In my summer physics class, I asked students the following question about motion online, in which they were asked to consider a wrong answer, explain why someone might think it, and then explain the incorrect answer away.
Here is a physics problem! Imagine you are driving a car along a long straight. Your friend in the passenger seat is recording where you are every hour of the trip.
Clock Reading Mile-marker 2:00 PM
20 3:00 PM
72 4:00 PM 124
5:00 PM ???
6:00 PM 228
Explain why your friend might reach the conclusion that the speed of the car at 3:00 PM was 24 miles/hour.
Then explain why that answer is probably not correct.
…
Here are different ways that students responded to the task:
Describe the Mistake Process and Explain the Correct Way
Someone might have tried dividing the mile-marker number (72) by the time on the clock (3pm). The answer cannot be correct because during each hour, the car has advanced to a mile marker 52 miles away. The time on the clock can only be used to determine the interval between time: 3pm-2pm =1hr and to find the speed, the initial mile-marker needs to be subtracted from the final mile-marker: v= (72mi-20mi)/1hr = 52mi/hr
Describe Mistake Only
Because they divided 72 by 3 thinking that they had traveled 72 miles in three hours.
Describe Mistake Process and Underlying Misunderstanding
If they confused 3:00 PM with 3 hours, they might be tempted to divide 72 by 3.
Show that Wrong Answer implies Something Else Incorrect
If the car were traveling 24 miles per hour at 3 pm then from 2:00 pm until 3:00 pm they would have only traveled 24 miles leaving them at mile marker 44 instead of 72.
Identify Hidden Assumption in Process and Explain why Wrong
If someone took the mile marker 72 and divided it by 3pm they would come to 24MPH however this would be assuming that the car had started at noon and had started at mile 0. however we do not know where the car started, only where it is now and where it was an hour ago.
Explain Correct but neither Describe or Explain the Mistake
I don’t know why someone would think that they were moving at 24 mph at 3 pm, but the time is given in increments of an hour. So obtain how fas they are moving you would subtract the mile marker number from the number of the previous mile marker of the previous hour. From 2 pm to 3 pm the car traveled 52 miles within the hour , so the car was traveling 52 mph.
State Correct Answer without Explanation
I don’t know why someone would think the speed of the car is 24 miles per hour. If you look at the data it’s clear that they are traveling at 52 miles/hour.
I’m curious: Which kinds of responses do you think are most productive? Which are the least? How could the task be re-written to encourage more of the productive kinds, and less of the unproductive kinds? OR What feedback would you give to students, or do in class the next day, as to encourage the more productive responses on later assignments?
Teach. Brian. Teach. 
I would be happy with any of the answers that explained the correct answer. Figuring out someone else’s errors is a good exercise for a teacher, but not necessary for a student. It is enough if they can figure out their own errors—diagnosing the weird errors that other people make is not necessary. (If a student would never mistake 3pm for 3 hours, it isn’t necessary tor them to realize that other people might.)
It’s not that they will mistake 3pm for 3 hours. I’m counting on the fact they probably won’t make that mistake here. But every student, will at some point treat a clock-reading as a time interval, or vice versa, and struggle to understand instantaneous from average, interval from points etc. By orienting to the ridiculousness now, it becomes a discursive tool for later. Having them think about the error is important, because it gives them a foothold for self-assessing that mistake later, when it will be more abstract, more entangled, and difficult for them to understand
This got me thinking. Like GSWP, it’s not crucial to me that my students are able to figure out someone else’s errors. The trouble is that my students have an awful time trying to figure out their own thinking — both because they have little practice, and because they find it humiliating to articulate their “before thinking,” being steeped in a culture of concealing mistakes.
I feel the need to demonstrate and scaffold the process. It’s much easier for them to notice a mistake that someone else made than their own, and much less embarrassing to talk about it, so it allows them to build comfort and competence in a low-stakes situation. Sometimes we play “find the teacher’s mistake,” which they relish. But sometimes an imaginary mistake made by an imaginary student yields the most honest comments.
That’s the best explanation I’ve heard yet for the value of this exercise.
Perhaps this was beyond the expectations of the assignment, but all of these responses are missing the fact that the instantaneous velocity at 3:00 cannot be determined from the average velocities during the preceding and following hours. I would think that this would be the underlying fallacy in the problem.
That was my first reaction, too, but I thought that the inappropriateness of expecting students to diagnose other people’s errors was a more general and less obvious problem than the difference between instantaneous and average velocity (which most physics teachers do a decent job of communicating).
So, for context, I use this question on the first assignment of the semester, after our first class (not about motion), in an online assignment. We haven’t talked about velocity or speed, let alone instantaneous vs. average. I think it would be inappropriate to draw that distinction here in the assignment. For more detail, the full question set can be found here:
https://teachbrianteach.wordpress.com/2012/05/16/reading-quizzes-2-0-day-1/
The question set is about teasing apart the distinction between time intervals and clock-readings, and positions and changes-in-position, in an everyday situation that supports such distinction making. The addressing the errors is to help make that easy-intuitive distinction making explicit, so that we can talk about the distinctions more abstractly and generally in class the following day.
In my opinion and experience, the question works well because it works at the level of the distinctions that students need to be orienting to now. Jumping to distinguishing instantaneous and average velocity without helping my students build these more fundamental distinctions would be futile.
In order of being asked…
1. I particularly like “Show that Wrong Answer implies Something Else Incorrect” since it provides a more concrete justification as to why the incorrect reasoning is in fact, incorrect. While it may not help provide a good avenue towards the correct reasoning, it effectively should rule out a common misconception.
2. The least helpful one to me is the correct answer with no explanation, which is ironic since it would be the easiest one to assign a grade to while also showing no real understanding of the concept.
3. I guess, in many ways the variations of their responses depend on the context in which the problem is given. Is it a test, a class problem, homework, etc. I’d be interested to see how those provide different results categorically, but that isn’t really what you asked. I’m not sure.
4. Encouraging that sort of thinking would require the teacher to focus on those mistakes and resolving them directly before even acknowledging the “correct” answer. However, this may be equally frustrating for students who understand the concept and those who are struggling.
Also, is it necessary to announce it as a physics problem?
I really like the idea of classifying these — very helpful for clarifying what exactly I am asking my students to do when they apply for reassessment.
The one that gave me the strongest “feeling” of student understanding was “Identify Hidden Assumption in Process and Explain why Wrong.” I think it’s because it’s the one that best satisfies the criteria “clear, consistent, causal.” Most of the answers are clear and relate coherently to other ideas; this one did the best job of explaining “what would cause someone to think this way.” I think I’m just frustrated with my lack of strategies for addressing magical thinking, so when I see someone demonstrate causal thinking (even if it’s about a student’s reasoning, rather than about physics), it makes me want to hug them.
I take the point from physjunkie, above, that “Show that wrong answer implies something else incorrect” is a useful strategy for ruling things out. Interestingly, that approach is the one that makes me cringe. It shows the student’s ability to be clear and assess consistency, which are important skills. I think I’m frustrated by it because my students often give this kind of response when I ask for a cause. The scenario posits a self-contradictory piece of argumentation. When you ask what would cause someone to think that way, the student replies that the contradictoriness of the argument causes the argument’s contradictoriness. Argh.