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Metacognition seems pretty universally positive. I'm wary of viewing it as such. Aside from the obvious criticisms like "you can't learn to ride a bicycle by thinking about and writing a 200 page treatise on riding bicycles" I'd like to know some concrete places where metacognition should be discouraged in mathematics education. Wittgenstein, for example, may have argued that we acquire the ability to do mathematics by implicitly learning what not to think about or reach for in a very unconscious way akin to natural language use. Even if one views Wittgenstein's views on mathematical practice as fringe, one might note that the typical (or perhaps I should say "ideal" in the sense of Reuben and Hersh) mathematician's aversion to philosophizing about mathematics reflects a general attitude against metacognition in mathematical practice…perhaps fearing one's never getting down to business because of being lured by the sirens of philosophy. Here, though I would like more concrete criticisms in the everyday practice of teaching.

Being an odd mathematician in terms of my attitude toward philosophy, I would love it if metacognition were universally good for students, as then we could just have them write in order to build in some continuity of concepts. I pause, though, when presented with a formalist critique akin to a Wittgenstein-like position above.

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    $\begingroup$ Metacognition can be faux-destructive when it inhibits movement toward goals... by seeing/declaring those goals as silly or stupid. (Teenagers routinely do this by implicity imposing the criterion of whether a given thing is connected to sex, drugs, or rock-'n'-roll, for example.) This can indeed create awkwardnesses in routine, cookbook math courses, for the obvious reasons... similarly in some of the "requirements" in the beginning of grad school. But, in fact, I claim "metacognition" is (eventually) mathematical methodology... self-management? Very mundane, after all? $\endgroup$ – paul garrett Apr 29 '16 at 22:19
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    $\begingroup$ I perform metacognition all the time, and I see many piers getting some intuitive/philosophical advice on how to "think". I think metacognition is fine for higher-level math students (students above their grade level) but possibly a bad thing for students at or below their grade level. Those students I will often find more doubtful of their abilities. $\endgroup$ – Simply Beautiful Art May 3 '16 at 0:42
  • $\begingroup$ It is certainly difficult to find counter-examples in literature. However, meta-cognitive strategies are well-known to have an effect size of 0.5 or higher (0.72 for "reciprocal teaching" according to John Hattie). This is well within the threshold of "must-do" teaching methods - i.e. you will see more positive outcomes if you look for ways to apply meta-cognition strategies rather than looking for places that it shouldn't be applied. $\endgroup$ – Marian Minar Jun 22 '16 at 14:58
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    $\begingroup$ Perhaps related or at least interesting: flavorsandseasons.wordpress.com/about, the Flavors and Seasons project about the experential aspects of doing mathematics. $\endgroup$ – J W Aug 11 '16 at 19:27
  • $\begingroup$ Possibly relevant post: matheducators.stackexchange.com/questions/7745/… $\endgroup$ – kcrisman Mar 11 at 13:57

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