The following guide is meant as an inspirational tool for instructors of a first calculus course (and their students!) wishing to do something more effective with their time than following a standard textbook. It consists of two components: a continuous sequence of (unedited) scanned lecture notes and a discontinuous sequence of inspirational outlines. This is not a hard-and-fast set of pre-cooked lectures but a partially ordered set of ideas with many gaps left for the user to fill. The underlying premise is that lectures should be treated more like an advertisement for the material rather than as a medium to transmit it directly to the students. Indeed, it is only by working amongst themselves that students will actually absorb new ideas and this can only happen if they are genuinely intrigued. This is reflected in the emphasis on student lead inquiry, basing each topic on a sequence of questions meant to be discussed and answered by the class. The other particular aspect of this guide is its attempt to give the course a coherent narrative in which our motivation to study mathematics comes from a need to “understand the world around us” (economics, biology, space, perspective in art or any other topic we may be interested in). In trying to acquire such an understanding, we should uncover a need to describe relationships between the various components of this “world”. Our interest in mathematics therefore stems from its standing as a universal language, allowing us to understand complicated relationships and explain them in a simple manner. Throughout the course, we are lead to explore different steps of such an understanding process. Starting from our “world” or “system” (e.g. economics) we proceed to extract relationships (e.g. supply and demand equations) which we interpret using functions. For the most part, these functions turn out to be differentiable and calculus is the tool we develop to analyze them. The underlying philosophy here is that highlighting this thought process over and over again is probably the best antidote to the student’s (mis)conceptions of calculus as a collection of random topics to be mastered disjointly.
Who is the guide for?
While the guide’s primary audience should be the instructors, many students would benefit from the insider knowledge they could gain by looking at it.
How should the guide be used?
Instructors can use this guide however they see fit. One possibility would be to read the inspirational outlines first, then take a look at the corresponding scanned notes before deciding a preferred approach for a given topic. Omitting to write down explicit learning objectives throughout the guide is a feature, not a bug: the idea is to inspire without constraint. There is a parallel set of documents outlining precise weekly learning goals which should be consulted by the instructors to make sure they maintain an adequate pace and cover the appropriate topics throughout the term.
How should users contribute to the guide?
Given that users may have diverging opinions or additional insights on the various approaches outlined throughout the guide, instructors are given the opportunity to leave their own mark on the documents in the form of annotations. These annotations should not be thought of as feedback but rather as a form of graffiti meant to emulate those insightful little hand written notes often found in the margins of second-hand textbooks.
Who contributed to the guide?
The guide was put together by Maxime Bergeron under the guidance of Warren Code and Mark Mac Lean as part of a flexible learning project. Its content is an amalgamated product of original work with various sources including: Spivak’s Calculus, Bartle & Sherbert’s Introduction to Real Analysis, Edwards’ Advanced Calculus of Several Variables, Robert Ghrist’s Funny Little Calculus Text, Math Stack Exchange threads, The Good Questions project at Cornell University and Mark Mac Lean’s notes for Math 104. The guide’s website content review and organization as well as its web design and annotation integration are supported by Hailan Chen (Instructional Designer) and Josefina Rosado (Web Designer) from CTLT respectively.