, Volume 49, Issue 4, pp 625–636 | Cite as

Cultivating mathematical skills: from drill-and-practice to deliberate practice

  • Erno LehtinenEmail author
  • Minna  Hannula-Sormunen
  • Jake McMullen
  • Hans Gruber
Original Article


Contemporary theories of expertise development highlight the crucial role of deliberate practice in the development of high level performance. Deliberate practice is practice that intentionally aims at improving one’s skills and competencies. It is not a mechanical or repetitive process of making performance more fluid. Instead, it involves a great deal of thinking, problem solving, and reflection for analyzing, conceptualizing, and cultivating developing performance. This includes directing and guiding future training efforts that are then fine-tuned to dynamically evolving levels of performance. Expertise studies, particularly in music and sport, have described early forms of deliberate practice among children. These findings are made use of in our analysis of the various forms of practice in school mathematics. It is widely accepted that mathematics learning requires practice that results in effortless conducting of lower level processes (such as quick and accurate whole number arithmetic with small numbers), which relieve cognitive capacity for more complex tasks. However, the typical training of mathematical skills in educational contexts can be characterized as drill-and-practice that helps automatize basic skills, but often leads to inert routine skills instead of adaptive and flexible number knowledge. In this article we summarize findings of studies which describe students’ self-initiated, deliberate practice in learning number knowledge and intervention studies applying deliberate practice in mathematics teaching, including technology-based learning environments aimed at triggering practice that goes beyond mechanical repeating of number skills.


Mathematics Education Instructional Design Conceptual Knowledge Procedural Knowledge Mathematical Skill 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was supported by the Academy of Finland Grant 274163 to the first author.


