Resetting the public’s understanding of how science works will be a big job, but a good place to start is with students who get science degrees. Unfortunately, most programs are full of didactic classes about scientific principles, with few if any requirements on the history and philosophy of science. Because many undergraduate science majors pursue careers outside of science, including medicine, a shift in curricula would ultimately produce a public that is more literate in the way that science works. This means making hard decisions about how to fit a broader, deeper perspective into curricula that are already jammed tight with the necessary basics. However, it’s urgent for scientists to make compromises in the way they teach for the greater good.
There is so much benefit one can gain from having a solid grounding in the philosophy of science. If you are a student pursuing science, see if you can find a course at your university. You won't regret it!