No Performance Benefit of Random Over Table-ordered Multiplication Training in Children: Does Memory Interference Between Facts Constrain Arithmetic Learning?

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Alice Boutros, David Müller, Jérôme Prado, Catherine Thevenot This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7139588/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract Interference between multiplication facts is often mentioned as a key source of children’s difficulty in learning and retaining multiplication tables. Shared operands and results (e.g., 6 × 4 = 24 and 8 × 3 = 24; 7 × 8 = 56 and 9 × 6 = 54) are indeed thought to create overlap between facts, leading to blurred representations in memory. One proposed way to reduce such interference is to practice facts in a mixed order across tables (e.g., 6 × 7 = 42; 3 × 8 = 24; 4 × 9 = 36) rather than table by table (e.g., 6 × 7 = 42; 6 × 8 = 48; 6 × 3 = 18). This approach was tested in the present study, in which sixth graders practiced multiplication facts from x 3 to x 9 tables either by table (N = 81) or in a mixed order (N = 62). Their performance on a multiplication fluency task was compared to that of a control group engaged in regular classroom activities (N = 71). Children in the mixed-order condition showed less improvement from pre- to post-test than those in the table-by-table condition and did not progress more than the control group. These findings show that randomly practicing multiplications facts may not be a productive pedagogical method for relearning. They more broadly suggest that the role of interference in multiplication learning may not be as central as commonly thought. Physical sciences/Mathematics and computing Biological sciences/Psychology Social science/Psychology Numerical cognition Arithmetic Training program Memory interference Retrieval from memory Multiplication facts Figures Figure 1 Figure 2 Figure 3 INTRODUCTION When individuals solve even simple arithmetic problems, they slow down and make more errors as problem size increases, a phenomenon known as the problem-size effect. In the case of single-digit multiplication problems, whose answers are typically learned by heart in childhood, this effect is generally attributed to interference among stored facts (Campbell, 1987; Campbell & Graham, 1985; De Visscher & Noël, 2014a). These interferences are thought to arise from the way multiplication facts are represented in an interconnected mental network, where shared digits across operands and products lead to substantial overlap (e.g., 6 × 7 = 42; 7 × 6 = 42; 6 × 8 = 48). While the degree of interference between facts does not strictly align with problem size, larger problems nonetheless tend to overlap more with one another than smaller ones (De Visscher & Noël, 2014a; De Visscher & Noël, 2014b). Retrieval of multiplication answers is therefore more difficult for larger problems than for smaller ones, resulting in a problem-size effect. Interference caused by shared surface features between problems are not the only type of interference that could explain the size effect. For example, Siegler proposed a model (1998) in which the relative difficulty of fact retrieval depends on the number and strength of incorrect answers that have become associated with a given problem during learning. The more often a wrong answer has been produced in the past, the more strongly it is activated during retrieval, making errors or retrieval failure more likely. Calculating the correct answer for large problems when retrieval fail requires more steps than for smaller ones, which increases the likelihood of errors and creates the problem-size effect. In Siegler’s model, interference arises from the activation of competing erroneous responses to a given problem, rather than from the overlap between problems. Although the models described above do not explain the problem size effect by referring to the same source of interferences (i.e. shared surface features versus competition between answers with different activation strengths), they are not necessarily mutually exclusive. However, favoring one over the other has implications for the design of training programs. Indeed, reducing interference between problems as opposed to reducing the number of competing answers associated with a single problem calls for different intervention approaches. If the number of competing answers associated with a single problem is to be reduced, drill practice combined with immediate feedback is likely the most effective approach. For example, if the answer 54 is produced for the problem 7 × 8, immediate correction to the correct answer, 56, followed by repeated practice of this association over time, should strengthen the link between the problem and the correct response, until competing answers are no longer activated or are too weak to interfere with accurate retrieval. This type of training has been shown to effectively improve children’s multiplication skills (e.g., Woodward, 2006). However, this approach is unlikely to reduce interference between problems. To address this issue, researchers have instead focused on manipulating the degree of overlap between problems during training. For instance, Heidekum et al. (2021) trained adults with low arithmetic skills using sets of either low-interference problems (e.g., 3 × 9 = 27 and 6 × 8 = 48) or high-interference problems (e.g., 6 × 7 = 42 and 6 × 8 = 48). The authors showed that high-interference problems were solved slower than low-interference problems, but only in participants with low arithmetic skills. In such participants, and after 5 days of training, the interference effect reduced but did not disappear. Nevertheless, for both individuals with low and high arithmetic skills, solution times of multiplication problems were shorter after than before training, whatever the condition. Dotan and Friedmann (2019) adopted a similar approach in a single-case study involving a woman who was particularly susceptible to interference effects. They found that when the patient rehearsed 4 multiplication facts per week, she learned more quickly and effectively when the facts were dissimilar rather than overlapping. Based on these findings, the authors called for a reconsideration of how multiplication facts are taught in elementary school, arguing that learning multiplication tables in a fixed, ordered manner may actually increase interference between facts. They investigated this possibility in a study involving 17 children aged 6 to 8 years who had not yet learned multiplication facts (Dotan & Zviran-Ginat, 2022). Over four weeks, the children learned four multiplication facts per week, with the level of interference alternating weekly between low and high. The results showed the emergence of a learning advantage for low-interference facts on the last training day, which was particularly apparent two days after the end of training. This effect of degree of interference between facts was still present, albeit smaller, 7 weeks after training has ended. Despite these promising results, the authors acknowledged that these results have several limitations, including a relatively small sample, a narrow range of arithmetic facts studied, and a sample that may not be representative of the broader population (children were recruited via social networks and children who could complete the task disproportionately consisted of children with higher attentional capacities). The authors therefore suggested that follow up studies should compare learning a wider range of multiplication facts organized within tables (high interfering condition) versus not organized within tables (low interfering condition) in a real pedagogical setting (i.e., in the