Experience with Mathematica in the teaching of a Calculus and Linear Algebra course

I once taught the calculus and linear algebra course called ZCA 110 to a class of around 80 students for a couple of semesters. For this kind of course sometimes it is rather abstract to explain certain concepts such as taking the limit of a function or the convergence of a series expansion to a function. Mathematica is a very powerful software package that can do many mathematical manipulations such as displaying the graphs of complicated function, performing algebraic and numerical integration and differentiation, expanding a function into its series representation, manipulating matrices and many more. This software is quite popular among researchers who need a convenient tool for visualisation, symbol manipulation or computationally inexpensive numerical programming. It has also proven to be quite a workable alternative approach to teach mathematics. There even exist Mathematica courses designed to teach mathematics that take advantages of the various functionalities of the software. Armed with some familiarity with Mathematica, I decided to make use of it in my Calculus and linear algebra course. I observe that students were generally amazed by the powerfulness of Mathematica. The abstract symbols on the textbook suddenly manifest themselves into vivid graphical form now visualisable on the screen. As a concrete example, when I taught series representation of a function, the student can now see it directly the graph of a series with a few terms converges gradually to the shape of the generating function when higher order terms are added. The visualisation on the screen helps to strengthen the understanding of the concept of “convergence of a series to its generating function”. Within the push of button I solved a calculus or algebraic problem from the textbook. I reckon that using Mathematica for teaching could be quite effective in making students to understand the otherwise abstract concepts. However I am also aware of the possibility that students the computational tool may divert the attention of the students away from the core mathematical concepts, resulting in over dependence on the software rather than their own brain. In addition, over demonstrating Mathematica in the class may also cause unnecessary confusion to some weak or techno-phobic students. Therefore I demonstrated the use of Mathematica only in a selected few lecture slots. In these slots I have successfully obtained the response I was after: the students are stunned, and paid all of their attention to the lecture.

To use Mathematica for teaching purpose, one has to be familiar with its syntax and a little bit of programming knowledge, on top of the mathematical concepts he/she want to demonstrate with Mathematica. This is a technical part which may be quite time consuming for those who are not computationally inclined. But once one learn up the syntax and get familiar with the programming logic, producing a few lines of codes to solve a differential equation or taking the limit of a Riemannian sum become a simple routine. At the end of the lecture I would upload the Mathematica codes used for demonstration during the class onto the course web site. Enthusiastic students then download these codes to reply on their own PC.

As a trick to attract students’ attention or to arouse their interest during the lecture, I would randomly pick a student by running a simple Mathematica code I wrote called “luckyone.m” on the projected screen. When the computer button was pressed, students would see the screen displaying the name of the “luckyone” selected randomly from the class’s name list. The randomly selected student picked by the computer random code would have to handle some question thrown by me. The atmosphere became interestingly excited when they see me running the code on the screen. Again, I gained what I wished for: their attention and aroused interest for my class.