Tag Archives: calculus

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.


Polar coordinates

More lazy email(ish) excerpts. This one from a conversation about polar coordinates, and my Calculus II students’ struggles with them last semester.

I don’t fully understand the usefulness of polar coordinates as traditionally presented. I feel like a great deal of the focus is on the pretty pictures we get when trig functions interact with polar coordinates. Cardioids and all that.

What do we gain from graphing this in PreCalc?

When I taught Calculus 2 this past semester, it was clear that my students were struggling to sort out differences between cartesian coordinates and polar coordinates. They knew how to convert between polar and cartesian coordinates, but they didn’t seem to know why one would do that, nor did they seem to see polar coordinates as a self-contained system. Polar coordinates were always (in their minds) in relation to cartesian coordinates.

My students struggled to think about an angle as an independent variable that could change (and correspondingly a radius as a dependent variable that could change).

They couldn’t view a function defined in polar coordinates as a dynamic relationship. They could identify points one at a time. They could make their graphing calculators display polar graphs. But they couldn’t think about the process of tracing out a polar graph. This seemed to be true even for students who could talk dynamically about cartesian graphs (increasing, decreasing, approaching an asymptote-this was terminology my students could apply to cartesian graphs, but not to polar ones).

The precise conceptual nature of the relationship between polar coordinates and cartesian parametric equations is unclear to me. My students saw some relationships that they couldn’t quite articulate. I’m interested in exploring this territory a bit.

For the record, I was flying blind through this material last semester. It was the first time I had taught polar coordinates in any serious way. I have never taught precalculus. I had never taught Calculus 2 before. So I kept bumping into obstacles that I hadn’t imagined would be there.

The lawnmower problem

David Peterson (@calcdave on Twitter) posted this lawnmower video to 101qs recently. It was love at first sight.

I needed a polar-coordinates-based assignment for my Calculus 2 students, so I pounced on it. The question they have been working on is, How long will it take to mow the lawn?

I read their work today. The following are some quotes from their writing.

“Establishing the polar function was difficult at first, until I thought about it as just a plain linear function.”

“I tried going on the treadmill to see what a comfortable walking speed for mowing would be.”

“Sorry for making this 13 pages. I really got into it.”

“Sometimes math needs a little touch up; this is when Photoshop is there to save the day.”

“The real real-world problem is how to convince your wife to upgrade mowers.”

“Rather than dealing with negatives and reciprocals, this paper will assume the lawnmower ‘un-mows’ the lawn from inside to out.”

“After realizing that the point on the outer ede of a circle has to cover more linear distance than a point near the center, angular velocity seems like it might have some flaws.”

I see in these excerpts students making mathematical connections that result from their struggles with the problem. I see them posing and refining mathematical models based on correspondence to the real world. I see them looking at this small slice of the world through a mathematical lens.

I am so proud of them.

NOTE: In original post, I did not know who had posted the video to 101qs. David Cox came through for me on Twitter. Credit given in revised post.

Course design question

Brian Frank observes:

Lots of physics majors get by with strong algebra skills.

Same story in Calculus, of course. And you can substitute “calculator” for “algebra” to describe a whole mess of other math courses.

Shouldn’t we be ashamed of ourselves when we see this happening? Shouldn’t we be designing courses (and course sequences) in which this is not possible? Isn’t it time we (as a field) stopped living by last century’s textbooks, allowing students to skate by on last century’s skills?

Should we be designing courses that require critical thinking; courses that require students to really, deeply learn the material?

I know Brian’s working on that problem. Shouldn’t more of us be reading his work and taking his example?

More than you bargained for

This semester has been out of control since January. One of many consequences is that I haven’t had time to write about my teaching. There’s a serious backlog of stuff to document; no time to type it up.

Today’s post is a bit of a mix.

Nat Banting wrote this week about a lovely moment when a student looked at a task from a new perspective.

The look on my face must have been priceless, because she started to laugh. The scene went on for quite a while. Slowly but surely, every student had approached the desk to see what was up. The student beamed as she explained…

Explained what? Go read it.

It got me thinking about similar episodes in my classroom; those moments when I take the time to ask instead of tell, and when my students’ ideas blow me away. The moments when my students teach me some mathematics.

Calculus 2 has had many of these moments this semester. We were studying approximate integration (about which, much more in future posts). We had a motto, “When you cannot integrate, you must approximate“. We had approximated with rectangles and trapezoids. We had built up the formulas for these methods based on students’ intuitions. And then it was time to deal with Simpson’s Rule.

If it’s been a while since you studied such things (or indeed if you never have), the basic idea is in the picture below:

We’re trying to find the area enclosed between the function above, the x-axis below, x=-1 on the left and x=1 on the right. Those rectangles give a pretty good approximation of that area. Each rectangle has a bit of extra area (above the function) and leaves out a bit of area; those roughly compensate for each other and the result is a good estimate.

The reason it’s not a perfect measure is that we are using straight lines (apologies to Chris Lusto) to approximate a curvy function.

So I asked my students what a reasonable solution to this problem would be. What curvy functions should we use to approximate f(x)? We all agreed that it would be desirable for these functions to have nice calculus properties, since that’s why we’re approximating in the first place (that original function doesn’t submit to the standard set of techniques for finding this area exactly).

We were building towards Simpson’s Rule, which is based on parabolas. Use parabolas as tops on those rectangles instead of horizontal line segments and you can get a really nice fit. Plus, parabolas are easy to integrate. Plus if you do a ton of complicated algebra, you can find a really nice formula so that you don’t even have to bother integrating (the pedagogical benefits of this are debatable; the calculational benefits are massive and undeniable).

So I asked. Not just rhetorically. I asked and I listened to their answers. They wanted to use sine (or cosine). And they wanted to use exponentials.

Of course. These are the things with the simplest antiderivatives. Polynomials get more complicated when we integrate. Sine, cosine and e^{x} pretty much stay the same so they’re easy to evaluate. It’s brilliant, right? The algebra won’t work out nicely, and you won’t end up with a clean and tidy rule. But who cares? We’re trying to learn some Calculus here; some ways of thinking mathematically about relationships among functions.

I made the evaluation of this integral by a sine-based version of Simpson’s Rule into an A assignment. Several students are working on that right now.

So much more to report. It’s been a productive semester. I’ll get on that in just a few more weeks.