classroom). This is precisely what we did in the present study, in which a total of 143 sixth graders practiced all multiplication facts from tables 3 to 9 during each of eight sessions over four weeks, either in the order of the tables they belong to or in a random order (referred to hereafter as the mixed condition). We chose to focus on sixth graders who had already learned multiplication tables in previous years (rather than to introduce these facts to younger children learning them for the first time) for two reasons. First, we needed participants with sufficient attentional and memory resources to handle a relatively large number of multiplication facts (i.e., 49 in our design) in a relatively limited number of sessions. Second, because learning multiplication facts in an ordered manner may help children grasp the conceptual basis of multiplication (Dotan & Zviran-Ginat, 2022), notably the concept of repeated addition (Park & Nunes, 2001), we were concerned that presenting facts in a random order could prevent younger children from developing this understanding. If interference between facts is a key factor underlying the difficulty of learning multiplication problems (or relearning them in our case as participants were 6 th graders) (Campbell, 1987; Campbell & Graham, 1995), then training multiplication facts in a mixed order should yield better outcomes than in a fixed-order table by table. However, if both types of training lead to similar improvements compared to a business-as-usual control group (n=71), this could suggest that interference between problems plays a less central role in multiplication (re)learning than previously assumed. This would suggests another source of difficulty, perhaps in the problems themselves and the history of errors associated with them (Siegler, 1987). METHODS Participants Only children who were present at both pre- and post-test were included in the study, resulting in a sample of 214 sixth graders (112 girls) from diverse socio-economic backgrounds. They ranged in age from 10 years and 2 months to 13 years and 4 months (M = 11 years and 9 months, SD = 5 months). One child had skipped a grade, two had repeated one grade, and one had repeated two grades. However, 94.4% of the sample were of typical age for sixth grade (i.e., between 10 years and 11 months and 12 years and 7 months) at the time of testing. Children were distributed across 10 classrooms in 7 schools located in central France and were randomly assigned to the control group (N = 71), the ordered by table group (N = 81) or the mixed group (N = 62). For all children, written informed consents were obtained from the legal guardians before the experiment and the study was approved by the Ethics Research Committee of the Faculty of Social and Political Sciences of the University of Lausanne (Decision number C_SSP_122022_00011). All experiments were conducted in accordance with the applicable guidelines and regulations of the country where the research was conducted. Material and Procedure All children were pre- and post-tested using a modified version of one of the arithmetic subsets from the French Kit (French, Ekstrom, & Price, 1963), retaining only the multiplication and excluding the subtraction problems (see Appendix). They had to solve as many column-form multiplications as possible within 2 minutes. Each problem involved multiplying a two-digit number by a single-digit number, either with carrying (e.g., 49 × 6) or without (e.g., 83 × 3). The problems were presented in lines, and children had to solve them from left to right without skipping any. The test was administered collectively in class by the teachers, who announced the start and monitored the timing, stopping all students after exactly 2 minutes. Children’s scores on the test corresponded to the number of correctly solved problems within this time limit. The pre- and post-tests were separated by approximately four weeks (Figure 1). In the control group, children continued with regular classroom activities between these two testing points. In both the ordered by table group and the mixed group, children practiced multiplication facts during 8 sessions between pre- and post-tests. The sessions were organized 2 or 3 times a week and lasted about 10 minutes. To avoid the time and logistical demands of having teachers work face-to-face with each individual child, children were paired into dyads. More precisely, during each training session, one child was quizzed by a classmate acting as a repetitor. The roles of repetitor child versus quizzed child were randomly assigned and remained fixed throughout the intervention. Only the data from the children who were quizzed are taken into account in the present study. The repetitor read each multiplication problem aloud from a list, and the quizzed child was instructed to respond out loud as quickly and accurately as possible. If the child took more than approximately five seconds to answer or gave an incorrect response, the repetitor immediately provided the correct answer written on the list after each problem. In cases where a repetitor was absent, the teacher was instructed to take their place. Each session involved practicing 49 multiplication problems, with the order of presentation depending on the experimental condition (ordered-by-table versus mixed). In the ordered-by-table condition, children practiced multiplication facts one table at a time, progressing from session to session from the 3 times table up to the 9 times table. On session 1, for example, they focused on the 3 times table. The facts (3 × 3, 3 × 4, 3 × 5, 3 × 6, 3 × 7, 3 × 8, and 3 × 9) were first presented in a random order, then repeated six more times, each time in a new random order, resulting in seven practice rounds of the same set of facts during that session. On session 2, the same procedure was applied to the 4 times table, and so on, up to the 9 times table on the eighth session. In the mixed condition, children practiced all 49 problems constructed with operands from 3 to 9 in a random order during each of the 8 sessions. As in the table ordered condition, each of the fact was presented 8 times per session. A different random order of the facts was used for each session. All sessions were implemented during regular class time and were supervised by the classroom teachers. The researchers were not present during the sessions, and their role was limited to designing the materials and providing training and written guidelines to the teachers. All completed pre- and post-test forms from the three conditions were returned to the researchers by the teachers, either scanned and sent by email or delivered physically by postal mail. RESULTS A 2 (Test: pre- and post) x 3 (Group: control, ordered and mixed) ANOVA, with Test as a within-subjects factor and Group as a between-subjects factor, was conducted on children’s scores on the fluency multiplication task. There was a main effect of Test, showing higher scores at post- (7.27) than pre-test (5.52), F(1, 213) = 68.22, η 2 p = .244 , p < .001. This effect was qualified by a significant Test × Group interaction, F (2, 211) = 5.79, p = .004, η²ₚ = .052. As shown in Figure 2, the improvement from pre- to post-test was greater in the ordered condition than in the mixed and control conditions, which showed similar, smaller gains. Unfortunately, as shown Figure 2, the initial pre-test scores of children in the ordered group were lower than those of children in the other groups. This discrepancy left more room for improvement in that group, which may have contributed to the observed effects. To address this issue, we conducted a second analysis in which children were matched on their pre-test scores. Specifically, we selected