How do I...? (change fontsize 2)

I've added a second page on changing the fontsize. The main purpose is to let you know that $latex \LaTeX$ gives you a lot of freedom to do what you want. That's true on many issues, but trying to learn (and remember) multiple methods complicates the learning process. In the future I'll focus on one method to accomplish a task. The second purpose in discussing another method for changing the fontsize is that it illustrates different methods might result in different output so as you learn, pick a method and stick to.

The new page is here. It's also on the sidebar.

It's obvious: all numbers are rational.

They aren't, of course, but students are always telling me how "obvious" something is, usually to justify omitting steps/reasons. I think it's essential students are aware that math has many results which are obvious and wrong; that's why we prove things. The Pythagorean Theorem gives you the opportunity to explain why people thought every number was rational.

Although we don't know who proved the Pythagorean Theorem, we do know when Pythagoras was alive mathematics was very different than today. "Numbers" to the Greeks meant 1,2,3,... and so on. Ratios, such as 3:5, were a relationship between numbers but if we consider the ratios as well then we could describe their number system as the nonnegative rational numbers. Negative numbers made no sense because length was never negative and the Greeks had no idea that the irrational numbers existed. [Updated: Initially the Greeks weren't sure whether zero was a number but by 130 they had a "zero like"-symbol which functioned as zero.]

It's easy to criticize now but that reasoning is very logical. In order to show that to my class, I created a very simple Sage interact demonstration. Here's the code running in the Sage Cell Server.

I've focused on the interval [0,1]. When it's divided into a lot of pieces (each of length 1/n for large n) it looks like every point in the interval will be labeled. By adjusting the slider you can show for larger values there's no space. Every number is rational...right?

I've put the screenshot above on the Sage Output page and the code is on the Python/Sage page.

Chess Basics: Discovered Attack

A discovered attack occurs when moving a piece results in an attack from another
piece. The piece that is moving could potentially attack another piece as well, in which case it is also a double attack. A discovered attack that results in a double attack or a check is a more powerful version of the discovered attack because it limits the opponent's choices.

I've written up the basics of discovered attacks; you can find them on the Chess page.

How do I...? (change fontsize)

Today's post continues to expand the How do I...? section on the LaTeX page. The eventual goal is to have the How do I...? section explain how a beginner can quickly find the LaTeX way to do something they're used to doing on a regular word processor.

Today's post briefly explains how to change the default fontsize in LaTeX. You can go directly to the page by clicking here.

Odds and Ends: June 22, 2012

I'm adding a section to the LaTeX page called "How do I...?" to help people trying to learn the basic LaTeX functionality they need to abandon the various WYSIWYG processors out there. The first item is on how to create a simple LaTeX document which you can find here. It assumes you've got some form of LaTeX on your computer, an IDE, and you know the basics of creating and compiling documents.

When I was in graduate school one of my teachers had a humorous posting on his door. I found it online recently; it's called "The Lesson" and it seems even funnier now that I'm teaching in a public school (the quote from the Pharisees). Perhaps you'll get some enjoyment out of it, too.

Pythagorean Theorem Proof

From my experience, most people choose a visual demonstration of the Pythagorean Theorem. That's a pretty good idea since it's all the proof a lot of students need. Wikipedia has an example here. [Update: There was a second link here earlier but it went to a page with a lot of processes running. One reader reported it's slowed down their computer so I've removed the link]. This site even has 93 proofs of the Pythagorean theoremPersonally, I like proof number 9 which uses a square with sides of length a+b.

But it 's geometry after all, so I wanted a proof of the result and I wanted it to match the visual proof. That's a little bit harder to find. A Khan Academy video walks you through the proof here. I've written up a proof of the result.

The proof is posted on the on the Handouts page and the diagram is posted on the Graphics page.

Anand's decline

I've already posted several times on the criticism leveled at Anand and Gelfand for their lackluster play in the latest World Championship match and their pointed responses: see here, here, and here. Both Anand and Gelfand have put Kasparov as public enemy number one for his comments. But you can't make a convincing case that Anand and Gelfand are playing world championship level chess and the problem is just Kasparov is out for revenge when so many players (mentioned in link 3 above, for example) have similar criticisms. It's even more apparent if you look at the live rating of the best players: Anand at number 5 (down 1 place) and Gelfand at number 16 (up 4). Anand struggled against someone not even close to the top 10.

So it's not too surprising to see Anand dial back his sharp criticism of Kasparov and to admit to some of the numerous complaints in a 2 part interview on the Chessvibes website. The defense is basically an admission of some problems (yes my play has declined) but the problems aren't as bad as you're hearing (because Gelfand prepared so well). Part 1 of the interview addresses lots of issues. His play in the Bundesliga: "I don't know, I just went nuts in that game, it's not even serious, my play. I just got annoyed and started making very angry moves... But the point is that I was trying to build some confidence with these games in the Bundesliga, but it didn't help very much." and "To pretend that these Bundesliga games actually were positive for me in some ways would be a bit much, but let's say they maybe had a silver lining.". On the rumors of Kasparov offering help to Gelfand: "But I did think Garry would offer his help. Not to get into details, but I think it's not a big secret that Garry and me are not the best friends anymore.".

