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So we had Just talked earlier about

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how the questions, if you expect - [You and I?]

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Yes you and I were talking about how

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when you pose a question if you want the kids to really

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think about it.

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And not guess what you want them to say.

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You have to phrase it in like

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an open ended manner, to where it allows them to think about

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verses a guessing game.

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And so yesterday, I mean I

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thought about the lesson, I planned

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about the lesson. We had talked

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about how we were going to

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use the video and

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how it would be, you know

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a tool in the classroom discussion.

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And low and behold,

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Carlos said it wasn't as bad as I thought it was

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but I don't know.

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Low and behold, I found myself

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one to many times, I don't even want to know how many

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I just found myself asking a question

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and then before I would even finish

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it I would realize how I was phrasing it was very much multiple choice.

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Like almost like the answers in the question

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if the heard it right.

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You know?

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And so I would catch myself,

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Because I'd... at least I did

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I guess at least I caught myself.

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But then when I would sit there and try to think

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of a better way to say it

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or more open-ended way of asking the question

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I'd just like couldn't. Like I didn't know how else

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to ask it.

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And I was just annoyed at myself.

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I ended up using a bunch of multiple choice

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type questions. And I

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didn't want to.

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And I was so conscious of it

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and yet I couldn't fix it.

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So then you became the centipede who [I was annoyed]

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trying to figure out which leg to move first. [Yes]

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[And so I mean...] And therefore not being able to walk [yes]

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So I mean it was just

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annoying to be so conscious of it

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and not be able to do anything about it.

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Well, How about it we sort of make that

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part of our planning addenda.

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That would be good.

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Yeah, that would be good.

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Yeah, another thing that

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you said that yesterday that

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the kids did a lot of shape thinking.

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Okay.

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Where do you think that came from?

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I think it came from, just the

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very early on activities

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where it was distance verses time.

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And speeding up was a steepness change.

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And You know, maybe direction change

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is a down, you know just

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I think it stems just from prior activities.

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And almost getting comfortable

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with that and programming in their mind that

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"oh so that means

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when it speeds up it's steeper" Or

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"When it bounces or whatever

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it has to go back up and down"

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And they don't really think about

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how it's changing

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on a smaller scale.

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They're just thinking of, almost

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broader things, like of speeding up

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is steep. You know direction changes

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a line slope change,

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or whatever.

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And so I think it came from just

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the older problems where

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steepness if a speed change or you know whatever.

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So then do you think that...

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Do you think that...

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Would it be fair to say

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they were taught to think that way?

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I think to an extent yes.

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To an extent Yes, I mean I think

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that it is something that

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they learned on their own.

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but it was also

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a correct answer at one point in time.

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You know at... depending on the situation

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they're right. You know?

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So, How were you speaking about speed?

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And the graphs and stuff?

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Were you speaking shape-talk?

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A little bit. Yeah, yeah.

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How might...if you, knowing what you know now about

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the trouble that they get into

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with shape thinking,

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that they end up giving graphs that

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just aren't totally incorrect

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relative to what they're trying to graph.

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knowing that now, how might you have

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done things differently at the beginning?

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Well, I think

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just in general almost

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it's going all the way back to the first activities.

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Where they were tracking

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clown's distance verses time.

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And he speeds up, you know?

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And...because we even said

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when clown speeds up what will

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your distance finger do?

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And they would say, "Oh, go up faster"

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or "Go down faster".

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Which is fine for like, that stage of it I think

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But when they start to do both at once,

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and they start to let them co-vary

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Um, I think, I'd probably

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should have, and could have talked about

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Okay, well now what? And not have

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"Make it go down steep" be a valid answer.

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Like, they would have to say

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"Well, you know, as this finger

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goes on, my distance finger needs to

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cover more distance as this finger ticks

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verses less distance when that finger ticks.

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Or even, you know, whatever.

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Something along those lines where they're

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thinking about changing and the distance.

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So keep the talk about the variables? [Yeah]

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Instead of about the graph? [Yeah]

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Because I mean,

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I let them say, "Oh, well this is going to be steeper."

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And so then it just became a - [Yeah]

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Instead of a

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[Right.]

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[Right] Instead of covariation.

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Another thing, we might have let them go

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We might have urged them to put the distance finger above the time finger

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[You think?] Well, here's what leads me to think that,

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When, in one of the graphs that

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you were talking about

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Yesterday? [uhm]

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When you were talking about,

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you and they were talking about

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change in distance with respect to a small change in time.

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You circled the graph.

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You should have circled the time and distance

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Yeah.

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There is nothing seeing - Up here there's no time

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and no distance. Because time is down here

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and distance is up here.

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And so it's hard to think about corvariation.

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If you think that what we're talking

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about is like

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being on a roller coaster that is following a track,

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[right]

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So - [So, if I wanted to talk about]

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that portion of the jump

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I should have like,

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gave, put a tick mark at

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that portion in time. [Right]

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And asked them about what distance doing.

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Yes, and then Say okay, so

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now, because distance is over

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here, distance is not over here.

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What this is, this is the covariation.

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But if you end up talking about the graph,

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without talking about these two, then that's not about

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covariation. [mhm]

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So, In fact this is what I'm going to do today.

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I'm going to bring them back to the variable axis.

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And I'm going... I'm not going to have them out here.

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Because, What's clear

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to me now is , that

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they don't see this magnitude

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and this magnitude, covariant.

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They see this finger tracing a graph.

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I think that's what is going on.

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I think that you're probably right.