# Guest Blog by Dana Ernst

This blog is part of our guest blog series. We ask other practitioners of IBL to share some of their ideas and tools so our audience can see what else is happening in the larger world of IBL. Enjoy!

Dana Ernst is an assistant professor in the Department of Mathematics and Statistics at Northern Arizona University in Flagstaff, AZ. We met him at the IBL workshop a few years ago, where we noticed that besides being a committed cyclist he knows a lot about using IBL in many different classes. Dana is co-author of the wonderful blog Math Ed Matters with Angie Hodge and writes his own personal blog. For example, check out this post that discusses how Dana encourages students to annotate their homework with colored pens while other students are presenting.

### Teaching Calculus 1 with a Focus on Student Presentations

This semester I am teaching the Honors Section of Calculus 1. It is important to point out that this is calculus for honors students as opposed to honors calculus. Mathematically, my students aren’t really any better than the students in the other sections (at least according to our department’s calculus readiness test). However, there are three aspects that set my students apart from the others. First, in general, the work ethic of my students is better. These are honors students; if I tell them to do something, most of them will just do it. Second, even though the mean and median scores on the readiness test are comparable to the other sections, I don’t have any extremely low achieving students. Lastly, I only have 24 students in my class this semester. It is worth noting that about half of my students have had calculus in high school, but this isn’t that different than some of the other sections.

We are using Brian Loft’s IBL calculus notes located at the Journal of Inquiry-Based Learning. Most class time on most days is devoted to students presenting their proposed solutions/proofs to the problems I assigned the class meeting before. I hardly ever lecture and when I do it is usually for 5-10 minutes. The lectures have been used to introduce a new topic or synthesize and summarize topics the students have already grappled with. A student’s first encounter with a new topic will usually occur when they are reading the notes and attempting to do the associated problems.

Loosely speaking, student presentations occur in one of two ways. On days when I feel like we can push the pace and covering a bunch of problems will be beneficial as opposed to detrimental, I will divide the class up into several small groups and assign each group a different problem to work on. The problems are almost always the ones they worked on for homework. The number of groups depends on the number of problems I would eventually like presented. Lately, the norm has been five small groups, each working on one of five problems. Thankfully, I have white boards on three walls in my classroom and there is room for the groups to spread out. The goal of each group is to come to consensus on what a solution could be and to write this solution on the whiteboard. In addition, each group needs to elect a spokesperson to summarize their group’s solution. Some problems are routine, but many aren’t. For example, one of the problems might be to prove the Product Rule. While the groups are chatting, my Peer TA and I wander around the room listening, chiming in occasionally, and answering questions. After some period of time (approximately 10-15 minutes), I bring the class back together and then have each spokesperson share out. During this time, students may ask questions while my job is to guide discussion and nudge students to ask the right questions. I’ll also take the opportunity to point out notation issues the students don’t notice and offer alternative approaches if they didn’t already come up. Moreover, I want to help students make connections with where we’ve been so far and where we are heading. We’ll move on after the class had agreed that the presented solution is valid.

On days when I feel like we need to slow the pace or discuss problems that I know will generate lots of discussion, I like to have students come to the board and present their individual proposed solution. The intention is for the presenter to write on the board and explain as they go. The audience can interrupt at any time to object or ask for clarification. My job during these presentations is the same as before. Again, we move on when we have consensus that the solution is correct.

So, how is this going? Excellent! It’s obvious that the students are learning a lot because I get to witness the evolution of their communication of mathematics. So far they are performing fantastically on exams and in the process we’re having a lot of fun! There has been zero push back from students and all of the feedback has been extremely positive.

As you can tell, I’m pretty excited about teaching this class. However, I’m also curious about a few things. Would everything be going so smoothly if I didn’t have the honors section? This is my second time teaching the class this way and both times I’ve had the honors section. I’ve wanted to try something like this in calculus for quite a while, but for various reasons, I held back until I had the honors section last fall. But what if I didn’t have the honors section? There’s no doubt that I would have to make some modifications if the class size increased (honors section or not). It’s easy for a student to hide in “large” class. I’ve taught calculus roughly 20 times and my perception is that I don’t have any more “top notch” students this semester than most others. So, it’s clear to me that the reason this is working is not because my students are all geniuses. On the other hand, in past semesters I’ve had plenty of students in my calculus classes that probably shouldn’t have been there (due to lack of preparation or maturity). I don’t really have any of these types of students in my honors section. How much of a difference does this make? Let’s explore this for a moment.

In a traditional lecture-based course, we can blissfully ignore our low-achieving students, except on exam days. However, in an IBL setting, all of the uncomfortable aspects of dealing with our struggling students are blatantly in our face. This is a feature of IBL, not a bug.

We are provided with an opportunity to respond to the needs of our students because the needs become clear to us. Take note that I am not advocating that we pander to our students! Rather I’m suggesting that we can make choices in the long term as well as decisions on the fly that maximize the benefit for individuals. I’ve found that this isn’t as hard as it sounds. One requisite is that we abandon the one-size-fits-all approach as much as possible.

