MathOverflow

Abstract: Why do people contribute content to communities of question-answering, such as Yahoo! Answers? We investigated this issue on MathOverflow, a site dedicated to research-level mathematics, in which users ask and answer questions. MathOverflow is the first in a growing number of specialized Q&A sites using the Stack Exchange platform for scientific collaboration. In this study we combine responses to a survey with collected data on posting behavior on the site. User behavior suggests that building reputation is an important incentive, even though users do not report this in the survey. Level of expertise affects users' reported motivation to help others, but does not affect the importance of reputation building. We discuss the implications for the design of communities to target and encourage more contributions.

Abstract: In formal education, learning mathematics is typically done by receiving direct instruction within the confines of a classroom. From first grade through graduate school, students are expected to learn mathematics primarily by being taught by instructors with previous knowledge of the subject. Research mathematicians, on the other hand, must rely on other methods; the mathematics they are trying to understand may not, as yet, be known to anyone else. Hence, they learn primarily through experimentation, self-directed study, and collaboration with peers. In recent years, these methods have been expanded to use modern tools and ideas. Research mathematicians initiated several successful large-scale online collaboration projects, such as the Polymath project and the MathOverflow website. In this chapter, we discuss these two projects, along with various other examples of online collaborative learning of mathematics. Our primary motivation is captured in the following question: why aren’t we all learning math this way? While a complete answer is beyond the scope of this work, we hope to at least stimulate a debate among a wide audience. The major part of our discussion is thus informal; we defer the contextualization of these examples within modern education research until the end of the chapter.

Abstract: The highest level of mathematics research is traditionally seen as a solitary activity. Yet new innovations by mathematicians themselves are starting to harness the power of social computation to create new modes of mathematical production. We study the effectiveness of one such system, and make proposals for enhancement, drawing on AI and computer based mathematics. We analyse the content of a sample of questions and responses in the community question answering system for research mathematicians, math-overflow. We find that mathoverflow is very effective, with 90% of our sample of questions answered completely or in part. A typical response is an informal dialogue, allowing error and speculation, rather than rigorous mathematical argument: 37% of our sample discussions acknowledged error. Responses typically present information known to the respondent, and readily checked by other users: thus the effectiveness of mathoverflow comes from information sharing. We conclude that extending and the power and reach of mathoverflow through a combination of people and machines raises new challenges for artificial intelligence and computational mathematics, in particular how to handle error, analogy and informal reasoning.

Abstract: This paper outlines a strategy for building semantically meaningful representations and carrying out effective reasoning in technical knowledge domains such as mathematics. Our central assertion is that the semi-structured Q&A format, as used on the popular Stack Exchange network of websites, exposes domain knowledge in a form that is already reasonably close to the structured knowledge formats that computers can reason about. The knowledge in question is not only facts – but discursive, dialectical, argument for purposes of proof and pedagogy. We therefore assert that modelling the Q&A process computationally provides a route to domain understanding that is compatible with the day-to-day practices of mathematicians and students. This position is supported by a small case study that analyses one question from Mathoverflow in detail, using concepts from argumentation theory. A programme of future work, including a rigorous evaluation strategy, is then advanced.