After spending a week grading AP exams, a weekend at Desmos HQ, and a few days at PCTM I’ve had a lot of time this summer to think about how I teach calculus, so when gnylbe asked for my thoughts/ideas on teaching calculus to high school students, I couldn’t resist sharing a few (literal) shower thoughts.
- Emphasize that calculus is the study of the infinitesimal and the infinite and ask students to be on the lookout for how both arise naturally throughout your time with them. A fun prompt for students is to ask them to start at zero and to find the next number. At this point, I’ve had some success talking with students about the countability of the rationals / uncountability of the reals with students who have only seen up through HS Algebra II. I’ve even asked how long are the rationals at this point and worked through showing that the rationals have measure zero with the aforementioned students.
- Emphasize the difference between the static nature of algebra and the dynamic nature of calculus. In algebra students may be prompted to solve problems such as
A car is traveling down the highway at 67 miles per hour. How far will the car travel in 2.5 hours?
Whereas in Calculus, students can be asked very different questions. Such as
The velocity of a car is given by
how far does the car travel between
and
hours. Assume
is in miles / hour and
is in hours.
If you are feeling fancy, you can even tie these questions into Desmos.
3. Where I like to start Calculus content is by taking everything back to slope. I have a conversation like this with my students
Remember your first time taking Algebra when you learned about lines? You learned about points, and plotting points, and connecting them using lines, then BAM you were hit with slope: Rise over run. Change in
over change in
. Average velocity. Rates. Dollars per item. So.Many.Applications. Point-Slope form. Slope-Intercept form. Then what? You multiplied your line by
doesn’t have slope! Sure it might not have the same slope everywhere, but I would argue that it does have slope.
From here I would run through parts of an activity like this one. And immediately after spend some time reviewing slope of lines and ask why the slope of a line won’t help us (yet)? (Hint: We don’t have two points to pick.)
Here we segue into secant lines, difference quotients, limits, and finally the definition of the derivative.
I’m not sure how much help this will be, but if I had to summarize my thoughts on all of calculus, they would go something like this:
Given enough time, I would be sure to teach my students that Calculus is much more than the mechanics that come from it. For instance, the limit shortcuts, and derivative rules, and integral properties are great and all, but they are not calculus, understanding the behavior of certain phenomena and describing it by using the language of calculus is calculus, the mechanics are just tools that allow us to do calculus. Much like saws, screws, and drills are not woodworking, but tools that help one to do woodworking.