What changes in Jerry does his mother perceive during their beach vacation in "Through the Tunnel"?

In "Through the Tunnel," Jerry's mother perceives that Jerry wants to become more independent, be more proficient at swimming like the older boys, and attain his rite of passage.
When Jerry and his mother come to the usual beach for the vacationers on their first day of vacation, she notices Jerry look toward a rocky bay and then back at the beach on which they have sat on previous vacations. She asks him if he would rather go somewhere else, but Jerry runs after his mother as though out of contrition for his desires.

And yet, as he ran, he looked back over his shoulder at the wild bay; and all morning, as he played on the safe beach, he was thinking of it. 

On the second day, Jerry's mother asks him again if he would rather go somewhere else. Jerry tells her he would like to have a look at the rocks over at the rather wild-looking bay. After some hesitation, his mother tells him,

Of course, Jerry. When you’ve had enough, come to the big beach. Or just go straight back to the villa, if you like.

Happy she gave him her approval, Jerry hurries to the wild beach and runs into the water. Initially, he feels lonely and looks back at his mother. Soon, however, Jerry ventures out to where some older boys were on some wild-looking rocks. They dive from a point into the sea that forms a pool among the rocks; then they emerge and swim around, pull themselves out, and wait to dive again in turn. Fascinated, Jerry watches. He then swims up to the rock and takes his place to dive, proud he can perform as well as the others.
When the boys dive down under the water and re-appear some distance away, Jerry realizes they must be passing through something under the water. He submerges himself, but cannot see exactly where they swim. He calls out, but the other boys gather their things and depart. Alone now, Jerry returns to the villa where he and his mother are staying. He goes to his mother and demands some swim goggles. As soon as his mother buys him goggles, Jerry runs off to the bay with them in his hands.
Jerry puts on the goggles and immediately submerges himself in his effort to discover the opening in the rocks where the boys have passed. After some time, Jerry discovers the hole of the tunnel through which the others have swum.

He knew he must find his way through that cave, or hole, or tunnel, and out the other side.

Jerry returns to the villa as he realizes he must learn to hold his breath for some time. Also, he must be able to propel himself through this tunnel as a rite of passage to adulthood. He practices holding his breath for hours. He looks at the clock one day after holding his breath and realizes he has held it for over two minutes.
When his mother announces that in another four days they will return home, Jerry decides to attempt his swim through the tunnel. After he submerges himself, he dives inside the hole in the rock. He swims for a while, then worries he will not succeed.

He must go on into the blackness ahead, or he would drown. His head was swelling, his lungs cracking. . . he feebly clutched at rocks in the dark, pulling himself forward, leaving the brief space of sunlit water behind. He felt he was dying.  

Finally, Jerry sees light and swims out through the tunnel, his rite of passage complete. Although his goggles are filled with blood, Jerry is satisfied because he has done what the other boys have. He returns to the villa, where he tells his mother he can hold his breath for two or three minutes.