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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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.
For example, while arguing against “discovery teaching,” Kirschner et al. (2006) unequivocally state that “the constructivist description of learning is accurate” (p. 78).
In the standard dictionary sense of repeated exercise in or performance of an activity so as to develop or maintain proficiency in it.
Also known as The Thinker, a famous bronze sculpture depicting a nude man who appears deep in thought despite visible muscle tension.
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.
Video recorded by Novell Bell of NYC.
For one comparison, see Watson and Mason (2006), on the pedagogical benefit of structuring varied exercises in mathematics education.
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.
Because no video or audio records were taken, this illustrative case was put together from observational notes, recollections, and discussions with the teacher.
The philosopher and historian of mathematics, Roi Wagner, called this “a commonplace” observation among mathematicians present and past (personal communication, 2017).
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.
Explanations are instead reserved for conveying what Hewitt (1999) might call the “arbitrary” aspects of the discipline.
“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.
E.g., ethanol and water. Alternatively, we might think of explorative practice as an emulsifying agent bringing together water and oil.
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For their highly constructive comments on earlier drafts, I wish to thank Hillary Swanson and José Gutiérrez.
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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
- Mathematics education
- Martial arts pedagogy
- Learning theory
- Direct instruction
- Embodied cognition
- Explorative practice