I’ve had similar thoughts about the same lesson, but having the kids graph those things by hand from scratch is pulling teeth–then again if I didn’t teach this right, maybe I didn’t teach plotting in polar well either.

Next time I have to teach this, I’ll probably have them match tables of values, functions, graphs, and descriptions of the graphs, kind of like that UK maths resource suggests.

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.

Devil’s advocate looming: while the students would get the benefit of the process of exploration, would they understand the “why” part any better? In this specific case at the very least I would include translation between rectangle and polar coordinates and some arrows showing the points of matching.

I know from working with preservice K-6 teachers this semester that pulling out the “why” (even on adding two numbers) is extremely hard and requires nurturing and practice far beyond what explorations provide. I’ve had decent luck getting them to make algorithm explanations with two columns (a How and a Why) but that’s entering into Way Too Much Time Spent To Be Practical in High School Class territory.

I dunno, as a student, I’m pretty sure that I’d have found “equation turns into flowery-thingy” interesting enough to look at and play with (and maybe get to make my own flowery-thingy, rather than just look at other ones.

Then, about a week later, maybe even on the test, although often later, I’d get the concepts. They’d just sort of dawn on me and all those things the teacher had said that seemed rather pointless (though were clearly important since s/he said them many times) would suddenly come into focus.

So, while I agree that just grinding through isn’t likely to stick, then again some grinding and some drawing or at least plotting and lots of repetition may oftentimes lead to delayed onset comprehension.

I’ve always thought you were a little too harsh on your 1st-year-teacher-self, second-year-teacher-self, and as not so subtly stated here, your preservice-teacher-self. Maybe it’s all by way of letting others know that even the great Dan Meyer was not so great once (even though I’m sure most of us on here would dispute that).

Personally I don’t think you’d be where you are now if you weren’t where you were then.

My goal, with teaching in general, isn’t so much for them to grasp the why now. It’s to grasp an appreciation for searching for the why now, so that when they’re my age they can’t stand but to go a day without trying to figure out why to something. I can’t, and it makes things awfully fun.

@ Jason:

I don’t know that Dan is implying that he would toss out these ideas, let the kids “explore,” and expect them to arrive at the “why” as a result. I believe in guided inquiry, and carry this out as often as I can/plan with my second graders. Giving the kids time to explore something gives them a frame of reference during the discussion of “why.” And I think it’s important to remember that understanding is a precursor to articulating that understanding. So…students will be able to understand “why” before they’re able to clearly explain the “why.”

John and Frank’s 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.

reminded me of Alan Schoenfeld’s paper When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematics Courses which might be of interest here.

That paper is pretty excellent. Thanks for the link.

One thing that was repeated a lot during my undergrad courses was “play with it.” While this is not something I could say without context to a group that includes high school males, that is what I asked my pre-calc students to do. All my students either had a graphing calculator or could borrow one overnight. Towards the end of class, we went over how to put the calculator in polar form (they already know what polar graphing is, we just hadn’t done much with it yet), and we look at a few graphs involving sin and cos. Then they take some time in class to play with it, and their assignment for the night is to play around with polar graphing and find some patterns.

The next day, we have a whole mess of patterns that have been found, and as a class, we can explore why those patterns exist. It works pretty well, though it does not rule out pseudoteaching without the next day exploration.

@Dan: Not disagreeing — I’ve run polar coordination exploration in class before — just being cautious. It didn’t read like a complete lesson to me yet.

So…students will be able to understand “why” before they’re able to clearly explain the “why.”

@Laura: This is the knotty bit I’d love to unravel: how do you tell when this happens? I’ve certainly done explorations where students had the patterns down as brainlessly as if I’d just told them a recipe. I’d love to avoid that, but sometimes it’s a hard thing to check. The standard approach is to add a wrinkle and recheck for understanding (like mentioned in the example, adding a negative) but that’s not foolproof.

I think they key here was the end realization: by facilitating performances during which students created (and then probably evaluated), a teacher would be able to accurately gauge understanding, which is the goal–not just performance or knowledge. I think this is where the task analysis folks miss the boat.

I believe students should be in a constant state of production. As one of the “straggling humanities teachers” still following along in the discussion, it’s easy for me: students are always writing, revising, conferring, and creating class-wide and personal language arts-based projects. For the example above, I might ask, “What were the students creating with these rose petal formation graphs?”

It’s tough to get kids to understand something if the final output is just graphs. Graphs for what purpose? The Teaching for Understanding (TfU) framework calls the steps along the way “understanding performances” because the students understands more deeply with each scaffolded output (from initial inquiry, to guided, to independent).

As you have already pointed out the aesthetic qualities of those rose petals, perhaps an art-based summative output wherein students had to manipulate the equations for intended results (i.e. students sketch first, plan out the art, then have to troubleshoot the math to make it work) might have engendered transfer.

Just a quick note on graphing by hand. Ever since I took a short course from Win Means at SUNY Albany on Mohr’s Circle (a graphical method for working with tensors), I have been much more likely to use sketch graphing and have a much greater appreciation for the potential of a simple sketch to convey useful and even reasonably precise quantitative information. When I realized I did not understand the functions above, I started in with SpaceTime on my iPad, but the finished graph lacked the information to show me how the petals connected to the trig function, especially since it converted all the coordinates to x, y pairs. I grabbed a handy envelope and went to work on a sketched polar plot, going from cos(theta) to cos (2theta) and within 5 or 10 minutes I had the basics figured out.

Dave

I love that you hold yourself to this amazing high standard: that every kid learns at a high level, every class.

I’m an old (humanities) teacher working with a new age group at a new school, and realize just how rarely I am able to get up to that level in my new setting. Whereas before I could convince myself that I was there a lot — now, hardly at all.

I’m looking, instead, at progress-over-time, when I feel like a failure. Here at six weeks into the semester they have certainly learned, even though I could hardly see it unfolding on a daily basis, and had so many pseudoteaching fails along the way.

Thanks for the reminder of exactly what I am striving for.