# Discrete mathematics for idiots?

January 31, 2012 7:17 AM Subscribe

Now that I'm back at college, I'm having serious trouble with an elementary course in Discrete Mathematics (for computer scientists). How can I complete this course?

Although I've always been very interested in math, this has unfortunately never been reflected in my ability to do math. Add the fact that it's been a couple of years since I've been to college or studied math, and you got the main reasons why I, halfway through the course, am failing hard. I've fail another discrete mathematics course that I took many years ago for pretty much the same reasons.

During lectures I have a hard time understanding what the teacher is saying because it's all gibberish to me and even note-taking is difficult because the pacing is too fast. So basically lectures are almost worthless to me (I've skipped the latest one because they've become so frustrating). Occasionally the teacher will refer to something as high school math but since it's unknown to be, I assume that I've never learned it or just forgotten it (it's been a decade since high school).

I find the textbooks non-pedagogical. Although I clearly have issues with math, most of them are incredibly abstract and non-engaging that mostly focus on theorems and their proofs. I know this is pretty standard for college math, but it makes things extra difficult for me. I lack the ability to see between the gap, i.e. between where the textbook explains a concept and presents a problem to be solved using reasoning based on the concept. To me the gap is an abyss and I can't see the connections between concept and problem-solving.

I'm honestly at a lost here. I get the feeling that none of this should be

Course Topics:

* set theory

* combinatorics

* Relations, functions

* Recursion, induction

* Graphs, trees, graph optimization, traversing/searching graphs

* boolean algebra

Although I've always been very interested in math, this has unfortunately never been reflected in my ability to do math. Add the fact that it's been a couple of years since I've been to college or studied math, and you got the main reasons why I, halfway through the course, am failing hard. I've fail another discrete mathematics course that I took many years ago for pretty much the same reasons.

During lectures I have a hard time understanding what the teacher is saying because it's all gibberish to me and even note-taking is difficult because the pacing is too fast. So basically lectures are almost worthless to me (I've skipped the latest one because they've become so frustrating). Occasionally the teacher will refer to something as high school math but since it's unknown to be, I assume that I've never learned it or just forgotten it (it's been a decade since high school).

I find the textbooks non-pedagogical. Although I clearly have issues with math, most of them are incredibly abstract and non-engaging that mostly focus on theorems and their proofs. I know this is pretty standard for college math, but it makes things extra difficult for me. I lack the ability to see between the gap, i.e. between where the textbook explains a concept and presents a problem to be solved using reasoning based on the concept. To me the gap is an abyss and I can't see the connections between concept and problem-solving.

I'm honestly at a lost here. I get the feeling that none of this should be

*this*difficult, but I have no idea on how to fix the situation. Ideally, I would like a Discrete Mathematics for Dummies type of resource that doesn't require strong math skills and that's very pedagogical.

Course Topics:

* set theory

* combinatorics

* Relations, functions

* Recursion, induction

* Graphs, trees, graph optimization, traversing/searching graphs

* boolean algebra

I've taught this exact course many times. What book are you using? I usually teach out of Ken Rosen's book, which students find reasonably down-to-earth.

The nature of this course is that there is a WIDE range of difficulty between the topics. Set theory and combinatorics are much easier than mathematical induction. It might help you to really hammer on the easiest parts, so that you have some foundation for the rest of the material.

posted by escabeche at 7:29 AM on January 31, 2012

The nature of this course is that there is a WIDE range of difficulty between the topics. Set theory and combinatorics are much easier than mathematical induction. It might help you to really hammer on the easiest parts, so that you have some foundation for the rest of the material.

posted by escabeche at 7:29 AM on January 31, 2012

One thing that may help is to understand how definitions work in mathematics. In mathematics, when we define a thing (be it "vertical asymptote" or "disjoint" or "relation" or whatever), we lay out the criteria that have to be met in order for the object under consideration to be called that thing. They are

There is one definition of relation. A relation is a set of ordered pairs. End of the story. If the object under consideration is not a set of ordered pairs, then it is not a relation. If it is, then it is.

A function is a relation where if an element appears as the first entry in an ordered pair, then it appears exactly once. For example, {(1,2), (a, hog), (george, jefferson)} is a function. The relation {(1,2), (a, hog), (george, jefferson), (george, foreman)} is not, since "george" appears twice as the first entry in an ordered pair.