  1. Araújo, D., Fonseca, C., Davids, K., Garganta, J., Volossovitch, A., Brandão, R., & Krebs, R. (2010). The role of ecological constraints on expertise development. Talent Development & Excellence, 2, 165–179.Google Scholar
  2. Baroody, A.J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–33). London: Erlbaum.Google Scholar
  3. Batchelor, S. (2014). Dispositional factors affecting children’s early numerical development (Doctoral thesis, Loughborough University, Leicestershire, United Kingdom).
  4. Batchelor, S., Inglis, M., & Gilmore, C. (2015). Spontaneous focusing on numerosity and the arithmetic advantage. Learning and Instruction, 40, 79–88. doi: 10.1016/j.learninstruc.2015.09.005.CrossRefGoogle Scholar
  5. Bojorque, G., Torbeyns, J., Hannula-Sormunen, M.M., Van Nijlen, D., & Verschaffel, L. (2016). Development of SFON in Ecuadorian Kindergartners. European Journal of Psychology of Education. doi: 10.1007/s10212-016-0306-9.Google Scholar
  6. Bonneville-Roussy, A., & Bouffard, T. (2015). When quantity is not enough: Disentangling the roles of practice time, self-regulation and deliberate practice in musical achievement. Psychology of Music, 43, 686–704. doi: 10.1177/0305735614534910.CrossRefGoogle Scholar
  7. Boshuizen, H. P. A., Schmidt, H. G., Custers, E. J. F. M., & van de Wiel, M. W. J. (1995). Knowledge development and restructuring in the domain of medicine: The role of theory and practice. Learning and Instruction, 5, 269–289.CrossRefGoogle Scholar
  8. Bransford, J. D., Barron, B., Pea, R., Meltzoff, A., Kuhl, P., Bell, P., Sabelli, N. (2006). Foundations and opportunities for an interdisciplinary science of learning. In K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 19–34). New York: Cambridge University Press.Google Scholar
  9. Brezovszky, B., Rodriguez-Aflecht, G., McMullen, J., Veermans, K., Pongsakdi, N., Hannula-Sormunen, M. M., & Lehtinen, E. (2015). Developing adaptive number knowledge with the Number Navigation game-based learning environment. In J. Torbeyns, E. Lehtinen & J. Elen (Eds.), Describing and studying domain-specific serious games (pp. 155–170). New York: Springer.CrossRefGoogle Scholar
  10. Brezovszky, B., McMullen, J., Veermans, K., Hannula-Sormunen, M., Rodríguez-Aflecht, G., Pongsakdi, N. & Lehtinen E. (submitted). The effects of the Number Navigation game-based training on primary school students’ arithmetic skills and knowledge.Google Scholar
  11. Bronkhorst, L. H., Meijer, P. C., Koster, B., & Vermunt, J. D. H. M. (2014). Deliberate practice in teacher education. European Journal of Teacher Education, 37, 18–34.CrossRefGoogle Scholar
  12. Brownell, W. A. (1944). When is arithmetic meaningful? Journal of Educational Research, 38(7), 481–498.CrossRefGoogle Scholar
  13. Côte´, J., & Hay, J. (2002). Children’s involvement in sport: A developmental perspective. In J. M. Silva & D. Stevens (Eds.), Psychological foundations in sport (pp. 484–502). Boston:Merrill.Google Scholar
  14. Davidson, J. W., Howe, M. J. A., Moore, D., & Sloboda, J. A. (1996). The role of parental influences in the development of musical performance. British Journal of Developmental Psychology, 14, 399–412.CrossRefGoogle Scholar
  15. Davis, J. T. M., Cullen, E., & Suddendorf, T. (2016). Understanding deliberate practice in preschool-aged children. The Quartely Journal of Experimental Psychology, 69, 361–380. doi: 10.1080/17470218.2015.1082140.CrossRefGoogle Scholar
  16. Degner, S., Lehmann, A. C., & Gruber, H. (2003). Expert learning in the domain of jazz guitar music. In R. Kopiez, A. C. Lehmann, I. Wolther & C. Wolf (Eds.), Proceedings of the 5th Triennial ESCOM Conference (pp. 384–388). Hannover:University of Music and Drama.Google Scholar
  17. Edens, K. M., & Potter, E. F. (2013). An exploratory look at the relationships among math skills, motivational factors and activity choice. Early Childhood Education Journal, 41, 235–243. doi: 10.1007/s10643-012-0540-y.CrossRefGoogle Scholar
  18. Ericsson, K. A. (2014). Why expert performance is special and cannot be extrapolated from studies of performance in the general population: A response to criticisms. Intelligence, 45, 81–103. doi: 10.1016/j.intell.2013.12.001.CrossRefGoogle Scholar
  19. Ericsson, K. A. (2016). Summing up hours of any type of practice versus identifying optimal practice activities: Commentary on Macnamara, Moreau, & Hambrick (2016). Perspectives on Psychological Science, 11, 351–354. doi: 10.1177/1745691616635600.CrossRefGoogle Scholar
  20. Ericsson, K. A., Charness, N., Feltovich, P. J., & Hoffman, R. R. (Eds.). (2006). The Cambridge handbook of expertise and expert performance. Cambridge: Cambridge University Press.Google Scholar
