The shuffling of mathematics problems improves learning
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In Rohrer and Taylor (2007), interleaved practice was more effective than blocked practice in both experiments.
“The results of both studies showed interleaved practice to be more effective than blocked practice.”
From Spacing and Interleaving Effects Require Distinct Theoretical Bases: a Systematic Review Testing the Cognitive Load and Discriminative-Contrast Hypotheses (2021) · cites this paper as Rohrer and Taylor (2007)
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Distributed practice in verbal recall tasks: A review and quantitative synthesis.
Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006) · Psychological Bulletin · doi:10.1037/0033-2909.132.3.354 · Cited by 1,888 · see citing works
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“…have advocated that learners space their study (Bahrick et al., 1993; Bjork, 1979, 1988, 1994; Bloom & Shuell, 1981; Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006; Dempster, 1989; Pashler, Rohrer, Cepeda, & Carpenter, 2007; Reynolds & Glaser, 1964; Schmidt & Bjork, 1992; Smith & Rothkopf, 1984).”
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“For this reason, many previous authors have advocated that learners space their study (Bahrick et al., 1993; Bjork, 1979, 1988, 1994; Bloom & Shuell, 1981; Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006; Dempster, 1989; Pashler, Rohrer, Cepeda, & Carpenter, 2007; Reynolds & Glaser, 1964; Schmidt &…”
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Self-citationSpacing effects and their implications for theory and practice
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“Test performance is generally superior after practice that is spaced rather than massed— a finding known as the spacing effect (e.g., Bahrick, Bahrick, Bahrick, & Bahrick, 1993; Bjork, 1979, 1988, 1994; Bloom & Shuell, 1981; Carpenter & DeLosh, 2005; Reynolds & Glaser, 1964; Smith & Rothkopf, 1984).”
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Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013) · Psychological Science in the Public Interest · doi:10.1177/1529100612453266 · Cited by 3,138 · see citing works
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“This pattern reversed, however, on the long-term retention test: Participants better retained the ability to solve the problems if those problems were practiced 1 week earlier in a mixed format—that is, spacing enhanced learning (Rohrer & Taylor, 2007).”
BackgroundSpaced Repetition Promotes Efficient and Effective Learning: Policy Implications for Instruction
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“Interleaving has strong benefits even when the problem types were quite different (Rohrer et al., 2014), compared with the earlier studies on mathematics problem solving (e.g., Rohrer & Taylor, 2007).”
BackgroundResultOptimising learning using flashcards: Spacing is more effective than cramming
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“One reason for this failure is that spacing has seldom been investigated using procedures that are directly applicable in educational settings (although there are exceptions, e.g. Rohrer & Taylor, 2006, 2007; Smith & Rothkopf, 1984).”
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“Spacing effects were found in two recent experiments in which college students learned a moderately abstract mathematics task (Rohrer & Taylor, 2006, 2007).”
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“In one study reported by Rohrer and Taylor (2007), college students learned to find the volume of four obscure solids by solving practice problems that were interleaved or blocked, and interleaving boosted subsequent test scores by a factor of three (d = 1.34).”
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Teaching the science of learning
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“For example, in a study with college students, Rohrer and Taylor (2007) found that shuffling math problems that involved calculating the volume of different shapes resulted in better test performance 1 week later than when students answered multiple problems about the same type of shape in a row.”
BackgroundInstructor and student knowledge of study strategies.
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“…the value of generating versus being given information (Slamecka & Graf, 1978), of interleaving versus blocking learning (Kornell & Bjork, 2008; Rohrer & Taylor, 2007), of static versus animated images (Mayer, Hegarty, Mayer, & Campbell, 2005), of testing versus restudying (Roediger &…”
BackgroundInterleaving Helps Students Distinguish among Similar Concepts
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“In one of these studies, Rohrer and Taylor (2007) taught college students to find the volumes of four obscure solids (Fig.”
