Both of these groups have literally never seen a problem like this worked out, and raise questions about mathematical tools and how to show and organize their work.

I think that questions about tools and questions about process go together, and I wanted to attend to how to show the work so that I could get to the point of asking the questions about why, making sure they just weren't doing things just by rote. What I was trying to provide for the first group is a model based on their bits and pieces of how they could show that work. For the second group who'd shown the work in a more organized way, it was a narrative about what that work might mean, mathematically, what the reasons might be underlying it. Some students get mixed up because they're forgetting about how to go about substituting in zero for x or y intercepts. They start from this very concrete process of finding the zero in a graph, and substituting it from there, to this quick "I'll just plug in zero," and they forget that there are reasons behind those zero choices that they figured out themselves.

I have to bring them back to that level of concreteness. It also comes up in this clip and others, where I have to tell kids to summarize their results in a little t-table, because they lose that representation and then they don't see the connection between all these numbers that they're figuring out and overall objective and what it might look like in a graphical representation.

So as the teacher, I'm thinking about am I going to have to attend to the big picture first and then the details and reasons, or the other way around? As a practitioner, I think about what's the big idea, and then what mathematics do kids need to figure out that big idea? In practice, different topics could work both ways and the teacher has to figure out which way helps students understand the whole picture.

I think that questions about tools and questions about process go together, and I wanted to attend to how to show the work so that I could get to the point of asking the questions about why, making sure they just weren't doing things just by rote. What I was trying to provide for the first group is a model based on their bits and pieces of how they could show that work. For the second group who'd shown the work in a more organized way, it was a narrative about what that work might mean, mathematically, what the reasons might be underlying it. Some students get mixed up because they're forgetting about how to go about substituting in zero for x or y intercepts. They start from this very concrete process of finding the zero in a graph, and substituting it from there, to this quick "I'll just plug in zero," and they forget that there are reasons behind those zero choices that they figured out themselves.

I have to bring them back to that level of concreteness. It also comes up in this clip and others, where I have to tell kids to summarize their results in a little t-table, because they lose that representation and then they don't see the connection between all these numbers that they're figuring out and overall objective and what it might look like in a graphical representation.

So as the teacher, I'm thinking about am I going to have to attend to the big picture first and then the details and reasons, or the other way around? As a practitioner, I think about what's the big idea, and then what mathematics do kids need to figure out that big idea? In practice, different topics could work both ways and the teacher has to figure out which way helps students understand the whole picture.