Seriously you guys what’s going on here? Now Liz Clark sends along a video of three cheese cubes melting in the microwave on the same plate. Spoiler: they melt in the opposite order of my video.

Cheese Cubes pt. II from Dan Meyer on Vimeo.

This isn’t funny anymore.

Related: Guided Inquiry and Surface to Volume Ratio

2011 Feb 27. Liz Clark writes back in, having taken the rotating tray out of the microwave and melted one cube at a time (per Matt’s recommendations). Our results now agree.

I threw twelve different blocks of cheese in the microwave last weekend. Here is video of five of them, 1-cm, 2-cm, 3-cm, 4-cm, and 5-cm cubes:

Cheese Cubes from Dan Meyer on Vimeo.

My goal is to use those data to predict how long it’ll take this guy to fully melt:

For fun, I also threw in two blocks with the same surface area and two blocks with the same volume, just to test out my two prevailing hypotheses, neither of which played out.

Here are the numbers in a Google spreadsheet. You could help me out by coming up with a model that fits the data well and (especially) explaining why your model makes sense in the world of microwaving cheese. I’ll post my own model as a spoiler in the comments. I have no idea why it works, though.

John Burk and Frank Noschese are running a pseudoteaching meme I encourage you to explore for yourself. Their definition:

Pseudoteaching is something you realize you’re doing after you’ve attempted a lesson which from the outset looks like it should result in student learning, but upon further reflection, you realize that the very lesson itself was flawed and involved minimal learning.

Recall also Deborah Loewenberg Ball:

I think we’ve all had the experience of giving so much structure and help that the problem becomes a simple routine problem when it wasn’t originally.

Scene

I don’t like to think about my student teaching year but for the sake of the exercise let me describe my PACT lesson, which, if you’re in California education, you know is the summative assessment of your worth as a preservice teacher. It was a precalculus class and the objective was an understanding of rose petal polar functions like r = 3cos(2theta). Students will be able graph the family. Students will be able to describe how each coefficient affects the graph. Etc.

As I recall it, I was up at the front of the classroom with a TI ViewScreen showing them graph after graph, asking them to determine what the a is doing to the graph of r = asin(3theta).

Then how does b affect the graph of r = 2sin(btheta)?

Then how do sine and cosine differ?

Then I assessed their knowledge of rose petal functions with a worksheet of graphing problems.

Breakdown

I thought my students understood the behavior of r = acos(btheta) on a deep level but they were only responding to superficial patterns in notation. (My supervisor positively thumped me for that one.) Many of you readers – even the straggling humanities instructors we haven’t yet scared off – can see from the graphs above that the petals double when b is even, that the petal length is equal to a but do any of you understand why? Do you know why the graph of r = cos(2theta) has four petals? Could you tell me why the tip of the first sine petal is always on the polar axis? (Not for nothing, the scaffolding in the worksheet also ensures I’m only getting pattern mimicry out of my students.)

Without that understanding, if I attach so much as a negative sign to that function, my students are toast. If I change a coefficient to a fraction, they’re toast. If I change sine or cosine to tangent, they’re toast. That inflexibility is an outcome of pseudoteaching.

Resolution

Tom Sallee:

Unless you understand what an algorithm is going to do, it isn’t going to make sense to you.

The students have to develop the algorithm themselves. Given a second chance at that mess, I’d get students in groups of three or four and let each student pick a member of the family of the functions – “Okay, you do r = 1cos(2theta). I’ll do r = 2cos(2theta). You do r = 3cos(2theta).” Rather than watch me mashing buttons at the front of the room, students would graph their functions by hand and then summarize their findings to each other and then the class. Maybe with a poster – your call. Questions from the teacher would then include everything in the breakdown above, everything I missed six years ago.

Karl Fisch reminds me in the comments that I’m giving a webinar a week from tomorrow for Education Week. That’s February 24, 4:00PM EST. The cost is $49 and if it helps you complete the cost-benefit analysis, I’m estimating you’ll see 30% new material on top of my Classroom 2.0 webinar from December – at least one never-before-seen WCYDWT and a couple of exercises for teachers.

For whatever else it’s worth, I’m not seeing a cent of your registration fee. If nothing else comes out of this, I hope to get a lot of smart people taking shots at my work. I’m also really grateful for the opportunity to quit fumbling around with blog posts and my mental bottle rockets for an hour, to sit down and see if I can get any of this to cohere.

Clay Reisler e-mails in a real tearjerker this morning. His class noticed that the meteorologist on the local Fox affiliate was charting the weather with unspecified intervals and axes, basically blaspheming the data representation gods.

So they e-mailed Fox and asked for a clean set of graphs to analyze (ie. no meteorologist cluttering up the frame) and Fox obliged! Then the students’ assignment was to compile a list of recommendations and e-mail them back to Fox. End of story, right? Yeah, except Fox took their recommendations to heart!

I hope you’ll head over to room 2001 and show those kids some love for speaking math to power.