"The answer is D because..."

Test corrections are the most valuable and productive use of my students' time I can think of.  Nowadays, I don't even return graded tests; rather, I give each student a blank copy of the test, along with notation about which questions they didn't earn full credit on.  They can see the original graded test (and their grade) only after getting all corrections checked off.

But test corrections are only useful if students explain their understanding thoroughly and correctly.

The first time we do test corrections, a few clueless (or intellectually lazy) students just write "I put C, but now I know the answer is D."  Um, really?  You call that a correction?  How does that help you understand the original problem?  More to the point, how does that convince *me* that you understand the original problem?  Begone, foul dwimmerlaik, and bear thy feeble "correction" with thee to the houses of lamentation.

Of course, there are teachers in other subjects who accept such baloney as a "correction."  Which is why many of us get pushback when we give credit for test corrections - parents and administrators and colleagues default to "oh, whoops, the answer was B, I've learned that now" in their own understanding of what "test corrections" mean.  No.  Duh.  They're much more than that.

However.  Even though most students and most physics teachers recognize that justification of the correct answer is required... it's still all too easy to accept a correction that doesn't truly show understanding.

A block of mass m is not touching a surface.  You pull up on the block with a tension T which is bigger than the weight of the block.  Which way is the block moving?

(A) down

(B) up

(C) the block can not move

(D) the block could be moving up or down.

"The answer is D because the net force is upward here."  How does this show understanding?  I mean, the net force is indeed upward.  And the answer is indeed D.  But has this student shown their personal understanding?  Or have they just quoted the answer their friend gave them with a vague handwave at a physics term?  Thing is, I don't know.  And so a better correction is necessary.

"The answer is D because the object could be slowing down or speeding up."  Well, this is also a true statement that doesn't show understanding.  How do you know the object could be speeding up or slowing down?  And what does that have to do with the direction of motion?

At this point, a student is likely to be getting frustrated with me.  They keep saying correct things, and I keep sending them back to the dungeons!  What do they have to do to get a correction checked off around here?!?

Answer: They have to start with a fact of physics.  

Legitimate starting points for all justifications include facts from our fact sheet (like "acceleration is defined as the change in speed every second"), problem solving procedures we've learned (like 1. free body diagram, 2. components, 3. write newton's second law in each direction), or equations (like "x = vot + 1/2at^2).  

I think we should all train our students that any other starting point at all, especially "The answer is D because..." is incorrect on its face.

"When an object speeds up, acceleration is in the direction of motion; when an object slows down, acceleration is opposite the direction of motion.  [These are facts from our fact sheet.]  Here the acceleration is upward, because the problem says the unbalanced force is upward, and acceleration is in the direction of the unbalanced force.  So the object could be moving up and speeding up, or moving down and slowing down.  D."

There's no way this student doesn't understand the problem.  There's no way that this student is merely parroting what a friend told them.  (At least, if they've obeyed the five-foot rule.  A friend might have told them what to do, but the student must have phrased the answer in their own words because they can't just copy.  And the five-foot rule is easy-peasy to enforce in class.)

Most importantly, this student has corrected a misconception. They likely originally said the object was moving upward because the unbalanced force was upward.  By quoting the facts, they are forced to confront how their intuition about how the world works is at odds with how physics actually works.  

How a wave moves - conceptual physics question

An alumna of my Conceptual Physics Summer Institute was having some trouble with picturing the answer to this question from waves problem set 9.  She asked if I could help.  Of course!  This is an important but very difficult question which helps students understand how a wave moves along a string.

The question, which I've adapted from an old New York Regents physics exam: 

If you assign this, do NOT let students ask you questions!  Make them show their own personal understanding without you giving them hints.  I'd either give this as a quiz question to ensure it's students' own work; or allow collaboration among classmates so that they argue with one another, but can't in their minds simply appeal to the authority that "teacher said to think about <foo>" or "teacher said the answer is <bar>, so I'll make up some random bologna for my justification."  

And so, most students will get this wrong on first attempt.  Let them.  They need to struggle, to make their own mistakes such that they care about the solution you show them for a greater purpose than just getting a problem done.

How do I explain the answer?  I first put the PHET waves simulation on screen.  My students are familiar with this simulation, having played with it for 10 minutes as a previous assignment, and having seen it on screen a few times.

I set the simulation to "oscillate" and "no end," as in the screenshot below.  I turn the "damping" slider all the way off.  I tune the frequency slider to 1 Hz, meaning the wave has a period of 1 second.

I show the students that the wave moves to the right, while the pieces of the string move up and down.  (Of course I've shown this before!  I still need to show them again to set the context for this particular problem.)  I ask, how far does a wave crest travel in 1 second?

After a discussion, the class usually agrees that the wave travels one wavelength.  Great.  

Then I ask, how far does the wave crest travel in a tenth of a second?  Since we just discussed how far the crest travels in one second, they pretty quickly come up with 1/10 of a wavelength.  

But what does that mean for this particular problem?  Students still have a tough time connecting the top picture to the correct answer.  Many still think that since the wave moves 1/10 of a wavelength, that the wavelength itself is now much smaller, making the wave extra-squiggly.  I know, I know, this makes no sense to us as physics teachers; but that's a very frequent misconception.  Here's how I bust it.

I pause the simulation, and circle several positions, as in the picture on the problem set:

Then, I move the animation forward a few frames.  Everyone sees immediately what the new wave looks like (and that the wavelength hasn't changed!):

This is a good a-ha moment for the class!  It's not merely okay for most of the class to get something wrong on a quiz or on an assignment - it's on occasion absolutely necessary in order to advance the class's collective understanding.