Dealing with the fear of being a boring teacher.

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Math, teaching

How I Teach Calculus: A Comedy (Limit Definition of the Derivative)

This post is a part a larger series documenting the changes I am making to my calculus course. My goals are to implement standards-based grading and to introduce genuine applications of the concepts being taught. I’m not suffering any delusions that any of this is all that ground-breaking, I just want to log the comedy that ensues:

Most common are attempts to space two kids with stopwatches as close together as possible. This, of course, plays right into my hands, but you can’t spring the trap yet.

On the first day of calculus I begin by introducing the limit definition of the derivative. This is a 100% break with tradition. Most calculus classes begin with a review of pre-calc and trigonometry. Let me just come out and say it, that’s ridiculous. First of all, if the kids didn’t retain it from pre-calc, why do you think your terse review is going to do any good? They need to see the material in the context of a new problem. Second, kids hate review. It’s like telling them you already think they’re stupid. I know I’m going to take heat in the comments for this position, and maybe this only works for my kids, but it works so that’s what I’m going to do.

By jumping straight to the limit definition of the derivative I have also circumvented an entire chapter discussing limits in general, continuity, and all sorts of other mathematical abstraction that means nothing to a 17 year-old. Why do you want to start there so badly anyway? They’re only learning about limits so they can use them the three times they show up later in the book (limit definition, end behavior, and area sums).

My goal is to motivate the necessity for learning more math. Why on Earth would anyone want to do this:

Limit Definition

without the proper motivation? A grade? Hah! Let’s not get into that here.

So the question I ask myself before I give any calculus lesson is: “Why on Earth would anyone actually go through the trouble of doing it this way?” There really is a rich set of useful problems that can only be solved using differential calculus, so why not present them (or, God forbid, let the kids think of them) to students and work your way out?

My rhetoric here is getting a bit lofty, and I hope my actual lessons don’t end up being a let down, but they sometimes are for me, too. It turns out that it’s hard to come up with real things for math to do when you’ve only been modeled the most sterile way of learning math.

So we begin with the speedometer. This is pretty standard, it’s even listed on page 56 of my book! Yay, we really can just stick to the book! Way to fit a chintzy connection in and then completely ignore it, book! Kids need to feel these connections.

So we begin class with the question, how do you know how fast you’re going? Invariably kids come up with the idea that you measure how long it took you to go a certain distance. I then pose the question: “Well how do you know how fast you’re going right now, at this very second?” After that I let the kids loose with office chairs. The instructions are:

  1. Push your friend so that his/her speed changes1
  2. Try to measure their speed at any single point

After this, nothing was planned. I know where I want them to go, but I can’t drop the math on them until I feel that their natural curiousity has brought us there. The kids return after trying many different methods. Most common are attempts to space two kids with stopwatches2 as close together as possible. This, of course, plays right into my hands, but you can’t spring the trap yet. They then run the office-chaired student by the timers and they attempt to calculate the speed.

Now that they’ve got the idea front loaded, let’s get on the ol’ information superhighway and check some things out. My kids asked how a car’s speedometer works, so that’s where we headed. We found:

After reading through that article, the kids found this picture:

Speedometer Parts

Image Credit: Unknown source

They described how it works (kind of) and then began to wonder. This is where I step in and the traditional math lesson begins. “What does the spedometer actually do?” I ask. It torques that little yellow rod based on how fast the magnet spins. The faster it spins, the bigger the torque, and therefore the needle deflects more until the spring can stop it. This is a physical representation of the derivative, but the kids don’t know what that is yet so don’t say it; they’re primed for the idea, though.

There’s no magic way to lecture. There’s best practice and all, but in the end talking is just talking. Without the lead up, they won’t have many hooks upon which to hang your nuggets of wisdom. The lecture goes through the modification of the slope formula like so:

slope - algebra to calculus

All the while developing this picture:

Limit Definition Graph

This is where the beginning of the lesson becomes crucial. Did they experiment with timing short intervals? How can we find the slope directly at x? By bringing the timing students closer and closer together. What would be the optimum distance to get the most accurate results? What? Zero distance you say? Interesting, that’s what the book says too. Enter the limit:

The LimiThe student have already physically attempted the limit process. They can remember it, and hopefully this translates into some kind of conceptual understanding.

