What breakthroughs has Jerry achieved by the end of "Through the Tunnel"? Consider his feelings about himself, his conquest of the tunnel, and his dependence on his mother.
In getting to the point where he is able to swim through the tunnel, Jerry has vastly improved his feelings about himself and he has greatly reduced his dependence on his mother.
When the story starts, he has very little self-confidence at all. You can see this in how he acts all stupid trying to make the local kids notice him. Even up to the last minute, Jerry is not really sure of himself. He is really planning not to try the swim, but all of a sudden he decides to do it.
Once he manages to swim through the tunnel, however, he is much more sure of himself. He has no need for the local boys. And he does not even have to...
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In the story "Through the Tunnel" going through the tunnel was a task that Jerry prepared himself to do. He knew it was going to be difficult, but he had challenged himself. When he went through the tunnel he had a very hard time. It was a difficult swim and he thought about turning around but he did not. After he had reached the other side, he realized he could accomplish great things. He had gained self-confidence.
Conquering the tunnel led Jerry to feeling more independent from his mother. He had taken a step away from her apron strings. He had performed a task that she would never realize the level of difficulty. Just having this knowledge made this experience highly significant.
All three of the issues you mentioned tie in together in helping Jerry to grow up.