When we come to the other side, I suspect that we will see how much has NOT been taught well. I suspect that much would be gained in discussions of the "A priori" and "A posteriori" if we take seriously that these terms represent the way that scholastic vocabulary developed in commenting on propter quid and quia demonstrations. Indeed, it is clear that early uses of the vocabulary were afoot in texts like Aquinas's commentary on the Posterior Analytics. One will find it in later scholastics as well (cf. the ol' Logic Museum on a passage from Bill of Ockham's Summa logicae). I'm sure that one finds it in the curriculum used by Kant before his so-called "Copernican Revolution."
I hate to be so negative, but it is a grave disservice (even to beginners in philosophy) to ignore these sorts of historical points—especially given the richness of the building blocks one gets from an Aristotelian perspective. Indeed, given that Aristotle stands behind so much of Western thought (even when he is rejected), it is a mark of unbecoming ignorance to wish to overlook the terminological links to the Stagirite and those who carried his work back into the existential situation of Europe from the Middle Ages on.
Just as much, it is part of the general tribal myth that is propagated by modernity, ever presenting itself as though it sprung from the brow of some thinker or some movement. Never look behind such matters! No, no! To do so will threaten the myths that many have crafted—myths that can be just as dangerous and deceptive as anything primitive. Indeed, I suspect that even though mixed with many negative elements, primitive man made his myths with some fear and trembling before something numinous and greater than himself. Alas, modernity (let alone so-called post-modernity) makes its myths so as to avoid any fear and trembling of this kind—let all things be made to man's measure alone!
Okay, I will stop venting my frustrations.... Still, go over to Wikipedia to see one kind of example of historical ignorance. Then, take a look at this post on Stanford to see a tour de force of academic "erudition" in typical Anglophonic fashion. See the Internet Encylopedia of Philosophy for similarly frustrating ruminations.
For now, see the remarks of Eduoard Hugon. I don't have time give a full translation of what is below. However, see the important bolded remarks below. They are, roughly:
Demonstration a priori does not coincide with demonstration propter quid, nor does demonstration a posteriori coincide with demonstration quia. Demonstration a priori proceeds through causes of any sort, whether proximate or remote; however, demonstration propter quid . . . through proper, immediate, and adequate causes. Hence, every demonstration propter quid is a priori; however, not every demonstration a priori is propter quid.
Demonstration a posteriori is only through an effect; however, demonstration quia is through an effect or through remote causes. Therefore, every a posteriori demonstration is quia, while it is not the case that every quia demonstration is a posteriori.
(The justification for reading "or" for "et" can be based upon remarks on p.383:) Demonstration quia, taking the particle 'quia' not as causal (i.e. meaning "because") but as it means 'that the thing is', proceeds either through a sign and effect or through remote, common, and inadequate causes.
Éduoard Hugon, Cursus Philosophicus Thomisticae, Vol. 1, Logica (Paris: Lethielleux, 1927).
From page 384: Demonstratio a priori est quae rem ostendit per causam ; a posteriori quae causam colligit per effectum. Unde in demonstratione a priori prœmissae sunt propositiones notiores quoad se , in demonstratione autem a posteriori pnemissae sunt notiores quoad nos, non quoad se, et hinc apparet demonstrationem a posteriori esse imperfectam. Demonstratio a priori non coincidit cum demonstratione propter quid, nec demonstratio a posteriori cum demonstratione quia.
Demonstratio enim a priori procedit per causas quascumque sive proximas, sive remotas; demonstratio autem propter quid, ut jam novimus, per causas proprias, immediatas, adaequatas. Hinc omnis demonstratio propter quid est a priori, non tamen omnis demonstratio a priori est propter quid.
Demonstratio a posteriori est solum per effectus, demonstratio autem quia est per effectus et per causas remotas. Omnis ergo demonstratio a posteriori est quia, at non omnis demonstratio quia est a posteriori.
From page 383: Demonstratio quia, sumpta particula quia, non ut est causalis, sed ut significat quod res est, procedit vel per signa et effectus, vel per causas remotas, communes, inadaequatas.