“How old is the shepherd?” – The problem that shook school mathematics
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Re: “How old is the shepherd?” – The problem that shook school mathematics
#2I don't think it reflects the fragility of the student's mathematical reasoning behaviour so much as their ability to play the meta-game of the test itself. They know they are expected to provide answer, and that they don't know how to reach for it - so they play the odds. I for one have used this technique, exploiting for example the fact that in school, correct answers tended to be round numbers, as an heuristic, for example.
I believe if "insufficient data" was stated as a possible answer (which the student's are not accustomed to), they would state that.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#3I think that, if they were trying to determine this outside the context of "here's a math problem from a teacher-type person", they would realize there's no way to know more easily. But, this still makes an important point, that the way we teach math doesn't look much like how we use it, which raises the question of whether we are really preparing students to use the math they are allegedly learning.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#4> 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?
Solvable: solution is a range.
All problems become unsolvable if you allow unknowns to creep in: "I have one apple in two hands. How many apples do I have?" is completely unsolvable unless you make assumptions about absolutely everything.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#5The obvious way to improve this is to make "insufficient data" a somewhat common answer to problems (others have mentioned this). Despite the apparent flippancy, I think this is a deep change, since despite its simplicity, it forces students to think about the problem solving process itself.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#6I don't think it reflects the fragility of the student's mathematical reasoning behaviour so much as their ability to play the meta-game of the test itself. They know they are expected to provide answer, and that they don't know how to reach for it - so they play the odds. I for one have used this technique, exploiting for example the fact that in school, correct answers tended to be round numbers, as an heuristic, f…
Exactly. Students say to themselves, "while questions can be tricky, they are seldom so tricky that no answer is expected. As far as I can recall even for tricky questions, an answer was ascertainable, I must be missing something, this is all I got, so here goes. A remote chance is better than no chance (answer) and at least I tried and didn't give up".
Re: “How old is the shepherd?” – The problem that shook school mathematics
#7If you want to know how a student will confront a real problem, you need to create a context where they won't try to outsmart whoever wrote the question.
It's entirely possible to solve a math problem using psychology instead of math, because math problems are made by people. We've all done a horrifying amount of schoolwork, and thus have a pretty good data set in our head from which to extrapolate.
Suppose I tell you that an algebra textbook question includes the numbers 32, 4 and 64 in the problem text. What's the answer? I'd bet most of us would guess 2 or 4.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#8I don't think it reflects the fragility of the student's mathematical reasoning behaviour so much as their ability to play the meta-game of the test itself. They know they are expected to provide answer, and that they don't know how to reach for it - so they play the odds. I for one have used this technique, exploiting for example the fact that in school, correct answers tended to be round numbers, as an heuristic, f…
There's nothing wrong with guessing an answer. What's horrifying is that the students actually try to solve it mathematically rather than just making up a guess. And they stick with it even if the results are unreasonable (one student guessed 625, for instance.) According to the article the students didn't even try ask the teacher if the question was misprinted or something.
Re: “How old is the shepherd?” – The problem that shook school mathematics
#9Obligatory link: A Mathematician’s Lament
http://www.maa.org/external_archive/devlin/LockhartsLament.p...
Re: “How old is the shepherd?” – The problem that shook school mathematics
#10> 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? Solvable: solution is a range. All problems become unsolvable if you allow unknowns to creep in: "I have one apple in two hands. How many apples do I have?" is completely unsolvable unless you make assumptions about absolutely everything.
Yes, minimum zero, maximum 73..
I don't know if it's really a bad thing for students to assume that their teachers are not going to mess with them by giving them an unsolveable problem, doesn't that just indicate that they trust their teacher? In the case of real world problems, there may or not be a specific correct answer, and I think students know that on some level. Maybe a better way to ask that question would be, "Suppose you know that there are 125 sheep and 5 dogs in a flock. Your friend Amelia asks you, 'How old is the shepherd?' What would be your response?" That way the hypothetical nature of the question is more obvious.