In which sentence does the teacher dislike the child?

If you chose the first sentence, you are right. This sentence emphasizes the child, whereas the other one stresses the whispering. In the second sentence, the possessive form child’s signals that whispering is a gerund, a verb form used as a noun. The writer of this sentence is following a traditional grammar rule: For subjects of gerunds, use possessive nouns or pronouns.
Usage experts disagree on the extent to which writers should adhere to this rule. Long ago, H. W. Fowler, who supported the rule, admitted that it was “on the retreat” (610). The American Heritage Book of English Usage claims that the rule has been broken by “respected writers for 300 years” and that the result “is perfectly idiomatic” (17). The same book’s usage panel supports the rule in simple sentences but tends to allow exceptions in complex constructions.
Nearly everyone agrees that exceptions must be allowed. As Patricia O’Conner says, this rule (like many others) is worth following “except when it leads you off a cliff” (47). Consider O’Conner’s example:

The possessive men’s and women’s sounds awkward, and since there is no possible confusion about the writer’s meaning, it’s best to allow an exception to the rule. But the rule still stands, says O’Conner, and in simpler constructions we should follow it: Basil objects to our [not us] kissing in public.
When using the possessive is awkward, you can sometimes recast the sentence: Basil doesn’t like to see men and women kissing in public. Here the emphasis is on the men and women, and kissing in public is a participial phrase modifying men and women. Because the -ing phrase does not function as a gerund, the possessive is not required.
Conclusion: Use a possessive noun or pronoun as the subject of a gerund unless doing so results in an awkward sentence. Then either recast the sentence or disregard the rule. When you choose to disregard the rule, though, make sure your meaning is clear.