Welcome to the third installment of the 4+1 Interview series. Today’s interview features **Dana Ernst**. Dana is a professor in the mathematics department at Northern Arizona University, a champion of Inquiry-Based Learning in mathematics, and an active writer about math and math education. I’ve known Dana for a couple of years, and he never fails to impress me with his clear-headed, positive-minded, student-centered approach to his work. His mountain biking exploits also inspire me to get up and exercise sometimes.

Enjoy the interview and make sure to catch Dana’s writing at his personal blog, the new Math Ed Matters blog (see below for more), on Twitter, and on Google+. If you missed the first two installments, you can click here for Derek Bruff’s interview and here for my interview with Diette Ward.

**1. You’re well-known as a vigorous proponent of Inquiry-Based Learning. Tell us what that term means to you and how you got started with using IBL in the classroom.**

Oh, man, that’s a tough first question! First off, there is no universal definition of inquiry-based learning (IBL). It likely means different things to different people. Furthermore, IBL manifests itself differently in different contexts. In fact, it is often the case that an IBL practitioner must modify his/her approach from one class to the next. Despite the difficulty in nailing down exactly what IBL actually is, I think there are a few features that most users of IBL can agree on.

In my opinion, for many students doing mathematics often means following the rules dictated by the teacher and knowing mathematics means remembering and applying these rules. However, the goal of an IBL approach is to challenge students to think like mathematicians and to acquire their own knowledge by creating/discovering mathematics.

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?

There are so many ways one could address this question in various contexts, which is the main reason that answering the “what is IBL?” question is so darn hard.

According to the Academy of Inquiry-Based Learning, IBL is a learner-centered mode of instruction. Boiled down to its essence, IBL is a method of teaching that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate; all those wonderful skills and habits of mind that mathematicians engage in regularly. 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.

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:

1. Guiding the acquisition of knowledge;

2. Validating the ideas presented. That is, students should not be looking to the instructor as the sole authority.

Said another way, an IBL approach emphasizes creation, communication, and critique of ideas.

These elements resonate with the definition of IBL used in an extensive study on the effectiveness of IBL headed by Sandra Laursen (Ethnography and Evaluation Research, University of Colorado, Boulder). This quasi-experimental study examined over 100 courses at four different campuses over two years. According to the study the “twin pillars” of IBL are deep engagement in rich mathematics and opportunities to collaborate. Classroom observation was used to verify that IBL classes were indeed different from those designated as non-IBL sections of the same course. 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. If you want to know more, watch this video of Sandra speaking at the 2012 Legacy of R.L. Moore Conference.

IBL has its roots in an instructional delivery method known as the Moore Method, named after R.L. Moore. 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 and consult outside resources. Variations of the Moore Method take many forms and are often referred to by the generic name: modified-Moore method. One significant modification that I make in my IBL classes is that I not only allow students to work together, I encourage it. For more detailed information, including history, of Moore and his method, check out A Quick-Start Guide to the Moore Method by Ted Mahavier, Lee May, and Ed Parker.

I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1998. My classes have always been interactive, but initially they were predominately lecture-based. Probably like most teachers, I modeled my teaching style after my favorite teachers. I was aware of the Moore method and IBL, but I had never experienced this paradigm as a student. By most metrics, my approach in the classroom seemed to be working. My teaching evaluations have been consistently high and I have received several teaching awards. However, prior to implementing IBL, I was suspicious that I could provide my students with so much more. Of course I want my students to learn the content of a particular course, but more importantly, I want my students to yearn for obstacles and to become independent of me. I want my students to experience the unmistakable feeling that comes when one truly understands something. Lecturing just wasn’t cutting it for me, but in my IBL classes, I think we’re getting close.

It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to consider using IBL. Carol’s workshop was about implementing a modified-Moore method approach in an undergraduate real analysis course. I do not remember the details of the workshop, but by the end, I was inspired to give IBL a shot. In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course, which happened to be an introduction to proof course at Plymouth State Unviversity. Perhaps surprisingly (since it was my first go), the course was a success and I was immediately sold on the potential impact that IBL can have on a student’s learning and character development. I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me. I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well. However, the IBL version was a vast improvement. Subsequently, I had students from all three variations in upper-level proof-based courses and the students from the IBL version were far and away more independent and, in general, better proof-writers.

**2. Do you have a sense that IBL works better in some subjects, or with some audience levels, than it does in others? In particular do you think IBL is something that should be reserved for more advanced subjects or students as opposed to, say, an intro Calculus 1 course?**

I think that IBL, not necessarily the Moore method, is the way to go at all levels of mathematics. What data we have is certainly pointing in this direction. That’s not to say that direct instruction is always bad or that there aren’t serious challenges to implementing IBL in some settings. Certainly, large classes present a whole host of challenges regardless of the approach that an instructor chooses to take. Most instructors likely won’t be successful adopting the Moore method in a large class. Yet, instructors (of large and small sections alike) can use small group work and techniques like think-pair-share to effectively drive inquiry. A flipped classroom (or inverted pedagogy) approach can also free up time for enriching in-class activities. However, as readers of Robert’s blog are well aware, one cannot simply move the lectures to video and then have students work on 1–25 odd from the back of the book. It is important that in-class activities feature creation, communication, and critique of ideas.

