Re: [Phys-l] Re. Simultaneity

[John Denker <jsd@av8n.com>]

On 05/22/2009 10:06 AM, Edmiston, Mike wrote:
> In case anyone is interested,

I'm interested.

> I placed the description of an exam
> question on my web space that deals with space-time diagrams.  The
> question allows me to see in a hurry if the students understand what
> they're doing or not.

> Although I have gone over this in class, I am sad to report that I
> generally find only 30% to 50% of my students can do this correctly on
> the final exam. 

........

> The next time I teach this course I
> will put more emphasis on "contours of constant value," 


Wow.  That suggests some pedagogical experiments to test some
hypotheses, to see what works best.

Case 0 is the original version, namely: 
  Students are provided with a blank space-time diagram. This means
  the x versus ct and x’ versus ct’ axes are drawn for them, but no
  events have been placed on the diagram. Students are asked to locate
  events A and B such that event A occurs before event B in the unprimed
  frame, but event A occurs after event B in the primed frame.

I remark that in this original formulation, the axes on the given
diagram are restricted to positive t, positive x, positive t', and
positive x'.

Case 1 is the same as case 0, except that in the given diagram, all
  four axes are pre-drawn to extend into negative territory.  That 
  is, they all meet at the origin and extend in both directions from 
  there.

My hypothesis is that students will do a lot better in this case, 
compared to case 0, for at least two reasons:  First of all, weak 
students always have trouble with _sequencing_ i.e. any situation 
where they need to perform a sub-task before proceeding with the 
main task.  Constructing the contours of constant t and constant t' 
is such a subtask.  Extending the axes makes this subtask unnecessary, 
because it provides places to locate event A and event B without 
constructing anything.  Secondly, even if the student doesn't have 
a general problem with sequencing, this particular subtask is easy 
to get wrong.

Case 2 is the same as case 0, except that in addition, students are 
  explicitly instructed to construct at least three contours of 
  constant t and three contours of constant t' (before being asked 
  to locate event A and event B).

My hypothesis is that students will do better in this case, even
better than in case 1.  The preliminary task of constructing the
contours gets them started down the right road.

Case 3 is the same as case 0, except that the pre-drawn spacetime
  diagram contains no axes, just a lattice of contours, i.e. 
  contours of constant t, constant x, constant t', and constant
  x' ... with a label on each contour.

My hypothesis is that students will do even better in this case.
It's sorta hard to get this one wrong, if they have any clue about
what spacetime events are.

Case 4 is the same as case 0, except that in preceding homework
  assignments, time and time again, students were required to 
  construct contours of constant value for each coordinate.

My hypothesis is that students will do better in this case compared
to case 0, but not quite as well as in case 3.  That's progress,
because this case is more demanding than case 3, and indeed no less
demanding than case 0.

I remark that case 3 and case 4 are how I approach the task when I
am doing the work myself (not assigning it to students).   In particular,
when the first sentence of case 0 refers to a  "blank spacetime diagram"
I automatically think of criss-crossing contours of constant t, constant
x, constant t', and constant x'.  I do not think of axes at all.  

I have an endless supply of spacetime graph paper with the contours
already drawn in.  That is to say, I have .pdf files and a printer.

  Tangential remark: The "blank spacetime diagram" provided in 
  connection with case 0 (i.e. axes only) is not something I would 
  heretofore have drawn under any circumstances.  I would not have 
  occurred to me.  On the other hand, having seen this example, I now 
  appreciate the teaching value -- or at least the testing value -- of 
  this version of the problem.  It tests whether the students can find 
  a way to disengage from the wrong path (axes) and then proceed down 
  the right path (contours).



The "contour" approach is not original with me.  Compare for example
Taylor & Wheeler _Spacetime Physics_ figures 9, 10, 26, 27, 28, and
especially 64, which show contours (in addition to axes).  Meanwhile 
Misner, Thorne & Wheeler _Gravitation_ takes the next step by suppressing 
the axes and using contours exclusively, as in figures 2.4, 2.5, 2.6, 
and many many others.  This is how I was taught, starting first term
freshman year in college.  I always thought it was the "normal" way of
doing business.