only those children who had identical pre-test scores across conditions. When the number of children with a given score differed between groups, a random selection was made to retain an equal number of children per group for that score, and the remaining children were excluded. This resulted in a final dataset of 49 participants per group (147 children in total), on which we conducted the same ANOVA as in the full-sample analysis. A 2 (Test: pre- and post) x 3 (Group: control, ordered and mixed) ANOVA conducted on this subsample showed a main effect of Test, with higher scores at post- (6.61) than pre-test (4.78), F (1, 146) = 54.11, η 2 p = .273 , p < .001. This effect was qualified by a significant Test × Group interaction, F (2, 212) = 5.53, η²ₚ =.071 , p= .005. As attested by Figure 3, the same pattern of results as for the whole sample was obtained, that is, higher multiplication score improvement after training for the ordered group than for the control and mixed groups. Once again, the improvement was the same for the control and mixed groups. Critically, the results obtained from the subsample of participants matched on pre-test scores could depend on the specific random selection. To address this concern, we replicated the matching procedure across four different random selections. This yielded a significant Test × Group interaction in three out of four cases (p = .005, p = .015, and p = .011), and a marginal effect in the fourth (p = .081). These findings indicate that the reported effects are not attributable to an arbitrary sample composition. DISCUSSION In this research, we aimed to determine whether training sixth-grade children on multiplication tables in a random order was more effective than training them table by table. The underlying idea was that presenting facts in a non-sequential order would reduce interference between them, thereby facilitating relearning. Our results show that it is not the case. Practicing the facts in a random order was less efficient than practicing them table by table. In fact, as attested by the lack of difference in the pre- versus post-test effect between the random and the control (business as usual) condition, random practice did not lead to any positive effect of the intervention. Reducing interference between facts by presenting them in a random order thus appears to be an ineffective strategy for boosting learning of multiplication tables, or at least relearning in our case, as children had already acquired these facts in the past. We can see at least three possibilities to explain these results. First, the random presentation may conflict with the structured network of associations that was already established in long-term memory in these children, which could explain the absence of any positive effect in the random condition. It is possible that younger children who have not yet begun to learn multiplication facts might still show beneficial effects of random practice, as stable table-like representations are not yet formed. However, as previously noted, and as also pointed out by Dotan and Zviran-Ginat ( 2022 ), such an approach should be adopted with caution. Random presentation of multiplication facts may indeed prevent children from grasping key conceptual aspects of multiplication (Park & Nunes, 2001 ), particularly its interpretation as repeated addition or, maybe more likely, from capitalizing on already stored facts to solve new facts (6 x 7 = 42 because it is 6 x 6 + 6, LeFevre et al., 1996 ). Another interpretation of our results, which is not necessarily incompatible with the previous one, is that reducing interference between facts may be less important for learning than previously assumed. This interpretation carries significant weight, as it challenges the widely held view that the problem-size effect in multiplication arises because larger facts are more susceptible to interference than smaller ones (e.g., Campbell, 1987 ). The difficulty of a given multiplication problem may indeed depend more on its intrinsic characteristics than on its relation to other facts. For example, it is well established that associations between operands and answers are harder to construct for large problems than for smaller ones, partly because large problems take longer to solve in the early stages of learning (Thevenot et al., 2001 ). Moreover, as previously discussed, larger problems are more likely to elicit incorrect responses, which may become associated with the problem alongside the correct answer, thereby reinforcing competing associations and hindering retrieval (e.g., Siegler, 1988 ). In fact, assuming that the intrinsic characteristics of multiplication problems are the primary source of their difficulty (Ashcraft & Guillaume, 2009 ), and that interference between problems play only a secondary role, could help explain some puzzling findings in the literature. For example, while size effects are mainly attributed to interference between problems, the absence of such effects is difficult to account for. Indeed, as interferences necessarily increase with the size of the problems, size effects should be consistently observed. In contrast, if the difficulty of a given problem reflects its specific learning history, such as nosiness due to strongly associated incorrect answers, multiple competing responses, or consistently long solution times, then targeted, repeated practice could reduce this difficulty. In such a case, even a large problem could eventually become as easy to retrieve as a smaller one if practiced enough, and size effects could vanish. As matter of fact, such absence of size effects has repeatedly been observed in adults for addition with sums 8, 9 and 10 (Bagnoud et al., 2021 ; Díaz-Barriga Yáñez et al., 2023 ; Poletti et al., 2023 ; Uittenhove et al., 2016 ) and was interpreted as reflecting constant retrieval times for problems with these sums (see however Chen & Campbell, 2018 ; Baroody, 2019; Thevenot & Barrouillet, 2020 for a debate about this interpretation). A final interpretation of our results is that learning multiplication facts in random order, rather than table by table, may simply not be sufficient to reduce interference between facts. Indeed, certain problems are frequently confused by individuals, such as 8 × 7 = 56 and 9 × 6 = 54 , despite belonging to different tables. While presenting problems in a random order across tables does reduce interference relative to table-by-table learning, it may not do so effectively enough. A more theoretically grounded randomization, based on models of fact similarity (De Visscher & Noël, 2014a , De Visscher & Noël, 2014b ), might be required to achieve a meaningful reduction in interference. Nevertheless, this interpretation appears unlikely, given that the mixed condition showed no positive effect at all, even though it should have, at the very least, produced a modest reduction in interference. To sum up and conclude, the results reported here are noteworthy for several reasons. From an educational perspective, they argue against learning (or at least relearning) multiplication facts in a random order, as this approach proved here less effective than practicing them table by table. As discussed above, it may be due to a specificity of our population (6th graders), for which a memory network already organized by table exists and may be resistant to restructuring, making random relearning not productive. Stated differently, it is possible that introducing facts in a random order conflicts with previously established representations, thereby limiting the benefits of such practice. However, from a theoretical point of view, our results may also suggest that interference between problems may be less critical than interference