Flipping Calculus: A First Adventure

I flipped my Calculus I class this fall. It was my first real attempt at the method (I made a half-hearted attempt in a multivariable class once), and I wanted to share some of my experience.

What is a flipped classroom and why would you do such a thing?

A flipped classroom inverts the traditional model of lecture and homework. In a flipped classroom, students focus on assimilating material outside of the classroom and spend class time working on problems.

When I first learned about the approach it made immediate sense to me. Although I think I’m a very good lecturer, even the best lecture is mostly a waste of time for most students. Students don’t have enough time to synthesize an idea in a lecture before we move on to a new one. Some students are at different levels and have different learning styles and you cannot cater one lecture to everyone. And most of it is already in the textbook anyway. What students really need to do is take a shot at the material themselves and come to class having identified where they are struggling and seeing if they can apply their understanding to problems. That’s the time when it’s most helpful to have the expert around.



Here’s how the class was designed. We use Hughes-Hallett, et al, which is a reform-minded but ultimately standard textbook. In the first semester, we cover up through the first part of the Fundamental Theorem of Calculus. This approach to the flipped class is adapted from a fine minicourse @RobertTalbert gave at MathFest 2015. After each class, I posted a “daily dispatch,” which generally included the following:

  • Summary of the day’s in-class activities
  • Elementary objectives (to be prepped for next class)
  • Advanced objectives (ideally reviewed but not necessarily mastered for next class)
  • Pre-class assignment (typically a reading, videos, and a few elementary problems)
  • Post-class assignment (online assignment largely intersecting with the in-class assignment)
  • Optional/recommended activities (additional videos, extra problems)
  • Announcements (upcoming exams, logistical nonsense)

Every class had a two lists of objectives: Elementary and Advanced. The idea was that students would focus on the elementary objectives at home and the advance objectives in class. To prepare these objectives, I would assign a reading, a couple of problems targeting elementary objectives and a recorded lecture or two, usually totaling 10-15 minutes. I recorded the lecture by writing on a tablet PC, so they would hear my voice and see my writing. It looks like a less polished version of what you see at Kahn Academy and similar sites (for some standard topics, I linked directly to external videos).

There would also be a post-class assignment where students would further develop the advanced objectives. Much of the post-class assignment would overlap with the in-class assignment.


First, a note on something I didn’t do, which is standards-based grading. In this model, you score students directly on their progress on the individual objectives.  @RobertTalbert does this in his calculus class. As my friend @ProfNoodleArms says, it is essentially the report card you get in kindergarten.

I like the idea in theory, but there are three reasons I didn’t try it. First, I think flipping the class was enough pedagogical intrigue for one semester.  Second, it sounds difficult to manage in practice.  I could probably get over that with some guidance. Finally, I’m not convinced it will make much difference. The A-F grading scheme is fairly coarse as it is, and the university forces me to map onto that scale anyway, so I’m not sure many students would be affected.  I can see where the objectives-oriented approach sells easier to the students by keying grades directly to those objectives, but I’m not sold yet.

My grading scheme was a more standard weighted average of chapter exams, quizzes, a final exam, online homework, and participation. The general design is that the exams are tough, but the other stuff should raise your grade a letter or so above your exam score.

We cover five chapters. There are three chapter exams, about one per month, plus one quiz each on chapter one (which is pre-calculus review) and on derivative calculations. The last chapter (integration) is covered exclusively on the final. The three chapter exams are averaged to weight better exams more heavily. The quizzes allow unlimited retakes, but with a maximum score that decreases with each sitting. I’ve done the derivatives test like that for a while (10 functions to differentiate, no partial credit) but just added the chapter one quiz (only three scores possible: Insufficient (0%), Sufficient (65%), and Proficient (100%, reduced by 10% per sitting)). This is basically a gateway exam model, and I will go all-in on this exam design next semester. This isn’t directly connected to the flipped class, but reflects the more objectives-oriented approach.

Graded homework is online and comprised the bulk of the post-class assignments. I assigned problems from the section online and much of class was spent working them. Not all of the problems I wanted to do are available online, so those were covered in class and did not get graded. There were also a few problems not done in class and was intended for students to practice.  My general philosophy of homework is that I try to assign good problems and require students to minimally engage the assignment, but it’s their responsibility to take it seriously. My rule is that anyone with 70% or more on the homework automatically gets 100%. I know students can get away with not doing nearly enough homework, but they will get eaten by the exams anyway. The online system is not perfect, but the instant feedback is valuable.


Notes and Observations

  • Once you get used to it, the recording is pretty easy to do.
  • Don’t get hung up on perfection for the recordings, and avoid getting involved in editing. These are time sinks. If you are recording short lectures (and you should be), it’s usually easier to just start over if things go wrong.
  • If you want to write on a tablet, you want active digitizer technology. In particular, writing on an iPad is clunky. I use a Fujitsu tablet PC, writing in Windows Journal or One Note.  (I just got a Samsung Tab phone that might work out, too.) I had to record the lectures in Panopto because that’s what my university wants, so I haven’t really explored software options beyond that. Panopto is pretty good, though, and it connects nicely to our course management system.
  • There is a trap I was warned of, and it’s a big one: it is very easy to accidentally overwork your students. This is because you have to invert your instincts on assignments. You want to give students some practice at home, but most of the mid-level homework should be done in class.
  • Lecturing online in short bursts is a great help to students and allows efficient delivery of content. You don’t have to waste half of the students’ time lecturing on prerequisite stuff like factoring polynomials, and you don’t have to waste the other half of the students’ time lecturing on more advanced topics they can’t handle yet, like proofs of the derivative rules. With lots of mini-lectures, students can get the content they need when they need it.
  • Don’t use the term “flipped classroom” or go into detail about why the class has the format it does.  Just explain the details in the syllabus and run with it.  Otherwise some students may get intimidated by the idea of a new format and some will decide on the first day that the methodology is the cause of all of their problems. Don’t give them a chance to make that excuse.

