Overheard from a professor in the math building: "What does this mean, 'grade on a curve?' What are they saying?"
I like math professors. Especially foreign ones.
Showing posts with label stereotypes of mathematicians that are true. Show all posts
Showing posts with label stereotypes of mathematicians that are true. Show all posts
Tuesday, May 10, 2011
Tuesday, February 16, 2010
Response to XKCD blog (WARNING: this is super nerdy)
The following is a response to a post on the XKCD blog.
The puzzle described is as follows:
XKCD claims there is such a strategy, and you can read it here. Basically what I'm explaining below is why this strategy doesn't really work, although it's tempting to say it does. To summarize: this strategy allows a dumb computer to keep getting the right answer more than 50% of the time if I don't change Alice's two numbers. But this doesn't really describe reality, in which I have to view the numbers Alice has given me as random variables.
Okay, here's my response.
I've thought about this one a long time now, and I think I get what's going on, and why this is so counter-intuitive. The actual result here is that, given A < B, I can write a computer program that picks one of these values to retrieve at random and then determines whether the one it didn't pick was higher or lower, such that this program has more than a 50% chance of success. The fact that I can use any old monotone increasing function and still get above 50% no matter what A and B are is kind of neat, but in the end it's really not a big deal.
We need to ask in what sense we're boosting the odds above 50%. What we actually have is this: given a fixed A and B, if we iterate this program over and over again, the computer will get it right more than 50% of the time. But if you think about it practically, this is a sad little program. Any human, after the first run-through of this game, would know the answer 100% of the time, so long as A and B never changed.
We need to remember that the idea of probability is this: given enough trials, I should be able to identify the probability that an event occurs by taking the ratio of number of times that event occurred to number of trials I ran. Since in real life Alice would be allowed to come up with new numbers for each trial, the numbers she picks should also be viewed as random variables, not as fixed constants.
Consider the following program:
1) Pick X_0 and X_1 such that P(X_0 < X_1) = P(X_1 > X_0) = 1/2
2) Pick B in {0,1} with probability 1/2
3) Let X = X_B
4) Pick T in (0,1) uniformly
5) If T is less than p(X), return "lower"; if T is greater than p(X), return "higher"
The numbers 0 and 1 essentially represent the abstract envelopes. Assuming X_0, X_1, B, and T are all independent, I believe one can show that in this case the probability of this program returning the correct answer is exactly 50%, no matter what p(x) is. And that, I think, mimics real life much better than the analysis given by xkcd. We really have to think of Alice's numbers as randomly generated, because we don't just get to use the same numbers over and over again.
As for the point about how there is no uniform distribution on the real line: this is correct, but we can still demand that the random variable X_0 - X_1 be such that P(X_0 - X_1 < 0) = P(X_0 - X_1 > 0) = 1/2. This seems fair; although Alice might have a different distribution in mind, in which case picking an envelope in an unbiased manner and then using xkcd's strategy is not necessarily to your advantage.
So essentially what I'm saying is that the result xkcd gives is counter-intuitive precisely because it's wrong. That is, it doesn't accurately represent a real-life situation in which you have to make a decision based on the information given. In real life, if given a choice between saying Alice has the larger number or saying Alice has the smaller number, just flip a coin. There really is no way to get better than a 50% shot.
The puzzle described is as follows:
Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.
Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?
XKCD claims there is such a strategy, and you can read it here. Basically what I'm explaining below is why this strategy doesn't really work, although it's tempting to say it does. To summarize: this strategy allows a dumb computer to keep getting the right answer more than 50% of the time if I don't change Alice's two numbers. But this doesn't really describe reality, in which I have to view the numbers Alice has given me as random variables.
Okay, here's my response.
I've thought about this one a long time now, and I think I get what's going on, and why this is so counter-intuitive. The actual result here is that, given A < B, I can write a computer program that picks one of these values to retrieve at random and then determines whether the one it didn't pick was higher or lower, such that this program has more than a 50% chance of success. The fact that I can use any old monotone increasing function and still get above 50% no matter what A and B are is kind of neat, but in the end it's really not a big deal.
We need to ask in what sense we're boosting the odds above 50%. What we actually have is this: given a fixed A and B, if we iterate this program over and over again, the computer will get it right more than 50% of the time. But if you think about it practically, this is a sad little program. Any human, after the first run-through of this game, would know the answer 100% of the time, so long as A and B never changed.
