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 It's been replaced by: JaxEdit.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.
  2. I've added a link to the American Mathematical Society's Google Plus page. The link is on the sidebar under Mathematics.
  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

ImplicitPlotDisconnectedIn 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.

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

CombProb4I'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.


ecalc1I'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.

eecalc2The 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.

Sunday reading: 3/2/14

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.

2. More typical of today's public school is this article where I found a lot of valid points: "The latest educational fad term is "STEM," which stands for a curriculum in the areas of science, technology, engineering and mathematics. GOOGLE reports over 174 million pages on "STEM education" alone. The problem is that adults, including some educators, still haven't figured out how to make peace with the "mathematics" in STEM. Everyone applauds classes in high-tech robotics as the sine qua non of a good STEM program, but ask them to explain how they integrate any mathematics content into the robotic curriculum and you may be surprised that most of these programs do not even work with mathematics teachers to legitimize themselves."...."The NASA brand has become synonymous with inspiring students, making them feel excited about what they know, and enticing them to learn more about STEM careers. The problem is that in our society, mathematics tends not to make people feel they are competent; it does not excite our emotions in a positive way, and often engenders a sense of dread. Any marketing expert will tell you that mathematics is the poison-pill for any brand, and so over the years NASA and other agencies have largely hidden their mathematics expertise from public view. ... There are no down-sides to students building robots out of parts from a box, and watching as these battery-powered critters scurry around the classroom floor. There is also little or no math involved in these curricula."....."But if you think that NASA is alone in some sinister plot to de-emphasize mathematics you are wrong. Virtually every federal agency that offers STEM resources to teachers does so by minimizing the mathematical content. They do this, for example, by creating middle and high school STEM activities that cover math skills far below what the students see in their corresponding math classes. The rush to make activities "hands-on" has seemingly created legions of resources that have students measure and plot and take a few percentages, but never delve into math concepts above grade eight such as linear equations, statistics and mathematical modeling.".  Personal comments: The "regular" Algebra 2 and Geometry classes I've taught have students who still haven't mastered their multiplication tables. They've been passed along through the system even though they don't know what they're doing. In one school I was at, the guidance counselors decided the math classes the student would take for the next year---in many cases over-riding the opinions of the math teacher who had just taught them. That meant students who complained enough would be put into honors classes (even though they didn't belong) and since there were too many students like that, the overall quality of the course suffered. Welcome to today's world of education; the emphasis is not on setting standards. It's more about placating parents and students.

3. Canada has problems that sound just like our math problems and they've identified discovery education as a culprit in why there math scores are going down: "Ontario’s curriculum, however, does not require students to memorize multiplication tables or learn basic algorithms such as long division. They are instead encouraged to break problems down into smaller portions to work through them.". Likewise, here you'll find: " “If you look at what’s been happening, predominantly over the last decade, there’s been an unprecedented emphasis on discovery learning,” said Donna Kotsopoulos, an associate professor in Wilfrid Laurier University’s education faculty and former teacher.

Robert Craigen, a University of Manitoba mathematics professor who advocates basic math skills and algorithms, said Canada’s downward progression in the international rankings – slipping from sixth to 13th among participating countries since 2000 – coincides with the adoption of discovery learning....Parents in Alberta, Ontario and British Columbia, for example, launched petitions over the Christmas holidays, calling on their governments to revamp curriculums with a greater emphasis on basic math skills.". Finally, this article summarizes the problem: "The one side says, “drill and kill.” The other says “drill for skill.” Basically, though, just about every mathematician and math education researcher who was interviewed for this story agrees that the perfect math class should have a mix of skills and problem solving. They just can’t agree on the amounts of each, when to add them, and what to skip."...."How does Shanghai do so well? They devote an average of 14 hours a week to homework (versus three for the Canadians) and 70 per cent have parents willing to pay for extracurricular math classes (versus 28 per cent in Canada). And those students who seem to spend so little class time on math also have teachers trained more rigorously and subject to greater supervision."....." top-performing Asian countries typically cover fewer subjects more deeply, especially in the early grades. A 2004 study found that Grade 1 teachers in Canada were expected to cover 18 topics versus just five in Hong Kong, where even textbooks may be hundreds of pages shorter.". Personal comments: Understand that the poor educational results our country has are inflated. They're pulled up by students who are getting extra tutoring outside of class to fill in the deficiencies of the educational system. Most students that I've encountered lack basic skills  (multiplication tables and fractions) and then the schools put calculators into their hands to avoid the drudgery of calculations; they never learn their arithmetic or algebra. But no big deal, pass them through the system and onto the college level. Complaints from parents are minimized, students can claim they're taking accelerated classes even when they lack the math skills of an accelerated student. Now the college has to figure out what to do with them.