## Math Without Math: Teaching Problem Solving to First-Year Students

Higginbottom discusses his experience teaching first-year college students, particularly those who might be “math-averse,” about mathematical problem-solving in a liberal arts context.

Mathematics sits comfortably within the tent of the liberal arts. Any student of the history of liberal education could make a neat list of the trivium and quadrivium and convince a skeptic of this fact within minutes. Mathematics has stood as one of the foundations of all types of education for millenia.

Mathematics is also one of the core STEM (science, technology, engineering, and mathematics) disciplines which have received so much attention within the last ten years. Popular wisdom states that students with a STEM major are poised to make a difference in the world and land a high-paying job directly out of college. The divide between these so-called “career-ready majors” and the liberal arts is growing in American culture, fueled perhaps by popular lore and the media. The implication, while not novel, is powerful—students hear that the liberal arts aren’t practical; they may be nice stimulation for the high-minded, but when citizens need to accomplish important tasks they turn to disciplines offering more technical expertise.

Recent interviews and surveys have refuted this tired cliché. In an article for Fast Company [2], Elizabeth Segran spoke to top officials at several technology companies. She writes of many CEOs with liberal education in their backgrounds and she quotes several executives who explain in detail how a liberal arts background can contribute to success in the technology sector. In the article, these specific benefits of a liberal education are paired with general reflections as well. The headline quote comes from Georgia Nugent, former president of Kenyon College: “It’s a horrible irony that at the very moment the world has become more complex, we’re encouraging our young people to be highly specialized in one task.”

In 2013 the Association of American Colleges and Universities released findings from a national survey of business and nonprofit leaders [1]. Participants in this survey showed strong support for the ideals of the liberal arts. Eighty percent of employers “agree that, regardless of their major, every college student should acquire broad knowledge in the liberal arts and sciences.” This appreciation for the liberal arts went beyond content, however; these leaders also valued the skills produced by training in the liberal arts. Ninety-three percent of those surveyed agreed that “a demonstrated capacity to think critically, communicate clearly, and solve complex problems is more important than [a candidate’s] undergraduate major.”

Mathematics shines as one of the liberal arts in which students gain these skills that employers find so valuable. In particular, a freshman seminar offers an opportunity to teach problem solving to students in a way that can impact the rest of their education, even if their remaining semesters are math-free.

### A First-Year Seminar

When tasked with designing a First-Year Seminar (FYS) at Washington & Jefferson College, I wanted subject matter connected to mathematics. However, because so many incoming students are math-averse, I wanted the course to be free of mathematical pre-requisites. So I focused on one of the fundamental skills of working mathematicians: problem solving.

I developed a course called Math Without Math: Unraveling Riddles and Solving Problems. My class attracted a mix of the mathematically-inclined and the mathematically-challenged; the word “math” and the phrase “without math” in the title of the course were chosen for just this purpose.

This is the way I introduced the class on my syllabus.

[While I have made significant additions and modifications, this course had its genesis with Dr. Randall Helmstutler and Dr. Keith Mellinger of the University of Mary Washington. All borrowing was done with permission.]

Do you like brain-teasers, puzzles, and logic games? Then this is the class for you! This is a course in problem-solving and critical thinking. We will learn a host of problem solving techniques and strategies, and we will use them to understand and solve numerous puzzles, problems, and riddles. All of these techniques are well-known to mathematicians, but we will work on problems that may only have a loose connection to mathematics at first glance. We will spend the majority of our time together working as a class and in smaller groups to discover these techniques and apply them to particular problems. There is a small amount of traditional mathematics in this class, and techniques from high school algebra will surface occasionally, but we will primarily be dealing with mathematical ideas instead of mathematical content.

On most class days, I introduced a topic to the class with a ten-minute mini-lecture or example. I then split the class into small groups to work on similar examples and advance the concepts. Students might have to find an optimal strategy for winning a game or determine which questions would lead to the answer to a riddle. Small group sessions usually ended with the students handing in a written summary of their work.

### Mathematical Problem Solving

On three occasions during the semester, students were responsible for solving a harder problem on their own. After choosing and solving a problem from a list I provided, they wrote a solution and gave an oral presentation to the class. I stressed the importance of a clear explanation that all students in the class could understand. As part of the assignment, students had to figure out either an extension or a generalization for their problem. For some students, this was their first encounter with applying a particular solution to a parallel or more general problem.

As a textbook for the class, I used The Art and Craft of Problem Solving, by Paul Zeitz ^{[3]}. The book is written as an introduction to mathematical problem solving and is aimed at students who have taken Calculus and who enjoy mathematics broadly. Many students in my class, however, met neither of these criteria and survived the course without incident.

In his book, Zeitz makes a helpful distinction between strategies for investigating problems (chapter 2) and tactics for solving problems (chapter 3). Strategies are content-independent approaches to problems. (Examples of strategies include “change your point of view” and “recast the problem.”) On the other hand, the tactics that Zeitz introduces bring students face-to-face with mathematical problem solving: in chapter 3 he discusses symmetry, the extreme principle, the pigeonhole principle, and invariants. While far from a complete list of the tools in a mathematician’s problem-solving toolbox, these tactics provide a gentle introduction to the practice. One benefit of studying these four tactics is that all of them have nice visual examples, so they can be explained to any student regardless of mathematical background. (The book goes into other tactics which are more mathematical in nature, like graph theory, combinatorics, and number theory. We skipped these later chapters.)

### Solving Problems in the Real World

After teaching the course twice, I made a concerted effort to bring “real world” problem solving into the class on my third attempt. As part of this effort, we spent more time as a class thinking carefully about the strategies for solving problems mentioned earlier. (I have included a list of the thirteen strategies from the first and second chapters of the book in an appendix.) Since these strategies have application far beyond the world of mathematical puzzles and games, I took several opportunities to make my students aware of their use outside of the classroom.

I assigned a final paper for which students had to investigate an example of problem solving in history. I encouraged students to consider examples like the capture of an elusive criminal, the building of an innovative building or bridge, or the negotiation of a peace treaty. As they learned about these problems and their solutions, students were to draw out a lesson about problem solving from their research. In particular, they were to comment on the use of any of the problem-solving strategies we discussed in class.

Toward the end of the course, I added two class days focused on non-mathematical problem solving. I assigned students each to read a Sherlock Holmes short story and to write a summary of the story highlighting the key ingredients of the solution to the problem as well as what broad problem-solving principles were evident. During class, I asked students working in small groups to agree on two problems that exist at Washington & Jefferson College, describe these problems, and brainstorm about the solutions. Who is responsible for the solution? Why? What attempts (if any) have been made to solve these problems in the past? How would you approach this problem if it were your responsibility?

### Conclusion

I try not to invest too much in students’ end-of-course evaluations, but the comments from the third iteration of the course were generally positive. In particular, many students thought their study of problem solving would benefit them in the future.

This First-Year Seminar is an example of how problem solving can be taught to college students in a forward-looking way. There may be more effective ways to instill this valuable skill, but this summary describes my best effort to date in which students would need little or no mathematical preparation.

#### Dr. Ryan Higginbottom

Dr. Ryan Higginbottom received his Ph.D. in Mathematics from the University of Virginia (2012) and his B.A. in Mathematics (with honors) from Bucknell University (1999). Currently an Associate Professor of Mathematics at Washington & Jefferson College, Dr. Higginbottom’s research focuses on the teaching of mathematics to undergraduate students. He is currently working on a project analyzing the use of Mastery Exams in Calculus I classes.