as the other boys leads him to physical and emotional distress, but his relationship with his mother is, at least for the moment, no longer the center of his world. Jerry is growing up, and there is a new distance in the relationship between him and his mother.

Jerry's relationship with his mother is a bit unusual in that he is an only child and she is a widow; thus, they are unusually attuned to one another's feelings.  She understands that at eleven years old Jerry is at the very beginning of the stage of development when children begin to separate from their parents and assert their own identity. Although it worries her, she begins the process of allowing him more independence.  

Jerry, the "man" of their tiny family, is attempting to find a way to balance looking after his mom, seen in his "unfailing impulse of contrition—a sort of chivalry" and his eagerness to attain the ability to swim through the tunnel.

As his desire to swim through the tunnel intensifies, so do Jerry's impulses toward independence.  When he realizes he needs swim goggles, he "nag[s] and pester[s]" his mother until he gets them.  Eventually, he stops asking his mother for permission to go to the beach where the the tunnel is; he simply leaves the villa before she does.  Once Jerry achieves his goal of swimming through the tunnel, he hides the evidence of his nosebleed and tears. 

In the story's conclusion, Jerry briefly reverts to a childish behavior when he calls her "Mummy" and blurts out what he has accomplished, clearly looking for her approval.  However, ultimately, he does not argue with her when she puts an end to his swimming on this vacation because he has already proved to himself that he is capable of doing what he sets out to do without her knowledge or consent. 

When the story begins, Jerry is anxious for some freedom from his mother's watchful and protective eye.  However, "Contrition sent him running after her."  He felt badly for wanting this independence and doesn't leave her on this first day of vacation.  This year, he's much less interested in the "safe beach" they've always frequented in the past; he wants to go to the "wild bay," alone.  

The next day, he gets his chance, and when he looks back at her beach, he feels "relieved at being sure she was there, but all at once very lonely."  He misses her but soon becomes distracted by some older "boys -- men to Jerry" who come along and take turns diving off the rock.  They eventually amuse themselves by swimming through a tunnel in the rock, and since Jerry cannot do it, he begins to clown around to refocus their attention on him.  It doesn't work, and they leave him, crying like a child.  Cried out, he "swam out to where he could see his mother.  Yes, she was still there, a yellow spot under an orange umbrella."  He seems to want to be free of her, but -- at the same time -- he wants to know that she is nearby.

As he begins to grow more confident in the water, he feels that his old beach "now seemed a place for small children, a place where his mother might lie safe in the sun.  It was not his beach."  And next time, when he goes, he does not ask her permission first.  It is at this point that "A curious, most unchildlike persistence, a controlled impatience, made him wait" to attempt swimming through the rock yet.  His ability to delay gratification provides evidence of his growing maturity (he was unable to do so earlier when he accosted her for goggles), as does the fact that it doesn't occur to him to ask her, anymore, if he can go to the bay without her.

In the end, after his experience in the tunnel has seriously frightened him, Jerry does return to her, still a child -- at least, for a while.  He calls her "'Mummy'" and clearly wants her approbation and praise for his new ability to stay underwater for three minutes.  When he tells her this news, "It came bursting out of him."  Jerry has obviously begun the process of maturing, though it seems that one's progress toward maturity is not a straight line, and we can see that in his fluctuating relationship with his mother.

I think you would benefit from looking at the story in much wider terms - certainly the Jerry at the end of the story is very different from the Jerry at the beginning, and you are right to identify that one way he has changed is in terms of his relationship with his mother. However, more generally, by the end of the story Jerry has gone through a journey from childhood to manhood, symbolised most stridently in his journey through the tunnel.

At the beginning of the story we are introduced to a character who is on the cusp of adolescense, and very clearly feels responsible for his mother due to their enforced intimacy. Yet despite his feelings of responsibility towards his mother, he nonetheless feels drawn to the "wild beach", which is away from the "safe beach" and his mother's attentive care. The wild beach here can be said to symbolise independence and life away from the protection of a parent figure - note how Lessing describes the two beaches to draw out this comparison.

His discovery of the tunnel and the challenge that the French boys set him through swimming through the tunnel spur Jerry on to train hard and eventually succeed in his attempt to go through the tunnel. Although certainly at the beginning of the story it is Jerry's need to be accepted by the older group of French boys that drives his desire to go through the tunnel, it is interesting that at the end of the story he no longer feels this is the case, as he is happy to go back home and spend time with his mother. This indicates that the tunnel was more about a process of self-acceptance and doing something to show he could do it for himself rather than for any other reason.

His relationship with his mother likewise has changed by the end of the story. Jerry deliberately witholds his triumph, only relating his ability to hold his breath. The dramatic irony in his mother's response ("I wouldn't overdo it, dear") indicates the independence that Jerry has achieved in his journey through the tunnel - he has now entered an arena where he has secrets from his mother and is able to engage in activities, dangerous activies, away from his mother's protection. He is no longer subject to her and has proved himself a man in his own right.

