The difficulty to evoke the spacing effect in mathematics: New findings and theoretical considerations

The difficulty to evoke the spacing effect in mathematics: New findings and theoretical considerations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report The difficulty to evoke the spacing effect in mathematics: New findings and theoretical considerations Anne Ludwig, Mirjam Ebersbach, Florian Scharf This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7747941/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We investigated whether practicing mathematics in a spaced manner, while additionally self-explaining worked examples, fosters lasting learning in mathematics in adults ( N = 208). The test after eight weeks revealed no main effects and no interaction, suggesting that the spacing effect for mathematics is not robust. We discuss how mathematics differs from other learning domains and how these differences need to be considered to evoke a spacing effect in mathematics. Physical sciences/Mathematics and computing Physical sciences/Physics Biological sciences/Psychology Social science/Psychology Figures Figure 1 Introduction Acquiring knowledge that is retrievable over long periods of time is a central aim of education. Lasting knowledge serves as prior knowledge, facilitating the comprehension and integration of new information into memory ( 1 ). Desirable difficulties in learning have been identified that make the learning process harder for learners but contribute to establishing lasting knowledge ( 2 ). One of these desirable difficulties is spaced practice, in which the total practice time is distributed across several sessions instead of practicing for the same amount of time in only one, massed session. Meta-analyses reported medium to large spacing effects that emerged for different age groups and learning materials (e.g., 3, 4, 5). However, one meta-analysis suggested the spacing effect to be smaller for more complex content than for simpler material ( 6 ). Mathematics present an example for such complex content, relevant in real educational contexts, professional, and private life. However, many students have problems to attain sufficient knowledge and comprehension of mathematics ( 7 ). One meta-analysis on the spacing effect in classroom learning revealed overall a small to medium spacing effect ( g = .46) ( 8 ). However, concerning mathematics, only six of the 15 included studies found a significant spacing effect—even though the authors concluded that “distributed practice appears generally effective in the math domain” (p. 12). Another meta-analysis on the effects of spacing and retrieval practice for math learning ( 9 ) reported a small spacing effect ( g = .28), but only ten of the 21 included effect sizes were significant. These results suggest that the spacing effect for math learning is smaller and does not emerge as reliably as in other domains, such as second language learning (e.g., 10: g = .80). Thus, spacing might require certain boundary conditions before acting as an effective instructional means to foster lasting learning of mathematics. One requirement might be the deeper, meaningful elaboration of the learning material during (spaced) practice to prevent learners from practicing a mathematical problem-solving procedure only in a superficial way. Asking learners to self-explain given worked examples, exemplary depicting the single solution steps, is a powerful means to evoke a deeper elaboration (e.g., 11). One study ( 12 ) introduced long multiplication to fourth graders in school, who afterwards self-explained worked examples (in addition to solving problems) during massed or spaced practice (i.e., three practice sessions with an inter-study interval of one day). Pure problem solving, which was also self-explained by the children, served as control condition. In a test after eight weeks, however, there were no main effects of spacing and worked examples, and no interaction—despite sufficient power. At least two reasons may account for the absent effects: ( 1 ) The inter-study interval was too short for the long retention interval (cf. 13), and ( 2 ) self-explaining also one’s own problem solving might have levelled out the effect of self-explaining worked examples. The present preregistered study ( https://osf.io/k3e6w ) tested these explanations in a controlled setting with adults, acquiring a new mathematical procedure. The inter-study interval was extended to two weeks, and only worked examples were self-explained, but no longer students’ own problem solving. Students’ working memory and long-term memory capacities, their specific prior knowledge, and general math achievement served as covariates. We expected main effects of spacing and worked examples under these optimized conditions and, most importantly, that spaced practice including self-explaining worked examples would result in the best test performance after eight weeks. Results Students’ test performance, separately for procedural and conceptual knowledge in each condition, is depicted in Fig. 1 . Two ANCOVAs were computed with heteroscedasticity-consistent (HC3) robust standard errors to check whether spacing, worked examples, and their interaction affected students’ procedural and conceptual knowledge in the test, while controlling for specific prior knowledge, math grade, long-term memory capacity, and working memory capacity (see Table 1 for detailed results). The categorical independent variables were effect coded, and the continuous independent variables were mean centered. Concerning procedural knowledge, the ANCOVA explained a significant but small amount of variance, F (7, 200) = 2.37, p = .024, R² = 0.08, R adj ² = 0.04. However, this was only due to the covariates specific prior knowledge and long-term memory capacity. Neither spacing nor worked examples significantly affected test performance, and their interaction was also not significant. Bayes Factor analyses, using Monte Carlo sampling and Jeffreys’ prior, largely confirmed these results. There was moderate evidence for the null effects of spacing and of worked examples, and anecdotal evidence for the null effect of their interaction. Concerning conceptual knowledge, the ANCOVA also explained a significant but small amount of variance, F (7, 200) = 2.91, p = .006, R² = 0.09, R adj ² = 0.06. Again, this was only due to the effect of covariates, mainly math grade, and neither spacing nor worked examples yielded a significant effect, and their interaction was not significant. Bayesian analyses revealed moderate evidence for the null effects of spacing and of worked examples, but no evidence in favor of or against the null effect of their interaction. Violin plots depicting students’ procedural and conceptual knowledge in the test (% correct), separately for each condition Note MAS + PS = Massed practice / Pure problem solving ( n = 51); SP + PS = Spaced practice / Pure problem solving ( n = 52); MAS + WE = Massed practice / Self-explaining worked examples ( n = 48); SP + WE = Spaced practice / Self-explaining worked examples ( n = 57). Table 1 Results of the ANCOVAs (one for each knowledge type), with corresponding Bayes factors indicating evidence in favor of the null hypothesis. Variables Procedural Knowledge Conceptual Knowledge F p \(\:{\eta\:}_{p}^{2}\) BF 01 F p \(\:{\eta\:}_{p}^{2}\) BF 01 Spacing 0.11 0.736 0.00 5.79 (4.29%) 0.00 0.949 0.00 6.72 (3.71%) Worked Examples 0.01 0.936 0.00 4.71 (4.34%) 0.08 0.774 0.00 4.71 (3.74%) Spacing × Worked Examples 3.69 0.056 0.02 1.00 (4.55%) 1.61 0.205 0.01 2.29 (3.79%) Note. df = 1, 200 for each knowledge type; BF 01 = Bayes factor (1–3: anecdotal evidence, 3–10: moderate evidence in favor of the null hypothesis). Percentage error estimates of Bayes factors in parentheses. Discussion The present study investigated whether spacing fosters lasting learning in mathematics under optimized conditions. Based on meta-analyses suggesting that the spacing effect in mathematics is not as robust as in other learning domains (e.g., verbal learning), we tested whether it can be evoked by cognitively enriching the (spaced) practice phases and by adjusting the inter-study interval to the long retention interval of eight weeks. Learners self-explained worked examples during practice (in addition to solving corresponding math problems), and the inter-study interval between the practice phases was set to two weeks. However, in the test after eight weeks, there was neither an effect of worked examples, compared to pure problem solving without worked examples, nor an effect of spaced compared to massed practice, and no interaction effect. The results were largely confirmed by Bayesian analyses with one exception that for procedural knowledge, the null effect of the interaction could neither be confirmed nor rejected. Probably, the sample size was too small to allow for a clear conclusion. However, from a practical perspective the potential interaction is no longer relevant because the effect would smaller than the desired effects of educational interventions ( 17 ). The findings align with other studies revealing no spacing effect in mathematics (e.g., 13; cf. 9). They furthermore suggest that the acquisition of more complex learning content might require other boundary conditions to benefit from spaced practice. On closer inspection, it becomes evident that learning mathematics differs from learning simpler content, such as vocabulary, for which a spacing effect appears robustly. First, mathematics comprises both conceptual (i.e., “knowing why”, also including facts) and procedural knowledge (i.e., “knowing how”). However, studies on the spacing effect in mathematics often involved only pure problem solving, requiring the retrieval of procedural knowledge, but not necessarily the retrieval of the corresponding concept (e.g., the formula). Vocabulary learning, in contrast, refers only to conceptual knowledge. Study-phase retrieval is one of the driving principles of the spacing effect (e.g., 15). Second, when practicing mathematics, learners usually solve different, isomorphic problems that refer to the same underlying concept, whereas in verbal learning identical items are presented repeatedly. Accordingly, practice in mathematics includes fewer item-based retrieval cues (e.g., the same digits or context). These two issues might contribute to the finding that the spacing effect is less robust for more complex learning contents, as long as practice is designed as described, because the retrieval processes, evoked by spaced practice sessions, differ. It might thus be promising to explicitly demand the retrieval of procedural and conceptual knowledge when trying to evoke a spacing effect for more complex material, and to provide learners with retrieval cues that work in a similar way as when presenting identical items, such as in verbal learning. Moreover, it seems necessary to assess procedural and conceptual knowledge separately as dependent variables, as in the present study, to uncover whether spaced practice of more complex learning material affects both types of knowledge in different ways. It is, for example, possible to solve an arithmetic task correctly in a rather automatized way without a grasp of the underlying concept. In turn, learners might know the concept but make a simple calculation error, which leads to an incorrect solution. To sum up, spaced practice is a learning principle easy to implement in real educational contexts. Identifying ways to make it effective for more complex learning material would support the acquisition of lasting knowledge in schools and universities. Methods Participants. The required sample size was computed for the ANCOVAs with three tested predictors (i.e., spacing and worked examples, and their interaction) and seven predictors in total by means of G*Power ( 16 ). To detect a small interaction effect of f² = 0.04 between spacing and worked examples, which corresponds to the minimum desired effect size for educational interventions of d = 0.40 ( 17 ), with α = 0.05, and a power of 1-β = 0.90, N = 265 participants were required. Given that prior knowledge and math grade explained on average 18% of the variance of the test performance in a similar study with children ( 12 ), the required sample size was conservatively adjusted according to ( 18 ) by the factor 1- R 2 = 0.82 to N = 217. University students ( N = 272) from different study programs were recruited. They participated voluntarily with informed consent and received either course credits or 25€. Only students who attended all three sessions were included in the analyses, resulting in a final sample of N = 208 (161 women, 43 men, 3 diverse, 1 not specified; age: M = 24.0 years, SD = 4.9). Study design and procedure. The experiment followed a 2 (practice schedule: spaced vs. massed) × 2 (consolidation condition: problem solving with self-explaining worked examples vs. pure problem solving) between-subjects design. Procedural and conceptual knowledge served as dependent variables. The study was conducted as computer-based experiment using Labvanced, with matrix multiplication as learning content. Students were randomly assigned to one of the four experimental practice groups ( n = 48–57). The first session took place in a group setting in the laboratory. Students first received two problems referring to matrix calculation to check whether they were already familiar with the procedure. This was the case for one student who was excluded from the experiment. Thereafter, students