How I Teach Calculus: A Comedy

I then show how the text book distills the idea down to a lifeless corpse, and I assign ungraded practice problems.

To act as a foil to my ongoing series on Standards-Based Grading, I would like to provide you all with a more in-depth study of how I think a math class should be run, and, of course, how that theory actually plays out with real, wriggling children. Being a trained science teacher, I was a bit flummoxed when I was given control of a couple of math courses at my school. I wasn’t bothered by the content so much as pedagogy. I’d never practiced teaching math during pre-service, so I was kind of at a loss for a center to gravitate towards.

I did what most would do in this situation: I thought back and parroted the instruction I had been given as a student. My initial classroom looked like this (in sequential order, daily):

  • Lecture New Material Daily (Try to be engaging)
  • Work problems students want to see from Homework
  • Give class time to do new homework (threaten to take away time, if wasted)
  • Grade homework sets for completion & accuracy
  • Give quizzes weekly
  • Give a few highly-weighted tests

Va – ni – lla.

At the end of last year, a few things hit me that I had a hard time dealing with, which almost caused me to leave teaching:

  1. My students have 0% ownership of their educations.   :(
  2. My students have no idea what they don’t know.   >:(
  3. My students are addicted to the memorize-regurgitate-repeat cycle.   }:-O!%^$!@

Then two things happened. I really got into the idea of Standards-Based Grading (alleviates 1 and 2), and then I read this and this (addresses 3).

My classroom has changed a bit:

  1. I present a genuine1 application of a new concept before ever using any math language.
  2. Students interact with the genuine application or simulation thereof. (may span days)
  3. I begin teaching, filling in blanks and understandings of how to address the application mathematically (healthy discussions usually ensue)
  4. I then show how the text book distills the idea down to a lifeless corpse, and I assign ungraded practice problems.
  5. Students perform as many practice problems as they deem necessary.
  6. An assessment (quiz/interview/tactile) is then given and graded via Standards-Based Grading.
  7. Students then choose if we continue on using their own ideas and excitations, or if we move on (within reason).
  8. A summative (cummulative) midterm and final are given to test retention and promote proper study habits.

And Mr. Cornally is happy.

Mr. Cornally is also playing with fire. Questions abound: How do I come up with real-world applications for material that mathematicians have intentionally abstracted beyond a high schooler’s recognition? How do I make sure we’re not entering activity mania? How do I ensure that students will actually do ungraded practice problems? What if nobody likes my new sweater?

It has been a year since I’ve made these changes, and here’s what has happened:

  • Copious Genuine Real-World Applications: This is the most difficult, because my goal is never to obscure a math skill, but also to tie it to reality. I have to remember that these kids do not have the years of application experience that I do. I’ve come up with a healthy list of applications. I plan to document each as these posts continue. Please stay tuned for the gory details of my new calculus curriculum. The kids have responded well, as long as I make the skill clear. The application gives them a handle for the skill that they can hold. It becomes a memory where they go to remember a specific idea. Not how I intended, but still helpful.
  • Activity Mania: When you let the physics teacher (me) teach calculus, you’re just asking for this problem. I definitely have to keep myself under control. We do one non-traditional activity (lab, simulation, whatever) per concept, and I limit myself. Students in high school calculus are for sure preparing for college, so I must keep collegiate math preparedness in mind (even if it is contrary to my instincts)
  • Ungraded Homework: Get the pitchforks, that’s blasphemy! If you change anything about how you teach math, do these two things: Adopt Standards-Based Grading, and STOP grading homework. You and I both know that homework is for practicing skills. Do athletes get medals for work outs? Musicians applause for rehearsal? Then why should students get ‘points’ for practice? There’s a whole lot of copying going on anyway, don’t kid yourself. So why not control the assessment through Standards-Based Grading. My experience with my students has been that they definitely do the problems. Some students don’t do all of them. They either don’t need to, or will learn the hard way to step it up. This teaches so many more important things than math. They become self-assessors (YAY!), they become self-reliant (YAY!!), and they might actually care if they get something wrong (YAY!!!). All of this hinges on your proper use of standards-based grading. By the way, have I mentioned standards-based grading yet?

First Lesson – FREE! The Limit Definition of the Derivative: Coming Soon.

Just kidding, they’re all free. Ready for Steak and Taters? I’m going to get you my play-by-play lessons for all of the major topics in calculus as we cover them. Stay tuned!

1: Genuine is a tough word, because there are many situations where an advanced technique could be used, but is more like nuking an anti hill. Genuine means that the concept you want to each is the BEST way to solve the problem. Rough.