“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.)
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.
These problems reveal a great deal about how we educate children.
Formal schooling imbues students with a static view towards mathematics.
Students believe that all math problems are well-defined, usually with a single right answer. They strongly associate mathematics with numbers, to the extent that they will instinctively derive numerical answers to problems regardless of the context. They are subservient to computational procedure and trust that accurate calculations will always lead them to relevant truths. They accept that confusion and ambiguity is a staple fixture of mathematics, willingly offering up solutions that are void of context, meaning or even common sense.
Undergirding these beliefs is a misplaced authority in school mathematics.
The authoritarian nature of formal schooling dictates that mathematical knowledge is fixed; not to be challenged by students.
Students are taught that their role is to consume knowledge, even if without understanding, and to acquire skills, even if without context. Students have limited space to develop intuitions or to explore concepts beyond the confines of the curriculum. Textbooks only reinforce the ambiguity of school mathematics with artificial story problems that often make no tangible connection with the real world.
And so the shepherd’s age must be deducible, because that is what was asked. This is school mathematics, where anything goes so long as it adheres to rote procedure and results in a numerical answer.
What is the alternative? How should we expect students to respond to the shepherd problem? The following hallmarks of mathematical reasoning and problem solving are a good place to start.
- Let students own the problem space
A mathematics problem is whatever you define it to be. Many students respond to the shepherd problem by tweaking its parameters, adding contextual information from which the shepherd’s age can then be calculated. This is fair game; mathematicians routinely add assumptions to create well-posed problems. Indeed, the mathematician’s craft is to constantly play with assumptions to derive the maximum truth from the minimum set of assumptions. This is a creative endeavour, well within the reach of schoolchildren. They should be given the space to change up the problem as it makes sense for them.
- Grant students intellectual autonomy
Students have a right to challenge knowledge; to call BS on superficial maths problems. Mathematical truths exist not as isolated facts, but as connected ideas. There is no mathematical knowledge without understanding. If students are unable to make sense of a problem, they should have the freedom to probe its assumptions and to challenge any suppositions that an answer exists. Teachers can retain their authority while empowering students to search for meaning and context in maths problems.
- Reflect on the process of problem solving
Mathematics is a journey; it is defined by process, not rigid outcomes. That process can not be reduced to a series of discrete computation steps. It is governed by a flow of reasoning that guides the thinker towards a solution. Problem-solving is often an unstructured, messy affair that requires several iterations of developing and testing assumptions. Error and ill-judgement are the most natural components of problem solving; they should be embraced. All mathematicians need pause to reflect on their problem solving strategies; to step back and retain full view of the big picture. Students must be afforded the same opportunities; their development as mathematical thinkers depends on this sense-making.
Reusser’s study is three decades old. That the findings still hold true today should alarm us. We can not afford to let another thirty years slip by, leaving our students — and the society they inherit — with mistaken impressions of mathematics. The promise of mathematics is that it helps us make sense of the world — let’s start by making better sense of the problems we expose our students to.
