PlanetMath Overview

I would like to write at least some words about PlanetMath as a potential or, by the time this book is fully edited and published, real case study or example of paragogy in practice. Right now, I will just write about my ideas.

My current "vision" for PlanetMath is that people will be able to log on and solve mathematical problems there, similar to Khan Academy, but with the twist that the solutions will be shared publicly, so that people can comment on each other's solutions, or ask for help when they run into trouble.

Furthermore, keywords from both problems and solutions will be automatically linked into the PlanetMath encyclopedia (which currently defines over 15000 mathematical terms). The thought here is that it is more or less impossible to solve a math problem when one does not know what the terms in the problem mean.

This was my strong impression as a beginning graduate student at the University of Texas, where the preliminary exams proved to be a very interesting but fatal obstacle for me. My response at the time was to compile a large catalog of definitions of terms from all of the past exams that were on file. I learned in practice that the raw definitions are not enough: one needs to know how to think about the definitions too.

Honestly, if I had seriously wanted to pass the prelims and get a Ph. D. in mathematics, I would have probably been better served by going to class regularly, doing homework to the best of my ability, asking questions to my peers and to my professors in office hours, much as I had done as an undergraduate. I was, however, taken by the idea that, even if I passed the exams, there would be more students in the future who would have to go through the same difficult process -- and I wondered if there mightn't be a way to significantly streamline the learning process, not just for myself, but for generations of mathematics students to come. I thought that would be a potentially bigger contribution than pushing myself through the degree.

Besides, I was often the only person who asked questions during class, which embarrassed me, and, although I had done reasonably well in my undergraduate courses, my peers at the university often seemed to be having a better time of it than me -- maybe because they were more used to the sort of disciplined work schedule; I wasn't sure.

In any case, I set off then on a very "alternative" course for learning mathematics, and if my ambitions in my current Ph. D. project pan out, I will start to see the vision I had then being fulfilled on PlanetMath -- about a decade after the initial decision to switch out of "mainstream" mathematics.

And so, we will need not only a massive collection of definitions, but a way for students who don't get it to speak up and say "Hey, this definition isn't really helping me solve the problem I'm working on -- I don't get it!" We also need helpful people who will respond to these requests for, er, help, and do something about them. I expect that time and time again, people will understand a definition in the abstract, but they won't know how to think about it, or they won't know what other definitions or theorems are supposed to come to mind, and why.

As time goes by, and as people ask lots of questions, and give lots of answers and advice, that PlanetMath will become an increasingly "complete" mathematical resource. "Encyclopedia" literally means "a total instruction", and I would love it if PlanetMath would become a mathematics encyclopedia in that sense, not just in the sense of being a reference resource.

I wonder if the same sort of educational and knowledge building model could apply in other domains. For example, we might imagine people on Wikiversity taking articles from Wikipedia and learning together by improving these articles. And some similar things have happened -- not necessarily by self-organizing on Wikipedia, but in classrooms (Piotr Konieczny's work).

There might be something going on with mathematics -- its abstractness, the presence of right and wrong answers -- that makes it easier to self-organize about than other domains of knowledge. Still, if I thought PlanetMath was the end of the road for me in my career, I'm not sure I would have been toiling away on it in one way or another for such a long time! My sense is that quite a variety of social and economic problems should be amenable to a similar sort of approach -- if only we can figure out the details. For the moment, I'm still trying to sort out the details of the PlanetMath case.