Using MATLAB as an Exploration Tool in Numerical Analysis and for Computational Thinking in the Sciences
Joan Weiss, Mathematics, Fairfield UniversityAfter completing my doctorate in 1979 I have been tenured by two institutions to be the lone Numerical Analysis instructor. So in forty years I have taught a Numerical Analysis course in over thirty semesters. Nearly every course has been a mathematics/computer science major upper level elective. My recent three offerings have been to a class of upper level undergraduate mathematics majors and master's mathematics graduate students. Initially I required the students to write Fortran or Pascal code for each numerical algorithm and/or use ISML routines. About twenty-five years ago Numerical Analysis texts started to be accompanied with the numerical algorithms on a disk or CD coded in Fortran and Pascal. The variety of software options has expanded to include C, C++, Mathematica, Maple and MATLAB and the coded numerical algorithms are now downloadable from the web. With the availability of the coded algorithms in my numerical analysis course students modify the code to solve problems.
Numerical analysts, when posed with a problem, will often analyze it graphically or numerically by writing a short script to generate or plot data in order to get an idea for how to solve the problem or what is causing the errors they might be experiencing. In the fall of 2018 in my Numerical Analysis course I initiated "MATLAB Explorations", short assignments by which I attempted to encourage my students to use MATLAB to explore mathematical concepts and tackle problems, i.e. to mimic numerical analysts, and use computation as a learning and investigative tool. I had planned to assign at least five "MATLAB Explorations" but only three were assigned: I. Exploring Taylor Polynomials with MATLAB; II. Use MATLAB to Explore the Numerical Limits of Your Computer; and III. Exploring Lagrange Polynomials with MATLAB. These didn't seem to accomplish my goal of encouraging students to use computation to investigate mathematical concepts and solutions to problems. Students did the basic assignments with little or no exploration. Students needed more direction on what and how to explore concepts using computation. I recently found "Building Self-Efficacy" on the SERC website. It provides excellent guidance to develop the types of explorations that should motivate students to develop computational skills useful in their discipline.
Hence for my spring 2020 sabbatical one of my goals is to develop a course that would educate science students in the use computation to solve problems in their discipline. I anticipate that each module of this course would be a problem in a science along with a computational method to solve the problem. I envision developing an interdisciplinary course, which presents problems from a variety of science disciplines and their computational solutions and/or scientific data set(s) that can be statistically analyzed or fit to a mathematical model. This course might be titled "Computational Skills with MATLAB for STEM Majors" and should prove useful to most science majors.