### by Francesca Fornasini, Astronomy

#### Teaching Effectiveness Award Essay, 2012

In Fall 2011, I was a GSI for AY 7A: Introduction to Astrophysics. For many students this is a challenging course, because it requires them to apply what they have learned in several different physics and math classes to understand the structure and evolution of stars. During the first couple weeks of the semester, many students approached me or emailed me asking if their answers were correct. My typical response was, “Why do you think that this is right?” Most of them, even the students who had done the problem correctly, immediately started doubting their answer as soon as I asked them a question. I realized that they had little or no confidence in their answers and that they did not have any strategies for assessing the reasonableness of their solutions. Therefore, I tried to incorporate into my discussion sections a variety of strategies to help my students test the reasonableness of their answers.

During discussion section, I would typically give students a worksheet of problems to work on in groups and then discuss as a class. After the first couple weeks, I began using the worksheet problems not only to improve their understanding of key concepts but also as examples of “answer-checking” strategies. When the problems had a symbolic solution, I asked them to check the limits of their answers and consider whether, in these regimes, their answers matched their intuition or what we covered in lecture. When the problems had a numerical solution, I asked them to check their units and then compare their answer to a relevant quantity they already knew or could easily look up; for example, when they had to calculate the radius of a 10 solar mass star, I told them to ensure their answer was larger than the radius of the Sun but much smaller than the average distance between stars. I noticed that some students lacked confidence in their answers because they were not sure that certain mathematical steps were “legal”; for example, some of them were uncomfortable finding the average square velocity < *v*² > of an ensemble of particles with a velocity distribution *f*(*v*) by calculating ∫ *v*²* f*(*v*)*dv*. In these situations, I asked the class to think of a similar but simpler problem that they could solve in a way they were more familiar with and in the new way that they had been taught so that they could compare the two answers; thus, to make students more comfortable with < v² > = ∫ *v*²* f*(*v*)*dv*, we came up with a discrete velocity distribution consisting of five *v* values and solved for <* v*² > by both using the equation and doing a standard average. Another common point of insecurity for students was knowing what assumptions to make when doing order-of-magnitude estimates and being aware of the assumptions implicit in certain equations derived in lecture. To address this problem, I occasionally came up with a Fermi problem (the class favorite was “How many balloons are required for the house in Pixar’s *Up* to fly in the sky?”), which involved course concepts but in which it was easier for students to consider all of the problem’s complexity, decide what approximations and assumptions were reasonable, figure out what equations they could use that fit with their assumptions, and check how much of a difference adding a certain element of complexity would make to their final answer. Finally, when students approached me asking, “Is this right?” in a one-on-one setting, I would always challenge them, “Persuade me.” This challenge initially frustrated some students, but by the end of the semester many of them thanked me, saying that having to defend their answers allowed them to find mistakes, discover incorrect assumptions, and gain confidence in their reasoning.

I had other indications that these strategies were successful. Several students told me or wrote on mid-semester evaluations that they benefitted from one or more of these strategies on their homework and even exams. When I bumped into one student at the astronomy department months after the course was over, she told me that the Fermi problems we worked through in section have benefitted her in other physics classes as well. Furthermore, when I asked my students in section “How could we check this result?” towards the end of the semester, my question was no longer met with silence. However, the strongest evidence of the success of these strategies was that the most common question about homework that I received changed from “Is this right?” to “My answer seems unreasonable because … Can you help me figure out where I’ve gone wrong? I think it might be …”