PCMI is over. There is a longer recap on its way as I keep puzzling through everything from the last few weeks. I’m also pretty excited about what my working group put together, and will be posting about that as well.

I had this odd few days between PCMI and Twitter Math Camp, and spent it backpacking in the Wind River Range in Wyoming. It was a great chance to reflect and try to figure out the meaning of life, mathematical structure, and conic sections. Not sure I made much progress on that front, but it was really beautiful — check it out!

I also have this handy satellite tracker so my parents don’t freak out when I do stuff like this. Here is my trip:

And here is a link to the interactive page (not sure how long it will be active). I think there’s some good math here — drawing functions to represent distance traveled, or altitude over time. Illustrating average rate — did my rate vary a lot, or did I just take breaks of varying lengths? The derivative and second derivative are pretty interesting, and require a good understanding of the context. Need to keep that one on the shelf, and maybe flesh it out some more later.

I am also almost finished my summer of driving:

There’s definitely some good math in there as well, probably about whether it was a good idea to buy this car used. Just one more drive to Leadville after Twitter Math Camp and I can hopefully put the keys away for a few days.

**John Holt**

Couldn’t put How Children Fail away though. If all of the secrets of life aren’t hidden in conic sections, they may be in that book. From the book:

There are sixteen kids in my math class. Four are poor students, one is fair; all the rest are exceptionally bright and able, with a good feel for math. They have all had place value explained to them many times.

The other day I asked, “Suppose I go to the bank with a check for $1437.50, cash it, and ask them to give me as much of the money as possible in ten dollar bills. How many tens will I get?” I wrote the number on the board. After some scrambling for pencils and scratch paper, answers began to appear. None were correct; most were wildly off. A few kids got the answer on the second or third try; most never got it.

I erased the original number from the board, and wrote $75.00. “How many tens will you get?” Everyone knew. I then wrote $175.00. “How how many?” This was much harder; a few got it, most did not. After a while, pointing to the digit 7 in 175, I asked, “What does this 7 tell me?” They said it meant that I had 70 dollars, or 7 tens. I wrote it on the board. Then I said, “How how about this 1?” They all said that it meant that we had a hundred dollars. Nobody said that it meant just as well that we had ten more tens. I said, “How many tens could we get for that hundred?” They all said 10. I pointed out that these 10 tens, plus the 7 they had already told us about, would give us 17 tens. I then wrote our first number – $1437.50 – on the board. We considered how many tens were represented by teach digit. The 3 told us we had 3 tens; the 4, that we had 40 more; the 1, that we had 100 more, for a total of 143 tens. I drew a circle around the digits 143 in the numeral 1437. By this time everyone was saying, “Oh, yeah, I get it; I see; it’s easy; it’s cinchy.” But I was skeptical, believing no longer in the magic power of “good explanations.”

This helped me clarify what I was thinking about last week with formative assessment. There is an art to asking questions like this that, while they seem on the surface not to be too complex, are really incredible tools for digging into what students understand and humbling me with the work I have left to do. I want to explore this idea some more.

**Twitter Math Camp
**Finally, I have just arrived at Twitter Math Camp. I spent a lot of time while I was hiking thinking about a recent post by David Wees, arguing that for teachers to really get something out of a conference, they need to both pick a specific topic and focus their attention on it, and take some time away from sessions to reflect and process what they’re learning. I think David makes a great point, and he helped me do some thinking around what I’m going to get out of Twitter Math Camp. Thinking back on TMC14, there was a ton of information, but a lot of it didn’t stick too well. In particular, there were lots of quick ideas that had me thinking “I have to do that this year” that I didn’t follow through on — and probably plenty more I have since forgotten.

My goals for TMC15:

**Have fun**. These are some of the best people around, and I want to enjoy my time with them.

**Make new connections**. I met so many great people last year, and I am excited to do that again. Some of those connections led to some great collaboration, and I hope to continue in that spirit.

**Learn something about teaching**. This is where David’s ideas have me thinking — I just don’t know if I have a clear idea of what I want to work on that I can use to focus which sessions I attend. I’m also curious about what I can do to increase the impact of individual sessions, both in terms of my own reflection down the road, and goals I set for myself. I’m going to be doing some more thinking about this the next few days, and trying to look at my conference experience through this lens.