What Can You Do With This: Estimating Coin Value

Adding the blog of Albany Area Math Circle to my Google Reader has added delight to my mornings. I'm seeing so many fun problems on there! Yesterday she linked to one here with a picture that begs you (well, me at least) to start computing.

Guess the total dollar value of the change in this box, and win a galley copy of Chad Orzel's soon-to-be-published book, How to Teach Physics to Your Dog.

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Edited on 8-17-09:
He's posted the answer. I was way low. As was the average answer, dubbed "the wisdom of the crowd". Orzel asks why. I'm thinking we tend to estimate low on money.

What would help us estimate better?

Mathsemantics, by Edward MacNeal, addresses this in one chapter. (I've assigned it as reading to many of my classes.) He talks about having a semantic web in your head that includes a few important numbers, like:

• population of the earth
• population of the U.S.
• population of your state
• radius of the earth
  

And then he recommends estimating often, committing to your estimate somehow, and then finding out the real value of what you estimated. For example, estimate your arrival time when you're in the car, tell the person next to you, and notice the time when you do arrive at your destination.

There are also books that give lots of examples of estimation problems that involve thinking step by step. I've skimmed through Guesstimation: solving the world's problems on the back of a cocktail napkin, by Lawrence Weinstein and John Adam, and Geekspeak: How Life + Mathematics = Happiness, by Graham Tattersall. (I liked Guesstimation better. Neither book was compelling reading, but Guesstimation will work well as a reference, whereas Geekspeak doesn't live up to its title at all.)

One classic of this type of problem is "How many piano tuners are there in New York City?" I've worked through this with many classes: What is the population of New York City? What proportion of households have pianos? How many more pianos are in the city? How long does it take to tune a piano? How often do the pianos get tuned? Put it all together, and if you estimated the pieces decently, you get an answer that's between half and double the true value, which is pretty good. [What does true value mean? How do we count people who work part-time, or who have another profession and tune pianos once in a while. Perhaps there is no exact right answer...]

I've done all this, and I'm still not that great at estimating. (I thought the visible layer of coins in that box was worth about $10, and that there were probably 6 to 10 layers like that in the box.)

Anyone have other ideas about how to help others, and myself, learn to estimate better? Anything you'd add to this if you used it in a class?