Cultivating mathematical skills: from drill-and-practice to deliberate practice
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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.
KeywordsMathematics Education Instructional Design Conceptual Knowledge Procedural Knowledge Mathematical Skill
This research was supported by the Academy of Finland Grant 274163 to the first author.
- 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
- 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
- Batchelor, S. (2014). Dispositional factors affecting children’s early numerical development (Doctoral thesis, Loughborough University, Leicestershire, United Kingdom). https://dspace.lboro.ac.uk/2134/17474.
- 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
- 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
- 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
- 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
- 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
- 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
- Ericsson, K. A., Prietula, M. J., & Cokely, E. T. (2007). The making of an expert. Harvard Business Review, 85, 1–8.Google Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.
- McMullen, J., Hannula-Sormunen, M. & Lehtinen, E. (submitted). Spontaneous focusing on quantitative relations as a predictor of rational number and algebra knowledge.Google Scholar
- Offner, C. D. (1978). Back-to-basics in mathematics: An educational fraud. The Mathematics Teacher, 71, 211–217.Google Scholar
- 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
- 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
- Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.Google Scholar
- 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
- 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.
- 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
- Wittmann, E. Ch. (2011). Vom Zählen über das “rechnende Zählen” zum “denkenden Rechnen“–mathematisch fundiert. Grundschulzeitschrift 248/249, 52–55.Google Scholar