Representational flexibility as evidence of sense making

I think that representational flexibility can be evidence of physical understanding.  By "representational flexibility" I mean students' ability to use representations in ways that differ (in major or minor ways) from their instructors' usage, or from their own past usage.  This may involve incorporating facts that were not previously mentioned in connection with this representation or exploring variations and representational conventions that have not previously been discussed or emphasized.


On Wednesday, UE2 students participated in energy theater about the diffusion of thermal energy from a cup of hot coffee.  On Thursday, they were asked to produce a diagrammatic representation of that energy theater.  Two of the three groups produced diagrammatic representations that closely mimicked the energy theater.  A third group demonstrated flexibility (in my opinion) by producing a diagram that was unlike the theater.

Here are the first two groups.  Notice that they drew exactly 13 T's (because there were 13 students in the energy theater), and they drew a single circle to represent the coffee cup.  (Or in one case, they drew the single circle twice; but both circles represent the cup.)

Here is the third group, table 7, which produced a markedly different representation.  Table 7 drew three circles -- one for the cup and two for thermometers that they used in an experiment.  This representation had been suggested and attempted for ET on Wednesday, but had been quickly discarded by the class and never acted out.  Table 7 also drew nine T's instead of thirteen.  (And drew six rows of circles instead of one or two.)

I suspected that Table 7's representation was a clue that the third group had a deep understanding of the physical situation.  Although I cannot quantify the depth of their understanding, I can at least verify that the construction of the new representation was associated with a sense-making conversation.  Below is a video of Table 7 constructing their diagram and beginning to engage in a sense-making conversation.  One teacher had invented the representation, and before the group can use the picture, he has to explain it to others.  At the beginning of the conversation, this teacher (Tim) explains his diagram.

Sherry: Oh but we, this isn't visualizing the energy theater model.
Gayle: It isn't?
Sherry: I mean it's not easily seen.
Tim: It is for me.
Gayle: It is for me too, so what, um,
Tim: Here is ... maybe I could explain why I drew it this way.
Sherry: Did you go home and do it last night?
Tim: No, I did it in class yesterday.
Sherry: Oh is that why you came up with the zero (???) two, every five minutes?
Tim: No, this is just an increment of time that's not defined.  Um, so here's the cup of hot water.  (Sherry: right.)  All of the thermal energy is right there at time zero, the beginning of the experiment, all the thermal energy is right here.  Few minutes down the road, whatever that few minutes might be, some of the thermal energy (Sherry: right, I mean, I can) travel, uh leaves the cup and is transferred to the air.  (Sherry: right, I can see)  And more and more is transferred away from the cup

The representation continued to generate conversation; for instance, this exchange about the end result of thermal equilibrium:

Tim: Well in the end, in the end, the room is actually hotter from that cup of tea.

Sherry: I mean what end are you talking about?  It will go off into Seattle's air.