I hate the formula for combinations. Here it is:
For n objects, the number of different ways to choose k of them is: n! / (n-k)! k!
For example, if I have 25 students and need to choose a committee of 5 of them, there are:
25! / 20! 5! possible committees.
The reason I hate the formula is because it gives virtually no clue about where it comes from or how it makes sense. Students forget it and misapply it because it doesn't fundamentally make any sense.
I always think of this kind of problem in this way: 25*24*23*22*21 / 5!
The numerator connects very straightforwardly to other counting problems. The 25*24*23 etc. represent how many students I could choose as the 1st student for my committee, the 2nd student for my committee, the 3rd student, etc. I have to divide because I'm over-counting the number of committees that are really different. For example, the committee with students {A, B, C, D, E} is really the same as the committee with students {B, C, A, D, E}--the order that I choose the students in doesn't matter. How much to divide by? Each committee I might choose is the same as 5! -1 other committees, because that's how many ways to re-order the same 5 people.
This is probably too quick an explanation to feel comfortable unless you've worked with other combinatorics problems recently--but my students have taken to it quite well after understanding the more fundamental types of counting it relies on.
This seemed to me a great leap forward in student understanding. I didn't tell them that there was a "formula for combinations" because I feared that they would be unwilling to just think about how the problem made sense. As I said, this worked out very well until I had to give them more elaborate problems:
How many 5-card poker hands have (exactly) 2 aces and 2 kings? This is a trickier problem to think through from first principals. It's a fairly long process with many subtle places to go wrong. It would be much easier to think about it in terms of a few independent choices:
(# ways to choose 2 aces)*(# ways to choose 2 kings)*(# of ways to choose another card), or
(4 choose 2)*(4 choose2)*(43 choose 1)
This is where the abstraction of (n choose k) would be handy. It would let students chunk the problem more easily.
This reminded me of something I read somewhere about quick thinkers. The author compared the mind to a pipe with water (thoughts) flowing through it. A lot of people think that fast thinkers are like pipes with faster running water; the thoughts come sequentially at a faster rate. This might happen sometimes, but far more common is for a quick thinker to have a wider pipe--they use abstractions so their thoughts simply contain more, even if they happen at the same speed as everyone elses.
The use of combinations and permutations seems to me a good example of this. A student /could/ reason their way, step by step to the correct solution, but that's many sequential steps. If they can think of the problem in terms of combinations and permuatations, each of their fundamental operations encapsulates several of the old sequential steps at once. This lets their minds arrive at the same place faster, and with less chance of error (or wandering, or boredom).
In my class, students who have an ability to focus their attention for longer periods didn't have trouble with the longer problems. It was students who "got lost" in the steps due to lapses of attention, or an inability to conceptualize the process as a whole, that had trouble.
Next time I'm going to try and achieve a best-of-both-worlds. I'll teach permuations and combinations my way, but then make sure they recognize each as a fundamental type of situation; and help them make the gestalt switch to breaking larger problems down into them, instead of building them from the ground up, as they're used to.
Sunday, May 17, 2009
Mental models
We had a department meeting last week in which we briefly discussed one of my list-item goals: to explicitly teach mental models instead of just procedures.
Some teachers wanted clarification about how I thought of a mental model. The short answers was: a mental model is what a concept means, instead of a procedure to get a certain kind of answer. This can be tricky because often students are looking for "the steps to get the answer"--they're not attuned to what something means. It's also common to accidentally "proceduralize" a question--that is, take a question that required applying one's understanding of the meaning of a concept, and instead turn it into a question that students can solve by following a procedure they've memorized. Typically this happens by giving the same type of problem repeatedly, allowing the students to notice the steps you follow to solve it. Then they can follow those same steps without really understanding where they came from or why they make sense.
Here's an example of this that occurred to me while writing questions for my final.
The Topic: What does the "end behavior" of a function refer to? Specifically, what does it mean to say (for example): as x->infinity, y-> 0; as x-> -infinity, y->0 ?
The original question: The target sort of question is to give students an equation, and have them describe its end-behavior. For example: Describe the end-behavior of f(x) = 1/x
I discovered that I had inadvertantly proceduralized this kind of question for students. The procedure most of them know at this point is:
Unfortunately, many of those same students were stumped when I asked the following:
Say that the function f(x) has the following end-behavior: As x->infinity, y -> 0. For each point, explain whether or not you think it's likely to be on the graph of f(x). [if you don't have enough information to know for sure, clearly explain why]
a). (0, 2) b). (99999, 0.9999) c). (9999, 9999) d). (9999, -0.00001)
It wasn't that students gave wrong answers (although many did). Many students didn't even know where to start. This is a clear indication that they don't really understand the /idea/ of end-behavior.
A similar phenomenon occured when I asked them the following:
Say the function f(x) has a horizontal asymptote at y = 3. What is it's end-behavior?
Many students didn't know where to start, because they had memorized procedures to /find/ the horizontal asymptote, without ever really thinking about what it means to be a horizontal asymptote, and how that idea is related to the idea of end-behavior.
So, to get back to mental models, what would the mental model(s) be for end-behavior? I think it's hard to capture what someone's mental model is exactly; it's whatever they use to think about what a question means.
When I see: "as x -> infinity, y -> 0". I always say out loud in my head "as the x-values get bigger and bigger, the y values get closer and closer to 0"
I also always picture a graph in my head something like this...
As the x-values of my points go to the right, the y-values of my points will get closer to 0 (the x-axis). In other words, as I keep plotting points, their height (the red dotted lines) will get shorter and shorter. (Actually I imagine an animation with the dots appearing from left to right, as the x-values get bigger)
Since the idea of end-behavior is a slight formalization of a very natural question to ask about a graph, I think a good way to introduce it would be to have students do their best to answer the natural question as best they can.
It could be a "describe the graph over the telephone"-type activity. Students could have a series of graphs, and their goal is to write a short (i.e. 1-2 sentence) statement explaining what the the trend at the "ends" of the graph are, so that someone else could re-create it only from the description. After discussing student descriptions, you could unveil the sort of description mathematicians have agreed upon in a context that makes sense.
This would also be a good lesson for a party game I like. Take a vertical strip of paper and fold it into (say) 4 sections. The first person in the group draws a graph. The second person in the group looks at the graph and describes its end-behavior. The paper is folded so the 3rd group member can only see the 2nd member's writing. The 3rd member has to draw a graph that matches the end-behavior described by the 2nd member, and so on. It's like "telephone" using paper, and switching back and forth between representations. At the end, the group checks to see if they ended with the same kind of end-behavior they started with.
To emphasize to students that they should be learning what the concept means, it would be good to give them assignments that require them to express this understanding directly--not use it to get an answer to another sort of problem.
Since the concept of end-behavior has a natural interpretation in terms of a progression, I think a nice assignment would be to have students create a short multi-panel comic strip explaining or illustrating the idea of end-behavior for someone in a younger grade (like 9th).
In my experience, some students won't produce anything useful, but many students will produce clearer explanations than anything I was likely to give; so re-distributing some of the better comic strips as "notes" would be a good follow-up.
Some teachers wanted clarification about how I thought of a mental model. The short answers was: a mental model is what a concept means, instead of a procedure to get a certain kind of answer. This can be tricky because often students are looking for "the steps to get the answer"--they're not attuned to what something means. It's also common to accidentally "proceduralize" a question--that is, take a question that required applying one's understanding of the meaning of a concept, and instead turn it into a question that students can solve by following a procedure they've memorized. Typically this happens by giving the same type of problem repeatedly, allowing the students to notice the steps you follow to solve it. Then they can follow those same steps without really understanding where they came from or why they make sense.
Here's an example of this that occurred to me while writing questions for my final.
The Topic: What does the "end behavior" of a function refer to? Specifically, what does it mean to say (for example): as x->infinity, y-> 0; as x-> -infinity, y->0 ?
The original question: The target sort of question is to give students an equation, and have them describe its end-behavior. For example: Describe the end-behavior of f(x) = 1/x
I discovered that I had inadvertantly proceduralized this kind of question for students. The procedure most of them know at this point is:
1). Plug in a very large x-value (or a few).This process usually yields correct answers to questions like the target.
2). If it's large then write: "As x->infinity, y-> infinity". If it's close to 0 write: "As x->infinity, y->0", etc.
3). Plug in a large negative value for x, etc.
Unfortunately, many of those same students were stumped when I asked the following:
Say that the function f(x) has the following end-behavior: As x->infinity, y -> 0. For each point, explain whether or not you think it's likely to be on the graph of f(x). [if you don't have enough information to know for sure, clearly explain why]
a). (0, 2) b). (99999, 0.9999) c). (9999, 9999) d). (9999, -0.00001)
It wasn't that students gave wrong answers (although many did). Many students didn't even know where to start. This is a clear indication that they don't really understand the /idea/ of end-behavior.
A similar phenomenon occured when I asked them the following:
Say the function f(x) has a horizontal asymptote at y = 3. What is it's end-behavior?
Many students didn't know where to start, because they had memorized procedures to /find/ the horizontal asymptote, without ever really thinking about what it means to be a horizontal asymptote, and how that idea is related to the idea of end-behavior.
So, to get back to mental models, what would the mental model(s) be for end-behavior? I think it's hard to capture what someone's mental model is exactly; it's whatever they use to think about what a question means.
When I see: "as x -> infinity, y -> 0". I always say out loud in my head "as the x-values get bigger and bigger, the y values get closer and closer to 0"
I also always picture a graph in my head something like this...
As the x-values of my points go to the right, the y-values of my points will get closer to 0 (the x-axis). In other words, as I keep plotting points, their height (the red dotted lines) will get shorter and shorter. (Actually I imagine an animation with the dots appearing from left to right, as the x-values get bigger)Since the idea of end-behavior is a slight formalization of a very natural question to ask about a graph, I think a good way to introduce it would be to have students do their best to answer the natural question as best they can.
It could be a "describe the graph over the telephone"-type activity. Students could have a series of graphs, and their goal is to write a short (i.e. 1-2 sentence) statement explaining what the the trend at the "ends" of the graph are, so that someone else could re-create it only from the description. After discussing student descriptions, you could unveil the sort of description mathematicians have agreed upon in a context that makes sense.
This would also be a good lesson for a party game I like. Take a vertical strip of paper and fold it into (say) 4 sections. The first person in the group draws a graph. The second person in the group looks at the graph and describes its end-behavior. The paper is folded so the 3rd group member can only see the 2nd member's writing. The 3rd member has to draw a graph that matches the end-behavior described by the 2nd member, and so on. It's like "telephone" using paper, and switching back and forth between representations. At the end, the group checks to see if they ended with the same kind of end-behavior they started with.
To emphasize to students that they should be learning what the concept means, it would be good to give them assignments that require them to express this understanding directly--not use it to get an answer to another sort of problem.
Since the concept of end-behavior has a natural interpretation in terms of a progression, I think a nice assignment would be to have students create a short multi-panel comic strip explaining or illustrating the idea of end-behavior for someone in a younger grade (like 9th).
In my experience, some students won't produce anything useful, but many students will produce clearer explanations than anything I was likely to give; so re-distributing some of the better comic strips as "notes" would be a good follow-up.
Monday, April 27, 2009
Lessons learned
Here are two problems I encountered while teaching problem-solving:
1). My students aren't attuned to what I'm trying to teach. When they see a problem, they're immediately filtering for "what are the steps for this problem type?" There's a lot of resistance to the idea that there is no algorithm to learn, and even more to the idea that we won't look at a single "type" of problem over and over again; instead each problem will look totally new every time.
2). Even when they get better at problem-solving, they feel adrift because there's nothing concrete they can point to and say "THIS is what I learned". As a result, they feel anxious and unsatisfied, because it's hard for them to know what they're doing or if they're getting better.
