Dúirt Max:

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I analyse "a" and "ag" as two forms of the same unit ("ag" becomes "a" when the object of the verbal noun precedes it).

That seems simple enough, even though it's not entirely accurate historically. I wonder if that would be a better way to present a and ag to learners? Could we test it here?

I gotta see this. Fire away.

>>That seems simple enough, even though it's not entirely accurate historically.

One of the first things we learn when we begin learning linguistics is the difference between "synchronic" and "diachronic".

Modern Irish syntax has to be analyzed synchronically.
The reason is so obvious that it might not be seen:
A language is nowhere but in its speakers' minds. And language enters the speaker's mind sychronically (babies don't have to learn old Irish before learning modern Irish). As a consequence, it is absolutely irrelevent to try and analyze the grammatical system of modern Irish in terms of its evolution through history.

This is the reason why I have always said that accusative, nominative or commom case did not exist in modern Irish. Not only is it true at the syntactical level, but it is also much easier for learners... (as a record: there is no mention of such cases in Learning Irish.)

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nominative or commom case did not exist in modern Irish

What do you call the noun forms that aren't genitive or dative or vocative?

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A language is nowhere but in its speakers' minds. And language enters the speaker's mind sychronically (babies don't have to learn old Irish before learning modern Irish).

Is fíor sin, gan dabht, ach tagann a lán rudaí nua a bhaineann le hábhar isteach in intinn duine nuair a théann sé ar scoil agus nuair a thosaíonn sé ag léamh agus ag scríobh. The written language informs the linguistic perceptions of literate people in an important way, too. If you insist that a in a dhéanamh is a variety of ag without being very careful how you phrase this, your students will reach a point when they realize from their reading of some not-very-old-at-all texts that a is really a variety of do and is not related to ag, and then they'll feel like they've been lied to.

I bhfocail eile, it strikes me that a purely synchronic description is pedagogically useful when a language has no prior written history. But when it does, you need to acknowledge that and work with it, too.

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"ag" becomes "a" when the object of the verbal noun precedes it

How does your rule account for a in these examples from Learning Irish:

Ba cheart duit a thíocht. (34.1.iii)

Bhí Cáit ag iarraidh a fháil amach [a dhéanamh amach / a shamhlú / etc.]cé a bhí ann aréir. (22.3)

Bhí mé sásta le simplíocht na rialach (ag go dtí go mbeidh cuspóir roimh an ainm briathartha, ansin a), ach feictear dom nach bhfuil sí chomh simplí sin anois.

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What do you call the noun forms that aren't genitive or dative or vocative?

I don't. But if I have too, I'd simply name them "nominal forms", or "unmarked forms", or "naked forms".

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The written language informs the linguistic perceptions of literate people in an important way, too.

That is correct... But for instance, the verbal ending -ent in French ("ils chantent") which comes directly from Latin -unt is, since it is not pronounced, taught as a spelling rule at school...

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If you insist that a in a dhéanamh is a variety of ag without being very careful how you phrase this, your students will reach a point when they realize from their reading of some not-very-old-at-all texts that a is really a variety of do and is not related to ag, and then they'll feel like they've been lied to.

1/ Either the texts are readable without any knowledge of the not-very-old-at-all form of the language, in which case it means that the syntactical system is changed so slightly (if at all) that the synchronic explanations still work; or they are not, in which case you simply have to learn another system.
(Bear in mind that the pronunciation evolves much quicker than the syntax and the spelling, therefore you can read older texts which have the same syntax as the modern tongue and thus understanding them perfectly without even knowing that the way you pronounce them is not the way they would have been pronounced at the time they were written.)

2/ I don't insist that "a" is a variety of "ag" (I agree, phrasing is very important)... I say that they are two forms of the same unit. But since "ag" is taking over "a", I think there is no harm in saying that "a" is a variety of "ag".

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How does your rule account for a in these examples from Learning Irish:

It doesn't...

"Ba cheart duit a thíocht."
"tíocht", "goil" and "beith" cannot stand alone. If so, then they are preceded by "a": "tá mé sásta súil" but "tá mé sásta a thíocht". (14.1.i)

"Bhí Cáit ag iarraidh a fháil amach [a dhéanamh amach / a shamhlú / etc.]cé a bhí ann aréir."
See explanation given right after the examples.

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ach feictear dom nach bhfuil sí chomh simplí sin anois.

I agree... this is surely why it is now being simplified...