As I have posted about before, I really want to do a problem-based unit this year in which students attempt to answer the question, "How far away is the horizon line?" I'm getting ready to start the unit in about a week, so I have been thinking about it a lot lately. Mostly, I have been thinking a lot about the idea of a line that is tangent to a circle and how students might conceptualize that. The "visual" that I get in my head when I think about this horizon line question is this:
This morning I was sitting around the house with my girlfriend, and I decided to see what visual she might come up with and what ideas she might have about tangent lines. First, I asked her to draw the visual that comes to mind when she thinks about the horizon line problem. This is what she drew:
Pretty fantastic, right?!?! Certainly more artistic than my picture! It was interesting to me how differently two people might be thinking about the same scenario.

Then I asked her to draw a circle. Then I asked her to draw a line that touched that circle in only one point. She drew line #1 below (I added the numbering to make some distinctions here). Our conversation went something like this:

ME: Tell me why you decided to draw it that way.
HER: Well, cause it would only touch the circle in one point.
ME: What would happen if you continued your line?
HER: It would cross the circle on the other side.
ME: So would that work?
HER: I guess not....Well, I was thinking about this (draws line #2).
ME: Why did you decide not to draw that one?
HER: It just seems like it would touch the circle in more than one point.
ME: What if we zoomed in? (I drew the picture on the right)
HER: Hmmm...not sure. I still feel more certain that line #1 would only touch in one spot.

To me, this was really interesting. I wonder about how students think. I wonder about their mental models. And, mostly, I wonder how much we actually listen to them and respond to how THEY think. It can be tempting to tell students about a tangent line in the context of this problem, but that would be a missed opportunity for rich discussion. Perhaps more importantly, it would be imposing a way of thinking on them that is incompatible with how they are currently thinking. You hear a lot of people say that they don't like math. I wonder how much of that is due to the fact that they have learned that math doesn't care about their ideas, that math is always right, and that they need to learn to think more like math.



09/17/2012 6:22am

The students know (believe) that you want a particular answer, while your girlfriend knows you're interested in her thinking. So you may get different sorts of answers because of that.

But it is fascinating to see what people draw. I was trying to help a friend with her son's homework last year. It involved similar triangles in a story problem, maybe how tall a tree was if its shadow was 7 feet long, and a 6 foot person's shadow was 3 feet long. I asked her to draw a person with their shadow. What she drew was nothing like what I would draw. The shadow came up at an angle - because she was drawing more realistically, using perspective. And the shadow wasn't just one line the way I'd draw it. That may have been the first time I realized that we might need to help students think about how to draw for math.

09/17/2012 2:23pm

Hi Sue...

I loved your last sentence about learning how to "draw for math." At the very least, it might be important for them to think about how certain depictions are more helpful in moving toward a mathematical model. The fact that this model isn't always as realistic is pretty interesting to me.

I also agree that, to a large extent, students feel the way you described: "the students know (believe) that you want a particular answer, while your girlfriend knows you're interested in her thinking." I think I would also argue that this is a really big problem in that needs to be considered. There are serious consequences, I think, when predetermined curriculum takes precedence over listening to how students are thinking. I think we need to listen to them more.

09/17/2012 7:15pm

I love the idea of asking students to draw a picture about such a seemingly unsurmountable question. It's so different than asking them to draw a picture for a question that is oh so clearly about right triangles. I'm curious about how this could be developed into a full lesson. I think this could be a powerful way of getting people to solve questions differently -- and very interestingly, make an engaging way for students to see how other people did things differently from them. They never really like sludging through other people's algebra, but I think pictures would help get them through it.

09/17/2012 10:12pm

Hi Meghan...

I'm thinking about launching the unit with something similar to what I did with my girlfriend. I'll probably pitch the question, have them all draw what they are envisioning, have them check out each other's drawings, ask them about the benefits of each, and then have them make an estimate about how far away they think it is. I imagine there should be some interesting visuals and discussion. I'll save the tangent line for a later lesson. What do you think?

Anna Geretz
09/21/2012 8:29am

I think having a whole lesson could be really cool. You could talk about modes of representation, and different ways of modeling. I had an astronomy class once where the professor brought in a few different kinds of maps, one was flat and square, one was flat but circular, one was a 3D plastic ball to represent the cosmos. He also brought in a black umbrella and proceeded to draw a map for us of a constellation on the inside of the umbrella, just to show how many different ways we can represent the same thing (here the night sky).
I thought it was really cool because he showed us that there are many ways to model the same thing and that each way of doing that has its own advantages and disadvantages. It's one thing to tell that to your class and another to show them.
I think that if a student is able to understand the material in more than mode of representation that they will have a much better grasp of it.
What do you think about a lesson plan in which the kids come up with a few different ways of showing, explaining, acting out the same material?

09/26/2012 9:02am

You will be opening a fun can of worms with your unit. Maybe find a great historical example of someone finding the distance horribly off from reality and examining the simple geometry vs the reality of measuring in the real world . Here is a little something to get you going!


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