By Kathleen Fowler, Jeffrey Humpherys, Eric Kostelich, Suzanne Weekes, and Lee Zia
At the 2016 Joint Mathematics Meetings, held in Seattle this January, SIAM hosted a panel in collaboration with the American Mathematical Society and the Mathematical Association of America. The panel focused on “Computing Across the Curriculum.” Motivated by the emergence of data science, industrial mathematics, and mathematical modeling as necessary workforce skills, faculty are now considering the challenges and benefits of incorporating computing into their courses. Lee Zia, Deputy Division Director for the National Science Foundation’s Division of Undergraduate Education (DUE), chaired the session.
To help frame the presentations and subsequent discussion, panelists Kathleen Fowler (Clarkson University), Jeffrey Humpherys (Brigham Young University), Eric Kostelich (Arizona State University), and Suzanne Weekes (Worcester Polytechnic Institute) considered several questions. (1) What does the phrase “computing across the curriculum” mean to you? (2) What forms does “computing” take in this characterization, and in what ways are “data science” or “big data” being addressed? (3) Describe the way(s) in which your department has incorporated computing within your curriculum, and the challenges you have encountered in implementing such a vision. (4) What opportunities to collaborate with other disciplines have you and your colleagues exploited?
Panel members described efforts to reinforce course content by integrating computing into pre-existing courses. Humpherys offered examples of teaching least squares paired with a focus on the numerical methods used to compute solutions, such as QR decomposition. This idea extends to analysis, linear algebra, optimization, probability/statistics, differential equations, control theory, etc., as well as approximation theory and theoretical computer science. Ultimately, students are able to code a simplified algorithm and compare output to an industrial strength version. Once they prove the concept computationally, they can and are expected to use the industrial strength algorithm. Humpherys encourages the use of the term “predictive analytics” over “big data,” observing that while the latter term is in vogue, it obscures the important point that mathematical analysis is useful in many settings. Even “small” examples lay the groundwork for understanding general applications of mathematical thinking.
Fowler considers the inclusion of computing components as a necessary method for helping students grow into innovative problem solvers. She requires computing with applied projects in her large-scale freshmen calculus courses. Although grading is a challenge, the trade-offs are worthwhile. Students gain early experience in writing technically, working collaboratively, and tackling open-ended problems. First-year students typically have some experience with Excel, and advanced students often use Python or Matlab, exploring how these programs can be used with modeling to approach real-world problems.
At Worcester Polytechnic Institute (WPI), computing across the curriculum is a priority. All students take calculus, which includes an hour in the computing lab with Maple exercises to reinforce or introduce calculus concepts. Non-faculty instructors and graduate teaching assistants lead these labs. Weekes says that all classrooms are equipped with computer projection systems, so faculty can readily demonstrate concepts using their favorite software. For example, differential equations faculty regularly use Matlab and Maple to demonstrate concepts like resonance and solutions to systems such as predator-prey models and the SIR model.
Weekes also spoke about higher-level courses such as linear programming and math modeling with ODEs, where students solve interesting problems and explore theory after an introduction to Matlab and access to functions such as linprog.m or pplane6.m. Some faculty use COMSOL and Maple in a lab section of a course entitled Boundary Value Problems to have students solve BVPs numerically and to, for example, demonstrate the collision of solitons. In Probabilistic Methods in OR, a colleague uses Python/NumPy for in-class examples, and students use a programming language of their choice to simulate Markov chains and implement the Metropolis Hastings algorithm.
Weekes emphasized that having excellent computing support resources at WPI has been key to making computing across their curriculum successful. In particular, some university staff offer training sessions to introduce students to scientific software applications; they do this outside of regular class time.
Kostelich suggested that, as motivation, interested faculty look at the January/February 2000 issue of Computing in Science and Engineering, edited by Jack Dongarra and Francis Sullivan, which presents a list of the “top 10 algorithms” of the 20th century;1 the list includes the Metropolis algorithm, the simplex method for linear programming, Quicksort, and the fast Fourier transform. Kostelich would add the Kalman filter, public-key cryptography, and shotgun genome sequencing to their list – but regardless of what one’s “top 10” might be, Kostelich argued that all mathematics undergraduates should have some in-depth exposure to a few of them.
The panel was well received and the panelists fielded questions regarding how to advance computing culture in audience members’ own home departments. In addition, each panelist spoke of his or her interactions with other disciplines; they all agreed that finding allies in different departments can help increase the number of students taking mathematics, which is ultimately good for home departments. Panel members also urged mathematicians to seek such allies in disciplines like economics and biology, in addition to the more traditional ones in engineering and the physical sciences. Many audience members voiced excitement about the notion of computing across the curriculum, but expressed simultaneous concerns about making it a reality. Conversation indicated that a shortage of graduate students to serve as TAs, lack of support from peers and department chairs, inadequate computing facilities, and outdated course offerings are all real hurdles.
To make a case for integrated computing in mathematics courses, Zia and an audience member also discussed the need for careful assessment. Such assessment could help promote change and demonstrate the benefits and trade-offs in this form of curricular improvement. Zia pointed out that DUE’s core funding program, Improving Undergraduate STEM Education (IUSE), is a natural place to seek support for such work. He added that mathematicians have not been as active in submitting proposals as their other disciplinary colleagues, and encouraged the field to engage in such efforts.
To this end, a major takeaway from the session was that efforts are needed to prompt faculty at a wide range of colleges and universities into integrating more computing in their curriculum. The mindset already exists among a majority of SIAM members (and the practices of many SIAM faculty members). With the formation of the SIAG on Applied Mathematics Education (SIAG/Ed), SIAM members have an opportunity to help share best practices, advice, and support with colleagues who have a genuine interest in making these changes. The 2016 SIAM Conference on Applied Mathematics Education, to be held September 30-October 2 in Philadelphia, is an ideal place to generate more discussion. The SIAM Education Committee is in the process of incorporating this theme into some of the proposed minisymposia for the conference.
1 An article summarizing the list appeared in Volume 33, Number 4 of SIAM News.
Kathleen Fowler is an associate professor of mathematics at Clarkson University and a member of the SIAM Education Committee. Eric Kostelich is President’s Professor of Mathematics at Arizona State University, where he runs a summer undergraduate research program. Jeffrey Humpherys joined the Department of Mathematics’ faculty at Brigham Young University in 2005. He also serves on the SIAM Education Committee, along with Suzanne Weekes, an associate professor in the Department of Mathematical Sciences at Worcester Polytechnic Institute. Lee Zia is the Deputy Division Director for the National Science Foundation’s Division of Undergraduate Education (DUE), prior to which he was a professor in the Department of Mathematics at the University of New Hampshire.