I agree with Lockhart in that math taught in schools right now focuses too much on rote procedures and facts, an algorithm of sorts. Student's don't ask "why" or even care to ask the underlying principles behind such compacted formulae. A lot of it comes from very beautiful proofs that teachers don't bother spending time on, hence diminishing possible opportunities that gives rise to some intuition for the students. However I do think rigor is a necessary component in mathematics as it is the foundation block that builds up other fields around us. Treating it too much like an art is too flowery. While teacher enthusiasm and creativity is important, one can incorporate context to topics being taught, engaging in fun discussions while maintaining the rigor so that we know math is math and music is music. We can allow math and art to rhyme, but there is still a clear distinction between them. Compared with Skempt (who was relatively more neutral on this issue), Lockhart greatly emphasizes on the need for a more open, curiosity-driven approach, where students and teachers alike can experience the joy of mathematical discovery. Skempt gives an outline of relational vs. instrumental approaches to teaching. I believe Lockhart greatly emphasizes on the relational aspect, though not in the context of rigor but in creativity and personal exploration.
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Thanks for this interesting response, Michael! I agree that Lockhart takes a polarized view -- and he isn't a K-12 teacher, and doesn't have much experience of schools. Skemp does take a much more balanced view of a similar topic. I do wonder what you mean by 'flowery' here though! When I think of our class math/art projects, for example, they embody as much rigour in their explanations as the search for a proof!
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