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We asked 101 high schoolers the following question:
There are 125 sheep in a flock and 25 dogs.
The question is an invitation to take a closer look at the kinds of mathematics that we are asking students to engage with in our maths classrooms today. What does it mean for us as educators when students give responses like 130 because 125 + 5 = 130 or 25 because 125/5 = 25? Moreover, what does it mean for us as educators when we expect these responses from students?
I first heard about the shepherd question through Robert Kaplinsky though the question has its origins based on research by Professor Kurt Reusser from 1986, possibly sooner.
The data was collected via an online survey (on account of school closures due to COVID-19) and was given to students in China earlier this year. Our initial goals were:
Of the 101 student who were surveyed:
At first, this data seems to be consistent with the results from Kaplinsky's experiment with the 32 eight graders, in which 75% of them gave numerical responses by using random addition, subtraction, division, or multiplication of 125 and 5. Upon closer examination, however, we see that of the 73 that gave numeric responses, 28 used random math procedures, thus not making sense of the problem, but 45 of those students gave some sort of reasoning independent of the problem to support their numeric responses.
Students that gave numeric responses by combining 125 and 5 via random math operations (not making sense):
Note that a couple of students pointed out that there seemed to be an issue with the problem, but proceeded to give an answer anyway.
Students that did not provide a numeric response (making sense):
One response in particular really blew me away (click to expand the image):
Not only did this student state that the question did not give enough information to provide a specific answer, they used what information was presented in the problem, along with sources to support their thinking, to deduce an age range for the shepherd! Wow. How can I get the rest of my students here??
This is the point where I began to see another category emerge... Students who provided a numeric response, but justified their answers outside the range of expected range numeric responses such as:
I was quite blown away by the number of students that treated this as a "trick" question and thus gave a wide range of responses, which ranged in creativity and depth of thinking. Like I mentioned, I found it difficult categorizing some of these responses, and found that after reading that Superstar response from above, my expectations rose (not necessarily a bad thing, but definitely made categorizing more difficult).
Some factors worth considering:
It is also worth noting that in a follow-up reflection activity with students, some pointed out the need for teachers to ask less ambiguous problems, a few attributed their responses to poor understanding of the problem due to language barriers, while a fair number mentioned the importance of practicing different kinds of problem solving to develop critical thinking skills. This, I think, is a step in the right direction.
International math educator who writes, occasionally.