Week 1 of the new semester is in the books, and with it the first week of the inverted calculus class. I am teaching two sections of this class, one that meets Monday/Wednesday/Friday and the other Tuesday/Thursday. It makes for tricky scheduling, but as I learned this week it also gives an opportunity for second chances, which is important if you don’t always get the in-class portion of the flipped classroom right.

People always seem to focus on the out-of-class experience when they talk about the inverted classroom. *How much time does it take to make the videos? How do I make sure my students do the Guided Practice?* But that’s not the hard part, nor is it the part where most of the learning takes place. The *in-class experience* for students is what makes the inverted classroom more than just a lab or a seminar course, and as the instructor, it’s both hard and crucial to get it right. Students have to be engaged but not overwhelmed. They have to be successful early on in the class meeting but challenged throughout. And the activities they do have to be rigorous, but focused so that all the essential concepts they need to learn get learned. All of this in 50 minutes. It’s not a very forgiving course design task.

So let me tell you how I failed at it this week, and what I did to fix it and keep fixing it.

The first two class meetings (Monday/Wednesday for one section, Tuesday and half of Thursday for the other) were focused on course overview and precalculus review activities. It wasn’t until the fourth meeting of the week – second half of Thursday for one section, Friday for the other, and yes that timing matters a lot – that we had our first real inverted classroom session. Students were tasked with doing this Guided Practice exercise. The class was scheduled to start with an Entrance Quiz over the “basic learning objective” list along with a syllabus-related quiz question. The plan was: 5 minutes on the quiz; 10 minutes to debrief the quiz, any issues from the Guided Practice responses, and open questions; and then 30 minutes working on this activity (the answers were added later), ending with 5 minutes of wrap-up and announcements.

Here’s what actually happened:

- The debrief of the quiz and Q&A took too long. I made too much out of some simple errors that didn’t occur very often on the Guided Practice. Result: We didn’t finish the Q&A time until 20 minutes past the hour and I am not sure that students got their questions answered.
- The first part of that activity was actually to work an activity from our textbook. The first part of
*that* activity is to calculate the average velocity of a ball with position function \(s(t) = 64 - 16(t^2 - 1)\) over *eight different time intervals*. I should have realized this is problematic for several reasons. First, even if you are fluent with the idea of average velocity, calculating eight average velocities with a non-trivial position function using a graphing calculator is going to take at least 10 minutes if you figure around one minute per calculation and assuming students don’t make an arithmetic/calculator error. Which some students did just because it happens sometimes. Result: By 30 minutes past the hour, with only 15 minutes left in the allotted time for work on the activity, the students had only engaged in basic arithmetic calculations and had not encountered a single essential idea that they needed to learn for the day. - The essential idea to which those calculations were tied was in parts (b)–(d) of that activity, which ask students to plot points on a graph, draw the lines connecting them, and make some observations ultimately leading to the discovery that the average velocity measures the slope of a secant line. Those parts partially involve students using a graphing tool to zoom in on a graph. This particular section meets in a computer lab, and so Geogebra is readily available for this. But some students aren’t yet fluent enough with Geogebra to use it in this way; others opted to use Geogebra but had to boot up their computers and find the software; others opted for graphing calculators but didn’t know how to zoom in on a point. Result: Lots of tech support, and questions about what the question was asking for, and we didn’t get anywhere close to addressing the main issue until about 10 minutes had passed… of the 15 that were remaining. And this was still in “part 1” of the day’s activity.
- When students moved on to part 2 of the activity, which was a structured approach to calculating the average velocity expression give by \(\frac{s(2+h)- s(2)}{h}\), only about 5 minutes remained in the time for the activity. And as anybody who’s taught calculus before knows, this calculation is where all hell breaks loose algebraically speaking. Many students could handle this, but many students were at sea. I made the spot decision to drastically shorten the final wrap-up session so we could have 3–4 precious minutes to work. Then I put the result for step (2) (calculating the expression \(s(2+h)\)) on the board, so students who were still struggling with the algebra could just work on it later and move on – but I wrote up \(s(2+h) = –16h^2 - 32h\) as the answer, which is the result for part (3) instead! So I had to go back a minute later, after a student had caught the error, and correct it, much to the well-justified sighs of the other students.
- Even the students who, to their benefit, ignored my attempts to help and worked on had to use the average velocity formula they got to calculate two average velocities in part (5), one of which uses a negative \(h\) value. What was the point of that second calculation? I have no idea.
- Then they got to part (6) which is horrendously ambiguously worded: “In your own words, what is happening as \(h\) approaches 0?” How is a student supposed to know how to answer that?

We had to stop with about 5 minutes remaining and I had to basically give a 4.9-minute lecture on the entire activity. You can guess how well that went. When they filled out the “muddiest point” form that asks them for the least clear topic from the day’s work, many students put: “Nothing”, which is ominous because there is no way everything in class was clear.