To address these problems, I'd like to give students a general algorithm for problem-solving. This will give them steps to follow when facing a new problem, but also be what I want them learn. I can introduce it early and have it appear (to start) on quizes as pure regurgitation. The goal is for the problem-solving steps to be so internalized as to become habits of mind (eventually). I can introduce heuristics a few at a time, and ideally have some accountability structure that will make students feel like they're making progress towards improvement on those heuristics.
Hopefully this will give them a cognitive reference point so they feel (and can explain to others) exactly what they're getting better at.
Thursday, April 9, 2009
Hypothetical v. Counterfactual
Jen told me a story about an episode of 30 days she saw in which an athiest lived with a fundamentalist Christian. The atheist was trying to explain why she was uncomfortable with the phrase "In God we Trust" on our currency. The Christian kept saying things like, "Well, that's normal; that's just what our money says, what's the problem?" She attempted to explain using a counter-factual: "Imagine if all our money said: There is no God. How would that make you feel?" Apparently the man seemed unable (or maybe unwilling) to consider this possibility. He just kept repeating things like: "But that's not what our money says."
Of course, this misses the whole point. But I have definitely had my share of this same experience in all sorts of situations. I ask someone to consider a scenrio to illustrate a point, and often get the reply back: "but things aren't that way" or "but things would never be like that".
This initially confused me; people should be good at this kind of reasoning since we all have to use hypothetical reasoning all the time. We imagine what if we did this or that, imagine the consequences and use that to guide our actions. So what's the difference?
My hypothesis is that the difference may lie in how well someone is able to imagine the alternative world. Hypothetical situations (what if I did X) may be generally easy for people to imagine. Counter-factuals that involve more drastic changes in the world (what if we lost WWII; what if there were a God? What if there were no God?) are, for some reason, harder for some people to imagine.
It's difficult for me to separate ability from inclination here; are they really unable to imagine a counter-factual situation in detail, or just unwilling to expend the effort to imagine it (they don't see the point, maybe)? My gut feeling is that usually the scenario they're being asked to imagine (in an argument, say) is both difficult and _unpleasant_ to imagine. These may be enough to make most people resist the attempt ("but it wouldn't be like that"), rather than really try and grasp the point.
People I've discussed this with in the past often identify this as a failure of logic or rationality--people are too dumb to get it. I now think it's a failure of imagination and of trust--a difficulty and discomfort with imagining things one finds unpleasant.
If we were going to rectify this kind of failure in school, the solution probably isn't a focus on the logical structure of such arguments. It probably will have more to do with creating safe environments to use, experience, and appreciate the legitimacy of the technique.
Of course, this misses the whole point. But I have definitely had my share of this same experience in all sorts of situations. I ask someone to consider a scenrio to illustrate a point, and often get the reply back: "but things aren't that way" or "but things would never be like that".
This initially confused me; people should be good at this kind of reasoning since we all have to use hypothetical reasoning all the time. We imagine what if we did this or that, imagine the consequences and use that to guide our actions. So what's the difference?
My hypothesis is that the difference may lie in how well someone is able to imagine the alternative world. Hypothetical situations (what if I did X) may be generally easy for people to imagine. Counter-factuals that involve more drastic changes in the world (what if we lost WWII; what if there were a God? What if there were no God?) are, for some reason, harder for some people to imagine.
It's difficult for me to separate ability from inclination here; are they really unable to imagine a counter-factual situation in detail, or just unwilling to expend the effort to imagine it (they don't see the point, maybe)? My gut feeling is that usually the scenario they're being asked to imagine (in an argument, say) is both difficult and _unpleasant_ to imagine. These may be enough to make most people resist the attempt ("but it wouldn't be like that"), rather than really try and grasp the point.
People I've discussed this with in the past often identify this as a failure of logic or rationality--people are too dumb to get it. I now think it's a failure of imagination and of trust--a difficulty and discomfort with imagining things one finds unpleasant.
If we were going to rectify this kind of failure in school, the solution probably isn't a focus on the logical structure of such arguments. It probably will have more to do with creating safe environments to use, experience, and appreciate the legitimacy of the technique.
Friday, April 3, 2009
"What's the next step?" v. "What do I need?"
Today I had an interesting experience that revealed a large gap between how my students and I think about the problems we've been doing. This difference corresponds to how you learn the algorithm for a specific skill vs how you tackle a problem you've never seen before.
If you're learning a specific skill (say, how to solve a quadratic equation), the process generally involves recognizing some cue that tells you what to do next, over and over again. The operative question is "What's the next step?"
If you have to solve a problem you've never seen before, you can't ask yourself "What's the next step?", because the whole point is that you don't know; maybe no one knows. So instead you have to ask yourself "What do I want to find out?", "What do I know already?", and "How can I start to connect the two?"
It's astonishing how few of the problems students get in school require them to ask themselves these three questions. Even if a problem starts out requiring these genuine problem solving questions, it's extremely easy for it to become "proceduralized"--either when I (or other students) give too many hints and reduce it to following-the-steps, or if we do the problem so many times that it becomes a "type" of problem the students recognize "the steps" for.
Students (usually) do a pretty good job listing what they know, and what they want to figure out; the whole trick is how to connect them together. You could "work forwards" by asking what else you can figure out with what you know. It might not seem directly relevant, but if you keep figuring out more and more things, eventually you may see a connection to what you want to know. You can also work backwards by identifying what key piece of information would let you solve your problem. This sets a new sub-goal to aim at.
My students seem to get this idea in general, but have trouble applying it. The specific phrasing of the questions I ask seem to make a big difference. To me, all these questions are equivalent:
For struggling students, it was difficult to get them to think of more than 1 thing they could try. I'd given them a diagram with two perpendicular lines, the equation of one of the lines, and a point on the other. I was hoping they'd suggest things to try such as finding the equation of the other line, finding the point of intersection, finding the x or y intercepts of either line, etc. None of these seemed to present themselves as salient pieces of information. I tried to prompt them by asking what kinds of questions they'd been asked before about situations like this, or what skills they remembered from algebra I or II about lines, points, or perpendicular lines, but without specific queues to trigger their memory, they weren't able to brain-storm a list effectively. At least not individually.
For my last block of the day we started as an entire class. I wrote the diagram on the board, and started a list with the two pieces of information we had (the equation of one line, and the coordinates of the point on the other), as well as the piece of information we were trying to find in the problem. Then we played a game where I would hand the pen to a student who would silently add a new "fact" about the diagram to our list of facts to figure out. They could add labels or additional points or lines to the diagram if they wanted. If they were really stuck, they could hand the pen to another student.
This seemed to work. Students were able to get a good list, extending or modifying ideas that were already in the list. It was interesting to notice that sometimes students would add a potentially relevant addition to the drawing (such as a line that made a promising-looking right triangle with existing lines), while a few would add a seemingly random line or point just for the sake of extending the list.
We had a discussion afterwards about which facts seemed more or less "helpful" in solving the problem. Again, as a class we were able to formulate a series of sub-goals to lead from the givens to the solution.
I think this general approach could work for small group problem solving. The main obstacle, I think, is actually getting the students to follow the process. As a whole class, they were happy to do it, but in small groups they tend to focus directly in on the problem, and seem to regard this sort of activity as long and circuitous.
Some ideas I'd like to try are these:
* Give them a situation and tell them to list, and find, as many things about the situation as possible (so there is no specific "problem" to solve).
* Scaffold the process by providing some of the steps. E.g. on the first problem, I provide the list of facts and they figure out how they connect together. Next time, we generate the list together first. After that, I prompt them to do it on their own. After that, I just give them the problem with no prompting to think about all the things they might try.
* Have the work on half-sized poster paper. Each sub-goal they formulate goes on a 3x5 card. They're reponsible (as part of their graded work) for creating and sequencing the cards on the paper (to show their problem-solving plan). After they have their plan, they can actually show the calculations on the paper.
With respect to how each student understands the specific self-questions involved in working forwards or working backwards, I think it might be best to have students say the idea in their own words, and we can just create a small collection of student phrasings. In my prior experience, this works much better than me trying to figure out the clearest way to express it, since student phrasings tend to make sense to other students. Having a diversity of phrasings will also increase the probability that one of them will catch.
Anyway, a lot of interesting issues to think about. More updates on the teaching of basic problem-solving later!
Students (usually) do a pretty good job listing what they know, and what they want to figure out; the whole trick is how to connect them together. You could "work forwards" by asking what else you can figure out with what you know. It might not seem directly relevant, but if you keep figuring out more and more things, eventually you may see a connection to what you want to know. You can also work backwards by identifying what key piece of information would let you solve your problem. This sets a new sub-goal to aim at.
My students seem to get this idea in general, but have trouble applying it. The specific phrasing of the questions I ask seem to make a big difference. To me, all these questions are equivalent:
- What could you figure out from here?
- What could you try to do next?
- What could you try to figure out next?
- What facts could you try and figure out about this situation?
- What information could you try and figure out next?
For struggling students, it was difficult to get them to think of more than 1 thing they could try. I'd given them a diagram with two perpendicular lines, the equation of one of the lines, and a point on the other. I was hoping they'd suggest things to try such as finding the equation of the other line, finding the point of intersection, finding the x or y intercepts of either line, etc. None of these seemed to present themselves as salient pieces of information. I tried to prompt them by asking what kinds of questions they'd been asked before about situations like this, or what skills they remembered from algebra I or II about lines, points, or perpendicular lines, but without specific queues to trigger their memory, they weren't able to brain-storm a list effectively. At least not individually.
For my last block of the day we started as an entire class. I wrote the diagram on the board, and started a list with the two pieces of information we had (the equation of one line, and the coordinates of the point on the other), as well as the piece of information we were trying to find in the problem. Then we played a game where I would hand the pen to a student who would silently add a new "fact" about the diagram to our list of facts to figure out. They could add labels or additional points or lines to the diagram if they wanted. If they were really stuck, they could hand the pen to another student.
This seemed to work. Students were able to get a good list, extending or modifying ideas that were already in the list. It was interesting to notice that sometimes students would add a potentially relevant addition to the drawing (such as a line that made a promising-looking right triangle with existing lines), while a few would add a seemingly random line or point just for the sake of extending the list.
We had a discussion afterwards about which facts seemed more or less "helpful" in solving the problem. Again, as a class we were able to formulate a series of sub-goals to lead from the givens to the solution.
I think this general approach could work for small group problem solving. The main obstacle, I think, is actually getting the students to follow the process. As a whole class, they were happy to do it, but in small groups they tend to focus directly in on the problem, and seem to regard this sort of activity as long and circuitous.
Some ideas I'd like to try are these:
* Give them a situation and tell them to list, and find, as many things about the situation as possible (so there is no specific "problem" to solve).
* Scaffold the process by providing some of the steps. E.g. on the first problem, I provide the list of facts and they figure out how they connect together. Next time, we generate the list together first. After that, I prompt them to do it on their own. After that, I just give them the problem with no prompting to think about all the things they might try.
* Have the work on half-sized poster paper. Each sub-goal they formulate goes on a 3x5 card. They're reponsible (as part of their graded work) for creating and sequencing the cards on the paper (to show their problem-solving plan). After they have their plan, they can actually show the calculations on the paper.
With respect to how each student understands the specific self-questions involved in working forwards or working backwards, I think it might be best to have students say the idea in their own words, and we can just create a small collection of student phrasings. In my prior experience, this works much better than me trying to figure out the clearest way to express it, since student phrasings tend to make sense to other students. Having a diversity of phrasings will also increase the probability that one of them will catch.
Anyway, a lot of interesting issues to think about. More updates on the teaching of basic problem-solving later!
Wednesday, March 19, 2008
Rationalization
On NPR just now there's a novelist discussing his new novel. The main character is a reporter, reporting on "sex tours", who intends to remain detached, but ends up participating in the tours himself.
Terry Gross reads several excerpts from the novel of the different justifications that the men on the tour give for their participation. One considers how many different services are considered legitimate: we pay people to carry our things, clean our houses, massage our bodies; so why the special standard for sex? (presumably the gentleman in question thinks there isn't an important difference). Another considers the argument that it's dangerous for women and reasons that there are lots of risky jobs in society: is being a sex worker riskier than being a police officer or fireman?