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“Rohrer and Taylor (2007) had students learn four kinds of geometry problems and practice with 16 additional problems, then were tested on eight new problems.”
BackgroundInterleaved practice improves mathematics learning.
Rohrer, D., Dedrick, R. F., & Stershic, S. (2015) · Journal of Educational Psychology · doi:10.1037/edu0000001 · Cited by 164 · see citing works
“Furthermore, several studies have shown that spacing can improve the learning of mathematics (Bahrick & Hall, 1991; Gay, 1973; Grote, 1995; Rohrer & Taylor, 2006, 2007; Yazdani & Zebrowski, 2006).”
Self-citationIncreasing Retention Without Increasing Study Time
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Artificial Intelligence in Education
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“Previous Studies of Interleaved Practice Four previously-published studies have compared the effects of interleaved and blocked mathematics practice (Le Blanc & Simon, 2008; Mayfield & Chase, 2002; Rohrer & Taylor, 2007; Taylor & Rohrer, 2010).”
BackgroundSelf-citationDistributed Learning: Data, Metacognition, and Educational Implications
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“…logic gate configurations (Carlson et al. 1989; Carlson and Yaure 1990), for new users of automated teller machines (Jamieson and Rogers 2000), and for solving mathematical problems as might appear in a class homework (Rohrer and Taylor 2007; Le Blanc and Simon 2008; Taylor and Rohrer 2010).”
The effects of interleaving versus blocking on foreign language pronunciation learning
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“…in free-recall tasks (e.g., multiplication facts; Don- ovan & Radosevich, 1999; Rea & Modigliani, 1985), there also is evidence demonstrating the benefits of distributed practice for learning more complex mathematical concepts, such as determining simple permutations (Rohrer & Taylor, 2006, 2007).”
BackgroundHow Retrieval Attempts Affect Learning
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“…1981); interleaved versus blocked practice (ie, mixing various problem types in one practice session versus massing practice of one problem type; Rohrer & Taylor, 2007); spaced versus massed practiced (ie, distributing practice across a longer time period versus cramming practice into a shorter…”
BackgroundA randomized controlled trial of interleaved mathematics practice
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“…in the introduction, the interleaved group in the present study likely received at least some blocked practice from their teachers before they received the experiment worksheets, and this appears to be true for the other math interleaving studies with one exception (Rohrer & Taylor, 2007).”
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“Interleaving enhances learning by alternating between different ideas or concepts, as opposed to the common practice of focusing on a single theme during a learning session (see for example Rohrer and Taylor, 2007).”
Research Commentary: The Effects of Spacing and Mixing Practice Problems
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“We excluded studies outside of these categories because they do not give any further insight into the spacing effect over long timescales beyond demonstrating that spacing is also beneficial for other types of material such as physics (Grote, 1995) and mathematics (Rohrer and Taylor, 2007).”
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“Although this approach may lead to more errors at the beginning and slow the learning process, it tends to enhance long-term retention and transfer of the material (Rohrer & Pashler, 2010; Rohrer & Taylor, 2007).”
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“…study in the context of category learning have shown interleaved benefits with as few as four categories with only four items presented in each category (e.g., Rohrer and Taylor, 2007, exp. 2; Taylor and Rohrer, 2010) and as many as twelve categories with ten items each (Zulkiply and Burt, 2013).”
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The Scarcity of Interleaved Practice in Mathematics Textbooks
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“higher on a final exam than students practicing math problems exclusively by type.(36) Similar results have also been found among adults learning to identify oil painters(20) or bird species.”
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“For example, understanding multiplication as repeated addition is cumbersome, when counting strategies for addition are still predominant ( Sherin & Fuson, 2005 ).”
BackgroundContexts, concepts and cognition: principles for the transfer of basic science knowledge
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“, 2013; Zulkiply and Burt, 2013), mathematical and clinical concepts (Rohrer and Taylor, 2007; Taylor and Rohrer, 2010; Zulkiply et al., 2012) as well as novel categories (Zulkiply and Burt, 2013; Carvalho and Goldstone, 2014a,b).”
