Joe: "Now wait a minute--what's the difference? Instead of adding, you just subtract it."
George: "No . . . we know the total force here is nine, right?"
Joe: "It's anything--call it anything."
George: "No, I mean according to the formula we developed, it's nine, right?"
George: "And that's force--I mean four--there on the other side (?) . . . . So what would you say the force of the string is?"
Joe: "It wouldn't be balanced so it's that minus that: those two minus the other one there."
George: "Or . . . that one minus nine, right? Four minus nine, which is five. So the force of this string is five. Right?"
Joe: "Yeah. That's what I say, you have to subtract to get the minus . . . . the weight is five."
George: "The weight or the force, or whatever it is. The torque?"
Joe: "So what you're saying is: left side minus right side equals X-- whatever X is . . . . so distance times weight equals torque here. Torque of the left side minus torque of the right side equals X- torque?"
Looking back over the previous section it is interesting to note the stages through which Joe and George construct their theory:
|1.||they analyze the direction of each torque;|
they notice the analogous function of the string and the right side of the|
balance with which they now have practical "hands on" experience;
|3.||they use their old formula to calculate the magnitude of the string's torque;|
they construct a general law abstracting from the procedure they used to|
solve one specific case.
The coordination of these four representational systems seems to be an essential component of physical understanding. (See Clement, 1977, for further explication of this idea.)
The four kinds of knowledge mentioned above can also be used to describe the progress Joe and George made in their attempt to understand the simple balance. First they built up a rough qualitative understanding, then Joe noted