When a lecturer in mathematics is appointed, this is normally on the strength of his or her research; not just on the fertility of his or her ideas, but also on the proven capacity to present them for publication according to the accepted norms. In mathematics itself the accepted norms generally have a deductive structure: definition, theorem, proof, corollary. This sequence has some degree of necessity about it. One needs terms from a definition in order to articulate a theorem, a proof to justify it and then corollaries to show its significance. Expounding mathematics according to this deductive sequence makes it as easy as can be for the logic to be checked. And where applications of mathematics are concerned the focus typically will be on a systematic presentation of the finished mathematical theory. Because these are the kinds of format in which the lecturer has established his or her expertise, it is not surprising that when lecturing or when writing a textbook, these are the formats a lecturer commonly uses. These formats are needed for convincing one's peers; they are rarely the most helpful for a student beginning to learn the subject.
In his Didactical Phenomenology of Mathematical Structures Hans Freudenthal (1983) distinguished between different phenomena in mathematics education: