Galileo s model of a falling body in VS .NET

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Galileo s model of a falling body
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Task Students are given a simple but defective program that simulates the motion of a dropped ball. The students are asked to improve it. This exercise has been run in many variations, usually in a few iterations of small-group design, interspersed by sharing and discussion in the full class. Students (and even teachers, performing the task as a part of professional development) engage in a fairly regular development involving (a) recognition of increasing velocity in a fall, (b) recognizing the regularity of the motion, (c) almost always producing the two models that were discussed by Galileo. This task has been replicated in many classrooms, from late elementary school through high school, including being run by teachers new to the idea of teaching motion, and new to the use of programming representations. Results generally validate similar developments and outcomes. However, none of the replications has undergone the kind of detailed analysis we reported above for designing Newton s laws. Of all our re-inventing activities, this one has had the most work in real classrooms, with real teachers.
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Figure 9.4 shows the initial faulty program. Figure 9.5 shows a normative model. In our experience, almost all editions have exposed a diversity of ideas and a fairly extended development, but almost always two competing models have emerged. One is the additive model, shown in Figure 9.5, and the other is the multiplicative model, shown in Figure 9.6. Students produce many variations in form and syntax that are identical to these, but we will not display them here. The multiplicative model is highly attractive. It has many intuitively plausible qualities, and the fall is dramatic in its speedup. The correct model is less dramatic. Galileo proved that the multiplicative model (which is equivalent to falling at a speed proportional to distance fallen) is impossible. His strategy was to show that, according to the multiplicative model, a fall of twice as far would take the same time, an impossible condition. An easier 235
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Figure 9.4. The starting prompt program with its visual result at left. Fd means move forward (downward), and dot means draw a dot. Reset and go (in the menu) are clicked on to reset and activate the simulation.
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Figure 9.5. The normative model (Galileo s model); a constant amount (acceleration) is added to speed each tick of the clock. Speed is thus proportional to time.
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Figure 9.6. The multiplicative model of a fall.
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New-Media Literacies counter to the multiplicative fall is that the model can t get started: Distance, and therefore speed, are zero at the start, so the falling object can t move at all. Another counter is more subtle. Reversing the model to make a toss, the multiplicative model becomes a dividing model, which never gets to the peak of the toss. However, our experience has been that these arguments are too subtle to regularly work with students, even high school students with a fair amount of scaffolding. So, we have come to be happy with convergence to these two models, using other work (such as taking data and matching to the models) to settle the issue. Steps along the way are informative of the work students actually do in constructing these models. For example, for elementary school students, the very fact of falling at an increasing speed is often problematic, and it takes some time to work through. Here, augmenting the task by feeling the impact of objects falling from different heights is usually convincing. Galileo also suggested that experiment. Another interesting development among students is toward an assumption of uniformity in the fall. Figure 9.7 shows the initial models produced by a high school student. I return in a moment to his first model, go, but notice that in both go and ho, the incremental (decremental) distances are non-uniform; they switch between an increment (decrement) of 1 and one of 2 half way through the motion. In our experience, almost all classes produce such non-uniform models (or step-wise models, where the object falls a constant distance for some number of steps, and then a new, greater constant distance for the same number of steps). Group discussion of the simplicity of a uniform model, or of a smoother one, have systematically won out in student discussion. Uniformity is an aesthetic consideration that Galileo emphasized, and it seems that groups of students can regularly come to appreciate its attractiveness in teacher-scaffolded discussion. Here we see important commonalty between scientists and students, not in the models that they have or produce, but in the aesthetics displayed in ultimate choice of model. Younger students sometimes include a phase of slowing down at the end of their falls. The rationale is surprisingly strong. If the ball stops at the bottom, surely it must slow down before it stops! Indeed, the ball does slow down, but only after it contacts the ground. That can be the subject of a good discussion of why and when a ball actually slows down in a drop. Representational considerations often arise in this modeling activity. The first model in Figure 9.7, go, shows a surprisingly common issue. When the program is run, the ball noticeably slows down at the bottom! However, the first time we saw this in a sixth-grade classroom, one of the pair of boys who produced it explained that it was intended to show speeding up. He said, essentially, we didn t want to show the ball speeding up, we wanted to show that it sped up. The problem in our interpretation was that, like many other 237
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