Joel,

Thanks for posting this information for us on mathedu.  I have watched the long
version (but not short version) of your screencast recording of your 
conference talk about using Tablet PC's in the teaching of mathematics.  I do have 
a few questions about some of the information that was discussed in this talk,
and I will ask my questions here since I believe others on mathedu might find
it worthy to read my post and your reply to it.

1.  You had mentioned some problems with recording screencasts of comments and 
symbols written with your digital pen and Windows Journal.  You had said that
sometimes the computer doesn't recognize your handwriting and handwritten
mathematics symbols.  I wonder what is going on here because, as far as I know,
software that records screenshots record exactly what is on the screen as is.  

2.  You had mentioned failures with having students watch the video lectures in 
advance and then discuss these ideas and ask questions in class instead of giving
a lecture.  This idea had failed because most students failed to come to class,
and the ones who had come often had not watched the videos and read the material
in advance and did not have any questions or only a very few of them.  The best
students would often arrive to class with the worksheets already worked out
rather than working them in class.  This idea in theory should work well because
it encourages students to think about the material for themselves, to find
their own connections and meanings and motivations for learning it, to learn
to think critically and creatively, and to avoid learning by being spoon fed 
everything.  Have you found a balance yet between lecturing and learning
by discussions and other constructivist/active learning approaches that 
works well with students who would not do the necessary preparation in advance
for class and with students who "overprepare"?  I caught some things you had
said about that, but I think you gave only a few details about this.  

3.  You had mentioned some of your thoughts on how much the recordings should
differ from lectures and class discussions and whether lectures should contain 
something that will be missed by students who do not attend, even if they watch 
the videos.  As you had said, we could either choose to add material to lectures
not in the videos or to record everything done in class so that those who miss
class can still get the material they missed.  This is an interesting question
because adding material to lectures and class discussions not in the recordings
might encourage more students to come to class than otherwise so that they will
not miss anything and also that they feel they gain something from being in class
that they would not gain otherwise.  Of course, there are students who know they
will gain something from being in class that they would not gain otherwise
regardless of which approach we choose to take.  However, many students do 
mistakenly believe that they gain nothing if they don't see the gains 
explicitly and immediately.  So it seems to make sense to many teachers
to help students see gains by making sure that some of these gains are explicit
and as immediate as possible.  What are your thoughts about this?  

However, these classes you teach appear to be mostly for math majors, so I find it 
surprising that this would be a major issue anyway because such students tend to be 
motivated enough to learn the mathematics without having to be motivated artifically 
through grades and other extrinstic motivators (I know there are exceptions, but
I would think these exceptions would be rare among students in such math courses
and would be a major issue only in those mathematics courses where most of the
students do not like mathematics and would not take the course if they did not
have to).      

I had also seen for the first time your screencasts on properties of open sets
in R^d and on an introduction to the theory of Riemann integration.  I do have
a few questions and comments about these screencasts: 

1.  In the Riemann integral screencast, what does NEB mean?  One of the opening 
slides mentions NEB in reference to the proofs and theorems on the exams.  
I had looked up this acronym on the Internet, and none of the meanings I had 
seen seem to fit.  

2.  On a related issue, are these your exams or departmental exams?  What do 
these analysis exams look like?  I'm a little curious to see how proofs arise
on these exams.  

3.  I haven't seen the notation ]a,b[ for open intervals before except maybe once.
The notation (a,b) for open intervals is the notation I have almost always seen.
Is this notation common in England since it appears to be rare in America?

4.  Perhaps you have done this in lecture rather in the screencast, so I could
be suggesting something you have already done, at least in lectures.  
It would be good for students to note that the proof of the intersection of
finitely many open sets is open breaks down in the case of infinite intersections
because the idea to use the minimum of the radii of the balls does not work
here: An infinite set of positive real numbers may not have a minimum, and
the infimum could be 0.  Either that can be said explicitly, or students could
be asked about that in class discussions or in homework to help see what is
going on.  Such thinking helps encourage students to check the hypotheses
of theorems carefully to be sure that any ideas they wish to try in a proof
do actually apply.  A common error among students (and sometimes mathematicians
as well) is to apply a theorem when it does not apply because one or more
hypotheses do not hold.  Finally, students who see this important observation
may better realize that generalizations do not always go as far as we would
hope or expect them to.  In short, if a result is true but not a particular
generalization of it, then it is good if the students see how that proof
breaks down when they try to use it to prove that particular generalization.
I'm sure doing something like this in every situation is too time-consuming,
but it is good, especially for students who are still learning about
proofs and learning to adjust to theoretical mathematics, to think about
such things as much as possible.  I'm sure you've done something like this
before, either with this theorem or some other theorems instead.  But I would
like to comment on it since it is worth mentioning and arises naturally
from that theorem because we see one generalization that works in one
case but not in another.  

5.  The Riemann integral screencast mentions that there are some discontinuous
functions that are Riemann integrable but does not mention an example since
it is tricky.  I'm wondering if you did mention such an example to them
in class or on a worksheet.  

Jonathan Groves