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Responding to students' questions is a critical part of teaching mathematics (Cavey, Whitenack, and Lovin 2006; NCTM 1991; Schifter 1996). A particular response may either stifle a student's inquiry or, ideally, stimulate his or her interest in mathematics. Although formulating responses that have the potential to engage students in developing new mathematical insights is challenging, we believe that this skill can be developed. We have found it helpful to view this challenge as a matter of recognizing the mathematics of a student's question—that is, the mathematics the student understands as well as the mathematics he or she may be ready to learn

The careful use of notation and language in the statement of both the definitions and problems of calculus can begin the process of making students mathematically literate while allowing them to enjoy working on challenging problems and applications without the aid of numerous examples. Engaging the students to participate in the precise use of the language inculcates a philosophy of careful use of language that benefits all students regardless of major. Not only will this benefit those few students who decide to major in mathematics, but the precision of language will especially benefit both majors and non-majors outside of their mathematical studies. This pedagogical piece exhibits anecdotal evidence that such an approach provides motivation to students, better preparation for “proofs” classes, and a desire to pursue higher mathematics. Examples of calculus problems presented in the traditional fashion alongside subtly more precise presentations illustrate the process.

We describe a two-semester numerical methods course that serves as a research experience for undergraduates without requiring external funding or modification of current curriculum. The first semester introduces traditional material and builds a proper set of tools that the students use in the second semester to approach a more research oriented problem. Our vehicle is an engineering problem associated with the hydro-dynamics of keel design which is used to introduce students to constrained optimization via a variation of the traditional isoperimetric problem of finding the curve with fixed end-points, fixed perimeter, and maximum area.