We need to remember that the idea of probability is this: given enough trials, I should be able to identify the probability that an event occurs by taking the ratio of number of times that event occurred to number of trials I ran. Since in real life Alice would be allowed to come up with new numbers for each trial, the numbers she picks should also be viewed as random variables, not as fixed constants.
Consider the following program:
1) Pick X_0 and X_1 such that P(X_0 < X_1) = P(X_1 > X_0) = 1/2
2) Pick B in {0,1} with probability 1/2
3) Let X = X_B
4) Pick T in (0,1) uniformly
5) If T is less than p(X), return "lower"; if T is greater than p(X), return "higher"
The numbers 0 and 1 essentially represent the abstract envelopes. Assuming X_0, X_1, B, and T are all independent, I believe one can show that in this case the probability of this program returning the correct answer is exactly 50%, no matter what p(x) is. And that, I think, mimics real life much better than the analysis given by xkcd. We really have to think of Alice's numbers as randomly generated, because we don't just get to use the same numbers over and over again.
As for the point about how there is no uniform distribution on the real line: this is correct, but we can still demand that the random variable X_0 - X_1 be such that P(X_0 - X_1 < 0) = P(X_0 - X_1 > 0) = 1/2. This seems fair; although Alice might have a different distribution in mind, in which case picking an envelope in an unbiased manner and then using xkcd's strategy is not necessarily to your advantage.
So essentially what I'm saying is that the result xkcd gives is counter-intuitive precisely because it's wrong. That is, it doesn't accurately represent a real-life situation in which you have to make a decision based on the information given. In real life, if given a choice between saying Alice has the larger number or saying Alice has the smaller number, just flip a coin. There really is no way to get better than a 50% shot.
Saturday, November 14, 2009
Natural Religion
I was hanging out with other math graduate students last night, having dinner and a beer before going to a Live Arts presentation featuring experimental computer-generated music.
Socializing with grad students is an eclectic combination of typical small-talk, pseudo-intellectual babble, and pure nerdiness. Amidst the pseudo-intellectual babble, religion is often a topic of interest.
One of my friends said something that I've heard enough times to trigger quite a number of thoughts on it: When faced with the question of whether there is any reason for living our lives, especially in any sort of moral way, it is "natural" to look to (or create) religion for answers.
I don't know what struggles this particular friend has with these questions, but I've noticed that in general this kind of statement is used to explain religion away. The implication can either be patronizing (as in, "Oh, well, it's fine that you believe that... it's perfectly natural") or condescending (as in, "How can you still believe that in this enlightened age?").
But the obvious question is, what else should one turn to? It sometimes seems like a blasphemous question in the academy, the church of secularism. I wondered aloud what would become of society if most of the world started to believe there really is no transcendent meaning to the world.
As if annoyed, my friends responded that the world has gotten on for a long time already, and it would surely continue on the way it is. This statement is highly presumptuous, and then again it is also rather depressing.
Presumptuous, because to say such a thing is to utterly fail to acknowledge the achievements of religious worldviews. For instance, to act as if the Christian way of thinking had nothing to do with the achievements of the Western world is, I dare say, shockingly stupid, though many people who act this way are not stupid.
Depressing, because it implies that not only is there no meaning, but there really has never been. The things we have accomplished as human beings have only come about because we wanted to do something with our little blips of existence in this meaningless universe. Under this philosophy, we humans would continue to do that even if we didn't believe in God or anything else, because we just can't help it.
In light of how depressing this idea is, I suppose religion is "natural." But I was musing on that word for a while, and I realized that there are really at least two senses in which we use it.
One use of the word "natural" is when we're saying something is easy, or obvious, as in, "It's only natural that he would think only of himself." It's the way we are when we're not trying very hard. Thus my drum line coach in high school used to yell at us, "Don't just do whatever feels natural! Do it right!"
On the other hand, we sometimes use "natural" in a more positive sense to mean that something feels right. When I started doing mathematics as a young child, it felt natural to me. For others, this could've been used in the first sense, meaning it came easily to me. But for me, it was more than that. I worked hard at math because it felt right.
You don't get good at anything just by taking it easy. You have to do more than what comes "naturally" to you. That's the first sense of the word. But surely when you find something you love to work hard at doing, it feels "natural" to you. You feel you are in your element.
