Between teaching mathematics to freshmen and spending most of my time learning mathematics, I’ve noticed this myself. When presented with a new result, the first inclination, especially depending on the authority of the source, is to believe it and figure there’s a valid proof of it. But occasionally the teacher realizes that they made a mistake and may even scold the students for not noticing since it is incredibly obvious (e.g. changing something like ||z—z_0|| to ||z—z_1|| between steps, even though a few seconds thinking reveals it to be a typo rather than a mathematical insight).
Sometimes (and for a few lucky people, most of the time) individuals are in a mental state where they are actively thinking through everything being presented to them. For me, this happens a few times a semester in class, and almost always during meetings with my advisor. And occasionally I have a student who does it when I’m teaching. But in my experience this is a mentally exhausting task and often leaves you think-dead for a while afterwards (I find I can go about 40 minutes before I give out).
All this leads me to a conclusion, largely from my experience with what behavior produces what effects, that in mathematics the best way to teach is to assign problems and give students clues when they get stuck. The problems assigned, of course, should be ones that result in the student building up the mathematical theory. It’s certainly more time consuming, but in the end more rewarding, in terms of both emotional satisfaction and understanding.
As someone who spends a lot of time on the student side of those math classes (and as the student in the class who almost always catches those typographical errors), I suspect that there are students who notice the error but don’t comment for social reasons (don’t want to interrupt, don’t want to be a know-it-all, don’t want to be publicly erroneous in a correction, etc.). Your solution of giving students problems, while an excellent teaching tool, is not a particularly good test for this phenomenon because it fails to distinguish between students who really do miss the errors because they assume you are right and the students who noticed but didn’t speak up, or those who simply weren’t paying attention in the first place.
I agree. I think the social-pressure aspect is even more exaggerated in business settings where there are not only no rewards for pointing out errors, but where you are often actively chastised for causing a team-member to lose face.
There are certainly situations in which the pointing out of errors is not socially appropriate, and doesn’t win you any friends.
When somebody’s telling a joke or an interesting anecdote, you’ll often find that nobody cares if the premises are correct. You’ll tend to get along better if you bite your tongue—even if it is the 500th time you’ve heard that “you only use 10% of your brain” (for instance).
However… I do tend to find that getting along with people that don’t want to know the truth is more energy-draining (for me)… just as I’m sure that if I let my own natural preference for truth take over… I’d be draining for them.
I find that “getting along with non-rational/truth-preferring people” is a tough skill… and involves a lot of compromise.
I’d love to see more articles on how to do this successfully (without going insane or compromising your values).
Also I’d like to point out that there really are situations in which you really do have to point out that somebody is just plain wrong… despite how uncomfortable it makes the other person feel.
That while the article is quite right that being patronising is not beneficial… there are many situations where “being right” is not about being patronising, but about making sure all the bases are covered.
This is often where IT-people clash with people such as their managers. Because really, sometimes code just can’t do what they’re asking, no matter how much they’d like us to “put on a can-do attitude”.
Similarly, clients can give ambiguous or flat-out contradictory requirements… and these errors must be pointed out, regardless of whether the person loses face by doing so. because IT have to make a profit just as much as the client does, and these kinds of errors are where later disputes arise. Nipping it in the bud by pointing out they’re wrong is the best thing for your long-term survivability here.
Of course—there are ways and means of doing so to make sure that egos aren’t bruised int he process… but that’s another article (or two), I’m sure. :)
The article isn’t about choosing the reinterpret the other person’s statements in a more favourable light.
It’s about not sweating the small stuff and not drawing attention your way and letting somebody else have fun without ruining it with detail that, in this social situation is not actually necessary.
Between teaching mathematics to freshmen and spending most of my time learning mathematics, I’ve noticed this myself. When presented with a new result, the first inclination, especially depending on the authority of the source, is to believe it and figure there’s a valid proof of it. But occasionally the teacher realizes that they made a mistake and may even scold the students for not noticing since it is incredibly obvious (e.g. changing something like ||z—z_0|| to ||z—z_1|| between steps, even though a few seconds thinking reveals it to be a typo rather than a mathematical insight).
Sometimes (and for a few lucky people, most of the time) individuals are in a mental state where they are actively thinking through everything being presented to them. For me, this happens a few times a semester in class, and almost always during meetings with my advisor. And occasionally I have a student who does it when I’m teaching. But in my experience this is a mentally exhausting task and often leaves you think-dead for a while afterwards (I find I can go about 40 minutes before I give out).
All this leads me to a conclusion, largely from my experience with what behavior produces what effects, that in mathematics the best way to teach is to assign problems and give students clues when they get stuck. The problems assigned, of course, should be ones that result in the student building up the mathematical theory. It’s certainly more time consuming, but in the end more rewarding, in terms of both emotional satisfaction and understanding.
As someone who spends a lot of time on the student side of those math classes (and as the student in the class who almost always catches those typographical errors), I suspect that there are students who notice the error but don’t comment for social reasons (don’t want to interrupt, don’t want to be a know-it-all, don’t want to be publicly erroneous in a correction, etc.). Your solution of giving students problems, while an excellent teaching tool, is not a particularly good test for this phenomenon because it fails to distinguish between students who really do miss the errors because they assume you are right and the students who noticed but didn’t speak up, or those who simply weren’t paying attention in the first place.
I agree. I think the social-pressure aspect is even more exaggerated in business settings where there are not only no rewards for pointing out errors, but where you are often actively chastised for causing a team-member to lose face.
’Nuff said
This was put up approvingly by two people on my friendslist.
Brilliant blogpost, and quite correct.
There are certainly situations in which the pointing out of errors is not socially appropriate, and doesn’t win you any friends.
When somebody’s telling a joke or an interesting anecdote, you’ll often find that nobody cares if the premises are correct. You’ll tend to get along better if you bite your tongue—even if it is the 500th time you’ve heard that “you only use 10% of your brain” (for instance).
However… I do tend to find that getting along with people that don’t want to know the truth is more energy-draining (for me)… just as I’m sure that if I let my own natural preference for truth take over… I’d be draining for them.
I find that “getting along with non-rational/truth-preferring people” is a tough skill… and involves a lot of compromise.
I’d love to see more articles on how to do this successfully (without going insane or compromising your values).
Also I’d like to point out that there really are situations in which you really do have to point out that somebody is just plain wrong… despite how uncomfortable it makes the other person feel.
That while the article is quite right that being patronising is not beneficial… there are many situations where “being right” is not about being patronising, but about making sure all the bases are covered.
This is often where IT-people clash with people such as their managers. Because really, sometimes code just can’t do what they’re asking, no matter how much they’d like us to “put on a can-do attitude”.
Similarly, clients can give ambiguous or flat-out contradictory requirements… and these errors must be pointed out, regardless of whether the person loses face by doing so. because IT have to make a profit just as much as the client does, and these kinds of errors are where later disputes arise. Nipping it in the bud by pointing out they’re wrong is the best thing for your long-term survivability here.
Of course—there are ways and means of doing so to make sure that egos aren’t bruised int he process… but that’s another article (or two), I’m sure. :)
I think the blog post was basically speaking in favor of the charity principle.
I don’t think I agree on that one.
The article isn’t about choosing the reinterpret the other person’s statements in a more favourable light.
It’s about not sweating the small stuff and not drawing attention your way and letting somebody else have fun without ruining it with detail that, in this social situation is not actually necessary.