It isn't always a common occurrence to have a distance and bearing to a particular location, but given my choice in airline for winter break, I had exactly that. So during our first class back from the break, I asked students to figure out where I was when I took this picture:
Students scrambled to open up various online maps and make sketches. The students settled on a range of answers. Then I showed them this:
We had a quick discussion about assumptions. Then students looked again and talked to each other while revising their answers. Once they were satisfied with their answers again, I shared the correct answer.
This led into a nice discussion of the mathematics exploration project that is submitted as the internal assessment of the IB mathematics courses. The students know that I take pictures and videos of this sort of thing all the time - it's a habit instilled in me by someone we all know. The students said though that they don't usually see math in the things around them, which is a problem given that the math exploration is supposed to come from them.
My recommendation, which comes partly from Dan's suggestions, is just to start. I told the students that any time they see something interesting or beautiful when they're walking around, to take a picture of it to review later. With some time between seeing it and reviewing it, they should ask themselves why it interested them. What is it that makes the picture beautiful? Are there patterns? Is it organized in an interesting way? I also shared my RSS reader on Feed.ly and how I save articles that interest me and tag them accordingly. This is how I find interesting ideas to share with the class - they should do the same to figure out what might be a good source of material for their work.
We have had several discussions in class about what this exploration will be about, but the emphasis has really been on something that interests them. Having students be curators of their own 'interesting-stuff' collection now seems the most obvious way to get them started.