Many definitions in mathematics are "fully crystalized". Sometimes the form of these definitions might be somewhat baffling to the uninitiated.
For example, the definition of a relation from $X$ to $Y$ as a subset of $X \times Y$ is extremely economical, and makes formulating and proving theorems about relations very efficient.
This same economy is a disadvantage when teaching the concept. I want my students to understand that "x is a brother of y" is an example of a relation on the set of people. In this context, saying that the relation "is a brother of" actually is a set of ordered pairs of people seems philosophically disturbing. I would rather speak of this set of ordered pairs as a kind of ledger sheet which records all of the brother relationships between people. For instance (Steve, Kayla) will be on that list because Steve (me) is the brother of Kayla (my sister), but (Kayla, Steve) will not be on this ledger sheet. If I accurately record all of the information about all pairs of people in this list, then I can replace the relation "x is a brother of y" with the equivalent relation "(x,y) is on the 'brother of' ledger sheet". Since any relation can be so translated, we skirt the issue of "actually" defining what a relation is by instead using this construction as a proxy.
I think that this kind of definition is quite common. We often use an easily available "proxy" for a concept to provide an economical definition. We then need to mentally translate each time we use the concept. This skill at conceptual translation is important to develop, but seems difficult to assess or help students to attain.
- Has this conceptual skill of translating between concepts and their proxies been identified and studied in the literature?
- What are the advantages and disadvantages of straying from the "official" definitions to allow easier assimilation? For instance, defining a relation instead as a two variable predicate might be "closer" to the students native conception of the concept.