followed a computer-based video tutorial (6 min) introducing matrix calculation. Immediately after the tutorial, the practice phase started. Students in the massed practice conditions completed all practice tasks (i.e., set 1 and 2) at once, whereas those in the spaced conditions completed half of the tasks during the first session (i.e., set 1) and the remaining tasks (i.e., set 2) in the second session two weeks later. In the pure problem-solving conditions, students had to solve eight matrix multiplication problems corresponding to the introduction. In the worked examples conditions, students were presented first with a worked example of a matrix multiplication (which they did not have to self-explain) and solved a corresponding problem to allow for practicing procedural knowledge. This problem was followed by another worked example that had to be self-explained by the students in writing, aiming at enhancing their conceptual knowledge. Self-explaining was prompted by four questions per example (e.g., “Why is 0 x 9 written at the position of the first row and second column?”). In total, students in the worked examples condition solved four matrix multiplication problems and self-explained four corresponding worked examples. All practice tasks were completed without further tutorials. However, students received corrective feedback on two problems (i.e., on the first problem of practice set 1 and 2, respectively) in form of the correct solution paths and solution. The second practice session in the spaced practice conditions, taking place two weeks after the first one, and the delayed test were completed individually on participants’ laptops or computers. To ensure that participants followed the instructions and did not use additional aids, the experimenter supervised them via the videoconference tool Zoom. Eight weeks after the final practice session, the delayed test was administered again online and supervised via Zoom. It was announced as a further practice session to prevent participants to prepare themselves, consisting of 14 tasks. After the test, students’ individual characteristics were assessed (except of their specific prior knowledge that had been assessed in the first practice session). Measures. The material can be found here: https://osf.io/ahqb9/files/osfstorage/68d4151072459ba6843c33b2 . Procedural knowledge in the delayed test was assessed by four pure problem-solving tasks similar to those in the practice phase and two self-explanation tasks concerning the procedure of matrix multiplication. The number of correct intermediate calculation steps and final solutions of the matrix multiplication problems served as indicator of procedural knowledge (max: 16 points). Conceptual knowledge was assessed by presenting students with two matrix multiplication tasks, each along with three options: (a) the correct solution steps and solution and (b) incorrect solution steps and solution and (c) the response option that this task is insoluble. Students had to choose the correct option, being informed that only one option applied, and to explain their decision in writing. In addition, students received one worked example marked as incorrect along with four questions, and one worked example marked as correct along with two questions, prompting students to self-explain the reasoning behind the solution steps. The number of correct answers in all these tasks served as indicator of conceptual knowledge (max: 18.5 points). Students’ procedural and conceptual knowledge performance was transformed into % correct. Students’ sociodemographic data, their general math achievement (self-reported final math grade at graduation), specific prior knowledge (their performance in the first practice session concerning procedural knowledge), working memory capacity (Digit Span Backwards test; 19), and long-term memory capacity (WMS-IV; 20) were assessed via computerized questionnaires and tasks. Data analyses. We note that the models we preregistered under the label “multiple regression” are equivalent to the reported ANCOVAs. Initially, we additionally aimed at exploring whether the learner characteristics potentially moderated the effects of spacing and worked examples. However, we decided to focus on the analyses of the preregistered hypotheses regarding main effects of spacing and worked examples, and their interaction, two ANCOVAs were computed (one for procedural and one for conceptual knowledge). Students’ specific prior knowledge, their general math achievement, their long-term and working memory served as covariates to enhance power. The alpha level was set to .05, the tests were two-sided. In the pre-registration, we erroneously corrected the alpha to .025 to account for the two analyses. However, as they refer to different dependent variables, this correction is not necessary. To validate the results, Bayesian analyses were conducted additionally, allowing for confirming null effects. All analyses were conducted using the software R (version 4.4.1) and RStudio ( 21 ). The used packages can be found here: https://osf.io/ahqb9/files/osfstorage/68d503f74dbd891a77432b63 . Prior to the main analyses, the assumptions of the ANCOVA were examined. Boxplots indicated no problematic outliers for grades, but five outliers for procedural knowledge, and three outliers for conceptual knowledge. These outliers were not excluded from the analyses, because there was no evidence to suggest that they resulted from measurement error, data entry mistakes, or other artifacts. We therefore considered these cases to represent genuine variation within the sample and retained them to preserve the integrity and generalizability of the results. Normality was evaluated with Shapiro–Wilk tests, Q–Q plots, and histograms. Results indicated significant deviations from normality for grades ( W = .95, p < .001), procedural knowledge ( W = .85, p < .001), and conceptual knowledge ( W = .95, p < .001). Skewness statistics suggested a slight negative skew for grades (-0.62) and positive skew for procedural (1.18) and conceptual knowledge (.82). Given the robustness of ANCOVA to moderate violations of normality and skewness, all variables were retained for subsequent analyses. Declarations Ethical approval for this study was obtained from the Ethics Committee of the Faculty of Human Sciences of the University of Kassel (Approval number: 202115). Funding This article is the product of the research unit “Lasting Learning: Cognitive Mechanisms and Effective Instructional Implementation” (FOR 5254, project no. 450142163), funded by the German Research Foundation (DFG: grant EB 462/4–1). Author Contribution A.L. developed and prepared the material, prepared the software, collected, curated and analyzed the data, created