Inquiry-based learning (IBL) manifests itself differently in different contexts. In particular, an IBL practitioner often modifies their approach from one class to the next. 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 think like mathematicians and to acquire their own knowledge by creating/discovering mathematics. IBL is a student-centered method of teaching that engages students in sense-making activities and challenges them to create or discover mathematics. Students are expected to actively engage with the topics at hand and to construct their own understanding. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, an IBL approach guides and mentors students through an adventure in mathematical discovery.

Students should—as much as possible—be responsible for guiding the acquisition of knowledge and validating the ideas presented. That is, students should not be looking to the instructor as the sole authority. In an IBL course, instructor and students have joint responsibility for the depth and progress of the course. While effective IBL courses come in a variety of forms, they all possess a few essential ingredients. According to Laursen and Rasmussen (2019), the Four Pillars of IBL are:

- Students engage deeply with coherent and meaningful mathematical tasks.
- Students collaboratively process mathematical ideas.
- Instructors inquire into student thinking.
- Instructors foster equity in their design and facilitation choices.

It is the responsibility of your instructor and class to develop a culture that provides an adequate environment for the last three pillars to take root. For additional information, check out my post What the Heck is IBL? that I wrote for Math Ed Matters.

Evidence in favor of some form of active engagement of students is strong across STEM disciplines. Freeman et al. (2014) 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 to active learning courses, and students in active learning courses outperformed students in traditional courses by 0.47 standard deviations on examinations and concept inventories. The following snippet from Freeman et al. (2014) captures the importance of utilizing active learning across STEM education:

The results raise questions about the continued use of traditional lecturing as a control in research studies, and support active learning as the preferred, empirically validated teaching practice in regular classrooms.

For IBL specifically, 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 (see Laursen et al. 2011; Laursen 2013; Kogan and Laursen 2014; Laursen et al. 2014). This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years.

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 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.
- 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.
- 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 were 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.

You can watch a short YouTube video of Sandra Laursen summarizing most of the recent research about inquiry-based learning here. The Conference Board of the Mathematical Sciences (CBMS) wrote the following in their position statement on active learning in 2016:

…we call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms.

In addition, the Manifesto of the MAA Instructional Practices Guide states:

We must gather the courage to advocate beyond our own classroom for student-centered instructional strategies that promote equitable access to mathematics for all students. We stand at a crossroads, and we must choose the path of transformation in order to fulfill our professional responsibility to our students.

Below is a list of IBL-related resources.

- The Academy of Inquiry-Based Learning is a hub for IBL in mathematics. I am a Special Projects Coordinator for AIBL.
- The IBL Blog focuses on promoting the use of IBL methods in the classroom at the college, secondary and elementary school levels. Sponsored by AIBL.
- AIBL’s YouTube Channel has a list of IBL-related videos.
- AIBL sponsors IBL workshops, which are excellent for anyone interested in learning to implement IBL in a college-level mathematics course.
- The Journal of Inquiry-Based Learning in Mathematics publishes university-level IBL course notes that are free, refereed, and classroom-tested.
- Inquiry-Based Learning SIGMAA is a special interest group of the Mathematical Association of America.
- 5 Elements of Effective Thinking by Edward B. Burger and Michael Starbird isn’t explicitly a book about IBL, but the book is written by two long-time practitioners of IBL and the themes of the book are relevant to anyone interested in teaching using student-centered approaches.
- A team of researchers at the University of Colorado led by Sandra Laursen have been working on the Inquiry-Based Learning in College Mathematics Project, which an extensive study on the effectiveness of IBL. The is quasi-experimental study examined over 100 courses at four different campuses using a longitudinal study that spanned two years.
- Math Ed Matters, which is an online column sponsored by the MAA. The column explores topics and current events related to undergraduate mathematics education.

In addition to the problem sequences available at JIBLM and the plethora of other notes scattered across the Internet, there are a few textbooks that are specifically designed for IBL. This list is far from complete.

