Bringing Inquiry to the First Two Years of College Mathematics

Abstract

In this editorial, we provide an introduction to the special issue on Inquiry-Based Learning in First and Second Year Courses. We also discuss the essential features of inquiry-based learning and provide a brief overview of the literature and evidence for its effectiveness.

Keywords:

• Inquiry-based learning
• IBL
• active learning

Inquiry-based learning (IBL) is a teaching paradigm that challenges instructors to involve students in the production of mathematics. For students, the challenges mimic those working mathematicians face: students should have the opportunity to experiment, solve problems, make conjectures, and to construct mathematical arguments. For instructors, the challenge lies in creating an environment where the students can have an authentic mathematical experience at a level appropriate for their development. An instructor must provide a good sequence of tasks and a positive, supportive group dynamic so that students can experience both successes and failures, each in a measure appropriate for fostering their growth as mathematicians.

There is a long tradition of using IBL methods in advanced courses where the accepted standard for argument is rigorous, axiomatic proof. In the United States, such courses tend to be of a reasonable size, have sufficiently flexible content goals, and possess a significant focus on proof-writing process. As a consequence, instructors can structure classes around student presentations and discussion of arguments students have made independently or in small groups. In contrast, it is often a challenge to bring an inquiry-based pedagogy into classes with a large number of students, prescribed content expectations, or a heavy computational focus.

Although IBL may be implemented in a variety of ways, there are core features that all IBL classes share. Among these features are the so-called twin pillars: deep engagement in rich mathematics and opportunities to collaborate (in and outside of the mathematics classroom) [3]. According to Laursen et al.:

The twin pillars reinforced each other: after struggling with a problem individually, students were well prepared to contribute meaningfully during class, and interested in the solutions that others proposed. Collaboration in turn motivated them to complete the individual work. It also made class enjoyable, encouraged clear thinking, and built communication skills.

The twin pillars characterize the role that the students play in an IBL environment. Rasmussen and Kwon [5], Rasmussen et al. [6], and Rasmussen and Wawro [7] describing a third pillar that focuses on instructor interest and use of student thinking. That is, instructors should inquire into student thinking.

No single teaching technique transforms a classroom into an IBL environment. Instead, instructors can place their courses on a broad spectrum of IBL implementations by paying attention to questions about who does the intellectual labor and choosing context-appropriate modes for student work. In roughly increasing levels of sophistication, an instructor can turn over responsibility for these kinds of things to the students:

• presenting new ideas;

• developing new ideas for discussion;

• critiquing the mathematics presented;

• making final determinations on the validity and completeness of an argument; and,

• asking questions, making conjectures, and setting the agenda for further investigation.

Evidence in favor of some form of active learning is strong across STEM disciplines. Freeman et al. conducted a meta-analysis of 225 studies of various forms of active learning, and found that students were 1.5 times more likely to fail in traditional courses as compared with active learning courses, and students in active learning courses outperformed students in traditional courses by 0.47 standard deviations on examinations and concept inventories [1].

For IBL specifically, there is a growing body of research that supports the claim that this pedagogical approach is effective for teaching mathematics and for fostering positive attitudes about mathematics. A research group from the University of Colorado Boulder led by Sandra Laursen conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students’ and instructors’ experiences of IBL (e.g., see [2] and [3]). This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years. The courses that were examined self-identified as IBL versus non-IBL and observations were used to verify that the IBL sections were indeed different. The IBL sections shared the following characteristics: students solve challenging problems alone or in groups; student solutions are shared and then analyzed, critiqued, and refined; class time is used for student-centered activities with students playing a leadership role; the course is driven by a carefully built sequence of problems, where the pace is set by the students’ progress; course goals emphasize thinking skills and communication while content “coverage” is less central.

Laursen et al. [3, 4] determined that IBL students reported higher learning gains than their non-IBL peers, across cognitive, affective, and collaborative domains of learning. In later courses, students who had taken an IBL course earned grades as good or better than those of students who took non-IBL sections. Non-IBL courses show a substantial gender gap: across the board, women reported lower learning gains and less supportive attitudes than did men (effect size 0.4–0.5). Women’s confidence and sense of mastery of mathematics, and their interest in continued study of math were lower. However, this gender gap was absent in IBL classes. IBL approaches leveled the playing field for women, addressing a problematic dynamic for women without disadvantaging men.