the figure, and wrote parts of the manuscript. M.E. developed the research idea, provided the funding, and wrote parts of the manuscript. A.E. and M.E. share the first authorship. F.S. provided substantial support concerning the statistical analyses. All authors read and approved the final manuscript. Acknowledgement This article is the product of the research unit “Lasting Learning: Cognitive Mechanisms and Effective Instructional Implementation” (FOR 5254, project no. 450142163), funded by the German Research Foundation (DFG: grant EB 462/4–1). The funder played no role in study design, data collection, analysis and interpretation of data, or the writing of this manuscript. We thank Freyja Pollack, Lena Ayse Roß, Lilli Berger, Nike Kliewe, and Sarah Ewert for supporting the data collection. Data Availability The data supporting the findings of this study, including experimental data collected via Labvanced and the data analysis scripts, are openly available on the Open Science Framework (OSF) at the following links: raw data: https://osf.io/ahqb9/files/osfstorage/68d50035c48a490ff1c94a14; data used for analysis: https://osf.io/ahqb9/files/osfstorage/68d40264af567ee6ce3c30a6; analysis scripts: https://osf.io/ahqb9/files/osfstorage/68d3fce148906622d4604301. All analyses were conducted using the open-source software R (version 4.4.1; R Core Team, 2024) in RStudio. The OSF repository provides full access to both the original data and the analysis code, ensuring transparency and reproducibility. References Baddeley, A., Eysenck, M. W., & Anderson, M. C. Memory (Psychology Press, 2009). Bjork, E. L., & Bjork, R. A. Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning (eds. Gernsbacher, M. A., Pew, R. W., Hough, L. 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11:10:38","extension":"html","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":69073,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7747941/v1/717d4cefd82c71e7f8000d1a.html"},{"id":94100582,"identity":"31606fba-f3ff-40d3-9736-c5b78011c0fb","added_by":"auto","created_at":"2025-10-22 11:10:38","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":193130,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eViolin plots depicting students’ procedural and conceptual knowledge in the test (% correct), separately for each condition\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eNote\u003c/em\u003e. MAS + PS = Massed practice / Pure problem solving (\u003cem\u003en\u003c/em\u003e = 51); SP + PS = Spaced practice / Pure problem solving (\u003cem\u003en\u003c/em\u003e = 52); MAS + WE = Massed practice / Self-explaining worked examples (\u003cem\u003en \u003c/em\u003e= 48); SP + WE = Spaced practice / Self-explaining worked examples (\u003cem\u003en\u003c/em\u003e = 57).\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7747941/v1/2b51d10d3796942069b395a3.png"},{"id":96331101,"identity":"0f24cef9-573a-4f5c-8880-7448a4ef9025","added_by":"auto","created_at":"2025-11-20 00:53:43","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":626455,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7747941/v1/c018ca38-6f1c-4467-8a0d-d42a71a2cb79.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The difficulty to evoke the spacing effect in mathematics: New findings and theoretical considerations","fulltext":[{"header":"Introduction","content":"\u003cp\u003eAcquiring knowledge that is retrievable over long periods of time is a central aim of education. Lasting knowledge serves as prior knowledge, facilitating the comprehension and integration of new information into memory (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e). Desirable difficulties in learning have been identified that make the learning process harder for learners but contribute to establishing lasting knowledge (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e). One of these desirable difficulties is spaced practice, in which the total practice time is distributed across several sessions instead of practicing for the same amount of time in only one, massed session. Meta-analyses reported medium to large spacing effects that emerged for different age groups and learning materials (e.g., 3, 4, 5). However, one meta-analysis suggested the spacing effect to be smaller for more complex content than for simpler material (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eMathematics present an example for such complex content, relevant in real educational contexts, professional, and private life. However, many students have problems to attain sufficient knowledge and comprehension of mathematics (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e). One meta-analysis on the spacing effect in classroom learning revealed overall a small to medium spacing effect (\u003cem\u003eg\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.46) (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e). However, concerning mathematics, only six of the 15 included studies found a significant spacing effect\u0026mdash;even though the authors concluded that \u0026ldquo;distributed practice appears generally effective in the math domain\u0026rdquo; (p. 12). Another meta-analysis on the effects of spacing and retrieval practice for math learning (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e) reported a small spacing effect (\u003cem\u003eg\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.28), but only ten of the 21 included effect sizes were significant. These results suggest that the spacing effect for math learning is smaller and does not emerge as reliably as in other domains, such as second language learning (e.g., 10: \u003cem\u003eg\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.80).\u003c/p\u003e\u003cp\u003eThus, spacing might require certain boundary conditions before acting as an effective instructional means to foster lasting learning of mathematics. One requirement might be the deeper, meaningful elaboration of the learning material during (spaced) practice to prevent learners from practicing a mathematical problem-solving procedure only in a superficial way. Asking learners to self-explain given worked examples, exemplary depicting the single solution steps, is a powerful means to evoke a deeper elaboration (e.g., 11). One study (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e) introduced long multiplication to fourth graders in school, who afterwards self-explained worked examples (in addition to solving problems) during massed or spaced practice (i.e., three practice sessions with an inter-study interval of one day). Pure problem solving, which was also self-explained by the children, served as control condition. In a test after eight weeks, however, there were no main effects of spacing and worked examples, and no interaction\u0026mdash;despite sufficient power. At least two reasons may account for the absent effects: (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) The inter-study interval was too short for the long retention interval (cf. 13), and (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) self-explaining also one\u0026rsquo;s own problem solving might have levelled out the effect of self-explaining worked examples.