or competitive retrieval among answers within a given problem to explain the relative retrieval difficulty of multiplication problems. This could explain why reducing interference between problems during relearning does not lead to efficient outcomes. Alternatively, or complementarily, although this seems unlikely, practicing multiplication facts across rather than within tables may nevertheless be insufficient to reduce interference between facts. Future research should examine these hypotheses more directly, potentially by comparing the effects of different frequency level of drill-based learning on high and lower interferent problems. Declarations Funding: There was no specific funding for this research Author Contribution A.B., J.P, & C.T. wrote the manuscriptA.B., prepare the figuresD.M. & C.T. conceived the experimentA.B ,D.M., & J.P. analyzed the resultsC.T. & J.P. supervised the analyses C.T. supervised the work Acknowledgement We would like to thank Catherine Simon and Franck Verdier as well as all the members of the Clermont-Ferrand IREM (Research Institute on Mathematics Teaching) for their precious collaboration. Data Availability The dataset used in this research is available upon request by contacting [email protected] . References Ashcraft, M. H. & Guillaume, M. M. Mathematical cognition and the problem size effect. In B. H. Ross (Ed.), The Psychology of Learning and Motivation (pp. 121–151). Elsevier Academic Press. (2009). https://doi.org/10.1016/S0079-7421(09)51004-3 Bagnoud, J., Dewi, J., Castel, C., Mathieu, R. & Thevenot, C. Developmental changes in size effects for simple tie and non-tie addition problems in 6- to 12-year-old children and adults. J. Exp. Child Psychol. 201 , 104987. https://doi.org/10.1016/j.jecp.2020.104987 (2021). Baroody, A. J. & Chen A commentary on and Campbell (2017): Is there a clear case for addition fact recall? Psychonomic Bulletin & Review, 25 (6), 2398–2405. (2018). https://doi.org/10.3758/s13423-018-1440-y Campbell, J. I. D. Network interference and mental multiplication. J. Experimental Psychology: Learn. Memory Cognition . 13 (1), 109–123. https://doi.org/10.1037/0278-7393.13.1.109 (1987). Campbell, J. I. D. & Graham, D. J. Mental multiplication skill: Structure, process, and acquisition. Can. J. Psychol. 39 (2), 338–366. https://doi.org/10.1037/h0080065 (1985). Chen, Y. & Campbell, J. I. D. Compacted procedures for adults' simple addition: A review and critique of the evidence. Psychon. Bull. Rev. 25 (2), 739–753. https://doi.org/10.3758/s13423-017-1328-2 (2018). De Visscher, A. & Noël, M. P. The detrimental effect of interference in multiplication facts storing: Typical development and individual differences. J. Exp. Psychol. Gen. 143 (6), 2380–2400. https://doi.org/10.1037/xge0000029 (2014a). De Visscher, A. & Noël, M. P. Arithmetic facts storage deficit: the hypersensitivity-to-interference in memory hypothesis. Dev. Sci. 17 (3), 434–442. https://doi.org/10.1111/desc.12135 (2014b). Díaz-Barriga Yáñez, A. et al. Neural evidence for procedural automatization during cognitive development: Intraparietal response to changes in very-small addition problem-size increases with age. Dev. Cogn. Neurosci. 64 , 101310. https://doi.org/10.1016/j.dcn.2023.101310 (2023). Dotan, D. & Friedmann, N. Reducing interference improves the memorization of multiplication facts in case of hypersensitivity to interference. J. Numer. Cognition . 5 (3), 400–430. https://doi.org/10.5964/jnc.v5i3.203 (2019). Dotan, D. & Zviran-Ginat, S. 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Poletti, C., Díaz-Barriga Yáñez, A., Prado, J. & Thevenot, C. The development of simple addition problem solving in children: Reliance on automatized counting or memory retrieval depends on both expertise and problem size. Journal of Experimental Child Psychology, 234 , 105710. https://doi.org/j.jecp.2023.105710. (2023). Siegler, R. S. Strategy choice procedures and the development of multiplication skill. J. Exp. Psychol. Gen. 117 (3), 258–275. https://doi.org/10.1037//0096-3445.117.3.258 (1988). Thevenot, C. & Barrouillet, P. Are small additions solved by direct retrieval from memory or automated counting procedures? A rejoinder to Chen and Campbell (2018). Psychonomic Bulletin & Review, 27 , 1416–1418. (2020). https://doi.org/10.3758/s13423-020-01818-4 Thevenot, C., Barrouillet, P. & Fayol, M. Algorithmic solution of arithmetic problems and operands: Answer associations in long-term memory. Q. J. Experimental Psychol. A: Hum. Experimental Psychol. 54A (2), 599–611. https://doi.org/10.1080/02724980042000291 (2001). Uittenhove, K., Thevenot, C. & Barrouillet, P. Fast automated counting procedures in addition problem solving: When are they used and why are they mistaken for retrieval? Cognition 146 , 289–303. https://doi.org/10.1016/j.cognition.2015.10.008 (2016). Woodward, J. Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learn. Disabil. Q. 29 (4), 269–289. https://doi.org/10.2307/30035554 (2006). Additional Declarations No competing interests reported. Supplementary Files Appendix.docx Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 03 Feb, 2026 Editor invited by journal 26 Nov, 2025 Reviews received at journal 23 Oct, 2025 Reviews received at journal 08 Oct, 2025 Reviewers agreed at journal 09 Sep, 2025 Reviewers agreed at journal 06 Sep, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers invited by journal 24 Jul, 2025 Editor assigned by journal 22 Jul, 2025 Submission checks completed at journal 20 Jul, 2025 First submitted to journal 20 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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CNRS UMR5292, Université de Lyon","correspondingAuthor":false,"prefix":"","firstName":"Jérôme","middleName":"","lastName":"Prado","suffix":""},{"id":491004005,"identity":"9a605408-e9e5-4fc3-ad3f-39f055b4cbca","order_by":3,"name":"Catherine Thevenot","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIie3PMWvCQBTA8QcH53Ll1hcO9CucFERQ+lk8Ah1FcHEqdomLkDXil3ByKvQg4BToKrRDRcjeRRwC9QUVRa5x7XB/CLwL+d3lAHy+/xmnB89DGbMAun166YqdCTsRzntE8B65HrjQx03+IHLysfr+eWv3JUgaom5fNqY7NRggyPnYSTALa80kx2EwZrXmLHoeBtHDUiX0Y/hlnURDyJWwaBaW0ZClZrEiIoho7LmJ3HJVEHkvSZH9EhF5NUE6BcpT6PoKRrYkvJLgetsKpkSSlNEwCg3d57FDRODaTWRsctzbFxNPXmnQTyZm6eZTFN26TNzkErtZizvf+3w+n6+iAy0mT5shju95AAAAAElFTkSuQmCC","orcid":"","institution":"Université de Lausanne","correspondingAuthor":true,"prefix":"","firstName":"Catherine","middleName":"","lastName":"Thevenot","suffix":""}],"badges":[],"createdAt":"2025-07-16 11:38:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7139588/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7139588/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87850271,"identity":"af9ff74e-d4a9-4628-aa3e-70a4f4ab936b","added_by":"auto","created_at":"2025-07-29 15:48:47","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":32160,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eSchematic representation of the study design for the experimental and control groups.