BackgroundEvidence for Cognitive Science Principles that Impact Learning in Mathematics
Booth, J. L., McGinn, K. M., Barbieri, C., Begolli, K. N., Chang, B. L., Miller‐Cotto, D., Young, L. K., & Davenport, J. L. (2017) · Elsevier eBooks · doi:10.1016/b978-0-12-805086-6.00013-8 · Cited by 52 · see citing works
“For instance, spaced practice is superior to massed practice for understanding abstract mathematical concepts, such as determining simple permutations (Rohrer & Taylor, 2006, 2007).”
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Interleaved practice in multi-dimensional learning tasks: Which dimension should we interleave?
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“…Schmidt & Bjork, 1992; Schneider, 1985; Simon & Bjork, 2001), and cognitively more complex tasks such as solving algebra problems in mathematics (Rohrer & Taylor, 2007; Taylor & Rohrer, 2010), troubleshooting (de Croock, Van Merriënboer, & Paas, 1998; Van Merriënboer, Schuurman, de Croock, &…”
BackgroundDeliberate Practice for Achieving and Maintaining Expertise in Anesthesiology
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The effects of practice schedule on learning a complex judgment task
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“…with relatively simple tasks are not necessarily effective for complex tasks as well (Wulf & Shea, 2002), earlier studies on complex mathematical problem-solving have found benefits of mixing or interleaving practice materials on post-practice performance (Rohrer & Taylor, 2007; Simon, 2008).”
BackgroundSpaced mathematics practice improves test scores and reduces overconfidence
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Self-citationEffects of spaced versus massed training in function learning.
McDaniel, M. A., Fadler, C. L., & Pashler, H. (2013) · Journal of Experimental Psychology Learning Memory and Cognition · doi:10.1037/a0032184 · Cited by 43 · see citing works
Spacing Effects on Learning and Memory ☆
Carpenter, S. K. (2017) · Elsevier eBooks · doi:10.1016/b978-0-12-809324-5.21054-7 · Cited by 45 · see citing works
Grospietsch, F., & Lins, I. (2021) · Frontiers in Education · doi:10.3389/feduc.2021.665752 · Cited by 43 · see citing works
Lipowsky, F. (2020) · Springer eBooks · doi:10.1007/978-3-662-61403-7_4 · Cited by 42 · see citing works
Computer‐based learning: Interleaving whole and sectional representation of neuroanatomy
Pani, J. R., Chariker, J. H., & Naaz, F. (2012) · Anatomical Sciences Education · doi:10.1002/ase.1297 · Cited by 39 · see citing works
“Prior work on the benefits of interleaving different types of material during learning encouraged examining this approach in a domain that featured complex spatial representation (Rohrer and Taylor, 2007; Kornell and Bjork, 2008; Taylor and Rohrer, 2010).”
BackgroundHarvey, A. G. (2022) · SLEEP · doi:10.1093/sleep/zsac026 · Cited by 38 · see citing works
Understanding the Distributed Practice Effect
Kuepper-Tetzel, C. E. (2014) · Zeitschrift für Psychologie · doi:10.1027/2151-2604/a000168 · Cited by 43 · see citing works
Codding, R. S., VanDerHeyden, A. M., Martin, R. J., Desai, S., Allard, N., & Perrault, L. (2016) · Learning Disabilities Research and Practice · doi:10.1111/ldrp.12120 · Cited by 37 · see citing works
“These spaced sessions have been shown to improve the retention of learned material among college students (Rohrer & Taylor, 2006, 2007).”
BackgroundEnhancing the Quality of Student Learning Using Distributed Practice
Wiseheart, M., Küpper-Tetzel, Weston, T., Kim, A. S. N., Kapler, I. V., & Foot‐Seymour, V. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108235631.023 · Cited by 36 · see citing works
When Does Interleaving Practice Improve Learning?
