A student group clarifies their approach to substituting for x; Cabana joins them to discuss their progress.
00:00 Entonces la d?
So, then, D?
00:04 Es 4… de aqui es 4x?
Is 4, from here is 4x?
00:07 -6 mas 2…
-6 plus 2.
00:16 Cual esta el numero?
What is the number?
00:20 Mm hm.
00:21 Menos dos? O 6?
Minus 2? or 6?
00:23 Porque 6 mas -2… será…
Because 6 plus -2.. will be...
00:31 Igual a 4. Mas 4x, menos 0.
Equals 4. Plus 4x, minus 0.
00:50 4x, que?
00:53 Aqui…tengo x … y tengo que empezar en las 6….
Here, I have x, and I have to start on 6...
01:02 Mm mm. Esperame. Aqui tienes que…
Mm hm. Hold on. Here you have that...
01:08 No mas usa.. tienes ese para x. Ya sabes que cuando hay un numero, cuando no nada mas tienes que poner…. Uno…y el 4.
You have to use... you have that for x. You already know that when you have a number, then you don't have to put anything else... 1 and 4.
01:30 Y ya. Y este la de x, tienes que hacer un t-table con los ceros a los lados? Y vas a poner 4.
And now. And the x, you have to do a t-table with the zeros on the sides? And you're going to put 4.
01:39 Like that?
01:41 Mm hm. Y vas a poner … si. Un 1, y un 4 a los lados. Porque son las x, las calculas como son en la grafica.
Mm hm. And you're going to put.. yes. A 1, and a 4 on the sides. Because they're x, you calculate them like they are on the graph.
01:53 Esta, aqui.
01:56 Y el cuatro, al otro lado.
And the 4, on the other side.
02:02 Y ahora, como para sacar y, tienes que volver a bajar estos numeros aca. Estos.
And now, to get y, you have to bring down these numbers here. These.
02:07 Ya esta, para que…
That's it, for what….
02:08 No. Esto. Del 0, esto, aqui. Pa' abajo. Ya tienes los puestos. Esto para aca. Tienen 0, no mas, x.
No, this. From the 0, this, here. Comes down. You already have their place. This goes here. You have 0, so then, x.
02:20 Ok. Escojan una de las 2 o 3.
Ok. Pick one from 2 or 3.
02:2 Dos. Gracias. Esto de quien es?
Two. Thanks. Whose is this?
02:29 De Leticia.
02:29 Ya lo califiqué, ¿verdad? Es perfecto.
I already graded it, right? It's perfect.
02:37 Ok. Este de quien es? Jesus? Ok. Esto esta todo correcto, lo que te falta es el primer paso, en este caso.
Ok. Whose is this? Jesus? Ok. This is all correct, what you're missing is the first step, in this case.
02:46 Cual es?
Which is it?
02:47 Que estas buscando aqui?
What are you looking for here?
02:48 El valor de… o sea..
The value of... that is...
02:51 Ah hah.
02:51 A substituir x.
To substitute x.
02:53 Aja, por que?
Uh huh. For what?
02:54 Por 0.
02:55 Exacto. Asi, lo puede ser puesto o encima, abajo, algo asi, pero tenemos que ver que estas substituyendo a 0.
Exactly. So, you can put it either above, below, something like that, but we have to see that you're substituting for 0.
03:02 Esto esta bien. Ahora si el proximo paso, no, se va a demostrar, yo se que abase de eso puedes encontrarlo a 6. Es todo correcto.
This is good. Now the next step, no, you're going to show -- I know that based on this you can find the 6. This is all correct.
03:09 Y esto, aqui, es perfecto. Aqui tienes la ecuacion, el 0, y las respuestas.
And this, here, is perfect. Here you have the equation, the 0, and the answers.
03:21 Mm hm.
03:24 Good, perfect, Jesus. Ok.
Good, perfect, Jesus. Ok.
03:33 Ok. Este es tuyo, Leticia?
Ok. This one is yours, Leticia?
03:39 Um… ah.
03:42 Ok. Esto esta correcto, pero, tampoco necessario.
Ok. This is correct, but unnecessary.
03:46 Porque aqui estas buscando… que estas buscando, aqui?
Because here you're looking for-- what are you looking for, here?
03:52 El y-intercept?
3:53 Mmm hm. Y… como, como se empieza?
Mm hm. And.. how.. how do you start it?
03:56 Cruzando x por cero?
Crossing x with 0?
03:59 Exacto. Si hubieras hecho aqui, que te diera?
Exactly. If you'd done that here, what would it give you?
04:05 Te diera la respuesta exactamente, verdad?
It would give you the exact answer, right?
04:07 0, el ultimo.
Or, the last.
04:09 Asi que ya es un numero, 0 es un numero tan facil, siempre te conviene poner el 0 primero y luego preguntar --
So it's already a number, 0 is such an easy number, it's always convenient to put the 0 first and then ask --
04:16 ok, cual es el algebra que debo hacer para resolver por x?