Now, I'm not saying there

Combinatorics is about counting things, usually in clever ways. How many 5-card hands are there from a standard deck of cards that contain exactly 2 jacks? How would you go about counting that? Choose the jacks first, or choose the other 3 cards first? It's up to you.

To hopefully give an answer to your question, I used the book "Discrete Mathematics with Graph Theory" by Goodaire and Parmenter when I was an undergraduate. It was a useful text, it may help you here.

The last thing to remember is that you are now beyond the methods of arcane symbol manipulation (which is what "college algebra" and its kin are all about). Most of the proofs you will have to do will follow almost exclusively from definitions and well-established theorems. When in doubt, just write down what everything means in words. For example, the exercise:

posted by King Bee at 8:00 AM on January 31, 2012

*not*like definitions of words, where there are many ways of thinking about what the word "subtle" means, and any number of definitions (not just the one in Merriam-Webster) might be acceptable.There is one definition of relation. A relation is a set of ordered pairs. End of the story. If the object under consideration is not a set of ordered pairs, then it is not a relation. If it is, then it is.

A function is a relation where if an element appears as the first entry in an ordered pair, then it appears exactly once. For example, {(1,2), (a, hog), (george, jefferson)} is a function. The relation {(1,2), (a, hog), (george, jefferson), (george, foreman)} is not, since "george" appears twice as the first entry in an ordered pair.

Now, I'm not saying there

*aren't*different ways of thinking about what a function is, or that some ways are better suited to certain situations. However, for your purposes, that's the definition. If someone hands you a relation and asks you if it is a function, then there is one thing to check.Combinatorics is about counting things, usually in clever ways. How many 5-card hands are there from a standard deck of cards that contain exactly 2 jacks? How would you go about counting that? Choose the jacks first, or choose the other 3 cards first? It's up to you.

To hopefully give an answer to your question, I used the book "Discrete Mathematics with Graph Theory" by Goodaire and Parmenter when I was an undergraduate. It was a useful text, it may help you here.

The last thing to remember is that you are now beyond the methods of arcane symbol manipulation (which is what "college algebra" and its kin are all about). Most of the proofs you will have to do will follow almost exclusively from definitions and well-established theorems. When in doubt, just write down what everything means in words. For example, the exercise:

For setsThis is an obvious statement to me, but maybe not to you; this is probably because you may not understand exactly what all those symbols mean. Write down what each of those symbol phrases means in terms of the definitions. You will begin to get better!AandB, prove thatA∩B=Aif and only ifA⊆B.

posted by King Bee at 8:00 AM on January 31, 2012

Before you go out to buy another book that's probably not going to help you much, take all the help you can get from people associated with the course. This may be the professor or TAs or other students. Does the professor have office hours? Go to them. Is there a TA-run help-room associated with the course? Go to it. Is there a place where other students work on problem sets for the course and/or study? Go there.

I am always shocked* at how students who are having trouble feel like they have to go at it alone. One of the best ways to learn something is to work through the material with someone else. When working with other students, you have to be careful that you're not a freeloader (for your own sake even more than everyone else), but if you're contributing and working hard to understand the material, collaborative learning experiences can be incredibly beneficial.

And

If you still decide you want a book, I suspect the Professor/TA can recommend a good one. They have lots of experience with this.

Good luck!

* I say I'm "shocked," but not really. As an undergraduate, it took me 2 years to learn that it was a good idea to work with other people and ask one-on-one questions to TAs and faculty. Unsurprisingly, my transition to doing this was correlated with a significant uptick in my grades.

posted by Betelgeuse at 8:03 AM on January 31, 2012 [4 favorites]

I am always shocked* at how students who are having trouble feel like they have to go at it alone. One of the best ways to learn something is to work through the material with someone else. When working with other students, you have to be careful that you're not a freeloader (for your own sake even more than everyone else), but if you're contributing and working hard to understand the material, collaborative learning experiences can be incredibly beneficial.

And

*please*: take advantage of any chances to talk to the Professor/TA one-on-one for this. As long as you're going in prepared to ask questions about the material and aren't there just to troll for answers, the Professor will be glad to see you at office hours.If you still decide you want a book, I suspect the Professor/TA can recommend a good one. They have lots of experience with this.

Good luck!