At this point the novelist said something that set me off; he said that this illustrates the problem with "words". He claimed that your moral sense has to be deeper than, beyond "words", because words can lead you astray (as they did his main character).
Let's re-phrase what he really seems to be saying: Often our reasons (those "words") are really rationalizations, and we can too-easily convince ourselves with spurious arguments. If your "moral compass" is immune to such reasoning, you can escape the danger of rationalizing.
This is perfectly true, but neglects the equally compelling danger of the alternative: having a moral sense that isn't sensitive to reason means you can never check whether your moral compass makes any sense. If rationalizing is pretending you don't see, dogmatism is never looking in the first place.
I don't want to overstate people's (myself included) ability to correctly reason--I think all evidence is that it's pretty poor. But I think it's the only weapon we've got. The danger of rationalizing isn't that we've done too much reasoning; the problem is that we haven't done enough. Rationalizations, if such they are, shouldn't be able to withstand closer scrutiny. The problem is that we stop as soon as we get to an answer we like and don't look at them as closely as we really should.
I don't see many good practical prospects for improving our situation. How do you compel people to consider more deeply? The problem is even worse than it may seem, since there are a lot of people who admittedly continue to do things they've judged that they shouldn't. These, perhaps, are the more honest among us who recognize they won't change, but at least don't try to rationalize their choices. What's to be done?
I have no idea, but I do know that a promising way to deal with individual weakness is through the social bonds formed in groups; there are plenty of "keep-each-other-strong" organizations in other areas; so why not these? Religious groups are particularly well-positioned to do this since they already have the infrastructure in place, so to speak.
The foundation of such a group would be a collective recognition that no one probably knows the truth about things, and our best shot at figuring anything out is through challenging, though respectful, dialogue and that everyone needs help to become who they wish they were.
Terry Gross reads several excerpts from the novel of the different justifications that the men on the tour give for their participation. One considers how many different services are considered legitimate: we pay people to carry our things, clean our houses, massage our bodies; so why the special standard for sex? (presumably the gentleman in question thinks there isn't an important difference). Another considers the argument that it's dangerous for women and reasons that there are lots of risky jobs in society: is being a sex worker riskier than being a police officer or fireman?
At this point the novelist said something that set me off; he said that this illustrates the problem with "words". He claimed that your moral sense has to be deeper than, beyond "words", because words can lead you astray (as they did his main character).
Let's re-phrase what he really seems to be saying: Often our reasons (those "words") are really rationalizations, and we can too-easily convince ourselves with spurious arguments. If your "moral compass" is immune to such reasoning, you can escape the danger of rationalizing.
This is perfectly true, but neglects the equally compelling danger of the alternative: having a moral sense that isn't sensitive to reason means you can never check whether your moral compass makes any sense. If rationalizing is pretending you don't see, dogmatism is never looking in the first place.
I don't want to overstate people's (myself included) ability to correctly reason--I think all evidence is that it's pretty poor. But I think it's the only weapon we've got. The danger of rationalizing isn't that we've done too much reasoning; the problem is that we haven't done enough. Rationalizations, if such they are, shouldn't be able to withstand closer scrutiny. The problem is that we stop as soon as we get to an answer we like and don't look at them as closely as we really should.
I don't see many good practical prospects for improving our situation. How do you compel people to consider more deeply? The problem is even worse than it may seem, since there are a lot of people who admittedly continue to do things they've judged that they shouldn't. These, perhaps, are the more honest among us who recognize they won't change, but at least don't try to rationalize their choices. What's to be done?
I have no idea, but I do know that a promising way to deal with individual weakness is through the social bonds formed in groups; there are plenty of "keep-each-other-strong" organizations in other areas; so why not these? Religious groups are particularly well-positioned to do this since they already have the infrastructure in place, so to speak.
The foundation of such a group would be a collective recognition that no one probably knows the truth about things, and our best shot at figuring anything out is through challenging, though respectful, dialogue and that everyone needs help to become who they wish they were.
Wednesday, February 27, 2008
argument and propaganda
Jen had posed two questions in an earlier post:
How is argument different from force, and why is it preferable?
Of course, argument can be seen as a kind of force; after all, you're trying to logically (instead of physically) compel someone to accept your position. But we think of argument as a legitimate means to convince someone, whereas pure rhetoric or propaganda represent an illegitimate means of "force" through psychological coersian. So, to re-phrase the question, what's the difference between convincing and coercing? What makes one legitimate and the other not?
The major difference between these has to do with the use of reasons. As a first pass, we could say that argument provides reasons, whereas propaganda attempts to bypass them. Ridicule (name-calling, stereotyping), for example, can be a way to get someone to dismiss a position without ever considering reasons for or against it. More generally, propaganda often functions by trying to associate a position directly with something that will be evaluated positively or negatively, in the hopes that these feelings will transfer.
This isn't quite sufficient, though. There are other forms of psychological coersian which don't it this model. Lying, for example, involves providing false information in the hopes that it will lead someone else's reasoning in the direction you want. Deliberate oversimplification would be another example; it also aims to influence the other person by using their reason.
I think the real unifying feature of these illegitimate techniques is their attempt to manipulate. In this sense, legitimate argument is an attempt to work with (cooperate with) someone's capacity to reason. The alternatives try to "work against" it. Let's flesh this out a bit more.
Providing reasons isn't the distinctive feature of legitimate argument, since the alternatives do that, too. Legitimate argument involves helping someone else's reasoning system do for itself what they would wish it to be able to do anyway. We do this, for example, by providing accurate and relevant information, or by pointing out logical inferences. It's very much like helping someone do a math problem. You can point out things they didn't see, but would have wished they could see.
This analogy works because both activities involve trying to find the truth about something. Because arguments are so often framed as a debate, it's easy to think the purpose of argument is to convince someone. The real purpose of legitimate argument is to help them see the same truth you do.
Let's consider some difficulties:
#1 "What if their reasoning system is so faulty (or, "different", let's say) that "helping it do what they would want to do with it anyway" violates your own standards of reason?" For example, how do you legitimately convince a fundamentalist Christian who only believes in the literal word of scripture that evolution is right?
One option here would be to accept that sometimes you just can't get what you want with legitimate argument. I think this is dangerous. For the above case, I would seek principals of reason you both do accept and leverage those. It will probably be a long (maybe life-long) conversation, because before you can tackle evolution, you might need to convince them (based on shared principals of reason) to modify how they reason.
Those who are severely mental abnormal are a much more difficult example. Patients whose perception of the world, or faculties of reasoning are radically different may, in the end, be unreachable by a shared process of reasoning. Just because someone's reasoning process is different, however, doesn't mean that it shares no commonalities with your own, or that you can't build commonalities.
#2: "Your characterization implies that if there is no external truth, there is no possibility for legitimate argument, since such argument involves helping them come to see this truth." For example, when discussing issues of personal taste, aesthetics, and arguably, morality, the "truth" might just be a certain way of looking at the world; and no one way is fundamentally better than any other.
These are interesting cases. Let's consider something definitely subjective, such as whether "chocolate is more delicious than vanilla." It's definitely wrong to say that a legitimate argument for this position involves helping the other person discover the truth about this statement. But a legitimate argument for this claim might involve helping someone see if the truth _for them_ is the same as the truth _for you_. You do this by describing aspects of your own experience that are decisive in the hopes that they may discover they agree. This need not be manipulative.
This same idea seems as if it extends to other aesthetic questions fairly easily. What about moral questions?
This idea is similar to that of "framing" an issue. Linguists have long thought that the specific words you choose to describe something carry with them a set of assumptions that will partially determine what makes "common sense". George Lakoff discusses the use of framing for political/moral questions in his books _Moral Politics_ and _Don't Think of an Elephant_.
At first glance, framing can look a lot like illegitimate manipulation. After all, the words you choose can establish "hidden" assumptions that will influence the other person's reasoning system.
Certainly, frames can (and are) used manipulatively in this way. The question is, can they be used as part of legitimate argument? We all frame issues all the time anyway; we can't help it--it's the way our brains and language work.
So if frames are always manipulative, this might put a serious kink in any hopes of purity.
I said that a frame carries with it certain assumptions. What are these assumptions? They represent a particular way of understanding a situation; a way of looking at the world. In fact, they underlie how we experience the world. I think they also contain a mix of objective and subjective elements.
This makes them difficult to know what to do with.
If frames were purely my subjective way of looking at things, then using a frame might be like testing out to see if someone else also finds that way of looking at the world natural. It would be directly analogous to the painting case.
Too often, however, the assumptions a frame brings have elements that could be checked objectively. The fact that a frame hides these, however, means that typically they are not.
I will leave the issue of framing for consideration in another post.
How is argument different from force, and why is it preferable?
Of course, argument can be seen as a kind of force; after all, you're trying to logically (instead of physically) compel someone to accept your position. But we think of argument as a legitimate means to convince someone, whereas pure rhetoric or propaganda represent an illegitimate means of "force" through psychological coersian. So, to re-phrase the question, what's the difference between convincing and coercing? What makes one legitimate and the other not?
The major difference between these has to do with the use of reasons. As a first pass, we could say that argument provides reasons, whereas propaganda attempts to bypass them. Ridicule (name-calling, stereotyping), for example, can be a way to get someone to dismiss a position without ever considering reasons for or against it. More generally, propaganda often functions by trying to associate a position directly with something that will be evaluated positively or negatively, in the hopes that these feelings will transfer.
This isn't quite sufficient, though. There are other forms of psychological coersian which don't it this model. Lying, for example, involves providing false information in the hopes that it will lead someone else's reasoning in the direction you want. Deliberate oversimplification would be another example; it also aims to influence the other person by using their reason.
I think the real unifying feature of these illegitimate techniques is their attempt to manipulate. In this sense, legitimate argument is an attempt to work with (cooperate with) someone's capacity to reason. The alternatives try to "work against" it. Let's flesh this out a bit more.
Providing reasons isn't the distinctive feature of legitimate argument, since the alternatives do that, too. Legitimate argument involves helping someone else's reasoning system do for itself what they would wish it to be able to do anyway. We do this, for example, by providing accurate and relevant information, or by pointing out logical inferences. It's very much like helping someone do a math problem. You can point out things they didn't see, but would have wished they could see.
This analogy works because both activities involve trying to find the truth about something. Because arguments are so often framed as a debate, it's easy to think the purpose of argument is to convince someone. The real purpose of legitimate argument is to help them see the same truth you do.
Let's consider some difficulties:
#1 "What if their reasoning system is so faulty (or, "different", let's say) that "helping it do what they would want to do with it anyway" violates your own standards of reason?" For example, how do you legitimately convince a fundamentalist Christian who only believes in the literal word of scripture that evolution is right?
One option here would be to accept that sometimes you just can't get what you want with legitimate argument. I think this is dangerous. For the above case, I would seek principals of reason you both do accept and leverage those. It will probably be a long (maybe life-long) conversation, because before you can tackle evolution, you might need to convince them (based on shared principals of reason) to modify how they reason.
Those who are severely mental abnormal are a much more difficult example. Patients whose perception of the world, or faculties of reasoning are radically different may, in the end, be unreachable by a shared process of reasoning. Just because someone's reasoning process is different, however, doesn't mean that it shares no commonalities with your own, or that you can't build commonalities.
#2: "Your characterization implies that if there is no external truth, there is no possibility for legitimate argument, since such argument involves helping them come to see this truth." For example, when discussing issues of personal taste, aesthetics, and arguably, morality, the "truth" might just be a certain way of looking at the world; and no one way is fundamentally better than any other.
These are interesting cases. Let's consider something definitely subjective, such as whether "chocolate is more delicious than vanilla." It's definitely wrong to say that a legitimate argument for this position involves helping the other person discover the truth about this statement. But a legitimate argument for this claim might involve helping someone see if the truth _for them_ is the same as the truth _for you_. You do this by describing aspects of your own experience that are decisive in the hopes that they may discover they agree. This need not be manipulative.