One of the biggest hurdles to a successful IBL class or any student-centered paradigm is the “coverage issue,” i.e., the set of difficulties that arise in attempting to cover a lengthy list of topics from a mandated syllabus. To ensure a course covers all the topics, the instructor often has to race through a topic before the students have had sufficient time to grapple with the material. I believe that this approach comes with a high cost. In my opinion, we should “cover” less and provide our students with an opportunity to probe deeper. What are they in our classrooms for anyway?

My experience with IBL has mostly been with proof-based courses, where I predominately take a modified-Moore method approach. My strategy has been to implement IBL in my proof-based courses first and then work my way down into the calculus sequence. Due to content pressure and class size, I’ve struggled to make my calculus classes a complete IBL experience. I would call my current approach in calculus IBL-lite. Yet, each semester I make at least small steps in the right direction.

**3. What kind of personality does it take for an instructor to pull off an IBL class successfully?**

Well, there are likely as many flavors of IBL as there are IBL practitioners. What works for me may not work for others. I’m much better at managing a classroom where individual student presentations form the backbone of the class than I am at managing small groups. Others prefer groups. If you try to replicate what R.L. Moore did, then personality likely plays a crucial role. What’s important is that each instructor figure out what works for them with the students they have. With the disclaimers out of the way, I will say that a typical IBL classroom is more unpredictable than the traditional model. If I lecture, I know exactly what is going to happen and how long it will take. My IBL classes are full of surprises and I rarely know exactly what is going to happen. As an IBL instructor, you have to be able to think on your feet, assess in real time, and be willing to adjust your game plan as needed; as they say in the Marines, “improvise, adapt, and overcome.”

My students come to me with 12+ years of experience that tells them what math class is going to be like. Yet, in an IBL class, students are asked to solve problems they do not know the answers to, to take risks, to make mistakes, and to engage in “fruitful struggle.” These are all very different from normal expectations. Students need to know what the instructor’s role is, expectations and goals need to be reiterated throughout the course, and students need to know that it is okay to be stuck and that we will support them in this endeavor. In order to create an environment where IBL can thrive, the instructor must develop a community of patience and trust and continually build on positive experiences.

**4. Could you mention a little about your new blog for the MAA, Math Ed Matters?**

Math Ed Matters is a (roughly) monthly column sponsored by the Mathematical Association of America. The posts are written jointly by myself and Angie Hodge (University of Nebraska at Omaha). The goal of the blog is to explore topics and current events related to undergraduate mathematics education. Posts aim to inspire, provoke deep thought, and provide ideas for the mathematics classroom. The passion that both Angie and I have for IBL color the column’s content. We are grateful to the MAA for providing us with the opportunity to share our musings.

**+1. What other question should I have asked you in this interview?**

How are you adjusting to your new job at Northern Arizona University?

My first year at Northern Arizona University is coming to a close. Well, it isn’t my first year at NAU. I finished my masters at NAU back in 2000 and then worked as an instructor for the 2000–2001 academic year in the Department of Mathematics and Statistics. Returning to NAU and Flagstaff is a dream come true for me and my family. I love working at NAU and Flagstaff is a wonderful place to live. By the way, Flagstaff is not the Arizona that most people imagine. Flagstaff sits at around 7000 feet above sea level in the middle of the largest ponderosa pine forest in the United States. The town is located just south of the San Francisco Peaks, the highest mountain range in the state of Arizona (the highest point is Humphreys Peak at 12,633 feet). The average temperature in July is 82 degrees F. We are surrounded by mountains, trees, and trails.

After leaving NAU in 2001, I worked for two years as a full-time math faculty at Front Range Community College in Boulder and Longmont, CO. I then started working on my PhD at University of Colorado at Boulder in August of 2003, and under the guidance of Richard M. Green, I finished in the summer of 2008. My first job post-PhD was at Plymouth State University, where I worked as an assistant professor for four years.

Each semester at PSU, I taught 3 or 4 different courses. My teaching duties always included one section of Calculus I or Calculus II and my remaining classes usually consisted of upper-level proof-based courses. Despite a relatively high teaching load, I managed to maintain a somewhat active research program, mentored undergraduate research students, and gave frequent math and math education related talks.

The teaching load at NAU is much lower, by about half, but the research expectations are much higher. This has been the most difficult aspect to adjust to. When I was at PSU, there wasn’t any serious pressure to publish. I did research mostly because I wanted to. If I had a busy few weeks, it didn’t stress me out that I wasn’t getting research done. I would just get to it later. However, at NAU the success or failure of my tenure application is going to depend on how many papers I publish. With a reduced teaching load, I’m often shocked to find that I don’t have more time to be productive research-wise. Just like at PSU I still get busy for a few weeks at a time and can’t get any serious work done on research. But now it really stresses me out!

As with many academics, it’s not exactly clear to me what a reasonable target is for my scholarly output. To complicate matters, my position is unique in that I am supposed to be active in both pure mathematics and mathematics education research. What is clear to me is that different people have different expectations about how this is supposed to play out in practice.

I enjoy the research that I am involved in and believe that there is value in publishing papers, but also find it frustrating that the “paper counting metric” has so much influence over my future. In particular, the pure mathematics papers that I write will likely “count” more than the math ed stuff that I do. But really how many people read my math papers? How much of an impact are they having? Meanwhile, the impact that blogging, giving talks and workshops about IBL, and mentoring undergraduate research is palpable. But this stuff just doesn’t count as much towards tenure. Don’t get me wrong; I love mathematics and have no intentions of discontinuing the pure math research that I do. However, I think academia has its values a little misplaced.