# Rematch!

With one round to go, Viswanathan Anand has won the right to challenge Magnus Carlsen for the next world chess championship in November! Anand, with 8 points, is 1.5 points ahead of Karjakin, Kramnik, Mamedyarov, Andreikin and Aronian. Anand vaults up more than 15 points on the Live Chess Ratings to claim 3rd place behind Carlsen and Aronian.

Needless to say, this wasn't supposed to happen. After a lackluster performance for the last 2+ years and a poor showing in the last world chess championship, it was clear Vishy Anand's play had declined. And is that really surprising? Anand has had a remarkably long career. Age catches up to everyone, even to one of the greatest natural chess talents in the history of the game.

Fresh off his stinging defeat by Carlsen at the world chess championship (0 wins, 3 losses), it seemed all but certain there would be a new challenger for the title. Aronian and Kramnik were logical favorites but they cracked under the rigors of competitive play. In fact, it seemed as if only Anand played well while the rest of the field played poorly/erratically. I consider this the most telling statistic of the tournament: After 13 round of play, only Anand has a plus score. Karjakin, Kramnik, Mamedyarov, Andreikin and Aronian had even scores and Mamedyarov, Svidler, and Topolov had minus scores. If you followed the games, there were enough blunders to go around: Kramnik blundered on move 7 in round 9 followed by a blunder in round 10 that eliminated him completely from contention.

Carlsen, with an enormous rating advantage, the title, and the jitters from playing in his first world championship out of the way has to be the comfortable favorite for the upcoming match, but as we've seen from this Candidates tournament, anything is possible.

# Sagetex: implicit plots with pgfplots

The utility of the sagetex package for plotting is on display with another proof of concept: implicit plots. I've added two templates to the Plotting with Sagetex page. Getting the list of points is more complicated with graphs which aren't functions and the code for the graph above works because the graph is in one piece. To make the graph look better, it's important to plot small points (around 0.17pt) because otherwise you'll see the points as "beads" connected by line segments. You can download the templates here.

# Sagetex: Combinatorics between sets

I've added another problem type to the (slowly) growing collection of randomized test problems with solutions. It consists of two problems of the form "how many functions are there from an m element set to an n element set?" and "how many of [those] functions are one-to-one?". It's problem type 4 on Sagetex: Combinatorics and Probability page; you can download the question and answer and insert into your randomized test with answer key.

# How do I...create a tightly cropped picture?

If you've got a figure that needs to be cropped then pdfcrop or Briss are great tools as I mentioned here. But if you're creating your own pictures you can use the standalone package to crop the pictures as they're being created. I've added a new page on creating a tightly cropped picture. You can find a link to it on the LaTeX page, the sidebar, or just click here.

Two small updates: first, I've added a template for using sagetex to create polar plots with pgf and TikZ. The screenshot is above; you can find the tex file on the Plotting with Sagetex page. Second, I've added an explanation for the probability of getting 3 of a kind. That's on the Handouts page.

# eCalc

I've made a change in my links: Picalc is gone; it's been replaced by eCalc (shown above) and the link is on the sidebar. This compact, handy calculator has a lot of features which are hidden from sight, giving the calculator a clean interface. The features are available through the "Menu" and "Side Bar" keys and the active features are shown at the top; the eCalc link gives comprehensive documentation. Pressing the "Menu" key lets you change the settings that are shown at the top. The angle choices are degrees, radians, and gradients. The coordinate system is either rectangular or polar. The number format can be either standard, fixed, scientific, or engineering. Last, but certainly not least, you can change the mode from a standard calculator to an RPN calculator. If you've ever had the chance to get comfortable with an RPN calculator, it's really helpful in speeding through calculations because you are able to put numbers and calculations on the stack and use them as needed.

Pressing the "Side Bar" key opens up some more options.

The options available from pressing "Menu" are shown above the calculator and the options available from pressing "Side Bar" are shown, naturally enough, on the side. I like the unit conversion feature; just enter a number in one of the fields and all the conversions are displayed.

This calculator has a lot more features including Complex Numbers, Constants Library, Online Solver (linear and polynomial), and Base Converter; you can read about them in the documentation,

# Explaining Probability

Combinatorics and probability are two of the more difficult subjects to learn and to teach. Difficult to learn because it's easy to make a come up with an answer in a logical way which is very, very wrong. It's difficult to teach because students steadfastly resist showing their work. Combining those two facts results in a mess unless the curriculum has a lot of time to get them trained properly. For almost any problem of reasonable difficulty and you can find your class has come up with 7 or 8 different answers and most answers aren't even close to the correct answer. When you ask how they got their answer they don't have much work to show and struggle to explain it. Breaking that down is problematic. Math IS difficult, especially when you have lousy habits.

I try to teach good habits, but getting "good" American students to do it "your" way is more difficult than the overseas students I've taught (not that they're excellent either). An important part of those good habits shows up when a problem uses the Fundamental Principle of Counting. IF you can get students to show steps (find the events, count the number of outcomes for each event, and give an example of what has been ascertained) they can get it. I've posted two examples on the Handouts page of showing your work in an organized fashion. It's the sort of process I want them to go through, too. The first is determining the probability of 1 pair in poker, and the second is determining 2 pairs. Determining two pairs causes problems for all but the best students so it's a decent example to use in class while you're teaching.

Some "math in the news" stories that I've read over the weekend.

1. Start with the famous Dr. Edward Frenkel: He laments that the math curriculum we have is boring and fails to show the beauty and utility of mathematics. In this article you have a video and podcast as well. Dr Frenkel connects math with hacking e-mail, changing the CPI calculations to raise taxes and cut benefits, and the financial crash in 2008. In this LA Times OP-ED piece, Dr Frenkel  puts the blame on a curriculum of studying mathematics that is more than a thousand years old. The LA Times article says, "For example, the formula for solutions of quadratic equations was in al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean geometry around 300 BC. If the same time warp were true in physics or biology, we wouldn't know about the solar system, the atom and DNA. This creates an extraordinary educational gap for our kids, schools and society.

If we are to give students the right tools to navigate an increasingly math-driven world, we must teach them early on that mathematics is not just about numbers and how to solve equations but about concepts and ideas." and leads to his point, "

Of course, we still need to teach students multiplication tables, fractions and Euclidean geometry. But what if we spent just 20% of class time opening students' eyes to the power and exquisite harmony of modern math? What if we showed them how these fascinating concepts apply to the real world, how the abstract meets the concrete? This would feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements — surely a more efficient way to spend class time than mindless memorization in preparation for standardized tests.

In my experience, kids are ready for this. It's the adults that are hesitant. It's not their fault — our math education is broken.".  Personal comment: Many students lack the basics (multiplication tables and fractions) needed for the course they're in and many math teachers (thanks to certification requirements that don't care about math qualifications) don't have that knowledge or appreciation of "power and exquisite harmony of modern math". Add on top of that the a curriculum that is stuffed full of too many topics (a "firehose approach" to learning). In my case I have to cover 1100+ pages of text for the accelerated class I teach. Dr Frenkel's approach is the right prescription IF the students in a class have the proper foundation and IF the teacher has the requisite knowledge and IF (a really big if) there was time in the curriculum. As such it wouldn't fly very well in a typical public school. The need for a teacher to cover the material so the students are prepped for the multiple choice state test that measures their knowledge (which then determines whether the teacher has done their job) precludes that.