# Resource: Internet Archive

Who doesn't like quality, free resources? The Internet Archive explains on their About page: "The Internet Archive is a 501(c)(3) non-profit that was founded to build an Internet library. Its purposes include offering permanent access for researchers, historians, scholars, people with disabilities, and the general public to historical collections that exist in digital format.

Founded in 1996 and located in San Francisco, the Archive has been receiving data donations from Alexa Internet and others. In late 1999, the organization started to grow to include more well-rounded collections. Now the Internet Archive includes:, and software as well as archived web pages in our collections, and provides specialized services for adaptive reading and information access for the blind and other persons with disabilities.".

What separates the Internet Archive from other resources is threefold: the quality of their resources, the range of resources (books, software, audio, video) and the ability to read through many of the books by flipping through the text (as well as downloading it in a variety of formats). Take a look at Kotov's "Grandmaster at Work"

Notice the 2 red boxes? The box at the bottom displays a variety of formats that you can download the book in. The box near the top right surrounds the "Full screen" button. Press that button to go into full screen mode.

Clicking on the right hand page flips the book forward, while the left hand page flips you back. So you can browse the resource online. Notice that the screenshot above has two more red boxes. The one in the top right hand corner will activate the voice reading of the book. The red square in the bottom left hand corner is a slider that can quickly get you to deep inside the book without flipping each page.

My only complaint is that the Search feature wasn't as helpful in finding the resources. I failed to find some books through searching "mathematics" which turned up in other unrelated searches.Here are some links to get you started:

Spivak: Calculus book, Supplement for the book, Dugopolski: Precalculus, Beginning and Intermediate Algebra,lots of CK-12 series books CK 12 AlgebraCK 12 Algebra II with Trigonometry, MOOCulus Sequence and Series Textbook, Advanced Math 2Python Programming, Soltis: What it takes to become a chess master, Botvinnik: Half a Century of Chess, Alburt: Test and Improve Your Chess, Kosikov: Elements of Chess Strategy, lots of old Schaum's books, and so much more! I've added the link to the Internet Archive to the sidebar.

Here are some stories which caught my eye the last week:

