Let me begin by stating that this is a really difficult question to answer! Inquiry-based learning (IBL) manifests itself differently in different contexts. In particular, an IBL practitioner often modifies his/her 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. In general, IBL is a student-centered method of teaching mathematics.

According to the Academy of Inquiry-Based Learning, IBL engages students in sense-making activities. Students are given tasks requiring them to:

- solve problems,
- conjecture,
- experiment,
- explore,
- create,
- communicate.

Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). These are known as the “twin pillars” in Sandra Laursen’s work on IBL.

Perhaps this is sufficiently vague, but I believe that there are two essential elements to IBL. 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.)

For me, the 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?

E. Lee May (Salisbury State University) may have said it best:

Inquiry-based learning (IBL) is a method of instruction that places the student, the subject, and their interaction at the center of the learning experience. At the same time, it transforms the role of the teacher from that of dispensing knowledge to one of facilitating learning. It repositions him or her, physically, from the front and center of the classroom to someplace in the middle or back of it, as it subtly yet significantly increases his or her involvement in the thought-processes of the 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 a bit controversial, but there’s no doubt that his approach to teaching has had a tremendous impact on how many approach active learning in mathematics. For additional information, check out my post What the Heck is IBL? that I wrote for Math Ed Matters. Also, see Why use IBL? at the Academy of Inquiry-Based Learning.

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.
- Math Ed Matters, which is an online column sponsored by the MAA. The column explores topics and current events related to undergraduate mathematics education.
- 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.
- The Moore Method — A Pathway to Learner-Centered Instruction by Charles A. Coppin, W. Ted Mahavier, E. Lee May, and Edgar Parker is an excellent book that provides an overview of what the Moore method is and how to implement it.
- 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.
- The Educational Advancement Foundation is a philanthropic organization that supports the development and implementation of IBL and the preservation and dissemination of the Moore method.
- The Legacy of RL Moore Project is primarily concerned with preserving the historical record, producing a comprehensive biography of R.L. Moore, and supporting research by others that relate to aspects of his life, work and influence.

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.
- 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.

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