From the book The Fourth Turning by William Strauss and Neil Howe:
"While mom's task was to tend to every complaint, dad's was to pay the bills without the kids' knowing where the money came from."
What's fascinating (ehhh, to a grammarian) about this sentence is that it is grammatically correct. But the reader gets thrown by the word kids' with its strange-looking s'.
What is it, exactly, that these kids are in possession of in this sentence?
It begins to make better sense if we reword it slightly to indicate the possession more directly. Here we'll rephrase "kids' knowing" into "the knowing of the kids" to give us:
"...without the knowing of the kids where the money came from."
Now this ought to look vaguely familiar to readers of this blog: this is a nominalization! And of course we know that the using of a nominalization is poor style.[1]
To improve readability and style, then, I would consider two possibilities:
1) "...to pay the bills without the kids' knowledge where the money came from" or,
2) "...to pay the bills without the kids knowing where the money came from."
The first option eliminates the nominalization problem by replacing the nominalized verb knowing with the noun knowledge. The second option, by removing the apostrophe, actually changes the sentence's grammar: the word knowing changes from something possessed into a gerund, which then makes kids knowing into a simple noun-gerund pair.
Since the second solution is the simplest and cleanest, and also removes that ugly apostrophe, I consider it the best option. Therefore:
"While mom's task was to tend to every complaint, dad's was to pay the bills without the kids knowing where the money came from."
[1] If you catch the grammar joke in that sentence a gold star for you!