Virtues of Presentation of FO Logic in Kleene's Mathematical Logic I refer to Stephen Cole Kleene, Mathematical Logic (1967 - Dover reprint : 2002).
What are the "pedagogical benefits" (if any) of the presentation chosen by Kleene, mixing Natural Deduction and Hilbert-style ?
Propositional Calculus - at pag.33 he refers back to formulas 1a-10b of Th.2 pag.15, which are usual Intro- and Elimination- rules for propositional connectives "rewritten" as axiom schemata.
Predicate Calculus - at pag.107 he add two axiom schemata : $\forall x A(x) \rightarrow A(r)$  (the A-schema) and : $A(r) \rightarrow \exists x A(x)$ (the E-schema), and the two rules - the A-rule : from $C \to A(x)$  to $C \to \forall x A(x)$ and the E-rule : from $A(x) \to C$  to $\exists x A(x) \to C$ , with x not free in C.
Then (Th.21 pag.118) he proves as derived rules the four standard rules of Intro- and Elim- for quantifiers.

One might argue that many will find it easier to write natural deduction proofs than axiomatic proofs.  Such a presentation may make it easier to find or write axiomatic proofs, since one can write the natural deduction proof first and then write an axiomatic proof from the natural deduction proof as a blueprint of sorts.
It might also lead us to think about different types of natural deductive systems.  For instance, we might look at a natural deductive system with the following rules (along with the deduction theorem maybe): 


*

*C$\alpha$$\beta$ $\vdash$ CC$\beta$$\gamma$C$\alpha$$\gamma$

*$\alpha$ $\vdash$ CN$\alpha$$\beta$

*CN$\alpha$$\alpha$ $\vdash$ $\alpha$


This system doesn't strike many as all that intuitive, but if you mix natural deduction and hilbert style systems, you can readily see how you come up with the notion of a system like this and others.
Also, perhaps Kleene wanted to suggest that classical logic has a special sort of structure in a sense, because such a mixing isn't quite so easily accomplished in say relevant logic or in Lukasiewicz three-valued or infinite-valued logic.
