«v. 1.8, 1/09, K. S. Cline Contents Anatomy of a Paper................................................ ...»
v. 1.8, 1/09, K. S. Cline
Anatomy of a Paper................................................................... 3
A Skimmable Paper................................................................... 7
The Team............................................................................ 10 A Timetable......................................................................... 14 The Time Trap...................................................................... 20 Useful References.................................................................... 22 Introduction The following is my own personal strategy of how to attack the MCM. It is certainly not the only way, and I make no claims that it is the best way. Participating in the MCM was the most fun, challenging, exciting, frustrating, and exhausting thing I ever did as an undergraduate.
I participated three years: The ﬁrst time, I was a barely out of Calculus, and the MCM hit me like a freight train. I was totally unprepared, utterly blindsided, and my team ended up with a Successful Participant ranking. I spent hours analyzing what went wrong, and plotting my revenge. The second year, we did better, but I still didn’t have a handle on things, and we didn’t know where to attack, ending up with an Honorable Mention. By the third year (’98), I was out for blood. I went over the previous years problems, studying the anatomy of a good paper, relentlessly preparing for the intellectual battle of a lifetime. This time my team clicked, and we annihilated the problem – receiving an Outstanding, the highest possible ranking. In the years since I graduated, I have coached teams both at the University of Colorado at Boulder 2000 - 2003, and at Carroll College 2004 - present, so I have seen a lot of teams succeed and others that didn’t do as well as they could. The advice that follows comes from my own personal experience both in the contest and as a coach. This is how I would attack the MCM today.
Kelly S. Cline 1 Anatomy of a Paper
1.1 Summary This is without a doubt THE most important part of the paper. The diﬀerence between an honorable mention and a successful participant is that the judges probably never read much more than the summary of an SP paper. The summary should be written LAST. Let me say that again: You should not write the summary until the rest of the paper is done. In an ideal timetable, all of Monday should be set aside for writing the summary.
The summary must brieﬂy hit all the main points and ideas of your paper. If you did
anything creative, it must be here. Further you should put numerical results in the summary:
“Our ﬁnal algorithm performed 67.5% better than a simple greedy algorithm, and 123.3% better than a random choice”.
Ideal summary length is hard. You must include all your main ideas in the summary, but brevity is VERY important. I would try to make the summary around half a page, deﬁnitely not more than 2/3.
The summary (and ideally the whole paper) should be written collaboratively, as a team.
Here’s my best advice on how to do this: Break up in to diﬀerent rooms and have each person sit down individually and write the best summary that they can. Set aside plenty of time for this, maybe an hour. Then, come back together and read the summaries out loud to each other (out loud is very important). After discussing them, set them aside and as a team write a new summary together up on a blackboard.
1.2 Introduction In the introduction, you want to restate the problem as you interpret it. Within many MCM problems, almost every team will ﬁnd a diﬀerent ‘problem’ to work on. After the contest if you read the papers from other teams, you will be amazed by how diﬀerently they approach the problem. Often they won’t even be working on the same thing!
So what you need to do in the introduction is to clearly explain how you interpreted the problem, and what you decided to work on. Writing the introduction and having the whole team read it can be a very good way to make sure that everyone agrees on what the problem is, and what needs to be done. The introduction is also a place to give a little more background on the problem, and show what you learned while researching it. Remember, the people reading your paper are math professors – they’ll get oﬀended if you don’t show that you understand the traditional textbook approach to your problem. Whether you chose to use a textbook method or something more creative, always mention the traditional methods in the introduction, so they know you did your homework.
The introduction can usually be written ﬁrst as a Friday project. It can help make sure that all the team members are in synch about what they are really working on.
1.3 The Model The purpose of a mathematical model is to predict how some real world system will behave, and to help you to understand it. The ﬁrst big section of your paper should be to describe your mathematical models. Most problems can be broken down into three parts: the models, the solutions, and comparison methods. You’re given some type of goal, and you’re asked to develop a method of achieving this goal: Finding a submarine, catching a prey dinosaur, get customers through an amusement park as quickly as possible, evacuating people away from a hurricane. The purpose of a mathematical model is prediction. Its purpose is to predict what will happen if you do diﬀerent things. For the submarine problem, you would describe the methods of modeling waves traveling through water. For the velociraptor problem, this models dinosaurs stalking and chasing each other. For the MRI problem, this part of the paper would describe how you created simulated data representing biological tissue. For the Hurricane Evacuation program, your model would predict how long the evacuation would take.
