I’m working on a paper that will be submitted for publication. It mostly records and expands on my NCTM presentation in Denver a while back.
Along the way, I decided to a bit of looking into the claims I have made about the standard view of what it means. I did it the lazy way. Google search on “Ready for Calculus”.
See for yourself. (Click on the images to be taken to original sources on the web.)
I did find one that had a different nature.
The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions… Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets… You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of function underlies everything that calculus considers.
This was nice, and if you’re at all interested in this topic, you should go read the essay in its entirety. (Don’t worry, it’s short.)
And now let’s all imagine how a community college developmental math program that took Exhibit D more seriously than Exhibits A—C would be different from the present-day state of affairs.
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.
One 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).
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.
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.
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.
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?