  21. Ericsson, K. A., Krampe, R., & Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363–406.CrossRefGoogle Scholar
  22. Ericsson, K. A., Prietula, M. J., & Cokely, E. T. (2007). The making of an expert. Harvard Business Review, 85, 1–8.Google Scholar
  23. Fuchs, L. S., Powell, S. R., Hamlett, C. L., Fuchs, D., Cirino, P. T., & Fletcher, J. M. (2008). Remediating computational deficits at third grade: A randomized field trial. Journal of Research on Educational Effectiveness, 1, 2–32.CrossRefGoogle Scholar
  24. Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences, 20, 89–100.CrossRefGoogle Scholar
  25. Gray, S. A., & Reeve, R. A. (2016). Number-specific and general cognitive markers of preschoolers’ math ability profiles. Journal of Experimental Child Psychology, 147, 1–21. doi: 10.1016/j.jecp.2016.02.004.CrossRefGoogle Scholar
  26. Gruber, H., Degner, S., & Lehmann, A. C. (2004). Why do some commit themselves in deliberate practice for many years–and so many do not? Understanding the development of professionalism in music. In M. Radovan & N. Dordević (Eds.), Current issues in adult learning and motivation (pp. 222–235). Ljubljana: Slovenian Institute for Adult Education.Google Scholar
  27. Gruber, H., Lehtinen, E., Palonen, T., & Degner, S. (2008). Persons in the shadow: Assessing the social context of high abilities. Psychology Science Quarterly, 50, 237–258.Google Scholar
  28. Hannula, M. M., Räsänen, P., & Lehtinen, E. (2007). Development of counting skills: Contributions from tendency to focus on numerosity and subitizing. Mathematical Thinking and Learning, 9, 51–57.CrossRefGoogle Scholar
  29. Hannula, M. M., & Lehtinen, E. (2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15, 237–256.CrossRefGoogle Scholar
  30. Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal for Experimental Child Psychology, 107, 394–406.CrossRefGoogle Scholar
  31. Hannula, M. M., Mattinen, A., & Lehtinen, E. (2005). Does social interaction influence 3-year-old children’s tendency to focus on numerosity? A quasi-experimental study in day-care. In L. Verschaffel, E. De Corte, G. Kanselaar & M. Valcke (Eds.), Powerful learning environments for promoting deep conceptual and strategic learning. Studia Paedagogica, 41 (pp. 63–80). Leuven: Leuven University Press.Google Scholar
  32. Hannula-Sormunen, M. M. (2015). Spontaneous focusing on numerosity and its relation to counting and arithmetic. In A. Dowker & R. Cohen Kadosh (Eds.), Oxford handbook of mathematical cognition (pp. 275–290). Croydon: Oxford University Press.Google Scholar
  33. Hannula-Sormunen, M. M., Lehtinen, E., & Räsänen, P. (2015). Preschool children’s spontaneous focusing on numerosity, subitizing and counting skills as predictors of their mathematical performance 7 years later at school a. Mathematical Thinking and Learning, 17, 155–177.CrossRefGoogle Scholar
  34. Hiebert, J., & LeFevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale: Erlbaum.Google Scholar
  35. Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38, 23–31.CrossRefGoogle Scholar
  36. Kucian, K., Kohn, J., Hannula-Sormunen, M. M., Richtmann, V., Grond, U., Käser, T., Esser, G., & von Aster, M. (2012). Kinder mit Dyskalkulie fokussieren spontan weniger auf Anzahligkeit. Lernen und Lernstörungen, 1, 241–253. doi: 10.1024/2235-0977/a000024.CrossRefGoogle Scholar
  37. Lampert, M. (2010). Learning teaching in, from, and for practice: What do we mean? Journal of Teacher Education, 61, 21–34.CrossRefGoogle Scholar
  38. Lehmann, A. C. (1997). Acquisition of expertise in music: Efficiency of deliberate practice as a moderating variable in accounting for sub-expert performance. In I. Deliège & J. Sloboda (Eds.), Perception and cognition of music (pp. 165–190). London: Psychology Press.Google Scholar
  39. Lehmann, A. C. (2002). Effort and enjoyment in deliberate practice: A research note. In I. M. Hanken, S. G. Nielsen & M. Nerland (Eds.), Research in and for music education. Festschrift for Harald Jørgensen (pp. 153–166). Oslo: Norwegian Academy of Music.Google Scholar
  40. Lehmann, A. C., & Ericsson, K. A. (2003). Expertise. In L. Nadel (Ed.), Encyclopedia of cognitive science Vol. 2 (pp. 79–85). London: Macmillan.Google Scholar
  41. Lehmann, A. C., & Gruber, H. (2014). Zielgerichtete Übung und Begabung. Zwanzig Jahre nach Ericsson, Krampe & Tesch-Römer (1993). In W. Gruhn & A. Seither-Preisler (Eds.), Der musikalische Mensch. Evolution, Biologie und Pädagogik musikalischer Begabung (pp. 87–107). Hildesheim: Olms.Google Scholar
  42. Lehmann, A. C., & Kristensen, F. (2014). “Persons in the shadow” brought to light: Parents, teachers, and mentors. How guidance works in the acquisition of musical skills. Talent Development & Excellence, 6, 57–70.Google Scholar