So we're getting some indication that Anand thinks his play wasn't the best. Part 2 of the interview has Anand taking blame for his quick draw in Game 12. I mentioned Kramnik being shocked at Anand offering the draw just when Gelfand had weakened his position so that things weren't equal anymore and, in addition, Gelfand was starting to run low on time: " Of course it was a mistake not to play on for a few moves, not because there's anything in the position, I think Black still has full compensation. I strongly disagree that I had some hope in the position. OK, he has to find a way to at least liquidate the queenside or exchange a pair of pawns and double somewhere there. At least he would have had to show something." followed by "So I have some mild regret and I can understand some of the criticism. Here you can really say there's no harm in us playing out a few moves." and "But, yes, it was a mistake. It was a wrong reflex as a result of just not adjusting in time. If fans complain that we stopped early, I respect that. I think it was a mistake.".

Anand also admits to his confusion during the rapid portion of the match, "I had the feeling I had nailed the draw, and then I got myself confused. First of all, it's just a trivial draw if I play a Kh5 somewhere, it's just a trivial draw. There were just a lot of things wrong. Both of us were hallucinating a lot. But still at the end of it, if you ask me, I would have to say that I was lucky. You can't pretend that there's some logic to all this." which he emphasizes again, "Yes, I was lucky, I can't argue with that.". For Game 4: "But my play in game 4 was ridiculous, there's no getting around that.". In addressing the frustration of chess players for the quality of play, "...in the press conferences you started to get this sensation, this frustration, and I could very easily imagine what the public was saying. I could even almost partly understand where it was coming from. But we were trying, we were just not getting very interesting positions because our preparation with Black was quite good. Neither of us was getting very interesting openings to do something with.

I could understand the criticism, but somehow I felt the main thing was to actually focus on the match. I mean, if you start playing for the gallery, it can get quite tricky." His response to Kamsky's criticism of too much preparation and not enough playing chess was odd: "There are arguments for 960; I don't think this is the only one. I'll put it this way: I think also Boris and me, our styles are particularly badly matched. We both tend to play in a certain way and prepare in a certain way which, if we're both well prepared, might not lead to very much. We showed this a little bit here. I tend to prepare my black opening well, he tends to do the same, we tend to defend well, and so on. I mean, with different players... I don't think chess is dead.". It sounds like he's rejecting the criticism of chess being dead because good preparation made the match dull...but the criticism is about too much preparation, which Anand seems to be acknowledging. The criticism also contends that the players took quick draws rather than play things out. That, too, he's also admitted earlier.

By the end of the interview, it sounds like Anand is finally admitting his problems. In his upcoming plans Vishy says, "The first thing is that I would like to do well in my upcoming tournaments. It's not only other people who have been disappointed about my tournament results last year. I'll play some tournaments this year and I understand that autopilot will not be good enough. I have to do something and in a way that's kind of my goal right now." followed by, " I understand that after three failed tournaments people are a bit fed up with my play but at least in this match I think I got the job done. I don't think I was playing particularly badly, I simply think Boris played well.".

So there you have it. Anand is at least starting to admit the problems everyone knows while clinging to the defense that it's not as bad as it looks because Gelfand was well prepared. Anand's defense that Gelfand's preparation is a big factor in his play looking badly is just feeble. Even after his good results Gelfand is still 100 points behind Carlsen. If Anand thinks Gelfand's defense makes him look worse then he'll be shocked how a stronger player like Carlsen who fights to win will be able to make his play look even worse.  I expect a difficult road ahead for Anand in trying to even get his rating above the 2800 mark.

Since Chessvibes has provided some excellent reading I've added it to the list of links on the sidebar.

Teaching Point: Exponents

If you took the quiz on exponents, it was designed to lead to a teaching point. By having questions with negative numbers for bases and even $latex (-\frac{1}{64})^{-\frac{1}{3}}$ forces your class to work with negative bases. So what to to do then when confronted with $latex (-1)^{\frac{6}{10}}$ ? They'll use the rules of exponents; specifically $latex a^{\frac{m}{n}}=(a^{\frac{1}{n}})^m=(a^m)^{\frac{1}{n}}$.