This same idea seems as if it extends to other aesthetic questions fairly easily. What about moral questions?
This idea is similar to that of "framing" an issue. Linguists have long thought that the specific words you choose to describe something carry with them a set of assumptions that will partially determine what makes "common sense". George Lakoff discusses the use of framing for political/moral questions in his books _Moral Politics_ and _Don't Think of an Elephant_.
At first glance, framing can look a lot like illegitimate manipulation. After all, the words you choose can establish "hidden" assumptions that will influence the other person's reasoning system.
Certainly, frames can (and are) used manipulatively in this way. The question is, can they be used as part of legitimate argument? We all frame issues all the time anyway; we can't help it--it's the way our brains and language work.
So if frames are always manipulative, this might put a serious kink in any hopes of purity.
I said that a frame carries with it certain assumptions. What are these assumptions? They represent a particular way of understanding a situation; a way of looking at the world. In fact, they underlie how we experience the world. I think they also contain a mix of objective and subjective elements.
This makes them difficult to know what to do with.
If frames were purely my subjective way of looking at things, then using a frame might be like testing out to see if someone else also finds that way of looking at the world natural. It would be directly analogous to the painting case.
Too often, however, the assumptions a frame brings have elements that could be checked objectively. The fact that a frame hides these, however, means that typically they are not.
I will leave the issue of framing for consideration in another post.
Sunday, February 24, 2008
Small Rant
It has apparently been widely reported in conservative (and other) media that in a focus group conducted by Fox news, 25 Obama supporters couldn't name a single one of his legislative accomplishments. I heard a group of people on the train today recount this fact (“discuss” would be too strong a word), shaking their heads and clucking their tongues: “...You would think that his supporters would know something about him.” And thus, the subject was dismissed; which, of course, was the point of the “news” story.
Of course, it would be easy to perform the same stunt with any candidate. I would have bet money that I could have found 25 McCain supports on the train who wouldn't have been able to name any of his legislative accomplishments; I doubt that's the way most Americans decide who to support. Indeed, I would be surprised if the certainty of the McCain supporters in question was based on their extensive legislative knowledge.
How big a factor should it be? Elections are about the future. Knowledge of a candidates' past accomplishments (and failures) will be valuable to the extent that they can inform your interpretation of the candidate's current self-presentation and plans. As such, their utility will vary greatly; but in the end I view their role as a supportive (almost secondary) one; as one of confirmation or correction of what should be primary which is what the candidate is presenting now.
But, as I alluded to above, these considerations give too much credit to a media stunt which doesn't even appear to try addressing such worthwhile questions.
For me, politics is about two things, neither of which (in my experience) come naturally to most people: hard research, and compromise. Doing a lot of hard thought and research is the only way to know what you should think about the difficult questions of politics (and they're pretty much all difficult). Compromise is the only way you make progress towards solving them.
If most people thought about politics this way, I think we would see more humility, curiosity, uncertainty, openness and goodwill. Take social security, for example. Most people are not economists, know none of the statistics that might be relevant, and haven't really thought very carefully about ideas like "personal responsibility" they're so ready to invoke; nonetheless, many people will confidently state a position, and caricature or ridicule those whose positions differ.
I have no idea what to think about social security; it's a question I haven't thought about yet, and which, quite frankly, seems pretty daunting. Far from being ready to dig in my heals, I'm actively searching for people who can make sense of their (or any!) position for me. Sadly, most people's views seem to bottom-out in something they heard someone say on tv that they didn't think about very much.
When push comes to shove, I don't think most people are really interested in doing the work required to construct a reasoned view for themselves. I don't blame them; it's a lot of work. But their certainty and demeanor is wholly inappropriate to their understanding. It's like having strong feelings about different interpretations of quantum mechanics that you're ready to vigorously defend, even though, when it comes right down to it, you don't really know very much about physics.
Despite this, I'm amazed at what people do to avoid conceding a point to "the other side." Compromise is often seen as (if not openly declared to be) weakness. The attitude I sense from most people is that politics isn't about finding compromises we can agree on, but about winning enough power to impose your ideas on everyone else. I remember in 2004 when an NPR correspondent asked a member of a republican think-tank whether Bush had any responsibility to the democratic 49% of the country, considering the narrow margin of his victory and the fact that his party now controlled both the house, the senate and the executive branch. His answer was along the lines of: "We won. Why on earth would we give-up power to the losers?"
This sort of attitude deepens divisions, clouds clear thinking, and, in my opinion, impedes the possibility of long-term progress. What kind of progress is it if it's just reversed 4 or 8 years later?
If politics should be about scholarship and compromise, what's gone wrong? I'm not totally sure, but the media seems to be fanning the flames. The way most stations cover politics differs little from the way they cover sports. It's about who's ahead, who's strong and who's weak, what strategies or tactics are successful. Politics is too difficult and important to be trivialized in this way.
Here's my advice for discussions of politics. Try to ask more questions than you answer. Try to figure out how your opponents' view makes sense to them. Find something you can agree with before finding something to disagree about. These dictates are harder to practice than preech, and I'd be the first to admit I'm not very good at them. But they're what I aspire to and would appreciate being reminded of as frequently as possible.
Friday, February 22, 2008
This might be too hard for me, pt. 1
Here is a problem we've been working on (originally taken from wu::riddles). Here is one way to describe it:
The blue lines are the solution. Obviously, it blocks all possible red lines (3 possible red lines shown), and the fact that it's symmetric makes it look pretty short. But how could we prove it was the shortest? In particular, we wanted to be sure there wasn't some weird scatter of short lines (with a shorter total length) that would also block all possible red lines.
(Here's an imagined weird scatter with few blocked red lines. Of course, a scatter like this could consist of a huge number of almost point-like line segments....so, pretty hard to think about on a case-by-case basis....)
One friend of ours suggested the following proof (actually "proof" since it turned out to be wrong). It's an interesting and useful form of argument, however.
Idea #1: Find the shortest solution for only some of the lines
Any solution that blocks all possible lines will at least have to block all the diagonal lines (in both directions).
If we can find the shortest solution that blocks just these lines, we'll know that a full solution has to be at least that long.
Let's divide the square into a bunch of parallel "slices"...
...and find the shortest solution for each "slice" that blocks only the diagonal lines from one side to the other (direction shown as the green arrow).
In this diagram, we have a slice (in black). We're only interested in blocking the green lines parallel to the one shown. The shortest solution for this slice will be a line perpendicular to the green ones we're blocking (such as the blue line shown).
So, if that blue line is the shortest solution for a slice, we can put that same solution together for all the slices...
As you can see, this blocks all the parallel diagonal lines in that direction (though, of course, it would let other lines through). But, if we're right so far, that's the shortest possible way to block those parallel lines (since for each little slice it was the shortest).
If we also want to block the diagonal lines in the other direction, we just repeat the procedure and overlay the results:
This is a scatter that we've constructed only to block the diagonal lines. BUT, we know that, for each direction, it does so with the shortest amount of drawing.
Since an actual solution (that blocks all the lines) will have to block those diagonal ones, and we know it has to be at least as long as this scatter to block the diagonal ones, the full solution will also have to be at least as long as this scatter.
But our "x" solution is exactly that long. It's easy to see why: instead of putting each little blue segment in a random spot, we line them all up along the diagonals.
You can imagine moving each bit perpendicular to its direction, and they line-up to make the x!
So, the "x" is the shortest possible way to block all the diagonals, but it also blocks all the other lines lines! Therefore, it must be the shortest full solution, because if there was another, shorter solution, we'd know it couldn't be blocking all the diagonal lines.
Since we thought the "x" solution was probably the best, we weren't very critical of the "proof" and thought we'd done a pretty good job!
Here's the problem:
This drawing is a shorter solution.
What went wrong with the original proof? Maybe lots of things, but one clear problem was how we broke the problem down into two smaller problems, and then re-combined them.
First we asked "what's the shortest way to block all the diagonal lines in one direction? Ok, let's keep that. Now let's do the same thing in the other direction to block all the other diagonal lines."
The problem is that one little bit of our "x" solution blocks lots of one type of diagonal (green), but none of the other type of diagonal (red)...
Whereas, a little piece of our 2nd solution can block both directions at once...
So maybe, even though our small pieces aren't the shortest for blocking lines in ONE direction or the other, they're the shortest for blocking blocking them together. Their overall blocking efficiency is better (we use this idea more in idea #3 below).
Idea #2: A Connected Solution is Always Shortest
The solution which beat the "x" is formed by adding two imaginary points so that, when you connect all the dots, they form 120-degree angles.
Someone pointed out that there's a theorem which says that this same method will produce the shortest path connecting any number of dots in the plane. So that seemed promising.
Of course, a shortest path connecting points is different than a shortest drawing that will block all possible lines, but it still seemed like a step in the right direction. Maybe we could connect the two problems together...
I hoped that this solution was, in fact, the shortest and we could prove it with a two-step argument:
1). If there's a "disconnected solution" (like a scatter), there will be a "connected solution" (continuous lines with no breaks) that's the same length or shorter.
2). Any "connected solution" must connect the vertecies of the square with each other.
3). Therefore, because of the shortest-path theorem, our solution will also be the shortest blocking solution.
However, we didn't get very far along this path until we found...
This was the shortest solution yet. It also showed the my old hope was wrong. I had hoped that the shortest solution would always be a connected one. Now we know that can't possibly be true.
Curious Student: "But wait! Isn't it possible that you could find an even shorter solution that is connected? Then you'd be right that the shortest solution is a connected one!"
Sadly, no. That theorem about the shortest connecting path implies that the shortest connected solution is the one we found. This disconnected solution is shorter. So if we find an even shorter solution than this, we know it won't be a connected one; it will have breaks in it.
Idea #3: Try A Smaller Problem. Measure Blocking Efficiency.
Since we didn't have a good way to analyze the square, we decided to try the simplest case we could think of: the equilateral triangle.
Here is the shortest solution we found for the equilateral triangle:
Once again we have the problem of how to prove this is the shortest.
Nick was hoping to use an idea similar to the last proof that didn't work. Here's the idea:
Any solution that blocks all lines will at least have to block all lines perpendicular to the three sides of the triangle.
If I have a random tiny line segment, I can measure how much of this blocking it does by measuring how long a "shadow" it would cast on each side of the triangle if I shined a light behind it (perpendicular to the wall) (and add them up).
Using these ideas, we hoped that we could show that our solution is the shortest by showing that it's not possible to have a random small piece that blocks more than a random small piece (of the same length) from our solution.
In other words, the orientation of the lines in our solution gives you the maximum amount of blocking possible. That would show that it was the shortest solution for lines perpendicular to the 3 sides. Since our solution also blocks the other lines, it would be the shortest overall solution (by an argument similar to our idea #1).
Unfortunately, we found that a small piece (in isolation) blocks the most when it's parallel to one of the sides; the pieces in our solution are all perpendicular.
This gave me a new idea, however. How much blocking a solution does depends on two things: how much blocking each little piece does, and whether or not any pieces are blocking the same lines (how much redundant blocking there is).
If you want to make your solution shorter, you could either try to change the orientation of your lines so that they block more, or you could move them around so that their blocking isn't redundant.
This led us to...
Idea #5: Measure Redundant Blocking
Here was my idea...
1). Find a measure of how much total blocking needs to be done.
2). Measure the maximum amount of blocking a given tiny line segment can do
3). This will let us calculate a theoretical shortest solution solution. If we assume no redundant blocking, the shortest solution will be...
(total blocking needed) / (how much blocking a unit of solution can do) = total units of length in solution.
4). Calculate the amount of redundant blocking in our proposed shortest solution.
5). Show (somehow) that exactly that much redundancy is required for any 100% blocking solution.
The really hard part of this is step 5.
But steps 1 and 2 are also tricky because they require us to measure blocking in some way. Before we were only measure how many of certain types of lines were being blocked. I wanted a way to measure all the lines being blocked.
How can we do it? Here was our idea...
