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A century ago E.H. Moore developed his "discovery method," which encouraged students to solve problems to form a conceptual framework rather than start with axiomatics. There is now a wealth of refined teaching techniques which emphasize mathematical inquiry, but relatively little awareness of them even among practitioners with similar philosophies but different methods.

By way of comparison, undergraduate curricula in the physical sciences have been placing ever-greater emphasis on demonstration and experiment. From interactive applets which illustrate concepts to lab classes to participation in active research labs to undergraduate theses which produce publishable results, first-hand experience with the scientific method is woven through a serious undergraduate science career. A science major not only learns a scientific body of knowledge but gains some sense for what research scientists do, and thus sees science as a dynamic pursuit.

The undergraduate curriculum in mathematics has generally not emphasized mathematical inquiry, for a number of reasons. Topics of current mathematical interest are typically difficult for even graduate students to approach. Doing exercises including proofs in advanced math courses gives students an impression of understanding why mathematics works. While there is some common understanding of a scientific method, there is no corresponding ``mathematical method.'' Including an inquiry component in a mathematics class generally means not being able to address as many topics. Some inquiry-based curricula at the K-12 level have unfortunately excluded formalism and rigor almost entirely, creating skepticism at the college level.

These limitations understood, a growing community of professors is incorporating inquiry more into the mathematics curriculum. Some find exclusive reliance on lecture limiting, especially considering that many aspects of definitions and proofs can be read by students on their own. Some are concerned with training future PhD's - how are students to be expected to encounter, assimilate, and then build on new mathematics if they have always worked with well-digested material? Some see the experience of learning the ``mathematical method'' as valuable not only for future research mathematicians but for anyone as part of a liberal arts education.

A list of inquiry-proof teaching techniques includes the following:
  • The Immersion Experience, as exemplified by the Ross and PROMYS programs. Students experience mathematics in ways close to that of a mathematician, as something that one can make for ones self. Students are given problems, including some open-ended ones suggestive of a body of theorems. They are encouraged to make conjectures and to generalize solutions they find. Only later does an instructor give explanatory lecture on these threads.
  • The Discovery Method, as pioneered by R. L. Moore, closely related to Inquiry-Based Learning. Letting students find and share the main arguments in a subject. Students are given definitions and statements of theorems, but not proofs, which they are to supply for themselves. Proofs are presented by students rather than the professor, who makes comments.
  • Working from Original Sources, and otherwise using historical resources, to develop topics as they were first developed. Rather than learn a streamlined, textbook version of material, see how the masters themselves grappled with concepts and put some things in final form while leaving other questions for further inquiry.
  • Mathematics Laboratory Experience, implemented for example at Mt. Holyoke College seeing "the mathematical method" through making conjectures and doing proofs. Students do experiments, by hand and by computer, and try to prove conjectures they make.
  • "Pre-work" and Worksheets, instead of or in addition to lecture, which can be particularly helpful in both "bridge courses" and courses for future teachers among others. Worksheets, often done in groups, have primary emphasis. Lecture, if it takes place, addresses common questions and ties together concepts which students have seen through problems. "Pre-work" assignments make the in-class worksheet time that much more effective, and further homework and exams encourage mastery.