In this way, I think at least a few of the world's religion could be considered quite "natural," in the second sense of the word. They make sense. Once you carefully consider their deepest and most powerful teachings, you can see how someone might believe such an idea.
Christianity, Judaism, Islam, Hinduism, and Buddhism (for example) all have ideas that really make sense of the world. In that sense it is "natural" to turn to one of them for answers to questions about the meaning of life. It makes sense.
Atheism, on the other hand, doesn't seem natural to me at all. It doesn't feel right. Is there really no transcendent meaning? What is the point, then?
But in the first sense, atheism is very natural, in the sense that it really is the most obvious answer. Is there a God? Well, I can't really see one, so I guess not. This is a very "natural" answer.
I imagine atheists would be extremely divided about this statement. I have read many opinions (from atheists) that say much the same thing that I am saying. Atheism is the obvious answer, and it's only this irrational desire for the second kind of "natural" that leads us to develop religion.
But I imagine other atheists would be deeply frustrated with my evaluation of their beliefs. Many atheists have had to really struggle to come to the conclusions about life they now hold. Given the culture in which they live, it is often a battle to leave the faith of one's youth and embrace a life of not knowing if there's really any point to it all.
I certainly don't want to diminish that struggle; I take it seriously. Yet I wonder if atheists take seriously enough the way in which religion has created a way for society to pass on the virtue of struggling for one's beliefs. "Seek, and ye shall find."
Christianity, in particular, does not assume that it is the default belief. (Many atheists, on the other hand, claim that atheism is the default belief.) Even where it is culturally dominant, Christianity admits and even insists that it is not the "natural" thing to believe.
This is why Christianity says faith is so important. True, "faith" can become a tool that the powerful use to brainwash the weak; but in its purest form "faith" is simply a statement of belief in something that is not obvious, yet is extremely important if it's true.
What if we lived in a society where struggle for one's beliefs was not considered a virtue? Sure, atheists are typically known for struggling to seek out knowledge now, in the culture in which we find ourselves, but what if it were different? What if no one felt the slightest urge to believe something that wasn't immediately apparent to all people?
Science would utterly collapse, just as much as any religion. Indeed, science is a religion, from a certain point of view. Although it easily passes as the true universal, uniting people from all cultures and all faiths, it really isn't any such thing.
Anyone who does math or science goes through a process of initiation, in which knowledge is revealed through many, many stages. Scientific knowledge is not natural in the first sense--it is not obvious--but it is natural in the second sense, in that once you see all the evidence, it begins to feel very natural, even if it's surprising.
To me, the truly atheist worldview has nothing of this second kind of naturalness that both scientific and religious knowledge have. Atheism just doesn't explain why. Worse, it almost seems to discourage asking such questions. All we can really know, it seems, is what.
Of course, my friends may continue to disbelieve in religion because religion lacks evidence, but I think that by this they mean only a certain kind of evidence, which does not include those indescribable connections a believer has with God.
For me, knowledge of God must begin with the heart, and all reason must subordinate itself to the connection found there. Surely this connection is not admitted as evidence by everyone, and perhaps with good reason--whose heart are we to believe, after all?
That is why I hesitate to insist upon how much I really know about God. I will say I know Him, but my mind is very limited in expressing this. Faith is a matter of the heart, and it is not always natural in the first sense. Yet it is natural in the sense of being beautiful, the way it is meant to be.
Socializing with grad students is an eclectic combination of typical small-talk, pseudo-intellectual babble, and pure nerdiness. Amidst the pseudo-intellectual babble, religion is often a topic of interest.
One of my friends said something that I've heard enough times to trigger quite a number of thoughts on it: When faced with the question of whether there is any reason for living our lives, especially in any sort of moral way, it is "natural" to look to (or create) religion for answers.
I don't know what struggles this particular friend has with these questions, but I've noticed that in general this kind of statement is used to explain religion away. The implication can either be patronizing (as in, "Oh, well, it's fine that you believe that... it's perfectly natural") or condescending (as in, "How can you still believe that in this enlightened age?").
But the obvious question is, what else should one turn to? It sometimes seems like a blasphemous question in the academy, the church of secularism. I wondered aloud what would become of society if most of the world started to believe there really is no transcendent meaning to the world.