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7139588/v1/0dac250acf5f02384e0ecb52.png"},{"id":87850272,"identity":"f7a9b28d-f182-42e5-9aa7-56c2aca235bd","added_by":"auto","created_at":"2025-07-29 15:48:48","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":65873,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eMean number of correct responses at pre- and post-test in the control, ordered, and mixed groups\u003c/em\u003e\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7139588/v1/17709daaa2e4e58cf8dd133c.jpg"},{"id":87851595,"identity":"91c36146-56a1-49c1-9c7b-34aeca73b255","added_by":"auto","created_at":"2025-07-29 15:56:48","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":63092,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eMean number of correct responses at pre- and post-test in the control, ordered, and mixed groups in matched samples\u003c/em\u003e\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7139588/v1/fbbf088c2b570eea436084c4.jpg"},{"id":87852193,"identity":"64d70d45-bde8-4494-8b93-3c05c3bcc38b","added_by":"auto","created_at":"2025-07-29 16:04:52","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":569904,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7139588/v1/f57922c3-ac5a-4fbe-be21-ff2506c51195.pdf"},{"id":87850277,"identity":"d499afe4-e803-42e1-894d-e498d9802844","added_by":"auto","created_at":"2025-07-29 15:48:48","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":125506,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-7139588/v1/ef1830b4ca820df332a665c2.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eNo Performance Benefit of Random Over Table-ordered Multiplication Training in Children: Does Memory Interference Between Facts Constrain Arithmetic Learning?\u003c/p\u003e","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eWhen individuals solve even simple arithmetic problems, they slow down and make more errors as problem size increases, a phenomenon known as the problem-size effect. In the case of single-digit multiplication problems, whose answers are typically learned by heart in childhood, this effect is generally attributed to interference among stored facts (Campbell, 1987; Campbell \u0026amp; Graham, 1985; De Visscher \u0026amp; Noël, 2014a). These interferences are thought to arise from the way multiplication facts are represented in an interconnected mental network, where shared digits across operands and products lead to substantial overlap (e.g., 6 × 7 = 42; 7 × 6 = 42; 6 × 8 = 48). While the degree of interference between facts does not strictly align with problem size, larger problems nonetheless tend to overlap more with one another than smaller ones (De Visscher \u0026amp; Noël, 2014a; De Visscher \u0026amp; Noël, 2014b). Retrieval of multiplication answers is therefore more difficult for larger problems than for smaller ones, resulting in a problem-size effect.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eInterference caused by shared surface features between problems are not the only type of interference that could explain the size effect. For example, Siegler proposed a model (1998) in which the relative difficulty of fact retrieval depends on the number and strength of incorrect answers that have become associated with a given problem during learning. The more often a wrong answer has been produced in the past, the more strongly it is activated during retrieval, making errors or retrieval failure more likely. Calculating the correct answer for large problems when retrieval fail requires more steps than for smaller ones, which increases the likelihood of errors and creates the problem-size effect. In Siegler’s model, interference arises from the activation of competing erroneous responses to a given problem, rather than from the overlap between problems.\u003c/p\u003e\n\u003cp\u003eAlthough the models described above do not explain the problem size effect by referring to the same source of interferences (i.e. shared surface features versus competition between answers with different activation strengths), they are not necessarily mutually exclusive. However, favoring one over the other has implications for the design of training programs. Indeed, reducing interference between problems as opposed to reducing the number of competing answers associated with a single problem calls for different intervention approaches. If the number of competing answers associated with a single problem is to be reduced, drill practice combined with immediate feedback is likely the most effective approach. For example, if the answer 54 is produced for the problem 7 × 8, immediate correction to the correct answer, 56, followed by repeated practice of this association over time, should strengthen the link between the problem and the correct response, until competing answers are no longer activated or are too weak to interfere with accurate retrieval. This type of training has been shown to effectively improve children’s multiplication skills (e.g., Woodward, 2006). However, this approach is unlikely to reduce interference between problems.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTo address this issue, researchers have instead focused on manipulating the degree of overlap between problems during training. For instance, Heidekum et al. (2021) trained adults with low arithmetic skills using sets of either low-interference problems (e.g., 3 × 9 = 27 and 6 × 8 = 48) or high-interference problems (e.g., 6 × 7 = 42 and 6 × 8 = 48). The authors showed that high-interference problems were solved slower than low-interference problems, but only in participants with low arithmetic skills. In such participants, and after 5 days of training, the interference effect reduced but did not disappear. Nevertheless, for both individuals with low and high arithmetic skills, solution times of multiplication problems were shorter after than before training, whatever the condition. Dotan and Friedmann (2019) adopted a similar approach in a single-case study involving a woman who was particularly susceptible to interference effects. They found that when the patient rehearsed 4 multiplication facts per week, she learned more quickly and effectively when the facts were dissimilar rather than overlapping.\u0026nbsp;Based on these findings, the authors called for a reconsideration of how multiplication facts are taught in elementary school, arguing that learning multiplication tables in a fixed, ordered manner may actually increase interference between facts. They investigated this possibility in a study involving 17 children aged 6 to 8 years who had not yet learned multiplication facts (Dotan \u0026amp; Zviran-Ginat, 2022). Over four weeks, the children learned four multiplication facts per week, with the level of interference alternating weekly between low and high. The results showed the emergence of a learning advantage for low-interference facts on the last training day, which was particularly apparent two days after the end of training. This effect of degree of interference between facts was still present, albeit smaller, 7 weeks after training has ended. Despite these promising results, the authors acknowledged that these results have several limitations, including a relatively small sample, a narrow range of arithmetic facts studied, and a sample that may not be representative of the broader population (children were recruited via social networks and children who could complete the task\u0026nbsp;disproportionately consisted of children with higher attentional capacities).\u0026nbsp;The authors therefore suggested that follow up studies should compare learning a wider range of multiplication facts organized within tables (high interfering condition) versus not organized within tables (low interfering condition) in a real pedagogical setting (i.e., in the classroom).\u003c/p\u003e\n\u003cp\u003eThis is precisely what we did in the present study, in which a total of 143 sixth graders practiced all multiplication facts from tables 3 to 9 during each of eight sessions over four weeks, either in the order of the tables they belong to or in a random order (referred to hereafter as the mixed condition). We chose to focus on sixth graders who had already learned multiplication tables in previous years (rather than to introduce these facts to younger children learning them for the first time) for two reasons. First, we needed participants with sufficient attentional and memory resources to handle a relatively large number of multiplication facts (i.e., 49 in our design) in a relatively limited number of sessions. Second, because learning multiplication facts in an ordered manner may help children grasp the conceptual basis of multiplication (Dotan \u0026amp; Zviran-Ginat, 2022), notably the concept of repeated addition (Park \u0026amp; Nunes, 2001), we were concerned that presenting facts in a random order could prevent younger children from developing this understanding.