- An Introduction to Proof via Inquiry-Based Learning by Dana C. Ernst (Northern Arizona University). I wrote these IBL course materials for an introduction to proof course. The first half of the notes are an adaptation of notes written by Stan Yoshinobu (Cal Poly) and Matthew Jones (California State University Dominguez Hills). I’d be thrilled if others decided to use these. Please let me know if you do.
- An Inquiry-Based Approach to Abstract Algebra by Dana C. Ernst (Northern Arizona University). IBL course materials for an abstract algebra course that emphasizes visualization and incorporates technology.
- Chapter Zero: Fundamental Notions of Abstract Mathematics by Carol Schumacher is a book designed for an IBL introduction to proof course. This excellent book is what I taught out of for my very first IBL experience. It also inspired much of the content of my An Introduction to Proof via Inquiry-Based Learning.
- A TeXas Style Introduction to Proof by Ron Taylor and Patrick X. Rault is an IBL textbook designed for a one-semester course on proofs that also introduces TeX as a tool students can use to communicate their work.
- Distilling Ideas: An Introduction to Mathematical Thinking by Brian Katz and Michael Starbird is another book ideal for an IBL intro to proofs class.
- Closer and Closer by Carol Schumacher is designed for an IBL undergraduate real analysis course.
- Number Theory Through Inquiry by David C. Marshall, Edward Odell and Michael Starbird is a wonderful IBL textbook for number theory.
- Euclidean Geometry: A Guided Inquiry Approach by David M. Clark (SUNY New Paltz) is written for an undergraduate axiomatic geometry course, is particularly well suited for future secondary school teachers.
- Topology Through Inquiry by Michael Starbird and Francis Su is a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students.

IBL has its roots in an instructional delivery method known as the Moore Method, named after R.L. Moore. Moore’s connection to IBL is controversial due to his troubling sexist and racist biases. However, there’s no doubt that his approach to teaching has had a tremendous impact on how many approach active learning in mathematics. In The Past, Present, and Future of the IBL Community in Mathematics, Laursen, Haberler, and Hayward summarize the issue succintly:

The connection between IBL and R. L. Moore, and his way of teaching, is important. To many long-time faculty members and instructors, this connection cannot be overstated. Without the early network of Moore’s former students, and without the financial and organizational efforts of Harry Lucas, the IBL community as it is currently formed would not exist. In 2016 at the Joint Math Meetings, Harry Lucas received special recognition for his Educational Advancement Foundation’s continuous efforts to promote active instruction of mathematics over the last two decades. This award was well-earned.

However, although the connection between IBL and Moore has played an important role in the group’s history, our analysis of the interview data suggests that it was also a barrier to further spread of IBL teaching. First, use of the name ‘Moore method,’ or even ‘Modified Moore method,’ during the early years did not provide insight into the nature of that approach. To understand the reference, instructors had to already know Moore or his academic descendants; the name did not describe the teaching. This issue is less salient now, as the term IBL has come into common use to describe the set of teaching approaches used by community members. Indeed, the use of this terminology, along with the expanded range of beliefs and specific classroom practices included under the inquiry umbrella, has coincided with the growth of the community in the past few years. Our studies show that this broadened conception of IBL is demonstrably supportive of new instructors as they decide whether to try IBL in their classrooms (Hayward, Kogan, & Laursen, 2016).

The second issue, however, remains a barrier to growth. Moore’s troubling sexist and racial biases are well known in the IBL Math community, and references to them are common in our interview data. Significantly, also common are stories about how the association of this teaching approach with Moore’s social biases has led some instructors to choose not to participate in IBL events, even though they may otherwise be interested in this teaching-centered community. Thus it is clear that, in today’s society, the symbolic connection between Moore and IBL is a problem for the spread of IBL. Our data suggest that failure to explicitly address the community’s history with Moore will allow this negative association to linger and may limit the growth of IBL in the future.

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