...inscribe our shape in a circle. Now, the lines you need to block can be seen as all the lines going from one arc of the circle to another.
Here's a picture of what I mean. The three arcs are in different colors. Any line starting in one arc and ending in another is a line we need to block.
This gives us a possible way to measure how much blocking a little line segment does: Imagine a light on the circle, shining onto the segment. Measure the length of the "shadow" it casts on the other side. This is a measure of how many lines it blocks from that point.
To find out how many total lines it blocks, imagine slowly rotating that flashlight around the circle and adding up all the arc lengths. When you've gone all the way around, this is the blocking measure of this segment.
Mathematically speaking, if we have a function to find the arc length, and integrate it from 0 to 360 (as the light source moves around the circle), this should allow us to measure it. We haven't made this integral yet because it seemed hard and we're hoping if we're clever enough we won't have to.
My great hope was that the total amount of blocking of a segment only depends on its length; not on its angle or location within the triangle. The reason this would be good is that a certain length of solution would directly yield a certain total length of blocking. Then, to find the shortest solution, all we have to do is find an arrangement which minimizes redundant blocking. So, instead of minimizing length of solution, we'd be minimizing redundancy of solution.
Unfortunately (for reasons I won't go into), it doesn't seem like that's going to work, though I'm not 100% convinced yet.
That's all I'm going to write for now....perhaps more later...
But first:
What's making this problem hard?
The difficulty we've been having in this problem is that we can't break it down into simple sub-problems we can solve independently.
A short solution depends on two things: the "blocking efficiency" of each little piece, and the amount of redundant blocking for the arrangement. The blocking efficiency seems to depend on the angle and location of each little piece; and the "blocking" of the pieces need to be balanced between all the directions that require blocking. The amount of blocking depends on how each piece is oriented with respect to every other piece; it's a global property of the solution. And so far we can't think of an easy way to divide all the possibilities into cases we can treat separately.
If anyone has new ideas, we'd love to hear them!
Draw a 1x1 square. In a minute, I'm going to take a ruler and draw a straight line that will "cut" accross some part of the square. It might be through the middle, or it might only be a small section of the corner; you don't know. You should draw anything you want inside the square so you can be certain that any line I draw must cross one of the lines in your drawing. The challenge is to do this with the shortest possible amount of drawing.The first, and most obvious, suggestion would be this:
The blue lines are the solution. Obviously, it blocks all possible red lines (3 possible red lines shown), and the fact that it's symmetric makes it look pretty short. But how could we prove it was the shortest? In particular, we wanted to be sure there wasn't some weird scatter of short lines (with a shorter total length) that would also block all possible red lines.
(Here's an imagined weird scatter with few blocked red lines. Of course, a scatter like this could consist of a huge number of almost point-like line segments....so, pretty hard to think about on a case-by-case basis....)
One friend of ours suggested the following proof (actually "proof" since it turned out to be wrong). It's an interesting and useful form of argument, however.
Idea #1: Find the shortest solution for only some of the lines
Any solution that blocks all possible lines will at least have to block all the diagonal lines (in both directions).
If we can find the shortest solution that blocks just these lines, we'll know that a full solution has to be at least that long.Let's divide the square into a bunch of parallel "slices"...
...and find the shortest solution for each "slice" that blocks only the diagonal lines from one side to the other (direction shown as the green arrow).
In this diagram, we have a slice (in black). We're only interested in blocking the green lines parallel to the one shown. The shortest solution for this slice will be a line perpendicular to the green ones we're blocking (such as the blue line shown).So, if that blue line is the shortest solution for a slice, we can put that same solution together for all the slices...
As you can see, this blocks all the parallel diagonal lines in that direction (though, of course, it would let other lines through). But, if we're right so far, that's the shortest possible way to block those parallel lines (since for each little slice it was the shortest).If we also want to block the diagonal lines in the other direction, we just repeat the procedure and overlay the results:
This is a scatter that we've constructed only to block the diagonal lines. BUT, we know that, for each direction, it does so with the shortest amount of drawing.Since an actual solution (that blocks all the lines) will have to block those diagonal ones, and we know it has to be at least as long as this scatter to block the diagonal ones, the full solution will also have to be at least as long as this scatter.
But our "x" solution is exactly that long. It's easy to see why: instead of putting each little blue segment in a random spot, we line them all up along the diagonals.
You can imagine moving each bit perpendicular to its direction, and they line-up to make the x!So, the "x" is the shortest possible way to block all the diagonals, but it also blocks all the other lines lines! Therefore, it must be the shortest full solution, because if there was another, shorter solution, we'd know it couldn't be blocking all the diagonal lines.
Since we thought the "x" solution was probably the best, we weren't very critical of the "proof" and thought we'd done a pretty good job!
Here's the problem:
This drawing is a shorter solution.What went wrong with the original proof? Maybe lots of things, but one clear problem was how we broke the problem down into two smaller problems, and then re-combined them.
First we asked "what's the shortest way to block all the diagonal lines in one direction? Ok, let's keep that. Now let's do the same thing in the other direction to block all the other diagonal lines."
The problem is that one little bit of our "x" solution blocks lots of one type of diagonal (green), but none of the other type of diagonal (red)...
Whereas, a little piece of our 2nd solution can block both directions at once...
So maybe, even though our small pieces aren't the shortest for blocking lines in ONE direction or the other, they're the shortest for blocking blocking them together. Their overall blocking efficiency is better (we use this idea more in idea #3 below).Idea #2: A Connected Solution is Always Shortest
The solution which beat the "x" is formed by adding two imaginary points so that, when you connect all the dots, they form 120-degree angles.
Someone pointed out that there's a theorem which says that this same method will produce the shortest path connecting any number of dots in the plane. So that seemed promising.Of course, a shortest path connecting points is different than a shortest drawing that will block all possible lines, but it still seemed like a step in the right direction. Maybe we could connect the two problems together...
I hoped that this solution was, in fact, the shortest and we could prove it with a two-step argument:
1). If there's a "disconnected solution" (like a scatter), there will be a "connected solution" (continuous lines with no breaks) that's the same length or shorter.
2). Any "connected solution" must connect the vertecies of the square with each other.
3). Therefore, because of the shortest-path theorem, our solution will also be the shortest blocking solution.
However, we didn't get very far along this path until we found...
This was the shortest solution yet. It also showed the my old hope was wrong. I had hoped that the shortest solution would always be a connected one. Now we know that can't possibly be true.Curious Student: "But wait! Isn't it possible that you could find an even shorter solution that is connected? Then you'd be right that the shortest solution is a connected one!"
Sadly, no. That theorem about the shortest connecting path implies that the shortest connected solution is the one we found. This disconnected solution is shorter. So if we find an even shorter solution than this, we know it won't be a connected one; it will have breaks in it.
Idea #3: Try A Smaller Problem. Measure Blocking Efficiency.
Since we didn't have a good way to analyze the square, we decided to try the simplest case we could think of: the equilateral triangle.
Here is the shortest solution we found for the equilateral triangle:
Once again we have the problem of how to prove this is the shortest.
Nick was hoping to use an idea similar to the last proof that didn't work. Here's the idea:
Any solution that blocks all lines will at least have to block all lines perpendicular to the three sides of the triangle.
If I have a random tiny line segment, I can measure how much of this blocking it does by measuring how long a "shadow" it would cast on each side of the triangle if I shined a light behind it (perpendicular to the wall) (and add them up).
Using these ideas, we hoped that we could show that our solution is the shortest by showing that it's not possible to have a random small piece that blocks more than a random small piece (of the same length) from our solution. In other words, the orientation of the lines in our solution gives you the maximum amount of blocking possible. That would show that it was the shortest solution for lines perpendicular to the 3 sides. Since our solution also blocks the other lines, it would be the shortest overall solution (by an argument similar to our idea #1).
Unfortunately, we found that a small piece (in isolation) blocks the most when it's parallel to one of the sides; the pieces in our solution are all perpendicular.
This gave me a new idea, however. How much blocking a solution does depends on two things: how much blocking each little piece does, and whether or not any pieces are blocking the same lines (how much redundant blocking there is).
If you want to make your solution shorter, you could either try to change the orientation of your lines so that they block more, or you could move them around so that their blocking isn't redundant.
This led us to...
Idea #5: Measure Redundant Blocking
Here was my idea...
1). Find a measure of how much total blocking needs to be done.
2). Measure the maximum amount of blocking a given tiny line segment can do
3). This will let us calculate a theoretical shortest solution solution. If we assume no redundant blocking, the shortest solution will be...
(total blocking needed) / (how much blocking a unit of solution can do) = total units of length in solution.
4). Calculate the amount of redundant blocking in our proposed shortest solution.
5). Show (somehow) that exactly that much redundancy is required for any 100% blocking solution.
The really hard part of this is step 5.
But steps 1 and 2 are also tricky because they require us to measure blocking in some way. Before we were only measure how many of certain types of lines were being blocked. I wanted a way to measure all the lines being blocked.
How can we do it? Here was our idea...
...inscribe our shape in a circle. Now, the lines you need to block can be seen as all the lines going from one arc of the circle to another.
Here's a picture of what I mean. The three arcs are in different colors. Any line starting in one arc and ending in another is a line we need to block.
This gives us a possible way to measure how much blocking a little line segment does: Imagine a light on the circle, shining onto the segment. Measure the length of the "shadow" it casts on the other side. This is a measure of how many lines it blocks from that point.
To find out how many total lines it blocks, imagine slowly rotating that flashlight around the circle and adding up all the arc lengths. When you've gone all the way around, this is the blocking measure of this segment.Mathematically speaking, if we have a function to find the arc length, and integrate it from 0 to 360 (as the light source moves around the circle), this should allow us to measure it. We haven't made this integral yet because it seemed hard and we're hoping if we're clever enough we won't have to.
My great hope was that the total amount of blocking of a segment only depends on its length; not on its angle or location within the triangle. The reason this would be good is that a certain length of solution would directly yield a certain total length of blocking. Then, to find the shortest solution, all we have to do is find an arrangement which minimizes redundant blocking. So, instead of minimizing length of solution, we'd be minimizing redundancy of solution.
Unfortunately (for reasons I won't go into), it doesn't seem like that's going to work, though I'm not 100% convinced yet.
That's all I'm going to write for now....perhaps more later...
But first:
What's making this problem hard?
The difficulty we've been having in this problem is that we can't break it down into simple sub-problems we can solve independently.
A short solution depends on two things: the "blocking efficiency" of each little piece, and the amount of redundant blocking for the arrangement. The blocking efficiency seems to depend on the angle and location of each little piece; and the "blocking" of the pieces need to be balanced between all the directions that require blocking. The amount of blocking depends on how each piece is oriented with respect to every other piece; it's a global property of the solution. And so far we can't think of an easy way to divide all the possibilities into cases we can treat separately.
If anyone has new ideas, we'd love to hear them!
Friday, February 15, 2008
What we expect of morality
When I try to think of the barest, most minimal requirements I have for something to be recognizable to me as morality, I end up with something like this: "Morality is a way of evaluating at least some possibilities of things we can do, or who and how we can be." Now, this is a pretty minimal specification; any evaluative scheme at all could fit this bill. But many people make much more substantial assumptions about what morality must be like. And they often do so without defending those assumptions (which is not to say that they should be defending them; it just means that I haven't seen much discussion of these ideas).
Here are some examples:
Probably there are more -- I'll keep collecting them here as I find them. The next questions are which of these make sense to include in our conception of morality, and how we would go about deciding that.
Here are some examples:
- Morality should be a simply-describable function. The inputs to the function will be actions (or perhaps biographies), and the outputs will be evaluations. The function itself will be describable by a fairly short list of rules.
- Morality involves publicly-expressed reasons. If we judge a certain thing morally good, then we must be able to give an intelligible account of why that thing is good, and what it would take to change that good thing into a bad thing. If we have moral judgments that we can't back up with reasons, then those are not real moral judgments at all, but rather psychological biases or distortions. (I take this to be more or less what Peter Unger thinks, after reading parts of Living High and Letting Die.)
