Calc in Context

The Calculus sequence at Hampshire College is innovative, relevant, and essential for students in the Natural Sciences. The curriculum has been designed to address a wide range of scientific questions and teaches the language of Calculus rather than simply techniques. The first semester is built around differential equations, emphasizing local linearity, approximations, and techniques. The second semester consists of connecting Euler’s Method to integration in a beautiful example of the Fundamental Theorem of Calculus, periodicity, dynamical systems, functions of several variables, and series. Over the next few years, I plan to absorb dynamical systems into the first semester, and find a more fluid path through the second semester topics.

Calculus in Context serves as a good foundation for my Mathematical Modeling Program. There are three specific main goals that I have for establishing this connection: (1) increase the modeling component, (2) collaborate with NS faculty to bring in specific examples from other courses, and (3) update the computational software and tools.

To increase the modeling component, I have developed techniques and materials to help the students understand mathematical models and assigned independent projects in the first semester of CiC.  The independent projects consisted of students finding an article that contained a differential equation model in an area of their interest, describing the model thoroughly, simulating solutions using Euler’s Method, and critiquing the biological accuracy of the model. The students studied all different types of biological models, economic models, and some even applied the models to other projects that they were working on.

Since mathematical modeling is ubiquitous in natural science, there are many ways to connect CiC to other NS courses. Although I am currently just highlighting the connections for my students in class discussion, I plan on developing more substantial modules. I anticipate co-teaching mathematical biology with different faculty and would like to use that opportunity to develop open-ended examples for the CiC sequence.

Since CiC was first developed with True Basic, a language that no longer is available, I have been experimenting with different software (Sage, R, Matlab).  While there are many built in features on mathematical software, I teach and advocate programming from scratch (mostly with defining objects and creating for loops). This way the the students get to understand the power of technology without sacrificing true understanding. When we run into challenges with the speed of computation, then it allows for a valuable discussion on what other methods we can develop. Although the actual software is irrelevant, it is more about the ability to construct algorithms, I will be obtaining Matlab licenses so that we can use it in the class.  I have chosen Matlab because it has incredible documentation, it’s the software I’m most familiar with, and is valued highly in industry and other research fields.  Since it can be challenging to learn a programming language, I have developed additional materials for the students.

One of the challenges with incorporating more applications and programming into courses is having the time to teach the basics and important theory. With limited class time, I have been seeking new pedagogical techniques to optimize my student’s understanding. One of the pedagogical techniques that is taking off is ‘classroom flipping’. Math classes are usually taught with lecture in class and problem sets for HW. I use lecture to mean introduce new concepts and include those who lecture interactively and even have lively classes. There are several fundamental problems with this structure: students frequently need extensive help on homework sets and many students don’t find lecture valuable. This could be because the student is too advanced and finds lecture boring or the student gets lost and then feels uncomfortable asking questions. It is important to point out that class lecture can be valuable for some students but does that structure challenge students to think creatively about the material. Currently there is an emergence of online resources; there are lectures on line, interactive workbooks, and even courses that have extensive materials associated with them (MOOCS). These materials allow students to work though at their own pace. I have been moving towards using these materials prior to covering the material in class. In the long run, I intend to produce my own online lectures for the students but, for the meantime, I’m using a variety of materials that are out there. Students read or work through those materials, complete a short assignment and then, in class, we discuss the material that students find challenging, students work in groups to help each other, and we tackle applied, complicated problems in a supportive classroom setting.