The two sax teachers I have learned the most from have a very similar teaching style. The teachers are Pete Ferrara and Chris Vadala(two of the greatest saxophonists ever); their teaching style is what I would recommend to any sax teacher. To begin with, let me outline a normal lesson:
We would begin with octaves, scales, and intervals. The octave drills stress tuning, and help the student acquire a better sense of pitch. The scales and intervals (arpeggios, etc.)help the fingers warm up in recognizable and often used patterns. By warming up with these patterns (both the finger warm ups and the octaves), the student practices the most tedious exercises with each lesson, without really feeling that they are tedious. It is like sneaking medicine into the "spoonful of sugar." No one wants to spend hours practicing only scales, but the ability to recognize and quickly play a diatonic pattern (that is, notes that follow a scale without skips) is invaluable. So the scales, etc., are used as "warm ups;" almost separate from the lesson itself.
Sometimes the warm ups would be followed by specialized finger drills. These would be short, yet intensive passages that cause the fingers to move very fast, or in unusual ways. An example of this type of exercise is going from low B-flat through low C-sharp and back in sixteenth notes. This would exercise only the pinkies on each hand. Again, these finger drills emphasized and embedded valuable patterns into the mind and fingers of the students.
Then would came the prepared pieces. These would be book work, or solos. Normally there would be at least two, one very melodic (exercising phrasing, breath control, and expression) and one would be technical (once again, exercising the fingers and pattern recognition). These were played through by the student, with the expectation to play the pieces as if they were in a performance situation (that means: no stopping!) And these pieces could always be assigned again for the following lesson, if more practice could make the performance better.
There would always be a jazz piece or two. In order to work often, a saxophonist has to be musically well-rounded. Jazz also offered a welcome change to the rest of the lesson. For me, it helped unwind any tension from missed notes or pieces that didn't sound as good as they did in rehearsal. These jazz pieces could be exercises in a book, solos, or even playing a saxophone part with a tape of a band.
