## What is IBL?

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.

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.

## The Moore Method

IBL has its roots in an instructional delivery method known as the Moore Method (sometimes referred to as the Texas Method), named after R.L. Moore. In 1966, Moore wrote:

That student is taught the best who is told the least.

In the words of J. Parker:

Robert Lee Moore (1882-1974) was a towering figure in twentieth century mathematics, internationally recognized as founder of his own school of topology, which produced some of the most significant mathematicians in that field. The 50 students he guided to their PhDs can today claim 1,678 doctoral descendants. Many of them are still teaching courses in the style of their mentor, known universally as the Moore Method, which he devised. Its principal edicts virtually prohibit students from using textbooks during the learning process, call for only the briefest of lectures in class and demand no collaboration or conferring between classmates. (Exceptions were Moore’s calculus and analytic geometry courses in which textbooks were used for setting problems. His doctoral students were allowed to refer to the literature mainly to ensure their theses were original.) It is in essence a Socratic method that encourages students to solve problems using their own skills of critical analysis and creativity. Moore summed it up in just eleven words: ‘That student is taught the best who is told the least.’

Loosely speaking, the majority of a Moore Method course consists of students presenting proofs/solutions that they have produced independently from material provided by the instructor. In a traditional Moore Method course, students are discouraged, in fact forbidden, to collaborate. Variations of the Moore Method take many forms and are often referred to by the generic name “modified-Moore method.” In particular, one modification I make is that I not only allow students to work together, I encourage it. The Moore Method or one of its modifications is typically associated with pedagogies including inquiry-based, discovery-based, student-centered, Socratic, and constructivist learning. For more detailed information, including history, of Moore and his method, check out A Quick-Start Guide to the Moore Method.

## IBL Resources

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.
- You can find my blog posts related to IBL here.
- Math Ed Matters is a (roughly) monthly column sponsored by the Mathematical Association of America and authored by me and Angie Hodge. The column explores topics and current events related to undergraduate mathematics education. Posts will aim to inspire, provoke deep thought, and provide ideas for the mathematics classroom. Our interest in and engagement with IBL color the column’s content.
- 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 an excellent for anyone interested in learning to implement IBL in a college-level mathematics course.
- The Inquiry-Based Learning in Mathematics Community on Google+ provides a forum for discussing IBL. This is a great place to ask questions.
- The Journal of Inquiry-Based Learning in Mathematics publishes university-level IBL course notes that are free, refereed, and classroom-tested.
- I have written some IBL course notes for an introduction to proof course. You can find the GitHub repository for the LaTeX source, by going here.
- 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. The Legacy Project hosts the Legacy of RL Moore Conference each summer.

## IBL Textbooks

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 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.