Good ways to engage in mathematics outreach? Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose children and adults to all the fun we are having.
What are some particular ways that people have found to do this? A couple come to mind. There are Math Circles in a number of cities, which welcome volunteers, and these are a fantastic resource. There are also clubs built around preparing middle and high school students for competitions, which are wonderful for many people.
What are some other ways people have found to reach out to the community at large? I would be particularly interested to hear ideas that "anyone can do" -- that don't require special knowledge or connections, or too much commitment (at least at first), and that I could try when I begin my first permanent job. I would also be very interested in any outreach that targeted adults, including faculty outside the math department.
Thank you!
 A: In Mobile we have a Math Circle. This is highly successful, but it is also a long-term, time-consuming commitment. 
This paragraph should be formulated as a question. It seems to me that just as there is art therapy and music therapy, there should be Math therapy. My belief (without strong experimental evidence) is that the act of engaging in mathematical activity --- whether it be as mundane as reciting multiplication tables, or solving sodoku, or as advanced as reading the mathematical literature --- exerts a calming influence upon the mind. Thus people with various mental incapacitations could benefit from the therapy of engaging in mathematical activities. 
The question then is, "How can the community, or should the community, develop mathematical therapy for adults in various levels of incapacitation? 
A: In the UK there are a lot of good programmes for engaging the public with mathematics, some of which have been touched on in other answers.  Here is a list of the ones I think have worked the best or been most innovative:
1) Royal Institution Masterclasses: a UK-wide series of masterclasses aimed at age 13 school pupils.  Classes are on Saturday mornings for between 5 and 10 weeks.  Each class has a different presenter and each class lasts for 2.5 hours with time for lectures, exercises and refreshments.  I've been involved in giving the Edinburgh and Glasgow masterclasses and they are a lot of fun for all involved.  The kids really appreciate getting to see some maths that they'd never encounter through school, and especially enjoy seeing the diversity and usefulness of the subject.
(In addition, a set of masterclasses gets broadcast on the BBC every Christmas!)
2) FunMaths Roadshow: a collection of 350 maths activities, each suitable for a particular age group between 5 and 20.  The resources are cheap and easy to make, as well as being easy to transport, so that they can be taken all around the local area and used with different groups of people.
3) Maths busking: a new phenomenon which has been surprisingly successful!  'Buskers' go out onto the streets and perform games and tricks which appear to be magic but which can be explained by mathematics. It's a great way to engage with people who would otherwise never attend a maths lecture or think of going to a science museum.
4) Mathematical walking tours: do a tour that opens people's eyes to the mathematics around your city.  This can be historical curiosities, geometrical constructions in the architecture or optimization in the town planning.  Such tours are currently being designed for Oxford and London, or get examples from Manhattan or this maths tourism blog.
Whatever you decide to do, just go ahead and do it with plenty of enthusiasm!  Good luck!
A: No one has yet mentioned exhibition material, science fairs and the Womens' Institute. (I have also been involved with Masterclasses (see the reply by J. Collins) but that sort of thing has already been discussed.) Some of you may know Bangor's Maths and Knots website. That started off as a physical exhibition that toured the UK with the Pop Math Road Show. I seems to have been a great success (1990). We tried to show that Mathematics was a natural extension of logical curiosity by going into the problem of classifying knots and the use of Analogy in mathematics (Look at Ronnie Brown's website for links to a paper we wrote on how the exhibition was made.)
I adapted that material for a talk to the UK Womens' Institute at their residential college. The theme was Mathematics a human activity. (Again I can give more details if someone wants to follow it up.)
Thirdly I prepared material for use at the Wrexham Scientriffic Science event. Here I used some puzzles but not just the usual ones.  I adapted material on optimisation from a course on OR and it was a great success. There were some people (adults) who spent over an hour discussing  problems on scheduling, knapsack problems etc. I did not provide the algorithms (which would spoil the fun).  The problems typically had a 'target' value, so if trying to maximise something anything over the target would be good.  I tried to get them to explain why they thought their method was possibly going to give a good answer and was very pleased at some of the systematic and logical approaches that were used. The puzzles were printed out from a powerpoint presentation and then put on laminated cards. (EDIT: Combinatorial puzzles seem to be easy to attempt and go down well.)
