At TMC14 the entertaining Steve Leinwand gave a talk about some of the trappings of math teaching. One section of the talk dealt with asking the simple question "about how much?" with a expression containing some oddball square roots.
At random around 11pm one night, that point resurfaced. A weakness I've noticed for a while is that students lack an understanding for the value of things like sqrt(2) or how you could use what you know to approximate sqrt(27). Or the value of 3/2 or how to estimate 1/3 of something. I had a conversation with another student once about the final price for a car, and they had no idea about how to approximate tax/title/license on top. After explaining it, they ask the poignant question "why does no one mention this in school?"
This year I think I'll do something about it. It's like a really focused version of Estimation 180 without all the pictures. I'm thinking I pose four situations a day and have a discussion about how these could be approximated to build their "oddball" number sense.
I probably wouldn't type them all in advance, this was an exploration exercise. Probably after a few months I could compile them here for those of you that would like a home version.
This could be fun. If you too have noticed a deficiency in number sense, don't complain about it or blame calculators, try to do something.