Assessing and Correcting

So was this a good idea? The short answer is yes, probably. I’m pretty sure the class is better, and I’m certain I didn’t make things worse.

The flipped classroom is not a magic bullet. I think freshman calculus has some pretty serious structural issues that no one methodology can fix — something I will be looking at more closely with the MAA’s study on this. But the flipped classroom allowed me to include lots more process in calculus, and that can only help.

I had a SGID (Small Group Instructional Diagnosis) performed for my class. This is a way to get anonymous feedback from students in a collaborative form. (I can’t recommend this method highly enough — I’ve had two classes SGIDed and got feedback that led directly to course improvements. In 15 years of teaching, I can’t recall a single time the standard end-of-term bubble evaluations made any difference.)

The SGID confirmed that the class was more or less achieving what I want.  You always get the “just tell us what’s on the exam” crowd, but otherwise students recognized why the method works the way it does. There were a number of insightful suggestions, but most of them I had already identified for correction or were just artifacts of it being a first run-through (e.g., postings sometimes went up late).

I am teaching Calculus I again in the spring — I requested it so I could get another iteration quickly — and I will use the method again. Here are some adjustments I’ll be making:

  • Better alignment between objectives and assignments. Previously, my classes were mostly lecture with some group activities. The group activities were related to the material, but were not really meant to practice so much as deepen understanding of certain topics. I fell into the trap of doing more of these kinds of activities and not fully considering that the students did not have a practice assignment for the mid-level problems. This is how you can fall into the accidental overwork trap I mentioned above. Later in the class I kept most of the activities on textbook problems and the online homework included many of these problems (further incentivizing the students to be productive in class). I will continue this, getting back to my old pattern of one “special” activity with the rest being more mid-level practice.
  • Revised objectives. I am going through my objectives list and reorganizing them quite a bit. There are no fundamental problems, but now that I’ve been through it I have a better sense of how to distinguish elementary from advanced objectives, how to write them to be more explicit and measurable, and what is realistic for an in-class assignment.  I’m sure I’ll be at least tweaking this list forever.
  • Greater accountability for pre-class activities. This is one of the method’s biggest challenges. It all falls apart if students do not come prepared. My plan right now is to implement short daily online quizzes testing the elementary content.  The trick is where to set the bar — low enough so that students don’t feel like I expect them to mastery coming to class, but not so low that they think they can knock of a few easy questions and consider themselves prepared.  I also hope this problem will partially solve itself with my aforementioned homework change as students realize that the more prepared they are, the less homework they’ll have on their own.
  • Exam Restructuring. This isn’t really a flipped class thing, but a direction I’ve been moving for a while. I don’t know that my chapter exams really help student learning much, but the quizzes with retakes are good assessments. It’s time to do the whole class like that. This will also solve a couple of logistical problems, like allowing more flexibility on when students may use notes or calculators. Also, the derivatives test makes testing the rest of that chapter a little awkward — but now there are simply two mini-exams for the chapter, one of which is the usual derivatives test.  There will be retakes offered for every exam, but you cannot take the next exam unless you meet a minimum score on the previous exams. Retakes are not the exact same exam, but are very similar. The maximum allowed score will decrease with each sitting, so that should keep the retake numbers under control. The trick to this will be managing all of it.
  • Re-record Some Videos. There are no fundamental problems with the videos I recorded, but in the dozens of posted videos, some are better than others — and I admit that some were hastily done. I expect to re-record about a third of them.
  • Getting Math In.  One problem in any calculus class is how much should you really worry about proofs? I argue that you are not doing math if you don’t justify everything, so I make sure everything has a reason. I don’t mean chase down every epsilon and delta, but there should be no “just trust the formula.” The problem is that it is often hard to assess this kind of thing. I want students to know why the chain rule works, but I don’t expect them to be able to write a proof. I need to refine the objectives to target these issues more specifically. (For the chain rule, I use interpretation in terms of units as a compromise position.)
  • Re-weight Participation I weighted participation far too heavily, thinking I would be formally evaluating it. That didn’t really happen because it never seemed necessary and so grades for this semester were a little inflated. Next time I will collect group work for selected assignments and grade them (lightly — I do not want these activities to be perceived as anything but formative). This will address another deficiency that the online homework creates, which is mathematical writing. The students need consistent assessments to write justifications for their solutions.

Should you try it?

I am an advocate for getting process into classes any way you can, and this is a good one. But you do have to be prepared to think about your class very differently. You also need to ensure that you are prepared to maintain the out-of-class learning component, which may require some unfamiliar technology.

I don’t think any method is right for everyone in every situation, but if you read this far, I encourage you to research the flipped classroom model further. The implementation I outlined is not the only way to go. Feel free to post in the comments or contact me if you want more details.

Bill Wood, @MathProfBill , is an associate professor of mathematics at the University of Northern Iowa.

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