How do we do proofs? (Part 2)

The screencast of the 2009-10 edition of How do we do proofs? (Part II) is now available from

http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/How-Proofs-II-0910/

Here some background knowledge of convergence of series is assumed. There is an associated handout available at

http://www.maths.nott.ac.uk/personal/jff/Teaching/Standard-Series.pdf

with some relevant background material, including some standard series and a bit about the comparison test.

Joel Feinstein

 

 

University of Nottingham YouTube maths playlist

The University of Nottingham YouTube channel (NottmUniversity) now has a Mathematics Playlist, which you can find at
http://www.youtube.com/user/NottmUniversity#g/c/A9721D7E1FB7CD34
So far you can find three of my movies/screencasts there.

See http://www.youtube.com/user/NottmUniversity#g/p for all of the Nottingham University YouTube playlists

Joel Feinstein

Proof by definition?

Students face many obstacles when they are trying to learn how to do proofs. I am trying to convince my students that at least SOME aspects of constructing proofs are relatively routine.

One fairly common type of proof that often gives trouble is what I like to call “proof by definition”. I have not invented this term, but many authors use it to mean something rather different. What I mean is that you are asked to prove something, and once you have substituted the definitions into the statements, there is either nothing or very little left to prove. Yet this kind of proof often gives students problems.

• Sometimes they feel they should be able to prove the result based on an intuitive idea of the relevant concept, instead of using the definition given to them.
• Indeed, sometimes students do not know the relevant definitions at all, but don’t realize that this is a major obstacle to proving the result.
• However, sometimes the fact that they have already finished once they have substituted in the definitions leads them to think that they haven’t done anything, and this is a more subtle problem.

For example, using my notion of absorption, on one question sheet I ask them to prove the following.

Let and let be a sequence of real numbers. Then

as

if and only if the following condition holds:

for all , the open interval absorbs the sequence .

Now, by the time you have substituted in the definitions, the two statements are either exactly or effectively the same, and you have finished. But this does not feel like a proof.

Perhaps the students are right to be uncomfortable with this? Consider the following “question and unsatisfacory answer”.

Question: working in , prove that is not in the interior of .

Unsatisfactory answer:
is not in the interior of because, for all ,
the open interval is not a subset of .

This time, the student has substituted in the definition, and can see that the statement is true, but the proof is NOT finished. Indeed, so far the student has said just a little more than “ is not an interior point because it is not an interior point”.

In this case, more is expected: the student should justify the claim by observing, for example, that is in the set but is not in the set .

So how are students supposed to know whether they have finished once they have substituted the definitions in and can see that the result is now obviously true?

In fact, there can’t be a definitive answer to this. After all, it is hard to argue with the statement (working in )
“Clearly is not an interior point of .”
Yet, this statement would not be acceptable as part of a proof of itself.

I feel that the difference between the two examples here is that, in the first, after substituting in the definitions, the two statements being compared end up EXACTLY the same. So, although it feels as if nothing has happened, nevertheless the proof is complete.

In the second example, after substituting in the definition, you arrive at a statement that looks to be clearly true. But appearances can be deceiving, and the student could ask whether this new claimed fact  is really significantly more clearly true than the original statement was.

Meanwhile, I am putting together a large collection of “proof by definition”  questions for students to practise on. These are (mostly) not supposed to be at all interesting! The idea is to get the students fluent in substituting in definitions, and then seeing how easy the rest of the proof can be.

See http://www.maths.nott.ac.uk/personal/jff/Teaching/More-Proofs.pdf for the current version. Perhaps there are large collections of similar routine proofs available on the internet? It  would surely be worth compiling a big collection as a universal student “practising routine proofs” resource.

Joel Feinstein

October 27 2009

How do we do proofs?

The second of my screencasts on how and why we do proofs,
How do we do proofs? (Part I), is now available at http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/How-Proofs-I-0910/
Part II will be in 2 weeks time.
Joel
20/10/09

Laptop performance

Well, I took my power cable to my lecture today and plugged in the power … or at least I thought I had!
However, I noticed most of the usual strange phenomena again, which was very disappointing. Only at the end of the lecture did I realize that I had been running on battery power again: I had plugged into an extension cable which was disconnected at the other end!
The most obvious effect of this problem visually during lectures is that when I finish writing one piece of text and then start a new bit, Windows Journal frequently draws unwanted straight lines connecting the end of the last word with the start of the new word. It looks as though the tablet “loses” the tablet pen for a moment, and then guesses that I must have been drawing a straight line during the lost period.
Oh well, I’ll try again tomorrow …
Joel Feinstein
19/10/09

Sennheiser solutions

Another technical post, this time on sound equipment.
I am still seeking a solution to my sound quality issues for my recordings. I am using a Sennheiser wireless transmitter and receiver (EW100G2) along with a Lavalier capsule microphone. The local support team have come up with various possibilities:
- Input level set too high on the laptop. This was certainly the case for my first recordings, but after that I reduced the microphone input sensitivity level on the laptop to avoid this. Nevertheless (as you can hear on my recordings) I was still getting pops and distortions.
- Interference? Other radio microphones in the area can cause interference. The local team reset my wireless frequency to a level that should be safely out of range of the other local microphones, but it did not solve the problem.
- Sensitivity (on the transmitter) and AF output (on the receiver). I had thought that I had tested this, but I may have underestimated the difference between my talking voice and my lecturing voice. I originally had the sensitivity set to -10 and the AF out level at -12. Alan Mintey has now suggested that I try sensitivity -20 and AF out -18. Let’s see if this helps!

