PlanetMath Outlook

Focusing on important problems is what Richard Hamming advocates (New School Economic Review, Volume 3(1), 2008, 5-26). This seems sensible in a research design, too, to keep coming back to ask "what is important here?".

With PlanetMath I'd like to demonstrate that people can learn a concrete and challenging subject via peer interactions on the internet. There is already significant evidence that people learn how to work with Free Software this way. So, what's important here isn't just learning, but also what happens next. If we build up a big repository of answers to mathematical questions, maybe we will be on the way to having some sort of mathematical AI. (See these notes by Jon Borwein.)

This might seem like a shocking claim: how do we move from a system for peer tutoring to a computer system that can solve mathematical problems on its own? Not without some serious work. But this has been part of my vision all along (better to be somewhat explicit about it).

The way education has worked in the past, in math, is that new generations of people keep learning the same things over and over again. Well, new cohorts of people, anyway: each year, the same standard curriculum is taught in high schools and colleges, more or less without exception across the world. But knowledge about the world has been growing at an alarming rate! Our understanding of major unsolved problems is growing -- but the problems are still problems.

Might it not make sense to give students tools to address real problems, as quickly as possible? In order to do this, education shouldn't be a series of made-up tests, but a series of real, meaningful challenges.

I don't know what evidence I'll be able to gather in the next year -- certainly nothing particularly interesting making the system "go live", and seeing how people bounce ideas off of one another there. (I'm thinking about Charlie's notes on "Frankenstein".) I'm not entirely sure I know how to do "science" about this, though, and I don't feel so happy about that.

The question I keep coming back to is "how can I detect when people are learning?" (Corneli and Ponti, submitted). If I can see from what microscopic events people learn, then I ought to be able to engineer systems so that those kinds of events come up more frequently. This is very different from engineering a large-scale test at the end of the semester that asks people to sum up what they have learned.

Maybe each problem is a sort of "test". I'm not sure if there is any other way to go about it than to have people start uploading problems and solutions and see what they get to in the end. It nags at me somewhat that there may be some copyright issues standing in the way. (We assume it is fair use to upload a given problem, even verbatim, from a textbook, though copying out all of the problems from a textbook isn't fair use.) Once we start to amass some material like this, we ought to be able to get into more detail about when learning is happening.

Another tedious obstacle is the programming work itself. My supervisors have suggested that I'm in a position to design my research studies under the assumption that all of the programming will go according to plan, but I'm still feeling hesitant about that. I don't know if I've given myself the assurances that I need. And I don't know how quickly the team I'm working with on the programming stuff is ready to progress. This has been the big hang-up for the last year or so. Maybe we're getting close -- but these "maybes" are excruciating.

From the research point of view it might help just to specify in detail what it would mean for "all the programming to have gone according to plan". I spend a lot of emotional energy being distressed that things aren't moving sufficiently, but probably not enough time crossing things off of the list. If the list itself was a bit more clear (and more clearly related to my research agenda), I might be much happier. Rather than feeling bitter that things aren't moving, I'd really do well to help them inch forward a bit.

It's hard to get my head out above the clouds to see the big picture sometimes. But years could go by with grief that the little things aren't working out and I'd never see the larger patterns, and could for this reason never spot the opportunities to make major gains somewhat quickly. In the coming days I'm going to try to go for the "big picture" approach.