Is there a price to being thoughtful?
My intro physics students did really well on my warm-up questions today–and they were some tricky graphing questions that I know students typically struggle with if they don’t understand a thing or two. I was pleasantly surprised both at how many and how quickly students converged on correct interpretations with good explanations of correct answers and good arguments against compelling wrong answers.
Despite this, I’m still worried about the exam students will have to take next week. The practice exams that have been posted by the lead instructor indicate to me that the exams are really about whether or not students can quickly crank through some plug-n-chug kinematics problems. Personally, I think it’s mostly one big math test with some minor applications to physics-like topics.
I think and hope that I have taught my students to be thoughtful, and I hope this is a benefit to them and not a liability. Consider with me this example:
This week, students were trying to decide if a vertically tossed ball with speed 25 m/s would be fast enough to get 30 meters into the air. Their instinct had them split about 50/50 yes and no. We worked it out together, and found that it gets 31.25 meters, which is just 12.5 m/s (average speed)* 2.5 seconds (time it takes to lose 25 m/s at a rate of 10 m/s per second). I then asked them to work out for themselves how fast the ball would be moving when it got to 30 meters. Before they got to work, they had to write their guesses at front board. The most common guesses were 2.5 m/s and 1.25 m/s. I’ll leave it to you to think why these were the most common guesses.
One way of solving the problem is to just plug into the equation vf² = vi² + 2 aΔx, and some groups went that route. However, the way two groups of students solved it was by thinking about something a student had said the class period before. One student (Ashley) had remarked after drawing some motion maps that the speed of the ball must be the same at the same position no matter if it was moving up and down. This was a really cool insight about the symmetry of free-fall motion. I celebrated this idea and made a minor big deal about it. On this day, these groups remembered this students’ idea and realized that the problem I was asking was nearly the same as different problem–finding the speed of a ball that simply drops 1.25 meters. They determined that such a ball should take 0.5 seconds to drop 1.25 meters. And then without any calculations or plugging into formulas, they new that this meant the ball would be moving 5 m/s. They knew this immediately, because they knew well what 10 meters per second per second meant for the acceleration due to gravity.
The thoughtfulness to first think about whether or not a ball thrown 25 m/s could possibly get up 30 meters. The thoughtfulness to realize that a ball achieving a height of 31.25 meters can’t be moving all that fast by the time it gets to 30 meters. The thoughtfulness to think about other classmates’ ideas. The thoughtfulness to attend to and find symmetries that make problems simple. The thoughtfulness to put down equations and rely on their understanding of what acceleration means. The thoughtfulness to wonder why your answer of 5 m/s is bigger than what you had thought. All of this is what I’m sure is not going to be assessed on this exam. And in fact, if my students decide to be this thoughtful on the exam, they might find themselves unable to finish in time… sigh.
Today, some of my students heard the phrase, “I think maybe you’re over-thinking this,” and I worried about them hearing this. If being interested in outcomes, bartering with your intuition, listening to others, borrowing ideas, transforming problems, and wondering about your result is over-thinking problems, than over-thinking problems is exactly where I want them to be.