The problem is that these laws apply for $latex a>0$ but it's not unusual for books to give problems where the base is negative. When it's time to go over the last problem, write this on the board:

$latex -1=\left((-1)^3\right)^\frac{1}{5} = (-1)^{\frac{3}{5}}=(-1)^{\frac{6}{10}}=\left((-1)^6\right)^\frac{1}{10}=(1)^\frac{1}{10}=1$

and ask your class to explain why -1 isn't equal to 1. From my understanding there are two ways to view $latex (-1)^{\frac{6}{10}}$. One way is to observe that since  $latex \frac{6}{10}=\frac{3}{5}$, it must be that $latex (-1)^{\frac{3}{5}}=(-1)^{\frac{6}{10}}$ and the "proof" above fails because the laws of exponents don't apply to negative bases. The problem is it forces students to check if they can simplify the exponent first and leaves them in doubt about how to work with negative bases in figuring out problems like $latex (-\frac{1}{64})^{-\frac{1}{3}}$. The second viewpoint, the one that I'm using, leads to this teaching point: If $latex m$ and $latex n$ are integers then $latex a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}$ provided $latex \sqrt[n]{a}$ exists. With respect to our problem, since $latex (-1)^{\frac{1}{10}}$ is not a real number $latex (-1)^{\frac{6}{10}}$ is undefined (with respect to the real numbers). Therefore the "proof" fails because $latex (-1)^{\frac{3}{5}}\neq(-1)^{\frac{6}{10}}$ and since $latex (-1)^{\frac{1}{10}}$ doesn't exist $latex (-1)^{\frac{6}{10}}\neq \left((-1)^6\right)^\frac{1}{10}$.

What happens if you're given  $latex (-1)^{.6}$ ? Under both interpretations it would be $latex (-1)^{\frac{3}{5}}$, the fraction in simplest form.

The .tex file and PDF can be found on the Teaching Points page. Your students should now understand: details matter!

Quiz on exponents

Details matter in math; it's a point I try emphasize to every class at various times during the year. But students don't care. Ask your students, for example, to state the Pythagorean Theorem. The best students I have will reply "$latex c^2=a^2+b^2$" and if I follow up with, "So what's the converse of the Pythagorean Theroem?" I can't get even get a reply. Ever. Which forces me to drop back and ask what $latex a, b$, and $latex c$ are and then gradually force the details out of them: $latex c$ is the hypotenuse? So you're talking about a triangle then? Any triangle?

It's a review ritual I've gone through a lot but, I think it's an important ritual to go through. Hopefully, by the end of the year, some of them will appreciate the details a little bit more. Exponents is a good place to show them why details matter. Here's a quiz on exponents that can help motivate the discussion.

Simplify as much as possible.

a) $latex -8^{1/3}$

b) $latex -5^{0}$

c) $latex 0^{1}$

d) $latex 0^{0}$

e) $latex -2^{6}$

f) $latex (-1/64)^{-1/3}$

g) $latex (a-b)^{0}$

h) $latex (-1)^{6/10}$

Be careful on this quiz. When you're ready, the quiz and answers are posted on the Handouts page as both tex and PDF files.

Best practice

Best practice, as this link explains, "...has been used to describe "what works" in a particular situation or environment. When data support the success of a practice, it is referred to as a research-based practice or scientifically based practice. As good consumers of information, we must keep in mind that a particular practice that has worked for someone within a given set of variables may or may not yield the same results across educational environments."

Keep that in mind as you work through the excellent article "Backtracking on Florida Exams Flunked by Many, Even an Educator". It's about a writing test (not math), but the way the public school system dealt with the problem was so memorable I thought it deserved to be posted here. Here are the key points; I've added my emphasis in red:

1. "In 2011, among the state’s 67 districts, Seminole (which serves 64,000 students, half of whom qualify for federally subsidized lunches) ranked third in math, fourth in reading and sixth in writing."

2. "The 2012 scores on the writing test — given to 4th, 8th and 10th graders — plummeted in all districts. Only 27 percent of Florida’s fourth graders were rated proficient, compared with 81 percent the year before. In Seminole, 30 percent were proficient, down from 83 percent."

3. "They could live with the results — that after 15 years of education reform, three-fourths of Florida children could not write. Or they could scale the results upward after the fact, an embarrassment, but people probably would not be so angry if they had good scores."

4. "So on May 15, Florida’s education commissioner, Gerard Robinson, held an emergency conference call with State Education Board members, while 800 school administrators from all over Florida listened in. The board voted to lower the cutoff to 3.

Presto! Problem solved. The proficiency rate for fourth graders was now exactly what it had been in the 2010-11 school year, 81 percent.

For 10th graders, the results actually improved, to 84 percent from 80 percent, meaning scores plummeted but proficiency increased."

5. "The Buros Center for Testing, the consultant the state pays \$100,000 to do annual audits, wrote that there was nothing to worry about, concluding that the scoring “was in keeping with the best practices of the profession.” (Imagine what the worst practices are.)

The audit referred to lowering the passing score to 3 as “equipercentile equating.”"

6. "Test scores are used to determine which third graders must be retained and which high school students can graduate. They determine a school’s report card grade, from A to F, as well as which teachers and principals will get bonuses and which ones will be fired."

7. "Two weeks after the writing results were announced, Broward County, the sixth-biggest school district in the nation, became one of 10 in Florida to pass the National Resolution on High Stakes Testing, calling for a reduced role of standardized tests in public education."

Best practices, indeed, but for what outcome? Certainly it's a best practice to make a failing school look okay and a best practice to get a bigger pay raise to someone who doesn't deserve it. Florida tax dollars down the  drain! It's doubtful, though, that it's a best practice if you're interested in actually providing a good education.