  43. Lehtinen, E., Brezovszky, B., Rodriguez-Afleht, G., Lehtinen, H., Hannula-Sormunen, M. M., McMullen, J., Pongsakdi, N., & Veermans, K. (2015). Number Navigation Game (NNG): Game description and design principles. In J. Torbeyns, E. Lehtinen & J. Elen (Eds.), Describing and studying domain-specific serious games (pp. 45–61). New York: Springer.CrossRefGoogle Scholar
  44. Lehtinen, E., & Hannula, M. M. (2006). Attentional processes, abstraction and transfer in early mathematical development. In L. Verschaffel, F. Dochy, M. Boekaerts, & S. Vosniadou (Eds.), Instructional psychology: Past, present and future trends. Fifteen essays in honour of Erik De Corte (pp. 39–54). Kidlington: Elsevier. (Advances in Learning and Instruction Series).Google Scholar
  45. Lehtinen, E., Hannula-Sormunen, M., McMullen, J., Brezovszky, B. & Jaatinen, M. (2016). Enhancing primary school students’ adaptive number knowledge with a computer game. Paper to be presented in the 2017 Biennial Meeting of the Society for Research in Child Development.Google Scholar
  46. Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47(3), 1–16.CrossRefGoogle Scholar
  47. Macnamara, B. N., Hambrick, D. Z., & Oswald, F. L. (2014). Deliberate practice and performance in music, games, sports, education, and professions: A meta-analysis. Psychological Science, 25, 1608–1618. doi: 10.1177/0956797614535810.CrossRefGoogle Scholar
  48. Macnamara, B. N., Moreau, D., & Hambrick, D. Z. (2016). The relationship between deliberate practice and performance in sports: A meta-analysis. Perspectives on Psychological Science, 11, 333–350. doi: 10.1177/1745691616635591.CrossRefGoogle Scholar
  49. Mattinen, A. (2006). Huomio lukumääriin: Tutkimus 3-vuotiaiden lasten matemaattisten taitojen tukemisesta päiväkodissa [Focus on numerosities: A study on supporting 3 year-old children’s mathematical development in day care]. Turku: Painosalama.Google Scholar
  50. McGaghie, W. C., Issenberg, S. B., Cohen, E. R., Barsuk, J. H., & Wayne, D. B. (2011). Does simulation-based medical education with deliberate practice yield better results than traditional clinical education? A meta-analytic comparative review of the evidence. Academic Medicine, 86, 706–711.CrossRefGoogle Scholar
  51. McMullen, J., Brezovszky, B., Rodríguez-Afleht, G., Pongsakdi, N., Hannula-Sormunen, M. M., & Lehtinen, E. (2016a). Adaptive number knowledge: Exploring the foundations of adaptivity with whole-number arithmetic. Learning and Individual Differences, 47, 172–181.CrossRefGoogle Scholar
  52. McMullen, J., Brezovszky, B., Hannula-Sormunen, M., Veermans, K., Rodríguez-Aflechta, G., Pongsakdi, N., & Lehtinen, E. (2017). Adaptive number knowledge and its relation to arithmetic and pre-algebra knowledge. Learning and Instruction, 49, 178–187. doi: 10.1016/j.learninstruc.2017.02.001.
  53. McMullen, J., Hannula-Sormunen, M. & Lehtinen, E. (submitted). Spontaneous focusing on quantitative relations as a predictor of rational number and algebra knowledge.Google Scholar
  54. McMullen, J., Hannula-Sormunen, M. M., Laakkonen, E., & Lehtinen, E. (2016b). Spontaneous focusing on quantitative relations as a predictor of the development of rational number conceptual knowledge. Journal of Educational Psychology, 108, 857–868.CrossRefGoogle Scholar
  55. McMullen, J., Hannula-Sormunen, M. M., & Lehtinen, E. (2013). Young children’s recognition of quantitative relations in mathematically unspecified settings. Journal of Mathematical Behavior, 32, 450–460.CrossRefGoogle Scholar
  56. McMullen, J., Hannula-Sormunen, M. M., & Lehtinen, E. (2014). Spontaneous focusing on quantitative relations in relation to children’s mathematical skills. Cognition and Instruction, 32, 198–218. doi: 10.1080/07370008.2014.887085.CrossRefGoogle Scholar
  57. Offner, C. D. (1978). Back-to-basics in mathematics: An educational fraud. The Mathematics Teacher, 71, 211–217.Google Scholar
  58. Pachman, M., Sweller, J., & Kalyuga, S. (2013). Levels of knowledge and deliberate practice. Journal of Experimental Psychology: Applied, 19, 108–119. doi: 10.1037/a0032149.Google Scholar
  59. Pachman, M., Sweller, J., & Kalyuga, S. (2014). Effectiveness of combining worked examples and deliberate practice for high school geometry. Applied Cognitive Psychology, 28, 685–692. doi: 10.1002/acp.3054.CrossRefGoogle Scholar
  60. Plant, E. A., Ericsson, K. A., Hill, L., & Asberg, K. (2005). Why study time does not predict grade point average across college students: Implications of deliberate practice for academic performance. Contemporary Educational Psychology, 30, 96–116.CrossRefGoogle Scholar