- Morality should judge my individual confrontation with possibilities in the world. If it is right for me to act a certain way, a change in the behavior of those around me can't make it wrong for me to act that way. (This seems absurd to me, but I hear it suggested by classmates, and see it expressed in a more limited domain as the "Compliance Condition" in Liam Murphy's "The Demands of Beneficence".)
- Morality applies only when our actions affect other people.
Probably there are more -- I'll keep collecting them here as I find them. The next questions are which of these make sense to include in our conception of morality, and how we would go about deciding that.
Wednesday, February 6, 2008
Biting the Bullet
A major reason I became attracted to philosophy were the clarity and certainty it appeared to offer. To my high school self, a major determining character trait in people was the possession of a very specific kind of intelligence, which I saw philosophy as exemplifying.
Those who I considered weaker and less intelligent were often uncertain what to believe, would believe things because they hoped them to be true, or, worst of all, didn't seem much interested in examining their beliefs. I imagined the philosopher as the exact opposite sort of person--someone who is constantly scrutinizing their beliefs to ensure they're held for compelling reasons, or not at all. Naturally, I saw myself as this latter sort of intelligent, tough-minded person.
Nothing let me publicly declare this allegiance as strongly as "biting the bullet." My earliest memory of biting the bullet was my far too joyful rejection of free will. I had heard arguments that free will was incompatible with determinism, which was a very difficult conclusion for most people I knew to accept. Part of my certainty that we didn't have free will came from the arguments themselves, but I think a large part of it came from how it let me set myself up as the sort of person I wanted to be. If someone didn't immediately embrace the unpalatable conclusion, it was easy for me to see them as either too stupid to grasp the arguments, or too weak to accept the conclusion. And, of course, my certainty was a clear indicator of my own superior intelligence and commitment to reason.
Now, I tend to lean in the opposite direction--I usually refuse to bite the bullet of a difficult conclusion because (or so I tell myself), its difficulty usually indicates that it represents something important, not to be overlooked or rejected. But I think this can also be seen as a sort-of biting the bullet. The psychological mechanisms are the same; I just have a new concept of what it really is to be tough-minded.
As my above descriptions may have revealed, I consider the sort of absolutist thinking involved in biting the bullet sort of juvenile. Biting the bullet isn't bravely following reason wherever it may lead, but desperately clinging to easy answers that might make you feel smart, or certain, but at the cost of ignoring important aspects of reality. A really tough-minded individual won't be seduced by too clean an argument, but will steadfastly accept logical tensions and conflicts because they reflect how reality really is. Once again, this kind of set-up re-inforces my feeling of certainty because it gives me a way to understand my conclusion as the one a really intelligent and tough-minded person would make. Biting the bullet is what those who are intellectually narrow or insecure do.
Having realized this, one response might be to re-double my efforts to believe things on the strength of the arguments themselves, rather than on how their structure lets me interpret myself. Knowing these twin pitfalls, I might re-examine my conclusions to look specifically for their biasing effects. For whatever reason, I haven't felt like this is the right way to look at things at all.
Instead, I've increasingly started to see philosophy as a normative activity. Accepting or rejecting free will, for example, isn't a matter of looking to the arguments to see what's most likely the truth. It's simply a choice to see the world one way or another; and each way has its own focus, blind-spots, dilemmas, and consequences. This leaves for me the question of how philosophy should be related to other disciplines (including those that I do see as pursuing specific forms of factual truth) and our everyday lives.
Those who I considered weaker and less intelligent were often uncertain what to believe, would believe things because they hoped them to be true, or, worst of all, didn't seem much interested in examining their beliefs. I imagined the philosopher as the exact opposite sort of person--someone who is constantly scrutinizing their beliefs to ensure they're held for compelling reasons, or not at all. Naturally, I saw myself as this latter sort of intelligent, tough-minded person.
Nothing let me publicly declare this allegiance as strongly as "biting the bullet." My earliest memory of biting the bullet was my far too joyful rejection of free will. I had heard arguments that free will was incompatible with determinism, which was a very difficult conclusion for most people I knew to accept. Part of my certainty that we didn't have free will came from the arguments themselves, but I think a large part of it came from how it let me set myself up as the sort of person I wanted to be. If someone didn't immediately embrace the unpalatable conclusion, it was easy for me to see them as either too stupid to grasp the arguments, or too weak to accept the conclusion. And, of course, my certainty was a clear indicator of my own superior intelligence and commitment to reason.
Now, I tend to lean in the opposite direction--I usually refuse to bite the bullet of a difficult conclusion because (or so I tell myself), its difficulty usually indicates that it represents something important, not to be overlooked or rejected. But I think this can also be seen as a sort-of biting the bullet. The psychological mechanisms are the same; I just have a new concept of what it really is to be tough-minded.
As my above descriptions may have revealed, I consider the sort of absolutist thinking involved in biting the bullet sort of juvenile. Biting the bullet isn't bravely following reason wherever it may lead, but desperately clinging to easy answers that might make you feel smart, or certain, but at the cost of ignoring important aspects of reality. A really tough-minded individual won't be seduced by too clean an argument, but will steadfastly accept logical tensions and conflicts because they reflect how reality really is. Once again, this kind of set-up re-inforces my feeling of certainty because it gives me a way to understand my conclusion as the one a really intelligent and tough-minded person would make. Biting the bullet is what those who are intellectually narrow or insecure do.
Having realized this, one response might be to re-double my efforts to believe things on the strength of the arguments themselves, rather than on how their structure lets me interpret myself. Knowing these twin pitfalls, I might re-examine my conclusions to look specifically for their biasing effects. For whatever reason, I haven't felt like this is the right way to look at things at all.
Instead, I've increasingly started to see philosophy as a normative activity. Accepting or rejecting free will, for example, isn't a matter of looking to the arguments to see what's most likely the truth. It's simply a choice to see the world one way or another; and each way has its own focus, blind-spots, dilemmas, and consequences. This leaves for me the question of how philosophy should be related to other disciplines (including those that I do see as pursuing specific forms of factual truth) and our everyday lives.
Monday, February 4, 2008
Moral Relativism
Lots of arguments in ethics try to persuade by analogy. They'll find a situation you agree with them about, and then try to convince you another situation is the same. Like this:
Once my moral intuitions conflict with those of others, it's easy for doubts about the whole process to creep in:
But, of course, most people don't really act in their everyday lives as if morality works like this. They may deny that anything is objectively right or wrong when they're arguing with you, and then try to convince other people to buy "cruelty-free" meat (or go vegetarian). Why? Because it's wrong to inflict suffering on animals, of course. Or they may try to convince their public representatives that it's unfair for them to be required to pay taxes to help the poor; that's money they earned, so they should be able to decide how to spend it. If neither of these examples move you, think for a minute about what cause really motivates you; I think we all have them.
There's more going on in our championing on a cause than simply wanting the world to be a certain way and trying to convince other people. We try to convince each other, not just to get our way, but because we think our way is the right way, and that other people shouldn't just do what we want, but should agree that we're right. If I think about why other people act against my moral causes, I don't think it's because they have different opinions which they have every right to; I think they're too confused or ignorant to see the truth, or too lazy and self-deluded to admit it!
This is just to point out that we treat moral questions differently than we treat questions of taste (like what the best ice-cream is). I can't stand heath ice cream. My wife loves it. I might think that's strange or funny, but I don't think she's confused or wrong about what's good; she's just got a different idea than I do. If I think water-boarding is an appalling practice and you don't, it's very hard for me to adopt the attitude that you just have a different preference than I do. I think that, for some reason, you cannot see or admit the real truth about the practice.
So, if we don't stop to ask ourselves, but covertly watch ourselves in action (to catch ourselves unaware, in our natural habitat), we don't look like we really believe that our moral intuitions are just our opinions. We believe they're really right and that other people should think so too!
Let's look at what research in moral psychology has to say about this. Several psychologists define moral judgements as judgements that we universally hold others accountable to. This is what distinguishes our thinking something's (morally) wrong from just thinking that it's unwise, imprudent, distasteful, etc. So they, too, recognize this fact about our everyday behavior.
Philosophers take this fact and ask "but are we right that there are moral laws that are really universal the way we seem to treat them?"
Psychologists instead recognize that we naturally seem to universalize certain of our judgements. This is different from thinking hard and deciding that there are certain universal principals that we should hold each other accountable to. Instead, certain judgments just strike us as being universally right or wrong.
Noticing this difference, psychologists ask "why is this a natural part of our psychology?" If it's not the result of a considered decision on each of our parts, the answer must lie deeper in a psychology we all share. One answer they've suggested is that we evolved to have this type of judgment because they are effective at helping us create and maintain stable social groups, which improved our chances of survival.
What makes our moral judgments effective in this capacity is merely the fact we think they apply universally; it seems that their content--what we actually think is right and wrong--could vary widely. In other words, you can form just as stable a community with one set of universal principals as with another; it's the fact of their being treated as universal that's important for the social effects.
And, indeed, there not only seems to be disagreement among cultures as to what kinds of behaviors are right or wrong, but also disagreement about what kinds of behaviors should be considered through a moral lens at all. For example, academic discussion of morality tends to focus on issues of fairness and harm, but in many cultures around the world there are strong moral judgements associated with eating, ones relationship to tradition or authority, menstruation, sexual practices, and cultural taboos.
What should we make of this disagreement? I am strongly inclined to think that most traditional cultures are simply mistaken in much of what they consider to be moral questions; but am I really right in this, or is it just an opinion? If menstruating women are prevented from entering the workplace in India, can I rightly condemn that practice as morally wrong?
Most people think it would be wrong to walk by a baby drowning in a puddle without doing anything.In my experience, if someone disagrees, they object that the situations are actually different: of course they should save the baby from the puddle, but the other case is importantly different somehow, so it's ok for them to keep their money. Then there ensues a back-and-forth about whether the differences between the two situations are really morally important differences.
So, what about babies elsewhere in the world whose deaths we (rich, blog-readers in the West) could easily prevent by giving a small amount of our resources?
It doesn't seem like it should matter whether you actually walked past the baby or not; in both cases, you are easily able to prevent the death of an innocent. If it's wrong for you not to in once case, it's wrong in the other case also.
So why aren't you sending more money in foreign aid??
Once my moral intuitions conflict with those of others, it's easy for doubts about the whole process to creep in:
I have one vague feeling, you have another. How can we really decide who's right? In fact, is there really a right answer, or is it just a matter of opinion?Most people I talk to are very quick to give up the fight for a real truth-of-the-matter in moral questions. They agree (and even vigorously argue) that morality is totally subjective. It's a mistake to try and find out what's really right and wrong. That would be like trying to find out what's really more delicious: chocolate or vanilla. There is no Truth independent of what people think it is. If I think chocolate is better, that's true for me. If you think vanilla is better, that's true for you. If you talk with me and convince me that vanilla is really better, then that also becomes true of me.
But, of course, most people don't really act in their everyday lives as if morality works like this. They may deny that anything is objectively right or wrong when they're arguing with you, and then try to convince other people to buy "cruelty-free" meat (or go vegetarian). Why? Because it's wrong to inflict suffering on animals, of course. Or they may try to convince their public representatives that it's unfair for them to be required to pay taxes to help the poor; that's money they earned, so they should be able to decide how to spend it. If neither of these examples move you, think for a minute about what cause really motivates you; I think we all have them.
There's more going on in our championing on a cause than simply wanting the world to be a certain way and trying to convince other people. We try to convince each other, not just to get our way, but because we think our way is the right way, and that other people shouldn't just do what we want, but should agree that we're right. If I think about why other people act against my moral causes, I don't think it's because they have different opinions which they have every right to; I think they're too confused or ignorant to see the truth, or too lazy and self-deluded to admit it!
