her out of that unfailing impulse of contrition -- a sort of chivalry."  The word "contrition" is a synonym for remorse, telling us that Jerry feels regretful and perhaps a little guilty for wanting to separate from her.  "Chivalry" used to refer to the appropriate and expected behavior of knights, indicating generosity, bravery, and politeness.  Now, people use it more often to describe gallantry on the part of a man toward a woman.  Therefore, we know that Jerry understands that, as a male, the social expectation is that he ought to be chivalrous toward his mother, a female.  He ought to treat her with courtesy and kindness, and the thought of abandoning her on their vacation seems to violate such a code.  Moreover, the fact that Jerry's father has died, leaving his mother a widow, may contribute even more to his sense of obligation to her.  He is "the man in the family" so to speak, and it is, thus, he may feel it his duty to take care of her.

For Jerry's mother, we see her trying to negotiate the line between being too protective of her young son and giving him too much freedom.  

She was thinking, Of course he's old enough to be safe without me.  Have I been keeping him too close?  He mustn't feel he ought to be with me.  I must be careful [....].  She was determined to be neither possessive nor lacking in devotion.  She went worrying off to her beach.

Certainly she is fulfilling the role expected of her: she is a devoted and conscientious mother.  She knows it is important to give her young son  freedom -- and she very much defers to his wants (typical of a society that privileges male prerogative).  She allows him to go to the "wild bay" in the first place.  Then, even when Jerry comes home with bad nosebleeds from holding his breath too long, she responds with, "'I shouldn't overdo things, darling, if I were you.'"  She doesn't order him to stop, she doesn't insist that he stay with her for the rest of the trip.  She likely allows him to call the shots not only because he is growing up but also because he is a boy and so her preferences, in society, would take a backseat to his.  Her deference to his choices helps illuminate the male privilege in this time and place.