As if annoyed, my friends responded that the world has gotten on for a long time already, and it would surely continue on the way it is. This statement is highly presumptuous, and then again it is also rather depressing.
Presumptuous, because to say such a thing is to utterly fail to acknowledge the achievements of religious worldviews. For instance, to act as if the Christian way of thinking had nothing to do with the achievements of the Western world is, I dare say, shockingly stupid, though many people who act this way are not stupid.
Depressing, because it implies that not only is there no meaning, but there really has never been. The things we have accomplished as human beings have only come about because we wanted to do something with our little blips of existence in this meaningless universe. Under this philosophy, we humans would continue to do that even if we didn't believe in God or anything else, because we just can't help it.
In light of how depressing this idea is, I suppose religion is "natural." But I was musing on that word for a while, and I realized that there are really at least two senses in which we use it.
One use of the word "natural" is when we're saying something is easy, or obvious, as in, "It's only natural that he would think only of himself." It's the way we are when we're not trying very hard. Thus my drum line coach in high school used to yell at us, "Don't just do whatever feels natural! Do it right!"
On the other hand, we sometimes use "natural" in a more positive sense to mean that something feels right. When I started doing mathematics as a young child, it felt natural to me. For others, this could've been used in the first sense, meaning it came easily to me. But for me, it was more than that. I worked hard at math because it felt right.
You don't get good at anything just by taking it easy. You have to do more than what comes "naturally" to you. That's the first sense of the word. But surely when you find something you love to work hard at doing, it feels "natural" to you. You feel you are in your element.
In this way, I think at least a few of the world's religion could be considered quite "natural," in the second sense of the word. They make sense. Once you carefully consider their deepest and most powerful teachings, you can see how someone might believe such an idea.
Christianity, Judaism, Islam, Hinduism, and Buddhism (for example) all have ideas that really make sense of the world. In that sense it is "natural" to turn to one of them for answers to questions about the meaning of life. It makes sense.
Atheism, on the other hand, doesn't seem natural to me at all. It doesn't feel right. Is there really no transcendent meaning? What is the point, then?
But in the first sense, atheism is very natural, in the sense that it really is the most obvious answer. Is there a God? Well, I can't really see one, so I guess not. This is a very "natural" answer.
I imagine atheists would be extremely divided about this statement. I have read many opinions (from atheists) that say much the same thing that I am saying. Atheism is the obvious answer, and it's only this irrational desire for the second kind of "natural" that leads us to develop religion.
But I imagine other atheists would be deeply frustrated with my evaluation of their beliefs. Many atheists have had to really struggle to come to the conclusions about life they now hold. Given the culture in which they live, it is often a battle to leave the faith of one's youth and embrace a life of not knowing if there's really any point to it all.
I certainly don't want to diminish that struggle; I take it seriously. Yet I wonder if atheists take seriously enough the way in which religion has created a way for society to pass on the virtue of struggling for one's beliefs. "Seek, and ye shall find."
Christianity, in particular, does not assume that it is the default belief. (Many atheists, on the other hand, claim that atheism is the default belief.) Even where it is culturally dominant, Christianity admits and even insists that it is not the "natural" thing to believe.
This is why Christianity says faith is so important. True, "faith" can become a tool that the powerful use to brainwash the weak; but in its purest form "faith" is simply a statement of belief in something that is not obvious, yet is extremely important if it's true.
What if we lived in a society where struggle for one's beliefs was not considered a virtue? Sure, atheists are typically known for struggling to seek out knowledge now, in the culture in which we find ourselves, but what if it were different? What if no one felt the slightest urge to believe something that wasn't immediately apparent to all people?
Science would utterly collapse, just as much as any religion. Indeed, science is a religion, from a certain point of view. Although it easily passes as the true universal, uniting people from all cultures and all faiths, it really isn't any such thing.
Anyone who does math or science goes through a process of initiation, in which knowledge is revealed through many, many stages. Scientific knowledge is not natural in the first sense--it is not obvious--but it is natural in the second sense, in that once you see all the evidence, it begins to feel very natural, even if it's surprising.
To me, the truly atheist worldview has nothing of this second kind of naturalness that both scientific and religious knowledge have. Atheism just doesn't explain why. Worse, it almost seems to discourage asking such questions. All we can really know, it seems, is what.
