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

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.

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Notes

  1. 1.

    Our present focus on “discovery” is a matter of convenience for the sake of a pedagogical argument. By discovery I do not also mean “minimally guided,” but merely that the learner is expected to make a discovery, in the standard dictionary definition of become aware of, or recognize the potential of.

  2. 2.

    For example, while arguing against “discovery teaching,” Kirschner et al. (2006) unequivocally state that “the constructivist description of learning is accurate” (p. 78).

  3. 3.

    In the standard dictionary sense of repeated exercise in or performance of an activity so as to develop or maintain proficiency in it.

  4. 4.

    Also known as The Thinker, a famous bronze sculpture depicting a nude man who appears deep in thought despite visible muscle tension.

  5. 5.

    Evidence used in this paper comes from data collected in New York City and the San Francisco Bay Area. They include semi-structured interviews, video data, and training notes. Other practitioners saw the author as an inquisitive member of the community rather than a scholar.

  6. 6.

    Video recorded by Novell Bell of NYC.

  7. 7.

    For one comparison, see Watson and Mason (2006), on the pedagogical benefit of structuring varied exercises in mathematics education.

  8. 8.

    According to Bernstein’s students, the two not only worked at the same institute, but played chess and were friendly. There is also some evidence that Bernstein assisted Vygotsky in at least one experiment with photography.

  9. 9.

    Because no video or audio records were taken, this illustrative case was put together from observational notes, recollections, and discussions with the teacher.

  10. 10.

    The philosopher and historian of mathematics, Roi Wagner, called this “a commonplace” observation among mathematicians present and past (personal communication, 2017).

  11. 11.

    Asking students to engage in practice while apparently withholding information from them requires trust or, as one anonymous reviewer put it, requires that students collude with the teacher—not exploring this dimension is a potentially significant limitation of present work.

  12. 12.

    Explanations are instead reserved for conveying what Hewitt (1999) might call the “arbitrary” aspects of the discipline.

  13. 13.

    “Discovery learning is often contrasted with didactic instruction, and given that choice, I vote for discovery” (Brown 1992, p. 168). While Brown noted problems with pure discovery, her explicit stance in this debate was for discovery.

  14. 14.

    E.g., ethanol and water. Alternatively, we might think of explorative practice as an emulsifying agent bringing together water and oil.

References

  1. Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47.

    Article  Google Scholar 

  2. 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 

  3. 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

  4. Anderson, M. L. (2010). Neural reuse: A fundamental organizational principle of the brain. Behavioral and Brain Sciences, 33(4), 245–266.

    Article  Google Scholar 

  5. Bamberger, J. (2013). Discovering the musical mind: A view of creativity as learning. New York: Oxford University Press.

    Google Scholar 

  6. Bamberger, J., & Schön, D. A. (1983). Learning as reflective conversation with materials: Notes from work in progress. Art Education, 36(2), 68–73.

    Article  Google Scholar 

  7. Barsalou, L. W. (2016). On staying grounded and avoiding Quixotic dead ends. Psychonomic Bulletin & Review, 23, 1122–1142.

    Article  Google Scholar 

  8. 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 

  9. 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.

    Article  Google Scholar 

  10. 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.

    Article  Google Scholar 

  11. Chemero, A., & Turvey, M. T. (2011). Philosophy for the rest of cognitive science. Topics in Cognitive Science, 3, 425–437.

    Article  Google Scholar 

  12. Dewey, J. (1944). Democracy and education. New York: The Free Press.

    Google Scholar 

  13. Draeger, D. F., & Smith, R. W. (1981). Comprehensive Asian fighting arts. Tokyo: Kodansha.

    Google Scholar 

  14. 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 

  15. 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.

    Article  Google Scholar 

  16. 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.

    Article  Google Scholar 

  17. Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435.

    Article  Google Scholar 

  18. Goldstone, R. L., Landy, D. H., & Son, J. Y. (2009). The education of perception. Topics in Cognitive Science, 2(2), 265–284.

