What I Learned This Summer

Summer teaching is a unique experience for many graduate students. For students in many disciplines, it may be the first, primary, or only chance to teach one’s own class. In addition to being an opportunity for graduate students to transition to instructor roles, summer courses also give students and instructors alike different opportunities than a Fall or Spring term.

For me, this summer presented the chance to develop new and exciting (to me, at least) materials for the course MATH 244, which is a course on differential equations geared mainly to engineers. Without changing the overall curriculum of the course, I decided to integrate computer-based exercises (small & large) to give students the chance not only to learn computer skills that accompany the mathematical material in the course, but to use computer-based work to aid them in learning the rest of the material.  The goal of these changes was to improve learning.  Although the course was not long enough to establish serious, long-term, in-depth skills, the experience should serve as a useful introduction to particular sorts of software and to computing skills.

an epidemiological model

I also believe the use of the computer modernizes some of the other instruction, and benefits students with algorithmic, visual, or kinetic learning styles. I believe the ability to manipulate mathematics and see significant visual output in real-time has a profound impact on how students understand concepts like the stability of equilibrium-solutions to differential equations.

I would also argue that this approach helped better organize the course, both logistically and in students’ minds, as components of the course related to computation and visualization were not segmented into awkward places throughout the term. By making computation and visualization more central, and more hands-on, students more easily integrated this material with the theoretical and non-computational methods.

an information flow model

This also effectively separated the computational and visual elements of the course from exams, where asking students to perform tasks better suited for a computer limits the assessment. Students were instead asked to delve into complex computer-based tasks over a longer period of time, as “projects”. (This, as a side-effect, provided some relief from high-stakes testing.)

Students used complex computational and graphical methods.

Student evaluations seemed to indicate that students perceived this part of the course as beneficial: Instructional survey scores for “I learned a great deal in this course,” “I rate the teaching effectiveness of the instructor as,” and “I rate the overall quality of the course as” were substantially higher than average. Individual comments reflected positively on the use of computers for assignments and for in-class demonstrations. Five respondents to the instructional survey (over 25% of the class) indicated that computer-based work was how “this course or the instructor encouraged [their] intellectual growth and progress.”

A population dynamics model

Of course, this was challenging for students, but despite the fast-paced nature of a summer course, most students did not feel overloaded with work. For every student who believed there was “extra work” due to the additional computer-based workload, there was another who realized that this was eliminating a significant amount of alternative work to be done without computer to cover the same material.

Here is an interactive demonstration similar to those used and produced by students. The Wolfram CDF Plugin is required.

Images used in this entry are used under fair-use guidelines. They are excerpts from student-generated work in the course described in this blog post.

Author: Kellen Myers

I am a Mathematics graduate student at Rutgers University. I am a Ph.D. candidate, which means at this point, my work is towards writing my dissertation. My advisor is Doron Zeilberger. My research interests are in combinatorics. A few specific interests are within Ramsey Theory, especially Ramsey Theory on the integers. This means I have an interest in both Ramsey Theory (on the whole), and on the theory of arithmetic structure in the integers. I would characterize my interest in mathematics, in general, as very broad. I enjoy the interdisciplinary nature of certain types of mathematics, and appreciate collaboration between mathematicians with diverse interests. I also take special interest in developing my skills as an educator and as a member of the academic community and workplace. I was the graduate coordinator for the DIMACS REU 2010-2013, and I participate in many other projects and endeavors to develop my skills as a teacher, a mentor, and an administrator.

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