Brian Katz: A Case for Teaching Inquiry

In this blog, Brian Katz is connecting beautifully the issues of equity and teaching using inquiry. He also promotes a special edition of PRIMUS which focuses specifically on inquiry-based learning.

A Case for Teaching Inquiry (by Brian Katz)

When I think back on graduate school, a recurring metaphorical image comes to mind: I'm walking along a plateau towards the edge that separates the high mesa from the low valley with an intimidating cliff. The symbolism is pretty simple. On the plateau, mathematics is the work of the classroom, in which virtually every task set before me was answerable with the tools I'd been told to pack in my bag. In order to continue past the cliff as a researcher, I needed to learn to fly or face a painful fall to the valley.

Contrary to this rather bleak image, I had many privileges that supported me through this challenge. My understanding of the concepts from my undergraduate curriculum was strong and solid. I did not have the kind of financial stressors that make continuing graduate study challenging. My parents both have advanced degrees, so they understood and supported further schooling, and I had sustained mentoring relationships with several faculty. I had some experience with research through an REU and an undergraduate thesis. I was generally confident and thought of myself as flexible and creative. When people looked at me or my CV, they assumed that I belonged in mathematics and would succeed. In short, I was aware that the cliff was coming and had plenty of provisions, but no amount of planning for flight could turn my arms into wings. In the end, I think I'm one of the lucky ones who learned to fly while falling.

As an educator, I want to change the landscape of the mathematical journey for my students because I think all people should feel empowered to ask and explore mathematical questions so that they can fly upwards from any spot, whether on the mesa or in the valley. I hold it as a self-evident truth that all people are capable of meaningful mathematical inquiry; however, I know that the supports needed to reach this goal depend on context, and I acknowledge that I need to build a diverse set of skills and a robust infrastructure to accomplish it.

I think that we, as a discipline and community of educators, can rise to this challenge by designing our curricula for equitable access to inquiry experiences and intentional coherence throughout those experiences. And yet, inquiry experiences are not completely commonplace yet. Even among colleagues who would like to infuse their teaching with more inquiry, I have seen people with concerns that not all students are capable of engaging in inquiry, with negative experiences from past experiences trying to teach with inquiry, and with even imagining what inquiry could be in their classrooms.

I have already declared that I believe all students are capable of inquiry, but I can also understand this confidence comes from my experience of teaching with inquiry across a wide variety of contexts. If you would rather not have simply to trust me, then perhaps you could trust the very people who express this concern. I've heard that inquiry works for the "strong" students, but not the "weak"; that it works for the weak but not the strong; and that it works for the middle group in a course but not the extremes. I've heard that it works for the future teachers but not the Math/Science/Engineering majors, and I've heard that it works for everyone but the teachers. I've heard it works in lower-division courses but not upper, and upper but not lower; service but not major and major but not service. Only in long class periods and only in short. Only in college, and yet we expect it in every elementary school classroom. Significantly, each of these commenters sees evidence that inquiry works in some greener pasture but hasn't seen it work in their context. Our beliefs about students' capacity for inquiry can support or undermine equitable access; rather than asking if students can participate in inquiry as we might design it, we must develop strategies that make space in our classrooms for the competencies students bring to us and help them grow.

The concern about fear of failure is powerful because changes to our teaching can threaten our very identities as educators, not to mention all of the practical challenges from student resistance to the time and energy needed for intentional re-design to access to professional support resources. I believe that when we, as teachers, subtly frame inquiry as unusual or separate from the rest of our course work, we invite students to resist or disengage. When inquiry is integrated as a consonant part of a course, we teach students that it is a normal part of doing mathematics, and I think that many of these fears melt away.

In some sense, awareness of multiple and diverse examples of teaching with inquiry can address all of these concerns, which leaves me with two related questions: what is inquiry, and how do we support its development in students?


In the service of this goal, Elizabeth Thoren and I proposed and edited a special issue of PRIMUS entitled "Teaching Inquiry" that contains 19 papers and two editorials and is organized around these two questions. Part I, entitled "Illuminating Inquiry", focuses on the nature of inquiry, from discussions of its theoretical foundations and generalizations across disciplines to descriptions and analyses of the experience of inquiry from the inside. Part II, entitled "Implementing Inquiry", focuses on approaches to offering inquiry experiences, from discussions of strategies to change student and instructor behaviors to descriptions and analyses of course design and project structures. Of course, a reader will find insight into both the nature of inquiry and approaches to achieving it in any paper in either part, and each part contains ideas for both instructors who have experience teaching with inquiry and those who are hoping to start.

Taylor & Francis, the owner and publisher of PRIMUS, has graciously agreed to make this entire special issue freely accessible for two months: 1/4/17 - 3/4/17. Please read and share! To pique your interest, here is the Table of Contents for the special issue.

Cheers, Brian Katz

PRIMUS: Illuminating Inquiry -- Table of Contents
  • It's All About Inquiry: A Cross-Disciplinary Conversation About Shared Foundations for Teaching
  • Teaching Inquiry with Linked Classes and Learning Communities
  • Teaching Inquiry to High School Teachers Through the Use of Mathematics Action Research Projects
  • Reflections on Transformative Experiences with Mathematical Inquiry: The Case of Christine
  • The Development of Teacher Knowledge in Support of Student Mathematical Inquiry
  • Constructing an Inquiry Orientation from a Learning Theory Perspective: Democratizing Access through Task Design
  • An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering
  • Student Perceptions of a Mathematics Major for Prospective Elementary Teachers with and Inquiry-Based Philosophy
  • Teaching Inquiry With a Lens Toward Creativity
PRIMUS: Implementing Inquiry -- Table of Contents
  • Ask Questions to Encourage Questions Asked
  • Turning Routine Exercises into Activities that Teach Inquiry: A Practical Guide
  • Teaching Students to Formulate Questions
  • Puzzle Pedagogy: A Use of Riddles in Mathematics Education
  • Encouraging Example Generation: A Teaching Experiment in First-Semester Calculus
  • To Each Their Own: Students Asking Questions Through Individualized Projects
  • Acting Like a Mathematician: A Project to Encourage Inquiry Early in the Math Major
  • Using Games to Engage Students in Inquiry
  • Teaching the Inquiry Process Through Experimental Mathematics
  • A Combinatorics Course with One Goals: Authentic Mathematical Inquiry

Note: This post draws heavily from the introductions to the special edition: introduction 1 and introduction 2