Of course, my friends may continue to disbelieve in religion because religion lacks evidence, but I think that by this they mean only a certain kind of evidence, which does not include those indescribable connections a believer has with God.
For me, knowledge of God must begin with the heart, and all reason must subordinate itself to the connection found there. Surely this connection is not admitted as evidence by everyone, and perhaps with good reason--whose heart are we to believe, after all?
That is why I hesitate to insist upon how much I really know about God. I will say I know Him, but my mind is very limited in expressing this. Faith is a matter of the heart, and it is not always natural in the first sense. Yet it is natural in the sense of being beautiful, the way it is meant to be.
Friday, September 25, 2009
Grading
This week has basically been consumed by teaching. I teach a section of Applied Calculus at UVA--it's my first year ever teaching a class. It's quite an experiment, really. Let's take a bunch of grad students, who have been admitted to the PhD program entirely because of academic merit in mathematics and not whatsoever on teaching merit, and throw them in front of some college freshmen (oh, I'm sorry, "first-years," as we say at UVA). Surely some learning will take place.
Tuesday was my students' first of three mid-term exams. I felt like it went well. There's nothing quite like grading an exam. Even in mathematics, it's highly subjective. The only time it isn't subjective is when the answer is entirely correct. I guess that does make it different from, say, grading a paper in philosophy. Is there ever an entirely correct answer in philosophy? This in itself is a philosophical question, of course... but the practical answer is, "no."
Subjectivity naturally comes into play, because for any given problem, a significant percentage of my students will have written something not 100% correct, and then I'm faced with the tricky problem of trying to measure how much understanding the answer communicates. How do you quantify understanding? Sometimes the whole process seems absurd.
I'm conditioned to expect a certain grade distribution, and I can kind of use this expectation to judge the validity of the grades I have come up with for my students. The truth is, though, I desperately try to find things that I can give credit for. Any signs of intelligent thought are rewarded with points.
Mathematics could be the most easily self-regulated discipline in the world. Every mathematician is highly critical by nature. One simply cannot succeed in mathematics without having this irrepressible urge to spot every important detail in a problem and verify it. Thus, if you tell any mathematician to grade a collection of calculus exams, his first inclination is simply to rip every answer to pieces.
Yet that same critical impulse generally causes me to turn inward and ask myself, what purpose would it serve to be hyper-critical of responses on a calculus exam? That combined with a genuine caring for my students as people causes me to try to find the best in every solution.
I never have to worry that I'm being overly generous. It is simply not physically possible for a mathematician to go above a certain threshold in giving credit for only partially correct solutions. The mistakes that appear in answers students give can range from humorous to painful, but in any case I just can't ignore them. No matter how much credit I want to give a student, I am forever bound by a mysterious force beyond my power to be more honest in my grading than perhaps my students would like me to be.
It's pretty brilliant, isn't it? The department never has to look over my shoulder to see if I'm actually being an ethical grader. They know that I am already constrained by a power far greater than any threats that bureaucracy might employ to keep me on track.
All of this is not to say that all of us grad students grade our students the same way. Sometimes we don't agree on what exactly the students are expected to understand about a problem. Math has this attractive quality to it because it is so objective, but setting learning goals for students (and then measuring success) is a more or less entirely subjective matter.
I could make it entirely objective, of course--each answer would be either right or wrong, no in between. But this probably would be the most unethical way to grade, because it would completely rob students of any way to communicate their ability to develop some sort of problem solving strategy, which is actually what we're trying to teach. Math, as it turns out, is not really about the answer.
When it comes right down to it, every academic discipline, math and the hard sciences included, are mostly about creating a dialog about something we wish to understand better. The reason we grade our students is to try to get them to make their ideas clearer, not just "right." There's something deeply personal about the whole thing, actually.
What's sad is how so many students go on asking what's going to be good enough--good enough for you, the teacher, to find them acceptable enough to label with an 'A' so they can go on and accomplish their own goals quite apart from anything you've told them. If only they realized that they'd really earn their 'A' if only they opened themselves up to actually communicating ideas.
I suppose it's really just the difference between being a consumer and thinking of your teacher as a supplier of a good, and being a person and thinking of your teacher as another person. If you're just a consumer, you can certainly get your 'A,' but only because our culture encourages grade inflation (education is subject to market forces, after all). If you really want to understand the material, you have to move from being a consumer to being a real person desiring real communication.