• The Intercept looks at "NO CHILD LEFT UN-MINED? STUDENT PRIVACY AT RISK IN THE AGE OF BIG DATA". From the article, "“What if potential employers can buy the data about you growing up and in school?” asks mathematician Cathy O’Neil, who’s finishing a book on big data and blogs at mathbabe.org. In some of the educational tracking systems, which literally log a child’s progress on software keystroke by keystroke, “We’re giving a persistence score as young as age 7 — that is, how easily do you give up or do you keep trying? Once you track this and attach this to [a child’s] name, the persistence score will be there somewhere.” O’Neil worries that just as credit scores are now being used in hiring decisions, predictive analytics based on educational metrics may be applied in unintended ways. Such worries came to the fore last week when educational services giant Pearson announced that it was selling the company PowerSchool, which tracks student performance, to a private equity firm for $350 million. The company was started independently; sold to Apple; then to Pearson; and now to Vista Equity Partners. Each owner in turn has to decide how to manage the records of some 15 million students across the globe, according to Pearson." • Reason.com reports on famous author Judy Blume warning about censorship in today's world. • Huffington Post has a piece on Dr. John Urschel, professional football player, on why more kids don't like math. • Microagrressions, which I mentioned in this post, are back again. Although I learned "America is a melting pot", that's now a color blindess microaggression because it denies a person of color's racial/ethnic experience. The College Fix can help get you up-to-date on on the latest witch hunt. From the article, "University of Wisconsin-Stevens Point officials have advised faculty that the term “America is a melting pot” is a racial microaggression. The common phrase was among a list of examples of so-called racial microaggressions used “as a discussion item for some new faculty and staff training over the past few years,” a campus official told The College Fix in an email. Other phrases on the list included: “You are a credit to your race,” “where are you from,” “there is only one race, the human race,” “I believe the most qualified person should get the job” and “everyone can succeed in this society if they work hard enough.”. Take some time and look at the lists from University of Wisconsin and University of California. Lots of the examples on the list are ambiguous in the sense that it presumes you know WHY a comment was made. So the example, to a woman of color about "I would never have guessed you were a scientist." is considered a microaggression. because it's assumed you said it because she's a woman which means it could be perceived as insulting her intelligence. Apparently it's okay to to say it to a white male, though, because it wouldn't be an attack on his intelligence....wait, what? Or if you've mistaken a faculty of color mistaken for a service worker then it assumes you've done it because they are of color and not because of how they were dressed, where they, or who they looked like. Heck, I've been mistaken for someone working in a store that I was shopping in for who knows what reason. Should I have been insulted? HOLDING AN OPINION that, "Affirmative action is racist" IS FORBIDDEN because it makes it seem like one group gets extra privileges. And the common practice of empathy, such as a someone saying, "As a woman, I know what you go through as a racial minority" is enough to cause a problem. How can a woman possibly know what racial discrimination is like. Your AMBIGUOUS ACTIONS are now under assault. "A person asks a woman her age and, upon hearing she is 31, looks quickly at her ring finger" is a problem because the reason WHY you did that was you thought "Women should be married during child-bearing ages because that is their primary purpose.". If faculty are being taught that examples like they've listed are transgressions then you get an indication of how today's young are looking at the world. It's a less tolerant, "he/she said this which made me feel ___, therefore they must pay the price". Imagine spending money to get an education and coming out less educated and less tolerant. ZeroHedge has a piece on how "hate speech" is used to destroy "freedom of speech". • There's an annoying piece that's getting a lot of play. From the Western Morning News we hear "Myth that men are naturally better at maths than women debunked". Now you'd think that such a conclusion would be based on some test scores which would show that women scored just as well (or better) than men. No such case. From the article, "US psychologist Dr Shane Bench, from Washington State University, who led a study that involved assessing the ability of men and women to predict their performance in maths tests, said: "Gender gaps in the science, technology, engineering and maths fields are not necessarily the result of women's underestimating their abilities, but rather may be due to men's overestimating their abilities." His team conducted two studies of 300 undergraduates who were asked to have their maths skill tested before guessing how well they had fared. In the first study, participants received feedback about their real performance before they were again asked to take a test and predict their scores. For the second study, the students only sat one test without receiving any feedback, and were questioned about any plans to pursue maths-related courses Across both studies, men were consistently found to overestimate the number of problems they solved correctly while women's appraisal of their own abilities was more accurate. After receiving feedback about how well they did in the first study, men were then better at estimating their scores in the second test.". Got that? With no information on actual math scores, what does this mean?? Suppose, for example, women scored 75% on the test and then estimated they scored about 75% whereas men scored 80% and estimated they scored 85%. Then the women are more accurate at gauging their performance, but since their performance is worse, how would that debunk the claim that men are better than women at math. And to make things worse, the research said "After receiving feedback about how well they did in the first study, men were then better at estimating their scores in