Beyond the direct expression of outrage, one purpose of protests is to sway public opinion. By that standard, the demonstrations against police violence that followed the killing of George Floyd in police custody appear to have been successful — at least by some measures: A Washington Post-Schar School poll released this past week found that 69 percent of Americans think Floyd’s killing signals a broader problem within law enforcement, compared with 29 percent who consider it to be an isolated incident.

That represents a significant change from 2014, when a Washington Post-ABC News poll found that only 43 percent of Americans felt that the killing of unarmed African American men in Ferguson, Mo., and New York City signified a broader problem (compared with 51 percent who thought they were isolated affairs). The new Post-Schar School poll also found that a large majority of Americans, 74 percent, generally back the protests — a trend that extended even to Republicans, 53 percent of whom support them. Echoing other commentators, Slate said the polling suggested that the Black Lives Matter movement “has made staggering gains in just two weeks.”

 There may be reasons for optimism among those who, like me, believe strongly in curbing police violence, but we should also be cautious in interpreting the polls. Declarations of a revolution in American consciousness are premature. For one thing, polls also reveal that a surprisingly high proportion of people thought that police behaved reasonably in response to the protests — despite the footage of the violent clearing of Lafayette Square, the shooting of journalists with pepper guns and the countless baton-beatings that police dished out.

The split in the polls on whether the protests were violent or peaceful is striking. The Post-Schar School survey found that 43 percent believe that the protests were mostly violent and that an identical share think they were mostly peaceful. Opinions diverge sharply by ideology, with 70 percent of liberals saying they were mostly peaceful and 60 percent of conservatives concluding they were mostly violent. In fairness, a majority (66 percent) thought that neither the police nor the protesters but “other people acting irresponsibly” were responsible for the violence.

Still, fully 50 percent of those polled say the police used an amount of force that was “about right” on peaceful protesters, compared with 44 percent who think they used too much. In the case of people who vandalized or looted, the public wanted the police to get tougher: 47 percent think more force was warranted. And in an ABC News-Ipsos poll, 52 percent agreed that — as Sen. Tom Cotton (R-Ark.) proposed — the military should have been sent to cities that saw violence and looting.

Clearly, what seemed obvious to many people on the left and center-left — that the police greeted protests against violence with violence — was not apparent to all. The political polarization evident in the reactions is also significant: As the debates continue, Americans may gravitate further toward “their” side’s view; the president will certainly try to make that happen by hammering on his “law and order” message.

Evaluating the protests is one thing. Then there’s the question of where to go from here. “Defund the police” has become a call to arms, admittedly one that is often misunderstood. It is certainly not a call to turn a blind eye to serious crime, though it would involve reducing police budgets. The idea is to redirect some money to social-service agencies that can handle certain issues better than the police — such as responding to calls involving mental health or interacting with homeless people. Still, cutting police budgets is deeply unpopular outside of Twitter, according to a recent Yahoo News-YouGov poll: Among Democrats, Republicans and independents alike, support for such a move languishes under 20 percent.

Those polling numbers may help explain why, to the disappointment of progressives, presumptive Democratic presidential nominee Joe Biden was quick to oppose defunding the police. To the extent that the Black Lives Matter movement embraces that slogan — Black Lives Matter D.C., for instance, added the words “Defund the Police” to the “Black Lives Matter” mural that Mayor Muriel E. Bowser (D) approved for a street near the White House — they may find themselves meeting substantial opposition. (On the other hand, support for more incremental police reforms is high: In the Yahoo-YouGov poll, two-thirds of American adults said they support banning neck restraints, and 80 percent wanted an “early warning system to identify problematic officers.”)

If the public backs away from its initial embrace of a civil rights cause, it would hardly be the first time. A Gallup poll in October 1964 found that Americans approved of the recently passed Civil Rights Act, which banned discrimination on the basis of race, color, religion, sex or national origin, by nearly 2 to 1. But there were concerns about implementation, with only 19 percent backing “vigorous” enforcement of the law. And Americans started to get cold feet almost immediately: The proportion who thought the Johnson administration was moving “too fast” on integration rose from 34 percent in a March 1965 Gallup poll to 45 percent in a May 1965 survey. Then, in 1968, voters tossed the party associated with the Civil Rights Act out of the White House.

In a 1964 article, “The Nature of Belief Systems in Mass Publics,” the University of Michigan political scientist Phil Converse made the case that few citizens possess a coherent belief system or ideology. We might consider ourselves more sophisticated today, but there are reasons to believe that Converse’s view still holds. People’s opinions often don’t add up in the way that those who spend a lot of time thinking about politics expect them to. A classic example: Polls consistently show low ratings for Congress in general but high marks for one’s own representative.

Perhaps Converse’s view explains some of the public-opinion data on policing. People see Floyd’s death as a sign of a widespread problem, but they can also watch police beat protesters and deem it largely justified. And when they consider radically reforming their own law enforcement agencies, the immediate instinct seems to be: No, thanks. The jumble of views presents a serious challenge for activists who want to seize the moment and end police misconduct.

Twitter: @ProfBenjamin