I don't like starting with all the details about limits - way too technical. (Nor do I want to start with weeks of review. Sam has some great ideas about quick reviews when needed.) I want to start with the beauty of the derivative idea. But I'll need materials to do that. (My blogging comrades will come to my aid there, all their lovely work is recorded in my google bookmarks.)

How do I approach the limits then? I've started thinking about how to bring them in at different levels, as we need them.

I'm gathering all my thoughts on a google doc. You can see my work, or you can email me (mathanthologyeditor on gmail) for edit access. I'd love to work together with others who are trying to move in the same direction.

Right now I'm working on getting all of my 50-some google bookmarks for calculus-related posts into my outline of the course. When I'm done, I'll make a version for others that focuses on the links.

My personal system is a modified SBG, I guess. I give tests on whole units, that are broken into sections for the major topics. Students can re-test on one section / topic in my office. Until now I've used percent grades, but I may put those in my gradebook and not on students' tests. Just let them know what went wrong and whether they've shown their mastery yet or not.

Right now I'm thinking 9 units. I'm not sure whether I'd test after every one. Maybe only after the 3rd, 5th, and 8th? Hmm...

Part I. Slopes and rates of change

- Exploring Slopes of curvy lines / Rate of change
- Playing with Limits (lighter treatment than most of the textbook stuff at this point)
- Exploring Derivatives (the 'rules')
- A more precise study of Limits
- Exploring Further with Derivatives (more 'rules', applications, graphing - this one's huge)

- Limits and Area
- Anti-derivatives (or "What's the big idea?")
- Exploring Integration
- Volume

Hi Sue,

ReplyDeleteI'll be teaching calculus, too, (after a 3 year hiatus). I taught basic limits to my precal class last year, and they seemed to have grasped it. Unfortunately, I have all my notes at school. I know I started out with graphical limits and kept stressing limits are where the function INTENDS to go and we trace it out with our left and right index fingers and such. We also choral chant reading out the notation and what each part means. Does this make any sense? After a bit of that (a day? 2 days?) we go to algebraic limits and I show them 3 cases. ..... anyway, this is getting long.

Are these the types of questions you were having, or was it of a different scope?

Different scope. (I can use those ideas later, when I'm thinking about how to teach limits. And I liked your summary of limits cases, which I noticed as I've been going through my calculus bookmarks.)

ReplyDeleteAt this point, I'm thinking about the order of topics. Why is calculus interesting to you? I want to start with something big like that.

I once explained to the person sitting next to me on a plane what calculus is, by showing him the basic problem of trying to find a tangent line to a curve. (Instead of begging the attendant for another seat, he told me I should apply for a teaching position in his community.)

I want to start with that sort of thing in class. That's what's real here. The limits historically came later, as mathematicians tried to make sure they were on solid ground. I want my students to see the need for limits.

I've added you. Check out the google doc I started and add your thoughts.

Hi Sue...I'm looking forward to following your project. In particular, I love that you are focusing on the big ideas of calculus.

ReplyDeleteI'm teaching a similar class next year except that I will have a group of high school seniors. Mine is a semester long class and I am hoping to use a problem-based approach in which each problem gives students an opportunity to investigate and unpack some of the issues that drove the development of calculus (limit, derivative, integral).

Some of my early ideas are:

1. Does pi exist? Using the famous Archimedean (is that a word?!?!) investigation for limits.

2. What did the trail look like? Giving students the image that you see at the end of a workout on the treadmill/stationary bike (in which the bar graph indicates difficulty, not slope of the hills) and have them recreate what the path might have looked like...derivative

3. Still thinking on a good one for integral??

Anyways, I'll check back and I would love to collaborate more. Good luck. I'll be excited to hear about your course!

Hi Bryan. Thanks for your comments. Have you looked at the google doc? Would you like permission to edit?

ReplyDeleteMaybe I should have made it wide open, like the Twitter math Camp doc. Then people can go write, right when they're inspired to do it.

I am and engineering student, and I must say ur blog is good! I also have a blog (about computer programs) and a web directory, would u like to exchange links MAth Mamma??

ReplyDeletelet me know

emily.kovacs14@gmail.com

Emily

I can barely keep up with all the math blogs. Sure can't add CS stuff into the mix...

ReplyDeleteHi, Take a look at my link on limits. I think it may be what you are looking for: direct but not too technical. http://www.zoomincalc.com/2011/08/02/limits-the-basics/

ReplyDeleteI also have 2 other videos on limits. They are under the category of limits on my site: http://www.zoomincalc.com By the way, I love your blog.

After a week on finding slopes (of secant lines) and average velocity, I think we're just about ready to approach the derivative.

ReplyDeleteI talk about how we're 'cheating'. We use 2 points and get a secant line when what we really want is the tangent line (which we only have one point for).

When we wanted the slope at x=.8 today, we used .8 and .799 and also .8 with .801. One student suggested we average the two slopes we get. I called it Vern's Hypothesis.

I think my students are going to

wantlimits by next Tuesday. When they do, I'll introduce it, and I will also point to your videos, Sue. Thanks.