Starting with the academic year 1941/42 Hindemith taught at Yale University in New Haven as Professor of the Theory of Music, as was his official title. Within his field of responsibility were composition and music theory, which he taught as two subjects independent of each other. His pupils were with him up to 16 hours per week, participating in instruction that almost had the character of a school.
His composition students were with him on Tuesdays and Thursdays from 9:00 AM - 1:00 PM. On Tuesday afternoons he taught «Traditional Harmony» to the theory students from 3:00-5:00 PM. He often extended these classes into the evenings in order to continue working with those who wanted to be in his composition class and therefore had to submit their own works to him. «Basic Principles» were taught on Saturdays from 9:00-11:00 AM. In this class Hindemith gave students a closer understanding of the ideas developed in the Unterweisung im Tonsatz (The Craft of Musical Composition), teaching them two and three-part writing. From 11:00 AM until 1:00 PM he taught the History of Music Theory.
Students got to know him as a sometimes merciless teacher. Louis Hemingway reported: «At the beginning I had the impression that he was unbelievably demanding and meticulously exacting: that the composition had to follow in his own style or he lost interest in it – for example when one lost track and stumbled upon something that could have been a bit more one's own style. I was often very frustrated in the beginning. During the course of the school year, however, without discussing it with us, he gave us more and more freedom in order to be able to go in our own directions.»
Many students, especially those were already advanced in composition, found it disagreeable that Hindemith insisted on teaching the fundamentals of harmony and composition to them as well.
Hindemith wanted to train his composition pupils to complete the compositional process in three set phases: firstly, the question as to what purpose the composition is supposed to fulfil, who is to play it and which instruments are necessary for this. In the next phase, the formal planning, outlining the design, and the first considerations concerning the guise of the musical material. The writing of the themes follows as the final phase of working out the composition.
Characteristic for Hindemith's own compositional process was his statement: «I never begin [when working out the composition] at the beginning; when I have my plan, I can work on any section – even starting with a fugue in the middle.»
Several textbooks resulted from Hindemith's with his pupils: already in 1942, the course in traditional harmony stimulated him to write the book A Concentrated Course in Traditional Harmony. Later followed Elementary Training for Musicians and Exercises for Advanced Students.