A student group has the conceptual pieces and is engaged in moving to the next step; Cabana works with them, trying to assess the extent to which they see the conceptual whole.
00:00 ¿El rectangulo? Ok, pero, pero primero ponganse de acuerdo en lo que estan tratando de buscar.
The rectangle? Ok, but, but first come to an agreement about what you're looking for.
00:10 Lo que es los x and y-intercepts. Right? ¿Con cuál van a empezar?
Which is the x and y-intercepts. Right? Which one are you going to start with?
00:13 Con….el….x minus 1.
Part of what's going on in this group is that for the most part they're understanding the processes to follow, but they're not being true about their business. That gave me the impression that they were doing something by rote and not seeing why they were doing what they were doing, so when they all say "OH!", that comes from the realization that "oh," this whole time we were doing all these things, but it was actually about this.
They did a whole bunch of math, but ended up getting the wrong answer because they were solving equations when they just learned it. They did random stuff, following haphazard rules that they think they've remembered. They're not thinking about the operations you choose when you're solving the equation and have that big idea underlying them. So I think that was going on a little bit for them- they factored, but didn't use parentheses, so they made some of the miscalculations themselves, but they weren't seeing the connections between the bits and pieces they were producing.
This happens a lot, especially when it comes to factoring. The fact that the multiplied forms of an equation are the same, I think, is a mystery to lots of kids, certainly that it can be represented by a length-times-width equals-area statement. Thoughts that most kids aren't asked to think about, which is unfortunate. Even bigger than that, the fact that you can choose whether to work with the multiplied form or the factored form, that feels really mysterious to lots of kids. They think there's something you're supposed to do.