At the beginning of the story, Jerry is repeatedly referred to as "the English boy" and is most concerned with fitting in with the older boys at the rocky bay. "They were big boys -- men to Jerry" and "To be with them, of them, was a craving that filled his whole body." Describing him as the English boy emphasizes his youth and immaturity, and his desire to fit in with the older kids also shows just how immature he is; they are children too, and yet Jerry sees them as "men" because of his even more youthful perspective.
They eventually abandon Jerry, and he runs to his mother to ask her for goggles so that he can attempt the feat that they did, swimming through the hole in the rock. He wants them "now, now, now! He must have them this minute and no other time." Jerry, like an immature child, is incredibly impatient and cannot delay gratification at all. It doesn't take long, however, for this to change. After several days of practicing at holding his breath and swimming, Jerry now looks back at the old beach he used to visit with his mother as "a place for small children [...]. It was not his beach." Now, he does "not ask for permission" to go to his rocky bay; only children ask permission, and Jerry is beginning to grow up. Further, although he thinks he could probably make it through the tunnel now, "A curious, most unchildlike persistence, a controlled impatience, made him wait." Jerry has developed some patience, some ability to look ahead and wait for what he wants. This is quite a departure from his desperation for goggles of a few days ago.
In the end, when his mother tells him not to swim any more today, "She was ready for a battle of wills," indicating that this is what she would typically expect from young, immature Jerry. However, "he gave in at once" to her and didn't fight her, indicating that he's developed a little more ability to reason and lost some of his former stubbornness (both associated with growing up and becoming more mature).