1999), EE are effective because they provoke impasses in which the learners are confronted with problems that they cannot solve immediately. If they first receive a CE and then an EE that exemplifies a different problem, they are likely to be prepared by the CE in finding and correcting the error and to do so more quickly. But the learners are still faced with an impasse as there is a problem that has to be solved first. However, if they first receive a CE and then an EE that exemplifies the very same problem the solution is already clear. Therefore, the learners will probably not deal with the EE in more depth. The beneficial reflection processes of EE according to the cascade theory are thus weakened. Therefore, the following hypothesis was formulated:

As previously described, it has not yet been investigated whether the order of more and less guided instructional methods influences the metacognitive variables JOL and JOL accuracy, although these are very important for self-regulated learning. Social-cognitive accounts on self-regulated learning suggest that academic achievement under these circumstances is determined by the quality of self-observation, self-judgment, and self-reaction (Zimmerman, 1989; Bandura, 1991; Broadbent and Poon, 2015). Accordingly, people monitor their own performance and compare them with certain standards. Depending on the result, they form judgments about their performance and possibly regulate their learning activities in order to meet those standards. When working on different types of examples, learners should therefore observe how well they are understanding the exemplified procedures, thereby gathering information about their own abilities and forming a more or less accurate self-judgment. Moreover, the belief-adjustment model (Hogarth and Einhorn, 1992) proposes that the order in which different belief-relevant information is processed is crucial for belief formation. Depending on consistency, complexity, and length of information sets as well as response mode, primacy or recency effects can occur. JOL can be seen as one kind of belief formation as someone forms an opinion of one’s own state of learning. The learning experience with different instructional methods itself can be seen as relevant information for this belief formation. However, it is impossible to use the belief-adjustment model as a guiding framework for concrete hypotheses in the case of this study. One might determine a priori the length of the information set and the response mode but it is difficult to suggest if CE and EE are consistent in the information that they convey for JOL and to determine beforehand how complex the processing of these examples is. Furthermore, due to the lack of research, it is unclear which information CE and EE themselves convey for JOL and its accuracy. But since the model suggests that there might be any effect of order at all the following two research questions were formulated:

The experiment followed a 2×2 between-subject design. The first factor represents the order (CE-EE vs. EE-CE) in which the CE and EE are presented. The second factor represents the congruence between the exemplified problems (same problems vs. different problems). Table 1 shows an overview of all experimental conditions. Before the study, an a priori power analysis was conducted using the software G*Power (Faul et al., 2007) for a 2 × 2 analysis of variance with a test power of 1-ß =0.80 and an alpha level of α = 0.05. Since no study has yet examined the effect of the order of CE and EE, the weighted average effect of the two studies that manipulated the order of CE and problem-solving tasks was taken as the estimated effect size: f = 0.34 (Van Gog et al., 2011; Leppink et al., 2014). The calculation resulted in a minimum sample size of n = 70.

The subjects were recruited via an e-mail distribution list of the Chemnitz University of Technology intended for study promotion. Current enrolment at a university was defined as a participation requirement to keep the sample as homogeneous as possible. Either eight euros or course credit was offered as compensation. A total of 86 students took part in the study. Two students were removed from the data set before the evaluation because they stated that they were not or no longer studying. Another student was removed because of technical difficulties during the survey that prevented the completion of one of the three learning units. The final sample size was therefore 83 students (72.0% female; age: M = 23.15; SD = 3.12). 89.15% of the students came from the [blinded for review]. 46.90% were enrolled in the Media Communication program, 29.62% in the Media and Instructional Psychology program, and 23.48% in other programs. Concerning the allowance, 69.88% opted for the course credit as compensation. Prior knowledge of the learning material of Vedic Mathematics was basically nonexistent. Only two participants stated that they had heard of the subject before, but were just like any other participant unable to solve any of the tasks in the subsequent pretest. Students were randomly assigned to one of the four conditions. There were no significant differences between the four conditions with regard to gender (p = 0.873), age (p = 0.649), university affiliation (p = 0.490), study program (p = 0.766), or prior knowledge (p = 0.585).

The learning environment in this study consisted of a slideshow on Vedic Mathematics. Vedic Mathematics is an alternative calculation method invented by Tirthaji (1965), which can be used to solve rather complex mathematical problems relatively quickly without using a calculator. Three of these methods, also called rules, were selected as learning topics for this study and each formed an independent learning unit within the learning environment. The rule of the first unit was called “Vertical of powers of 10 and crosswise,” which concerned the multiplication of multi-digit numbers near powers of 10. The rule of the second unit was called “Vertical and crosswise,” which concerned the multiplication of any two-digit numbers. Lastly, the rule of the third unit was called “Division by 9 with remainder,” which concerned tasks where any number needs to be divided by nine.