This is just to point out that we treat moral questions differently than we treat questions of taste (like what the best ice-cream is). I can't stand heath ice cream. My wife loves it. I might think that's strange or funny, but I don't think she's confused or wrong about what's good; she's just got a different idea than I do. If I think water-boarding is an appalling practice and you don't, it's very hard for me to adopt the attitude that you just have a different preference than I do. I think that, for some reason, you cannot see or admit the real truth about the practice.
So, if we don't stop to ask ourselves, but covertly watch ourselves in action (to catch ourselves unaware, in our natural habitat), we don't look like we really believe that our moral intuitions are just our opinions. We believe they're really right and that other people should think so too!
Let's look at what research in moral psychology has to say about this. Several psychologists define moral judgements as judgements that we universally hold others accountable to. This is what distinguishes our thinking something's (morally) wrong from just thinking that it's unwise, imprudent, distasteful, etc. So they, too, recognize this fact about our everyday behavior.
Philosophers take this fact and ask "but are we right that there are moral laws that are really universal the way we seem to treat them?"
Psychologists instead recognize that we naturally seem to universalize certain of our judgements. This is different from thinking hard and deciding that there are certain universal principals that we should hold each other accountable to. Instead, certain judgments just strike us as being universally right or wrong.
Noticing this difference, psychologists ask "why is this a natural part of our psychology?" If it's not the result of a considered decision on each of our parts, the answer must lie deeper in a psychology we all share. One answer they've suggested is that we evolved to have this type of judgment because they are effective at helping us create and maintain stable social groups, which improved our chances of survival.
What makes our moral judgments effective in this capacity is merely the fact we think they apply universally; it seems that their content--what we actually think is right and wrong--could vary widely. In other words, you can form just as stable a community with one set of universal principals as with another; it's the fact of their being treated as universal that's important for the social effects.
And, indeed, there not only seems to be disagreement among cultures as to what kinds of behaviors are right or wrong, but also disagreement about what kinds of behaviors should be considered through a moral lens at all. For example, academic discussion of morality tends to focus on issues of fairness and harm, but in many cultures around the world there are strong moral judgements associated with eating, ones relationship to tradition or authority, menstruation, sexual practices, and cultural taboos.
What should we make of this disagreement? I am strongly inclined to think that most traditional cultures are simply mistaken in much of what they consider to be moral questions; but am I really right in this, or is it just an opinion? If menstruating women are prevented from entering the workplace in India, can I rightly condemn that practice as morally wrong?
Friday, January 25, 2008
Ways of getting people to do what you wish they would.
Let's say I have to go to the store for something, and I really want Dave to come with me. He says he doesn't want to.
There are a variety of ways in which I can try to get him to come anyway. (This list is probably not exhaustive.)
I feel, as I expect most people do, that argument seems like the "best" of these methods, in some sense; the others should be resorted to only in exceptional circumstances, and only when argument is for some reason impossible. But this leaves us with a couple of important questions.
There are a variety of ways in which I can try to get him to come anyway. (This list is probably not exhaustive.)
- Argument. I could start tallying all the reasons he should want to come to the store with me (the duties of partnership, the fact that he will enjoy the trip outside, etc.)
- Bribing. I could promise to buy him a chocolate croissant if he comes.
- Lying. I could tell him that he will enjoy the trip outside, even though I'm quite sure that he won't.
- Force. I could tickle or punch him until he stops resisting, and cram him into the car.
I feel, as I expect most people do, that argument seems like the "best" of these methods, in some sense; the others should be resorted to only in exceptional circumstances, and only when argument is for some reason impossible. But this leaves us with a couple of important questions.
- What is distinctive about argument? How does it differ from bribing, and from force?
- In what ways, or for what reasons, is argument preferable to the other methods?
Sunday, January 13, 2008
Chanterelle hunt: a success!
Dave and I went to our top-secret chanterelle grounds in Portola Valley. We found a bunch of great stuff, including various mushrooms, a half-dozen well-camouflaged salamanders, and a giant male deer ~40 feet away from us!
You can see some pictures here.
You can see some pictures here.
Sunday, January 6, 2008
What's The Point of a Debate?
After the republican debate, the tv station airing the debates said some analysis would follow. Their "analysis" consisted of asserting who "won" or "lost" the debate (with little to no reference to the content of the debate), and employed extensive use of sports and boxing metaphors. They might as well have been saying: "Did all those issues confuse you? Let us tell you what to think about the candidates."
What I would prefer the media do is the following:
1). Re-cap the major logical flow of the debate (since that can be unclear in the moment)
2). Try to identify what are the right questions to ask from here.
For example, the commentators summarized the argument between Obama and Clinton with the phrase "experience vs. change" (which itself arguably confuses and obfuscates more than it clarifies). Then they proceeded to discuss how "strong" or "weak" each candidate's performance was (not even how strong or weak their positions were). What I wish the commentators had said was something like this:
"Clinton said she stands on her experience. Her position seems to be that experience is necessary to convert words and ideas into useful action. Edwards, on the other hand, seemed to be suggesting that too much experience could hinder a president's ability to make real change, because they'd be too entrenched in the system. The next step in this debate is for both candidates to explain exactly what kinds of experience are important to a presidency and why. What, exactly, does a president need to do to get support for their program? What can cause their efforts to fail? What personal qualities, or expertise is required for them to succeed?"
Without this focus on what the candidates are actually saying it is, as one commentator approvingly described the event, "pure theater."
What I would prefer the media do is the following:
1). Re-cap the major logical flow of the debate (since that can be unclear in the moment)
2). Try to identify what are the right questions to ask from here.
For example, the commentators summarized the argument between Obama and Clinton with the phrase "experience vs. change" (which itself arguably confuses and obfuscates more than it clarifies). Then they proceeded to discuss how "strong" or "weak" each candidate's performance was (not even how strong or weak their positions were). What I wish the commentators had said was something like this:
"Clinton said she stands on her experience. Her position seems to be that experience is necessary to convert words and ideas into useful action. Edwards, on the other hand, seemed to be suggesting that too much experience could hinder a president's ability to make real change, because they'd be too entrenched in the system. The next step in this debate is for both candidates to explain exactly what kinds of experience are important to a presidency and why. What, exactly, does a president need to do to get support for their program? What can cause their efforts to fail? What personal qualities, or expertise is required for them to succeed?"
Without this focus on what the candidates are actually saying it is, as one commentator approvingly described the event, "pure theater."
Democratic Debates
Last night Jen and I watched the democratic debate in New Hampshire.
The following isn't quite the exchange that took place between them, but I wanted to present my re-phrased version of what I thought were their strongest arguments.
Hillary's major argument has always been experience. I was never terribly moved by it before (because I think we've had, and can have, effective presidents without prior experience), but below is a version of it that gave me pause:
Hillary: All the democratic candidates want change, but we need someone who can deliver it. If you want change, look to what changes each candidate has already made. I have a 35 year record of it. I don't have to remind you of the importance of what the next president will need to do for the country and the world. Obama and Edwards are great people, with real passion and good ideas; the question you need to ask yourselves is, are you willing to take the gamble?
Are we willing to risk a president who can't bring his good ideas into reality? What our country needs is a president who can turn words into action, and I am that president.
What I think would be effective about this (for me) is that, even if you're not totally convinced about the role of experience, the "gamble" argument trades on a widespread conviction that we can't afford anything like that last 4 years.
Here is my re-phrasing of Obama's response:
Obama: Hillary is absolutely right that she's made some important changes in her career. But the kind of changes we need can't be made by the strength of a single person. As Hillary should know, large-scale changes can only be done with broad coalitions.
I don't think Obama could elaborate in a debate about why he's better suited for this task. Here's why: it seems to me that building a broad coalition requires charisma, and it requires that people like and and trust you. I don't believe (though I'm willing to be convinced) that Hillary is widely liked or trusted. Therefore she's going to have a lot more trouble building coalitions.
The following isn't quite the exchange that took place between them, but I wanted to present my re-phrased version of what I thought were their strongest arguments.
Hillary's major argument has always been experience. I was never terribly moved by it before (because I think we've had, and can have, effective presidents without prior experience), but below is a version of it that gave me pause:
Hillary: All the democratic candidates want change, but we need someone who can deliver it. If you want change, look to what changes each candidate has already made. I have a 35 year record of it. I don't have to remind you of the importance of what the next president will need to do for the country and the world. Obama and Edwards are great people, with real passion and good ideas; the question you need to ask yourselves is, are you willing to take the gamble?
Are we willing to risk a president who can't bring his good ideas into reality? What our country needs is a president who can turn words into action, and I am that president.
What I think would be effective about this (for me) is that, even if you're not totally convinced about the role of experience, the "gamble" argument trades on a widespread conviction that we can't afford anything like that last 4 years.
Here is my re-phrasing of Obama's response:
Obama: Hillary is absolutely right that she's made some important changes in her career. But the kind of changes we need can't be made by the strength of a single person. As Hillary should know, large-scale changes can only be done with broad coalitions.
I don't think Obama could elaborate in a debate about why he's better suited for this task. Here's why: it seems to me that building a broad coalition requires charisma, and it requires that people like and and trust you. I don't believe (though I'm willing to be convinced) that Hillary is widely liked or trusted. Therefore she's going to have a lot more trouble building coalitions.
Review of four candidates on healthcare
Apparently, the Democratic plans are all similar because they have all taken counsel from MIT economist Jonathan Gruber, who is at the center of an group of analysts who agree that the best way forward is with minimal disturbance to current players (Working Californians, July 2007) Others (not including any of the Dem front-runners) prefer a single-payers system. (Huffington Post, October 2007)
Obama's Plan
(in this section, text not in [brackets] is taken directly from Obama campaign materials, found here)
Clinton's Plan
(in this section, text not in [brackets] is taken directly from Clinton campaign materials, found here)
Edwards's Plan
(in this section, text not in [brackets] is taken directly from Edwards campaign materials, found here)
Paul's Plan
(in this section, text not in [brackets] is taken directly from Paul campaign materials, found here)
Obama's Plan
(in this section, text not in [brackets] is taken directly from Obama campaign materials, found here)
- Guaranteed eligibility. No American will be turned away from any insurance plan because of illness or pre-existing conditions.
- Comprehensive benefits. The benefit package will be similar to that offered through Federal Employees Health Benefits Program (FEHBP), the plan members of Congress have. The plan will cover all essential medical services, including preventive, maternity and mental health care. [This plan will be available to all Americans.]
- Subsidies. Individuals and families who do not qualify for Medicaid or SCHIP but still need financial assistance will receive an income-related federal subsidy to buy into the new public plan or purchase a private health care plan.
- National Health Insurance Exchange: The Obama plan will create a National Health Insurance Exchange to help individuals who wish to purchase a private insurance plan. The Exchange will act as a watchdog group and help reform the private insurance market by creating rules and standards for participating insurance plans to ensure fairness and to make individual coverage more affordable and accessible. Insurers would have to issue every applicant a policy, and charge fair and stable premiums that will not depend upon health status. The Exchange will require that all the plans offered are at least as generous as the new public plan and have the same standards for quality and efficiency. The Exchange would evaluate plans and make the differences among the plans, including cost of services, public.
- Mandatory Coverage of Children: Obama will require that all children have health care coverage.
- Reducing Costs of Catastrophic Illnesses for Employers and Their Employees: Catastrophic health expenditures account for a high percentage of medical expenses for private insurers. The Obama plan would reimburse employer health plans for a portion of the catastrophic costs they incur above a threshold if they guarantee such savings are used to reduce the cost of workers' premiums.
- Lower Costs by Modernizing The U.S. Health Care System. [A variety of measures, including mandatory electronic recordkeeping, disease management programs, emphasis on prevention, and incentives for using proven best practices, are purported to reduce an average family's annual costs by "up to $2,500".]
