I teach introductory physics. I don't teach math. I am not trained as a math teacher nor am I well read in math education. So I guess I'm not too upset with myself that this happened:

The class in question is the lowest level of physics that we teach. Almost every kid in the class needs it to graduate. 50% of the class has a diagnosed learning disability - and consequently the class is co-taught with a learning specialist.

These kids have never really experienced a ton of success in school. So we go out of our way to create a learning environment that is as engaging as possible, underpinned with solid pedagogy.

One of the skills I'd like the kids to have is the ability to find the slope of a line on a graph (with data table). Today - before I asked them to calculate the slope I asked them if they know what "slope" meant. I don't normally take student silence to mean that they don't know - more than they are unwilling to be the first to share. However, this seemed different.

So I waited (I'm big into "wait time") and finally one kid blurted out, "You mean y = mx +b ?" This got nods and murmurs from the other kids. So I asked, "Which part of y = mx + b represents the slope?" More silence...more waiting.

New strategy...I drew the diagram shown below on the board.

And asked them, "Which of these do you think has the largest slope?"

After some more waiting...one of the kids volunteered, "They are all straight so they have the same slope!"

Um...what?

I definitely didn't see that one coming. Is this part of a math misconception with which I'm not familiar?

After polling a bit deeper I heard one of the kid say - they all have a slope of 1; don't they?

I was thinking - what is going on here? Clearly they're working from a play book which I've never seen. Where does this stuff come from?

So what's the point? I didn't know that kids had this issue - and I wouldn't have known if I hadn't been doing the Socratic Dialogue thing. I was asking questions and WAITING some answers.

What did I do? I wanted to connect the vocabulary word "slope" to something that they have a concrete feel for. I asked them if there was an everyday word that we could connect with slope. [Granted I don't think that slope is that tough of a word but it has been used for them in a math class context so they are confused because of the abstract nature of their math classes]. I asked who of them has ever been hiking. They all raised their hands. Then I asked which of the lines above would be the most difficult to hike up. I actually drew little hikers on the graph.

They agreed that the blue would be the hardest to climb. Why? It is the steepest. This was a consensus answer. When I asked them to rate the slopes again one of the girls said, "Wait...slope is just steepness?"

Eyes wide open - mine!

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## 3 comments:

Teaching geometry for 4 years and I'm well aware that most students struggle with slope. Even if they can give you the correct numerical answer, they have no idea what they're actually saying.

That's why I made slope such a focus of my attempt at modeling in geometry this year. We spent all of last week just trying to nail down everything we know about slope (remember, this is a concept they supposedly learned in algebra).

The quote I kept repeating is that slope is just a description of a line's direction. I know that's not the greatest functional definition, but I'm trying to start small before I ask them to transfer to practical uses.

Clearly slope is much more difficult for students than I'd previously thought. I wonder what kind of constructivist resources there are out there to help the kids make sense of it.

I don't know Don... Got any hills?

3 times on this one too... grumble grumble..

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