I've been debating this with myself. Should my first blog post after an extended break be about a teaching mistake? Doesn't everyone want to get inspiration for good teaching, rather than bad? Nah. We all need to learn from each other's mistakes, and we should discuss them rather than pretend we never make any.

So, with that in mind...

The aim of the lesson was to introduce students to derivatives of trig, exponential, and logarithmic functions.

I decided to use Desmos for an investigative activity (technology! and inquiry-based math!).

I created a graph with instructions, and shared with my students.

In case you're too lazy to click that link and check out the instructions, here's a summary:

- Check out the graph of f(x)= sin(x). What do you think the derivative will look like?
- Ask desmos to graph derivative. Huh. What IS that function do you think? Graph your guess to see if it is correct. If it's correct, write down your conclusion. If not, try again.
- Repeat with cos(x), tan(x), e^x, and ln(x)

Once students got the hang of the instructions, they engaged very actively with the task and were more or less successful with guessing the derivative functions. At the end of 30 minutes or so we could summarize the findings. Everyone was happy. It

*felt*like an awesome lesson.
But it was crap. I realized it a while later, out of nowhere. There was nothing to trigger conceptual understanding, no variety of thinking and problem-solving. Instead of consulting desmos to ask for derivatives, students would have been better served by using their formula booklets, which is what they'll have on exam,

I'm writing this lesson up as a warning example of the seductive nature of technology and inquiry-based teaching. Both have their place, of course, but only as tools for obtaining specific goals relevant to learning.

In this case, what I should have done, is to build on student understanding of the derivative as a gradient function by assigning them to groups, one per function, and having each group work old-school with paper and ruler to draw tangents, estimate gradients, and plot the derivative function based on these values. Then they could have tried to identify the formula of that function. How do you beat paper and ruler for an activity like this? You don't.