Monday, September 19, 2016

NY Times: Stop Grading to a Curve

An excellent article by Adam Grant, professor at the Wharton School of the University of Pennsylvania:
The more important argument against grade curves is that they create an atmosphere that’s toxic by pitting students against one another. At best, it creates a hypercompetitive culture, and at worst, it sends students the message that the world is a zero-sum game: Your success means my failure.

Read the full article here.

Monday, September 5, 2016

Epsilon-Delta, Absolute Values, Inequalities

Working through the famed "baby" Rudin, Principles of Mathematical Analysis. (Which was not the analysis book I used in grad school: we used William Ray's Real Analysis).

First thing that dawns on me is that I'm shaky on following the formal proofs of various limits. I decide that I need to brush up on my epsilon-delta proofs, and go back to do exercises from Stein/Barcellos Calculus and Analytic Geometry. I find that I get stuck outside of simple limits of linear functions (say, with a quadratic).

Second observation is that these proofs use extensive algebra involving absolute value inequalities, and my trouble is that, in turn, I am shaky with these to the point of being blind to a lot of very basic facts. Kind of frustrating that I teach basic algebra as a career but as soon as as absolute values and inequalities enter the picture I'm nigh-helpless.

Third observation is that this explains the copious section in college algebra and precalculus texts on absolute value inequalities (which are skipped in the curriculum at our community college). Previously my intuition had failed to see the use for these; but it's precisely the skills you need in analysis to write demonstrations involving the limit definition.

A few key properties on which I had to brush up (and would argue for preparatory exercises if I was teaching/scaffolding in the direction of analysis):
  • Subadditivity: \(|a+b| \leq |a| + |b|\). Equivalent to the triangle inequality. 
  • Partial Reverse Triangle Inequality: \(|a| - |b| \leq |a - b|\). A bastardized name of my design. It follows from the preceding because \(|a| = |a - b + b| \leq |a-b| + |b|\), and then a subtraction of \(|b|\) from both sides. Gets used at the start of almost all my limit proofs. 
  • Multiplicativeness: \(|ab| = |a||b|\). Which leads to manipulations such as: multiplying both sides of an equation by a positive number allows you to just multiply into an absolute value; squaring both sides can be done into the absolute value, etc. 

More at: Wikipedia.

Discussion of general limit exercises: StackExchange.

Friday, August 26, 2016

Crypto Receipts for Homework

An interesting idea to a problem I've also experienced (student claiming they submitted work for which the instructor has no record): Individual cryptographic receipts for assignment submissions.



Friday, August 19, 2016

Link: Everything is Fucked, The Syllabus

By Prof. Sanjay Srivastava, a proposed course on the overall breakdown of science in the field of social psychology:


Friday, August 12, 2016

Natural Normality

Normal curve in flag sticker water-leak (upside-down), 2016:


Monday, July 25, 2016

Teaching Math with Overhead Presentations

At our school, we currently have a mixture of classroom facilities. Some classrooms just have classic chalkboards (and nothing else), while other classrooms are outfit with whiteboards, computer lecterns, and overhead projectors. Well, actually: All of the classrooms on campus have the latter (for a few years now), excepting only the mathematics wing, which has been kept classical-style by consensus of the mathematics faculty.

That said, it's been one year since I've converted all of my classes over to use of the overhead projector for presentations. In our situation, I need to file room-change requests every semester to specially move my classes outside the math wing to make this happen. But I've been extremely happy with the results. While this may be very late to the party in academia as a whole, this identifies me as one of the "cutting-edge tech guys" in our department. Note that I'm using LibreOffice Impress for my presentations (not MS PowerPoint), which I need to carry on a mobile device or network drive. My general idiom is to have one slide of definitions, a theorem, or a process on the slide for about 30 minutes at a time, while I discuss and write exercises on the surrounding whiteboard by hand.

As a recent switchover, I wanted to document the reasons why I prefer this methodology while they're still fresh for me; and I say this as someone who rather vigorously defended carefully writing everything by hand on the chalkboard in the past. Here goes:

