Everybody knows what Immaculate Conception is. They must because they reference it all the time. Except they often get it wrong. People will jokingly allude to Immaculate Conception when describing someone being born without their parents having, you know, done it. But that's not Immaculate Conception, it's Virgin Birth, which the Bible says occurred in Jesus's case. Immaculate Conception refers to Mary—that's Jesus's mom, for the uninitiated—having been born (after being conceived the traditional way) without original sin. The confusion is enough to drive an ex-Catholic to prayer.

A similar mix of certainty and cloudiness surrounds "broken windows" policing, which has been debated hotly since Eric Garner of Staten Island died after police officers—including one who had an arm around Garner's neck—wrestled him to the ground to arrest him for selling "loosie" cigarettes. Some have questioned whether "broken windows" policing that focuses on minor infractions like illegal cigarette sales is appropriate. Others confidently defend "broken windows."

But broken windows is a more nuanced theory than some popular renderings suggest. The evidence for its value is mixed. And the connections between the theory and the policing that sometimes bears its name can be pretty weak.

The theory was most prominently articulated by James Q. Wilson and George Kelling in a 1982 article in the Atlantic Monthly. More recently, Kelling put the theory in simple terms: "just as a broken window left untended is a sign that no one cares and leads to more broken windows, so disorderly behavior and conditions left untended are signs that no one cares and lead to fear of crime, serious predatory crime, and urban decay – in sum, minor offenses matter."

On its face, the logic is appealing. And the mechanisms at work make sense, too. Criminals see minor disorder being tolerated and so feel emboldened. Decent people are turned off by the chaos and stay off the street or even move out of the neighborhood, yielding more and more space to lawbreakers. And the theory's association with New York City's dramatic reduction in crime during the 1990s seems to be solid evidence that it works.

Only it's all more complicated than that. How closely related are disorder and serious crime? If one goes up or down, must the other? How does a crackdown on "disorder" avoid becoming the kind of anti-urban sterilization that Jane Jacobs lamented as fatal to cities? Do all minor offenses matter? Is some disorder (e.g., people using abandoned houses as hang-out spots) worse than others (e.g., the guys who loiter on the corner with open containers in hand) when it comes to fomenting crime? If visible disorder is the issue, how does someone surreptitiously selling cigarettes become a target?

And how closely does broken windows policing actually reflect broken windows theory? Over the years, some proponents of intense low-level enforcement have justified it for reasons other than its effect on disorder. Some like the fact that busting people for, say, turnstile jumping means cops get a chance to do warrant checks and perhaps haul in more serious criminals. Others thought it productive to remove all offenders from street circulation for a while so as to disrupt the more serious criminal activity of the felonious few. A few even justified the focus on "zero tolerance" and "quality-of-life" policing as a way to keep cops active and productive, rather than stuck in a responsive posture—especially as the number of felonies for them to respond to dropped.

Given all the permutations of the theory, it's no surprise that the evidence on whether it works is decidedly mixed (for instance, check out this, this and this). Different police departments have adopted the theory in unique ways, and with varying degrees of competency.

Also complicating the empirics: Public streets aren't laboratories, where you can control all the variables to isolate the effect of a single strategy. At the same New York City was implementing broken-windows policing, it was seeing changes in the market for crack cocaine, new patterns of real-estate development, the 1990s economic boom and reduction in poverty, demographic shifts in age distribution, and policy initiatives outside the NYPD that might also have reduced crime.

Meanwhile, as Molloy College professor and former NYPD Captain John Eterno pointed out to me, Kelling himself warned of the "real dangers" of the broken-windows approach," which is "active and interventionist, involving police intimately in neighborhoods as they attempt to solve problems before they worsen," meaning it "must arise out of a broader base of authority and wider network collaboration."

That base of authority has broken down in many neighborhoods, Eterno says in an email—something Bratton realizes and is attempting to address (while also, as CityLimits.org was first to report, doubling down on arrests for a large number of low-level crimes). But the commissioner faces a major obstacle.

"After 12 years of policing in one way, many officers have no idea how to police any other way. The main need is to get back to enforcing the 'spirit of the law' where necessary," Eterno writes. "They have to get away from being an 'army of occupation' and 'policing by the numbers.' Officers need to understand that they actually have discretion and do not have to do everything to be reflected in their activity reports (arrests and summonses). Sometimes not writing a summons or making an arrest is the absolute best action to take."

One irony, of course, is that that was one of the chief critiques of the "stop-and-frisk" approach: that so many police encounters ended without a summons or an arrest. But there were, one might say, other issues with that strategy.