Sunday, March 27, 2011

Complexity as Pedagogy

Learning an academic subject usually goes like this. First you have to get used to a new language and ideas expressed in that language. At first you're quizzed on meaning, but you're increasingly required to actively use the new concepts (using a different part of your brain to do so). When you get far enough along, you can begin to critique work using the new ideas. Ideally you critique your own work as you produce it. I think of this as the analysis and creativity cycle. The former demands use of the language (terms, ideas, grammar and syntax), whereas the latter depends on insight, trial and error, and imagination.

The problem is that it usually takes a long time to get any good a new language before you can effectively be creative in it. Take speaking in a foreign language, for example. You can't just make up words and sentences that 'sound German' and hope for it to be meaningful. A German friend gave a good example of this in reverse, when his sister tried to order a sandwich in New York:
Using "Ich will einen Hamburger bekommen"  (I want to get a hamburger.) to create the English approximation: "I will become a hamburger." 
But 'will' and 'become' mean very different things in the two languages, and there's no way to approximate the knowledge.
You have to understand how verbs are consummated and whatnot...then you can express yourself.

All this preparation is a hindrance if you just want to introduce a student to a subject. One of my professors, David Kammler, exposed me to the interesting idea in math that it should be taught like MacArthur's island-hopping campaign in the Pacific in WWII. That is, don't try to do everything, just teach strategic material that an able student can use as basis to explore other parts as needed. This has the effect of lessening the language burden.

There is an advantage to that. Being creative is like play, or it can be. It reinforces the language and gives students the thrill of discovery. This is how games work. If you make the learning curve too steep for beginners, they may not even make it through the rule book. Moreover, it's not necessary to have a lot of language/rules to have an interesting game. Chess, for example, or Backgammon, or Poker. The trick is to create a tight language that is unambiguous, and thereby allow a platform for exploration that can be easily checked by the rules.

Math and computer science are particularly amenable to this approach. My colleague Soumia Ichoua has a grant to do operations research activities, and part of it is an outreach project for middle or high school students. We are working with the university's Upward Bound program to coordinate logistics, and have created several types of logistical games for the kids.

The idea is to create a simple system that has these three properties:
  1. It's easy to understand the rules and check them. (low complexity rules = simple descriptive language)
  2. It's difficult to solve the problem. (high complexity solution = perfect solution method has a long description, and is unlikely to be found.)
  3. There is a wide range of possible solutions. (expressive language, allowing heuristics to be effective)
As it turns out, there is a major class of algorithmic challenges in computer science that has this property. They are in a class called NP. One example is the traveling salesman problem, where you try to find the shortest circuit to visit a list of destinations. The rules are very simple: add up the length of each leg (transit from one location to the next) in the circuit. The total distance is the sum. Finding the shortest path is very hard, however. So it's a perfect environment to employ creativity in a real discipline-based problem without having to learn a bunch of language and rules first.

The image above is taken from Wolfram Alpha, and shows a minimal distance circuit for the given points. The space of possible solutions is vast, but humans can use heuristics to find reasonably good solutions.

One of the game boards was created by my daughter, and is shown below. There's a story behind it about King Lolly, who needs to keep his people fed in winter, and involves moving food markers around from the grainery and the castle to the villages. There are several scenarios to keep it interesting.

Other logistics games include air-dropping supplies into a remote region, delivering hot dogs to hot dog vendors in a city, and planning for natural disasters. The last one is the focus of Dr. Ichoua's research, which has her visiting FEMA to get real data.

A draft of the rules for Lollyland is found below. The plan is to use undergraduate helpers to assist rising 9th and 10th graders in pairs. We are developing assessments for cognitive and non-cognitives.


Food travels from the castle and keep to the villages over the roads. If you have to move the food through the castle, the hungry people will eat half of whatever you send through (rounded up). Each food marker is enough to satisfy one hungry villager.

Winter in Lollyland

In the small kingdom of Lollyland, King Lollygaster has to plan well to feed his people during the winter. Being wise, he has constructed a grain tower in his castle and another one at a distant keep. Between these two sources, he must distribute emergency food if the winter is harder than expected. Besides the town surrounding the castle, which has its own store and can take care of itself, there are four large villages that have no such protection. It is these that the kind must provide for in time of famine.

Scenario 1.

The ground stayed frozen so long that the crops were put out late. The king must distribute food to the four villages to keep them from starving. He isn’t good with numbers, so he puts you in charge and entrusts you to get his people fed.

Set up: 25 food in the castle, and 11 in the keep. Each village gets 9 hungry people.

  1. Can all the villagers be fed?
  2. How many ways are there to distribute the food to feed them?
Scenario 2.

After many happy years King Lollygaster became too frail to rule effectively, and turned the kingdom over to his son Lollyfright, who promptly dumped the old man down a well. Lollyfright neglected to lay up much in the way of food storage, assuming the winter would be mild. You are now the Minister of Happiness, and tasked with the following situation:

Set up: 18 food in the castle and 8 in the keep. Each village gets 9 hungry people. All of the roads are closed due to lack of interest in clearing snow from them. Choose four roads to open and deliver food; the others will remain closed.

  1. What is the most number of villagers that can be fed?
  2. What are the best roads to open?
Scenario 3.

After the disaster, the merchant class of Lollyland dethrone Lollyfright and send him into exile. They form a governing council and put you in charge of planning for the next famine.
Set up: Assume there will be 7 hungry villagers in each village, the populating having declined of late. With this decline there is less ability to clear roads of snow. Assume that you can open only three roads in the worst case.

Question: How much food do you need to put at the castle and the keep?

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