Last year's class introducing logarithmic and exponential differentiation was a bust. I tried to include it as an application of implicit differentiation, but I knew afterwards then, and still believe now that doing so was an incredibly horrible idea. There's no way students are going to 'see' an application of an abstract concept like implicit differentiation better...by using it in another abstract concept. I've accepted that, and vowed this year to do a much better job.
I also had a shocking moment yesterday when a Calculus student came to me after school and asked me 'what is the derivative?' We had started the unit with a conceptual development of the derivative using limits and average rate of change, and had since moved to applying differentiation rules, so we were deep in that process - power rule, quotient rule, product rule, chain rule...really the primary 'rules' section of any Calculus course. I was taken aback by the comment - had I really stopped emphasizing the definition of the derivative in our class activities? In a way, yes. We had been writing equations for tangent lines and graphing them, but we hadn't seen the limit definition (which I've been impressed by students remembering) in a little while. This proved that not only did I need to do a better job with logs and exponential functions, but that a little conceptual basis in that process would be useful.
I always like using Geogebra as a tool to pre-load information I am about to give students - what is about to happen? What should my result look like when I do this on pencil and paper? The graphing capabilities make it really easy to do this and set this up - I created this file and made it look the way I wanted in a few minutes.
You can direct download the file here.
These were the instructions I gave students:
Sketch what you would expect the derivative of y = 2^x to look like. Then click the 'Show Derivative Function' to graph the actual derivative. How close were you?
How would you expect your sketch to change for the derivative of y = 3^x?
Graph and make a prediction of the graph of the derivative of y = 2^-x. Check and see how close you were using the Geogebra tool.
Can you adjust the slider value for a so that the derivative is the same as the function itself? Use the arrow keys to adjust the slider more precisely.
Go through this same process to sketch the derivative of y = ln(x) in a new Geogebra window. Create this by going to the 'File' menu and selecting 'New Window'.
It was really great seeing students predicting what the derivative would be, and then using the applet to confirm what they thought. There were lots of good conversations about scale factors and reflections, and some of them pretty much nailed what the general forms were going to be. This made the algebraic derivation a piece of cake - they knew where it was headed.
I've been really getting into the idea of standard based grading, and have been doing a form of it through my quizzes for a while, but it is still a small component of the overall grade calculation. While their grades aren't being calculated any differently at the moment, I shared that this list would make a really good tool as we prepare for the unit exam on derivatives next week, and most started going through on their own and deciding what they needed to work on.
I'm still getting caught up after a couple very busy weeks, but I really like how this group in Calculus has been developing and maturing as math students in only a couple months. Their questions are more directed: 'I don't understand this application of the chain rule' compared to 'I don't get it'. Their written work is detailed and clear, making it easy to locate errors. As a group, they get along really well, and class periods are filled with moments of furious productivity and camaraderie as well as humor and smiles throughout.
It was raining hard all day. I watched some students walk into class, look outside at the afternoon sky, and sink into their chairs, clearly feeling a bit down. I told them it was perfect Calculus weather - why not sit inside and do some differentiation?
Probably not what they had in mind. By the end of class, everyone left the classroom looking much more positive than when they walked in, and at least feeling good about the work they had in front of them.