How to address students with incorrect reasoning when you can't identify their exact mistake?

A projectile is thrown upward at an angle.  At the instant the projectile is at the peak of its flight, is the projectile gaining speed, losing speed, or neither?  Justify your answer.

A student responds, "The path followed by a projectile is a parabola.  At the peak of motion, the slope of that parabola is flat, or zero -  the change in speed is zero.  The projectile neither gains or loses speed."

Right or wrong?

Wrong.

"Wait, what?" says the student.  "Everything I said is true.  Where's my mistake?"

Deconstruct the student's response sentence by sentence, and you'll find that yes, everything they said is in fact true:

* The path followed by a projectile is a parabola.  Essentially a fact.  

* At the peak of motion, the slope is zero.  Yes.  Draw a tangent to the trajectory at the top - horizontal line.

* The change in speed is zero.  That is the correct answer, as indicated in the key the teacher read online.

* The projectile neither gains or loses speed.  That's what "change in speed is zero" means.

"So if you can't find my mistake, you can't count it wrong."  [Cue body language of rolled eyes, upraised palms, indicating that the student is dealing with a lesser being, someone on a lower social and intellectual plane.  Their body says, "I can't wait until I have the money and the power to properly avenge myself upon this plebe."]

It's tough not to cave in to a smart student making this kind of complaint... when you know in your experience that the student is wrong, but you can't explain the error clearly, and the student is starting to get frustrated at what they perceive as your intransigence.  It's tempting to say, "Okay, you must be right, the key is wrong."  Or, "Well, I see your point.  Since we are both so confused, this must be a bad question.  I'll throw it out."

Don't do that.

But also don't do the "Shut up, I'm the teacher, I'm always right, sit down" routine.  No matter how polite you are, just pursing your lips and firmly ending the argument will be perceived as an over-the-top exercise of authoritay.

What do you say?

"Look, I can't pinpoint the error in your answer right now, but I'm pretty sure it's there.  I need to look carefully not just at each statement, but at the logical connections between each statement.  I'll look at this tonight, and I'll see if I can give you a better explanation tomorrow."

Then if you're still struggling, email a colleague, a mentor, or me.  You'll figure it out.

A couple of teaching points: 

* This kind of argument over physics is healthy and important.  That's how scientists operate - they argue through difficult problems.  However, be sure that the argument stays respectful at all times.  The teacher shouldn't pull out the authority card; but at the same time, the student can't be displaying disrespectful body language.  The argument is not about points or grades.  The argument is about physics.  

* This sort of conversation is exactly why I recommend starting out your AP Physics 1 class with AP Physics B-style calculational questions.  A student generally doesn't argue if the answer key says "20 m/s" but their answer says "12 m/s".  To them, numbers have an authority that mere words do not.  The best way to avoid fights with students about subtle issues with explanations is to first build trust in your competence and fairness using problems with unambiguous numerical answers.  As you build in verbal response questions gradually over the year, the students will get used to discussing physics not points.  They will get used to the idea that they are supposed to answer in writing the first time, not come to you later trying to explain what they really meant.

* The beauty of teaching a full-year class is that there's always time to think overnight.  Nothing is so urgent that it needs an answer right now.  (Contrast that with baseball umpiring, where "let me think for a while about whether you're out or safe" doesn't win friends.)  Take the time.  Very often, you'll discover that the student who was so upset today doesn't really care any more tomorrow - teenagers live in the moment.

* Oh, what exactly is the error in this student's logical connections?  I'll save that for this later post here.