I’m lucky to be able to teach part-time in addition to my work at the Center for Innovative Education, and right now I’m reflecting on my just-completed year. I’m gathering my thoughts and notes on things that went well, things that decidedly didn’t, and all that somewhere-in-between stuff I want to improve.
One of the things I discovered this year was the universe of dedicated, insightful educators on Twitter (I suppose I discovered this world much the way De Soto discovered the Mississippi River or Columbus the New World – which is to say it was known to plenty of others already, just not to me, until a lucky encounter with @JohnBerray at DeeperLearning2017). In any event, one of those insightful educators on Twitter is Robert Kaplinsky, a math teacher with a great website and blog. A few months ago, I read his post, “How Old is The Shepherd?”. (Stick with me. This will tie in.)
He asked thirty-two eighth graders this question: There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
Now, one doesn’t know the first thing about the shepherd. Yet 75% of the eighth grade students he asked gave a numerical answer. Check out the video where some of the students explain their thinking.
I thought of my fifth and sixth grade math students as I watched that video. I didn’t think they would answer the same way.
So I tried it.
…and they did.
A quick note: As suggested by Kaplinsky, I kept the exercise anonymous. My prediction going in was that most students would tell me the question was ridiculous. I didn’t want anyone who wrote a number to be embarrassed, so anonymity made sense to me. I asked students not to show others their answers and we all crumpled our papers and had a “snowball” fight to mix them up before tallying responses.
The snowball fight, by the way, was a huge hit. Students wanted to crumple and throw all their papers. Homework problems? Permission slips? Newsletters? Birth certificates? Let’s crumple and throw it all! Just kidding about the birth certificates.
But I digress.
The results from my classes were eerily similar to the results in the video above. Nine of my twenty-six students responded that the shepherd was 25 years old, showing the impeccable long division they had performed to arrive at the answer. Twelve others gave numerical answers such as 26, 52, 65, 74, 75, 130, etc. Most of the answers were numbers that could be a person’s age. Clearly, the student who answered 130 added the given numbers, and was either unfamiliar with longevity norms, or simply preferred addition over the other operations. No one multiplied and produced a 625-year-old shepherd. One student wrote, “The shepherd is either 10 or you’re talking about the dog and this is a trick question.”
I desperately wanted to know why the shepherd would be ten, not to mention how the dog would be the tell of a trick question. These are the downsides of anonymity.
Exactly two students expressed that they knew the question couldn’t be answered: “It doesn’t say anything about the shepherd,” wrote one. “Impossible,” wrote another. But even the student who wrote “impossible” hedged his or her bets, adding to the answer, “or maybe 25?” A third student left the paper blank (refusing, I imagine, to engage in the nonsense, but not quite confident enough to spell it out). A fourth wrote, “Dead.” I clumped that student in with the other healthy skeptics.
I watched them as they answered the question. There was no shortage of puzzled looks and quite a lot of furrowed concentration. Nearly all valiantly tried to give me the numerical answer they assumed I wanted and expected.
Perhaps this question didn’t feel so different from other math questions they’ve encountered over the years. A lot of problems go to great lengths to provide “real-world” context and end up asking incongruous questions.
Case in point: Scientists discovered the fossil of a huge African crocodile that was more than 40 feet long. So far so good. I’m intrigued. But wait for it: About how many door widths are equal to the length of the crocodile? (That’s a real question. From our math book.)
I asked my students what they had been thinking as they answered the shepherd question. My favorite response was: “I added the sheep and the dogs and then divided by 5. I don’t know what that means.” (Me either, buddy.) I recognized myself in the student who feared not getting credit without providing an answer (Let me please you!). I saw that many students were thinking well outside the parameters of the problem, trying to create the context they needed. At least one was ready to question my mental state.
And so, for several weeks now I’ve been thinking about the shepherd problem and what it might mean – and this is where it ties back in to what I’d like to do better next year. I want to be better at creating a classroom culture where I am not posing questions and waiting for students to please me with the answers they assume I’m safeguarding, one where students feel more comfortable challenging the very premise of a question.
I’ve been reading about numberless word problems, which seem like a great idea for getting kids thinking about what’s really happening rather than being flummoxed by the numbers; and I came across the idea of providing only numbers and having students design problems that could fit them. Maybe something like: 54 books. 12 students. What could the question be? Given that information, I’m quite certain no student will come up with “How old is the librarian?”. One advantage to this style of question is that there are many possible answers; another is that the process of creating their own questions may give students a better feel for how to read and understand questions they come across (even when the question asks them to measure a fossil in door widths).
The cultural shift I’m looking for is bigger than word problems, of course, and I won’t get it all right in a single year. Still, the shepherd problem could become an annual exercise, one way in which I measure my progress toward this classroom culture goal. Year One: 22 out of 26 tried to give me the shepherd’s age. Year Two? Find out next spring.
Photos by Kristin Blais.