On leaving my building: What sort of symmetry is there on the building's door?
On the subway: For what percent of a year have my wife and I been married?
In the elevator: The weight capacity is listed as 2500 lbs or 13 people. How much are they assuming that each person weighs?
In all these cases I was intrigued to figure out the answer, and I felt satisfaction when I did.
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(In the elevator on my way home I also saw this, booo NYC.)
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What's remarkable about these questions is how easy they all are for me. They are not very challenging. And I really like math! I do it in my free time. I teach it. That makes me for sure in the top 1% of mathy people on the entire planet. And I know way tougher math than lines of symmetry, percentage conversions and division.
You might have thought that lots of questions occur to me over the course of the day of varying difficulties. But that's not what's happened for me over the past two days. There seems to be an intimate connection between the questions that strike me as interesting and the questions that I'm capable of answering successfully.
I'm essentially asking questions that I've seen asked before.
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To temper this a little bit, some tough questions did occur to me later in my day. I'm doing some work with a 7th Grader as part of an independent study, and she's really into infinite series. While preparing for our time together, I asked myself whether I could measure the speed at which a sequence of decreasing fractions approaches zero. That was tough for me, and I didn't know how to take it. I tried to directly compare consecutive elements in the sequence, but that didn't really seem so informative.
I gave up really easily. And then I caught myself, and then I tried again. And then I gave up again when it wasn't really yielding anything.
Why did I find this question interesting? I think, for a moment, I thought that it would be easy to answer, and that I would have a bit more valuable knowledge under my belt. When it turned out that I didn't have easily accessible knowledge, I more or less gave up, and then it took all of what I know about learning to get back to work. And then I still gave up.
I have two take-away thoughts: (1) I need to work harder and (2) Questions rarely stay interesting if the road forward is unclear.
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Danielson says that questions are evidence of learning. Maybe part of the reason why is that we usually ask questions when we're fairly secure that we could figure out the answer.
Or not? Again, thanks for kicking around these ideas with me while I try to make sense of it all.
I've been rereading Willingham's Why Don't Students Like School? His first point says pretty much what you discovered: "People are naturally curious, but we are not naturally good thinkers; unless the cognitive conditions are right, we will avoid thinking." Curiosity is fragile. Solving problems brings pleasure, so we tend to enjoy problems we are pretty sure we can solve, but we give up easily if it looks like work.
ReplyDeletehttp://what-if.xkcd.com/
ReplyDeleteI like to see what he thinks, I like to read the question and try it myself, and it makes me think of more applications
I posted my thoughts at the link below...what I'm curious about is the difference between interest (I want to know) vs engagement (I will stick with this and finish it)
ReplyDeletehttp://fivetwelvethirteen.wordpress.com/2013/11/15/interest-vs-engagement/
Michael, I think you'd really love the Math Circle Institute that happens the second week of July each year. I think it will stretch your conceptions of what's interesting and what's doable. Your post made me think of this because you sound like me a few years ago.
ReplyDeleteSince I got involved with math circles, I have found myself attracted to much harder problems. I spent much of one winter break figuring out Pythagorean triples, and much of another figuring out how they put together the game of Spot It. The Spot It question 'interested me' for about a year before I finally, with the help of a less math-inclined friend, truly began to tackle it.