I understand what you’re saying here and when I teach calculus I ask my students to show their work. A couple of comments:
No, the purpose of “showing your work” is not for The Man to keep you down. It’s to demonstrate that you didn’t just copy your answers from someone else (or from a calculating device). It can be tedious, yes, but so long as students do copy answers I don’t grudge the teachers this method of making that more difficult.
I agree that this is reasonable given how educational institutions are structured (in particular, given how much emphasis is currently placed on evaluation) but think that there’s a general problem of focus on evaluation being detrimental to quality of education. My experience teaching calculus discussion sections has been that my students’ focus on getting good grades takes up so much of their attention that they have little attention to bring to the task of actually learning the material. On the flip side circumstances force instructors to devote so much of their time to ensuring that grading is fair that their ability to focus on actually teaching is markedly impaired. I see the common focus on deterring cheating as part of this dynamic. This is not to say that I have a solution, I’m just saying that’s Something’s Wrong :-).
So this justification only applies if you’re always correct or the teacher is entirely unable or unwilling to teach (granted, sometimes true, but not always). Also that you’re not proving anything—because if you can’t write down your reasoning, why on earth do you expect anyone to believe your conclusion? That it’s difficult is not sufficient reason not to do it.
It seems likely to me that Thurston is Generalizing From One Example here. He’s legendary for having consistently made obscure and apparently ungrounded statements which have turned out to be fully justified. One could imagine such a background making it easy to forget that not everybody is the same way.
William Thurston’s “geometrization theorem” concerning structures on Haken
three-manifolds is another often-cited example. A grand insight delivered with
beautiful but insufficient hints, the proof was never fully published. For many
investigators this unredeemed claim became a roadblock rather than an inspiration.
The full proof of the geometrization theorem for Haken manifolds was later published along the lines that Thurston had originally suggested. Thurston responded to Jaffe and Quinn in his essay titled On proofs and progress in mathematics.
I agree about the overall structure of educational systems vis a vis grades, although there is apparently some evidence that being tested periodically helps one retain information.
It seems likely to me that Thurston is Generalizing From One Example here.
Actually, in fairness, I don’t think he’s generalizing; I think he’s observing one example. My interpretation of the statement as prescriptive may not have been intended. Of course, if it’s to be included on this list, it should be expected to be interpreted as prescriptive.
He’s legendary for having consistently making obscure and apparently unjustified statements which have turned out to be fully grounded.
I wonder whether my lack of preconceptions about the source of the quote was helpful or harmful here. (My guess is “no.” ;))
I agree about the overall structure of educational systems vis a vis grades, although there is apparently some evidence that being tested periodically helps one retain information.
Thanks, I hadn’t seen the things that the linked NY Times article discusses before.
Actually, in fairness, I don’t think he’s generalizing; I think he’s observing one example. My interpretation of the statement as prescriptive may not have been intended.
As you remark, equating “show your work” with “reject mental insights and alternative approaches” is too strong. My suggestion was that to the extent that he’s drawing such an equivalence, he’s likely to be generalizing from one example.
But in line with what you say above, the essay that the quote is from was written as a personal reflection rather than a careful analysis and so the quote is most properly viewed as an offhand remark reflecting on his own experience.
Of course, if it’s to be included on this list, it should be expected to be interpreted as prescriptive.
I cited Thurston for a description of his own experience with school math.
I wonder whether my lack of preconceptions about the source of the quote was helpful or harmful here. (My guess is “no.” ;))
Sure, makes sense. I was just giving some background in case you’re curious. I personally found the essays that I linked above well worth reading.
I cited Thurston for a description of his own experience with school math.
Understood, but if it’s included on a list of quotes about math education the overall thrust of which is to show what’s wrong with it, the implication is that what he describes is one of the things which is wrong with it.
I hope it’s clear that my last line there wasn’t any kind of rejection of your explanation. I was just musing which way the bias ran. And I still think this is a worthwhile post overall.
Understood, but if it’s included on a list of quotes about math education the overall thrust of which is to show what’s wrong with it, the implication is that what he describes is one of the things which is wrong with it.
Thanks for pointing this out, you might be surprised to know that it didn’t occur to me! I’ll bear this point in mind and add some sort of disclaimer if I incorporate the quote into a top level post.
I understand what you’re saying here and when I teach calculus I ask my students to show their work. A couple of comments:
I agree that this is reasonable given how educational institutions are structured (in particular, given how much emphasis is currently placed on evaluation) but think that there’s a general problem of focus on evaluation being detrimental to quality of education. My experience teaching calculus discussion sections has been that my students’ focus on getting good grades takes up so much of their attention that they have little attention to bring to the task of actually learning the material. On the flip side circumstances force instructors to devote so much of their time to ensuring that grading is fair that their ability to focus on actually teaching is markedly impaired. I see the common focus on deterring cheating as part of this dynamic. This is not to say that I have a solution, I’m just saying that’s Something’s Wrong :-).
It seems likely to me that Thurston is Generalizing From One Example here. He’s legendary for having consistently made obscure and apparently ungrounded statements which have turned out to be fully justified. One could imagine such a background making it easy to forget that not everybody is the same way.
In Jaffe and Quinn’s “Theoretical mathematics″: Toward a cultural synthesis of mathematics and theoretical physics the authors say
The full proof of the geometrization theorem for Haken manifolds was later published along the lines that Thurston had originally suggested. Thurston responded to Jaffe and Quinn in his essay titled On proofs and progress in mathematics.
I agree about the overall structure of educational systems vis a vis grades, although there is apparently some evidence that being tested periodically helps one retain information.
Actually, in fairness, I don’t think he’s generalizing; I think he’s observing one example. My interpretation of the statement as prescriptive may not have been intended. Of course, if it’s to be included on this list, it should be expected to be interpreted as prescriptive.
I wonder whether my lack of preconceptions about the source of the quote was helpful or harmful here. (My guess is “no.” ;))
Thanks, I hadn’t seen the things that the linked NY Times article discusses before.
As you remark, equating “show your work” with “reject mental insights and alternative approaches” is too strong. My suggestion was that to the extent that he’s drawing such an equivalence, he’s likely to be generalizing from one example.
But in line with what you say above, the essay that the quote is from was written as a personal reflection rather than a careful analysis and so the quote is most properly viewed as an offhand remark reflecting on his own experience.
I cited Thurston for a description of his own experience with school math.
Sure, makes sense. I was just giving some background in case you’re curious. I personally found the essays that I linked above well worth reading.
Understood, but if it’s included on a list of quotes about math education the overall thrust of which is to show what’s wrong with it, the implication is that what he describes is one of the things which is wrong with it.
I hope it’s clear that my last line there wasn’t any kind of rejection of your explanation. I was just musing which way the bias ran. And I still think this is a worthwhile post overall.
Thanks for pointing this out, you might be surprised to know that it didn’t occur to me! I’ll bear this point in mind and add some sort of disclaimer if I incorporate the quote into a top level post.