Which of the two sentences below is correct?
Why do you insist so much on your child learning your native language?
Why do you insist so much on your child's learning your native language?
I've seen either, and both can be accepted as correct. Here's a quote from the Wikipedia article on possessives (Although beware of all of the linguist jargon, https://en.m.wikipedia.org/wiki/English_possessive#Actions):
"When possessives are used with a verbal noun or other noun expressing an action, the possessive may represent either the doer of the action (the subject of the corresponding verb) or the undergoer of the action (the object of the verb). The same applies to of phrases. When a possessive and an of phrase are used with the same action noun, the former generally represents the subject and the latter the object. For example:
"When a gerundive phrase acts as the object of a verb or preposition, the agent/subject of the gerund may be possessive or not, reflecting two different but equally valid interpretations of the phrase's structure:
are grammatically correct.
However, some grammarians may argue that, since the gerund is nominal, the second form with the subject of the gerund in the possessive form is preferred. It should be noted that the possessive will mainly be used only when personal pronouns and names are involved, as can be read in the passage that follows from Longman's Living English Structure by W. Stannard Allen:
What is NOT correct is the form used in the title:
(*) insist on somebody to do something
The other possible pattern with the verb insist is insist that somebody (should) do something.