## Modelling the Golden Gate Bridge

This implementation was observed in a class of 35 students by the end of March. This class was familiar with the « Jigsaw Classroom » organization that had already been put into operation for collaborative work on a lesson. The students were mostly average achievers. The contents at stake in physics and mathematics had been taught to students in previous lessons. The phases have been video recorded, and interviews were conducted with 3 students after phase 3.

a)

b)

**Figure 1: answers in the first phase**

In the first phase, most students sketched a suspension bridge without suspensors (figure 1a). They generally explained the non-horizontality of the main cable by the weight of the deck, but agreed with the false explanation of the shape by the length of the suspensors. They understood from the video that the weight of the deck "bends" the cable, but they did not link the shape of the cable with the uniform repartition of the weight, thanks to the suspensors. Isolated students showed a better understanding, writing that the shape is the consequence of the forces (or tensions) on the cable (figure 1b). In the rest of the first phase, after overcoming instrumental difficulties with the apparatuses , the students correctly recorded the angles and the intensity of forces. They recognized the static equilibrium law, but they generally did not compute the components. Especially the fact that the horizontal component is the same on all the three dynamometers in the apparatus with two weights did not appear. In the classroom discussion, before dividing the class in groups for phase 2, the teacher insisted on the decomposition of a vector in components and on the variations of the components of the tension in the cable at stake in the next phases. He also repeated that the goal is to study mathematically the shape of the cable.

For the group work in phases 2 and 3, I report on a series of four groups observed doing each task in phase 2, and on one group in phase 3 bringing together students observed in phase 2, leaving for further work a comparison to other students in the whole class. In phase 2, students observed doing task A mainly succeeded, while difficulties were observed for students doing other tasks. Students doing task B started by sketching a bridge with a lot of suspensors, not allowing to consider segments. They were prompted by the observer to limit to 4 suspensors. They took time to find the coordinates of the anchoring point, and had difficulties to use the formula given for the slope of the segments and the distance between suspensors in order to calculate the coordinates of the next point. Actually, this calculation involves several parameters related to data of the bridge, a common situation in physical sciences but not in mathematics. Students doing task C took time to enter the algorithm in Casyopée. Nothing or a wrong display appeared on the screen, because of small mistakes. They could correct only when the observer helped them to analyze the algorithm. They identified the parameter n as related to the number of suspensors and proposed the value 83 (the number of suspensors in the Golden Gate Bridge). They considered that this value is "close to infinity" and that is why the curve did not appear as a collection of segments, in contrast to small values of n. When the observer explained that H is a tension, they get aware that increasing the value of this parameter "straightens" the cable, and found a suitable value. Students doing task D found a formula for the vertical tension, but had difficulty to interpret the fact that the tension is in the direction of the tangent to the curve.

In the group of phase 3, each student explained her task and her work in the preceding phase. The video recording shows that other students listened attentively and asked for further explanation. The parameter H was identified by students as playing a role in each task; for instance when a student who did task C did not remember the effect of increasing H, confusing with the "height of the cable", the student who did task A corrected him, saying that it is a tension and then increasing should "straighten" rather than "slacken" the cable. The same student helped to overcome the difficulty met by the student who did task D to find the direction of the tangent to the curve and then the derivative of the function, saying "you just integrate the quotient of V and H". Going further in task D, the students were confused by the parameter H in the denominator, some proposing a Ln(H) in the antiderivative. They could achieve task D, using Casyopée. In contrast, the unfinished task B, and the connection with the algorithm in task C were not discussed.

Three students were interviewed after phase 3, as a method to further evaluate what connections students made between working spaces during the group work. They stressed that the situation was more complex than usually ("we had to connect a lot of different things") and that they were "not used to mix physic and mathematics". Commenting the first phase, they showed how their awareness of the structure of a bridge progressed: they mentioned the role of the suspensors and made the link between a suspension bridge and an arched bridge with regard on how the deck is supported. They made also the link between the apparatus with two weights and a suspension bridge "with two suspensors". They still had difficulties in considering the slopes of the segments in task B in order to find the coordinates of the suspension points. However, they correctly interpreted the algorithm of task C, and were able to connect the evolution of H, and *x *and *y *respectively to task A and B. They did not show clear awareness that the function of task D was a limit of the continuous piecewise function of Task C. From graphical evidence they thought that it was more or less the same function for big values of n. Visualization is then the way students connected the algorithmic and the mathematical function working spaces. The observer asked to explain why the gradient in a point of the curve is the quotient of V and H. The expected answer was that the tension has the direction of the tangent, but the students simply wrote without more explanation. It seems that the first equality is common in the physics course, and that the second derives from the definition of the components in task A. Thus students made a connection between the statics space and the mathematical function space without explicit consideration of a limit.

**Figure 2: the connections made by students between working spaces**