# The shuffling of mathematics problems improves learning

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## Abstract

In most mathematics textbooks, each set of practice problems is comprised almost entirely of problems corresponding to the immediately previous lesson. By contrast, in a small number of textbooks, the practice problems are systematically shuffled so that each practice set includes a variety of problems drawn from many previous lessons. The standard and shuffled formats differ in two critical ways, and each was the focus of an experiment reported here. In Experiment 1, college students learned to solve one kind of problem, and subsequent practice problems were either massed in a single session (as in the standard format) or spaced across multiple sessions (as in the shuffled format). When tested 1 week later, performance was much greater after spaced practice. In Experiment 2, students first learned to solve multiple types of problems, and practice problems were either blocked by type (as in the standard format) or randomly mixed (as in the shuffled format). When tested 1 week later, performance was vastly superior after mixed practice. Thus, the results of both experiments favored the shuffled format over the standard format.

## Keywords

Mathematics Practice Distribute Mass Block Mix Interleave Spacing## Notes

### Acknowledgments

This research was supported by a grant from the Institute of Education Sciences, US Department of Education. We thank Kristina Martinez and Erica Porch for their assistance with data collection.

## References

- Bahrick, H. P., Bahrick, L. E., Bahrick, A. S., & Bahrick, P. E. (1993). Maintenance of foreign-language vocabulary and the spacing effect.
*Psychological Science, 4*, 316–321.CrossRefGoogle Scholar - Bjork, R. A. (1979). Information-processing analysis of college teaching.
*Educational Psychologist, 14*, 15–23.CrossRefGoogle Scholar - Bjork, R. A. (1988). Retrieval practice and the maintenance of knowledge. In M.M. Gruneberg, P.E., Morris, & R.N. Sykes (Eds.),
*Practical aspects of memory II*(pp. 391–401). London: Wiley.Google Scholar - Bjork, R. A. (1994). Memory and meta-memory considerations in the training of human beings. In J. Metcalfe & A. Shimamura (Eds.),
*Metacognition: Knowing about knowing*(pp. 185–205). Cambridge: MIT.Google Scholar - Bloom, K. C., & Shuell, T. J. (1981). Effects of massed and distributed practice on the learning and retention of second-language vocabulary.
*Journal of Educational Research, 74*, 245–248.Google Scholar - Carpenter, S. K., & DeLosh, E. L. (2005). Application of the testing and the spacing effects to name learning.
*Applied Cognitive Psychology, 19*, 619–636.CrossRefGoogle Scholar - Carson, L. M., & Wiegand, R. L. (1979). Motor schema formation and retention in young children: A test of Schmidt’s schema theory.
*Journal of Motor Behavior, 11*, 247–251.Google Scholar - Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis.
*Psychological Bulletin, 132*, 354–380.CrossRefGoogle Scholar - Christina, R. W., Bjork, R. A. (1991). Optimizing long-term retention and transfer. In D. Druckman & R. A. Bjork (Eds.),
*In the mind’s eye: Enhancing human performance*(pp. 23–56). Washington DC: National Academy Press.Google Scholar - Dempster, F. N. (1989). Spacing effects and their implications for theory and practice.
*Educational Psychology Review, 1*, 309–330.CrossRefGoogle Scholar - Driskell, J. E., Willis, R. P., & Copper, C. (1992). Effect of overlearning on retention.
*Journal of Applied Psychology, 77*, 615–622.CrossRefGoogle Scholar - Fitts, P. M. (1965). Factors in complex skill training. In R. Glaser (Ed.),
*Training research and education*(pp. 177–197). New York: Wiley.Google Scholar - Foriska, T. J. (1993). What every educator should know about learning.
*Schools in the Middle, 3*, 39–44.Google Scholar - Gilbert, T. F. (1957). Overlearning and the retention of meaningful prose.
*Journal of General Psychology, 56*, 281–289.CrossRefGoogle Scholar - Glencoe (2001)
*Mathematics: Applications and Connections—Course 1*. New York: Glencoe-McGraw Hill.Google Scholar - Grote, M. G. (1995). Distributed versus massed practice in high school physics.
*School Science and Mathematics, 95*, 97–101.CrossRefGoogle Scholar - Hall, J. F. (1989).
*Learning and memory*, 2nd Ed. Boston: Allyn & Bacon.Google Scholar - Jahnke, J.C., & Nowaczyk, R. H. (1998).
*Cognition*. Upper Saddle River: Prentice Hall.Google Scholar - Kester, L., Kirschner, P. A., & Van Merriënboer, J. J. G. (2004). Timing of information presentation in learning statistics.
*Instructional Science, 32*, 233–252.CrossRefGoogle Scholar - Krueger, W. C. F. (1929). The effect of overlearning on retention.
*Journal of Experimental Psychology, 12*, 71–78.CrossRefGoogle Scholar - Mayfield, K. H., & Chase, P. N. (2002). The effects of cumulative practice on mathematics problem solving.
*Journal of Applied Behavior Analysis, 35*, 105–123.CrossRefGoogle Scholar - Pashler, H., Rohrer, D., Cepeda, N. J., & Carpenter, S. K. (2007). Enhancing learning and retarding forgetting: Choices and consequences.
*Psychonomic Bulletin & Review*(in press).Google Scholar - Postman, L. (1962). Retention as a function of degree of overlearning.
*Science, 135*, 666–667.CrossRefGoogle Scholar - Radvasky, G. (2006).
*Human memory*. Boston: Pearson Education Group.Google Scholar - Rea, C. P., & Modigliani, V. (1985). The effect of expanded versus massed practice on the retention of multiplication facts and spelling lists.
*Human Learning: Journal of Practical Research & Applications, 4*, 11–18.Google Scholar - Reynolds, J. H., & Glaser, R. (1964). Effects of repetition and spaced review upon retention of a complex learning task.
*Journal of Educational Psychology, 55*, 297–308.CrossRefGoogle Scholar - Rittle-Johnson, B. & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other?
*Journal of Educational Psychology, 91*, 175–189.CrossRefGoogle Scholar - Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process.
*Journal of Educational Psychology, 93*, 346–362.CrossRefGoogle Scholar - Rohrer, D., & Taylor, K. (2006). The effects of overlearning and distributed practice on the retention of mathematics knowledge.
*Applied Cognitive Psychology, 20*, 1209–1224.CrossRefGoogle Scholar - Rohrer, D., & Taylor, K. (2006). The effects of overlearning and distributed practice on the retention of mathematics knowledge.
*Applied Cognitive Psychology, 20*, 1209–1224.CrossRefGoogle Scholar - Schmidt, R. A., & Bjork, R. A. (1992). New conceptualizations of practice: Common principles in three paradigms suggest new concepts for training.
*Psychological Science, 3*, 207–217.CrossRefGoogle Scholar - Smith, S. M., & Rothkopf, E. Z. (1984). Contextual enrichment and distribution of practice in the classroom.
*Cognition and Instruction, 1*, 341–358.CrossRefGoogle Scholar - VanderStoep, S. W., & Seifert, C. M. (1993). Learning ‘how’ versus learning ‘when’: Improving transfer of problem-solving principles.
*Journal of the Learning Sciences. 3*, 93–111.Google Scholar