Carvalho, P. F., & Goldstone, R. L. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108235631.017 · Cited by 35 · see citing works
“Blocked practice can also improve verbal learning. For example, Carpenter and Mueller (2013) showed that non-French speakers learned orthographic-tophonological mappings in French (i.e., “-eau” and the corresponding sound /o/ in the words “bateau,” “carreau,” and “corbeau,” and “-er” and the sound /e/ in the words “ad…”
BackgroundResultInfluentialMixing topics while studying does not enhance learning.
Hausman, H., & Kornell, N. (2014) · Journal of Applied Research in Memory and Cognition · doi:10.1016/j.jarmac.2014.03.003 · Cited by 33 · see citing works
The mediating effect of context variation in mixed practice for transfer of basic science
Kulasegaram, K. M., Min, C., Howey, E., Neville, A. J., Woods, N. N., Dore, K. L., & Norman, G. R. (2014) · Advances in Health Sciences Education · doi:10.1007/s10459-014-9574-9 · Cited by 33 · see citing works
Jonsson, B., Kulaksiz, Y. C., & Lithner, J. (2016) · International Journal of Mathematical Education in Science and Technology · doi:10.1080/0020739x.2016.1192232 · Cited by 36 · see citing works
“However, the reason that it is quick and reliable is that it is designed to CONTACT Bert Jonsson bert.jonsson@umu.se © Informa UK Limited, trading as Taylor & Francis Group avoid meaning; therefore, it does not give rise to any deeper understanding of the principles of mathematics.”
BackgroundInfluentialStrategies to Improve Learning and Retention During Training
Roediger, H. L., III, Nestojko, J. F., & Smith, N. S. (2019) · Oxford University Press eBooks · doi:10.1093/oso/9780190455132.003.0014 · Cited by 32 · see citing works
Undesirable Difficulty Effects in the Learning of High-Element Interactivity Materials
Chen, O., Castro-Alonso, J. C., PAAS, F. G. W. C., & Sweller, J. (2018) · Frontiers in Psychology · doi:10.3389/fpsyg.2018.01483 · Cited by 32 · see citing works
Spaced Retrieval Practice Imposes Desirable Difficulty in Calculus Learning
Lyle, K. B., Bego, C. R., Ralston, P. A. S., & Immekus, J. C. (2022) · Educational Psychology Review · doi:10.1007/s10648-022-09677-2 · Cited by 31 · see citing works
“…synonymous with desirable difficulty (e.g., Bjork & Bjork, 2011; Bjork et al., 2013; Clark & Bjork, 2014), but, in fact, several studies have found no negative effect of spacing on practice math tasks (Barzagar Nazari & Ebersbach, 2019; Ebersbach & Barzagar Nazari, 2020a; Rohrer & Taylor, 2007).”
BackgroundSuzuki, Y., Yokosawa, S., & Aline, D. (2020) · Language Teaching Research · doi:10.1177/1362168820913985 · Cited by 33 · see citing works
“The results of this study showed an advantage for interleaved practice over blocked practice in L2 grammar learning, which corroborates many of the previous studies in cognitive psychology (Mayfield & Chase, 2002; Rohrer, 2012; Rohrer & Taylor, 2007; Rohrer et al., 2014; Taylor & Rohrer, 2010).”
ResultAn In Vivo Study of Self-Regulated Study Sequencing in Introductory Psychology Courses
Carvalho, P. F., Braithwaite, D. W., de Leeuw, J. R., Motz, B. A., & Goldstone, R. L. (2016) · PLoS ONE · doi:10.1371/journal.pone.0152115 · Cited by 32 · see citing works
“The way study materials are sequenced influences learning [8,9,14,20,34,36,37,42].”