"Through the Tunnel" opens with a boy and his widowed mother vacationing at a beach resort to which they have come frequently. Because he has no father, Jerry's mother, perhaps, feels more protective of her son than a married woman would, yet she is "determined to be neither possessive nor lacking in devotion." And, while Jerry  feels the loss of his father, he thinks that he must be more attentive to her, acting from an "unfailing impulse of contrition--a sense of guilt about his father's death?--a sort of chivalry." At any rate, they seem devoted to each other in their solitariness at this beach.  For, when Jerry first swims out in the ocean a good ways, he searches for his mother on the beach:

There she was, a speck of yellow under and umbrella that looked like an orange peel.

On the other hand, his mother's devotion is divided between her maternal protectiveness--"Why, darling, would you rather not come with me?"--and her burgeoning understanding that eleven-year-old Jerry needs to be given some independence. For, after he asks her for swim goggles without giving an explanation, she 

gave him a patient, inquisitive look as she said casually, "Well, of course, darling."

Yet, she still remains quite protective because she immediately becomes concerned when Jerry returns after successfully swimming through the tunnel, achieving his "rite of passage" to maturity.  "How did you bang your head?" she asks, and when he dismisses this injury, Jerry's mother becomes "worried." However, the mother does begin to recognize Jerry's maturation as she says to herself, "Oh, don't fuss! Nothing can happen. He can swim like a fish." Still, she cautions him not to "overdo it." This recognition of the mother is reinforced by Jerry's feeling that his mother's warning is "no longer of the least importance." Clearly, their relationship has grown from protective mother and child to an understanding mother and maturing son.

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.

We are presented with a very awkward relationship between the protagonist of this excellent short story and his mother as Jerry is on the threshold of adulthood and his mother struggles to work out how best to give him limited independence whilst trying to ensure his safety. Note how we see a conflict between them right at the beginning of the story when she spots Jerry looking at the "wild beach," and how "contrition" forced him to follow her to the "safe beach." Perhaps the most revealing paragraph we have that addresses their relationship comes when the author tells us some background information about the mother and Jerry:

He was an only child, eleven years old. She was a widow. She was determined to be neither possessive nor lacking in devotion. She went worrying off to her beach.

Thus we can see that the mother, as a single mother, is desperately trying to raise Jerry well by herself, and wants to try and get that balance between giving him independence whilst at the same time letting Jerry know that he is deeply loved. Thus it is that she lets him go to the "wild beach" by himself.

Jerry is eleven years old, and is the child of a widowed mother. He is respectful of his mother and does not to want to worry her in any way. For instance, even though he’d rather spend his time on the “wild and rocky bay,” he stays with his mother in the safe bay. It seems that the two have a close relationship, as his mother is discerning of his thoughts. She notes his disinterest in the safe beach and, not wanting to be overly possessive, lets him venture out to the wild-looking bay when he asks her permission to do so. She thinks that Jerry is “old enough to be safe without her”; this is proof that she would like her son to develop independence in his life. However, she still worries about the best kind of upbringing to give her son—she is “determined to be neither possessive nor lacking in devotion."

Jerry loves his mother and feels the need to always be around her. For instance, when she allows him to go by himself to the wild-looking beach, he “almost runs after her again feeling it unbearable that she should go off by herself.” However, at eleven years of age, he is at a stage in life where he is transitioning into adulthood—hence, his need for independence and his search for self-identity.

At the “wild-looking bay” he meets a group of older boys and yearns to be a part of them. When they accept him into the group, he “is happy to be with them.” Note that, after a long time spent on the wild-looking beach, Jerry swims out to a place from which he can see his mother. Her presence offers his some sort of reassurance. After time spent with the older boys in the wild beach, he resolves to learn to swim through the tunnel in the water, just like they did. After some practice, he finally makes it and shares the exciting news with his mother.

Jerry's growth and developing maturity is very much reflected in his relationship with his mother.  Early in the story, Jerry is concerned about pleasing her, and the idea that he might disappoint her causes him to feel a great deal of "contrition," or guilt.  In fact, it "sent him running after her," even after she'd given him permission to leave her and go to the wild bay that so entices him.  Later, when he wants goggles, Jerry is still quite childish, demanding them "now, now, now!  He must have them this minute and no other time.  He nagged and pestered until she went with him to a shop."  His inability to delay gratification is yet another example of Jerry's lack of maturity early on.

However, with his repeated practice and work, Jerry begins to grow.  His old beach "now seemed a place for small children [...].  It was not his beach."  Now, instead of being overly concerned about disappointing his mother, Jerry "did not ask for permission."  Children ask for permission, and children remain with their mothers.  Jerry, though, is no longer a child.  In fact, he's described as having a "curious, most unchildlike persistence" now in his pursuits.  In the end, Jerry still longs for his mother's approval, but he has begun the process of growing up.