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How Students Learn: History, Mathematics, and Science in the Classroom
A few students say, “The water on top pushes down.” Several others add, “The water below pushes up. The water on the sides pushes sideways.”
I now ask, “So, how do we explain the observation that the scale reading is less?” Several students are now constructing an explanation. I give them a few minutes to work on their explanations in small groups and then ask them to share their conclusions: “The water pushes up and the water pushes down. But the push up is greater than the push down, ‘cuz it is deeper.” Some students have it, but others are still struggling. If students do not volunteer consideration of the comparison between the pushes, I may ask the question, “Why should the push from below be larger? Why does that make sense?” Several students respond, “Because the deeper we go the bigger the push.”
At this point, several students have represented the application of our recently derived ideas with words. In the interest of deepening the understanding for all students, I suggest they represent the situation with pictures, using arrows to show the directions of the forces and varying the lengths of the arrows to show the magnitude (size) of the forces. I ask each group to take white board and a marking pen and draw such a picture of the submerged metal cylinder. After a few minutes, we compare diagrams and have members of each group describe their drawings and explain the situation. By now, nearly all the groups have drawn the cylinder with a larger arrow up than down. Each of these arrows, they say, represents the size and direction of the push by the water on that part of the cylinder.
Building an Analogy to Understand the Benchmark Experience
Now that it appears the students understand the weighing-in-water situation, I direct them back to the weighing-in-a-vacuum situation. “So, what does all of this tell us about the situation of weighing under the bell jar? If we had a really accurate instrument, what do you think would have happened to the scale reading and why?” A few students begin to construct an analogy: “Weighing in air would be like the weighing in water.” I ask, “How so?” One student responds, “The air around the world is kind of like an ocean of air. Down here is like being deep in the ocean of air. On a mountain air doesn’t press as hard.” Another adds, “Air can push in all directions, just like water. So if water can push up and down on the cylinder, so can air.” “But air doesn’t push as much [hard], so you don’t get as big a difference,” says another. With some guidance from me, the students build an analogy: “Weighing in the water is to weighing out of the water (in air) as weighing in the ocean of air is to weighing out of the air, that is in a vacuum.” I ask, “So what would happen to the scale reading in the vacuum if we had a very accurate instrument?” One student responds, “The scale reading would be more.”