I’m not claiming any magic here, I’m only trying to say that kids need succinct and enjoyable experiences that point to specific content standards. Now you can show them specific examples of specific functions. Show them a difficult function, show them easy ones, but always tie it back to the idea of arriving at instantaneous information for fluidly changing values. That is what calculus is. This lesson isn’t perfect, but I’m sure you’ll hit the central theme of calculus home much easier than if you lecture on the first day about when to use calculus before they even know what the hockey sticks you’re talking about.

Here are the example functions that I run through and why:

Root Function

This function gets at rationalizing a root that has an algebraic expression. This is a must see for a budding calculon.

Simple Functions

These two provide the same derivative, and allow the student to see how a constant is treated.

inverse x

This function reviews common denominators with algebra.

These three (four) examples allow me to simultaneously show how to work through the limit definition, but they also allow me to dredge up very commonly misunderstood algebra techniques in context of the new material. The old feelings (boredom, hatred, etc…) associated with the problems they’ve attempted before will be forged anew as they attempt this new process…

The way that the speedometer works will be a central piece of the puzzle as I try to get the students to understand instantaneous rates of change. I’ve even toyed with building speedometers: perhaps next year.

So, they learned a skill just like a normal class, but they also have somewhere to put it. The way that you connect the math skills to their experiences on the office chair really is key. Don’t worry so much about your examples, and worry more about the students generating context. Plus, watching a bunch of students timing themselves on office chairs is pure comedy.

1: If this is too vague, and you want to make sure they do what you want, relax, I mean have them push each other with a bathroom scale, and tell them the scale has to read the same value during the whole push. This opens up a whole new can of worms and questions, which is awesome.

2: Usually just the stopwatches that every kid has on their cell phone.

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10 thoughts on “How I Teach Calculus: A Comedy (Limit Definition of the Derivative)
  • Doug says:

    Thanks for showing a great example with bringing some rich context into a calculus class. I have maybe a dumb question though… How is the speedometer a physical representation of a derivative? Is it the idea of taking a speed (rotational) and translating it to a distance (needle on the scale)?


    • Shawn says:

      @Doug: I suppose it’s more the position of the needle’s point that is the derivative. If the car is moving at a constant speed, the needle stays put, if the car accelerates, the needle move with constant angular speed. If the car changes it’s acceleration evenly, the needle accelerates evenly. Here’s a sweet youtube video I use to teach both physics and calc.

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  • GS Chandy says:

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  • Sue VanHattum says:

    >After that I let the kids loose with office chairs.

    Brilliant! I wonder if I can get enough office chairs. I know the hallway we can use. I teach at a community college, and on the first day, I have over 40 students sometimes. Hmm…

  • [...] Shawn Cornally: So the question I ask myself before I give any calculus lesson is: “Why on Earth would anyone actually go through the trouble of doing it this way?” There really is a rich set of useful problems that can only be solved using differential calculus, so why not present them (or, God forbid, let the kids think of them) to students and work your way out? [...]

  • CalcDave says:

    Heh, well, as I think about it, I guess I emphasize that more in the “review” part at the beginning of the year. I don’t think mine have as big a problem with replacing x with t, though, especially if I leave y as the vertical axis.

    Using things like Sean Sweeny’s ( or Dan Meyer’s ( “graphing stories, they get a decent sense of how those go.

    • Shawn says:


      Thanks for those links! Yea, by skipping review, we build up to those kinds of things as we develop material. It really is a trade off, and I can totally see the right review working for the right kids. Thanks again for the comments; these kinds of conversations are the reason I started blogging.

  • CalcDave says:

    I’m curious why (when the “lecture” part begins) you use Delta y over Delta x. I’d think Delta y over Delta t would fit better with their speed understanding, right?

    I guess you’re using the physical proximity of the stopwatch bearers as Delta x, but I think it would make more sense to me to measure distance traveled over shorter amounts of time.

    Maybe I don’t understand the concept the way it is presented here (like why do you need more than one stopwatch?).

    • Shawn says:

      Thanks for the comment! I totally agree with you, t would be a much more appropriate variable, but for my kids the use of x is so ingrained in them as the left-right axis, that I stick with it on the first day. The double stopwatches is also a student comfort thing. They try all sorts of different stuff, and generally the kids who aren’t pushing/driving like to both have stopwatches, but really only one is necessary. Sometimes they both time the event and compare, sometimes only one kid does the timing.

      The left-right is x and up-down is y thing has started to bother me as a teacher. They really get flustered when we work outside of that paradigm. Do you have an suggestions for combating that little addiction? I’d love to hear them.