Advantages of Overhead Presentations in a Math Class

  1. Saves time writing in class. (I think I recoup at least 20 minutes time in one of my standard 2-hour classes.)
  2. Additional clarity in written notes. (I can depend on the presentation being laid out just-so, not dependent on my handwriting, emotional state, time pressures, etc.)
  3. Can continue to face forward & speak towards the students most of the time.
  4. Don't have to re-transcribe the same material every semester. (Which simply seems inefficient.)
  5. Can have additional graphics, tables, and web links that are time-constrained by hand.
  6. Easy to go back and pull up old slides to review at a later time. We may do this within one class period, after "erasure", or across prior lectures. For extremely weak community-college students who can't seem to remember any content from day to day, we frequently must pull up slides and definitions from earlier classes; and this gives confidence when students in fact disbelieve that the material was covered earlier. Actually, this is most critical in the lowest-level remedial courses (the first course I tried it on out of desperation one day, with great success and student responsiveness). 
  7. I can carry around the same slides on a tablet in class. This allows me to assist students while they work on exercises individually and I circulate. Similar to the prior point, if one student forgets, needs a reminder, or was absent in a prior class, I can show them the needed content in my hand without distracting the rest of the class. 
  8. Greatly helps reviewing for the final exam.  I can quickly pull up any topic in its entirety from across the entire semester. (Again, trying to jog students' memories by showing it again in the exact same layout.)
  9. Having the computer overhead allows demonstration of technical tools, like navigating the learning management system, using certain web tools, a programming compiler, etc.
  10. Able to distribute the lecture material to students directly and digitally. 
  11. Enormously reduces paper usage. All of my former handouts, practice tests, science articles, etc., which used to generate stacks of copied paper are now shown on the overhead instead (on made available on the learning management system if students want a copy later). 
  12. Having written expressions in the LibreOffice math editor, it makes it compatible to copy-paste into other platforms like LATEX, MathJax, Wolfram Alpha, etc.
  13. If for some reason I transition to a larger auditorium for a particular presentation (e.g., someone organized a "classroom showcase" of this type last fall), I'm ready to go with the same presentation available on a larger screen.
  14. Less to erase (saves clean-up time). There's actually a mandated policy to clean the board after one's class at my school, and people have gotten very testy in the past if one suggests not doing that. 
I only anticipated maybe the first 5 of these items above before starting to switch my classes over; once I saw all the very important side-benefits, I became a complete devotee, and transitioned all my classes. I have a hard time seeing that I'd ever want to go back now. That said, I need to be careful about a few things:

Problem Areas of Overhead Presentations in a Math Class

  1. Boot-up time. I need to get into class about 10 minutes earlier to boot-up all the technology -- the computer itself, the network login, directories, starting LibreOffice mobile, opening the files, starting the browser if needed, etc. Our network is very slow and it's a bit unpleasant waiting for things to open up. Meanwhile, students are eager to ask questions, so overall this is a somewhat awkward/anxious moment in time. It would be better if our network were faster, and/or LibreOffice were installed on the local machines (which to date has been turned down by IT).
  2. Presenter view doesn't work on our machines. This is normally a sidebar-view visible to the presenter which shows notes and the next slide to come, etc. Unfortunately our machines weren't supplied with a separate monitor feed to make this work. During the lecture I have to keep my tablet up and manually synchronized on the side to keep track of verbal notes and so forth. 
  3. Making slides available to students is imperfect. I can just give LibreOffice files, because most students don't have it, or possibly the technical confidence to install it. I can provide it as PDFs, but then students usually print them out one-giant-slide-per-page, resulting in everyone with a huge ream of paper. I've directed people to print them (if they must) 6-slides-per-page, but most can't follow that direction. At the moment, LibreOffice cannot directly export a multi-slide-per-page PDF. (And even the normally printing facility has a bug.)
  4. Lecterns have an external power button, and unfortunately I tend to lean on the lectern, press up against the button, and shut off power to the entire system by accident. This is hugely embarrassing and triggers another 10-minute boot-up sequence.
  5. Some lecterns also have a very short power-saving time set (like a default of 20 minutes; note that's less time than I expect to spend on a single slide, as stated above). Evey day I need to remember to set the power-save mode in those rooms to 60 minutes when I start.
  6. Finally, based on my use-case, the handheld presentation clicker which I purchased is not as useful as first expected. It presents another tech item which has a several-minute discovery/boot-up sequence to get started, and I tend to press it by accident or pinch it in my pocket and advance the slide past what I'm talking about. For me I've found it better to discard that and just step to the keyboard when I need to advance the slide (again, just a few times per half-hour).

Nonetheless, all of these warning-notes are fairly minor and eminently manageable; the advantages of using the overhead in my classes pay dividends many times over. Among the top benefits are time-savings, clarity, ability to go back, opening up web access and other materials in class, digital distribution and compatibility, etc. In contrast, here is a question on MathOverflow from late 2009, where the responding mathematicians all seem very adamant about using traditional blackboards. At this point, I really don't grok any of these arguments for not using the overhead. 


Friday, July 22, 2016

Paul Halmos on Proofs

Paul Halmos on mathematical proof:
Don't just read it; fight it!