Bizarrely, my transmitter got into a state today where it was periodically peaking every few seconds (even with the microphone detached and the unit left alone on my desk). This went away when I switched from -10 to -20, and (as you might expect?) did not come back when I tried switching back to -10.

I am beginning to wonder whether using a wireless microphone setup introduces too many additional complications. There are so many places in the chain where something can go wrong. If I had a simpler stereo microphone setup (with a sufficiently long extension lead!) perhaps that would eliminate some of the problems?

Anyway, watch this space to see if these settings solve the problems!

Joel Feinstein

October 12 2009

Mathematical curiosities and treasures from Professor Stewart’s cabinet

Mathematical curiosities and treasures from Professor Stewart’s cabinet

This talk by Professor Ian Stewart FRS looks like it should be good. See

http://royalsociety.org/event.asp?id=8613&month=11,2009

Note that they plan to provide a live webcast of the talk at
http://royalsociety.org/page.asp?id=4110

Joel Feinstein
October 8 2009

Screencast of “Why do we do proofs?”

Unfortunately I am still having sound quality issues, but people may still be interested in looking at the screencast I have made of this year’s edition of “Why do we do proofs?”
This is available at:
http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/Why-Proofs-09-10-b/Why-Proofs-09-10-b.html

Joel Feinstein
7/10/09

Camtasia solutions

This is another technical post, following on from the Toshiba solutions post: I think that the Camtasia project deserves a post of its own.
Note that I am using Camtasia Studio 6 (Academic version).
Here are some of the issues I have been dealing with. More will follow (no doubt!) in comments on this post.

• One issue that I have not mentioned before is the following. When Camtasia is recording, the mouse pointer on my display flickers very rapidly, and this is a problem when I am using (say) the hand tool to point at things during lectures. Interestingly, this flickering is NOT present on the resulting screencast video, so students may have a better experience watching the screencast than they do in the original lecture. I think I may have found the explanation now on the page
  http://forums.techsmith.com/showthread.php?t=6171
  I’ll report back on whether the problem is cured by turning off “layered” capturing, as suggested.
  Earlier I found the page http://forums.techsmith.com/archive/index.php/t-1155.html
  which suggested turning off display acceleration during capture.
  Update Oct 4 2009: I originally thought that disabling display acceleration hadn’t helped me, but this was a “failure to actually change settings” error on my part (see comments below). In fact this method does pretty much work for me (I can put up with intermittent flickering) and this may be my safest option.
• As I mentioned before, I have been struggling to get high quality audio recordings, which is disappointing given that I have some relatively expensive equipment (a Sennheiser kit).
  Perhaps it would have been a good idea to look properly at http://www.techsmith.com/learn/camtasia/editing/edit-setup-recording.asp which is the official tutorial! Still, I don’t think I saw all of the issues below discussed there. However, you should consult this tutorial for more explanation of some of the menus and features discussed below.
  I have also  had some helpful suggestions from Alan Mintey Television & Video Production Manager, Nottingham University IS Learning Team). He suggested using the Audio Setup Wizard under Voice Narration. The interesting thing  here is that, at least on my machine, some very similar menus have rather different sets of options.
  If I fire up Camtasia Recorder directly (as you can) without going via the full Camtasia Studio, then in the advanced audio options (under tools->options->audio) the set of possible recording formats and qualities available is far smaller than those available via the corresponding menu in the Audio Setup Wizard.
  Moreover, when I fire up Camtasia Recorder directly, the audio settings always revert to a relatively low quality default setting.
  Here the moral appears to be: DON’T start Camtasia Recorder directly. Instead, you should run the full Camtasia Studio program and click on “Record the Screen”. This again fires up Camtasia Recorder, but it preserves the audio settings that you have previously chosen. Moreover, the full set of audio settings is then available from the tools->options->audio->advanced settings, instead of the highly restricted set of options I found when going to Camtasia Recorder directly.
  In the end, though, I think that the main cause of the distortion in the sound quality on my first few screencasts was that I had the microphone input settings on the laptop at too sensitive a level. I have now experimented with this, and I think that I have just about found the right level. I have tried using the Auto-Adjust Volume tool in the Audio Setup Wizard, and it may be that this does give results with least possible distortion, but it also appears (to me) to give recordings that are too quiet. So I am going for a slightly higher sensitivity level for now.
• I have now modified my “Joel medium” production preset (as discussed in my earlier comments in the Toshiba Laptop Solutions thread) so that, in addition to producing the mp4 screencast and an mp3 file, it also produces an m4v file (suitable for ipods). I  have hit two snags, though.
  • The additional file downloads (mp3 etc.) available in addition to the screencast are shown on the web page produced by Camtasia. However, these are listed at the bottom of the page, under the movie frame, and may disappear off the bottom of the screen if the viewer is not operating at high resolution. (This happens to me even on my 19 inch display at home, which is currently set to 1152 x 864.) Perhaps this is the price I pay for going for an 800×600 movie, instead of something smaller?
  • Not all web/media servers are set up to serve m4v files. I had a problem with this, but fortunately this has been resolved with the help of the learning team. (Thanks Sally!)
• Watch the comments on this post for further developments.

Joel Feinstein, October 2nd 2009

Revision Quiz on first year analysis

I gave a revision quiz to my second-year analysis students yesterday on first-year material. The screencast is available at
http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/MAN-quiz-29-9-09/MAN-quiz-29-9-09.html
These are early days with my use of Camtasia, and I don’t think I have quite optimized the sound quality yet. (I’ll have more to say on this in my comments on the Toshiba tablet PC solutions post.)
Joel Feinstein
September 30 2009