Last Wednesday at UC Berkeley in Alan Schoenfeld's class on mathematical thinking and problem solving, Kim Seashore wrote the following paragraph on the board:
Eric is standing at the end of a line of fifty sheep, waiting to be sheared. He is hot and impatient. Each time a sheep is sheared and Becky, the sheep shearer, turns to put the wool away, Eric sneaks around the next two sheep in line.
"What question am I going to ask next?" Seashore asked us. We thought for a moment and then shared out responses. Here are a few:
- How many sheep will be behind Eric when it's finally his turn to be sheared?
- How many sheep were sheared before Eric?
- How many kilograms of wool will the sheep yield?
- How many sweaters can you make out of them?
- What's the significance of skipping two? Why two instead of three?
- Will the other sheep get mad?
- What if the 49th sheep had the same idea after seeing Eric skip ten more sheep and started skipping three every time? Who gets sheared first?
"Great," Seashore said. Then she had us categorize those questions:
- Which can we answer?
- Which can't we answer?
- Which need more information to be answered?
Then she asked us to work for awhile on a question that interested us and was answerable. One person took up "how much wool?" and she asked him to be explicit about his assumptions. After ten minutes we grouped ourselves and explained our work to other people.
A Few Notes On This Scene
- "What question am I going to ask next?" isn't the same question as "What question interests you here?"
- Why fifty sheep? How was that number chosen? Fifty sheep was short enough that some students determined how long Eric would wait to be sheared by simulating the entire problem. What is gained or lost by describing a line of 1,000 sheep?
- Asking students to generate their own questions is risky. Seashore encouraged us to pursue our every whim even though the "kilograms of wool" question was going to involve very different mathematical thinking than any of the others. I don't know how she planned to reconcile that difference. ¶ My approach is to sample the room for questions and take +1′s for each. (ie. "Is anybody else interested in Sam's question?") This reveals a hierarchy of student interest which we handle in order. ¶ Meanwhile, I am in contact with teachers who ask their students to generate questions only to coerce them down to the one they (the teachers) originally wanted to pursue. This interaction will only pay off negative dividends, as far as I can tell. These classes would be much improved if the teacher would simply ask a concise question that she knew in advance would be of some general interest to her students. Most questions asked in math class are neither concise nor of much interest to the students so we're already way ahead of the game.
- Abstraction was nine tenths of the work. In answering, "how long will it take Eric to get sheared?" I had to represent the problem with variables and build a model out of them. This was, by far, the hardest work of the problem. Moreover, no one I spoke with chose the same independent variable that I did.
- Your textbook would abstract the problem for your students.
Be less helpful.
2011 Sep 20: Bowen Kerins locates the original text of the problem, which mercifully leaves the hard work of abstraction to the student.
A line of 50 sheep makes me wonder why I would ever have to use variables to represent the problem.
A line of 1,000 sheep makes me wish I had an easier problem – one I could actually act out.
What number of sheep will motivate me to model a simpler case and look for patterns? What number of sheep will force me to generalize and move from concrete models to abstract thinking, without stepping over the boundaries of the story?
Shearing a line of 1,000 sheep? Eric will be waiting a very long time.