“I study how children are doing in relation to their parents,” said sociologist Siwei Cheng of New York University, the study’s lead author. “And America is not doing that well, compared to other Western countries.” In Canada, for instance, the chances of a low-income child entering the middle class are twice as great as they are in the U.S. The question that Prof. Cheng sought to answer is whether Americans still believe in the American dream. “Are Americans really that optimistic about mobility?”

Prof. Cheng and Fangqi Wen, a postdoctoral fellow at the University of Oxford, polled the attitudes of 3,077 American adults. Each participant was asked to consider the prospects of a child whose family’s income was in a specific percentile, compared to all American families. A computer spat out a randomly generated income rank, and the participant would estimate how much a child growing up in such a family would earn as a 40-year-old. The next step was to compare subjects’ perceptions to what up-to-date tax data tell us about the actual earnings of someone from such a family.

This comparison revealed a disconnect: Americans underestimate the future earnings of children from poor families and overestimate the future earnings of children from middle and upper class families. “The reality is that there is indeed a [mobility] gap, by international standards. But what people have in their mind is a larger gap, in terms of how rich and poor kids will do. They’re pessimistic about equality of opportunity,” said Prof. Cheng. To be precise, “the American public perceives the gap in economic outcomes between children from rich and poor families to be twice as large as it actually is.”

The researchers also discovered some surprising demographic divides. College-educated adults estimated a larger opportunity gap than those without a degree. Liberals were more pessimistic than conservatives, younger people more pessimistic than those over 30, and those earning between $30,000 and $100,000 a year more pessimistic than everyone else. In other words, middle-class, educated Americans see less reason for hope about social mobility than the rest of the population.

So much for the American dream.

The study doesn’t try to identify the sources of this pessimism, but we can hazard some guesses. It could be that the squeeze on middle-class jobs and incomes over the past generation has undermined middle-class faith in advancement. In addition, the suspicion of inequality in American society may lead people to assume that the children of the wealthy are getting a free pass, while the children of the poor must be even more hobbled by their reduced chances.

Whatever the reason, if Americans no longer believe that children who start near the bottom can make it to the top—or even to the middle—a good first step is to reconcile our attitudes with the data. There is more upward mobility in America than most of us think, but unless we want to start calling it the Canadian dream, there’s still a long way to go.