I understand why many students can't move this direction. I'm actually quite okay with most students in my class not caring about math. I'll do my best to be a supplier of a good--maybe if they work hard, I can even supply 'A's for them. There is, after all, a certain level of dignity in that kind of transaction.
But being a teacher, while also being a student, has caused me to reflect that what's really behind all those commands to "show your work" is a desire for something more personal. I wonder if my students will ever get that.
Tuesday was my students' first of three mid-term exams. I felt like it went well. There's nothing quite like grading an exam. Even in mathematics, it's highly subjective. The only time it isn't subjective is when the answer is entirely correct. I guess that does make it different from, say, grading a paper in philosophy. Is there ever an entirely correct answer in philosophy? This in itself is a philosophical question, of course... but the practical answer is, "no."
Subjectivity naturally comes into play, because for any given problem, a significant percentage of my students will have written something not 100% correct, and then I'm faced with the tricky problem of trying to measure how much understanding the answer communicates. How do you quantify understanding? Sometimes the whole process seems absurd.
I'm conditioned to expect a certain grade distribution, and I can kind of use this expectation to judge the validity of the grades I have come up with for my students. The truth is, though, I desperately try to find things that I can give credit for. Any signs of intelligent thought are rewarded with points.
Mathematics could be the most easily self-regulated discipline in the world. Every mathematician is highly critical by nature. One simply cannot succeed in mathematics without having this irrepressible urge to spot every important detail in a problem and verify it. Thus, if you tell any mathematician to grade a collection of calculus exams, his first inclination is simply to rip every answer to pieces.
Yet that same critical impulse generally causes me to turn inward and ask myself, what purpose would it serve to be hyper-critical of responses on a calculus exam? That combined with a genuine caring for my students as people causes me to try to find the best in every solution.
I never have to worry that I'm being overly generous. It is simply not physically possible for a mathematician to go above a certain threshold in giving credit for only partially correct solutions. The mistakes that appear in answers students give can range from humorous to painful, but in any case I just can't ignore them. No matter how much credit I want to give a student, I am forever bound by a mysterious force beyond my power to be more honest in my grading than perhaps my students would like me to be.
It's pretty brilliant, isn't it? The department never has to look over my shoulder to see if I'm actually being an ethical grader. They know that I am already constrained by a power far greater than any threats that bureaucracy might employ to keep me on track.
All of this is not to say that all of us grad students grade our students the same way. Sometimes we don't agree on what exactly the students are expected to understand about a problem. Math has this attractive quality to it because it is so objective, but setting learning goals for students (and then measuring success) is a more or less entirely subjective matter.
I could make it entirely objective, of course--each answer would be either right or wrong, no in between. But this probably would be the most unethical way to grade, because it would completely rob students of any way to communicate their ability to develop some sort of problem solving strategy, which is actually what we're trying to teach. Math, as it turns out, is not really about the answer.
When it comes right down to it, every academic discipline, math and the hard sciences included, are mostly about creating a dialog about something we wish to understand better. The reason we grade our students is to try to get them to make their ideas clearer, not just "right." There's something deeply personal about the whole thing, actually.
What's sad is how so many students go on asking what's going to be good enough--good enough for you, the teacher, to find them acceptable enough to label with an 'A' so they can go on and accomplish their own goals quite apart from anything you've told them. If only they realized that they'd really earn their 'A' if only they opened themselves up to actually communicating ideas.
I suppose it's really just the difference between being a consumer and thinking of your teacher as a supplier of a good, and being a person and thinking of your teacher as another person. If you're just a consumer, you can certainly get your 'A,' but only because our culture encourages grade inflation (education is subject to market forces, after all). If you really want to understand the material, you have to move from being a consumer to being a real person desiring real communication.
I understand why many students can't move this direction. I'm actually quite okay with most students in my class not caring about math. I'll do my best to be a supplier of a good--maybe if they work hard, I can even supply 'A's for them. There is, after all, a certain level of dignity in that kind of transaction.
But being a teacher, while also being a student, has caused me to reflect that what's really behind all those commands to "show your work" is a desire for something more personal. I wonder if my students will ever get that.
Wednesday, July 1, 2009
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