The learning material contained a total of nine slides. Each unit contained three slides, one with direct instruction on the field of application and the methodology of the respective rule (see Appendix A), one slide with a CE, and one slide with an EE. The examples constituted the stimulus material. Which example was presented first and if the exemplified problems were the same or different was therefore dependent on the experimental condition. In addition, self-explanation prompts (“Try to understand why the examples are correct or incorrect and try to remember the correct solution steps for the test”) were inserted above the examples, as studies show that these are crucial for the learning effect of CE and EE (e.g., Chi et al., 1989; Große and Renkl, 2007; Hilbert and Renkl, 2009).

The study was conducted via an online conference system. After a short welcome and introduction, the participants were distributed into separate conference rooms and asked to share their browser windows with the investigator. This was to ensure that the subjects were exclusively engaged in the study and that no attempts at deception could take place. The participants accessed the learning environment via a link. Figure 3 illustrates the procedure with all important stages for all four conditions. After some demographic information, they were asked about their previous knowledge. On the next page, they were introduced in a few sentences to the topic of Vedic Mathematics and informed about the procedure of the study. Then the learning phase according to the experimental condition began. First, the unit “Vertical of powers of 10 and crosswise,” then the unit “Vertical and crosswise” and finally the unit “Division by 9 with remainder” were completed. Each unit contained a total of five pages. On the first three pages, the learning material was presented, first one page with explicit textual instruction and then two pages with one example each. If the CE or EE was presented first and also if the second example exemplified the same problem as the first example depending on the condition. For these three learning pages, the processing time was limited to 2 min per page. First, to ensure a high level of difficulty and variance in the results, and second, to avoid subjects spending so much time on a slide that they struggle to understand and consequently skip the rest of the slides. In the run-up to the study, the 2 min proved to be sufficient for independent testers to receive and elaborate on all the information on the slides at their leisure. Learners had the option to move on to the next page before the 2 min elapsed by pressing buttons labeled “Next” or “I have understood everything.” On the fourth page, the JOL_immediate for the respective rule was collected and on the last page, the problems_immediate for the respective rule was presented. After going through all three units, participants reported their mental load and mental effort. Then the JOL_delayed was collected. Finally, a block with the problems_delayed followed. After that, the experiment was finished.

A MANOVA with mental load and mental effort as dependent variables showed a marginally significant main effect for the factor order, Wilk’s Λ = 0.94, F(2,78) = 2.52, p = 0.087, η_p² = 0.06. However, there was neither a main effect for the factor congruence, Wilk’s Λ > 0.99, F(2,78) = 0.03, p = 0.968, η_p² < 0.01, nor an interaction effect, Wilk’s Λ > 0.99, F (2,78) = 0.19, p = 0.827, η_p² = 0.01. Separate ANOVAs were then conducted for the factor order. There was a significant medium effect on mental load, F (1,79) = 4.50, p = 0.037, η_p² = 0.05. The EE-CE group (M = 34.00, SD = 5.66, N = 41) reported higher mental load on average than the CE-EE group (M = 31.07, SD = 6.70, N = 42). There was no significant effect on mental effort, F (1,79) = 0.51, p = 0.479, η_p² = 0.01.

A MANOVA with learning performance from the first, second, and third learning unit as dependent variables showed neither a main effect for the factor order, Wilk’s Λ = 0.99, F (3,77) = 0.29, p = 0.836, η_p² = 0.01, nor a main effect for the factor congruence, Wilk’s Λ = 0.99, F(3,77) = 0.21, p = 0.887, η_p² = 0.01. Furthermore, there was no interaction effect, Wilk’s Λ = 0.93, F(3,77) = 2.05, p = 0.117, η_p² = 0.07.

In total, six mediational analyses were conducted with two different mediators (mental load or mental effort) and three different dependent variables (problem_delayed for Learning unit 1, 2 or 3). Different from the other analyses, only the problem_delayed score was set as dependent variable to ensure antecedence in time of measurements since the immediate problem score was collected before the measurement of mental load and mental effort.

For JOL_immediate, a MANOVA was conducted with the corresponding variables for the first, second, and third learning unit. There was no effect for the factor order, Wilk’s Λ = 0.98, F(3,77) = 0.54, p = 0.659, η_p² = 0.02. An ANOVA was performed for the JOL_delayed. It showed no effect for the factor order, F(1,79) = 1.85, p = 0.178, η_p² = 0.02.

For JOL accuracy_immediate, a MANOVA was conducted with the corresponding variables for the first, second, and third learning unit. There was no effect for the factor order, Wilk’s Λ = 0.98, F(3,77) = 0.59, p = 0.625, η_p² = 0.02. An ANOVA was conducted for the JOL accuracy_delayed. There was a small, significant main effect for the factor order, F(1,79) = 5.52, p = 0.021, η_p² = 0.02. As indicated by confidence intervals, both the CE-EE (M = −23.26, SD = 20.37, N = 42, 95% CI [−29.67, −16,86]) and the EE-CE group (M = −12.44, SD = 21.16, N = 41, CI [−18.99, −6,03]) underestimated their performance on the problems_delayed. However, the EE-CE group underestimated their performance to a lesser extent and their estimation was closer to the actual performance.