  61. Rahkamo, S. (2016). The road to exceptional expertise–A case study of the collective creativity of five Finnish multiple Olympic gold medalists. Espoo: Aalto University publication series doctoral dissertations, 257/2016.Google Scholar
  62. Rathé, S., Torbeyns, J., Hannula-Sormunen, M. M., De Smedt, B., & Verschaffel, L. (2016a). Spontaneous focusing on numerosity: a review of recent research. Mediterranean Journal for Research in Mathematics Education, 18, 125–141. doi: 10.1080/10986065.2016.1148531.Google Scholar
  63. Rathé, S., Torbeyns, J., Hannula-Sormunen, M. M., & Verschaffel, L. (2016b). Kindergartners’ spontaneous focusing on numerosity in relation to their number-related utterances during numerical picture book reading. Mathematical Thinking and Learning, 18, 125–141. doi: 10.1080/10986065.2016.1148531.CrossRefGoogle Scholar
  64. Redshaw, J., & Suddendorf, T. (2013). Foresight beyond the very next event: Four-year-olds can link past and deferred future episodes. Frontiers in Psychology, 4, 404.CrossRefGoogle Scholar
  65. Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge in mathematics. In R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition (pp. 1102–1118). Oxford: Oxford University Press. doi: 10.1093/oxfordhb.Google Scholar
  66. Rittle-Johnson, B., Schneider, M., & Star, J. (2015). Not a one-way street: Bi-directional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587–597. doi: 10.1007/s10648-015-9302-x.CrossRefGoogle Scholar
  67. Sakakibara, A. (2014). A longitudinal study of the process of acquiring absolute pitch: A practical report of training with the ‘chord identification method’. Psychology of Music, 42, 86–111. doi: 10.1177/0305735612463948.CrossRefGoogle Scholar
  68. Sella, F., Berteletti, I., Lucangeli, D., & Zorzi, M. (2016). Spontaneous non-verbal counting in toddlers. Developmental Science, 19, 329–337.CrossRefGoogle Scholar
  69. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.Google Scholar
  70. Suddendorf, T., Brinums, M., & Imuta, K. (2016). Shaping one’s future self. The development of deliberate practice. In S. B. Klein, K. Michaelian & K. K. Szpunar (Eds.), Seeing the future: Theoretical perspectives on future-oriented mental time travel (pp. 343–366). London: Oxford University Press.CrossRefGoogle Scholar
  71. Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50(1), 29–47.CrossRefGoogle Scholar
  72. Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM–The International Journal on Mathematics Education, 41, 541–555. doi: 10.1007/s11858-009-0195-3.CrossRefGoogle Scholar
  73. Tournaki, N. (2003). The differential effects of teaching addition through strategy instruction versus drill and practice to students with and without learning disabilities. Journal of Learning Disabilities, 36, 449–458.CrossRefGoogle Scholar
  74. Van den Heuvel-Panhuizen, M., Elia, I., & Robitzsch, A. (2014). Effects of reading picture books on kindergartners’ mathematics performance. Educational Psychology, 1–24. doi: 10.1080/01443410.2014.963029.
  75. Van Gog, T., Ericsson, K. A., Rikers, R. M., & Paas, F. (2005). Instructional design for advanced learners: Establishing connections between the theoretical frameworks of cognitive load and deliberate practice. Education Technology Research and Development, 53(3), 73–81.CrossRefGoogle Scholar
  76. Van Hoof, J., Degrande, T., McMullen, J., Hannula-Sormunen, M., Lehtinen, E., Verschaffel, L., & Van Dooren, W. (2016). The relation between learners’ spontaneous focusing on quantitative relations and their rational number knowledge. Studia Psychologica, 58(2), 156–170.CrossRefGoogle Scholar
  77. Verschaffel, L., & Greer, B. (2013). Domain-specific strategies and models: Mathematics education. In J. M. Spector, M. D. Merrill, J. Elen & M. J. Bishop (Eds.), Handbook of research on educational communications and technology (fourth edition) (pp. 553–563). New York: Springer Academic.Google Scholar
  78. Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359. doi: 10.1002/cbdv.200490137.CrossRefGoogle Scholar
  79. Wheatley, G. H. (1991). Constructivist perspectives of science and mathematics learning. Science Education, 75, 9–21.CrossRefGoogle Scholar
  80. Wittmann, E. Ch. (2011). Vom Zählen über das “rechnende Zählen” zum “denkenden Rechnen“–mathematisch fundiert. Grundschulzeitschrift 248/249, 52–55.Google Scholar

Copyright information

© FIZ Karlsruhe 2017

Authors and Affiliations

  1. 1.Department of Teacher EducationUniversity of TurkuTurkuFinland
  2. 2.Department of Educational ScienceUniversity of RegensburgRegensburgGermany

Personalised recommendations