Calculus Fail


I’ve been beating myself up, and it’s only day 4 of school. It’s sad because I just want this year to be the most fantastic year ever, and I wanted it to start so positively. But I’m feeling sad about my classes. I am okay being a teacher centered teacher for my Algebra II class. I really am. We have a curriculum that we are following, and we don’t have too much time to dawdle. Also, the kids are younger, so I feel okay keeping them mostly reigned in. And my MV Calculus class is going to be relaxed, though more challenging to teach than last year, because there are only two students (gasp!). That is a nice combination of student and teacher directed.

However, my calculus classes are a different story. I don’t have a set curriculum, which allows me a lot of freedom. I want to make sure that these students leave understanding calculus. I want them to see what makes calculus cool. What makes math cool.

So I promised both sections of my calc kids on the first day that my goal was to make math understandable to them. And I secretly promised myself that day that I would make math more interesting than they’ve ever seen it before.

It’s day 4 of teaching, and I feel like I’m flopping already. My classrooms are depressing (no sunlight in one; loud sounds of recess floating through the window in the other). I haven’t made one interesting lesson or one group/partner activity. I’ve just been up at the stupid SmartBoard pointing, talking, asking questions, going over homework. We’re just reviewing. And honestly, I don’t even really know where the students are in terms of what they know and what they don’t know. I call on random people, I walk around when they’re working on problems practicing in class, and still: not much clue. That’s not good.

I want to feel okay letting go this year and shift from having a teacher centered classroom to having times when the room is student centered. Where I’m not the one talking for most of the class. And I feel if I talk about that goal here, it’ll force me to keep it in mind. And be slightly more accountable.

As Alison Blank (@pvnotp) said on Twitter: “Maybe just try to be student-centered a little more often – like set aside one class every two weeks where you switch it up.”

Baby steps. And I’m going to try that. Even if it means something as small as playing a review game with students, or having some sort of hard problem challenge we spend the whole period solving together (like I do sometimes in MV Calculus), or making a guided worksheet to lead students through a concept. I should also remember that I can mix things up by asking for different forms of homework, instead of book homework, worksheets, etc. I can ask students to write a letter to their math-illiterate uncle explaining a concept we’ve been working on in class, or create a quiz of their own, or write a formal solution to a challenge problem. I can have students each work on different problems and make a SmartBoard presentation of their solutions for the class — and grade their presentations. Or even have students research the practical applications of calculus.

My brain to itself: Okay, Mr. Shah, keep these things in mind as things you can do instead of traditional classwork and traditional homework. And you just came up with these in the last 5 minutes. Imagine what you could come up with if you gave yourself 10 minutes, or (egads!) 15 minutes?

So I’m going to try to experiment a little this year in calculus. Be slightly daring. Put my foot in the water.

Optimism! Glimmers of hope!

If you want to see why I’m so dejected at the moment, you can see my SmartBoard presentation for my calculus class.

Setup. We’re in Algebra Boot Camp and we’re learning about rational functions before we start on the Limits unit. Up to this point, these kids have reviewed holes and vertical asymptotes, and have just started thinking about the domain of rational functions.

It’s not that the SmartBoard is bad, exactly. I actually think it’s pretty well thought out and organized. But you can see what my class would look like, by looking through it. (FYI, this particular lesson on domain, x- and y-intercepts, horizontal asymptotes, and sign analyses takes more than a day to go through. It will take 2 days to teach and 1/2 a day to pull it all together.)

I know I shouldn’t beat myself up too much. It’s only day 4. But I am. I’ve just been in a bit of a teacher funk. I’ll get out of it. All I need is some kid to say that they’re actually learning something in my class, and that they’re excited about it. I’ll get that.

Important Note. I don’t mean this to be a pity party. I don’t want pity comments – please. I only posted this because this is a place for reflection, thoughts, emotions, whatever. An archive of how I’m feeling today, so I can look back later and see how I’m evolving.