BackgroundEquivalent but not the Same: Teaching and Learning in Full Semester and Condensed Summer Courses
Walsh, K. P., Sanders, M., & Gadgil, S. (2019) · College Teaching · doi:10.1080/87567555.2019.1579702 · Cited by 26 · see citing works
“Research suggests that this sort of interleaving is more beneficial for knowledge retention (Rohrer and Taylor 2007; Kornell and Bjork 2008).”
BackgroundNemeth, L., Werker, K., Arend, J., & Lipowsky, F. (2020) · Learning and Instruction · doi:10.1016/j.learninstruc.2020.101354 · Cited by 27 · see citing works
DYNAMIC INTERPLAY BETWEEN PRACTICE TYPE AND PRACTICE SCHEDULE IN A SECOND LANGUAGE
Suzuki, Y., & Sunada, M. (2019) · Studies in Second Language Acquisition · doi:10.1017/s0272263119000470 · Cited by 27 · see citing works
“Because cognitive psychologists have revealed the benefits of interleaved practice in a variety of learning contexts (e.g., Kornell & Bjork, 2008; Rohrer & Taylor, 2007; Rohrer, Dedrik, & Burgess, 2014; Shea & Morgan, 1979), it may be possible to enhance L2 learning by manipulating practice…”
BackgroundDobson, J. L. (2011) · Advances in Physiology Education · doi:10.1152/advan.00039.2011 · Cited by 32 · see citing works
“Interleaving has been shown to benefit motor (18), baseball batting (8), and badminton (7) skills as well as the ability to both perform math problems (15) and recognize specific artistic styles (11).”
BackgroundPerceptual learning of multiple talkers requires additional exposure
Luthra, S., Mechtenberg, H., & Myers, E. B. (2021) · Attention Perception & Psychophysics · doi:10.3758/s13414-021-02261-w · Cited by 26 · see citing works
Nemeth, L., Werker, K., Arend, J., Vogel, S. T., & Lipowsky, F. (2019) · Frontiers in Psychology · doi:10.3389/fpsyg.2019.00086 · Cited by 26 · see citing works
“Several studies – mostly laboratory studies – have shown that spacing has a positive effect on the learning outcomes of students – also for mathematics (e.g., Grote, 1995; Cepeda et al., 2006; Rohrer and Taylor, 2006, 2007; Dunlosky et al., 2013).”
BackgroundGottlieb, M. A., Riddell, J., & Crager, S. E. (2016) · Annals of Emergency Medicine · doi:10.1016/j.annemergmed.2016.04.003 · Cited by 25 · see citing works
The advantage of mixing examples in inductive learning: a comparison of three hypotheses
Guzman-Munoz, F. J. (2017) · Educational Psychology · doi:10.1080/01443410.2015.1127331 · Cited by 22 · see citing works
“Rohrer and Taylor (2007) illustrate the idea of distributed retrieval with a task of learning to solve mathematical problems.”
BackgroundDistributed Practice: Rarely Realized in Self-Regulated Mathematical Learning
Barzagar Nazari, K., & Ebersbach, M. (2018) · Frontiers in Psychology · doi:10.3389/fpsyg.2018.02170 · Cited by 23 · see citing works
“The number of problems that were practiced and the total practice time were the same in each condition, yet students of the distributed practice condition outperformed students of the massed practice condition 1 week after practice (Rohrer and Taylor, 2007) as well as 4 weeks after practice (Rohrer and Taylor, 2006).”
BackgroundThe Relative Merits of Explicit and Implicit Learning of Contrasted Algebra Principles
Ziegler, E., Edelsbrunner, P. A., & Stern, E. (2017) · Educational Psychology Review · doi:10.1007/s10648-017-9424-4 · Cited by 23 · see citing works
“Interleaved processing has been found to be beneficial for learning mathematical formulas and rules (Rohrer & Taylor, 2007; Taylor & Rohrer, 2010).”