I used the puzzle presentation as well in various Masterclass/Mathclub situations in Wales, Canada and ireland and always got a good response. 
A: In the past, I have run several workshops at elementary and secondary schools. The aim was exactly, as you so nicely put it, to show the kids how much fun we are having (although the first time I did that, I hadn't even started university yet). So let me tell you a personal story.
My experience, which comes from dealing with three different schools in two different countries, is that you don't need any connections at all! In all three instances, I just contacted the head masters of the schools and told them "how about I do weekly maths/logic afternoon workshops for your kids" and they always gratefully and immediately accepted. It was always my choice for what age group, how long, how often, and about what specifically the workshops were supposed to be. So go right ahead, pop into the next school you pass on your way and ask to talk to the headmaster.
As for reaching grown ups, the first workshops I conducted were for 9-10 year olds, I myself was mere 19 years old. I always gave the kids "homework" and they always came back the next week, telling me how their entire family, including older siblings and parents, enjoyed solving the puzzles. At the end of the workshop (that is, when I decided to end it), the headmaster told me that she had received several requests from the parents, whether it would be possible to initiate a similar workshop for them. She indicated that the school was willing to provide a room and the material that I would need. Unfortunately, I was about to start my civil service, so this never came about, but it obviously would have done if had so chosen.
One more thing: the kids didn't let me go after the last session. They literally clung to my legs and kept asking whether I would do the same thing the following year (which I couldn't because I was about to begin my undergraduate studies abroad). The moral is that these workshops have been one of my most gratifying mathematical experiences, so I recommend this to anybody who considers it, even out of completely selfish considerations. Please help yourself liberally to the material that I used and that you can find on my website. Please also let me know by email if you do go ahead with this. I will be very grateful if you could share your material and ideas with me.
A: the park city programs in summers are excellent opportunities to meet and interact with high school teachers. I am also impressed with "the wisconsin idea", that university research should affect the lives of the community.  They have an outreach department that does more than cure corn blight, and actually communicates with local school teachers.   I hope this idea may take hold here in my state, where we have a top ranked university math education graduate school, but poor quality math education overall in our pre-college school system.
Twenty years ago or so I also volunteered to teach advanced calculus at a local private high school.  We used the book by marsden and tromba and it worked well for those students ready for it (most of them).  I wrote a summer proposal into my NSF grant to fund these students, a few other younger ones, and their teacher Steve Sigur, to be paid to go through Mike Spivak's appendix to his Calculus (on the real numbers, rigorously) with me, while I again volunteered the free month of my summer for this.  One of my students was Jeff Brock, now a professor at Brown in topology.  Another of my students, Jonathan Manson, became a Phi Beta Kappa in physics at Harvard, and took a PhD at Illinois.  Another student went to Chicago and took the Spivak calculus class there.
I also monitored the students in preparing various math projects, on Galois theory (that student went to MIT and is now a math teacher at a private school), cyclotomic polynomials, and Mandelbrot theory and eigenvectors.
When my son was in second grade I colored the faces of some polyhedra and gave a lecture to the kids on the relations between vertices edges and faces, and one little girl (who later became an aeronautical engineer) guessed Euler's formula.
You can do a lot as long as you are willing to volunteer your labor.  It would be nice also to earn some income at this, but that seems harder.  From my experience, although funding is minimal for high school students, that is precisely when you need to magnetize them to mathematics, or they may be lost forever.  It takes real persistence to keep loving math through 12 years of boring instruction.
A: There are a number of people who explain mathematics through videos and then post them on a widely used video viewing site such as YouTube.  The Khan Academy (which got the notice of Bill Gates and Google) is a good example of this.