This special issue gathers together work on the challenges and opportunities met when bringing an IBL mindset to teaching first- and second-year college mathematics. In particular, these papers exhibit uses of the twin pillars of IBL and show how instructors can inquire about their students’ thinking in a variety of courses common in early college study.

The issue begins with a discussion of design principles for creating and revising IBL activities by Green and von Renesse, whose general questions are situated by sharing examples of the process from the calculus sequence. One can see how the twin pillars for student work inform their design structure and also how they reflect on the nature of student thinking throughout. The importance of the instructor’s inquiry into student thinking is highlighted there, as it is the first question addressed in their work.

The next paper by Shelton shares a use of the Process-Oriented Guided Inquiry Learning format for group work, which is a tightly structured way to manage the opportunities for collaboration. There are two activities, one on the Squeeze Theorem and one on the Fundamental Theorem of Calculus, and each is followed by student reflections on the learning process. These reflections form the basis of Shelton’s inquiry into student work.

Another example of an IBL activity for the calculus sequence is exhibited by Ekici and Gard. Their approach is much more open, taking several weeks of class time for the students to discover the basic transcendental functions and their properties. In this case, the twin pillars are highlighted throughout, as the students engage in explorations and discussion of some deep mathematics. The instructor’s inquiry is inside the talk moves and discussion management practices the instructors use to guide student thinking.

Although the calculus sequence is often “the” mathematics pathway required for all STEM majors, there are many other lower-level courses that can be impactful on both society’s views of mathematics and students’ future careers, especially when using an IBL approach. Piercey and Militzer share their process in bringing IBL methods to a quantitative reasoning course designed for business students, with an emphasis on modeling. Their careful attention to student beliefs about the nature of mathematics explicitly informs their lesson design. Because of this, they are able to bring their students into mathematics so successfully as to show a sharp increase in student retention rates compared to the traditional college algebra course they replaced.

Toews shows us how he implements IBL in an introductory statistics course. This course has several interesting features including a computational focus, exploratory work using a statistical software package, pair programming, and regular lab reports. Though these structures are unusual for mathematics courses, Toews brings them out as powerful and context-appropriate manifestations of the twin pillars.

The next paper by White describes a short unit constructing the natural numbers from the Peano axioms in an IBL presentation-style appropriate for students making the transition to rigorous axiomatic mathematics and illustrates a way to implement IBL in a week or two of class time. This small unit could provide an instructor new to IBL a simple way to try it out without making a huge commitment of class time.

The final paper by Cooper, Bailey, Briggs, and Holliday highlights the importance for instructors who wish to use IBL methods to attend to student beliefs about what mathematics is and what constitutes learning in mathematics. For any “new” teaching environment, student buy-in is essential and can be hard to earn. Students’ prior beliefs must be accounted for whenever making a choice in course structure that takes students out of their comfort zones. Many of the earlier papers in this volume address these questions specifically in context, and we hope the reader notes some of the successful strategies on display.

Additional information

Notes on contributors

Dana C. Ernst

Dana C. Ernst is a professor of mathematics at Northern Arizona University. His primary research interests are in the interplay between combinatorics and algebraic structures. In addition, he is passionate about mathematics education and his scholarly activities include topics in this area with a specialization in inquiry-based learning. Lastly, he is an avid cyclist and drinks a copious amount of coffee.

Theron Hitchman

Theron Hitchman is a professor of mathematics at University of Northern Iowa. His interests include inquiry-based learning, undergraduate research, and a mix of geometry, topology and dynamics, especially knot theory. When not doing mathematics, he can be found enjoying soccer or telling ridiculous stories with his children.

Angie Hodge

Angie Hodge is a professor of mathematics at Northern Arizona University. Her research interests are in mathematics education with specialty areas in inquiry-based learning and gender equity in the STEM disciplines. In her free time, she enjoys running ultra marathons and all too often eats cereal for dinner.