I’m the kind of teacher who broods over classes that don’t go well. I literally lay awake that night wondering what I was thinking, what I did wrong, and what I could do for the other section that meets on Friday. In the process, I think I came up with a better framework for in-class activities (not that it would take much to improve what I started with) that rests on three ideas.

**First**: *The in-class activities have to focus on the essential concepts for the unit.* This reminds me of how peer instruction classes are built – you first identify the 3–5 essential concepts for the day and design everything in the class around those. I realized that the activity that I used in the Thursday section was backward; I had them doing lots of activities – more than they had time for, it turns out – that I hoped would lead to an essential concept. But I needed to start with the concepts. For this unit, I thought the main ideas were:

- The average velocity on \([a,b]\) is measured by the slope of the line connecting \((a, s(a))\) to \((b,s(b))\).
- Instantaneous velocity is calculated by constructing a table of average velocities over shorter and shorter time intervals.
- Instantaneous velocity can also be calculated by setting up and simplifying an expression for average velocity and letting \(h \to 0\).

**Second**: *The in-class activity should include only those calculations that support learning the essential concepts.* The activity I had made had far too many redundant calculations that added no value to the learning experience. To understand the first concept above, students do not need to make eight different average velocity calculations – they need to make *one* such calculation. And then they need to put that same, single calculation on a graph. And then they need to be asked to extract the main idea. To get to the second main concept, students don’t necessarily need to make *any* calculations but rather just observe some calculations that have already been made. In this way students can have a lot more time to focus on the one place in this unit where calculations can be problematic, which is in the third concept and in simplifying that difference quotient.

**Third**: *The activity needs to have checks built in that allow all of us to know whether students are actually acquiring the concepts I want them to acquire.* This means a few questions – maybe one for each concept – that are precisely phrased and force students to state what they have learned, which I can then assess on the spot. The activity only had one such question: Question (7) in part 2, and we were so pressed for time that not every student made it that far.

Serious instructional designers and professors are probably facepalming at this point because these three points are so obvious in retrospect. But hey, sometimes I can be a little dense.

After writing down all these ideas and setting my Friday alarm back a little to give me some time in the early morning to do a major revision to the activity, this activity is what I came up with. Here are some of the ways I tried to fix it and how it turned out when I gave it to the Friday class.

- The activity is organized into three sections, each of which addresses a major concept.
- The first part can be completed with just two calculations and leads directly to the observation I want students to make – that average velocity is the slope of the secant line. When students did this in class, every group finished in under 5 minutes and the main concept was obvious to them. (They needed a little help in the fill-in-the-blanks item to put “\((a, s(a))\)” and not “\((0.4, s(0.4))\)”.)
- The second part has only two calculations, and those are focused on the concept – the two time values are already very close together. And students can skip those altogether if they need to, because the table in question (3) contains the answers. I realized that I didn’t need students to
*construct* the table in question (3) in class; by the time they got there, they’d already done three average velocity calculations and know whether they are getting them right, so I can assume that given sufficient time, they could construct this table on their own. What I *really* needed to know is whether they can extract the right information from a correctly-constructed table. And that’s what the question in that item is all about. To make sure they don’t just give an answer that sounds right, I asked them a “sensitivity analysis” question in part (4) – how would you redo the above process to get the instantaneous velocity at another time value? - The third part is mostly unchanged from the first handout. I did put the result of the \(s(2+h)\) calculation on the form itself, so students should try to get it, but if they were stumbling, I wanted to assign it for out-of-class work rather than get sidetracked on algebra. I cut the number of average velocity calculations down to one, and the interval is already at \(h = 0.001\) to give students the idea of what they should be thinking in questions (6) and (7) – which is no longer ambiguous.

The results of this revised activity were outstanding. First of all, I made sure to keep my debriefing of the quiz and the Q&A session on a tight leash – 10 minutes maximum for everything. So we were able to start the activity by 10 minutes after class started. Students finished with 10 minutes remaining in the class, although some students who were struggling with algebra were given some hints about it and then told to work on it outside of class. We had a solid 7–8 minutes to debrief the activity, highlight the main results, and have unhurried time to reflect and fill out the “muddiest point” form. That’s the way every in-class session in an inverted class ought to go.

And what of the folks in the Thursday section? I’ll be amending my mistakes with them by building in some review work into their activities this week that the MWF section won’t get. If any of them are reading this, I’m sorry! I’m fixing it.

The moral of this story is that if you are going to invert your classroom, you need to be prepared for moderated failures, just like we ask students to be prepared, and these failures can come from any source – and whether or not you are experienced with the course design. But also as we expect from students, you need to be ready to learn from your mistakes and get better. This kind of vulnerability is one of the things I like so much about the inverted classroom, though, so I consider that a feature and not a bug.