\u003c/p\u003e\u003cp\u003eThe present preregistered study (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://osf.io/k3e6w\u003c/span\u003e\u003cspan address=\"https://osf.io/k3e6w\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e) tested these explanations in a controlled setting with adults, acquiring a new mathematical procedure. The inter-study interval was extended to two weeks, and only worked examples were self-explained, but no longer students\u0026rsquo; own problem solving. Students\u0026rsquo; working memory and long-term memory capacities, their specific prior knowledge, and general math achievement served as covariates. We expected main effects of spacing and worked examples under these optimized conditions and, most importantly, that spaced practice including self-explaining worked examples would result in the best test performance after eight weeks.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eStudents\u0026rsquo; test performance, separately for procedural and conceptual knowledge in each condition, is depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Two ANCOVAs were computed with heteroscedasticity-consistent (HC3) robust standard errors to check whether spacing, worked examples, and their interaction affected students\u0026rsquo; procedural and conceptual knowledge in the test, while controlling for specific prior knowledge, math grade, long-term memory capacity, and working memory capacity (see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e for detailed results). The categorical independent variables were effect coded, and the continuous independent variables were mean centered.\u003c/p\u003e\u003cp\u003eConcerning procedural knowledge, the ANCOVA explained a significant but small amount of variance, \u003cem\u003eF\u003c/em\u003e(7, 200)\u0026thinsp;=\u0026thinsp;2.37, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.024, \u003cem\u003eR\u0026sup2;\u003c/em\u003e = 0.08, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eadj\u003c/sub\u003e\u003cem\u003e\u0026sup2;\u003c/em\u003e = 0.04. However, this was only due to the covariates specific prior knowledge and long-term memory capacity. Neither spacing nor worked examples significantly affected test performance, and their interaction was also not significant. Bayes Factor analyses, using Monte Carlo sampling and Jeffreys\u0026rsquo; prior, largely confirmed these results. There was moderate evidence for the null effects of spacing and of worked examples, and anecdotal evidence for the null effect of their interaction.\u003c/p\u003e\u003cp\u003eConcerning conceptual knowledge, the ANCOVA also explained a significant but small amount of variance, \u003cem\u003eF\u003c/em\u003e(7, 200)\u0026thinsp;=\u0026thinsp;2.91, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.006, \u003cem\u003eR\u0026sup2;\u003c/em\u003e = 0.09, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eadj\u003c/sub\u003e\u003cem\u003e\u0026sup2;\u003c/em\u003e = 0.06. Again, this was only due to the effect of covariates, mainly math grade, and neither spacing nor worked examples yielded a significant effect, and their interaction was not significant. Bayesian analyses revealed moderate evidence for the null effects of spacing and of worked examples, but no evidence in favor of or against the null effect of their interaction.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cem\u003eViolin plots depicting students\u0026rsquo; procedural and conceptual knowledge in the test (% correct), separately for each condition\u003c/em\u003e\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eNote\u003c/strong\u003e\u003cp\u003eMAS\u0026thinsp;+\u0026thinsp;PS\u0026thinsp;=\u0026thinsp;Massed practice / Pure problem solving (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;51); SP\u0026thinsp;+\u0026thinsp;PS\u0026thinsp;=\u0026thinsp;Spaced practice / Pure problem solving (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;52); MAS\u0026thinsp;+\u0026thinsp;WE\u0026thinsp;=\u0026thinsp;Massed practice / Self-explaining worked examples (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;48); SP\u0026thinsp;+\u0026thinsp;WE\u0026thinsp;=\u0026thinsp;Spaced practice / Self-explaining worked examples (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;57).\u003c/p\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003e\u003cem\u003eResults of the ANCOVAs (one for each knowledge type), with corresponding Bayes factors indicating evidence in favor of the null hypothesis.\u003c/em\u003e\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"9\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eVariables\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"4\" nameend=\"c5\" namest=\"c2\"\u003e\u003cp\u003eProcedural Knowledge\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"4\" nameend=\"c9\" namest=\"c6\"\u003e\u003cp\u003eConceptual Knowledge\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e\u003cem\u003ep\u003c/em\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\eta\\:}_{p}^{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e\u003cem\u003eBF\u003c/em\u003e\u003csub\u003e\u003cem\u003e01\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e\u003cem\u003eF\u003c/em\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e\u003cem\u003ep\u003c/em\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\eta\\:}_{p}^{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003e\u003cem\u003eBF\u003c/em\u003e\u003csub\u003e\u003cem\u003e01\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSpacing\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.11\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.736\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e5.79 (4.29%)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.949\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e6.72 (3.71%)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eWorked Examples\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.936\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e4.71 (4.34%)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.08\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.774\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e4.71 (3.74%)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSpacing \u0026times; Worked Examples\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.69\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.056\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.02\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e1.00 (4.55%)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e1.61\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.205\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e2.29 (3.79%)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003ctfoot\u003e\u003ctr\u003e\u003ctd colspan=\"9\"\u003e\u003cem\u003eNote. df\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 200 for each knowledge type; \u003cem\u003eBF\u003c/em\u003e\u003csub\u003e\u003cem\u003e01\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;Bayes factor (1\u0026ndash;3: anecdotal evidence, 3\u0026ndash;10: moderate evidence in favor of the null hypothesis). Percentage error estimates of Bayes factors in parentheses.