Author: Joe Corneli
PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. I’ve described my involvement with the project briefly in my “Learn Math(s) the Hard(er) Way” project proposal, and at considerably greater depth in my thesis! This article summarizes the main design ideas. It gets a little technical, but don’t worry, there’s not too much math…
In short: I lumped the different activities that people could do on PlanetMath.org into 5 categories (see the table below). More or less this table just means that on PlanetMath, people write articles and link these articles to other articles, add comments, ask questions, make corrections, and connect problems and solutions to expository material. They also deploy heuristics for solving problems — and they make and join groups.
The five categories (Context, Engagement, Quality, Structure, and Heuristic) come from reflecting on the 5 paragogy principles, and comparing them with the Martin Nowak’s 5 rules for the evolution of cooperation, then clustering the actual activities that people can do on PlanetMath (as well as some new planned activities) into these categories. I also drew inspiration from the pattern and heuristic “language” we developed in the peeragogy project. I started by clustering our pattern language diagram into 5 segments, like this:
The “key” that shows how things fit together is as follows:
- Context ~ Changing context as a decentered center. ~ Kin selection
- Engagement ~ Meta-learning as a font of knowledge. ~ Direct reciprocity
- Quality ~ Peers provide feedback that wouldn’t be there otherwise. ~ Indirect reciprocity
- Structure ~ Learning is distributed and nonlinear. ~ Spatial selection
- Heuristic ~ Realize the dream if you can, then wake up! ~ Group selection
The analogies are not perfect, and are meant to help inspire, rather than to constrain, thoughts on the learning/platform design. It’s important to remember that Nowak’s formalism is meant to be general enough to describe all different kinds of collaboration –
In a “kin selection” regime, we are working in a “generational” modality; we are looking at what is “related”, and this helps to define that which is “unrelated” — the other.
On PlanetMath, the most important senses of “relatedness” apply to elements of the subject domain. Topics that are linked to one another in the encyclopedia are related. These links can either be implicit term references (which are spotted by PlanetMath’s autolinker), or more explicit connections added by authors, readers, or editors. Such links can build an implicit context for a “newcomer” who approaches a given topic.
In a “direct reciprocity” regime, we “learning about ourselves” in practice, usually in a social context.
One of the key legacy features of PlanetMath is that every object in the system is “discussable”. You can ask a question about an encyclopedia article, for example, and this will go into a common pool of questions. One of the driving ideas behind the site’s (re)design is that every question should help us improve the site, for example, by pointing out a place where the original expository article could be improved. Of course, at the most basic level, we hope that the questions receive good one-off answers (providing a benefit to the initial question-asker). Even the most simple question is a “constructively critical” question. On the level of site semantics, it would be good to keep track of which questions have been answered, and which have not. Questions can be “mutated” into corrections, requests, or mathematical problems to solve.
In an “indirect reciprocity” regime, we are building something that may be useful later on.
Another important legacy feature of PlanetMath is that, unlike Wikipedia, articles are not generally open to the public to edit (though some are). Rather, the typical process of “crowdsourcing” takes place through a corrections mechanism. From an analytical perspective, we might expect corrections to be one of the key ways in which site authors learn from one another. In a sense, the opportunity to get corrections or suggestions pointed out later might be one of the biggest incentives for writing an article in the first place! Offering a correction to someone else is, of course, a way to point out one’s own knowledgability (as such, a sort of flip-side of asking questions). Certain behaviors can help one develop a good reputation (though PlanetMath does not model this very explicitly)… and perhaps even more importantly, a high-quality resource “emerges” from such one-to-one interactions.
In a “spatial selection” regime, we are again defining an “inside” and “outside”, and looking for ways in which the structures that we have identified can fit together.
One of the features that the legacy version of PlanetMath lacked was any sort of support for “problem solving behavior” — which, in mathematics, is actually a pretty essential thing. Rather, the site was set up as a “reference” tool for people who solved problems elsewhere. By moving support for problems, solutions, and reviews onto the PlanetMath site itself, we expect not only to open the “marketplace” up to new kinds of learners (i.e. people working at a more basic level than encylopedia authoring OR people working at a fairly advanced level who are more interested in applications than in theory), but also to get significant improvements to the core knowledge resource itself (the encyclopedia). This is because “an article without an attached problem” is not a very practical article from a learning or application standpoint. Similarly, “a problem without a solution” is lacking something, as is “a solution without a review”. Building support for this, and support for people to structure/stage problems with problem sets should help make the site a much more practically useful learning tool.
In a “group selection” regime, we are building “sets” of activities and patterns (milestones, roles) which can then act as “selectors” for behavior. (This is why I’ve combined it with the catch-all “heuristic” category.)
Another historical weak point of the legacy site was support for “teams.” Thus, for example, one effort to improve PlanetMath’s coverage of topics in Real Analysis foundered – because there was no way to gather a critical mass to this project. There are social, technical, and knowledge aspects to this problem. Co-working requires people to be able to join groups, and it requires the groups to be able to structure their workflow. In some sense this is similar to an individual’s work being structured by the use of heuristics. A person’s choice to apply this strategy instead of that one, or to join this group instead of that one, is in the end a somewhat similar choice.
These notes have shown how the paragogical principles, supplimented with very general theories of collaboration, and some practical observations as examined in the Peeragogy Handbook, can help design a space for learning, which is itself a “learning space” in the sense of knowledge building. Although the case study has focused on mathematics learning, similar reflections would apply to designing other sorts of learning spaces (e.g. to the continued development of the Peeragogy project itself!).
The discussion continues: Reliving the history of mathematics as a peeragogical activity
Doug Breitbart: It occurred to me that you could add a learning dimension to the site that sets up the history of math as a series of problems, proofs and theorems that, although already solved, could be re-cast as if not yet solved, and framed as current challenges which visitors could take on (clearly with links to the actual solutions, and deconstruction of how they were arrived at, when the visitor decides to throw in the towel).
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