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Very thoughtful.  Thanks for highlighting the "misunderstanding" issue, as well as that of "in the context of the problem at hand."  Both should have been on my list of "sticking things."  Understandings are indeed always only valid or invalid in the context of particular problems.  And that shines a very useful light on the meaning of "misunderstanding": it exists only in the context of particular problems (often those on the mind of a teacher).  I like a lot too your offering of conversation as the appropriate way to deal with this.  See some thoughts of Alice's in a different but related context.