\u003c/td\u003e\u003c/tr\u003e\u003c/tfoot\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe present study investigated whether spacing fosters lasting learning in mathematics under optimized conditions. Based on meta-analyses suggesting that the spacing effect in mathematics is not as robust as in other learning domains (e.g., verbal learning), we tested whether it can be evoked by cognitively enriching the (spaced) practice phases and by adjusting the inter-study interval to the long retention interval of eight weeks. Learners self-explained worked examples during practice (in addition to solving corresponding math problems), and the inter-study interval between the practice phases was set to two weeks. However, in the test after eight weeks, there was neither an effect of worked examples, compared to pure problem solving without worked examples, nor an effect of spaced compared to massed practice, and no interaction effect. The results were largely confirmed by Bayesian analyses with one exception that for procedural knowledge, the null effect of the interaction could neither be confirmed nor rejected. Probably, the sample size was too small to allow for a clear conclusion. However, from a practical perspective the potential interaction is no longer relevant because the effect would smaller than the desired effects of educational interventions (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThe findings align with other studies revealing no spacing effect in mathematics (e.g., 13; cf. 9). They furthermore suggest that the acquisition of more complex learning content might require other boundary conditions to benefit from spaced practice. On closer inspection, it becomes evident that learning mathematics differs from learning simpler content, such as vocabulary, for which a spacing effect appears robustly. First, mathematics comprises both conceptual (i.e., \u0026ldquo;knowing why\u0026rdquo;, also including facts) and procedural knowledge (i.e., \u0026ldquo;knowing how\u0026rdquo;). However, studies on the spacing effect in mathematics often involved only pure problem solving, requiring the retrieval of procedural knowledge, but not necessarily the retrieval of the corresponding concept (e.g., the formula). Vocabulary learning, in contrast, refers only to conceptual knowledge. Study-phase retrieval is one of the driving principles of the spacing effect (e.g., 15). Second, when practicing mathematics, learners usually solve different, isomorphic problems that refer to the same underlying concept, whereas in verbal learning identical items are presented repeatedly. Accordingly, practice in mathematics includes fewer item-based retrieval cues (e.g., the same digits or context). These two issues might contribute to the finding that the spacing effect is less robust for more complex learning contents, as long as practice is designed as described, because the retrieval processes, evoked by spaced practice sessions, differ.\u003c/p\u003e\u003cp\u003eIt might thus be promising to explicitly demand the retrieval of procedural \u003cem\u003eand\u003c/em\u003e conceptual knowledge when trying to evoke a spacing effect for more complex material, and to provide learners with retrieval cues that work in a similar way as when presenting identical items, such as in verbal learning. Moreover, it seems necessary to assess procedural and conceptual knowledge separately as dependent variables, as in the present study, to uncover whether spaced practice of more complex learning material affects both types of knowledge in different ways. It is, for example, possible to solve an arithmetic task correctly in a rather automatized way without a grasp of the underlying concept. In turn, learners might know the concept but make a simple calculation error, which leads to an incorrect solution.\u003c/p\u003e\u003cp\u003eTo sum up, spaced practice is a learning principle easy to implement in real educational contexts. Identifying ways to make it effective for more complex learning material would support the acquisition of lasting knowledge in schools and universities.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003e\u003cb\u003eParticipants.\u003c/b\u003e The required sample size was computed for the ANCOVAs with three tested predictors (i.e., spacing and worked examples, and their interaction) and seven predictors in total by means of G*Power (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e). To detect a small interaction effect of \u003cem\u003ef\u0026sup2;\u003c/em\u003e = 0.04 between spacing and worked examples, which corresponds to the minimum desired effect size for educational interventions of \u003cem\u003ed\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.40 (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e), with α\u0026thinsp;=\u0026thinsp;0.05, and a power of 1-β\u0026thinsp;=\u0026thinsp;0.90, \u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;265 participants were required. Given that prior knowledge and math grade explained on average 18% of the variance of the test performance in a similar study with children (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e), the required sample size was conservatively adjusted according to (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e) by the factor 1-\u003cem\u003eR\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.82 to \u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;217.\u003c/p\u003e\u003cp\u003eUniversity students (\u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;272) from different study programs were recruited. They participated voluntarily with informed consent and received either course credits or 25\u0026euro;. Only students who attended all three sessions were included in the analyses, resulting in a final sample of \u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;208 (161 women, 43 men, 3 diverse, 1 not specified; age: \u003cem\u003eM\u003c/em\u003e\u0026thinsp;=\u0026thinsp;24.0 years, \u003cem\u003eSD\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4.9).