* I say I'm "shocked," but not really. As an undergraduate, it took me 2 years to learn that it was a good idea to work with other people and ask one-on-one questions to TAs and faculty. Unsurprisingly, my transition to doing this was correlated with a significant uptick in my grades.

posted by Betelgeuse at 8:03 AM on January 31, 2012 [4 favorites]

First, I would like to stress that Betelgeuse's answer above is very important:

In addition, I would recommend asking questions on math.stackexchange. There are a lot of helpful people there.

On the other hand, I don't recommend trying to get a book and learning from that. Realistically speaking, that isn't a viable way of getting through a challenging math course at the undergrad level, and I would strongly recommend against that.

posted by Frobenius Twist at 8:10 AM on January 31, 2012

**go talk to your professor and ask questions**, no matter how simple or basic you think they may be. That's what his/her office hours are for. Speaking as an academic in math, one of the biggest mistakes I see my students making is that they think they're supposed to do it all themselves. Your professor will be happy to help out. This is your single biggest resource.In addition, I would recommend asking questions on math.stackexchange. There are a lot of helpful people there.

On the other hand, I don't recommend trying to get a book and learning from that. Realistically speaking, that isn't a viable way of getting through a challenging math course at the undergrad level, and I would strongly recommend against that.

posted by Frobenius Twist at 8:10 AM on January 31, 2012

you've got to make contact with the teacher immediately. tell him what you're saying here. you're not going to be able to wing it on your own.

posted by facetious at 8:10 AM on January 31, 2012

posted by facetious at 8:10 AM on January 31, 2012

This book is pretty good for this kind of stuff. Especially learning different methods of proof and how to analyze a proof/theorem to help you prove other things (which, if I'm understanding you correctly, is the "gap" you're talking about). It assumes you know nothing about logic and mathematical proof. I think you're finding it abstract because the material

Ask the TA for help for sure, but, better than that, see if you can join a study group. Talking about how stuff works really, really, really helps you figure out all the ins and outs. I've never been able to learn math strictly from textbooks without discussing the problems with colleagues (I say this as someone who took a lot of graduate level math).

posted by bluefly at 8:11 AM on January 31, 2012

**is**abstract.Ask the TA for help for sure, but, better than that, see if you can join a study group. Talking about how stuff works really, really, really helps you figure out all the ins and outs. I've never been able to learn math strictly from textbooks without discussing the problems with colleagues (I say this as someone who took a lot of graduate level math).

posted by bluefly at 8:11 AM on January 31, 2012

Sorry, I should have mentioned that I'm taking this course in Sweden. The required reading are two books in Swedish.

Regarding course resources, there are no TAs nor a study center where I can go to get help with problems. I know that another student in the class has asked if she could come by his office for some Q&A time but no luck here. All in all, he doesn't seem very approachable and kinda makes you feel dumber than you already feel.

I've been looking at some khan academy lectures and although they seem good, there are some topics missing and there's no connection between the various videos. But I'll keep looking, thanks for the reminder.

posted by Foci for Analysis at 8:15 AM on January 31, 2012

Regarding course resources, there are no TAs nor a study center where I can go to get help with problems. I know that another student in the class has asked if she could come by his office for some Q&A time but no luck here. All in all, he doesn't seem very approachable and kinda makes you feel dumber than you already feel.

I've been looking at some khan academy lectures and although they seem good, there are some topics missing and there's no connection between the various videos. But I'll keep looking, thanks for the reminder.

posted by Foci for Analysis at 8:15 AM on January 31, 2012

*I know that another student in the class has asked if she could come by his office for some Q&A time but no luck here.*

I'm sorry, are you saying that the professor doesn't have specified office hours when he's available for questions and discussion of course material? Is that the norm in Sweden? Check the syllabus to make sure.

*kinda makes you feel dumber than you already feel*

Yeah, that's how asking questions makes you feel. It's unpleasant. But there's no other way. You gotta go through the feeling dumber to get to the feeling smarter.

posted by escabeche at 8:22 AM on January 31, 2012 [3 favorites]

With your added information, I'm not sure the book I recommended would be good -- if you truly are learning on your own, it's not that helpful (it supplements a class but is not intended for solo use). I would try speaking to other students in the class and see if anyone else is struggling and try to study together.

