*By Dana C. Ernst*

As regular readers of this blog know, I am passionate about inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has had a profound impact on my life as a teacher. Actually, scratch that. It has had a profound impact on my life!

When I started teaching, I mimicked experiences I had as a student. I tried to emulate my favorite teachers. Because it was all I really knew, I lectured. And this seemed to work out just fine. By standard metrics, I was an excellent teacher. Glowing student and peer evaluations, as well as reoccurring teaching awards, indicated that I was effectively doing my job. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to: (i) heavily depend on me to be successful, and (ii) retain only some of what I had taught them. In the words of Dylan Retsek:

*"Things my students claim that I taught them masterfully, they don’t know."*

Inspired by a desire to address these concerns, I began transitioning away from direct-instruction towards a more student-centered approach. The goals and philosophy behind IBL resonate deeply with my ideals, which is why I have embraced this paradigm.

While there is variation in practice, IBL courses typically provide students with experiences that differ from traditional lecture-based courses. In many mathematics classrooms, doing mathematics means following the rules dictated by the teacher and knowing mathematics means remembering and applying these rules. However, an IBL approach challenges students to create/discover mathematics. According to the Academy of Inquiry-Based Learning, IBL is a method of teaching that engages students in sense-making activities. Students are given tasks requiring them to conjecture, experiment, explore, and solve problems. Rather than showing facts or a clear, smooth path to a solution, the instructor guides students via well-crafted problems through an adventure in mathematical discovery.

I believe that there are two essential elements to IBL. First, students should (as much as possible) be responsible for guiding the acquisition of knowledge, including the pace at which this happens, and second, they should be responsible for validating the ideas presented. That is, students should not be looking to the instructor as the sole authority. In most IBL courses, student-led presentations and small group work form the backbone of the course. In general, the majority of class time is spent on these types of student-centered activities, which provide ample opportunity to discuss and critique ideas that arise from a group problem or student-presented solution.

One guiding principle of IBL is the following question:

*Where do I draw the line between content I must impart to my students versus the content they can produce independently?*

While instructors might give a mini-lecture to introduce or summarize the day's work, the instructor's main role is not lecturing, but rather to foster a safe environment, facilitate discussion, and redirect as necessary. In an IBL course, instructor and students have joint responsibility for the depth and progress of the course.

A research group from the University of Colorado Boulder lead by Sandra Laursen has conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students’ and instructors’ experiences of IBL. This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years. The courses examined self-identified as IBL versus non-IBL. Classroom observation was used to verify that IBL classes were indeed different from those designated as non-IBL sections of the same course. The following is a list of characteristics that the IBL sections shared:

- Students solve challenging problems alone or in groups; share solutions; analyze, critique & refine their solutions
- Class time is used for these student-centered activities
- Students play a leadership role
- Activities change often
- Course is driven by a carefully built sequence of problems or proofs, rather than a textbook
- Pace is set by students' progress through this sequence
- Course goals usually emphasize thinking skills & communication; content “coverage” is less central
- Instructor serves as "guide on the side" not "sage on the stage"—manager, monitor, summarizer, cheerleader

On average over 60% of IBL class time was spent on student-centered activities including student-led presentations, discussion, and small-group work. In contrast, in non-IBL courses, 87% of class time was devoted to students' listening to an instructor talk. In addition, the IBL sections were rated more highly for a supportive classroom environment and students conveyed that engaging in meaningful mathematical tasks while collaborating was as essential to their learning.

Below is a brief summary of some of the outcomes of Laursen et al.'s work:

- After an IBL or comparative course, IBL students reported higher learning gains than their non-IBL peers, across cognitive, affective, and collaborative domains of learning.
- IBL students’ attitudes and beliefs changed pre- to post-course in ways that are known to be more supportive of learning, compared to students who took the non-IBL sections.
- 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, despite having "covered" less material.
- On a research-based test of students' ability to evaluate proofs, IBL students showed evidence of greater skill in recognizing valid and invalid arguments and of the use of more expert-like reasoning in making such evaluations. The volunteer sample consisted of only high-ability students; no instructors gave the test to all students during class time.
- Non-IBL courses show a marked 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, was lower. This difference appears to be primarily affective, not due to real differences in women’s mathematical preparation or achievement.
- This gender gap was erased in IBL classes: women’s learning gains were equal to men’s, and their confidence and intent to persist similar. IBL approaches leveled the playing field for women, fixing a course that is problematic for women yet with no harm to men.
- When sorted by prior achievement, the grades of most students (IBL and non-IBL alike) dropped in subsequent courses as course work became more difficult. But grades of initially low-achieving students who had taken the IBL course rose 0.3-0.4 grade points, unlike their low-achieving, non-IBL peers, and unlike their higher-achieving classmates.

This work determined that there are two "twin pillars" of IBL that may explain the student outcomes, namely (i) deep engagement in rich mathematics and (ii) opportunities to collaborate. Here, deep engagement refers to individual and group effort in tackling meaningful problem-solving tasks that are not merely "busy work." Collaboration is a key component as students learn from explaining their ideas and trying to understand others. 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."*

IBL is not a magic bullet, but the experiences that I have had watching students transform into independent learners is why I am so passionate about it. I want students to have life-changing experiences! Learning the content of mathematics is just a bonus.

One of my principal goals is to make my students independent of me. I want them to experience the unmistakable feeling that comes when one really understands something thoroughly. In the words of Carol Schumacher:

*"When one can distinguish between really knowing something and merely knowing about something, that individual will be on his/her way to becoming an independent learner."*

I’m not terribly picky about the particular flavor of IBL or active learning one chooses to employ, but it is becoming increasingly clear to me that if we want to produce life-long independent learners, then the twin pillars need to form the foundation for the pedagogical approach we choose to take.

**References**

[1] Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study.

*Journal of Research in Mathematics Education*, 45(4), 406-418.

[2] Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics.

*Innovative Higher Education*, 39(3), 183-199. DOI 10.1007/s10755-013-9269-9.

[3] Laursen, S. L. (2013). From innovation to implementation: Multi-institution pedagogical reform in undergraduate mathematics. In D. King, B. Loch, L. Rylands (Eds.),

*Proceedings of the 9th DELTA conference on the teaching and learning of undergraduate mathematics and statistics*, Kiama, New South Wales, Australia, 24-29 November 2013. Sydney: University of Western Sydney, School of Computing, Engineering and Mathematics, on behalf of the International Delta Steering Committee.

[4] Assessment & Evaluation Center for Inquiry-Based Learning in Mathematics (2011).

*Evaluation of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics*. (Report to the Educational Advancement Foundation and the IBL Mathematics Centers) Boulder, CO: University of Colorado, Ethnography & Evaluation Research.Further resources can be found here.