Certainly, with Hemingway's minimalist style, it is difficult to specifically assess the relationship between father and son in "A Day's Wait." Nevertheless, the reader may infer that the father does indeed love his son dearly, and Schatz must feel close to him, although he is a reserved child. And, perhaps, he is much like his father who is coolly rational, although he lacks his father's life experience.
Hemingway's opening line indicates Schatz's iron self-control:
He came into the room to shut the windows when we were still and bed and I saw he looked ill.
For, most children in Hemingway's time would wake their parents, particularly the mother, and tell her that they do not feel well. Then, when his father--not the mother--perceptively asks Schatz what is wrong, the nine-year-old replies that he merely has a headache and is "all right." After the father dresses and goes downstairs, he checks the boy's forehead and realizes that the child is running a fever; so, he calls the doctor. [In those days, doctor's made house calls.]
Conscientiously, the father makes notes on the instructions of the doctor. Then, he watches over his child and reads to him in order to distract the boy from his discomfort. It is Schatz's reticence, however, which causes the misunderstanding between father and son. For, the father does not realize that Schatz has mistakenly comprehended his temperature as measured by the Celsius scale rather than the Farenheit. Then, he attempts to be brave and spare his father the agony of watching him die. Later, he refuses to let anyone into his room, "You mustn't get what I have." Never does the father understand that Schatz has a misconception about his temperature, probably because of the similarity of him with his son, who are both rational because he probably expected his son to explain if something else bothers him.
Because of Schatz's brave determination to spare his father--whom he obviously loves dearly--worry, the situation escalates and becomes one that traumatizes the boy, causing him in the future to break down easily. The father, who cares deeply for his son, does not read the nine-year-old's feelings well enough perhaps because the boy is young than one who would be so courageous.