“How old is the shepherd?” — The problem that shook school mathematics

How to liberate students from authoritarian maths instruction

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

Got it yet? Of course you haven’t. There is no way to discern the age of the shepherd from the size of his flock. No subtle trick or manipulation can rescue this problem from its absurd framing. We can all see through it, right?

Now consider that, according to researchers, three quarters of schoolchildren offer a numerical answer to the shepherd problem. In Kurt Reusser’s 1986 study, he describes the typical student response:

125 + 5 = 130 …this is too big, and 125–5 = 120 is still too big … while … 125/5 = 25 … that works … I think the shepherd is 25 years old.

Remarkable. In their itch to combine the numbers presented to them, students negotiate three solutions. They show some awareness of context in dismissing the first two candidates. But a 25-year-old farmer is plausible enough for students to offer it up as their answer. The calculations are correct, but they are also irrelevant. Common sense has deserted these students in their pursuit of a definitive answer.

The study has been repeated several times, each one showing students’ propensity to reach for an answer at all costs, however absurd the reasoning (Robert Kaplinsky’s 3-minute montage shows the multitude of ways in which students arrive at a numerical answer. It is not pretty viewing.)

No clues here.

Anyone with access to schoolchildren can replicate these observations. I put the question to a group of three sixth graders. Oh, the horror that unfolded. The students appeared to be lulled into a stupor, losing sight of the problem at hand as they mined through computational methods with consummate ease. They served up a platter of answers, ranging from 5 to 625. In the ultimate betrayal of reasoning and common sense, one student insisted on 25 remainder 2 (I haven’t yet figured out where this one came from. Neither has he, for that matter. Any suggestions?)

The shepherd problem exposes the fragility of student’s mathematical reasoning. It is not an isolated example.

Take the following problem, also from Reusser’s study:

Yesterday 33 boats sailed into the port and 54 boats left it. Yesterday at noon there were 40 boats in the port. How many boats were yesterday evening still in the port?

According to Reusser, only one student from a sample of 101 fourth and fifth graders correctly deemed this problem insoluble. Equally alarming, only five students cast doubt on their numerical answers after being asked to comment on their method. For the rest, even a moment of reflection could not fix their faulty reasoning.

Then there’s my nephew’s menacing homework problem, which he was intent on solving even after being convinced it was impossible.

And many more.