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. ;)
Praxis of Reflection 
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Ooh I know this one. This is what I do about it: the first couple of times that I show sine/cosine graph I make sure it’s theta on the “x”-axis, and write sin(theta) (or cos(theta) instead of y on the “y”-axis. When they are completely OK with that, I say why not call it x and y instead and write it up in function notation and all.
THEN I bring back the unit circle and remind them how we used x for cos and y for sine, and have them think about this one minute until they agree that that is something completely different from what we’re doing now, and the confusion is stemming from the fact that x and y are names/labels of variables which could signify whatever we want. Basically I point out that it’s very important to have a clear understanding of what is the independent and what is the dependent variable in any given situation.
Students seem to be OK with this explanation in the groups I’ve tried so far, but I haven’t tried it with the honors and maybe they’ll think of some further objections.
I didn’t even think of relabeling the axes! I love this way of introducing it. Thanks so much for sharing!
For the first misconception, what I did at one point was rather than label things pi, 2pi, etc. I wrote the labels as 3.1415, 6.2820, etc. This way they see that it’s pi and 2pi but don’t have to grapple at first with some letters being variables and others being constants.
I learned of a very interesting misconception last week while doing transformations of periodic functions. When deciding whether or not a sine graph was reflected she said “I thought that if the graph was below the x-axis then it was reflected.” Though it’s wrong, I can totally see the sense-making going on.
Yeah, I think that’s the route I’ll go next year. What I did this time was to show them the same graph on GeoGebra, so they could see the integer labels on there as I pointed out that the x-intercepts were where multiples of pi would fall. That helped, but next year I’ll ease them into it more. :)
One thing I really love about teaching math is that I get to keep solving puzzles, both mathematical ones and the puzzles about what’s going on inside the students’ minds. It’s so interesting to see how they’re thinking and to try to identify places where their understanding has made a wrong turn so that you can go back and help them get on track. (I did that when teaching history, too, but the content’s so different that the process of evaluating student thinking isn’t the same.)
I think I’ll start with putting degrees on the x-axis and then switch to radians. Is that bad? Or maybe start with radians, but make sure they are comfy with measuring radians in pi/4, etc, then figure out which scale to use based on the properties of the function (as in “we want the main characteristics such as intercepts, max and min etc to be clearly visible”).
I second your opinion of math teaching as constant puzzle-solving. I teach psychology too, and as you say teaching social sciences is a completely different thing.
Hi Julia – I’m not sure why I can’t reply directly to your last comment, but I don’t have a reply link there.
Anyway, I don’t know that it would be bad to start with degrees necessarily, but I’m thinking that once we’re talking about trig functions, it’s nearly always in radians. It’s an easier graph to make, for one thing (if you want your x and y scales to be the same), and then there’s the fact that the derivatives of the functions aren’t the same if the angles are in degrees.
So I don’t think I would want to go that route, but rather to ensure greater familiarity with the unit circle and what the trig ratios really mean. I didn’t lay that foundation as well this year as I should have.