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

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## 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.

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