\u003c/p\u003e\u003cp\u003e\u003cb\u003eStudy design and procedure.\u003c/b\u003e The experiment followed a 2 (practice schedule: spaced vs. massed) \u0026times; 2 (consolidation condition: problem solving with self-explaining worked examples vs. pure problem solving) between-subjects design. Procedural and conceptual knowledge served as dependent variables. The study was conducted as computer-based experiment using Labvanced, with matrix multiplication as learning content. Students were randomly assigned to one of the four experimental practice groups (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;48\u0026ndash;57).\u003c/p\u003e\u003cp\u003eThe first session took place in a group setting in the laboratory. Students first received two problems referring to matrix calculation to check whether they were already familiar with the procedure. This was the case for one student who was excluded from the experiment.\u003c/p\u003e\u003cp\u003eThereafter, students followed a computer-based video tutorial (6 min) introducing matrix calculation. Immediately after the tutorial, the practice phase started. Students in the massed practice conditions completed all practice tasks (i.e., set 1 and 2) at once, whereas those in the spaced conditions completed half of the tasks during the first session (i.e., set 1) and the remaining tasks (i.e., set 2) in the second session two weeks later.\u003c/p\u003e\u003cp\u003eIn the pure problem-solving conditions, students had to solve eight matrix multiplication problems corresponding to the introduction. In the worked examples conditions, students were presented first with a worked example of a matrix multiplication (which they did not have to self-explain) and solved a corresponding problem to allow for practicing procedural knowledge. This problem was followed by another worked example that had to be self-explained by the students in writing, aiming at enhancing their conceptual knowledge. Self-explaining was prompted by four questions per example (e.g., \u0026ldquo;Why is 0 x 9 written at the position of the first row and second column?\u0026rdquo;). In total, students in the worked examples condition solved four matrix multiplication problems and self-explained four corresponding worked examples. All practice tasks were completed without further tutorials. However, students received corrective feedback on two problems (i.e., on the first problem of practice set 1 and 2, respectively) in form of the correct solution paths and solution.\u003c/p\u003e\u003cp\u003eThe second practice session in the spaced practice conditions, taking place two weeks after the first one, and the delayed test were completed individually on participants\u0026rsquo; laptops or computers. To ensure that participants followed the instructions and did not use additional aids, the experimenter supervised them via the videoconference tool Zoom.\u003c/p\u003e\u003cp\u003eEight weeks after the final practice session, the delayed test was administered again online and supervised via Zoom. It was announced as a further practice session to prevent participants to prepare themselves, consisting of 14 tasks. After the test, students\u0026rsquo; individual characteristics were assessed (except of their specific prior knowledge that had been assessed in the first practice session).\u003c/p\u003e\u003cp\u003e\u003cb\u003eMeasures.\u003c/b\u003e The material can be found here: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://osf.io/ahqb9/files/osfstorage/68d4151072459ba6843c33b2\u003c/span\u003e\u003cspan address=\"https://osf.io/ahqb9/files/osfstorage/68d4151072459ba6843c33b2\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e. Procedural knowledge in the delayed test was assessed by four pure problem-solving tasks similar to those in the practice phase and two self-explanation tasks concerning the procedure of matrix multiplication. The number of correct intermediate calculation steps and final solutions of the matrix multiplication problems served as indicator of procedural knowledge (max: 16 points).\u003c/p\u003e\u003cp\u003eConceptual knowledge was assessed by presenting students with two matrix multiplication tasks, each along with three options: (a) the correct solution steps and solution and (b) incorrect solution steps and solution and (c) the response option that this task is insoluble. Students had to choose the correct option, being informed that only one option applied, and to explain their decision in writing. In addition, students received one worked example marked as incorrect along with four questions, and one worked example marked as correct along with two questions, prompting students to self-explain the reasoning behind the solution steps. The number of correct answers in all these tasks served as indicator of conceptual knowledge (max: 18.5 points). Students\u0026rsquo; procedural and conceptual knowledge performance was transformed into % correct.\u003c/p\u003e\u003cp\u003eStudents\u0026rsquo; sociodemographic data, their general math achievement (self-reported final math grade at graduation), specific prior knowledge (their performance in the first practice session concerning procedural knowledge), working memory capacity (Digit Span Backwards test; 19), and long-term memory capacity (WMS-IV; 20) were assessed via computerized questionnaires and tasks.\u003c/p\u003e\u003cp\u003e\u003cb\u003eData analyses.\u003c/b\u003e We note that the models we preregistered under the label \u0026ldquo;multiple regression\u0026rdquo; are equivalent to the reported ANCOVAs. Initially, we additionally aimed at exploring whether the learner characteristics potentially moderated the effects of spacing and worked examples. However, we decided to focus on the analyses of the preregistered hypotheses regarding main effects of spacing and worked examples, and their interaction, two ANCOVAs were computed (one for procedural and one for conceptual knowledge). Students\u0026rsquo; specific prior knowledge, their general math achievement, their long-term and working memory served as covariates to enhance power. The alpha level was set to .05, the tests were two-sided. In the pre-registration, we erroneously corrected the alpha to .025 to account for the two analyses. However, as they refer to different dependent variables, this correction is not necessary. To validate the results, Bayesian analyses were conducted additionally, allowing for confirming null effects.