Archives For calculus

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.

The fall semester starts in a couple days and I’ll be teaching Calculus 1 (for like the 15th time) and our undergraduate Abstract Algebra course. Despite my relatively low teaching load, I’ll also be advising 3 undergraduate research students (on two different projects) and 2 masters thesis students. Combined with the fact that I’m still frantically trying to prepare brand new IBL materials for my Abstract Algebra class, I expect a very busy semester. For this reason, I decided not to mess with the format of my calculus class very much. I think it has room for improvement—namely ramping up the IBL aspect of the course—but this will have to wait until a later semester. I feel a little guilty about this, but I’m no use to anyone if I’m trying to do too much.

Notice that I said that I wouldn’t mess with the format “very much.” I am going to make some small changes. In previous semesters, I always devoted one class period a week to students presenting problems at the board. This has always worked well for me, but last semester I tried something else that I don’t think was as successful. So, I’ve decided that I’ll return to presentation days. In the past, I had a fairly nebulous way of assessing student presentations. I want to encourage students to present, so I make it worth something. But on the other hand, I don’t want it to be a high stakes thing. A class typically benefits more from the discussion surrounding a less than perfect solution to a problem than they do from a presentation that is flawless. So, I encourage students to share what they have. Of course, I don’t want students putting crap up on the board on a regular basis either. In a small class, this isn’t very hard to manage, but in a class with 45 students (which is what I currently have enrolled in my calculus class), it gets a little trickier. I’ve been thinking about how to manage this for a few weeks.

The latest version of WeBWorK—which we use for our online homework platform—has a new feature called “Achievements.” You can read more about this here. This basic idea is this:

Instructors now have the ability to create and award “Math Achievements” and “Math Levels” to students for solving homework problems and for practicing good WeBWorK behavior. In a nutshell, students can earn achievements by meeting preset goals. For example, they might earn an achievement for solving 3 homework problems in a row without any incorrect submissions, or for solving a problem after taking an 8 hour break. Earning achievements and solving problems earns students points and after a student gets enough points they will be given a new “Math Level”.

I have zero experience with the WeBWorK Achievements, but I thought I would give it a try. I don’t want to make earning them mandatory and I don’t want to offer extra credit either. So, I’ve been passively brainstorming how to handle them.

Northern Arizona University now requires faculty to take daily attendance in all freshman-level courses. However, how we take this data into account is up the instructor. It just has to count for something. The past couple semesters, I’ve been diligent about taking attendance, but I’ve always been a little bit vague about how it does or does not impact a student’s grade. Policies like, “you’ll lose a letter grade if you miss X number of classes,” drive me nuts.

Yesterday, I decided it was time to sort out what the plan should be for presentations, WeBWorK Achievements, and attendance. I had read a short article about gamification in education (I can’t remember which article) recently and I thought, “hey, why don’t I just gamify this stuff that I’m not sure what to do with.” In general, I’m not a fan of offering points for things that students should naturally do (and I’m also sure I have tons of counterexamples to what I just said), but I’m going for it anyway. Maybe this is a horrible idea. Here is what I currently have on their syllabus.


As per university policy, attendance is mandatory in all 100-level courses, and in particular, I am required to record attendance each class session. Daily attendance is vital to success in this course! You are responsible for all material covered in class, regardless of whether it is in the textbook. Repeated absences may impact your grade (see the section on Achievements). You can find more information about NAU’s attendance policy on the Academic Policies page.

Presentations and Participation

Throughout the semester, class time will be devoted to students presenting problems to the rest of the class. In addition, we will occasionally make use of in-class activities whose purpose is to either reinforce/synthesize previously introduced concepts or to introduce new concepts via student-driven inquiry. If necessary, these activities will be explicitly graded.

I expect each student to participate and engage in class discussion. Moreover, I will occasionally ask for volunteers (or call on students) to present problems at the board. No one should have anxiety about being able to present a perfect solution to a problem. In fact, we can gain so much more from the discussion surrounding a slightly flawed solution. However, you should not volunteer to present a problem that you have not spent time thinking about. Your overall participation includes your willingness to present, engagement in and out of class, and consistent attendance record. Your grade for this category will be worth 8% of your overall grade and will be based on Achievements (see below).


This semester I’ve decided to “gamify” good student behavior. Here is the gist. I’ve generated a list of items that I deem good student behavior. Every time you achieve one of the items on the list, you earn some points. The more points you earn, the higher your Presentation and Participation grade (worth 8% of your overall grade in the course). So, how do you earn achievement points? Here’s a list.

Points Description
5 Stop by my office sometime during the first two weeks of classes. This is a one time offer and stopping by to just say hello is fine.
2 Stop by my office hours to get help. This goes into effect after the first visit and you can earn points for this up to 4 times.
1 Attend class, arrive on time, and stay the whole class meeting.
1/2 Attend class but arrive late.
1/2 Attend class but leave early.
2 Attend a review session offered by our Peer TA. You can earn points for this multiple times.
4 Stop by the Math Achievement Program to get help. This is a one time offer and you must get a “prescription” form from me in advance for a tutor to sign.
2 Find a typo anywhere on the course webpage, homework, exam, etc. These points are first come, first earned, but there is no limit to the number of times you can earn points for this.
5 Volunteer to present a problem to the class on presentation days.
3 Agree to present a problem to the class on presentation days after I’ve called on you.
2 Earn a non-secret Achievement in WeBWork. You can see list of possible non-secret Achievements by clicking on the appropriate link in the sidebar after logging in to WeBWorK.
3 Earn a secret Achievement in WeBWork. These shall remain a mystery.
2 Post a useful resource such as a video or link to a math-related website on our course forum.
2 Post a relevant question on our course forum.
2 Post a useful response to a question on the course forum that does not just give an answer away.
5 Earn at least an 8/10 on your highest score for a Gateway Quiz.

Important: Any time I feel you are taking advantage of the spirit of this, I reserve the right to take away an achievement point.

To calculate your grade for the Presentation and Participation category, I will divide your Achievement points by the maximum number of Achievement points earned by a student and then convert to a percent.

Feedback is extremely welcome. I’ll let y’all know how it goes.

Edit: One thing that I forgot to add is that I have a Peer TA for 10 hours per week. She attends class and has access to the course forum, etc. So, I’ll let her do most of the point tracking. Otherwise, I’d have trouble with the bookkeeping. Also, I decided to use the highest number of Achievement points earned by a student to calculate a percentage for each student. In the comments below, Strider suggested that I use the average of the top 3 instead. I like this idea, but I think I’ll use the top 5.