Shecky reminded me of this set of 4 videos that James Tanton did. I'm not sure what #2 means (watched the video a while back, forgot the details), but the rest sounds great to me.
This might be a good place to direct my students in the first week. Maybe I'll give them some choices, and ask them to tell me what they think of at least one website.
Mary O'Keeffe wrote earlier this summer about doing Guerrilla Math Circles at a local park. She did a Pascal's Triangle question, making a big triangle with sidewalk chalk. I like the idea of grabbing the attention of random people in public. This video on Maths Busking* looks like something similar.
Malke wrote a great post about her daughter's take on The Cat in Numberland, pointing to a fabulous review of it [pdf] I hadn't seen before. It goes into great depth on the pedagogical issues involved.
While I was at the Math Circle Teacher Training Institute (just last week? is that possible?!), I ran a math circle for the top level kids, mostly 12-16. I had them analyze Spot It. (Posts here, here, and here; can you see any progression?) There were 9 kids, in 3 groups of 3. Each group approached it differently, and one young man (call him Trevor) became absolutely obsessed with the problem. It was a delight to get to work with this group. Our institute participants were there to observe me, and I seduced them into thinking about the problem too. A few of them were able to multi-task well enough to give me some good feedback. One noticed that Trevor and I got into a conversation that excluded the other kids (oops!). But the other kids watched us for a few moments and then decided to get back to work on their own ideas! (Invisibility achieved, though indirectly.)
The adults, who were working on the floor in the back of the room, made a bigger dent into the problem, and spurred me to think more about whether cards with 5 or 7 pictures per card can be made with the process I came up with in January. (5's, yes. 7's, I'm not sure yet. This question has connections to something called finite field planes, which I intend to explore soon.) Bowman Dickson was in that group and wrote a great post about their thinking. He also wrote about the origami he learned to make. (I meant to make some origami, but never found the time - dang it!)
[Note added on 11/30/12: Perhaps 5's cannot be made. The procedure Bowman's group used, which I had also used, does not work for 5. A student apparently made a deck using trial and error. I wonder if I can replicate that.]
Dan McKinnon seems to be thinking about number theory on his blog, mathrecreation:
[For] these kinds of sequences (generated by polynomials with Integer coefficients) - their terms are either always even, always odd, or alternate between even and odd values (e.g. you won't get a sequence that goes "even, even, odd, .." or some combination other than the three possibilities mentioned). Can you see how you can show that this is true using similar arguments to the ones used here for last-digits?
- The math behind a beautiful building
- "The hard bigotry of poverty" (re school 'reform')
- Learning time with Scaredy Squirrel
- The calculus behind an eyebrow
- Surface area for the kindergarten crowd
- A liar and truthteller puzzle (where the number involved can't be determined)
- Yoshimoto cube models (check out the videos at the bottom of the page)
- Letters to a new teacher (Bowman also got this going, amazing! I haven't had time to write one yet, but maybe in a few weeks, when I'm thinking about my courses...)
- Getting students to ask more detailed questions
- Teaching kids computer programming by having them 'program' each other to dance
- Inventions Festival
- Pairs of cities with the same population
- Getting better at teaching
- Show that the blue and red parabolas intersect at right angles
- Ten hours with Erica (in which Rebecca changes Erica's math life)
- Stephen Lazar's 2011-2012 Teaching Portfolio (I'd like to do this some summer)
* Busking is a British term meaning performing for donations.