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIf interference between facts is a key factor underlying the difficulty of learning multiplication problems (or relearning them in our case as participants were 6\u003csup\u003eth\u003c/sup\u003e graders) (Campbell, 1987; Campbell \u0026amp; Graham, 1995), then training multiplication facts in a mixed order should yield better outcomes than in a fixed-order table by table. However, if both types of training lead to similar improvements compared to a business-as-usual control group (n=71), this could suggest that interference between problems plays a less central role in multiplication (re)learning than previously assumed. This would suggests another source of difficulty, perhaps in the problems themselves and the history of errors associated with them (Siegler, 1987).\u003c/p\u003e"},{"header":"METHODS","content":"\u003cp\u003e\u003cstrong\u003eParticipants\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOnly children who were present at both pre- and post-test were included in the study, resulting in a sample of 214 sixth graders (112 girls) from diverse socio-economic backgrounds. They ranged in age from 10 years and 2 months to 13 years and 4 months (M = 11 years and 9 months, SD = 5 months). One child had skipped a grade, two had repeated one grade, and one had repeated two grades. However, 94.4% of the sample were of typical age for sixth grade (i.e., between 10 years and 11 months and 12 years and 7 months) at the time of testing. Children were distributed across 10 classrooms in 7 schools located in central France and were randomly assigned to the control group (N = 71), the ordered by table group (N = 81) or the mixed group (N = 62). \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFor all children, written informed consents were obtained from the legal guardians before the experiment and the study was approved by the Ethics Research Committee of the Faculty of Social and Political Sciences of the University of Lausanne (Decision number C_SSP_122022_00011). All experiments were conducted in accordance with the applicable guidelines and regulations of the country where the research was conducted.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMaterial and Procedure\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll children were pre- and post-tested using a modified version of one of the arithmetic subsets from the French Kit (French, Ekstrom, \u0026amp; Price, 1963), retaining only the multiplication and excluding the subtraction problems (see Appendix). They had to solve as many column-form multiplications as possible within 2 minutes. Each problem involved multiplying a two-digit number by a single-digit number, either with carrying (e.g., 49 \u0026times; 6) or without (e.g., 83 \u0026times; 3). The problems were presented in lines, and children had to solve them from left to right without skipping any. The test was administered collectively in class by the teachers, who announced the start and monitored the timing, stopping all students after exactly 2 minutes. Children\u0026rsquo;s scores on the test corresponded to the number of correctly solved problems within this time limit.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe pre- and post-tests were separated by approximately four weeks (Figure 1). In the control group, children continued with regular classroom activities between these two testing points. In both the ordered by table group and the mixed group, children practiced multiplication facts during 8 sessions between pre- and post-tests. The sessions were organized 2 or 3 times a week and lasted about 10 minutes. To avoid the time and logistical demands of having teachers work face-to-face with each individual child, children were paired into dyads. More precisely, during each training session, one child was quizzed by a classmate acting as a repetitor. The roles of repetitor child versus quizzed child were randomly assigned and remained fixed throughout the intervention. Only the data from the children who were quizzed are taken into account in the present study. The repetitor read each multiplication problem aloud from a list, and the quizzed child was instructed to respond out loud as quickly and accurately as possible. If the child took more than approximately five seconds to answer or gave an incorrect response, the repetitor immediately provided the correct answer written on the list after each problem. In cases where a repetitor was absent, the teacher was instructed to take their place.\u003c/p\u003e\n\u003cp\u003eEach session involved practicing 49 multiplication problems, with the order of presentation depending on the experimental condition (ordered-by-table versus mixed). In the ordered-by-table condition, children practiced multiplication facts one table at a time, progressing from session to session from the 3 times table up to the 9 times table. On session 1, for example, they focused on the 3 times table. The facts (3 \u0026times; 3, 3 \u0026times; 4, 3 \u0026times; 5, 3 \u0026times; 6, 3 \u0026times; 7, 3 \u0026times; 8, and 3 \u0026times; 9) were first presented in a random order, then repeated six more times, each time in a new random order, resulting in seven practice rounds of the same set of facts during that session. On session 2, the same procedure was applied to the 4 times table, and so on, up to the 9 times table on the eighth session. In the mixed condition, children practiced all 49 problems constructed with operands from 3 to 9 in a random order during each of the 8 sessions. As in the table ordered condition, each of the fact was presented 8 times per session. A different random order of the facts was used for each session.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAll sessions were implemented during regular class time and were supervised by the classroom teachers. The researchers were not present during the sessions, and their role was limited to designing the materials and providing training and written guidelines to the teachers. All completed pre- and post-test forms from the three conditions were returned to the researchers by the teachers, either scanned and sent by email or delivered physically by postal mail.\u003c/p\u003e"},{"header":"RESULTS","content":"\u003cp\u003eA 2 (Test: pre- and post) x 3 (Group: control, ordered and mixed) ANOVA, with Test as a within-subjects factor and Group as a between-subjects factor, was conducted on children\u0026rsquo;s scores on the fluency multiplication task. There was a main effect of Test, showing higher scores at post- (7.27) than pre-test (5.52), F(1, 213) = 68.22, \u003cem\u003e\u0026eta;\u003c/em\u003e\u003cem\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e\u003cem\u003ep =\u003c/em\u003e .244\u003cem\u003e, p\u0026nbsp;\u003c/em\u003e\u0026lt; .001. This effect was qualified by a significant Test \u0026times; Group interaction, \u003cem\u003eF\u003c/em\u003e(2, 211) = 5.79, \u003cem\u003ep\u003c/em\u003e = .004, \u0026eta;\u0026sup2;ₚ = .052. As shown in Figure 2, the improvement from pre- to post-test was greater in the ordered condition than in the mixed and control conditions, which showed similar, smaller gains.