I many friends who are getting PhD's in education. Recently, one of my friends made the comment that so much work is being done to come up with new methods for teaching, and questioned why these methods are not being used in the classroom. I thought about this and responded with the question, "how complex is the problem of education?"

I've been thinking about that for a few days now. I mean, at a basic level, it seems that the problem of educating a group of students can be seen as a transshipment problem. There is a single supplier, the teacher, and a set of demands, given by the students. The item being transported, though, is information and information doesn't seem to meet the standard metrics of this problem.

For instance, how do you know if the information has been passed from teacher to student? How do we know if a student knows the information? One way of measuring this is through a test, but tests have their own biases and its very difficult to encapsulate all the information on the test. Plus, how do we know they learned the information vs just memorizing it? A second metric would be some type of a paper/writeup of what they learned. Again, this has its pros and cons, what if the student learned, but isn't a good writer? What if the student enjoys one part of the material more than others, and thus ignores some part of the lesson? Another metric could be based on the set of questions a student asks, but what about shy students or students who don't know they don't understand the material yet? I've also seen Other metrics such as games and having the students teach class posed.

Assuming that we have some 'black box' which will tell us whether or not the student has learned the information, there is still the concept of passing information. Unlike passing materials (like a box), when I pass information to someone, I do not lose that information. In fact the opposite may happen, where I actually have more understanding of the information. So the standard supply and demand constraints of a transshipment problem do not apply. One could get around this by assuming that the teacher has an infinite supply of information, and thus is able to supply any number of students. It then becomes a transshipment problem where the teacher wants to find the a way of presenting the information so that each student learns the material. This also brings up a second question though about the teacher. If the teacher is limited in how much they know about the topic, then they are definitely limited in how much they can say about it, and the number of ways they can teach it.

But this brings up another difference between education and the transshipment problem, time. Generally, classes have some fixed length, say 50 minutes. So teachers need to find a set of ways to pass information to a classroom of, say 27 students, such that each student receives this information that takes no longer than 50 minutes. In today's world, this is relaxed a bit because in addition to office hours, teachers are able to provide teaching methods by way of the internet and email, but this only works for those who are acceptable to those methods of teaching.

So I'm wondering about this problem. Where is it in the polynomial hierarchy? Is it in the polynomial hierarchy? How much relaxations like "we'll use the metric of tests or standardized tests to judge how much a student has learned" help simplify the problem? What is lost in this doing things this way?

These are just some thoughts on the subject. I'm really curious about how well this has been studied in terms of general complexity though. I do see a journal which has sparked my interest. Its called Complicity: An International Journal of Complexity and Education. I think I'll give a few articles a read in my spare time.

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