"Which is exactly what you don’t want your programmers learning."
Is "learning" gerund or present participle in the above sentence?
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It is a participle predicatively adjected to programmers. If it were a gerund, it would have to be capable of being interpreted as a noun substantive, which it is not.
It may help to expand the sentence into
Which is exactly what you don't want your programmers to be learning.
What you don't want is a situation where
Your programmers are learning ...
Here, learning is clearly a predicative adjective. The fact that what is the object of learning qua verb does not prevent the latter from also having the function of an adjective.
Note that, as far as the distinction between participle and gerund is concerned, one needs only consider that participles are adjectives while gerunds are substantives. If the progressive aspect associated with present participles is here absent, this does not mean that learning is not a present participle, but rather that the progressive aspect does not always follow present participles.
In a comment, John Lawler wrote:
Both. 'Present participle' is the name of the verb form ending in -ing. 'Gerund' is the name of a particular complement construction that uses the present participle form of the verb. And learning in the example sentence is a gerund, the direct object of want. Since want can also take infinitive complements, what you don’t want your programmers to learn would also be grammatical, though it wouldn't use either gerund or participle form.
The OP's sentence is unnecessarily complicated in that it involves a fuse relative construction. So I'll use a simplified example, (1a):
(1a) You don't want your programmers [learning it].
This is a raising construction in that your programmers is the object of the verb want, syntactically but not semantically. That is, in (1a) it is not your programmers that you don't want but the accusative case your programmers shows that your programmers somehow is the syntactic object of the verb want. This is because syntactically your programmers raises to the matrix clause as object even though semantically it is subject of the subordinate clause learning it. Hence, the raising construction.
The same raising construction is found in (1b):
(1b) You don't want your programmers [to learn it].
Note that (1a) is virtually equivalent in meaning to (1b) but not to (1c):
(1c) You don't want your programmers [to be learning it].
(1a)'s learning doesn't have a progressive meaning whereas (1c)'s learning does.
This can be proved by the fact that, while (2a) is grammatical, (2c) is ungrammatical because the verb know cannot be used in the progressive aspect.
(2a) You don't want your programmers [knowing it].
(2b) You don't want your programmers [to know it].
(2c) *You don't want your programmers [to be knowing it].
Therefore, knowing in (2a) doesn't have a progressive meaning.
With that said, let's address the question. The grammatical nature of learning is determined by the usage of the matrix verb want. The verb want doesn't normally allow a V-ing phrase (e.g., learning it) as complement:
(3a) *You don't want [learning it].
(3b) You don't want [to learn it].
Even when your programmers is kept as object, the verb want doesn't allow a V-ing phrase when the verb is not negated:
(4) *You want your programmers [learning it].
What all this means is that the verb want cannot be taking learning it as object, and therefore that learning it is not a gerund in (1a). Since it's not a gerund, it has to be a participle.