*By Priscilla Bremser, Contributing Editor, Middlebury College*

Somehow, over the last 600 years or so, mathematics has moved from the core of the liberal arts disciplines to entirely outside. We’re all used to this; a “liberal arts math” course is understood to serve non-STEM majors, for example. The reasons for this shift are interesting to ponder (see [1] and [2]), but in this post I suggest that we consider some of its unfortunate present-day implications. It’s also worth considering the broader aim of a liberal arts approach, which transcends disciplinary boundaries.

A well-known exposition of the liberal arts ideal appears in a fifteenth-century treatise of Vergerius, in which he advocates studies “worthy of a free man.” While Vergerius lays out specific areas of study, including the “mathematical arts” –the quadrivium of arithmetic, geometry, astronomy, and music — he opens with the importance of a liberal education to character development. From early on, then, the liberal arts ideal goes beyond eschewing the vocational; it values sustained engagement with abstract concepts as central to the capacity to live a good life.

Although I have bristled at a smorgasbord interpretation of liberal arts, which misses the point in its focus on breadth with little attention to depth, equally concerning is the suggestion that some areas of study are more worthy of inclusion. In a recent report of the Association of American Colleges and Universities, which calls itself “A Voice and Force for Liberal Education in the 21^{st} Century,” “[t]he term ‘liberal arts’ is used … as a description for majors in the humanities, arts, and social sciences.” The common misperception of mathematics as solely practical may have contributed to its exile; it certainly adds to the challenge of teaching. It may be obvious to me that the study of Calculus benefits one’s overall capacity to engage with the world, but it’s not always evident to my students.

When Vermont first adopted the Common Core State Standards for Mathematics, a veteran teacher-trainer told me, “I worry that because the Content Standards take up so many more pages, people will focus on them, and then this won’t work. The Practice Standards are key.” Since then, I’ve heard two of the lead writers of the CCSSM say that they worry about too much focus on the Practice Standards, and that they’re meaningless without the Content Standards. Conclusion: both are important, and they work in concert.

Analogously, our understandings of the liberal arts are about content and practice. Both are important, and they work in concert. We can’t develop our students’ intellectual capacities without carefully considering content, and we can’t rely on content alone to prepare them for the challenges they’ll face confronting questions we can’t even imagine right now. (I don’t remember Fortran, but the value of learning to write computer programs endures.)

What does this say about the “Liberal Arts Mathematics” course? Steve Strogatz has been tweeting about the one he’s been teaching, using materials from the Discovering the Art of Mathematics project. Jessica Lahey wrote about his course for *The Atlantic. * What I’ve seen suggests that it is indeed possible to engage students, even those who start the course with apprehension or even fear, in “authentic mathematical experiences,” as the project intends. For this audience, figuring out how to produce a scalene triangle with one cut of the scissors is an exercise in plane geometry, a foundational topic. It’s also an exercise in making mistakes and persevering, something that may not have been encouraged in their earlier mathematics courses. The authenticity is evident in both the geometric content and the exploratory approach.

What does it mean for our mathematics majors if we insist that mathematics is still one of the liberal arts? For one thing, it is one way I remind myself, and my students, that too much emphasis on coverage risks losing the longer-lasting lessons of careful, detailed analysis. Those lessons go beyond the specifics of any particular course. The other day I heard a student in our common room scoff at the notion that math is just about numbers; “it’s about logical thinking,” he said. From now on I’m going to ask explicitly for that sort of metacognition in my upper-level classes. What have your mathematics classes had in common? When have you noticed advances in your capacity to think abstractly? How have your upper-level math courses changed your thinking about the earlier ones? Asking students to look for coherence and connection will, I believe, make it more likely that their mathematical studies will have a lasting impact on them as members of society, not just as workers.

We mathematicians understand that our discipline involves creativity, beauty, and abstraction as well as precision and utility. An education worthy of a free person should include active, meaningful experience with all of those elements. I will continue to speak up about the historical importance of the quadrivium, the “mathematical arts,” in the liberal arts. At the same time, I will affirm the value of mathematical practice to intellectual development for all present-day students, not just those majoring in mathematics and the sciences. Finally, and most important, I want every course I teach to reflect the centrality and the value of mathematics, content and practice, to a modern liberal education.

[1] Grant, Hardy. Mathematics and the Liberal Arts. *The College Mathematics Journal*, **30**, No. 2 (Mar., 1999), 96-105.

[2] Grant, Hardy. Mathematics and the Liberal Arts. *The College Mathematics Journal*, **30**, No. 3 (May, 1999), 197-203.

Many thanks to Luisa Burnham, Ben Braun, and Julian Fleron for their assistance.

The liberal arts aspect of mathematics that I always emphasize is that it offers a window to the entire world. I’m thinking of the following types of things. The history of math is global and ancient, involving cross-cultural interactions and developments — this is built into our language even, e.g. algebra = al-jabr, “algorithm” is a derivative of Al-Kwarizmi, our use of numerals originate from India, etc. Current discussions about mathematical practitioners, who does math, who has access to it, etc, immediately leads to cultural studies, gender and women’s studies, sociology of education, etc. Questions about how we teach and learn math is an entrance to questions about cognition, neuroscience, psychology, etc. Applications of math make us “look out” at the world, at how we use engineering, biology, chemistry, physics, etc, and through this we can be simultaneously amazed at what we can accomplish, and also reflective about ethical and moral considerations in how human endeavors impact the world. Anyway, the idea that “mathematics is a window to the world” in deep ways is, for me, at the heart of the mathematical experience in a liberal arts context.

To add to your list, the philosophy of mathematics can be used as a more precise domain in which to explore very general philosophical questions. Much of Wittgenstein’s thought went through “play testing” in the philosophy of math, for example. If you want a particularly challenging topic (at least for me), consider asking “in what ways (if any) are ‘mathematical truths’ different from ‘moral truths’?”; hopefully you won’t have to assign Parfit’s ‘On What Matters’ as supplementary reading for your math classes.