In this sentence is singing a verb or a gerund?
Look at the children whom you can hear singing.
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Singing is a present participle in the case (following @tchrist ’s suggestion) of “I hear children singing.”
It would be a gerund, i.e., serving in lieu of a noun as direct object of verb hear, if you got rid of the children: “I hear singing.” But in “I hear children singing,” the direct object of the verb is children, and singing is an adjectival modifier thereof, a common function for participles.
A partially analogous construction is available using the past participle, as in “I heard the aria sung,” though the use of the past participle makes the construction passive where the present makes it active: the child sings, the aria is sung.
In comments, John Lawler wrote:
The direct object of hear in I hear (the) children singing is the gerund clause (the) children singing.
- Try a cleft on it: (The )Children singing is what I hear;
- or passivize it: Children singing (sweetly) was heard by everyone in the room.
Both possessive and accusative gerund complementizers are perfectly normal. There's another sense of children singing that means the same but has a different construction; from children (who are) singing, and that's plural. But if you add sweetly to the gerund, you get (The )Children singing sweetly is what I hear, and that's unquestionably sentential and singular.
"Children singing" by itself is ambiguous between modified plural noun and singular clause; but one can prefer one interpretation, especially in a context where it doesn't make a difference.