the second test.". I'm not sure how this "research" proves anything. It only seems to show that, without feedback on performance, women are better at appraising their performance than men. But with feedback on performance (which is what happens in the real world as students get feedback on each test throughout the semester) men are better at appraising their performance. But none of this has to do with mathematical expertise. • The PC climate claims another high school teacher. Reason.com has the report, "An Illinois high school teacher was fired after stepping on the American flag to prove a point about free speech. The teacher, Jordan Parmenter, had been using a flag as a pointer during class on May 15. At least one student accused him of being disrespectful toward the national symbol, so Parmenter dropped the flag on the ground and stomped on it, according to WGNTV.com. Word quickly spread, and soon enough, demonstrators appeared outside Martinsville Junior-Senior High School. Parmenter wrote a letter of apology, but the school board voted 6-0 to fire him....The school board had a golden opportunity to show kids that honoring the values the flag represents is more important than honoring the flag itself. Instead, they imparted a different lesson: that no act of defiance goes unpunished by the government. Perhaps that’s an important lesson as well.". Beware the angry mob. • The PC climate claims a college teacher as well. The Advocate has the story of an LSU professor, Teresa Buchanan, fired for using salty language. The teacher is fighting back with a lawsuit. From the article, "She said the university is trying to dictate how she teaches and in the process is impinging on her academic freedom. “The occasional use of profanity is not sexual harassment,” Buchanan said. “Nor is the occasional frank discussion of issues related to sexuality, particularly when done in the context of teaching specific issues related to sexuality.” LSU spokesman Ernie Ballard declined comment Friday on Buchanan’s dismissal, saying it’s a personnel matter and involves possible litigation. Buchanan was fired even though a committee of five faculty members that presided over an 11-hour dismissal review hearing held on March 9 recommended that she keep her job. While the committee found that her adult language and humor violated university policies that protect students and employees from sexual harassment, it found no evidence Buchanan’s comments were “systematically directed at any individual.” The committee recommended she be censured and agree to quit using “potentially offensive language and jokes” that some found offensive." # Sagetex: Definite Integrals I've added two definite integrals to the Sagetex: Integrals page. The first problem creates two random parabolas in different directions and the area between the two curves must be calculated. This requires them to find the intersection points as well to set up the integral properly. The second integral gives a random exponential in e along with (random) endpoints of integration. Here are some issues that caught my eye this past week: • USA Today reports "Texas is decriminalizing students' truancy": "Gov. Greg Abbott has signed into law a measure to decriminalize unexcused absences and require school districts to implement preventive measures. It will take effect Sept. 1. Reform advocates say the threat of a heavy fine — up to$500 plus court costs — and a criminal record wasn't keeping children in school and was sending those who couldn't pay into a criminal justice system spiral. Under the old law, students as young as 12 could be ordered to court for three unexcused absences in four weeks. Schools were required to file a misdemeanor failure to attend school charge against students with more than 10 unexcused absences in six months. And unpaid fines landed some students behind bars when they turned 17."
• There's a wrinkle in a story from my last post. A teacher who was reported to have been removed from his position because he read from Mark Twain was actually removed for making an inappropriate joke (relating to a Mark Twain passage). LA Times has the details: "In his first interview since he was pulled from his fifth-grade class, Esquith told The Times on Monday that controversy stemmed from a joke he made in the classroom. He said he quipped with students that if he could not raise enough money for the annual Shakespearean play, they would all have to perform their parts naked like the king in Mark Twain's "The Adventures of Huckleberry Finn." After another teacher complained, he said he explained the context of the joke to his principal at Hobart Boulevard Elementary. The principal, he said, told him he had nothing to worry about. Nonetheless, Esquith was removed from the classroom in April."
• EducationWeek reports on the L.A. Unified budget has reduced the spending on police. This was a victory for The Dignity in Schools Campaign which, "..demanded that the school district, which is the second largest in the country, redirect $13.1 million in funds it had planned to spend on policing practices during the 2015-16 school year into jobs and programs aimed at improving school climate. (The district is still budgeting about$54 million for school police from other parts of its budget.) Though the district school board adopted the revised budget, campaign organizers don't yet know how much of the redirected money will go toward their specific funding recommendations, which include using $8 million for restorative justice measures like technical assistance and staff training, as well as$5 million for hiring prevention and intervention staff in alternative schools to create counselor-student ratios of 1 to 50. Such investments have been proven to positively transform school climate, whereas school-based policing has not, said Ruth Cusick, an education rights attorney at Public Counsel."
• Huffington Post's piece "Meet the 63rd Black Woman in American History with a Physics Ph.D." provides a glimpse into "the challenges faced by marginalized communities in science".
• Put this on your radar: Reason.com has an update on a case making its way through the legal system: "A little over a year ago, a group of nine California students with the help of the activist group Students Matter won an amazing victory in California Superior Court in the case of Vergara v. California.As I reported at the time:

