# Update: April 21,2014

Several small changes:

1. A sidebar link to an online LaTeX compiler wasn't giving access to the online compiler so the that link was removed. It's been replaced by: JaxEdit. It's "an online LaTeX editor with instant preview" according to the website. You can find the link on the sidebar under LaTeX.
3. There's a link to the Aperiodical on the sidebar under Mathematics. Make sure to check out the monthly Carnival of Mathematics.
4. As I mentioned earlier, it's Math Awareness Month and mathaware.org is posting a new video each day. Richard Wiseman's video "The Prediction" is far and away my favorite; my students enjoyed it as well. I've added the link to my Other page as well. If you haven't seen it, make sure you take a look.

# Sagetex: Implicit disconnected plots with pgfplots

In an earlier post I put together some code to illustrate how sagetex could be used to create an implicit plot, provided the plot was connected. If the graph is disconnected, such as a hyperbola, then the code would only plot 1 of the pieces. I've modified the code to handle disconnected implicit plots, such as the screenshot above. The template is posted on the Plotting with Sagetex page.

# Sagetex: Inclusion-Exclusion

I've added another problem to the Sagetex: Combinatorics/Probability page. It's the same problem as appears on the Problems page: How many numbers between 1 and N are divisible by primes $p_1,p_2,p_3$? The screen shot is shown. The problem has been done for two primes as well as three primes.

# Odds and Ends: April 5, 2014

1. I've added explanations on the probability of 4 of a kind and a full house. You can download the files from the Handouts page.

2. April is Math Awareness Month. Head on over to the Math Aware website and check out the Activity Calendar. With each day that passes a video and activity is revealed. As the website says, "Each page also includes activities for engaging with the underlying mathematical ideas at a variety of levels, with challenge questions, written explanations, and references.".

3. This website shows off the Aurial handwriting font. It looks great and, if you take the time to look around you'll notice some nice $\LaTeX$ 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.

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.