The results of this study are limited in the sense that there are a few alternative explanations for the findings. Regarding the insignificant main effect of order, it can be argued from a productive failure perspective that not all conditions for the occurrence of an order effect were fully met. The productive failure effect has so far been found mainly in problem-solving tasks and direct instruction, but not in EE and CE. EE resembles problem-solving in that solutions have to be generated independently, but this usually applies only to a part and not to the whole task. Although, Kapur (2014) also conducted studies on vicarious failure in which learners, instead of generating solutions themselves, were presented with incorrect solutions by classmates and asked to evaluate them. This condition was found to be less beneficial than the classic productive failure problem-solving – direct instruction condition but still more beneficial than the direct instruction – problem-solving condition. Therefore, EE should at least in principle meet the criteria for a productive failure effect. Furthermore, unlike in the above-mentioned productive failure studies, we did not start with the exploratory method but with a short direct instruction phase. This was necessary so that the subjects had minimal prior knowledge of the topic and were able to recognize errors in the execution of the Vedic rules. However, this should not have prevented a true productive failure effect, since the EE themselves were followed by another form of guided instruction, namely the CE.

Secondly, the findings might only extend to learning environments that include EE with rather few errors. As the EE used in this study only include two errors the difference between EE and CE was rather small. This might have attenuated a possible effect and explain why no effect was found in contrast to former studies where the difference between the instructional methods used was more significant (e.g., Van Gog et al., 2011; Kapur, 2014). Future studies could include more errors in the EE to test this alternative explanatory approach.

Thirdly, it is possible that an effect of order did not occur because learners only focused on the CE when learning and tended to ignore the EE, for example, because the EE was judged to be too complex and difficult (Van Gog and Paas, 2008). This explanation is also supported by the finding that learners invested less time in learning with the EE (M = 59.87, SD = 35.52) than with the CE (M = 99.56, SD = 25.98), although EE should actually cause a higher cognitive load than CE and learners thus need more time to process them (McLaren et al., 2016). This finding also contradicts other studies that have compared the learning time of CE and EE (e.g., McLaren et al., 2016). In these studies, however, learners were also asked to correct the EE in writing. In this study, this was avoided to keep the conditions comparable. As a result, the implemented self-explanation prompt may have lost its effectiveness. Although collecting qualitative data was beyond the scope of this study, it might be recommendable for future studies to conduct post-hoc interviews with some participants and learn about their experience within the learning environment.

Regarding the significant indirect effects of order via mental load on learning, it must be carefully noted that a statistically significant mediation model does not ensure automatically that there actually is a causal relationship between the variables, as this assumption has to be met by the study’s method as well (Rohrer et al., 2022). In this study, however, only the predictor order had been manipulated experimentally, not the mediator mental load. Hence, the results might only be seen as an indication for a mediated model of effects. Future studies with a more suitable methodological framework are necessary to substantiate this finding. This could also help to shed light on the somewhat contradicting finding that there was no significant total effect of order on learning but a significant indirect effect via mental load.

In addition, the results of the study are limited concerning the interaction hypothesis between order and congruence. On the one hand, the null effect might be attributed to the low power of the study. The observed effect size was lower than expected and the value of p of the interaction almost reached marginal significance. Therefore, future studies with bigger sample sizes may find significant results. On the other hand, it can be assumed that there was no interaction effect because the learners did not have enough time to remember the several individual steps of the CE. As a result, and contrary to the original intention, the subjects in the congruent CE-EE condition might have failed to recognize the difference between the examples (i.e., the errors) early. This explanation is supported by the descriptive statistics of the learning units. In the first and simplest learning unit, where the steps could possibly be remembered more easily, the mean values of the groups tended to correspond to the postulated pattern. This was not the case in the other learning units. To test this alternative explanation, instead of two errors, only one error could be embedded in the EE in future studies. Or examples could be used that are clearer and do not require remembering so many calculation steps. Another possibility would be to use EE with concrete error signaling.

Moreover, as there has not been much theoretical and empirical work on the effect of order and congruency on metacognitive judgments, this study’s explanations for the significant findings on JOL accuracy need further elaboration. More data is needed to check on potential effects and their mechanisms. Using the belief-adjustment model (Hogarth and Einhorn, 1992) as a framework for explaining order effects on metacognitive judgments might be recommendable for future studies. Moreover, theoretical work is needed on the question of how CE and EE in isolation influence JOL and in case the order effect was indeed caused by recency bias why CE promote better JOL accuracy.

This is the first study to experimentally investigate the effect of the order of correct and erroneous worked examples on learning performance as well as to examine whether congruency of exemplified problems functions as a moderator. Overall, the results show no total effect of order on learning but still indicate that there is an indirect effect. Presenting correct worked examples first and erroneous worked examples second resulted in lower mental load, which in turn was associated with better learning performance, thereby supporting the assumptions made by the expertise reversal effect and contradicting the assumptions made by the productive failure approach. Congruency did not moderate the impact of order on learning, challenging cascade theory as an explanation for erroneous worked example effects. Furthermore, order of examples significantly influenced JOL accuracy as learners estimated their learning performance more accurately when erroneous worked examples were presented first. Further studies are needed to test the limitations discussed above.