First, experimental verification. During graduate studies at the University of Washington, I had the privilege of working as a teacher's aide under the direction of Lilian McDermott's physics education group. They spend a lot of time in careful interviews with college students both before and after physics lectures, trying to draw out precisely how the students understand various things they encounter in the real world, like speed, velocity, electricity, forces, momentum, etc. This means actual interviews with individual students, paying attention to what they are saying, carrying on an attentive and probing conversation about their understanding of a specific physics topic, taping and transcribing the whole conversation, trying afterward to understand the what the student was thinking of, and then repeating the process with many other students. I have observed the same results myself in a less scientific manner by guiding students through units of McDermott's workbook "Physics by Inquiry," which uses Socrates' method (a lot of carefully posed questions challenging the reader to think through specific examples) to help the reader reason through a full understanding of particular topics.
The surprising result of McDermott's research is that even when people talk about scientific concepts like speed, velocity, forces, or momentum, they as a rule understand them in a much different way than scientists do. What a scientist means by velocity is much different than what the average person means. In other words, everyone carries around a mental model of the world - for instance they have ideas about falling objects. Everything falls, and that is because of gravity. Usually these ideas are not that precise, and it's easy for a scientist to ask questions that puzzle others. (And vice versa, but that's another story.) For instance, consider a ball on top of a table, which is pushed, and moves off the side of the table and falls. If you are beside the table and observe the trajectory of the ball as it moves past you, off the table, and toward the ground, what sort of curve does it make? Take a minute and think about it; draw a picture of the table and the floor (looking from the side so that the ball is moving past you not towards you), and then draw the trajectory of the ball.
Experimental evidence with people answering this question shows that a large fraction of college students (and by extension most adults) answer this question incorrectly. In fact they do have some correct ideas about falling (everything falls, roughly how long, gravity), but the overall mental model of falling and how it works is incorrect. Given a few simple but precise questions - obvious ones not head in clouds ones - almost everyone will give an incorrect answer to one or more, revealing an incorrect understanding of falls. Some will feel a bit puzzled or uncertain as they answer, others not; some will be able to give long and detailed explanations of their reasoning and evidence and predictions, others not - but irregardless they will be consistently wrong. This despite daily and long term experience with falling objects, including one's own body.