This distinction was brought to mind in a technical writing course. The teacher answered every “why” question by telling us to look at another example of the principle. For a science-bred boy this was maddening. Just tell me what principle was violated by the turned-in example; but the teacher was unable to. At first, I thought the teacher was just being stubborn. Slowly I came to realize, she just didn’t organize her thoughts that way, the axiomatic way.
In science and many academic subjects, the teacher presents the students with a set of principles, then uses deductive logic to demonstrate how they can explain facts that are observed.
In creative courses, the teacher presents the students with a set of examples of excellent works in the subject area. The students are to discover for themselves what is essential in the examples that they can use to create their own works.
Of course, many teachers use a blend of these techniques, but the extreme positions highlight the strengths and weaknesses of the axiomatic and the discovery methods.
Axiomatic teaching promotes faster mastery of existing understanding.
Discovery style promotes creation of new, personally meaningful insights.
Axiomatic teaching deadens insights.
Discovery style fails when masses of prior information must be understood.