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.
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Paul:
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.
13 Comments
Paul
September 19, 2011 - 2:22 pm -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.
Bowen Kerins
September 19, 2011 - 3:58 pm -Paul’s right — the number of sheep is pretty tricky to pick here, and definitely part of good problem design here. I would pick 110 or 200 sheep if teaching to a high school audience, while 50 or 80 might be better for an elementary or middle school audience. I think 1,000 sheep could intimidate some students while moving others to the abstraction too quickly (inviting mistakes in executing the abstraction).
I also feel it’s important to pick a number of sheep that is 1 less than a multiple of 3, which kicks in the other thing I see as important if an algebraic approach is taken. How many students might answer 16 1/3 to this problem because they’re just solving for x and not contextualizing the solution?
“Eric the Sheep” comes from an Australian curriculum, and this page shows the problem being used in Grades 1, 3, 5, and 7:
http://www.blackdouglas.com.au/taskcentre/045eric.htm
Fortunately, this curriculum does NOT abstract the problem for students — the task is presented simply and directly. There are also some nice extensions about rule changes, asking students to come up with new situations (but in a relatively structured scenario).
The task along with some of its extension problems can also be found in the Algebra component of Annenberg’s “Learning Math”:
http://www.learner.org/courses/learningmath/algebra/session1/part_b/index.html
Matt McCrea
September 19, 2011 - 4:25 pm -Thoughts on starting with a small number – I like Bowen’s suggestions for each grade level – then making the follow-up question a larger number? It seems this would be especially helpful in lower grades where developing algebraic thinking would require a bit more scaffolding, though this approach isn’t necessarily more helpful. Or is it?
Christopher Danielson
September 19, 2011 - 6:17 pm -Dan writes:
Per our conversation over in my place, I find this to be of particular interest. What was the point of the example lesson? If it was to model learning mathematics through challenging tasks that admit a variety of approaches, then the plans for reconciling the differences among the various problems being worked on would be of primary importance. It’s going to be hard to have a group of students learn much from each other’s work if they are struggling to wrap their minds around each other’s problems.
But there may be dozens of different reasons for doing Eric the Sheep with graduate students.
Allison Krasnow
September 19, 2011 - 9:17 pm -I remember when Kim used this problem at least 10 summers ago as part of a Bay Area Math Project summer institute. Hearing that she’s still using it, with a new angle, as part of a totally different class, makes me realize that I don’t reuse good problems enough. My teaching could evolve while using the same original problem, but posed in new ways, reflecting how I am growing as an educator. Thanks for posting this…it made for good food for reflective thought for me.
luke hodge
September 20, 2011 - 1:34 pm -From one perspective I can see this problem involving an abstraction — introducing variables and/or an equation to solve/model. From another perspective I think this problem is really about removing abstraction. We have abstract procedures for dividing numbers with no context, but here we are presented with a concrete reason and meaning of division — it is quicker than subtracting 3 a bunch of times.
I don’t see the benefit in explicitly using a variable(s) or an equation to solve this problem. Viewing this problem as a division problem seems much more natural. Why introduce all that abstraction if you don’t need it?
Also, judging from the class questions, it appears that there are a few students in the mathematical thinking and problem solving class that are trying desperately to avoid mathematical thinking and problem solving.
morrowmath
September 20, 2011 - 2:27 pm -I noticed that many of the commentators focused on the number 3, as in “a multiple of 3” and “subtracting by 3 a bunch of times.” I watched a class of sixth graders tackle this problem as part of a lesson study recently. The “3-ness” of the problem was not readily to apparent to any of the groups, although it was to one or two students.
Part of the process of mathematizing a problem or context is recognizing the salient mathematical features. In this case, a change of 3 sheep. Because, as Bowens points out, it’s 3 sheep, but the step nature of the context means that an answer like 16 1/3 can be generated, which makes no sense as an actual answer. What the average student needs, is to recognize a change of 3 first, and why. And we can’t tell them it’s there.
The generalizations to larger numbers of sheep and the divisibility issue are only relevant if the student can see “3”.
luke hodge
September 20, 2011 - 3:33 pm -morrowmath: I think many students would need at least a subtle hint (suggestion to draw or act it out) to “find” the 3 and most would benefit from a very concrete demonstration of why 3 is important. How did the teacher react to the student’s difficulties in the class you observed?
morrowmath
September 20, 2011 - 4:11 pm -I don’t think that this is a “difficulty,” but rather part of the reasoning process. One group’s table saw a repeated +2 pattern and represented it this way (the number of sheep jumped by Eric):
2+2+1
2+2+2
2+2+2
2+2+2+1
2+2+2+2
2+2+2+2
The only 3 in this problem is the repeated similar groups of 3 expressions.
This is 6th grade, and they were able to verbally generalize this problem. In discussion, the teacher did ask for conclusions that led to thinking about the changes that happened around a multiple of three, but the students’ reasoning did not lead there. Sometimes you never know where a lesson or mathematical exploration will lead.
luke hodge
September 21, 2011 - 4:26 pm -morrowmath: Sure, you could call it part of the reasoning process, grappling, etc. But are you saying that these students were able to generalize the problem in the sense of being able to fairly quickly determine how many sheep are sheared before Eric for large numbers of sheep? I would not have guessed many students could do that without seeing that Eric moves forward three spots each time a sheep is sheared. How did they do it?
Chera G.
September 22, 2011 - 6:19 pm -This is an interesting math question. I loved how the students were able to formulate their own questions based on what they thought was going to be asked next. I also loved that they were able to chose the question that best interests them and describe their assumptions. The teachers I know from my own experiences and colleagues do not open up the door for creativity and higher level thinking. Allowing students to chose their own questions and then solve them requires independent thinking and reasoning that fosters growth. I would love to do something similar in my classroom. It’s very interesting that no one chose the same variable.
Matt Skoss (@matt_skoss)
October 13, 2011 - 3:37 am -What is also also interesting with this problem from the Maths 300 resource bank (http://maths300.esa.edu.au) is that the problem subtly changes if:
– Eric sneaks forward x sheep, then the shearer grabs a sheep
versus:
– the shearer grabs a sheep and then Eric sneaks forward x sheep.
Further on Maths 300, I’m interested in seeing how we can use Twitter tags (I use #maths300) to share our pedagogical ‘a-ha’ moments when using lessons from the Maths 300 library.