Unexpected Misconceptions

My trig students are now learning about the graphs of trig functions. (Yes, I’m behind.) I started off by talking about periodicity, and then I talked about how the values of sine and cosine repeat once we get all the way around the unit circle and start going around again. I then presented the idea of f(x)=sin(x) and started getting them to tell me what values I would plot.

They were confused. I figured out two major reasons for their confusion. The first one, I think I’ve addressed, but the second is, I think, still causing befuddlement.

Reason #1: I put a graph up there and used crazy things like pi/2, pi, 3pi/2, and 2pi as my notch marks on the x-axis. They’re used to integers there, and to think that I could divide my axis so that each notch represented pi/2…well, that was just insane to them. I think it relates to the way they don’t really like to think of pi as a constant, since they can’t express its value exactly in any way other than a weird Greek symbol. But I think I dealt with this one okay, though I might mention it again today just to make sure they caught on.

Reason #2: This is best expressed by giving a quote from a student.

“Why are you saying ‘sin x’ when sine is y?”

The student was confused because, on the unit circle, she had memorized that the y-value is the sine of the angle (which we’ve been calling theta). All that relating of the values on the unit circle to right triangles? Yeah, that didn’t work for her. She just memorized that “sine is y.” I tried to show her that we had been saying that sin(theta)=y on the unit circle, and that now instead of theta, x is the angle of which we’re taking the sine for our function…but I don’t think it clicked for her. Well, I thought it had, but then when I said that we could also do a function f(x)=cos(x), she said, “But then cos(x) would be y, and I thought sine was y?”

So I’m still working on that one. But I’m making myself notes, so that next year I’ll be able to anticipate these misconceptions and hopefully prevent them by changing my earlier instruction. I’m sure new ones will pop up to surprise me then. ;)

Cool Stuff!

E was confused about marginal distribution on her quiz. She came in for a reassessment a couple of days ago…and was still confused, so her score didn’t change. I explained the concept to her again, and she seemed to get it.

We have homeroom at the end of the day. As soon as homeroom was over today, E came running into my classroom.

“Mrs. Dean!” she said. “I made a contingency table in homeroom!”

I said, “Great! Is it on the board in Mr. C’s room, then?”

“Yes – come see!” I was supposed to be going to a meeting, but I figured it could wait a minute or two, so I walked down to Mr. C’s room with her. As we walked, she continued: “Mr. C took a poll – somebody had this toy thing, and we were trying to decide whether it’s an evil fairy or an alien. So I said that we could break down the results by girls and boys. And I got the marginal distribution part and everything!”

We got to the classroom where, sure enough, she’d drawn this:

She pointed out that while there were some votes for alien, ALL of the girls voted for evil fairy. She also pointed out the marginal distribution that she’d written at the bottom. I asked her what percent of the people who voted for evil fairy were boys, and although she couldn’t calculate the percentage in her head, she knew that it was 2/12.

So I changed her score. Because she knows it, and I know she knows it. This wasn’t a scheduled reassessment that I generated for her; she saw an opportunity to use what she’d learned, and then she drew my attention to it because she knew it was a demonstration of her understanding. And that? Is awesome.

Edited because apparently writing a post quickly makes me leave verbs out of my sentences…sorry ’bout that.

Wait, what?

I hate it when kids seem to really get something, it works great for them, they can do the problems…and then they ask me a question that shows me they missed the point of the whole thing.

My calc kids are taking a quiz over the precalc review stuff. One of the questions asks them to find a natural logarithm regression equation for a set of data. Should be no problem – they’ve been doing great with that.

But one of them just came up and said, “I found the equation, but when I look back at the x-values from the data we were given, it doesn’t have the right y-values.”

Now, I know it’s not the stats class, but still, I didn’t realize that they didn’t know what a regression equation is all about. I’m glad to realize it now, but I hate that I was just having them find regression equations without understanding what they were doing. Sigh.

So, what to do about it? I think I need to be more careful, more deliberate, about making sure they understand concepts that I think should be prior knowledge for them. I need to stop assuming that they know something because they can execute an algorithm; that doesn’t help them learn, and it will end up causing me frustration down the line when I want them to build on a concept they never had to start with.

Habits of Mind

This post is coming after a Twitter discussion with @jybuell, who is reading about the 16 Habits of Mind as described by Arthur Costa and Bena Kallick in this book (let me know if I linked the wrong book, Jason). It’s actually a collection of four books that were originally published separately. My school has been using these books as part of our School Improvement Plan, and I’ve read all four, though it’s been a while.

Briefly, the idea is that regardless of content, we should be teaching our students how to think. We want them to develop a habit of persisting when they don’t get something right away, not a habit of giving up. We want them to develop a habit of thinking flexibly, not a habit of wanting to know The One Way That Things Must Be Done. We want them to develop a habit of communicating with clarity and precision, not a habit of talking about, you know, um, stuff. (Wikipedia has an entry that lists all 16 of the HOM, for anyone who wants to see the list.)

Let’s look at that last one I mentioned and see a specific example. I once had a student who, in the middle of a class discussion, was on the verge of making a very good point, but was struggling to articulate it. He stumbled over his words a bit and then abruptly turned to me and said, “Mrs. D, can you do that thing where you take what I said and make it sound smart?” I was amused, but at the same time, a little ashamed. Although I use rephrasing a student’s ideas as a way to make sure that (1) I’m listening carefully and (2) they know I’m listening, I shouldn’t be robbing them of the opportunity to make themselves “look smart.”

What can I do instead? I can ask questions to encourage clear, precise communication, returning the responsibility for the communication to the student. When they give me something clear and precise, I acknowledge it, using the language of the HOM. Students are great mimickers, and once I started doing that, they did the same to one another. With an environment established where one can’t get away with unclear ideas, students correct themselves. Some take longer to speak, pausing as they internally make sure that what they’re about to say is clear and precise. Others will continue to speak out quickly, but will immediately follow up with, “No wait! I meant [more precise wording].”

Just for the record, I don’t have this down – I tend to go in cycles where I let my own bad habits (taking over for others) resurface. But when I’m being consistent, I do see these sorts of results.

I keep the HOM posted on a bulletin board or somewhere else in the room; the visual reminder is good for both me and the students. I remind them that these are habits they can use in any class, not just mine. It helps that the other teachers at my school are also working with this same idea, but even without that, I can ask them what they’re reading in English class and how they have (to continue the same example) used clear and precise language when talking about that particular novel.

One of my favorite habits of mind is metacognition, or thinking about your thinking. Sometimes during or after a discussion or activity, I will say, “Okay guys, let’s do some metacognition. Which habits of mind did you need to use to complete this task? Which habits of mind did Suzie just demonstrate? Can we try responding with wonderment and awe in this situation?” The more frequently I am doing that, the better responses I get, because they’re more comfortable with thinking about their thinking. And that is pretty awesome to see. :)