Good papers generally contain a series of models, starting very simple and working toward more complex and realistic models. You should always try to ﬁgure out a way to create a ﬁrst model so simple that you can solve it on paper. Generally, more complex modeling will occur on the computer, so the challenge is to translate the computer work into words, and justify each step. In order to create these models for continuous problems, I would recommend having a clear understanding of how to solve diﬀerential equations, however many continuous problems have not involved this. For the animal population problems, you would want to be able to write diﬀerential equations describing the relationship between the predator and prey populations, then numerically integrate them. Know what a partial diﬀerential equation is – know the wave equation and the diﬀusion equation and what they mean. With this knowledge in hand, the concrete slab problem would have looked almost like an old friend!
Remember, this is the mathematical MODELING competition, so do not gloss over this section. It may be simple: For the grade inﬂation problem, this section might simply involve simulating the actual grades for a class, perhaps using a random distribution, then using some method to skew them upwards due to inﬂation. In general, for the discrete problem, you want to be familiar with how to generate random number sets with diﬀerent properties – this can be very useful in constructing the sets to test your solution methods with. The computer person should be setting up these models Friday morning, and so this section should be roughed out on Friday or Saturday.
1.4 The Solutions (PLURAL!) The second BIG section of the problem. Here, we describe our methods of trying to achieve the goal: We will attempt to protect our stunt person by cushioning their fall with four cardboard boxes. We will attempt to undo the eﬀects of grade inﬂation by ranking the diﬃculty of diﬀerent classes, and giving diﬃcult classes a greater weight in GPA calculations. This is the section that actually describes how we solve the problem. In the submarine problem, this is our algorithm that takes the simulated data from waves traveling through water, and uses this data to guess at the position of the submarine. In the velociraptor problem, this is our algorithm which mathematically states how the raptor tries to catch the thescelosaur, and how the thescelosaur tries to get away.
(Occasionally a problem will explicitly give you the solutions they want you to test. In the Escaping a Hurricane problem, the solutions were to reverse lanes on the freeways and possibly the surrounding roads. Other times, no solution is really needed at all: The Deep Impact problem simply asked teams to predict the results of a large asteroid impacting Antarctica.) You MUST have more than one solution. Let me say that again: MORE THAN ONE SOLUTION. In order to show that you have a brilliant method of ﬁnding submarines or cross-sectioning gridded MRI data, you need a baseline, something to compare your solution with. You want to start with the simplest, most obvious algorithm possible, then gradually build on it, reﬁning it until get to your best solution.
Often for the discrete problem, the simplest solution may be merely to make random choices. For the meeting scheduling problem, you might want to have one algorithm which just randomly makes up the schedules. Then when you compare your better solutions with it, they look good!
You want to show that you’ve explored the problem thoroughly, and that you’ve tried many diﬀerent approaches. Even if you started with your best algorithm, then tried a bunch of blind alleys, in the paper you want to present things as if you started with the dumbest most basic solution, then gradually reﬁned it and ﬁnally arrived at your best solution.
What if you tried a more sophisticated solution method, which didn’t work well? Put it in the paper! Show all the angles you tried, even if your best solution is not the most complex and interesting one. In real life, that happens very often!
1.5 Solution Comparison Methods Usually the problem will state very clearly what the goal is, so it makes your algorithm testing methods fairly easy. For the submarine problem, your model is to create a simulation of sound waves propagating through water, bouncing oﬀ a submarine, then being received by an array of microphones. This data is passed to your various solution methods which all take a guess at where the submarine actually is. All you have to do is ﬁnd out how far each algorithm was from the mark and you have an easy method of comparison.
However, you usually have to make some decisions in how you compare the results of your solutions. In the MRI cross-sectioning problem, you can compare your algorithms’ estimate of tissue density with the actual density created by your models for each of the thousand or so points. But do you just average the variance? Maybe you should look at RMS error? Are you concerned with making sure that no point is drastically wrong or that the overall error is small?
With a lot of problems there will be many ways to compare your diﬀerent solutions, and there’s good reason to use more than one method to evaluate them. Evaluation methods should be one area of brainstorming that you keep working on all weekend.
1.6 Results Here, you need to actually present the results of the testing. This section should be very focused, because you’ve described everything else. If possible, you want a lot of data to back up your conclusions. Try a variety of models, and use them all to predict the results of a variety of solution methods. In general, you’re going to end up with a lot of parameters to play with – in models, your solutions, and comparison methods. Try to explore as much of this parameter space as possible. You want to show that you’ve taken a mature approach to the problem, and probed all aspects of it as best you could.
The speciﬁcs of data presentation are diﬃcult. If you can make graphs, by all means do so.