Judge Rolf M. Treu reasoned that the challenged teacher rules—regarding permanent employment status, dismissal procedures, and a "last in first out" rule for layoffs—do indeed damage California children's constitutional right (on the state level) to an education. He wrote that the challenged statutes "cause the potential and/or unreasonable exposure of grossly ineffective teachers to all California students" and "to minority and/or low income students in particular, in violation of the equal protection clause of the California constitution."

Naturally, the losers appealed, and Judge Treu stayed actual enforcement of his ruling pending appeal. Today, the Students Matter side filed their brief in the appeal process in the Court of Appeal for California, 2nd appellate district....In a press conference call this morning announcing the brief, lawyers on the Students Matter side say they still need to wait for the teachers side to file its response brief and then await an actual court date. Once the hearings are over, though, a decision must come within 90 days, but that could still be a very long time away--more's the pity for California public school students.". There's a decent video that's posted on the page.

• The Norway Chess Tournament 2015 ended with victory for Topalov. The tournament was marred by some blunders and a short draw between Anand and Topalov in the final round. But most newsworthy is what Chessbase reports here,"...this is easily the worst tournament ever played by Carlsen after obtaining his GM strength."
• The 43rd Sparkassen Chess Meeting Dortmund 2015 has begun; it features players such as Kramnik, So, Hou Yifan, Naiditsch.
• American chess lost an icon recently. The NY Times has a piece on Walter Browne who passed away in Las Vegas at the age of 66.
• Michael Krieger posts on "Salt “Black Markets” Emerge in Indiana School System as Students Seek to Avoid Bland Michelle Obama Lunches"

# Problem: "Puzzle math"

I've added the following puzzle to the Problems page: For the 8 squares below (corners aren't included) squares are adjacent up/down/left/right/diagonally. Fill in each square with a number from 1 through 8 (one time each) so that adjacent squares don't contain consecutive integers. I found the problem here. You can reason it out logically but I ended up using graph theory to get the answer.

Lots of stories this week:

