The highlight of the week was our presentation from Peter Anderson of the Mathematics Collaborative. I think that it was profoundly helpful. It made me realize that part of a teacher's job is to instill passion in his students.
Beyond, the pedagogical aspects of our interactive lesson, I grew in my knowledge of physics. The first part of the presentation, asked us to determine, which of several sports balls would roll down an incline fastest. My initial answer was wrong. I was correct that it related to the ball's moment of inertia but wrong in calculating that it related to the ball's mass. For posterity, the correct law of motion for a ball rolled down an incline without slipping (or spinning) is
\[ v_{\text{max}} = \sqrt{\frac{gh}{\frac{1}{2} + \frac{\alpha}{2}}} = \sqrt{ \frac{2gh}{1+ \alpha} } \]
and its acceleration is
\[ a = \frac{g \sin \theta}{1 + \alpha} \]
Thus as far as content knowledge, under the ideal conditions presented in a physics class, the coefficient of the moment of inertia determines which ball rolls fastest. Thus, a golf ball and bowling ball will roll equally fast and faster than a basketball or tennis ball, because the former are solid spheres while the latter are hollow spheres. (\( \alpha = \frac{2}{3}\) for a hollow sphere and \(\alpha = \frac{2}{5}\) for a solid one.)
We were posed two additional questions. These are pertinent, not only because they test content understanding, but invite students that solve them to understand the limitations of the conditions used in a physics class and can lead them to engage the activity by wondering what real life properties of rolling balls lead to real world outcomes.
We were asked which ball would roll furthest and which would roll for the longest time. The answer: all the balls would roll equally far and for the same amount of time. In particular, all the balls will roll indefinitely because the work that friction does on each is zero.
We were asked to then predict how long our chosen ball would take to roll down a 15 meter incline whose angle is 3 degrees above the horizontal. I used my derived formula and calculated about 9 seconds. The actual ball didn't even make it down the incline. In part the surface was very uneven. It contained large horizontal grooves. In any case, the exercise showed us the limitations of calculations and real life scenarios. I thought that the exercise was good because it made me think about what needed to happen to make the equation more true. I realized, simply, that increasing the angle would make the prediction better.
The experiment made me think about the limitations of derivations. I would have, however, also enjoyed a setup that confirmed physics calculations. Using a smooth, even surface, angled to overcome the "activation energy" of rolling, would be striking because we could verify calculations made by hand.
Pedagogically, the most striking part of our learning activity is Mr. Anderson statements, that we should re-evaluate the way in which we teach and that we should respect student inquiry--"Respect their questions." Sometimes that means that teachers should not just provide answers. Instead, we should focus on building curiosity. Upon reflection, the central theme of our activity is doing our best to instill passion through curiosity. I thought back to the circumstances that interested me in math as a kid and realized that beyond teaching to standards and maintaining coherent content knowledge, teachers should strive, as much as possible, to instill passion that arises from interesting problems that have fascinating solutions. I began to think that an excellent teacher is one who is able to expertly interweave standards, content, and curiosity. We should make sure that no children are left behind--they have a right to be exposed to the content they need for later success and content that satisfies educational standards. But, in order to help students grow as people, we should use that extraordinary opportunity we have to craft lessons that intermingle all these characteristics. One day, in some class, a student may connect deeply with an activity that gives rise to long term sustained scholarship. I feel, especially after this week's activity, that teachers should strive to expose students to as many of such experiences as possible.
I enjoyed how Mr. Anderson concluded our activity--that we should re-evaluate how we teach. His conclusion was memorable and stuck with me. I am still getting used to the idea that pedagogy sometimes trumps content. This activity exemplified that point. I feel that there is still a missing ingredient that I hope to master, which is gaining student participation. For example, I was able to get a lot out of the Mathematics Collaborative activity because I was actively engaged in its content. That is one area I hope to work on which is designing lesson plans that create fascination and engagement.
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