My own course used an edition of this book which was chosen over other popular texts in this subject for its "easy to readness." The Rosen book is also pretty popular for this material.

posted by bluefly at 8:23 AM on January 31, 2012

My own course used an edition of this book which was chosen over other popular texts in this subject for its "easy to readness." The Rosen book is also pretty popular for this material.

posted by bluefly at 8:23 AM on January 31, 2012

Came in to say what Betelgeuse said. I'm sure you know this but: don't just ask for answers, but come in with specific questions about how you're approaching problems and you're more likely to get helpful guidance. Ask about questions you got wrong on assignments and tests as well; I was always tempted to just pretend exams were over and done with after I took them.

posted by dismas at 8:28 AM on January 31, 2012

**On preview**: If your professor's not helpful, that's a problem, but it sounds like other students also realize that. I'd try to set up a study group with them. Even if you feel like you're far behind, they probably won't mind having you around if you make an effort and try to contribute. I realize that if you feel overwhelmed by the topics that figuring out 'how to contribute' is difficult - come to study sessions having made an honest attempt at the problems and being willing to talk about things you tried, even if they got you nowhere.posted by dismas at 8:28 AM on January 31, 2012

*I'm sorry, are you saying that the professor doesn't have specified office hours when he's available for questions and discussion of course material?*

Yes. Professors don't have specified office hours here. I guess it's assumed that they should just be available to you but that's not really the case here.

There's another, equally confused, student that I might form a study group.

posted by Foci for Analysis at 8:36 AM on January 31, 2012

I've only been a professor in the US, so I don't know what the norm is in Sweden, but I would assume that if you approach your professor after class and mention that you're having trouble, you should be able to make an appointment. Study groups are great, and I would highly recommend them, but they are no replacement for going to the source.

As for feeling dumb, it may or may not make you feel any better to know that

posted by Frobenius Twist at 8:47 AM on January 31, 2012 [2 favorites]

As for feeling dumb, it may or may not make you feel any better to know that

**never ends**in math. I've been a professional mathematician -- postdoc, visiting professor, grad student, etc -- for over a decade, and math makes me feel stupid all the time. I'd describe mathematics as a cycle with two phases: feeling stupid, feeling less stupid, and then back to feeling stupid again.posted by Frobenius Twist at 8:47 AM on January 31, 2012 [2 favorites]

One other trick (and this depends on your institution's resources and setup): Is there a location where most people tend to study? (Like, is there a math or computer science commons, or a particular library or lab or something along those lines?) Try to work there if it's not too distracting. I found as a math student in undergrad that the cool math kids all did their work there and informally discussed problems with anyone who happened to be around. Again, YMMV.

posted by dismas at 8:47 AM on January 31, 2012

posted by dismas at 8:47 AM on January 31, 2012

Thanks, bluefly, for the recommendation. Should be able to pick up

Regarding making an appointment with the professor, I would need to get my confusion in order first because right now I'm not even sure what I'm confused about. ;)

posted by Foci for Analysis at 8:57 AM on January 31, 2012

*Discrete ... Applications*tomorrow.Regarding making an appointment with the professor, I would need to get my confusion in order first because right now I'm not even sure what I'm confused about. ;)

posted by Foci for Analysis at 8:57 AM on January 31, 2012

This is a long shot, but do you have any friends or acquaintances who are a class or two above? I'm informally tutoring someone in a stats class below me because it gives me a chance to thoroughly review prior concepts and solidify my understanding of them, while helping a friend digest the material their first time around.

posted by smirkette at 9:01 AM on January 31, 2012

posted by smirkette at 9:01 AM on January 31, 2012

Unfortunately no, smirkette.

posted by Foci for Analysis at 9:07 AM on January 31, 2012

posted by Foci for Analysis at 9:07 AM on January 31, 2012

I think going in with specific questions is

The inaccessible professor is unfortunate. I think forming a "study group" with the other student is a great idea. If there's a class email list, sending out a general invite might encourage some more people to come out; if the professor is as opaque and inaccessible, I can guarantee you that there are lots of others that need the help.

