# Odds and Ends, February 12, 2015

1. I've added another math article template to the LaTeX page. This one uses the Classic Thesis style.
2. Three chess packages were added to the LaTeX page.
3. The Problem page has another problem, due to Leibniz; he demonstrated that $\sqrt{6}=\sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}$.
4. Carlsen tied with Naiditsch at the Grenke Chess classic with Magnus winning on tiebreaks. Chessbase has the report here.
5. Have you ever gotten tired of keeping your students off the portable distraction devices. It might be addiction! If you saw the recent viral photo of someone checking their cell phone while missing the giant humpback whale nearby or saw the report on Japanese teenage girls spending 7 hours on their phones you will have no trouble believing Science Daily's article on the use of mobile technology by children. "The authors question whether heavy device use during young childhood could interfere with development of empathy, social and problem solving skills that are typically obtained by exploring, unstructured play and interacting with peers....."'Until more is known about its impact on child development quality family time is encouraged, either through unplugged family time, or a designated family hour," added Radesky."
6. Voice of America asks, "Is America in a Math Crisis?". Well known mathematician and author Albert Posamentier asserts, “Elementary school teachers in this country, and Europe as well, are part of that general population and consequently they bring that dislike of mathematics subconsciously, sometimes consciously at other times, to the classroom and as a result the teaching mathematics at the elementary school level lacks motivation, lacks enthusiasm.  The enthusiasm of the teacher  is extremely important in turning kids on to the subject matter". Let's also remember that 1/3 of high school math teachers don't have a degree in mathematics. The problems aren't just limited to elementary school.
7. A 10 year old Chinese girl made the Wuhan Evening News for her poem on the tortures of math: “Math is the root of death and it makes life like hell. It wears out children and worries parents. It expels the vitality of school and slowly takes away life.”. Needless to say, I disagree.
8. An 8th grader who through the American flag out of a classroom window is now in serious trouble: From the article, "The principal called the school resource officer with Rio Arriba County, but because it is a federal offense they referred him to the FBI. “I want to report it to them because it is a federal law, so it’s in their hands,” Archuleta said. Last week, he suspended the 14-year-old last week for 10 days, but he is recommending long-term suspension or expulsion."

# Handouts: Stern-Brocot/listing rationals

In earlier posts I mentioned the Stern-Brocot sequence and how it can be used to create another sequence that lists the positive rationals in lowest form exactly one time each. There was a lot of information on the subject so I put together a PDF containing that information plus a little bit more--it's posted on the Handouts page.  In addition to the Numberphile, the link to the Caulkin, Wilf paper, the tree of rationals, and Sage Code I've included an algorithm along with a dry run (shown above) along with some other information from the Caulkin-Wilf paper and some sample questions/tasks that can help flesh out a lesson to your class:

The "Fact" box above as well as the box containing the Sage Code below were created using the tcolorbox package.

So, if you plan on using the Stern-Brocot sequence there is now 1 resource containing all the components of the Stern-Brocot sequence.

Here are a few things that caught my eye recently:

• The Grenke Chess Classic 2015 has one round to go with Carlsen and Naiditsch tied for first and several others in striking distance. In the even of a tie there will be blitz games to decide the winner of the tournament. The games stream here. (Click "Live")
• In the last post there was an item about the shift in the debate over Common Core. Rather than deal with constant criticism the new approach is to say Common Core is about standards, not content. That's misleading from my experience because standards drive content. So schools look to buy Common Core aligned material which help set the level of difficulty and subject matter. The problem of 4 ways to subtract caused a backlash because it was showing up in the textbooks. And as RT notes in this video,  Common Core has legible handwriting taught in kindergarten and first grade only. This skill, no longer being required at higher levels is replaced with other content. As a result, many schools have stopped teaching cursive. The result are kids who can't read or write cursive. It's a piece with a lot of interesting arguments.

# Python/Sage: Stern-Brocot sequence

In the last post I mentioned Numberphile's excellent video on the Stern-Brocot sequence. The explanation of how the sequence is formed is clear enough but extra time will be needed to make sure students know how the sequence is formed. That would mean discussing the algorithm for how the sequence is formed, using the algorithm to create the sequence and then getting them to calculate some terms of the sequence on their own. I put together a quick Python program to create some terms of the sequence and then show the sequence of positive rational numbers. Having a list of both sequences will help to catch any mistakes you might make from, say, getting distracted/interrupted fielding questions along the way. The code is posted on the Python/Sage page.

Here are some other events that caught my eye recently.

1. The 13th Tradewise Gibraltar Masters tournament has ended with Hikaru Nakamura in clear first place followed by David Howell second and a big tie for 3-11. Now all eyes are on Grenke which will continue with round 4. You can follow the game here. (Click Video)

2. I stumbled onto a PhD Dissertation on "Students' Errors in Graphing Calculator Based Precalculus Classes". It's worth looking at some of the misconceptions and examples that caused students to struggle.

