Dear Jonathan,
Let me take these points in turn. This does make for a rather long post, for which I apologize to everyone!
> 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.
>
There is no problem with the screenshots here. The issue was whether the computer could automatically improve my handwriting for me! If I select some of my handwriting and ask Windows Journal to recognize it and convert it to "typed" text, it usually does quite well on the English. However, it doesn't realize that the maths isn't English, so it tends to convert it to strings of random symbols that have a somewhat similar shape to the original pen strokes.
> 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.
>
As this is the first year that I have used screencasts, the previous experiment was done with my earlier system involving annotated slides and audio podcasts from the preceding year. I tried this with students in year 3 and in year 4 two years ago, and you can see full details of my experiences in my first case study, "Using a tablet PC and audio podcasts in the teaching of undergraduate mathematics modules", adapted from a case study appearing in Giving a Lecture, Exley and Dennick (April 2009), in the revised form available online at http://www.maths.nottingham.ac.uk/personal/jff/Papers/pdf/podcasting.pdf
I may try this again for 3rd-year and 4th-year students, but using screencasts, once I have a full set available for my modules. However, this will have to wait a while as the latest course review is currently working its way through, and so these modules are all about to change.
I think that using screencasts from the previous year may help to resolve some of the problems that got in the way when students were expected to use just the annotated slides and audio from the preceding year. However it will be a while before I have a chance to try this.
On the worksheets side, it looks as if it is best to give the students something completely new to work through in class, based on the material under discussion, and with a good range of questions from relatively routine questions to more challenging questions. That way, no-one who is making an effort should be either completely lost or bored!
Attendance at my classes this year was again down to 50% for the last few weeks of term. But many of my colleagues also report that attendance levels at their classes are down to 50-60%, so perhaps my screencasts/annotated slides/audio podcasts are not really to blame. Attendance already suffers if printed lecture notes are available online, for example. However, it does seem clear that some of my students do allow themselves to fall a long way behind in the belief that, as all the classes are available in full online, they can catch up whenever they want to. This belief is, perhaps, not always realistic.
> 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?
>
This is a very tricky issue. If you look at the feedback on my current screencasts at http://explainingmaths.wordpress.com/feedback/ you will see that students who want to revisit a tricky point in a class and hear my comments on that point again find the recordings very helpful. I am reluctant to remove this option from the enthusiastic students in order to try to force the less enthusiastic students to attend classes.
Now suppose that I try a system based on the previous year's screencasts. Should I make screencasts of the discussion classes/workshops available too? I did make (audio) recordings available on my previous attempt. I am undecided, but mostly I would still rather make more options available to the enthusiastic, rather than try to force the less enthusiastic to attend. Unfortunately, this probably does not lead to the best average exam performances from the class!
> 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).
>
My second year mathematical analysis module this year is for 183 students out of the whole year of about 260 maths students (approx 150 single honours + 110 joint honours students). For most of the students the module is optional, but nevertheless the set of options available is not so large that everyone really wants to be there! Also, the module is compulsory for our mathematical physics students: in the short term, it is not completely obvious what the applications of this material are in mathematical physics. Perhaps a little more motivation on this specific point would be helpful.
> 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.
>
NEB stands for "Not examinable as bookwork". I invented this acronym as far as I know. Students are, of course, very keen on knowing what might turn up in the exam, and are not very happy with the answer "all of the material". So I try to indicate that some material is not regarded as a "routine" part of the module. Although anything goes in the unseen portions of exam questions, I do have portions of exam questions which I regard as bookwork. NEB material will not be required in "bookwork" portions of exam questions.
> 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.
>
My exams are, so far, VERY traditional in nature. I often ask students to state some of the standard definitions and/or results, to work with examples to establish their properties, or give their own examples exhibiting various combinations of properties. These may be bookwork examples, or they may have to work with or think up new examples for themselves. I may ask them for some bookwork proofs, or I may ask them to prove some unseen results for themselves. (And so on!)
I have not yet looked into significantly modifying the nature of my examinations or assessed coursework. This is another major project of its own!
> 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?
>
It may be French in origin? Generally I prefer (a,b), but this can clash with ordered pair notation (especially in the proof that every open set in the real line is a countable union of open intervals!).
> 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.
>
In the screencast on open sets, the last portion is devoted to an example of a sequence of open sets whose intersection is not open. However, I can't remember whether I had time to talk about the infimum of infinitely many positive real numbers issue on this occasion. I think I did have time to say something about "things that can go wrong in the limit" or something like that.
If you look at the extra question sheet associated with the sessions on "How and why do we do proofs" you will see several other problems of this type, including the elementary but instructive example of a sequence of finite sets whose union is not finite.
> 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.
The screencast does mention the characteristic function of a single point set at one point, and this example is dealt with in more detail on the question sheet.
Best wishes,
Joel
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