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I have been teaching Woodwind instruments for several years now, ever since I got my Grade 8 Clarinet as a 13 year old. Since then I've learnt more than my 13 year old self knew there was to learn. The most valuable thing that I've learnt is that music should always be understood as a whole when learning and that a piece of music shouldn't be played without understanding at least some of the context surrounding it. I believe that the Clarinet or whichever instrument should have theory learnt alongside it, along with aural skills and even a bit of music history. This gives the player a better understanding of the music and also empathy with what the composer was hoping to achieve.
For example, if a pupil was playing a Brahms sonata the I would teach them who the Clarinettist was that Brahms had written the music for, because that Clarinettist (Richard Muhlfeldt) was the reason that Brahms quit retirement, as he was so inspired by the sound that Muhlfeldt created. I would teach them where the harmony changes are that so dramatically alter the mood of the music and teach them to spot these changes for themselves. This should be integrated into the learning of the notes and proper technique.
The point of this is that learning the Clarinet, Saxophone or Flute is not simply learning that instrument, but learning Music as a whole. I believe it should be taught as such to deepen the understanding of the instrument and also because it makes the learning a more fun and rewarding experience.