However if you have ideas on activities/games that work for you, things that break “the teacher introduces an idea –> teacher asks questions to develop idea –> teacher goes through example applying idea –> teacher asks students to practice a few problems –> start over” cycle, I would love for you to throw those in the comments below.



  1. Do they already know the terms domain, range, x-intercepts, and y-intercepts?

    If so, give them a graph of a rational function, have them find the requested information. Then compare with a partner/group. Then call on someone to write the answers on the board. Say nothing. Ask another student if they agree. Then ask why.

    Then do another one from a graph. Then from a table (this may be tougher if they aren’t used to it).

    Challenge them to write the function that generated each graph. Then give them just the equation and ask them to find the same information.

    You may only get through 3 or 4 problems, but you may have a much better idea of what they know when they walk out.

    That is how I’d do it. (and did it last year). I am in no way saying it is the best way. It works for me and my students. (that is pretty much what I do every day, no matter the topic. I just introduce terms as we need them and clean up notation along the way).

    1. After I finished rational functions and sketching, I gave no instructions and gave 4 graphs of rational functions and said “find the equations.” Like you suggested. I loved watching them work on it. Working backwards. I could see their brains whirring — how do I get a vertical asymptote here? What about a hole? Does the y-intercept match up.

      It was great.

      We’ll see what they came up with when they took it home to finish it for homework.

  2. Like you said in the earlier post, it’s hard to begin the year with the rapport from last year. They need to get to know you as much as you get to know them. Teacher-centered is usually more comfortable for both parties to start out and then let them gel together for the more student-centered stuff to come.

    There’s tons of fun (slightly off-topic) things to talk about with infinity that my kids love when we talk about that sort of thing. I never really did like KWL charts for some stuff, but it can be interesting with some concepts like infinity when you can turn them on their head with the “are you SURE you know that?”

    I also mix it up with different “kinds” of functions when talking about domain/range. “You already have an idea of functions like y = x^2, but a function is just something for which a single input always has the same output. For example, here is the ‘last name function.’ I walk in to this machine and out of the other side it spits out ‘Petersen.’ What is the domain of this function? Could you put in the number 1? A rock? What is the range?”

    I guess those things are still teacher-centered and get a little more off-topic than what you’re going for here, but the kids tend to like that and their eyes open to the idea that these objects (functions, infinity, etc.) are more in depth than they’re used to thinking about in math classes.

    (Edit note: page 18 “thing” –> “think”)

    1. I guess I can clarify about the infinity a bit. Things like, “I love you infinity.” and then the other person says, “I love you infinity + 1.” Who loves whom more? “ewww. He’s dating a freshman.” What does he say back about age difference? “Yeah! Well, when I’m 100 and she’s 96, it won’t matter!” Is that true? Then work into the problems with a simple x/x, then change it to (x+1)/x, then mix it up a bit with (x^2 + 2)/x and how does that change things? Again, nothing revolutionary here, but they can get sucked in a little more.

  3. “It will behoove you to understand going into class what exactly Mr. Shah wants from you, which is an attentive, honest, and interested student. It has taken me six months to realize what Mr. Shah really wants from you; he wants you to ultimately be a good person in the world.”

    BAM! Check that out. Someone wrote that about you. In fact, someone wrote that about past you, and chances are that present you is a way better teacher. Get some sleep, wake up tomorrow and kick some butt.

    If tomorrow, you still need to get some active, fun learning immediately, here are some things you can try:

    1) Go crazy with word problems. Last year my Calculus class took awhile to pick up related rates problems. Doing them over and over was getting so dry that I was getting bored. So, for one class I threw a picture of a radar up on the board, wore a headset and started yelling about how the instruments in my plane were malfunctioning, and if they couldn’t figure out what my speed needed to be in 5 minutes as a group, I was going to crash into the mountains. I couldn’t provide them with any help, of course, as I was merely an airplane pilot.