\u003c/p\u003e\u003cp\u003eAll analyses were conducted using the software R (version 4.4.1) and RStudio (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e). The used packages can be found here: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://osf.io/ahqb9/files/osfstorage/68d503f74dbd891a77432b63\u003c/span\u003e\u003cspan address=\"https://osf.io/ahqb9/files/osfstorage/68d503f74dbd891a77432b63\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003ePrior to the main analyses, the assumptions of the ANCOVA were examined. Boxplots indicated no problematic outliers for grades, but five outliers for procedural knowledge, and three outliers for conceptual knowledge. These outliers were not excluded from the analyses, because there was no evidence to suggest that they resulted from measurement error, data entry mistakes, or other artifacts. We therefore considered these cases to represent genuine variation within the sample and retained them to preserve the integrity and generalizability of the results. Normality was evaluated with Shapiro\u0026ndash;Wilk tests, Q\u0026ndash;Q plots, and histograms. Results indicated significant deviations from normality for grades (\u003cem\u003eW\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.95, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;.001), procedural knowledge (\u003cem\u003eW\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.85, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;.001), and conceptual knowledge (\u003cem\u003eW\u003c/em\u003e\u0026thinsp;=\u0026thinsp;.95, \u003cem\u003ep\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;.001). Skewness statistics suggested a slight negative skew for grades (-0.62) and positive skew for procedural (1.18) and conceptual knowledge (.82). Given the robustness of ANCOVA to moderate violations of normality and skewness, all variables were retained for subsequent analyses.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eEthical approval for this study was obtained from the Ethics Committee of the Faculty of Human Sciences of the University of Kassel (Approval number: 202115).\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThis article is the product of the research unit \u0026ldquo;Lasting Learning: Cognitive Mechanisms and Effective Instructional Implementation\u0026rdquo; (FOR 5254, project no. 450142163), funded by the German Research Foundation (DFG: grant EB 462/4\u0026ndash;1).\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eA.L. developed and prepared the material, prepared the software, collected, curated and analyzed the data, created the figure, and wrote parts of the manuscript. M.E. developed the research idea, provided the funding, and wrote parts of the manuscript. A.E. and M.E. share the first authorship. F.S. provided substantial support concerning the statistical analyses. All authors read and approved the final manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThis article is the product of the research unit \u0026ldquo;Lasting Learning: Cognitive Mechanisms and Effective Instructional Implementation\u0026rdquo; (FOR 5254, project no. 450142163), funded by the German Research Foundation (DFG: grant EB 462/4\u0026ndash;1). The funder played no role in study design, data collection, analysis and interpretation of data, or the writing of this manuscript. We thank Freyja Pollack, Lena Ayse Ro\u0026szlig;, Lilli Berger, Nike Kliewe, and Sarah Ewert for supporting the data collection.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data supporting the findings of this study, including experimental data collected via Labvanced and the data analysis scripts, are openly available on the Open Science Framework (OSF) at the following links: raw data: https://osf.io/ahqb9/files/osfstorage/68d50035c48a490ff1c94a14; data used for analysis: https://osf.io/ahqb9/files/osfstorage/68d40264af567ee6ce3c30a6; analysis scripts: https://osf.io/ahqb9/files/osfstorage/68d3fce148906622d4604301. All analyses were conducted using the open-source software R (version 4.4.1; R Core Team, 2024) in RStudio. The OSF repository provides full access to both the original data and the analysis code, ensuring transparency and reproducibility.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBaddeley, A., Eysenck, M. W., \u0026amp; Anderson, M. C. \u003cem\u003eMemory\u003c/em\u003e (Psychology Press, 2009).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBjork, E. L., \u0026amp; Bjork, R. A. Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning (eds. Gernsbacher, M. A., Pew, R. W., Hough, L. M. \u0026amp; Pomerantz, J. R.) 56\u0026ndash;64 (Worth, 2011).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eCepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., \u0026amp; Rohrer, D. Distributed practice in verbal recall tasks: A review and quantitative synthesis. \u003cem\u003ePsychol. 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Vienna, Austria: R Foundation for Statistical Computing; \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.R-project.org/\u003c/span\u003e\u003cspan address=\"https://www.R-project.org/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e (2024).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":true,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7747941/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7747941/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe investigated whether practicing mathematics in a spaced manner, while additionally self-explaining worked examples, fosters lasting learning in mathematics in adults (\u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;208). The test after eight weeks revealed no main effects and no interaction, suggesting that the spacing effect for mathematics is not robust. We discuss how mathematics differs from other learning domains and how these differences need to be considered to evoke a spacing effect in mathematics.\u003c/p\u003e","manuscriptTitle":"The difficulty to evoke the spacing effect in mathematics: New findings and theoretical considerations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-22 11:10:34","doi":"10.21203/rs.3.rs-7747941/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"12830347-6a9c-49bf-9328-de27a79cc0f8","owner":[],"postedDate":"October 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":56490534,"name":"Physical sciences/Mathematics and computing"},{"id":56490535,"name":"Physical sciences/Physics"},{"id":56490536,"name":"Biological sciences/Psychology"},{"id":56490537,"name":"Social science/Psychology"}],"tags":[],"updatedAt":"2025-11-20T00:53:21+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-22 11:10:34","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7747941","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7747941","identity":"rs-7747941","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}