3. Politifact weighs in on the Texas governor's claim, "it takes "more than a minute" to teach a student "how to add nine plus six."". Lost in the claims and counter claims (Common Core is standards not content) is the fact that the new materials which are aligned to common core have this bizarre new math. Remember, for example, the 4 methods of subtraction? It's in the books that teachers have to teach out of and wasn't in books before Common Core.

4. An excellent web page on Using Writing in Mathematics.

# Counting the Rationals

In The Most Important Lesson of High School Math I mentioned that I thought the goal of the high school math curriculum should be to get students to the point that the rational numbers have measure zero (while avoiding such technical speak). The video I linked to (How Big is Infinity) took students through Cantor's diagonal argument that the positive rational numbers can be listed. That list, however, has different forms of the same number (4/2=2/1=8/4=...). But I frecently found a post by Numberphile on the Stern-Brocot sequence. This sequence has a nice property: take the ratio of consecutive terms  $\frac{a_n}{a_{n+1}}$ to create another sequence which lists every positive rational in lowest terms exactly once. A brief paper by Caulkin and Wilf has more of the mathematics. I've put together a tree diagram that was used to explain how the rational numbers are created. You can find the PDF and tex file on the Graphics page. The Stern-Brocot sequence and the rational sequence resulting from the ratio of consecutive terms would be a good example to put into your coverage of sequences.

Some other points of interest:

• I found a minor mistake in the code for the last post. If you look at the screen shot I've got y_coords and y1_coords used in the same IF statement. I've fixed the file and uploaded it.
• I've started a list of LaTeX packages that are important/useful to me when I'm using LaTeX; it's on the LaTeX page. There's been a big increase in packages over the years and I'm finding new packages that weren't around years ago. Since I can't keep always remember the package or the commands, I'm listing the packages which are linked to the CTAN documentation.
• The Gibraltar tournament is into round 7 of 9 and Nakamura is in the lead. Meanwhile, the Grenke Chess Classic 2015 has started. It's got Carlsen, Anand and many other top players. You can follow the games by clicking on the link and then pressing the "Video" tab.

# Inquiry Based Learning

Inquiry Based Learning (IBL) is a teaching method "that engages students in sense-making activities.  Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate... all those wonderful skills and habits of mind that Mathematicians engage in regularly.". The website I linked to has lots of information on running a class using IBL. While I don't think it's practical to run a standard classroom that way because the curriculum couldn't be covered, I do think it's an important part of teaching. With 1100+ pages of material that needs to be covered for my accelerated precalculus class, finding time for IBL is not always so easy. Most of the time I have to resort to bonus problems in order give an IBL type problem. But finding these more nuanced problems isn't always easy. In an earlier post, I gave a link to the problems that Phillips Exeter Academy posts on their website. The IBL site I linked to above has lots of links to helpful resources. The goldmine, for me is Art of Mathematics website which has 11 free PDF books that you are free to download. The books have titles: Art & Sculpture, Calculus, Dance, Games & Puzzles, Geometry, Knot Theory, Music, Number Theory, Patterns, The Infinite, Truth, Reasoning, Certainty, & Proof. There's something for everybody. And the materials are well thought out. If you look around the site, there's plenty of useful information to get you started on the road to IBL. One of the links is to a site for IBL Calculus.

I've added Exeter's problems and the Art of Mathematics books to the mathematical links on the sidebar.

Here are some things that caught my eye:

# Odds and Ends: Nov 2, 2014

I've added a new problem to the Sagetex: Limits page. The screen shot above shows the general form of the question: Evaluate the limit: $\displaystyle \lim_{x \to 1}\frac{x^{7}-1}{x^{3}-1}$.

The Problems page has also had a trig problem added to it: Prove $\frac{\pi}{4}=4\arctan\left(\frac{1}{5}\right)-\arctan\left(\frac{1}{239}\right)$. This problem is intended for the best of an accelerated class. I would expect to use it in a weekly (optional) problem competition.

Here are some current events that caught my eye:

• The Daily Signal has "10 Reasons Why Common Core  Should Spook You This Season"
• It doesn't matter what metric you use to judge a teacher; the new metric will result in some teachers doing worse, some staying the same, and some improving. And in the US that means lawsuit. The Washington Post has an article on a 4th grade teacher in NY who has been penalized by recent changes. It seems like spooky reason number 10 from the Daily Signal: The one size fits all, top down approach has some experts designing a model that will quantify teacher effectiveness. The top down approach means that there is an industry of "experts" that live off of providing a plan or model to "solve" a problem which is then "implemented" on everybody, resulting in all the deficiencies being brought to light which leads to a backlash and the push for a "better" solution. Poltifact has an article which, thankfully, shows US test results versus the world; In math we've gone from 18th (out of 28) against industrialized countries (2000 data) to 27th (out of 34) for 2012 data. A decade of time shows the top down government model has done poorly but don't expect accountability for being terrible at what you do in government. How much money was wasted to get that level of performance?