In computability theory, it is often necessary to prove some particular function is a "computable function". Until the 1960s, this was most commonly done by actually demonstrating a formal algorithm for the function in a kind of pseudocode, or giving a set of recursion equations. Needless to say this style of presentation was heavily symbolic and conveyed little intuition about why the function was defined the way it was.
The more modern style of presentation relies instead on having a good sense of the closure properties of computable functions, and identifying a large class of basic computable functions (the "primitive recursive functions"). So one can simply explain how to obtain the function at hand from primitive recursive functions using operations that preserve computability. This style of proof allows for much more detailed exposition of the intuition behind the definition of a computable function. Everyone in the field understands how, in principle, to take this kind of proof and obtain a formal algorithm, if it is ever necessary.