    Article  Google Scholar 

  19. 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 

  20. 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.

    Article  Google Scholar 

  21. 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 

  22. 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).

  23. Kamii, C. K., & DeClark, G. (1985). Young children reinvent arithmetic: Implications of Piaget’s theory. New York: Teachers College Press.

    Google Scholar 

  24. Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38, 1008–1022.

    Article  Google Scholar 

  25. Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299.

    Article  Google Scholar 

  26. Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45–83.

    Article  Google Scholar 

  27. 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.

    Google Scholar 

  28. Kimble, G. A., & Perlmuter, L. C. (1970). The problem of volition. Psychological Review, 77, 361–384.

    Article  Google Scholar 

  29. Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work. Educational Psychologist, 41, 75–86.

    Article  Google Scholar 

  30. Kirsh, D. (2013). Embodied cognition and the magical future of interaction design. ACM Transactions on Computer-Human Interaction (TOCHI), 20(1), 3.

    Article  Google Scholar 

  31. Kleiner, I. (1988). Thinking the unthinkable: The story of complex numbers. Mathematics Teacher, 81, 583–592.

    Google Scholar 

  32. Knorr, W. R. (1975). The evolution of the euclidean elements. Dordrecht: Reidel.

    Google Scholar 

  33. Lee, H. S., & Anderson, J. R. (2013). Student learning: What has instruction got to do with it? Annual Review of Psychology, 64, 445–469.

    Article  Google Scholar 

  34. Llinas, R. (2002). I of the vortex: From neurons to self. Cambridge: MIT Press.

    Google Scholar 

  35. Lockhart, P. (2009). A mathematician’s lament. New York: Bellevue Literary Press.

    Google Scholar 

  36. Martin, A. (2007). The representation of object concepts in the brain. Annual Review of Psychology, 58, 25–45.

    Article  Google Scholar 

  37. Nathan, M. J. (2012). Rethinking formalisms in formal education. Educational Psychologist, 47(2), 125–148.

    Article  Google Scholar 

  38. 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 

  39. O’Regan, J. K., & Noë, A. (2001). A sensorimotor account of vision and visual consciousness. Behavioral and Brain Sciences, 24, 939–973.

    Article  Google Scholar 

  40. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.

    Google Scholar 

  41. Piaget, J. (1977). Psychology and epistemology: Towards a theory of knowledge. New York: Penguin.

    Google Scholar 

  42. 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.

    Article  Google Scholar 

  43. 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 

  44. 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.

    Article  Google Scholar 

  45. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18, 253–286.

    Article  Google Scholar 

  46. 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.

    Article  Google Scholar 

  47. 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.

    Article  Google Scholar 

  48. 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.

    Article  Google Scholar 

  49. 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 

  50. 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 

  51. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematica Society, 30(2), 161–177.

    Article  Google Scholar 

  52. Tobias, S., & Duffy, T. M. (2009). Constructivist instruction: Success or failure?. New York: Routledge.

    Google Scholar 

  53. Tomasello, M. (2008). The origins of human communication. Cambridge: MIT Press.

    Google Scholar 

  54. Trninic, D. (2015). Body of knowledge: Practicing mathematics in instrumented fields of promoted action. Unpublished doctoral dissertation. University of California, Berkeley.

  55. 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.

  56. 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.

  57. Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge: M.I.T. Press.

    Google Scholar 

  58. 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.

  59. Vygotsky, L. S. (1997). Educational psychology. (R. H. Silverman, Translator). Boca Raton, FL: CRC Press LLC.

  60. Wagner, R. (2017). Making and breaking mathematical sense. Princeton: Princeton University Press.

    Google Scholar 

  61. 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.

    Article  Google Scholar 

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Acknowledgements

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

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Correspondence to Dragan Trninic.

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Trninic, D. Instruction, repetition, discovery: restoring the historical educational role of practice. Instr Sci 46, 133–153 (2018). https://doi.org/10.1007/s11251-017-9443-z

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Keywords

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