\u003c/p\u003e\n\u003cp\u003eUnfortunately, as shown Figure 2, the initial pre-test scores of children in the ordered group were lower than those of children in the other groups. This discrepancy left more room for improvement in that group, which may have contributed to the observed effects. To address this issue, we conducted a second analysis in which children were matched on their pre-test scores. Specifically, we selected only those children who had identical pre-test scores across conditions. When the number of children with a given score differed between groups, a random selection was made to retain an equal number of children per group for that score, and the remaining children were excluded. This resulted in a final dataset of 49 participants per group (147 children in total), on which we conducted the same ANOVA as in the full-sample analysis.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eA 2 (Test: pre- and post) x 3 (Group: control, ordered and mixed) ANOVA conducted on this subsample showed a main effect of Test, with higher scores at post- (6.61) than pre-test (4.78), \u003cem\u003eF\u003c/em\u003e(1, 146) = 54.11, \u003cem\u003e\u0026eta;\u003c/em\u003e\u003cem\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e\u003cem\u003ep =\u003c/em\u003e .273\u003cem\u003e, p\u0026nbsp;\u003c/em\u003e\u0026lt; .001. This effect was qualified by a significant Test \u0026times; Group interaction, \u003cem\u003eF\u003c/em\u003e(2, 212) = 5.53, \u0026eta;\u0026sup2;ₚ =.071\u003cem\u003e, p=\u003c/em\u003e.005. As attested by Figure 3, the same pattern of results as for the whole sample was obtained, that is, higher multiplication score improvement after training for the ordered group than for the control and mixed groups. Once again, the improvement was the same for the control and mixed groups.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eCritically, the results obtained from the subsample of participants matched on pre-test scores could depend on the specific random selection. To address this concern, we replicated the matching procedure across four different random selections. This yielded a significant Test \u0026times; Group interaction in three out of four cases (p = .005, p = .015, and p = .011), and a marginal effect in the fourth (p = .081). These findings indicate that the reported effects are not attributable to an arbitrary sample composition.\u003c/p\u003e"},{"header":"DISCUSSION","content":"\u003cp\u003eIn this research, we aimed to determine whether training sixth-grade children on multiplication tables in a random order was more effective than training them table by table. The underlying idea was that presenting facts in a non-sequential order would reduce interference between them, thereby facilitating relearning. Our results show that it is not the case. Practicing the facts in a random order was less efficient than practicing them table by table. In fact, as attested by the lack of difference in the pre- versus post-test effect between the random and the control (business as usual) condition, random practice did not lead to any positive effect of the intervention. Reducing interference between facts by presenting them in a random order thus appears to be an ineffective strategy for boosting learning of multiplication tables, or at least relearning in our case, as children had already acquired these facts in the past. We can see at least three possibilities to explain these results.\u003c/p\u003e\u003cp\u003eFirst, the random presentation may conflict with the structured network of associations that was already established in long-term memory in these children, which could explain the absence of any positive effect in the random condition. It is possible that younger children who have not yet begun to learn multiplication facts might still show beneficial effects of random practice, as stable table-like representations are not yet formed. However, as previously noted, and as also pointed out by Dotan and Zviran-Ginat (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), such an approach should be adopted with caution. Random presentation of multiplication facts may indeed prevent children from grasping key conceptual aspects of multiplication (Park \u0026amp; Nunes, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2001\u003c/span\u003e), particularly its interpretation as repeated addition or, maybe more likely, from capitalizing on already stored facts to solve new facts (6 x 7\u0026thinsp;=\u0026thinsp;42 because it is 6 x 6\u0026thinsp;+\u0026thinsp;6, LeFevre et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1996\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eAnother interpretation of our results, which is not necessarily incompatible with the previous one, is that reducing interference between facts may be less important for learning than previously assumed. This interpretation carries significant weight, as it challenges the widely held view that the problem-size effect in multiplication arises because larger facts are more susceptible to interference than smaller ones (e.g., Campbell, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1987\u003c/span\u003e). The difficulty of a given multiplication problem may indeed depend more on its intrinsic characteristics than on its relation to other facts. For example, it is well established that associations between operands and answers are harder to construct for large problems than for smaller ones, partly because large problems take longer to solve in the early stages of learning (Thevenot et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Moreover, as previously discussed, larger problems are more likely to elicit incorrect responses, which may become associated with the problem alongside the correct answer, thereby reinforcing competing associations and hindering retrieval (e.g., Siegler, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e1988\u003c/span\u003e). In fact, assuming that the intrinsic characteristics of multiplication problems are the primary source of their difficulty (Ashcraft \u0026amp; Guillaume, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), and that interference between problems play only a secondary role, could help explain some puzzling findings in the literature. For example, while size effects are mainly attributed to interference between problems, the absence of such effects is difficult to account for. Indeed, as interferences necessarily increase with the size of the problems, size effects should be consistently observed. In contrast, if the difficulty of a given problem reflects its specific learning history, such as nosiness due to strongly associated incorrect answers, multiple competing responses, or consistently long solution times, then targeted, repeated practice could reduce this difficulty. In such a case, even a large problem could eventually become as easy to retrieve as a smaller one if practiced enough, and size effects could vanish. As matter of fact, such absence of size effects has repeatedly been observed in adults for addition with sums 8, 9 and 10 (Bagnoud et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; D\u0026iacute;az-Barriga Y\u0026aacute;\u0026ntilde;ez et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Poletti et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Uittenhove et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) and was interpreted as reflecting constant retrieval times for problems with these sums (see however Chen \u0026amp; Campbell, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Baroody, 2019; Thevenot \u0026amp; Barrouillet, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2020\u003c/span\u003e for a debate about this interpretation).