A research mathematician turned teacher has a word problem for you: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?”
Most adults would quickly come to the conclusion that there is not enough information to find an answer. Not most students.
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.
He says those findings are the direct result of the type of problems we ask students to solve during their travels through school math.
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.
So, what’s the alternative to our current standard math curriculum, featuring repetitive pages of calculations and ambiguous word problems with one right answer? The writer has some excellent ideas. However, they would require completely reimagining the way we teach math.
At the core of those changes is an emphasis on understanding process rather than finding answers.
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 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.