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In my opinion the real understanding involves well developed
conceptual understanding linked with mathematical apparatus in
physics.
... he continued that this "hand waving" is not enough, and of
course students need also to master the topics; by this he meant the
mathematical mastery.
I think that he identifyed conceptual understanding with hand waving.
Sometimes there may not be any good way of explaining a piece of
physics for high school students (e.g. invoking quantum mechanics or
Maxwell's equations is pretty meaningless for them);