Historical Intro to the Tangent Line

I mentioned when I started this blog that this is my first year teaching math, but that I spent the last seven years teaching history. So it’s probably not a surprise that I find the history of mathematics interesting, though I’ll admit I have a LOT to learn in that area. I’ve pulled a few lessons from this CD-ROM, called Historical Modules for the Teaching and Learning of Mathematics, which I highly recommend. And then I developed a lesson for my calculus students as an introduction to The Derivative.

For full disclosure, you guys are actually getting the edited version. What I gave the students was overly ambitious and overlooked the fact that ellipses and hyperbolas have more complicated derivatives than parabolas, and I was really just going for something simple to ease them into things. I made some on-the-spot changes during the presentation of the lesson, and now I’ve come back to the lesson and changed it to what I really ended up doing.

All that said, here’s the lesson.

Some things to note about the execution of it…

• I didn’t actually say the word “derivative” until they got to the very end. Their textbook doesn’t give them the word until the next chapter, but hey, they found it! Why not tell them that’s what they were doing?
  
• Your students need to be familiar with GeoGebra in order to make this thing work. Mine ended up having to use the Van Schooten worksheet to find their tangent line slopes because they couldn’t actually do what I’d asked them to – this was the first time I’d told them to do something other than graph a function on GeoGebra, and it was over their heads. At the same time, they did learn a lot about what it can do and what doesn’t work! :) I’m hoping to have next year’s group prepared by working with GeoGebra a good bit while I have them in trig/analyt this year.
  
• Don’t tell your students you just ordered a book on the history of calculus unless you want them to look at you like you’re insane.
  
• My students didn’t know what “analytic” meant or how it was different from “geometric.” My own understanding of geometric solutions to problems has been greatly increased as I’ve done my thesis (on Descartes’ La Geometrie), so hopefully I was able to explain it to them well. I am much more of an analytic thinker than a geometric one.

My goal in using this was to have the historical setting provide a context for studying the mathematical concept. I think it worked pretty well. In any case, staring at the board where they’d done a lot of the work together and where I’d circled an expression and written “derivative,” one student said,

You know, I really do think I get this. It looks complicated, but it’s really not that hard – it’s just slope.

Wait, what?

I hate it when kids seem to really get something, it works great for them, they can do the problems…and then they ask me a question that shows me they missed the point of the whole thing.

My calc kids are taking a quiz over the precalc review stuff. One of the questions asks them to find a natural logarithm regression equation for a set of data. Should be no problem – they’ve been doing great with that.

But one of them just came up and said, “I found the equation, but when I look back at the x-values from the data we were given, it doesn’t have the right y-values.”

Now, I know it’s not the stats class, but still, I didn’t realize that they didn’t know what a regression equation is all about. I’m glad to realize it now, but I hate that I was just having them find regression equations without understanding what they were doing. Sigh.

So, what to do about it? I think I need to be more careful, more deliberate, about making sure they understand concepts that I think should be prior knowledge for them. I need to stop assuming that they know something because they can execute an algorithm; that doesn’t help them learn, and it will end up causing me frustration down the line when I want them to build on a concept they never had to start with.