### More Resources and Facts about IBL, active learning and the like...

A 2014 article in the Proceedings of the National Academy of the Sciences summarized the results of a meta-analysis of 225 prior studies on active learning. Here is what the researchers found: Students in these studies who were in classes that were predominately taught using direct instruction were 55% more likely to fail their courses than their counterparts in classes that focused on active learning! But there’s more: Not only does active learning, on average, increase the learning outcomes for all students, but it produces the greatest gains in the students who struggle the most, including those who are from historically underrepresented populations in mathematics.

Results from the recent work of Sandra Laursen et al. resonate with the conclusions of the PNAS study. Laursen’s research team has conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students’ and instructors’ experiences of IBL. This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years. The courses examined self-identified as IBL versus non-IBL. Classroom observation was used to verify that IBL classes were indeed different from those designated as non-IBL sections of the same course.

My blog post The Twin Pillars of IBL discusses Laursen’s work in more detail, but there are three points I’d like to share here:

- 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, was 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.
- When sorted by prior achievement, the grades of most students (IBL and non-IBL alike) dropped in subsequent courses as course work became more difficult. But grades of initially low-achieving students who had taken the IBL course rose 0.3-0.4 grade points, unlike their low-achieving, non-IBL peers, and unlike their higher-achieving classmates.

My own personal experience in teaching IBL courses bears all of this out. While I don’t really have any low-achieving students this semester, I have had three females in my calculus class tell me that because of their experience in the course they are now thinking about adding a mathematics major. And we’re not even halfway through the semester yet!

In a recent blog post, Robert Talbert had this to say:

So I am convinced that, knowing what we know now about teaching and learning, that the use of active learning in the classroom is an ethical issue and not an academic freedom issue.

I strongly encourage you to go read his whole post.

Having the honors section initially inspired me to take my current approach to IBL in the class, but now that I’ve seen it in action, I really want to know how things will play out in one of our typical calculus sections. It’s clear to me now that I have no choice but to give it a go and see what happens.

Many people often associate IBL with some variation of the Moore Method. Since student presentations play such a prominent role in my approach to IBL, I’m not helping to dispel this misconception. However, it is important to emphasize that IBL encompasses so much more.

In my view, there are two key features of IBL. Students should as much as possible be responsible for:

1. Guiding the acquisition of knowledge and

2. Validating the ideas presented. (Students should not, that is, be looking to the instructor as the sole authority.)

There are many ways to implement these features and student presentations are just one vehicle.

While I do believe there is great value in having an individual present his or her work, it’s not actually the reason I choose to structure a class around student presentations. I’m a pretty good lecturer if I do say so myself.

When I’m at the board presenting, students diligently copy down everything I write and believe everything I say. However, when a student is presenting, the students in the audience are naturally cautious of what is being said. This causes them to engage with the material being presented in a very different way. Moreover, the mistakes the students make while presenting provide the context for great class discussion. We get to wrestle with our mistakes in a productive way, showcasing them as part of the learning process.

One of my goals is to destigmatize mistakes and one way to do this is put them front and center. By honoring productive failure, our students can come to see that mistakes can be good.

Certainly, reading books, articles, and blog posts about the Moore Method will give lots of ideas and guidance for managing student presentations in the classroom. I often write about IBL on Math Ed Matters and on my personal blog and many of these posts discuss student presentations. For example, check out this post that discusses how I encourage students to annotate their homework with colored pens while other students are presenting.

You might also find the syllabus for this semester’s calculus class and a recent syllabus for my introduction to proof class useful. Both syllabi contain a rubric that I use for grading student presentations. However, it is important to point out that the scores I write down for a student are meant to keep me honest and motivate the students to take the presentations seriously. I can’t tell my students that we will value productive mistakes that arise from a good faith effort and then penalize them for making them.

Each student will have an average presentation score and I will never give him or her a grade for the presentation portion of his or her overall grade that is below this average. However, if a student presented often, took risks, and generated good class discussion, then I am definitely going to give him or her a better grade than his or her average presentation score. Moreover, it’s much more important that a student attempt a few challenging problems as opposed to cherry picking the easiest ones all semester.

I also encourage you to check out TJ Hitchman’s blog, where he regularly writes about his approach to IBL using student presentations in his Euclidean Geometry class.

### Experience with IBL in Other Courses

For the last few years, all of my classes have incorporated aspects of IBL to various degrees, some more than others. The following is a list of courses that I have taught using what I would consider to be a “full-blown” IBL experience.

- Calculus I. As with most universities, this is a service course consisting of a variety of majors.
- Introduction to Mathematical Reasoning. This is a course for math minors and math majors and focusing explicitly on problem solving as opposed to content.
- Foundations of Mathematics. This is our introduction to proof course for math majors. I consider this course to be my IBL forte. I’ve written a set of open-source notes for this course that several people have utilized.
- Number Theory. The last time I taught this course I was working at Plymouth State University in New Hampshire. The course is for mathematics majors.
- Abstract Algebra. This is your typical groups-first introduction to abstract algebra. I’ve also written a set of open-source notes for this course that I would love have more people to try out.
- Real Analysis. As with Number Theory, the last time I taught this course was a few years ago at PSU. The audience for this course is junior and senior math majors.