In the episode below, two students (Jared and Lee) used different approaches to solve this problem. I will discuss Jared‘s approach in this example. Jared started with the basic resistance equation and then converted it into an integral form as shown in Figure 6.10. When he completed this equation, the interviewer prompted him to explain his thinking. Below is a transcript from the conversation between Jared and the interviewer.
Figure 6.10 Equation set up by Jared.
Interviewer: Can you guys talk about what you are trying to do before you continue?
Jared: Uh, I took this and converted to this right here which is ( ). And then multiply by d, uh, small distance x, each of the little resistivity? It‟s all those little pieces going up together. And so it‟s the integral from 0 to L, of e to the negative x over L, dx.
Interviewer: Can you explain more about this part, the integral part? ( ) d[] ∫ d⎕ . Function Differentiation Integral-Differential relation integral- sum across
Symbolic space Math notion space
Blended space Substitution
Differentiation
67
Jared: Basically this part right here (drew a rectangular box on the equation as shown in Figure 6.10) is the , resistivity, and the dx is the length, and uh, since it‟s not just multiplied by one small length, it‟s from 0 to L, so you get the whole length. Interviewer: Uh huh.
Jared: And you are also adding up all of the resistivities together.
Jared explained that each small piece was represented by ―dx‖ and by using integration, he was adding ―all those little pieces going up together‖. Jared seemed to view the use of integral in this physical situation as adding up small quantities. By saying ―I took this and converted to this right here‖, Jared mapped the basic resistance equation and the summation idea into the structure of the basic resistance equation. He interpreted the variable of integration dx as ―little pieces‖ and the integral part in the rectangular box (Figure 6.10) as ―adding up all of the little resistivities.‖ The expression in the rectangular box is neither a complete nor correct expression without dx. By saying ―adding up all of the resistivities together‖ he seemed to have understood that an integral represents a Riemann sum. However, he did not seem to realize that the resistivity represents the property of this material and the resistivity at different points could not be added up in this physical scenario.
The blending diagram (Figure 6.11) represents how Jared constructed his solution. There are three mental spaces involved in this blending process: symbolic space, math notion space, and physical world space. The symbolic space contains basic resistance equation, the integral template∫[], and the differential notation d[]. The math notion space contains the idea of differential as a small amount of a quantity, and the integral as a sum. The physical world space contains the cylindrical resistor.
68
Figure 6.11Blend under „Equation Mapping‟ frame.
Students seemed to make the following connections across the input spaces. First, they seemed to associate the idea of an integral as a sum in the math notion space with the integral template ∫[] in the symbolic space. They also seemed to relate the idea of the differential as a small amount of quantity from the math notion space to the differential notation d[] in the symbolic space. When composing the blend, I speculate that the students mapped the structure of the basic resistance equation ( ) into the blended space. They also mapped the idea of the integral as a sum and the integral template into the blended space to construct an integral of the resistivity function. Further, they seemed to map the idea of the differential as a small amount of quantity from the mathematical notion space, the differential notation d[] from the symbolic space, and the cylinder from the physical world space into the new space to construct the differential dx. Jared seemed to recruit an organizing frame -- the structure of the resistance equation -- primarily from the symbolic space.
I call this frame the ‗Equation Mapping‘ frame, because it refers to the fact that students tried to use the structure of this basic equation and map it to the physical situation. The structure of their solution strongly relied on this equation rather than the physical situation or the physics concepts. In our problem, students were given the basic resistance equation which could be used when resistivity was constant. The ‗Equation Mapping‘ organizing frame differs from the two organizing frames I described above. Unlike the first two blends, Jared recruited the structure of
( ) ∫ d[] Integral is sum Differential is a small quantity Symbolic space Blended space
Physical world space
Small distance Summing up little resistivities
69
the equation from the symbolic mental space but also combined the notion of ―integral as a summation‖ and the physical picture of a cylinder to construct the solution.