\u003c/p\u003e\u003cp\u003eA final interpretation of our results is that learning multiplication facts in random order, rather than table by table, may simply not be sufficient to reduce interference between facts. Indeed, certain problems are frequently confused by individuals, such as \u003cem\u003e8 \u0026times; 7\u0026thinsp;=\u0026thinsp;56\u003c/em\u003e and \u003cem\u003e9 \u0026times; 6\u0026thinsp;=\u0026thinsp;54\u003c/em\u003e, despite belonging to different tables. While presenting problems in a random order across tables does reduce interference relative to table-by-table learning, it may not do so effectively enough. A more theoretically grounded randomization, based on models of fact similarity (De Visscher \u0026amp; No\u0026euml;l, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014a\u003c/span\u003e, De Visscher \u0026amp; No\u0026euml;l, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2014b\u003c/span\u003e), might be required to achieve a meaningful reduction in interference. Nevertheless, this interpretation appears unlikely, given that the mixed condition showed no positive effect at all, even though it should have, at the very least, produced a modest reduction in interference.\u003c/p\u003e\u003cp\u003eTo sum up and conclude, the results reported here are noteworthy for several reasons. From an educational perspective, they argue against learning (or at least relearning) multiplication facts in a random order, as this approach proved here less effective than practicing them table by table. As discussed above, it may be due to a specificity of our population (6th graders), for which a memory network already organized by table exists and may be resistant to restructuring, making random relearning not productive. Stated differently, it is possible that introducing facts in a random order conflicts with previously established representations, thereby limiting the benefits of such practice. However, from a theoretical point of view, our results may also suggest that interference between problems may be less critical than interference or competitive retrieval among answers within a given problem to explain the relative retrieval difficulty of multiplication problems. This could explain why reducing interference between problems during relearning does not lead to efficient outcomes. Alternatively, or complementarily, although this seems unlikely, practicing multiplication facts across rather than within tables may nevertheless be insufficient to reduce interference between facts. Future research should examine these hypotheses more directly, potentially by comparing the effects of different frequency level of drill-based learning on high and lower interferent problems.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding:\u003c/h2\u003e\u003cp\u003eThere was no specific funding for this research\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eA.B., J.P, \u0026amp; C.T. wrote the manuscriptA.B., prepare the figuresD.M. \u0026amp; C.T. conceived the experimentA.B ,D.M., \u0026amp; J.P. analyzed the resultsC.T. \u0026amp; J.P. supervised the analyses C.T. supervised the work\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eWe would like to thank Catherine Simon and Franck Verdier as well as all the members of the Clermont-Ferrand IREM (Research Institute on Mathematics Teaching) for their precious collaboration.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe dataset used in this research is available upon request by contacting [email protected].\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAshcraft, M. 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Q.\u003c/em\u003e \u003cb\u003e29\u003c/b\u003e (4), 269\u0026ndash;289. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.2307/30035554\u003c/span\u003e\u003cspan address=\"10.2307/30035554\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e (2006).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Numerical cognition, Arithmetic, Training program, Memory interference, Retrieval from memory, Multiplication facts","lastPublishedDoi":"10.21203/rs.3.rs-7139588/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7139588/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eInterference between multiplication facts is often mentioned as a key source of children\u0026rsquo;s difficulty in learning and retaining multiplication tables. Shared operands and results (e.g., 6 \u0026times; 4\u0026thinsp;=\u0026thinsp;24 and 8 \u0026times; 3\u0026thinsp;=\u0026thinsp;24; 7 \u0026times; 8\u0026thinsp;=\u0026thinsp;56 and 9 \u0026times; 6\u0026thinsp;=\u0026thinsp;54) are indeed thought to create overlap between facts, leading to blurred representations in memory. One proposed way to reduce such interference is to practice facts in a mixed order across tables (e.g., 6 \u0026times; 7\u0026thinsp;=\u0026thinsp;42; 3 \u0026times; 8\u0026thinsp;=\u0026thinsp;24; 4 \u0026times; 9\u0026thinsp;=\u0026thinsp;36) rather than table by table (e.g., 6 \u0026times; 7\u0026thinsp;=\u0026thinsp;42; 6 \u0026times; 8\u0026thinsp;=\u0026thinsp;48; 6 \u0026times; 3\u0026thinsp;=\u0026thinsp;18). This approach was tested in the present study, in which sixth graders practiced multiplication facts from x 3 to x 9 tables either by table (N\u0026thinsp;=\u0026thinsp;81) or in a mixed order (N\u0026thinsp;=\u0026thinsp;62). Their performance on a multiplication fluency task was compared to that of a control group engaged in regular classroom activities (N\u0026thinsp;=\u0026thinsp;71). Children in the mixed-order condition showed less improvement from pre- to post-test than those in the table-by-table condition and did not progress more than the control group. These findings show that randomly practicing multiplications facts may not be a productive pedagogical method for relearning. They more broadly suggest that the role of interference in multiplication learning may not be as central as commonly thought.\u003c/p\u003e","manuscriptTitle":"No Performance Benefit of Random Over Table-ordered Multiplication Training in Children: Does Memory Interference Between Facts Constrain Arithmetic Learning?","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-29 15:48:43","doi":"10.21203/rs.3.rs-7139588/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-03T05:23:18+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-11-26T05:15:49+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-23T04:38:47+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-08T12:30:56+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"337934489940460791199516666824022165626","date":"2025-09-09T09:18:23+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"88008714108570840933932662770304098468","date":"2025-09-06T17:26:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"67133527192455092867631239328034459122","date":"2025-08-05T19:55:36+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-24T12:54:28+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-22T11:22:48+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-07-20T07:12:13+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-07-20T07:09:47+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9bbc1289-b937-4740-97e5-e60df56b202d","owner":[],"postedDate":"July 29th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":52256219,"name":"Physical sciences/Mathematics and computing"},{"id":52256220,"name":"Biological sciences/Psychology"},{"id":52256221,"name":"Social science/Psychology"}],"tags":[],"updatedAt":"2026-03-16T20:23:39+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-29 15:48:43","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7139588","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7139588","identity":"rs-7139588","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}