This past Thursday was the first day of student presentations in my calculus class. It went awesome! The plan is to spend one whole class period each week having students present and discuss problems. The remaining three classes each week will be predominately spent on direct instruction and some small group work. I was emotionally prepared for a crash and burn session as it is often the case that the first day of student presentations in my calculus classes can be a bit rough. I’m totally okay with this as it always improves. But Thursday didn’t leave much room for improvement.

Class started by me asking for volunteers to present their proposed solutions to the problems from their Weekly Homework assignments. I anticipated being met with silence while students stared at their desks. The plan was to then gently nudge people to present. Instead, I had more people volunteer than we had problems. That wasn’t a problem I was expecting. I think I have a good group of students, so I maybe I had nothing to do with this. However, I am suspicious that the heavy marketing that I did helped a lot. Also, I’m curious how much my Achievement Points experiment played into the number of volunteers. Whatever the case, the students were ready to rumble.

In case you are interested, here is the general format for student presentation days, which will typically occur on Thursdays. A few days before the students are to present, I assign a Weekly Homework assignment. You can find the first one that the students did by going here (it’s not terribly exciting). For the most part, the problems on the Weekly Homework assignments are a subset of the material covered the previous week. I’m a big fan of having the students circle back on concepts as much as possible. The students also have Daily Homework assignments that consist almost entirely of problems from WeBWorK.

When the students arrive to class on presentation days, they are supposed to have completed or done their best to complete all of the problems from the Weekly Homework. Upon entering the room, students should grab a colored felt tip pen. I have a box of pens in a variety of colors. The students love the sky blue and purple ones and hardly anyone ever chooses the red ones. During class students can annotate their work only with their felt tip pen. My approach to this is nearly identical to what is described in my Math Ed Matters post located here. The big picture is that I want students to be able to process what they have on their paper as other students are presenting. However, I also want to make sure that I know what work students had done before they entered the room.

One difference between what I describe in the Math Ed Matters post is how I grade the homework that students are presenting problems from. In my other classes where presentations play a more prominent role, the students are presenting problems from their Daily Homework and it is these problems that they annotate with the felt tip pens. In this case, I grade the Daily Homework with a $\checkmark-$, $\checkmark$, $\checkmark+$ system. But in my calculus class, since the students are presenting problems from their Weekly Homework, the students are annotating these problems. Moreover, since students are revisiting topics from the previous week, I feel comfortable weighting the Weekly Homework in calculus more and grading it harshly.

There is no penalty for students using the felt tip pen and I encourage them to annotate to their heart’s content. However, when grading the assignment, the annotations are more or less ignored. That is, they are graded on the work they had done before class. The students need some coaching with how to use the felt tip pen, but they seem to dig it.

At the beginning of class, I usually write down all the problems that I’d like to see presented and then I ask for volunteers. In the case that more than one person wants to present a particular problem, the student with the fewest presentations has priority. If there is no priority, I try to choose one of the volunteers at random.

Depending on the number of problems and their difficulty, I will either have the students all come to the board at once to write down their solution while I bounce around the room answering questions or we’ll have one student come to the board at a time to present their proposed solution. In the first case, after the solutions are on the board, we will discuss each problem. In most cases, I’ll have the student that wrote down their proposed solution lead a discussion with the whole class about the problem and their solution.

Here is where my approach likely differs from many others. I don’t want all the solutions to be correct. I’d rather have students make mistakes. Ideally, I’d like to see a mixture of correct answers, answers with small mistakes, and answers with huge errors. It’s not that I want students to screw up. What I want is to have something to talk about. One of the greatest advantages of doing student presentations is that the audience should take on the role of skeptic. This makes the student engage with the material in a much different way than if they watch me. I’m an authority and usually the students just believe everything I say. I encourage the students to be willing to share what they have. It’s a low stakes endeavor for them. Just being willing to present is what they get credit for. (This is different in my upper-level proof-based courses.) On the other hand, I don’t want students to present total crap either.

This semester I have 48 students enrolled in calculus. I’d prefer a smaller class, but I can make it work. The goal is to get as many students to the board as possible. On Thursday, 9 students presented. I’m always concerned about how many students in class are engaged on presentation days. However, I’m confident that it is many orders of magnitude more than when I lecture (and I like lecturing and I think the students like it, too).

On Thursday, most of the solutions where flawless, but we also had a few that led to excellent discussions. In each of these cases, the presenters did a good job at fielding questions and comments from the audience and from me. This works because I make it a point to develop a community of trust. The audience has to behave appropriately. I encourage students to clap after each presenter and they clapped loudest after the two students that were at the board the longest due to flawed solutions. One of these students had told me before class that she did not understand function transformations (last week we were still reviewing precalculus). I encouraged her to volunteer for a problem related to function transformations, so that she could show us what we had. Indeed, she did have some misunderstanding, but in my view, this was the most beneficial presentation.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.