We're growing as expository writers as we explore elegant mathematical topics.
Although not always apparent to students, the canon of topics introduced in a standard precalculus course (exponentials, logarithms, trigonometry, and complex numbers) leads in a variety of delightful directions; they do not exist solely as a set of functions with which to apply the theory of calculus. One of the main purposes of this course has been to enjoy and explore these elegant topics in their own right, prior to seeing them in other contexts. At the same time, my intent is for students to review and reinforce their knowledge of these topics on their way to using them in the service of calculus.
For example, we have discovered a plethora of ways in which the mathematical constant e arises. This value, approximately equal to 2.718281828, appears when considering problems related to compound interest, when playing a guessing game against our computers, even in as innocent a setting as figuring out how many random numbers it takes, on average, to obtain a sum greater than 1. Along the way we have reviewed laws of exponents and stumbled upon one of the most legendary relationships in all of mathematics relating e, pi, and i.
The centerpiece of this course, however, has been the lab reports that students complete every other week. My goal is to introduce students to the art and craft of writing a self-contained, extended discussion of a single main result. We have talked about how to organize a two to three page article, how to use LaTeX to typeset mathematical expressions, how to use proper notation to assemble a coherent proof, and how to incorporate figures and examples into a discussion to illuminate a result for the reader. It is an ambitious undertaking for my students, but they are rising to the challenge beautifully and already (as I peruse the second set of lab reports coming in) making significant strides in their ability to communicate effectively. I anticipate adding many of these polished lab reports to the corresponding student portfolios.
-- Sam Vandervelde