Jim True-Frost as Roland Pryzbylewski in “The Wire.” (HBO)

Over the course of 2014, I have spent a lot of time thinking about the stories Americans tell themselves about police and the gap between those stories and the actual experiences of people who encounter police every day.

How did we get from the ideal of Sheriff Andy Taylor (Andy Griffith) in Mayberry, a man who saw no need for a gun, to the nightmarish fantasies of former Ferguson police officer Darren Wilson and Florida neighborhood watchman George Zimmerman, who saw their fellow citizens as antagonists rather than constituents? How did we reach a place where “In the Heat of the Night” seems optimistic in its portrayal of a confrontation between white cops and a black man that doesn’t end with that man dead or injured?

And yesterday, after a grand jury declined to indict officer Daniel Pantaleo, whose use of a chokehold killed Eric Garner, and Cleveland reporters published their investigation of Tim Loehmann, who shot and killed 12-year-old Tamir Rice, I found myself thinking of Roland Pryzbylewski, the young police officer on “The Wire” played by Jim True-Frost.

Pryzbylewski, or Pryz, as he’s known to fellow officers, is the kind of officer we almost never see in mass media: a bad one.

When he arrives on a special detail dedicated to investigating a major drug ring in the first season of “The Wire,” Pryz already has a serious demerit on his record: For some reason he shot up his own cruiser in a panic, and then filed a false report that suggested he had been under attack. He is considered emotionally unstable, and his assignment to the detail is supposed to be a way to dump him on a black officer, Lt. Cedric Daniels (Lance Reddick), so no other district will have to deal with him, and also to get him out of the way in a manner that will not offend his father-in-law, Maj. Stan Valchek (Al Brown), who is a district commissioner.

Echoes of Przy’s previous police work show up in the opening paragraphs of Adam Ferrise’s story on Tim Loehmann’s stint with the Independence, Ohio, police department before he joined the force in Cleveland.

“A Nov. 29, 2012 letter contained in Tim Loehmann’s personnel file from the Independence Police Department says that during firearms qualification training he was ‘distracted’ and ‘weepy,’ ” Ferrise reported. ” ‘He could not follow simple directions, could not communicate clear thoughts nor recollections, and his handgun performance was dismal,’ according to the letter written by Deputy Chief Jim Polak of the Independence police. … ‘I do not believe time, nor training, will be able to change or correct the deficiencies,’ Polak said.”

Like Pryz, Loehmann is white. And like Pryz, prior emotional displays that raised doubts about his ability to handle guns appropriately did not prevent him from staying on a police force where he would ultimately do greater damage.

In “The Wire,” Pryz and several other young officers do a stupid thing. They get drunk, go late at night to one of the housing projects that is a target of their investigation and hassle the residents there, forcing a man to strip on the sidewalk and dumping out another’s laundry. When a 14-year-0ld tweaks them by lounging on the hood of Przy’s car, Pryz clocks the boy with the butt of his pistol. The child ends up losing the sight in one eye.

And this will not be the last time that he uses his weapon inappropriately. In the third season of the show, Pryz neglects to identify himself when he responds to a distress call and shoots and kills a plainclothes African American officer.

In real life, Loehmann shot Tamir Rice just seconds after arriving on the scene where Rice had been spotted playing with a toy gun. An officer at the scene estimated Rice’s age at 20, unable to recognize him as a child.

In “The Wire,” Pryz is removed from street work in the first season only after he survives an internal affairs investigation with some coaching from Daniels. A grand jury has yet to hear the case against Loehmann, but it seems likely that he will join Darren Wilson and Daniel Pantaleo in escaping indictment. Wilson has since resigned from the Ferguson police department.

David Simon, the creator of “The Wire,” extended empathy to drug lords and corrupt union officials in his masterful series. And while “The Wire” is also one of a minority of police procedurals (including “The Shield”) to take a hard and extended look at police corruption and police brutality, Simon does something doubly unusual with Pryz. After an unpromising start, Pryz gets to earn back the audience’s respect.

Once he is off the street, he shows a talent for investigations that are based in research rather than contact with suspects. And after leaving the Baltimore police department, Pryz becomes an unusually dedicated math teacher, caring for African American teenagers who might have been targets of his violent temper when he worked in law enforcement.

Simon is making an important distinction here. Pryz is neither a monster nor a racist. He has redeeming personal qualities and something to offer his community. But from the very beginning of Pryz’s police career, it has been abundantly clear that he is temperamentally unsuited to carry a firearm and that he should not be invested with the authority to use it on behalf of the government. The failure to heed these clear signs has tragic results for Baltimore residents, and for the Baltimore police department itself.

For Tamir Rice, for Eric Garner and for Michael Brown, it is long past time for police departments across the country to come to the conclusion that Pryz himself reaches when he quits the force after killing another officer. If policing is to be truly honorable, meaningful work that serves our communities, we have to acknowledge that it’s hard to do well, and that there are people other than serial killers and bigots who are incapable of doing it.