November 10, 2010 – 11:52 am
I’ve been thinking a lot about that question the last few weeks. I teach math to three different “ability groupings” of kids, and yet in many ways they have similar sets of reactions in grappling with a difficult problem.
When they first hear the problem, there generally is interest and excitement, especially if I have chosen a good problem. (By the way, by “problem”, I don’t mean a routine exercise, but rather something that requires them to think, to use what they have previously learned but in a different way.) Students generally start talking among themselves about what they might do, or they raise their hands to ask clarifying questions and to propose a plan of attack. At this point, things have a really good feel– students are engaged, and they are anticipating solving a thorny problem if they just make a sincere effort. And indeed, that is often how things go– students spend a few minutes trying an approach or two, make a mini-breakthrough, and then solve the problem.
But often, after 5-10 minutes of thinking, students find themselves at an apparent dead end. Continue reading →
November 10, 2010 – 11:47 am
I assigned the following problem last week in my class:
Giuseppe likes to count on the fingers of his left hand, but in a peculiar way. He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on.
One day his parents surprise him by saying that if he can tell them some time that day what finger the number 1,234,567 would be, he can have a new sports car. Giuseppe can only count so fast, so what should he do?
Here’s how it went down:
Some of my students realized instantly that counting up to 1,234,567 on their fingers wouldn’t be a very effective use of their time (or Giuseppe’s!). Others counted up to about 180 until they “abandoned ship” on the brute force method. Continue reading →
November 9, 2010 – 11:20 pm
This is one of my favorite problems:
You’re planning a huge party for tomorrow, which will include a toast exactly 24 hours from this moment. You have 1000 bottles of wine, but one of them is contaminated with a slow-acting poison that will kill any living thing within 24 hours of being ingested. You happen to have 10 altruistic mice on hand who have volunteered to test the poison. How many bottles of wine can you safely serve at the toast?
I’ve given it to a number of classes, ranging in age and strength, and it’s produced wonderful discussions every time. Here’s a reconstruction of how many of these have gone.
Right off, several students come up with the idea of splitting up the 1000 bottles evenly among the 10 mice. When one of the mice dies, they explain, you would know that the poisoned bottle was among the 100 that it drank, and so the remaining 900 would be safe to serve.
Continue reading →
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