- $55-60 billion annually, after fully phased in, to be paid for by efficiency improvements and repealing the Bush tax cuts for those making over $250K/year. (Obama Health FAQ)
- Does not mandate coverage. Obama's plan does not require that everyone have health insurance. Though insurers will be required to cover everyone, and the government will provide an unspecified level of subsidy for poor people not covered under Medicaid or SCHIP, economists estimate around 15,000,000 Americans will choose to remain without insurance. (The New Republic June 2007; Cohn blog post)
- Claims about efficiency gains seem contentious. Jonathan Cohn, who I trust knows more than I do about this, calls the proposal "detailed and well thought-out" in this area (The New Republic June 2007). But the Obama campaign's numbers for administrative overhead in healthcare are at the very highest end of what's out there; and John Mongan has convinced me that replacing medical recordkeeping systems is very hard to do (although I mostly remember arguments about doctor personalities, which may matter less in national policy than in private campaigns). I remain unconvinced that these predictions are realistic.
Clinton's Plan
(in this section, text not in [brackets] is taken directly from Clinton campaign materials, found here)
- The Same Choice of Health Plan Options that Members of Congress Receive: Americans can keep their existing coverage or access the same menu of quality private insurance options that their Members of Congress receive. In addition to the broad array of private options that Americans can choose from, they will be offered the choice of a public plan option similar to Medicare.
- Individuals: will be required to get and keep insurance in a system where insurance is affordable and accessible.
- Insurance and Drug Companies: insurance companies will end discrimination based on pre-existing conditions or expectations of illness and ensure high value for every premium dollar; while drug companies will offer fair prices and accurate information.
- Reducing Costs: By removing hidden taxes, stressing prevention and a focus on efficiency and modernization, the plan will improve quality and lower costs.
- Provide Tax Relief to Ensure Affordability: Working families will receive a refundable tax credit to help them afford high-quality health coverage.
- Limit Premium Payments to a Percentage of Income: The refundable tax credit will be designed to prevent premiums from exceeding a percentage of family income, while maintaining consumer price consciousness in choosing health plans.
- $110 billion, to be paid for by efficiency improvements and repealing the Bush tax cuts for those making over $250K/year. (Full, detailed breakdown in Clinton Health Plan)
- The only substantial criticism I found is that even with the mandate, some people -- perhaps ~1.5%, or 4.5 million -- will go uninsured, given realistic levels of subsidies. (Washington Post blog, November 2007)
Edwards's Plan
(in this section, text not in [brackets] is taken directly from Edwards campaign materials, found here)
- Requiring businesses and other employers to either cover their employees or help finance their health insurance.
- Making insurance affordable by creating new tax credits, expanding Medicaid and SCHIP, reforming insurance laws, and taking innovative steps to contain health care costs.
- Creating regional "Health Care Markets" to let every American share the bargaining power to purchase an affordable, high-quality health plan, increase choices among insurance plans, and cut costs for businesses offering insurance.
- Once these steps have been taken, requiring all American residents to get insurance.
- Edwards claims he will "use [his] power as president to take [Congress's] healthcare away" (TV ad, cited by factcheck.org) unless his plan is passed. This seems like an empty threat, since Congressional healthcare is provided for by legislation, and the only legal way to revoke it would be with more legislation. Written campaign materials make the weaker claim that he will "submit legislation" to achieve this end (which, presumably, would be voted down by Congress). And Edwards has defended this plan, saying "The most powerful tool that the president has is the bully pulpit. And that means making the case to America, submitting legislation to support exactly what I just said, and then making the case to America in any place--any congressional district or any state where a senator is opposing it--saying `your senator, your congressman is defending their health care at the same time that they're not providing health care for you.'" (Face the Nation, quoted in Mike Kuykendall's blog)
- Maybe I'm just tired, but I didn't see this on the webpage or in the detailed plan PDF. Elsewhere, "Campaign estimates cost to be $90-$120 billion a year. Would finance the plan by rolling back tax cuts for those earning more than $200,000 a year." (KFF)
- This appears to me to be esentially the same as Clinton's plan, though less specific (and with the addition of the Health Care Markets, and the Congressional bribe). So, again, the only criticism is that an individual mandate is less likely to provide truly universal care than a single-payer system would be.
- Announced in Feb 07, Edwards was the first of the candidates with identical plans. Obama announced in May, and Clinton May-Sep. (KFF)
Paul's Plan
(in this section, text not in [brackets] is taken directly from Paul campaign materials, found here)
- Making all medical expenses tax deductible.
- Eliminating federal regulations that discourage small businesses from providing coverage.
- Giving doctors the freedom to collectively negotiate with insurance companies and drive down the cost of medical care.
- Making every American eligible for a Health Savings Account (HSA), and removing the requirement that individuals must obtain a high-deductible insurance policy before opening an HSA.
- Reform licensure requirements so that pharmacists and nurses can perform some basic functions to increase access to care and lower costs.
- not provided.
- This plan seems unlikely to substantially alter the current situation.
Healthcare Background Info
I spent today researching health care in preparation for a discussion about the presidential candidates. Here are my findings.
Healthcare Spending
We're spending a lot, and the spending rate is growing.
- In 2005, the U.S. spent $2 trillion on health care, which is 16 percent of GDP and $6,697 per person. (KFF August 2007)
- Health care costs have grown on average 2.5 percentage points faster than U.S. gross domestic product since 1970. (KFF August 2007)
- Health care costs are also rising in other OECD countries, at somewhat slower rates (KFF January 2007)
Two other surprising figures.
- Almost half of health care spending is used to treat just 5 percent of the population. (KFF August 2007)
- In 2005, nearly half of US health expenditures were public (just under $3,000 per capita). US public health expenditures per capita (in terms of USD PPP) are among the top in the world (behind only Luxembourg and Norway) (OECD)
Where the money goes.
(from KFF)
A note: "Program Administration" here means marketing and billing by health insurance plans. Other administrative costs are included in other categories. One study on 1999 data estimated total administrative costs at $1059 per capita (or almost 31% of total expenditure that year), compared with $307 per capita (for 16.7% of total expenditure) in Canada. (NEJM August 2003) A response in the same journal says this gap is overestimated by 25%. (NEJM August 2003) Also, much of the gap appears to be attributable to Canada's public health insurance program; Canadian private insurers have slightly higher administrative costs than their American counterparts, but make up a much smaller share of the health market.
Potential reasons for increasing health costs.
Potential reasons which are clearly not large factors (at least not directly).
The Uninsured
Total US population: 301,000,000 (CIA World Factbook July 2007)
Uninsured population: 46,500,000 (Kaiser Commission on Medicaid and the Uninsured. "The Uninsured and Their Access to Health Care," Oct. 2007)
Uninsured as proportion of total population: 15.4%
Uninsured as proportion of nonelderly population: 18%
Estimated undocumented immigrant population: 11,500,000 (Pew Hispanic Center Factsheet April 26, 2006)
Estimated uninsured undocumented population: 7,800,000 (inference from above and below)
Estimated uninsured as proportion of undocumented population: 68% (RAND Corporation November 2005)
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A Problem with Individual Health Insurance Markets
From my somewhat dusty memory of a lecture by Peter van Doren; similar points are made in an article by Paul Krugman. The problem stems from the fact that, unlike in many other insurance markets, many of the costs being insured against in health insurance are known in advance.
Imagine that there is only one health insurance company in the US, offering full-coverage plans to everyone at the average annual health costs per capita (plus a little extra for overhead). These plans cost *a lot* more than the healthiest Americans expect to get back in terms of benefits, so most of the healthiest don't participate. The very sick, on the other hand, are eagerly joining up. Now, the company is charging average rates and providing care for a group with higher-than-average costs.
The company can increase rates, but the same thing will happen -- the healthiest of the old participants now prefer to take their chances in exchange for much lower average costs, while the sickest are eager to participate. This cycle will not end unless the insurance company is allowed to exclude the sickest from buying its coverage. Insurers have no incentive to insure the sick, unless the sick are included in a large, varied population as part of a group plan (in which case their higher costs are offset by healthy participants with lower costs).
General Resources
Kaiser Family Foundation (KFF)
OECD Health data
RAND health
A collection of links to Paul Krugman's NYT articles on health
An econ professor draws and discusses various charts about national health spending.
Healthcare Spending
We're spending a lot, and the spending rate is growing.
- In 2005, the U.S. spent $2 trillion on health care, which is 16 percent of GDP and $6,697 per person. (KFF August 2007)
- Health care costs have grown on average 2.5 percentage points faster than U.S. gross domestic product since 1970. (KFF August 2007)
- Health care costs are also rising in other OECD countries, at somewhat slower rates (KFF January 2007)
Two other surprising figures.
- Almost half of health care spending is used to treat just 5 percent of the population. (KFF August 2007)
- In 2005, nearly half of US health expenditures were public (just under $3,000 per capita). US public health expenditures per capita (in terms of USD PPP) are among the top in the world (behind only Luxembourg and Norway) (OECD)
Where the money goes.
(from KFF)
A note: "Program Administration" here means marketing and billing by health insurance plans. Other administrative costs are included in other categories. One study on 1999 data estimated total administrative costs at $1059 per capita (or almost 31% of total expenditure that year), compared with $307 per capita (for 16.7% of total expenditure) in Canada. (NEJM August 2003) A response in the same journal says this gap is overestimated by 25%. (NEJM August 2003) Also, much of the gap appears to be attributable to Canada's public health insurance program; Canadian private insurers have slightly higher administrative costs than their American counterparts, but make up a much smaller share of the health market.
- increases in expertise and technology have made more (and more expensive) care possible (KFF August 2007)
- US population is aging (KFF August 2007)
- US population's health is worsening (Michael Pollan claimed in a recent interview that our health care spending has been inversely correlated with our spending on food during the last half century)
- improving insurance coverage encourages higher rates of care access (KFF August 2007)
Potential reasons which are clearly not large factors (at least not directly).
- medical malpractice suits: Total costs of defense, awards, and settlements is less than 1% of total health expenditure in US. [This figure does not include malpractice insurance costs, which appear to total $2 billion, less than .5% of total health expenditure (Healthcare Financial Management 2002 )]. This is not far out of step with other countries' malpractice costs. (Health Affairs 2005)
- greater basic access, no queues: The US has less healthcare availability (in terms of hospital beds, doctors and nurses, and MRI and CT scanners per capita) than the OECD median. Procedures for which some countries have queues (i.e. some elective surgeries) account for only 3% of US expenditures. (Health Affairs 2005)
The Uninsured
Total US population: 301,000,000 (CIA World Factbook July 2007)
Uninsured population: 46,500,000 (Kaiser Commission on Medicaid and the Uninsured. "The Uninsured and Their Access to Health Care," Oct. 2007)
Uninsured as proportion of total population: 15.4%
Uninsured as proportion of nonelderly population: 18%
Estimated undocumented immigrant population: 11,500,000 (Pew Hispanic Center Factsheet April 26, 2006)
Estimated uninsured undocumented population: 7,800,000 (inference from above and below)
Estimated uninsured as proportion of undocumented population: 68% (RAND Corporation November 2005)
A Problem with Individual Health Insurance Markets
From my somewhat dusty memory of a lecture by Peter van Doren; similar points are made in an article by Paul Krugman. The problem stems from the fact that, unlike in many other insurance markets, many of the costs being insured against in health insurance are known in advance.
Imagine that there is only one health insurance company in the US, offering full-coverage plans to everyone at the average annual health costs per capita (plus a little extra for overhead). These plans cost *a lot* more than the healthiest Americans expect to get back in terms of benefits, so most of the healthiest don't participate. The very sick, on the other hand, are eagerly joining up. Now, the company is charging average rates and providing care for a group with higher-than-average costs.
The company can increase rates, but the same thing will happen -- the healthiest of the old participants now prefer to take their chances in exchange for much lower average costs, while the sickest are eager to participate. This cycle will not end unless the insurance company is allowed to exclude the sickest from buying its coverage. Insurers have no incentive to insure the sick, unless the sick are included in a large, varied population as part of a group plan (in which case their higher costs are offset by healthy participants with lower costs).
General Resources
Kaiser Family Foundation (KFF)
OECD Health data
RAND health
A collection of links to Paul Krugman's NYT articles on health
An econ professor draws and discusses various charts about national health spending.
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