Going in a small group to his office (no more than 2 or 3) might be a good strategy. The Prof. can feel like he's being "more efficient" and it might drive home the fact that there's more than one student that's having trouble with the class.

posted by Betelgeuse at 9:09 AM on January 31, 2012

*great*, but I think it's totally fine to go in without specific questions, as long as it's clear that you've been trying to grapple with the material. The thing that we (as Professors) really dislike is when students come in just trying to get the "right" answers on problem sets without any understanding of the material. We understand that students can get in over their heads -- we've been through at least 8 years of post-HS schooling, so chances are that's happened to us a lot more than it's happened to you.The inaccessible professor is unfortunate. I think forming a "study group" with the other student is a great idea. If there's a class email list, sending out a general invite might encourage some more people to come out; if the professor is as opaque and inaccessible, I can guarantee you that there are lots of others that need the help.

Going in a small group to his office (no more than 2 or 3) might be a good strategy. The Prof. can feel like he's being "more efficient" and it might drive home the fact that there's more than one student that's having trouble with the class.

posted by Betelgeuse at 9:09 AM on January 31, 2012

Take a look at the Schaum's Outline Discrete Mathematics book, which is likely to be example-heavy and exercise-based. I found Schaum's books much more accessible than standard university textbooks when I was struggling my way through the second year of my maths degree.

The Amazon "Look Inside" preview (at that link) includes the full contents pages and enough of the text to give you some idea of the writing style - hopefully enough to let you judge whether it'll work for you.

posted by ManyLeggedCreature at 9:12 AM on January 31, 2012

The Amazon "Look Inside" preview (at that link) includes the full contents pages and enough of the text to give you some idea of the writing style - hopefully enough to let you judge whether it'll work for you.

posted by ManyLeggedCreature at 9:12 AM on January 31, 2012

As for this:

I get this as a professor

I loved Froebenius Twist's comment above. I've been studying mathematics now for 12 years, and even with a PhD and all that jazz, we still get stumped all the time. We know what it feels like to be clueless, and we don't want to make you feel worse about it. That feeling is actually essential to the learning process.

If I were you, I would just show up to the professor's office and ask if you can chat about some problems you're having with the course. The worst thing that can conceivably happen is that he'll send you away.

posted by King Bee at 9:19 AM on January 31, 2012 [1 favorite]

*All in all, he doesn't seem very approachable and kinda makes you feel dumber than you already feel.*I get this as a professor

**all the time**. If you come to my office to ask me questions about the course I'm teaching, I will do my best to help you. I am not out to make you feel dumb. Virtually no professor is. Those who get off on making their undergraduate students feel dumb are actually few and far between, although the perception from a lot of students is that its probably closer to 50%.I loved Froebenius Twist's comment above. I've been studying mathematics now for 12 years, and even with a PhD and all that jazz, we still get stumped all the time. We know what it feels like to be clueless, and we don't want to make you feel worse about it. That feeling is actually essential to the learning process.

If I were you, I would just show up to the professor's office and ask if you can chat about some problems you're having with the course. The worst thing that can conceivably happen is that he'll send you away.

posted by King Bee at 9:19 AM on January 31, 2012 [1 favorite]

I'm taking discrete math right now

posted by spikeleemajortomdickandharryconnickjrmints at 9:25 AM on January 31, 2012 [1 favorite]

*for fun*. Really. MeMail me if you want to try a virtual study group.posted by spikeleemajortomdickandharryconnickjrmints at 9:25 AM on January 31, 2012 [1 favorite]

OK, now I understand the office hours situation better. It's a sad fact that some professors are just not that interested in talking with confused students (which is why the US system of having formal times laid out for this purpose is a good one, in my view!) But: does your institution have graduate students? They are much more approachable, much more interested in developing the craft of teaching, and they really need money. To the point that it does not take a lot of money to capture your attention. Ask at the math department office whether there are any graduate students around who tutor for cash. Here at UW that would give you a list of about thirty names.....

posted by escabeche at 9:32 AM on January 31, 2012

posted by escabeche at 9:32 AM on January 31, 2012

spikeleemajortomdickandharryconnickjrmints, thanks, I'll keep that in mind!

escabeche, Haven't thought about paying someone to be my tutor but it's a solution that I'll keep in mind. A couple of thousand bucks would actually be worth getting a decent grade.

posted by Foci for Analysis at 9:45 AM on January 31, 2012

escabeche, Haven't thought about paying someone to be my tutor but it's a solution that I'll keep in mind. A couple of thousand bucks would actually be worth getting a decent grade.

posted by Foci for Analysis at 9:45 AM on January 31, 2012

I just did a math course that was well beyond my abilities, and what got me through it was a really great TA. I would go through the homework and do my best to figure out how to at least approach each problem. The first few times we got together I basically needed to be guided through how to solve everything - by the end of the class I had a much better grasp and was able to do ~60% of the problems on my own.

