Calculus is a beautiful and powerful story, with historical necessity driving it. Starting with limits (a supremely technical and possibly alienating topic) seems guaranteed to turn droves of students away from what could be their favorite math class ever. About a year ago, I read the first 6 pages of Morris Kline's

*Calculus: An Intuitive and Physical Approach*, and was entranced. (The link goes to the Google book, which allows you to read all of those first 6 pages. The rest of the book didn't work quite so well for me, but it may for you.) That, along with a bit of a mini-history in Matt Boelkins' book, helped me set the stage during the first week. I assigned those 6 pages for students to read as homework this weekend in both calc I and II. (And I wish I had made a handout asking them a few questions about it. It's done now - I'll assign it on Monday.)

Here's the mini-history from Boelkins' preface (page v):

Several fundamental ideas in calculus are more than 2000 years old. As a formal subdiscipline of mathematics, calculus was first introduced and developed in the late 1600s, with key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz. Mathematicians agree that the subject has been understood rigorously since the work of Augustin Louis Cauchy and Karl Weierstrass in the mid 1800s when the field of modern analysis was developed, in part to make sense of the infinitely small quantities on which calculus rests.

In both calc classes, I talked about the 3 phases of this history:

- ancient history, in which Archimedes (and probably many others) figured out areas of simple curved shapes like the circle,
- the 1600's, in which Newton and Leibniz developed the calculus, and fought over it, and
- the 1800's, in which Cauchy and Weierstrass built a logical foundation for the parts that Newton and Leibniz had left pretty fuzzy. (I told them this was where the textbook treatment of limits belonged, and that we'd build up to it after playing with the deep ideas developed by Newton and Leibniz.)

In Calc I, I also described the two main problems calculus was invented to solve - finding slopes of curvy lines, which is the same as finding rates of change and instantaneous velocity, and finding areas. I had them graph y=x

^{2}, draw in a tangent line at x=2, and estimate the slope. I heard lots of good conversations. Toward the end I asked them what definitions they already knew for tangent. They don't know the definition of tangent we use for calculus yet, but that definition just makes precise our intuition. (Unlike limits, which do not really connect with any prior intuitions.) They gave me the circle and trig definitions, and we discussed how to define tangent in this new situation. We did not arrive at anything momentous, but the issue is in their heads now, hopefully simmering.

I loved starting this way. In the calc II class I had planned for them to find the area of a circle on the first day, but left out the best part - cutting up a circle. So their project extended to day 2, and I also used it in calc I on day 2. Fawn Nguyen's work on this with middle school students was my inspiration. Although my students know much more mathematical content, I think this is a tremendously valuable lesson in mathematical thinking. How many people notice this important difference between the two circle formulas they've memorized, C= 2πr and A=πr

^{2}? C= 2πr is a simple algebraic revision of the

*definition*of π (since π is defined as the ratio of circumference to diameter), and A=πr

^{2}is a provable

*theorem*. Here's my handout (after revisions):

Circumference and Area of a CircleThinking about Definitions and ProofsYou probably know the ‘formulas’ for circumference and area of a circle. (Do you have any tricks for remembering them?)But where did they come from?Part I.The number pi isdefinedas the ratio of circumference to diameter: π=C/DRegardless of the size of the circle, and of the units of measure used, the circumference will always be pi times the diameter. (Why?) From this definition, we get the more usual form: C= 2πrPart II.Now let’s try to find the area of a circle. We’ll pretend we don’t know any ‘formulas’.1. Start by drawing a circle as carefully as possible on the centimeter graph paper on back. What is its radius?2. Make a guess, just by looking, for the area of your circle. Guess: _________3. Now come up with the best estimate you can for the area of your circle. Describe how you did it:Part III.Once you’ve gotten to this point, get a (coffee filter) circle from me and fold it through the center as many times as you can. Cut on the folds. Arrange the pieces in a way that helps you re-think area. Are you seeing anything interesting?Part IV.Can we turn any of our estimates into aproof?

I saw some eyes lighting up as students saw the circle turning into a lumpy rectangle, and came up with area = height (radius) times width (half the circumference), on their own! Then we got to discuss whether we had a proof yet. (No, it's not a rectangle. We'll need something like limits to turn this into a proper proof.)

On Wednesday and Thursday, we worked through the first projects in the Boelkins text. My students need lots more detailed instructions than he provides. I'm going to modify this quite a bit for my next class. But I do like it.

______

There are always a few students in Calc I who have taken the course before. I was talking to two of them in my office, and discussing the need for limits. We've been taking the slope of secant lines, where the two points are very close. We want to bring them crashing together. So we're looking at the limit as the distance between them (h) goes to 0 of the slope. As I explained, I realized that all the pictures with limit questions use x instead of h, and never have the hole in the function at 0. I drew my own picture, and helped them see the connection of limits to the slopes we're trying to find. When I address this in class next week, I hope to have a good drawing of this. I can't believe I never noticed this lack before.

Next Week:

- More practice with finding tangents through numeric approximation and algebraic work,
- Starting to define the idea of limit,
- Defining the derivative,
- Sketching graphs based on function and derivative values, and
- Sketching the graph of the derivative of a graphed function.

> As I explained, I realized that all the pictures with limit questions use x instead of h, and never have the hole in the function at 0.

ReplyDeleteOMG. Thank you! This is a great example of a connection that is clear to teachers at a level so below the radar we never surface it and show it.

Thanks, Dan, for confirming my reaction. OMG, can't the textbook authors see how helpful it would be to do some limit questions this way?

ReplyDeleteI'm curious how my limit lessons will go.

Deriving the definition of pi could be done in 5th grade (after learning to divide decimals) and estimating the area of a circle using circle wedges could be done in 6th (after learning to find the area of a parallelogram)

ReplyDelete