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Curriculum Guide to Majors in the Mathematical Sciences

Mathematics as a Liberal Art

The value of a mathematical education to a future member of society goes well beyond the mathematics itself.  The content and skills conveyed by mathematical training have for millennia been part of a classical education.  Logic was among the three fundamental subjects of the trivium; arithmetic and geometry were among the four arts of the quadrivium.  (The other two, music and astronomy, have mathematical connections themselves.)   The Platonic ideal for logically persuasive reasoning has long been mathematical.  For instance, in writing the Declaration of Independence, Thomas Jefferson looked to Euclid for the persuasive form of an argument.   Mathematics is an essential tool in science, technology, business, and elsewhere.   At its core,  mathematics is a way of thinking that enriches all human activity, even where mathematical content does not explicitly appear.  Majoring in mathematics is preparation for life, regardless of one’s intended career. 

Mathematics is an art—both in the aesthetic sense and in the sense of reasoned living. Many students choose a mathematics major not as career preparation but for its own sake, because they happen to be successful at it or are interested in it.  Society benefits from college graduates who are generally educated in higher mathematics, whose lives and social activities are influenced by their understanding of mathematics and, through it, of interesting aspects of history and culture.  Beyond career and employment issues, “pure” mathematics majors are parents, aunts, uncles, volunteers in schools, tutors, voters in elections, school board members.  Pure mathematics courses, including those driven mainly by aesthetic concerns, can help prepare students to become valuable citizens, all of  whose contributions are augmented by skills and habits of mind developed through learning mathematics.

The cognitive goals laid out in the Overview to this Guide aim to foster effective thinking that is useful for everyone.  Thinking clearly, producing and following logical arguments,  and learning  to distinguish sound reasoning from malarkey are skills of universal value.  Working with hypotheticals and investigating their consequences are powerful strategies for approaching problems of all sorts.   Understanding the role of quantifiers in argument helps people see a complicated world with detail and precision.   Learning to explore the unknown by examining it from several different points of view is an effective skill in any domain.  Mathematics can teach students the value of productive persistence, learning from mistakes, and turning present errors into future successes.   Students who major in mathematics become better writers, better speakers, better thinkers, and better members of society.

All benefits of mathematical training notwithstanding, it is painfully clear that few leaders in our society are well trained in mathematics.  This deficiency has many causes, some of them beyond our control.  But we mathematics educators share the blame.  It is easy to encourage and cultivate students who excel at all things mathematical—those who seem destined for graduate school, and may remind us of ourselves.  We should continue to encourage these students, but we should also work to attract and nurture a cadre of people who combine mathematical training with broader humanistic skills.  We should encourage students who love mathematics, but may struggle with it.  For these students, too, a mathematics major can be a great choice.  They need to hear us acknowledge that the world needs more people who, like them, approach mathematics with love, not fear. 

Curriculum and Pedagogy

This Guide makes specific content suggestions for diverse student audiences, including those planning graduate study in pure mathematics, prospective secondary mathematics teachers, and those planning careers in mathematical areas as varied as actuarial science, statistics, or financial mathematics. 

The content needs of students not planning to pursue mathematics professionally are less specific.  The broad-based mathematical training suggested by the Overview document will serve these students well, and the core mathematical subjects traditionally described as “pure” are especially rich vehicles for developing thinking skills that are rare and of special value to the student and to society.  

One of our profession’s important challenges is to make the mathematical experience successfully  foster these uplifting steps toward better thinking.  Mathematics can help transform students into better thinkers—but only if we allow and help students develop and practice those thinking skills.  Mimicking procedures is not enough.   Students must experience for themselves what it means to understand mathematical ideas deeply. They must feel the exhilaration of mathematical insight. They must learn and appreciate productive struggle.   They must learn to articulate mathematical ideas clearly, orally and in writing.  Students who develop their thinking skills through mathematical investigation will become the clear-thinking lawyers, business people, artists, leaders, and citizens that our world so urgently needs. 


Carol S. Schumacher, Kenyon College

Michael Starbird, The University of Texas at Austin