The problem was that I didn't even understand simple things, things that textbooks and professors take for granted to the degree that they don't even know how to explain them. The only way to figure that stuff out is to sit with someone and tell them you don't understand, and then have them explain it to you in a few different ways until something clicks. Don't feel stupid - it's not your fault you don't know this stuff - but it will take time and work to get those basic concepts.

In north america, there is a strong sense that you deserve to have access to help either from the professor or a TA. But the reality is that they might not have time to give you what you need. Asking for tutoring from a grad math student is a good idea. You might ask the professor if they have a recommendation. Look for someone who is both smart and friendly - having a sympathetic TA helped me to loosen up a little, which also helped my ability to learn the material.

posted by ianhattwick at 9:46 AM on January 31, 2012 [1 favorite]

The problem was that I didn't even understand simple things, things that textbooks and professors take for granted to the degree that they don't even know how to explain them. The only way to figure that stuff out is to sit with someone and tell them you don't understand, and then have them explain it to you in a few different ways until something clicks. Don't feel stupid - it's not your fault you don't know this stuff - but it will take time and work to get those basic concepts.

In north america, there is a strong sense that you deserve to have access to help either from the professor or a TA. But the reality is that they might not have time to give you what you need. Asking for tutoring from a grad math student is a good idea. You might ask the professor if they have a recommendation. Look for someone who is both smart and friendly - having a sympathetic TA helped me to loosen up a little, which also helped my ability to learn the material.

posted by ianhattwick at 9:46 AM on January 31, 2012 [1 favorite]

Often grad students in a department will offer tutoring. I did that for two semesters of calculus-based physics - I think I paid ~$30/hour for an hour a week, but this was 10 years ago now. You might need more time as well, but it may not be as expensive as you think it will be.

posted by clerestory at 11:04 AM on January 31, 2012

posted by clerestory at 11:04 AM on January 31, 2012

You may find some success with googling some combination of "class notes," "discrete math," "combinatorics," "[insert other class topic here] filetype:ppt, and/or site:.edu.

Your campus library is another resource with a whole lot of books about math. Is this a "proof-based" course? If so, you might want to track down resources on proof writing too.

If you can't follow what the professor is doing, at least try to write down terms, proofs, and major conceptual ideas mentioned in class. Then use a variety of sources to reconstruct the material covered in class for your own study guide. (This is most helpful if the professor actually tests what he "taught.")

The great thing about study groups is that you can bounce ideas off each other. Explaining a concept to someone else is really how you understand a concept.

posted by oceano at 7:46 PM on January 31, 2012

Your campus library is another resource with a whole lot of books about math. Is this a "proof-based" course? If so, you might want to track down resources on proof writing too.

If you can't follow what the professor is doing, at least try to write down terms, proofs, and major conceptual ideas mentioned in class. Then use a variety of sources to reconstruct the material covered in class for your own study guide. (This is most helpful if the professor actually tests what he "taught.")

The great thing about study groups is that you can bounce ideas off each other. Explaining a concept to someone else is really how you understand a concept.

posted by oceano at 7:46 PM on January 31, 2012

I loved discrete and would also be glad to help if you have questions!

posted by joshuaconner at 9:39 PM on January 31, 2012

posted by joshuaconner at 9:39 PM on January 31, 2012

I just want to say hang in there, keep going to the lectures. My experience with discrete maths was that various topics were quite self-contained, and the start of each topic was a fresh start.

I also found a lot of the theories were quite easy to grasp once you took them out of their formal pyjamas.

For both these reasons, I found this subject more than others rewards group study.

posted by spongeboy at 3:26 AM on February 1, 2012

I also found a lot of the theories were quite easy to grasp once you took them out of their formal pyjamas.

For both these reasons, I found this subject more than others rewards group study.

posted by spongeboy at 3:26 AM on February 1, 2012

Thanks joshuaconner, that's very kind of you!

posted by Foci for Analysis at 11:25 AM on February 2, 2012

posted by Foci for Analysis at 11:25 AM on February 2, 2012

This thread is closed to new comments.

posted by thelonius at 7:22 AM on January 31, 2012