Ok, what algebra do I need to use to solve for x?
04:19 O por y. Ok? Pero, lo tendrias que factorizar igual, porque si substituye 0 por y aqui, eso si no puedes resolver sin los factores.
Or for y. Ok? But you have to factor it equally, because if you substitute 0 for y here, then you can't solve it without the factors.
04:29 Ok. Y este paso tambien te falta.
Ok, and you missed this step as well.
04:32 Abase de esto, no puedes aguantar la respuesta todavia.
Based on this, you're not going to be able to get to the answer yet.
04:38 Porque le falta substituir 0.
Because you missed substituting 0.
04:39 Aha. Sin el 0, mmm hmm.
A ha. Without the 0, mm hmm.
04:45 Porque si no se preocupa que estan siguiendo el mismo procedimiento, siempre el mismo, sin saber las razones,
Because if you're not mindful about following the same procedure, always the same, without knowing the reasons,
04:50 Sin saber porque estan haciendo lo que estan haciendo? Ok?
Without knowing why you're doing what you're doing, ok?
04:53 Asi que si este fuera en un examen no te hubiera dado un A+, por esto aqui. Esto esta todo correcto, pero por esto no hubiera sido un A+. Ok?
So if this had been an exam, I wouldn't have given you an A+, for this here. It's all correct, but because of this it wouldn't have been an A+. Ok?
05:02 Para que te sabes eso.
So that you know that.
05:07 Y tambien es de Leticia, aja… ok.
And this one's also Leticia's, a ha. Ok.
05:11 En cual estamos, la e?
Which one are we on, E?
05:16 En cual estamos?
Which one are we on?
05:19 La e. Pero de Dulce. Seguiria, entonces. La f.
E, but Dulce's. Let's go on then, to F.
05:27 2e, donde esta?
2e, where is it?
05:29 Esto es la e…
This is the E.
05:30 A, ya lo se. Pero ya tiene un A+, no se por que.
Ah, I know. But this already has an A+, I don't know why.
05:37 Gerardo, get to work.
Gerardo, get to work.
05:43 Asi substituye…
05:44 Mm hmm. Ok. Esta todo perfecto tambien. Excellent, you guys. Nice job. Ya que tienes, asi? Ok. Sigan lo de atras, y ya. Ya les puse el A+.
Mm hmm. Ok. This is all perfect too. Excellent, you guys. Nice job. You have it, like this? Ok. Go on to the back, and done. I already gave you an A+.
This is an interesting grouping of kids, because Leticia can really wrangle a group of people, especially these three boys. Eduardo is across from Edgar. He had previously been unsuccessful in Algebra, not because he couldn't handle the math, but because he couldn't be bothered to do the bits and pieces that would help him be successful. He just didn't see the point.
I remember feeling like my job was to make this kid feel as smart as I can possibly manage so that he does all these things for his essential intellectual accomplishment, so that he brings some pride into who he is as a student. It wasn't always easy and it wasn't always successful, but I think it was a different stance than what he was used to being, which was being nagged at and scolded because he could be so much better than what he was producing.
It's pretty rare that I orchestrate the kid's seats, but I usually kept Alfonso close to the overhead if I wanted attention, or so I could see whether he was listening to me or not. Also if I was in the back of the room because if the kid was presenting, he was right in my line of vision. So this group has three boys with very, very different needs and patterns of interaction. Leticia was the perfect person to orchestrate all of that-- I don't know how apparent that is in this video, but she brings them all together. She is the most unabashedly vocal.
When I speak with this group about generating "A+ work," this comes from an early experience, where they were given a project and, for the most part, we got mediocre results back except for one girl who was so amazing that it transformed our sense of what the project could be about. I remember the girl's name and I remember her take on it, which was very much from this standpoint of connections from representations. That example became the norm for what we were looking for in groups. It's not that we held up her project as, "Make it look like this", but "here's some elements of what she did that make it A+ work."
That became part of our toolbox as a department in terms of alerting kids to what we expected of them. We framed A+ work as being within the grasp of every kid in the classroom. It had teachers showing off wording and showing off thinking, which I think is where I was being nitpicky with her. There was something where her train of logic was missing in her work, and I wanted to be public about that. I think I was also trying to push her to be more communicative. I knew that she cared about being successful and getting a good grade, but was just too scared to go get it if it wasn't already accessible to her, which is, I think, the opposite of how we usually think about kids. I usually want to find the strengths in their work. There are sometimes when, especially with kids that are successful but impetuous, where I have to point out what they can get from others. What they can get from thinking about this learning agenda that I've set for them.
The last thing I would say about A+ work is that I feel like there are lots of things that as teachers we're not aware of needing to attend to. We know that we need to attend to classroom culture. If we don't, the kids will do it for us. We also need to attend to the classroom definition of what it means to be smart, because if we don't, the students will. Closely related to that, I think we have to attend to the definition of exemplary work. By holding up "A+ work" as a common standard that is available to any kid, for me it removes that exemplary distinction from being available only to the so-called smart kids in the room.