    2) Have a group assessment (quiz/test). These are surprisingly awesome. I’ve actually had my students take a group final for calculus the last two years with great success. I was scared at first, but after a decent amount of lecturing (“This test will be very difficult, if you don’t study normally, you’re only letting yourself down. If you don’t study for this test, you’re letting your entire group down”) they have gone very well.

    3) Competition. Whether it’s a simple game of review Jeopardy or a rousing game of war with limits, competition is wonderful to motivate and almost anything can be turned into a game.

    4) Assign groups of students a different example to learn from the book. (Hey, they have to be able to figure things out from their books for college, don’t they? And besides, you’re there to help.) Have each group do a mini lesson to explain their example to their peers. This could work especially well with review, as they’ve (hopefully) seen it before.

    If you wake up and you’re not feeling your slides, feel free to be impulsive and do anything you can think of to get the kids working together and discussing things in a group.

    Good luck! (Even though you don’t need it.)

  4. I try to do some sort of hands-on project once a week, to break the flow of lecture/practice and let the students take control of their learning. Anything from combining art with math in studying tilings of the plane, to using oatmeal to compare the volumes of pyramids and prisms, to throwing a ball in the air and timing its descent to see the quadratic effect of gravity, to using pieces of spaghetti to draw a sine graph (thanks @k8nowak). I have not taught calculus, so don’t have any immediate ideas there, but I encourage you to try new things!

  5. Hey Sam,

    First of all, don’t sweat it – it’s only day 4! Secondly, try some problem-based approaches. You don’t have to go all in the way I have, but give them some problems to work, an exploration or a lab that they have to work through together which gives you a chance to watch them work rather than direct them. If you aren’t sure what to do, there is a series of calculus books by MAA with labs and applications that are pretty handy (I can send you exact titles if you are interested), and for a little more structured exploration I recommend a book by Paul Foerster. It’s an exploration book designed to go along with his textbook, but I have been using it independently for years.

    Good luck!

  6. Sam,

    First off, let me say that I love your SMART board presentation…”check yo’ self before you wreck yo’self.” *smile* As for the teacher-centered/student-centered questions, that is something that has been a goal of mine for this year, too.

    Something that I suggested to Anne (@msgregson) at one point was a partner review/practice activity…you have 2 forms of similar style problems…partners get different forms, solve a problem, then the switch and check each other. This could possibly work with new concepts, too, where students take the responsibility of (your quote-ish) “Making something they don’t know look like something they know”. Pulling the class together at the end can bring up some conversation about how the partner thing worked, what mistakes they saw in each other. You can also partner with someone, esp. if there’s an odd number of students.

    You inspire me with your reflections and transparency. Keep it up!

  7. I have done a lot of thinking recently about teacher vs student centered classrooms. My classes are almost entirely student-centered, but I’m constantly finding I have to make cuts to what gets covered because we’re spending so much time presenting answers and discussing concepts.

    In our calculus class, we won’t cover sign analysis, but we will probably look at asymptotes vs holes pretty soon. This is what I’m thinking we’ll do: I’ll give them a table with several functions, and columns for recording holes, asymptotes, etc. They’ll fill out the table with their groups using calculators, record any patterns they find, and share out. As long as I pick the functions carefully, I won’t have to explain what it is that makes some functions have holes and others have asymptotes – they’ll figure it out.

    In general, I try to tell the students as little as I can – the less information they start with and more carefully I choose the starting information, the more the students get out of coming to big conclusions on their own. Plus, they often end up making some amazing connections that I never would have thought of if I was straight up lecturing!

    Ooh – one last thing that helps. Whenever possible, I try not to answer their questions directly. So, I answer “Why is that point located at (pi/2, 1)” is answered with, “Good question! Can someone help explain why that point is (pi/2, 1)?”

  8. @Mr Sweeney, Nike, Jim, melanie, Kate: Thanks. I am going to try hard to breathe life into my calculus class. With your help. I’ll credit you if it works, and blame you if it doesn’t!

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