Anderson, S., Jamniczky, H. A., Krigolson, O. E., Coderre, S., & Hecker, K. G. (2019) · npj Science of Learning · doi:10.1038/s41539-019-0050-4 · Cited by 23 · see citing works
Cognitive Load Theory, Spacing Effect, and Working Memory Resources Depletion
Chen, O., & Kalyuga, S. (2019) · Advances in educational technologies and instructional design book series · doi:10.4018/978-1-5225-9833-6.ch001 · Cited by 23 · see citing works
Ebersbach, M., & Nazari, K. B. (2020) · Frontiers in Psychology · doi:10.3389/fpsyg.2020.00811 · Cited by 22 · see citing works
“The dependent variable was the performance in the final test tapping procedural knowledge (i.e., solving permutation tasks; Rohrer and Taylor, 2006, 2007).”
BackgroundMethodInfluentialKassa, A. K., Tessema, N., Habtamu, A., Girma, B., & Adane, Z. A. (2023) · Water Practice & Technology · doi:10.2166/wpt.2023.168 · Cited by 22 · see citing works
Exploring the prospects of using artificial intelligence in education
Liu, Z., & Yushchik, E. V. (2024) · Cogent Education · doi:10.1080/2331186x.2024.2353464 · Cited by 22 · see citing works
“This model encompasses teaching approaches (Bereiter & Scardamalia, 1989), developmental and learning practices (Rohrer & Taylor, 2007), cognitive load on students, and the provision of feedback (Mayer & Moreno, 2003).”
Making a lecture stick: The Effect of spaced instruction on knowledge retention in medical education
Timmer, M. C. J., Steendijk, P., Arend, S. M., & Versteeg, M. (2020) · Medical Science Educator · doi:10.1007/s40670-020-00995-0 · Cited by 20 · see citing works
“The beneficial effects of spaced learning on retention have been shown for a variety of learning tasks concerning factual knowledge [10], conceptual knowledge [11, 12], and procedural knowledge [13].”
BackgroundExploring the necessary conditions for observing interleaved practice benefits in math learning
Mielicki, M. K., & Wiley, J. (2022) · Learning and Instruction · doi:10.1016/j.learninstruc.2022.101583 · Cited by 22 · see citing works
Turning Coffee Time into Teaching Moments Through Bite-Sized Learning for Adult Learners
Ho, Y. Y., Yeo, E.-Y., & Wijaya, D. S. B. M. (2022) · The Journal of Continuing Higher Education · doi:10.1080/07377363.2021.2024000 · Cited by 23 · see citing works
“Rohrer and Taylor (2007) showed that when undergraduate participants were made to practise permutation calculation in a single session (massed presentation) or in two different sessions a week apart (spaced presentation), participants in the spaced practice conditions achieved higher accuracy rates…”
Background
- Authors
- Rohrer, D., & Taylor, K.
- Affiliations
- University of South Florida (US)
- Corresponding author
- Rohrer, Doug
- ORCID
- Rohrer, Doug
- Venue
- Instructional Science
- Published
- 2007-04-19
- Volume
- 35
- Issue
- 6
- Pages
- 481-498
- ISSN
- 0020-4277 (ISSN-L)
- 1573-1952
- OpenAlex
- W1966731838
- Cited by
- 357
- References
- 33
- Citations by year
- 2026: 22 · 2025: 26 · 2024: 35 · 2023: 29 · 2022: 23 · 2021: 26 · 2020: 31 · 2019: 35 · 2018: 20 · 2017: 21 · 2016: 22 · 2015: 16 · 2014: 16 · 2013: 11 · 2012: 9
- FWCI
- 5.64
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Instructional Science (publishedVersion)
- Digital Commons - University of South Florida (University of South Florida) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Visual and Cognitive Learning Processes
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Keywords
- mathematics learning, practice questions, spaced practice, massed practice, mixed practice, blocked practice, mathematics textbooks
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