I sometimes here casual phrases in English like a hipping and a hopping (a hippin and a hoppin).
How exactly does this fit into English grammar and what is the history of the construct?
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If you have access to the Oxford English Dictionary (e.g. through your local library), this is covered under a, prep.1. The OED considers it to be a variant of the preposition on. It includes both a number of senses where it introduces a regular noun — quotations include "Those fat and fair Objects that make their mouths run a-water so" from 1664 and "He was here a Sunday" from 1996 — and a number of senses where it introduces a gerund, with quotations going as far back as the 1200s.
Grammar-wise, you can mostly think of it as an explicit participle marker; whereas "going" can be either a gerund ("Going there is fun!") or a participle ("We're going there!"), "a-going" can only be a participle ("We're a-going there!").
Per the OED, it can also sometimes have a passive sense (e.g. "abuilding" meaning "being built"); but then, the bare present participle can also sometimes have a passive sense (called the "passival"), so I guess that's still in line with its being an explicit participle marker of sorts.