Realism
"The Man Who Was Almost a Man" uses a raw approach to realism to illustrate the societal implications of Dave's coming-of-age tale.
In fact, Richard Wright is rightly regarded as a forefather of the so-called social realism school of literature. Social realism takes a realistic (duh) approach to depicting the trials and tribulations of the working classes, illustrating hardships typically ignored by society at large. This approach is very present in the story, with class and race-based power structures made evident through characters like Mr. Hawkins.
These socially-minded insights are delivered in the form Dave's coming-of-age journey, though. Dave's struggles and failures tell us a lot about the society he inhabits, and what we see isn't all that pleasant. Although Dave doesn't seem to have grown much at first, he eventually makes the choice to leave town, rejecting the racist and classist power structure that threatens to destroy his potential. If that's not becoming a man, then we don't know what is.