Here's something to ponder:
In sports, it can be hard for people to participate in a sport if they're not in the elementary-to-college sports pipeline. Many sports have recognized this and formed recreational clubs: there are running clubs, Ultimate Frisbee leagues, bike groups, master's swim groups. These groups are vital because they contribute to peoples' quality of life in terms of fitness and social contact, and because they provide a community of supporters for the young and the elite in the sport. Community clubs sponsor scholarships for talented participants, provide coaches and staff races or other competitive events, provide economic support to the businesses that sell equipment or training. An enthusiastic community of amateurs helps lift the quality of the entire sport. Sports that don't have this community involvement have trouble succeeding in some aspects above. In the US, for instance, there is not to my knowledge a javelin-throwing and shotput-tossing community on the level of, say, Ultimate Frisbee or soccer. Other countries give a lot more opportunities for track and field and so they are more successful on an Olympic level in javelin and shotput, among other things.
Translate to mathematics. How do we involve and engage adults in mathematics? We have math circles or math team for kids and students. We have math majors for college students, and Pi Mu Epsilon and the Putnam. What about the 30-year-old who likes math but never was a math major or didn't even go to college? What about the 55-year-old who really likes solving certain kinds of puzzles or playing logic games but didn't see math as an option when she went to college and kind of wishes she'd learned more? How can people out of school be engaged in mathematics on a recreational level?
Even more audacious, how can people not in school or academia be engaged in math on a research level, even a small one? A friend commented to me recently that she could not begin to imagine my world and what I do on a daily basis -- she just has no idea what the process of math research feels like, looks like, is like. Maybe that will begin changing as people who have done REUs graduate and go into professional life. That doesn't help the people I mentioned above.
I know people who read all of Brian Greene's books and feel like they get an idea of what goes on in modern physics. They are fans of physics: they support it, are interested in what people do with it, read about the Higgs boson, support physics funding by government entities, staff the physics club at the local high school. They are the community boosters for physics. I have not met as many people who feel that way about math. I feel like there aren't as many math books that allow that "in," with some notable exceptions. How could we open up the world of math research to nonprofessionals to a tiny degree so that we, too, could have a community support network for mathematics?