Decoding mistakes in Calculus

This year I am using mistakes to prompt discussion in class and make homework dissection more engaging. In my AP Calculus AB class, I use Active Calculus as the text, and many of its activities and exercises are perfect for an Activity Builder conversion. Such was the case when I assigned this activity to work on section 1.3 Ex 1 in Active Calculus.

The very first slide has students drawing some secant and tangent lines on a function. Most students in all three sections did pretty well with it, but I saw a few common errors I wanted to address. So channeling my Talk Less course from TMC16, I made up these two slides.

day-13-ab-errors_1day-13-ab-errors_2

Each table had a minute to respond to the prompt. The best discussion happened with the second slide when I asked students to conjecture on what the student was thinking when he drew the green line (which was supposed to be tangent). When I started asking this question in the first of 3 sections, I didn’t really have a good idea myself.

Students in each class proposed that the student thought tangent means it can only intersect the function ONCE (like with a circle). The student may have thought this was the only way to accomplish such a definition. I’d never considered that. It led to a great wrap up and clarification.

Here’s another slide I used from that activity.

day-13-ab-errors_3

There was one more unexpected observation from that activity. After the exercise from the text was done, I built in a few slides at the end where students would connect elements from the limit definition of the derivative to its traditional graphical counterpart. It started with an animated version (slide 8) and then asked questions about specific terms and where they link up. On slide 9, I asked where the two points for the AROC were. Most answered in the way I would have expected, by selecting the 2 points in their one spot. But then I got this.

Screen Shot 2016-09-08 at 9.42.29 PM.png

Exactly one student in each section had a response like this. And again, students were able to put themselves in that one student’s shoes and realize that in fact, that second point is not static and can be in many different places. I would have never done that on my own.

 

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3 thoughts on “Decoding mistakes in Calculus

  1. Jasmine

    I’m loving that you are starting to build some Activity Builder activities based on Active Calculus! I used Active Calculus for the last few months of calc last year (as we talked about at TMC) and I’m using it as the backbone for our course this year. I just finished an intensive 3 week algebra bootcamp w/my calc kids (they had some tough teacher turnover in the last two years, so needed some extra TLC and guided review before we launched into calc content). We’re going to start using Boelkins the week of 9/26, and I’m collecting additional activities to use with my students.

    Some questions about your class:
    What other resources to you use to supplement Boelkins? Do you use SBG in your class? If so, do you have a skills list? How do you structure your time w/kids vs. what they do for homework? How much time is lecture/guided work vs. time to work in groups/alone during class? Do you do other activities/projects with them? Do you do any take home tests or all in class? Are your kids also preparing for the AP? What additional resources do you give them for that preparation? In what ways does formative assessment happen in your classroom? Do you do any sort of daily warmup or daily quiz? Do you do many activities or other ways to practice/solidify content? How do you structure grades?

    Reply
    1. Dave Sabol Post author

      I have been making a few ABs for Active Calculus — maybe I will collect them and post somewhere on here. They are very straightforward to the text, but they can still save some time for others. In many cases, it is just far better to get them on the desmos dashboard and get an overlay to see how it all looks.

      >What other resources to you use to supplement Boelkins?

      I have some resources from before I started using Boelkins that I still go back to. I also use MathXL for skill building. As the year progresses, I will use quite a few AP FRQs in class. You know, I hadn’t thought of it yet, but I think Activity Builder may be a nice platform for FRQs.

      >Do you use SBG in your class? If so, do you have a skills list?

      No and no, but I am trying to build a good skills list so I can use it to be more transparent about where we are and where we are headed. Plus if I have a good skills list in place, it will make it easier for me to switch to SBG in the future.

      >How do you structure your time w/kids vs. what they do for homework? How much time is lecture/guided work vs. time to work in groups/alone during class?

      Considering the depth of the Boelkins activities and exercises, it’s not uncommon for most of a class to be dedicated to HW review (2-3 activities/exercises a night). They are in 3-4 student groups, so a good portion at the start is collaborative in checking work and getting it on the board. In a week (200 minutes) I would say that about 110 minutes is HW review, 20 minutes group checkin/HW review, 30 minutes summative assessment, and the rest guided discovery activities. Keep in mind that that 110 minutes for HW review as a whole lot of tangents (pun intended) on reinforcing key concepts that come up.

      >Do you do any take home tests or all in class?

      Most of my unit tests are split between in class (objective) and take home FRQs (my own) that require a lot more time and justification.

      >Are your kids also preparing for the AP? What additional resources do you give them for that preparation?

      They are. I have found some good resources through the College Board listserv over the years. Boelkins doesn’t quite hit all the points in the same language or depth that the AP wants (but not in too many places) so there is some work to be done making sure they are on the same page as the test.

      >In what ways does formative assessment happen in your classroom? Do you do any sort of daily warmup or daily quiz? Do you do many activities or other ways to practice/solidify content?

      My classes are so so short (40 minutes, yikes) so I struggle with setting up good open and close routines. But as intro to new topics I do like to find good activities to intro concepts. I think it’s key to find creative ways to intro topics so that students will (hopefully) forever link that topic to the connection we make in class.

      >How do you structure grades?

      Relatively traditional — weighted categories. 50% test, 36% quizzes, rest is HW and other in class work.

      Reply
  2. Pingback: Stars of the Week Vol 6 #SOTW – Insert Clever Math Pun Here

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