# Why use inquiry-based learning in math for liberal arts courses

**Written by:** Dr. Phil Hotchkiss with Dr. Julian Fleron

### Featured student work:

**Video:** Proud Student Sharing**Painting:** The Infinite**Poem:** Looking Deeper at $\sqrt{-1}$**Song:** A Mathematics Student's Lament**Quote:** On the first day...

Based on surveys and mathematical autobiographies, it is clear that the experiences in previous mathematics classes for most of our mathematics for liberal arts (MLA) students has been largely painful and frustrating and we suspect this is true for many MLA students; it is very unlikely that these students will be successful and rethink their opinions about mathematics in a traditional lecture course. The main themes we hear from our students are that they do not like mathematics, it feels disconnected from their lives and they do not have high expectations for themselves in this class. These are among the reasons why inquiry-based learning (IBL) is perfect for MLA classes.

The goals of MLA courses can be lofty. Particular skills such as computing derivatives, performing T-tests or optimizing linear programs are not required outcomes. Instead, the content can involve some of the subject's great ideas, famous problems, recent advances, etc. Larger cognitive and metacognitive goals—reading, writing, reasoning, problem-solving, reflection, personal responsibility—can take precedence. IBL supports this broader content and these goals in ways that are deeper than a standard lecture approach.

Public policy documents universally assert the need for mathematics pedagogy to be less lecture driven and less procedural. However, in 2005, a staggering 79% of MLA courses at two-year colleges were taught using the "standard lecture mode" and by 2010 this had risen to 85%!! (2010 CBMS report)

IBL provides a format in which students learn mathematics by actively **doing** mathematics instead of just passively listening to the teacher present the mathematics. This approach requires them to invest more of themselves in doing mathematics and their learning; MLA students will rise to the responsibility of taking charge of their learning when we allow students the space to do mathematics and become mathematicians. As a result, their work, understanding, and confidence improve. Moreover, there is a more positive energy in the classroom because the students actually enjoy coming to class doing the work. Despite their professed dislike for mathematics, they become interested in the topics and want to find answers to the questions. In fact, a few days into the course many of the students discover, much to their surprise, that they are so engrossed in the mathematics, they are unaware that class time is over.

IBL enables MLA students to experience, and hence, appreciate mathematics in an entirely different way—one that is more indicative of the subject's true nature. They begin to understand why someone would find mathematics to be beautiful, interesting and worthy of intellectual study. In a liberal arts education, this is important in its own right. But it also has broader implications. If a student's memory of their last mathematics class is a positive one, when they are parents they will be more likely to have a positive influence on their children's perceptions of mathematics. Moreover, many MLA students will eventually take on roles in which they will be making important decisions about mathematics and their memories of their mathematics classes will influence their decisions.

Many MLA students are humanities students, majoring in English; music; art; theater; history; and communications, among other subjects; and they are used to being actively involved in their classrooms. The collaborative aspect is an essential part of their learning and the small group work used in IBL classrooms provides a familiar setting to work on and learn mathematics.

MLA students are also used to creating something as an important part of their learning such as poems, songs, stories, art, or posters, and the IBL environment allows students to use their creative ways to learn mathematics.

Using IBL in MLA classes provides opportunities for learning about both mathematics and teaching. In an IBL setting, MLA students can draw on their own expertise and interests, to come up with interesting and successful approaches to mathematical problems that have not occurred to the instructor. By walking around and listening to the student groups, the instructor gets a better understanding about what students understand, why students may be struggling and ways to improve the course for next time.

At the conclusion of Mathematical Explorations, our university's mathematics for liberal arts course, students create "a reflective piece" on their experience in the course. They write in their own voice, using whatever vehicle seemed most suitable to express their experience. We have received videos, visual art, narratives, letters, journals, poems and songs. **Shane Murray's poem** is exceptional in the way that it weaves substantive mathematics into what is essentially a lyrical poem. **Joshua Loell's wonderful song** stands on its own, but is also notable because of its larger connection—the title and context play off of "A Mathematician's Lament" by Paul Lockhart which was required reading in the course.

Although it is an unpopular course with many faculty (according to the 2007 CBMS report, 55% of MLA students at four-year colleges are taught by faculty who are neither tenured nor tenure-track) we find that using IBL to teach MLA is among the most rewarding courses that we teach. The students usually arrive tense and disinterested, having been disenfranchised from meaningful mathematical experiences. Engaged as a central part of the mathematical experience by IBL, the students leave the course empowered.

And you can hear this in our student's own words:

"On the first day of Math Explorations... [our teacher] told his students that he was giving us a mathematical paintbrush, and that he wants us to paint. That he was giving us a mathematical tambourine; to make our own music. And this is exactly how class should be taught... He gave us the opportunity and it was up to us to create something amazing and beautiful with what he had given us. And that is exactly what we did. The fact that it was never easy to find an answer to the problem made me want to find it so much more. I would sit at my table with my three classmates and work so hard on one problem until we got the guts to ask for help. But we were so personally determined to get the answer ourselves that we almost did not want to hear what we were doing wrong. And even when we asked for help, we were never

toldthe answer. Rather we were given suggestions to turn different corners to possibly find the answers there. It taught us to think for ourselves. And it taught us so much more than just the way to get the answer to certain problems. The worst feeling in the world was to look up at the clock and realize that class was over seven minutes ago and you still have not figured out the solution. Yet the best feeling was when you figured out a solution and could not stop smiling for the rest of class because you had figured out that one question that had been pushing on your shoulders since the first time you read it.

—DAoM Student