By Reva Kasman
Mathematics Department

Reva Kasman

It would be fair to say that I did not exactly ride into the world of undergraduate education on the liberal arts bandwagon.  I grew up in Toronto, and while much of Canadian television may be heavily influenced by its imposing neighbor to the south, Canadian higher education is still closer in spirit to the European model than the American.  Students can specialize in their major fields early and with fervor – after one required course in each of the “social science” and “humanities” categories, I shamelessly took almost no electives.  I reveled in the sheer quantity of math I could take post-high school, having finally finished proving how well-rounded I was.

My first encounter with the American educational system came when I moved to Chicago for graduate school.  I must confess that my initial reaction to a general education core was a smug mixture of amusement and astonishment, maybe accompanied by just the tiniest dose of disdain.  All I saw were the drawbacks.  Weren’t dedicated math majors being done a disservice when they were forced to divide their energies between one or two math courses and two or three courses in psychology, political science or music?  What was the benefit of insisting that a theater major who had painfully scraped through high school math pass algebra before being allowed to graduate?  Were my students really fulfilling a university requirement when they took bowling?

Thus, my graduate student self would probably have been shocked this past January to find me at the national meeting of the Mathematics Association of America organizing a special session on Mathematics Courses for the Liberal Arts Student.  For two days I watched presentations with titles like “Games and Critical Thinking,” “Mathematical Ways of Reasoning and Knowing Through Geometry,” and “Teaching Quantitative Reasoning with the News”.  Course topics ranged from Renaissance paintings to airplane flight, democracy to investment banking.  This was not your mother’s math course, if your mother happens to be a math-phobic French and Italian major (like mine).

Requiring a mathematics or quantitative reasoning course as part of a liberal arts education is not a new concept, but many of the innovative ways that students are able to fulfill this requirement are.  Previously students would likely have taken a course covering traditional math content, such as precalculus.  The class would have been filled with a mixture of students needing the material as a prerequisite for further math study, and those taking it solely to satisfy a terminal general education requirement.  The instructor may have found it difficult to meet the needs of such disparate audiences in one class, unsure of whom to prioritize.  The entire course experience was almost certainly reminiscent of the high school classes that, for too many of these students, had felt mechanical and uninspiring.  Surely this was not what anyone had envisioned when they included a mandatory math component for graduation.

The goals of Salem State’s core curriculum include developing “competence in problem-solving, critical thinking, and abstract reasoning.”  But there is also an aspiration to “encourage creativity and natural curiosity.” Do traditional math courses at the lower level, such as precalculus and algebra, fulfill the first set of goals?  Possibly.  Do they inspire creativity and curiosity?  Probably not, especially in the many students who did not find math to be captivating and exciting before college.  And yet, most math professors choose their field because they find the subject creative and beautiful, quirky and practical, astonishing and subtle.  So the real challenge is, how do we convey these sentiments to non-math majors? How do we capture the interest of bright students who participate eagerly in their humanities classes, but enter a math classroom as though being audited by the IRS?  How do we satisfy yet another requirement of our core’s mission, that of “illuminat[ing] how inquiry is conducted in the various disciplines” without perpetuating the popular myth that mathematics is accessible only to a select few technical geniuses?

Modern incarnations of the Math for Liberal Arts course are the result of significant efforts by career mathematicians to imaginatively meet these goals and engage our (perhaps justifiably) critical and reluctant students.  And if I can trust the reports of the speakers at my recent conference, we are starting to succeed, and we’re having a delightful time in the process.  More importantly, it seems that many of our students are too.

For better or for worse, there is no such thing as a standard curriculum for a liberal arts math course.  My own course, MAT 120, which I have taught several times in the past three years, is motivated by my desire to showcase a variety of characteristics of mathematics through topics which students are unlikely to have encountered previously in their formal education.  Despite the depth of our work, the technical prerequisites are minimal, and all our calculations can be done on a cheap calculator from Walgreens.  The atmosphere in the class is dynamic, and active explorations are used in almost class meeting.  I want my students to understand that they are “doing math” – and from a mathematician’s perspective, that involves asking questions, making and testing conjectures, experimenting with examples, and arguing (logically, of course!) with each other.

The message that this is not going to be the same old math course comes early.  On the first day of my course students participate in small group discussions about the results of a fictional election to move the capital of Tennessee.  In our scenario, the most popular choice for the new capital is Memphis, but paradoxically, Memphis also has the highest number of last place votes.  Through the lesson students engage in dialogue about what it means to say that democratic elections reflect the will of the people.  They question their assumptions about the equity of popular votes, and together we think about what role mathematics could have in supporting and articulating our perceived notions of fairness.  For the next few weeks we study mathematical voting theory – an active field of research mathematics that is not just philosophically engaging but also relevant – for instance, Burlington, Vermont now uses a modification of Instant Run-off Voting (a technique studied in our course) to choose their mayor, and the Heismann Trophy winner is based on a Borda Count.

Each subsequent topic highlights different features of mathematics, both pure and applied.  We see how a simple and familiar children’s puzzle about tracing a picture without lifting your pencil can provoke deep questions in the area of graph theory.  In the culminating project for this unit, students minimize the cost of an Amazing Race-style trip around the world which they design themselves.

A unit on geometry and frieze patterns has students noticing reflectional and rotational symmetry everywhere they turn – car hubcaps, tattoo designs, even the façade of the Sullivan Building!  This topic also leads to musings on the role that mathematics plays in art:  Is there such a thing as mathematical art?  Does scrutinizing the mathematical nature of a painting or design somehow sully its beauty, or can the analysis actually increase our admiration?

Finally, we investigate coding and cryptography, learning the basic mathematical notions behind both the ISBN codes on our books and the encryption of our credit card data when we buy books online.  (Thankfully, though, the tools to hack an Internet security program are beyond the scope of this course!)

MAT 120 has become one of my favorite courses to teach, and watching my students develop positive attitudes about mathematics (as well as their own capacity to do mathematics) has been extremely rewarding.  So, you ask, have I completely recanted my graduate student beliefs about the relative value of a general education core?  I suppose that I have not recanted them so much as I have broadened my point of view.  I certainly do not regret the amount of mathematics that I was able to learn as an undergraduate by having the freedom to fill my schedule with courses from within my major.  But I also believe strongly in all of the values espoused through our core curriculum, and I have come to appreciate the role that general education courses can play in instilling these values in our students.  At their best, courses in Math for Liberal Arts help students learn to trust their intellectual and creative instincts, and to become confident in their own ability to solve unfamiliar problems outside their usual comfort zone.   I would not be so arrogant as to suggest that mathematics is the only vehicle for imparting these lessons, but maybe some perspectives are best observed from the view of your neighbor’s window.

This article is part of ASpect’s March 2010 issue on the core curriculum.