T: Some of the things that we need to remember [gestures to screen]. Parallelograms are quadrilaterals that have opposite sides equal and parallel. So remember that on a parallelogram, the sides that are opposite of each other [gestures to parallelogram on screen] will be the same length and will be parallel to each other. On both sides. That's how you can identify all that's a parallelogram. So is a square a parallelogram? Would a square be considered a parallelogram?Granted, this is a short excerpt from a longer class in which lots of good teaching and learning happened, but this little exchange jumped out at me as a great example of what seems a typical teaching-as-explaining moment. It's difficult to infer any evidence that the students have understood the teacher's explanation; to the contrary, the students seem to be playing the Yes/No Guessing Game. I'll also add that though it may not be apparent in the transcript, the teacher's questioning becomes rather aggressive. Starting when she asks "So is a square a parallelogram?" the first time, when students respond with silence, she repeats her question in a more aggressive tone, signalling that, in her view, the answer to her question should be obvious. I wonder how and to what extent this influences the ways in which students respond to her line of questioning.
SS: [silence]
T: Opposite sides are equal, and they're the same length, and they're parallel to teach other. So, on a square, are their opposite sides not parallel? [silence] Just answer that. Are their opposite sides parallel?
S: [timidly] Yes?
T: They are. Are they the same length?
SS: [confidently] Yes.
T: Yes, because a square is the same all the way around, correct? So is a square a parallelogram?
SS: Yes.
T: How about a rectangle?
SS: No.
One final curricular thought: it seems that her students have learned about squares and rectangles before moving up the hierarchy to the more general case of parallelogram (and earlier, she makes a reference to moving sequentially through a curriculum). To my way of thinking, it makes more sense (particularly in the context of thinking about area, which is what this unit is about), to start with the general case (parallelogram) and then to understand squares and rectangles as instances of a parallelogram. Can anybody make an argument for the ordering this teacher was following?
I always think of this as a "questioning so I can continue with the lesson" interaction, versus a "questioning so I can understand what you know" interaction.
ReplyDeleteYep. Most kids don't get it that way.
I based my geometry curriculum on the Modeling philosophy, meaning I tried to make the content discover-able by students whenever possible. The question of order was the first and most important issue I faced and I settled on "specific --> general" as opposed to "general --> specific" because I felt that the specific content is easier and more approachable.
ReplyDeleteGiving students a generic parallelogram and asking them to make conclusions about it can be overwhelming, as the basic properties aren't immediately obvious. Additionally, students are often more familiar with the specific, especially in this case in dealing with squares. As we expand beyond squares, we can investigate what properties that were true for squares are also (or no longer) true for rectangles, rhombi, and parallelograms?
I understand that traditional curricula are structured "general --> specific," but those sequences also tend to be based on the "teacher as teller" model, which is at least logically consistent. What I mean is that were I to teach geometry as a lecture based class, it would make sense to start with parallelograms and work toward squares.