# Instruction, repetition, discovery: restoring the historical educational role of practice

- 101 Downloads
- 1 Citations

## Abstract

This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that “direct instruction” and “discovery” are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetition-without-repetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.

### Keywords

Mathematics education Martial arts pedagogy Learning theory Direct instruction Discovery Embodied cognition Awareness Drill Explorative practice Exercise## Notes

### Acknowledgements

For their highly constructive comments on earlier drafts, I wish to thank Hillary Swanson and José Gutiérrez.

### References

- Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning.
*Educational Studies in Mathematics,**70*(1), 27–47.CrossRefGoogle Scholar - Abrahamson, D., Shayan, S., Bakker, A., & Van der Schaaf, M. F. (2016). Eye-tracking piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action.
*Human Development,**58*(4–5), 218–244.Google Scholar - Abrahamson, D., & Trninic, D. (2015). Bringing forth mathematical concepts: Signifying sensorimotor enactment in fields of promoted action.
*ZDM Mathematics Education*,*47*(2), 295–306. https://doi.org/10.1007/s11858-014-0620-0 - Anderson, M. L. (2010). Neural reuse: A fundamental organizational principle of the brain.
*Behavioral and Brain Sciences*,*33*(4), 245–266.CrossRefGoogle Scholar - Bamberger, J. (2013).
*Discovering the musical mind: A view of creativity as learning*. New York: Oxford University Press.CrossRefGoogle Scholar - Bamberger, J., & Schön, D. A. (1983). Learning as reflective conversation with materials: Notes from work in progress.
*Art Education,**36*(2), 68–73.CrossRefGoogle Scholar - Barsalou, L. W. (2016). On staying grounded and avoiding Quixotic dead ends.
*Psychonomic Bulletin & Review,**23,*1122–1142.CrossRefGoogle Scholar - Bernstein, N. A. (1996). On exercise and motor skill. In M. L. Latash & M. T. Turvey (Eds.),
*Dexterity and its development*(pp. 171–205). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings.
*Journal of the Learning Sciences,**2*(2), 141–178.CrossRefGoogle Scholar - Bucher, D., & Anderson, P. A. V. (2015). Evolution of the first nervous systems—What can we surmise?
*Journal of Experimental Biology,**218*(4), 501–503.CrossRefGoogle Scholar - Chemero, A., & Turvey, M. T. (2011). Philosophy for the rest of cognitive science.
*Topics in Cognitive Science,**3,*425–437.CrossRefGoogle Scholar - Dewey, J. (1944).
*Democracy and education*. New York: The Free Press.Google Scholar - Draeger, D. F., & Smith, R. W. (1981).
*Comprehensive Asian fighting arts*. Tokyo: Kodansha.Google Scholar - Fletcher, J. D. (2009). From behaviorism to constructivism: A philosophical journey from drill and practice to situated learning. In S. Tobias & T. M. Duffy (Eds.),
*Constructivist theory applied to instruction: Success or failure?*(pp. 242–263). New York: Taylor and Francis.Google Scholar - Foster, C. (2013). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts.
*International Journal of Mathematical Education in Science and Technology,**44*(5), 765–774.CrossRefGoogle Scholar - Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., et al. (2014). Active learning increases student performance in science, engineering, and mathematics.
*Proceedings of the National Academy of Sciences,**111*(23), 8410–8415.CrossRefGoogle Scholar - Freudenthal, H. (1971). Geometry between the devil and the deep sea.
*Educational Studies in Mathematics,**3,*413–435.CrossRefGoogle Scholar - Goldstone, R. L., Landy, D. H., & Son, J. Y. (2009). The education of perception.
*Topics in Cognitive Science,**2*(2), 265–284.CrossRefGoogle Scholar - Gresalfi, M. S., & Lester, F. (2009). What’s worth knowing in mathematics? In S. Tobias & T. M. Duffy (Eds.),
*Constructivist instruction: Success or failure?*(pp. 264–290). New York: Routledge.Google Scholar - Haken, H., Kelso, J. A. S., & Bunz, H. (1985). A theoretical model of phase transitions in human hand movements.
*Biological Cybernetics,**51*(5), 347–356.CrossRefGoogle Scholar - Hewitt, D. (1999). Arbitrary and necessary. Part 1: A way of viewing the mathematics curriculum.
*For the Learning of Mathematics,**19*(3), 2–9.Google Scholar - Howison, M., Trninic, D., Reinholz, D., & Abrahamson, D. (2011). The mathematical imagery trainer: from embodied interaction to conceptual learning. In G. Fitzpatrick, C. Gutwin, B. Begole, W. A. Kellogg, & D. Tan (Eds.),
*Proceedings of the annual meeting of CHI: ACM Conference on Human Factors in Computing Systems*(CHI 2011), Vancouver, May 7–12, 2011 (pp. 1989–1998). ACM: CHI (CD ROM).Google Scholar - Kamii, C. K., & DeClark, G. (1985).
*Young children reinvent arithmetic: Implications of Piaget’s theory*. New York: Teachers College Press.Google Scholar - Kapur, M. (2014). Productive failure in learning math.
*Cognitive Science,**38,*1008–1022.CrossRefGoogle Scholar - Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning.
*Educational Psychologist,**51*(2), 289–299.CrossRefGoogle Scholar - Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure.
*Journal of the Learning Sciences,**21*(1), 45–83.CrossRefGoogle Scholar - Kiefer, M., & Barsalou, L. W. (2013). Grounding the human conceptual system in perception, action, and internal states. In W. Prinz, M. Beisert, & A. Herwig (Eds.),
*Action science: Foundations of an emerging discipline*(pp. 381–407). Cambridge: MIT Press.CrossRefGoogle Scholar - Kimble, G. A., & Perlmuter, L. C. (1970). The problem of volition.
*Psychological Review,**77,*361–384.CrossRefGoogle Scholar - Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work.
*Educational Psychologist,**41,*75–86.CrossRefGoogle Scholar - Kirsh, D. (2013). Embodied cognition and the magical future of interaction design.
*ACM Transactions on Computer-Human Interaction (TOCHI),**20*(1), 3.CrossRefGoogle Scholar - Kleiner, I. (1988). Thinking the unthinkable: The story of complex numbers.
*Mathematics Teacher,**81,*583–592.Google Scholar - Knorr, W. R. (1975).
*The evolution of the euclidean elements*. Dordrecht: Reidel.CrossRefGoogle Scholar - Lee, H. S., & Anderson, J. R. (2013). Student learning: What has instruction got to do with it?
*Annual Review of Psychology,**64,*445–469.CrossRefGoogle Scholar - Llinas, R. (2002).
*I of the vortex: From neurons to self*. Cambridge: MIT Press.Google Scholar - Lockhart, P. (2009).
*A mathematician’s lament*. New York: Bellevue Literary Press.Google Scholar - Martin, A. (2007). The representation of object concepts in the brain.
*Annual Review of Psychology,**58,*25–45.CrossRefGoogle Scholar - Nathan, M. J. (2012). Rethinking formalisms in formal education.
*Educational Psychologist,**47*(2), 125–148.CrossRefGoogle Scholar - Nemirovsky, R., & Rasmussen, C. (2005). A case study of how kinesthetic experiences can participate in and transfer to work with equations.
*Proceedings of PME,**29*(4), 9–16.Google Scholar - O’Regan, J. K., & Noë, A. (2001). A sensorimotor account of vision and visual consciousness.
*Behavioral and Brain Sciences,**24,*939–973.CrossRefGoogle Scholar - Papert, S. (1980).
*Mindstorms: Children, computers, and powerful ideas*. New York: Basic Books.Google Scholar - Piaget, J. (1977).
*Psychology and epistemology: Towards a theory of knowledge*. New York: Penguin.CrossRefGoogle Scholar - Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization.
*Mathematical Thinking and Learning,**5*(1), 37–70.CrossRefGoogle Scholar - Roth, W.-M., & Thom, J. S. (2009). Bodily experience and mathematical conceptions: From classical views to a phenomenological reconceptualization.
*Educational Studies in Mathematics,**30*(2), 8–17.Google Scholar - Schmidt, R. A., & Bjork, R. A. (1992). New conceptualizations of practice: Common principles in three paradigms suggest new principles for training.
*Psychological Science,**3,*207–217.CrossRefGoogle Scholar - Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer.
*Journal of Educational Psychology,**103,*759–775.CrossRefGoogle Scholar - Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction.
*Cognition and Instruction,**22,*129–184.CrossRefGoogle Scholar - Sfard, A. (2007). When the rules of discourse change, but nobody tells you—making sense of mathematics learning from a commognitive standpoint.
*Journal of the Learning Sciences,**16*(4), 567–615.CrossRefGoogle Scholar - Spiro, R. J., & DeSchryver, M. (2009). Constructivism: When it’s the wrong idea, and when it’s the only idea. In S. Tobias & T. M. Duffy (Eds.),
*Constructivist instruction: Success or failure?*(pp. 106–123). New York: Routledge.Google Scholar - Taber, K. S. (2010). Constructivism and direct instruction as competing instructional paradigms: an essay review of Tobias and Duffy‘s constructivist instruction: Success or failure?
*Education Review*. https://doi.org/10.14507/er.v0.1418.Google Scholar - Thurston, W. P. (1994). On proof and progress in mathematics.
*Bulletin of the American Mathematica Society,**30*(2), 161–177.CrossRefGoogle Scholar - Tobias, S., & Duffy, T. M. (2009).
*Constructivist instruction: Success or failure?*. New York: Routledge.Google Scholar - Tomasello, M. (2008).
*The origins of human communication*. Cambridge: MIT Press.Google Scholar - Trninic, D. (2015). Body of knowledge:
*Practicing mathematics in instrumented fields of promoted action. Unpublished doctoral dissertation*. University of California, Berkeley.Google Scholar - Trninic, D., & Abrahamson, D. (2012). Embodied artifacts and conceptual performances. In J. v. Aalst, K. Thompson, M. J. Jacobson, & P. Reimann (Eds.),
*Proceedings of the international conference of the learning sciences: Future of learning*(ICLS 2012) (Vol. 1, pp. 283–290). Sydney: University of Sydney/ISLS.Google Scholar - Trninic, D., & Abrahamson, D. (2013). Embodied interaction as designed mediation of conceptual performance. In D. Martinovic, V. Freiman, & Z. Karadag (Eds.),
*Visual mathematics and cyberlearning*(Mathematics education in the digital era) (Vol. 1, pp. 119–139). New York: Springer.Google Scholar - Varela, F. J., Thompson, E., & Rosch, E. (1991).
*The embodied mind: Cognitive science and human experience*. Cambridge: M.I.T. Press.Google Scholar - von Glasersfeld, E. (1983). Learning as constructive activity. In J. C. Bergeron & N. Herscovics (Eds.),
*Proceedings of PME*(Vol. 1, pp. 41–69). Montreal: PME-NA.Google Scholar - Vygotsky, L. S. (1997).
*Educational psychology*. (R. H. Silverman, Translator). Boca Raton, FL: CRC Press LLC.Google Scholar - Wagner, R. (2017).
*Making and breaking mathematical sense*. Princeton: Princeton University Press.CrossRefGoogle Scholar - Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making.
*Mathematical Thinking and Learning,**8*(2), 91–111.CrossRefGoogle Scholar