There are 125 sheep in a flock and 25 dogs.
How old is the shepherd?
- To collect some data regarding sense making in mathematics amongst our high schoolers (grades 10 – 12)
- To analyze the data and assess what this means for us as educators. Are there differences in responses amongst the different grade levels? Are there gaps in student learning that we need to address? If so, how might we begin to address those gaps?
- To use this activity as a way to begin a dialogue with students and teachers about sense-making in mathematics
- 25 stated there was not enough information to answer the problem, or did not supply a numerical response based on the constraints identified in the problem
- 74 gave numeric responses
- 2 students did not answer the question (e.g. one student wrote “nice question”)
Students that did not provide a numeric response (making sense):
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:
- 125 + 5 = 130
- 125 – 5 = 120
- 125/5 = 25
- Or other such random combinations of numbers. Note that no one responded with 125*5 = 625 as they seemed to realize this may be ridiculous age for a shepherd to be. Although one student did provide a response of 956000 through a series of random additions and subtractions.
I referred to this new category of responses as “supported guesses.” To be honest, I had difficulties categorizing some of these responses as “making sense” after seeing the superstar example from above, but ultimately decided that anything other than random math was a step in the right direction, although you will probably agree with me that some responses seem to employ more evidence of reasoning than others:
Some factors worth considering:
- The responses were collected via an online survey. Would results have differed if this was done face to face? Did students try looking up the problem before attempting to solve it?
- Students feeling like they did not have intellectual autonomy; not wanting to question the questioner due to respect for authority.
- The old “my math teacher is asking me this, so I must calculate something” trick.
- Cultural factors may be at play here. Perhaps students have seen some version of this problem before, thus accounting for the varied supported guesses observed.
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.