Stevo’s Forcing Class Fall 2012 – Class 3

 (This is the third lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the second lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement.)

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Stevo’s Forcing Class Fall 2012 – Class 2

(This is the second lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the first lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement.)

(Also, there are bound to be typos. I would appreciate anyone who could point them out in the comments and I will promptly fix them.)

Continue reading Stevo’s Forcing Class Fall 2012 – Class 2

Is Algebra Necessary? – Andrew Hacker

Today on The Sunday Edition, on CBC Radio 1, host Michael Enright interviewed Andrew Hacker, about his controversial views on mathematics education. Basically, he thinks that demanding that all high school students master Algebra is “an onerous stumbling block for all kinds of students”. Listening to the interview this morning made me angry. I don’t get riled up easily, but my blood was boiling this morning.

Here is the link to the radio interview. (It starts around the 6:30 mark and is about 24 minutes long.) Here is the article Andrew Hacker wrote for the New York Times.

I intend to write a detailed response to the claims of Andrew Hacker later this week, but first let me share some reckless assertions that pissed me off. (These are paraphrases.)

“Should we be learning mathematics for learning’s sake? Don’t get me wrong, I’m a liberal-arts professor; I teach multi-disciplinary courses. However, I think that we should be focusing on learning about mathematics: the history of mathematics, the philosophy of mathematics. Not the dry stuff.”

“Hacker: 1.7 million Americans enter American universities every year, all required to have some high school mathematics. How many do you think choose to major in mathematics?

Micheal Enright: I don’t even want to guess.

Hacker: 15 Thousand. This tells me that even though teachers have had the chance to show the students the ‘beauty of mathematics’ they just aren’t getting it. [Goes on to imply that it’s because this is too hard to do.] Why are we training all of these students in high school when so few are going into STEM [science, technology engineering and mechanics] careers?”

“Do people need to learn about Fermat’s Theorem [sic] or Rye-man’s Hypothesis [sic]?”

I agree that something is broken with the way we learn, teach and perceive mathematics, especially in middle years and high school. This is a conversation we must have. However, it is troubling when the social scientists try to use their irrelevant authority to persuade the masses, leveraging the common man’s bias against mathematics.

I’m mad. I don’t like when people piss on what I love, and I especially dislike when they do it while tricking the public. As people we need emotions: love, anger, joy, contempt; but as a people we need thought, analysis, reason and mathematics to universally prevail.

Again, I plan on writing a cool-headed, but strong reply, and this is just me venting. Until then, let me leave you with this, taken from JFK’s famous speech:

“But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic? Why does Rice play Texas? We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.”

Stevo’s Forcing Class Fall 2012 – Class 1

(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)

Summary of class 1.

  • Discuss examples of ccc posets whose product is not ccc.
  • Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
  • Display the differences in chain conditions.

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Hindman’s Theorem write-up

It came to my attention that Leo Goldmakher had written up notes for a lecture I gave in August 2011 on the proof of Hindman’s Theorem via ultrafilters. The notes are quite nice so I thought I would share them.

Here is a link to the notes (pdf) and here is Leo’s website.

The lecture I gave follows the papers:

  • “An Algebraic Proof of van der Waerden’s Theorem” by Vitaly Bergelson, Hillel Furstenburg, Neil Hindman and Yitzhak Katznelson. (L’enseignement Mathematique, t. 35, 1989, p. 209-215)
  • Ultrafilters: Some Old and some New Results” (pdf) by W.W. Comfort. (Bulletin of the AMS, Volume 83, Number 4, July 1977)

The Battery Problem – Math for my Mom

(After writing some posts directed at other mathematicians, here is one for everybody.)

I’ve never actually done this.

I was sifting through some old issues of Crux Mathematicorum last Friday. For those of you who don’t know, this is a wonderful magazine that contains tons of math questions generally like those you would see in a math contest or olympiad, and the difficulty ranges from elementary school to undergraduate. In the September 2009 issue, I stumbled upon the following nice problem originally from the 2005 Brazilian Mathematical Olympiad. It is one of those problems that is mathematical in flavour and doesn’t need any previous math knowledge to begin thinking about the problem. For me, a nice problem is one that rewards you for thinking about it and can be attacked from many different angles.

So here’s the problem as stated:

We have four charged batteries, four uncharged batteries, and a flashlight which needs two charged batteries to work. We do not know which batteries are charged and which ones are uncharged. What is the least number of attempts that suffices to make sure the flashlight will work? (An attempt consists of putting two batteries in the flashlight and checking if the flashlight works or not.)

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