• A MUST READ article by John Bohannon: "I Fooled Millions Into Thinking Chocolate Helps Weight Loss. Here's How". From the article, "It was, in fact, a fairly typical study for the field of diet research. Which is to say: It was terrible science. The results are meaningless, and the health claims that the media blasted out to millions of people around the world are utterly unfounded." and the key to generate bad conclusions from good data is "...If you measure a large number of things about a small number of people, you are almost guaranteed to get a “statistically significant” result. Our study included 18 different measurements—weight, cholesterol, sodium, blood protein levels, sleep quality, well-being, etc.—from 15 people. (One subject was dropped.) That study design is a recipe for false positives.". And the p-value is instrumental in the deception, "It’s called p-hacking—fiddling with your experimental design and data to push p under 0.05—and it’s a big problem. Most scientists are honest and do it unconsciously. They get negative results, convince themselves they goofed, and repeat the experiment until it “works”. Or they drop “outlier” data points.". This article is a great resource if you teach statistics.
• Magnus Carlsen won a 3 board blindfold (with clock) exhibition. The video is posted on Chessbase.
• Caruana and Nakamura earned their place in the upcoming Candidates tournament to determine the next challenger for the World Chess Championship by taking the top two places at Khanty-Mansiysk 2015.
• The editor in chief of the Lancet, one of the world's most prestigious medical journals has stirred up some controversy. "Dr. Horton recently published a statement declaring that a lot of published research is in fact unreliable at best, if not completely false. “The case against science is straightforward: much of the scientific literature, perhaps half, may simply be untrue. Afflicted by studies with small sample sizes, tiny effects, invalid exploratory analyses, and flagrant conflicts of interest, together with an obsession for pursuing fashionable trends of dubious importance, science has taken a turn towards darkness.. Yes, statistics isn't really math.
• Forbes has one of those REALLY annoying posts by someone who really lacks basic knowledge about what they're writing about--something all too common in mainstream media where even robots now generate worthless content. Although posed as a question "Should We Stop Teaching Calculus in High School?" the author is clearly saying "yes". "The list of high school math courses in the U.S. hasn’t changed for decades. My daughters are taking the same courses I took long ago: algebra, geometry, trigonometry, and calculus. These are all fine subjects, but they don’t serve the needs of the 21st century.". But the author goes on to say, "...the vast majority will never use calculus again. And those who do need it – future engineers, physicists, and the like – can take it in college.". So the courses do serve the needs of the 21st century. The author makes the point that we are awash in data today so the author asks, "What math courses do young people really need? Two subjects are head-smackingly obvious: computer science and statistics.". Huh? Who believes computer science is math? And anyone reading this blog knows (e.g. see here and here) that statistics isn't math either. Yes, theoretical stats is basically analysis but the statistics he's talking about (confidence intervals, p-values, etc) isn't. Whether (see the links) you want to look at schools having a "department of math and statistics", or that bigger schools have a separate statistics department, or that AMSTAT news says statistics isn't a subfield of math, or as is mentioned in the second link that statistics books and teachers have been presenting p-values incorrectly. If you believe stats is math then please explain what other branch of math teaches you the wrong way to do something as has been done with p-values? There is a different reasoning process for stats. But back to the article. With respect to statistics, "Most high schools don’t offer either one. In the few schools that do, they are usually electives that only a few students take.". Let's mention that statistics is covered in Common Core. The Common Core Standards are posted here and this government site notes, "The recently adopted Common Core State Standards for Mathematics (CCSSM) contain a large amount of statistics in the middle and high school grades and some at the elementary school level.". Not being a math teacher, this author is unaware how much things have changed: more statistics, mathematical proofs in geometry have been largely removed (I even had to teach probability(!) in my geometry classes), and 4 ways to subtract (which has confounded parents) are some noticeable changes. Assuming that because his daughters are taking the same courses he took decades ago means there hasn't been any change in content is, at best, sloppy journalism. To be clear, let me agree that computer courses in high school would be great. Python is such a natural choice that could be useful--but you don't sacrifice core math classes for that; you eliminate or consolidate less relevant courses to make room for it. The removal of most proof content from geometry is tragic because proofs are the essence of mathematics. As Alfred Renyi said, "A mathematician is a device for turning coffee into theorems". The author opting for computer science and stats as math classes while neglecting discrete math is, I suspect, based in the ignorance of not knowing discrete math is the math of computer science. The author continues, "Convincing schools to give up calculus won’t be easy. I imagine that most math educators will scream in protest at the mere suggestion, in fact. In their never-ending competition to look good on a blizzard of standardized tests, schools push students to accelerate in math starting in elementary school, and they offer calculus as early as the tenth grade. This doesn’t serve students well: the vast majority will never use calculus again.". There's some truth in here but since he hasn't taught high school he can't properly interpret what's happening. High schools get awarded numerical scores according to a formula for "performance" (which often get translated into star ratings for the school). Admin look at how they can increase scores (to make their performance look better). More students taking AP exams means a higher score is given, regardless of how poorly the students do. That's why school admins get teachers to encourage students to sign up for AP classes and that's why schools often pay for the student to take the AP exam--it's an easy way to raise the school's score. The consequence is that it's commonplace for students who struggle with fractions to be taking AP Calculus. But no matter, just require a graphing calculator to give students a chance. The child feels smart, the parents feel proud, admin performance improves, and some business makes a lot of money selling expensive calculators when you can buy a laptop computer for $200. The only problem is the child still has poor math skills that they'd be put to shame by a typical student from another country at a lower grade level that has a fraction of the resources but has parents making sure kids learn multiplication tables and basics and not giving them a calculator to use as a crutch at lower levels. But the issue isn't about producing quality in the US, so it's no surprise we never get it. They're looking to maximize performance under the rules they've been given so wasting taxpayer money on improving the school's score gets the admin credit for improving school quality even though no real quality has taken place. Same thing with attendance. Some schools have poor scores due in part to poor attendance. Since that poor attendance is, in cases, predictable (before Christmas break) schools often have incentives (such as a drawing for a free computer) for students who attend. This expenditure of taxpayer money has nothing to do with quality. There's so much to criticize in this article but you get the idea. His argument that "here’s a simple fix: get rid of high school calculus to make way for computer programming and statistics" is nonsense. • The controversy of why women don't perform as well as men in chess, which I first raised here, continues with a new study (authored in part by women). The data itself is interesting: "“Hard” sciences such as physics and statistics on average have a larger gender gap than social sciences and humanities – no surprise there. However, this is only part of the story. According to 2013 data from the National Science Foundation in the United States, there is large variation within each category: whereas women earned only 19% of PhDs in physics and 18% in computer science, they earned no less than 54% of PhDs in molecular biology – amounting to gender parity. Within the humanities and social sciences, those numbers ranged from 78% in art history and 72% in psychology to a dismal 27% in philosophy and 28% in music theory and composition.". I think the conclusions drawn off the data are suspect, to put it lightly. "The key claims of Leslie and Cimpian’s paper are: 1. that fields vary in the degree to which its practitioners believe that innate ability (“genius” or “brilliance”) is required for success; and 2. that society often promotes the notion that men have greater innate abilities than women.". If women were doing well in those fields then I don't see how they would be "bluffed" out of continuing. It makes more sense to me that they weren't doing well, didn't like it, found their "passion" somewhere else (as jobs opened up in other more desirable fields), etc.. The fact is that women used to hold a large percentage of computer science job. The claim that women would somehow give up pursuing a job in these lucrative white collar fields because society has told them men have greater innate abilities seems demeaning to women and contrary to the general rise of women in the workforce over the past 40 years. Women have faced harassment in many areas and haven't quit. More believable to me is that the radical change in the field took computer science out of their comfort zone: computer science today is so much different than back in the 80's. Note also that the study separates Statistics from math (as it should) and the resulting percentages for the two are quite different. • The chronotope blog comments on the (included) recent John Oliver video on student test in high schools. Make sure to check out the video! • Ravens guard John Urschel analyzes the extra point rule change in football. • The Hindu notes on the passing of chess legend Anand's mother. Grandmaster R.B. Ramesh has a fitting quote, "Without Anand it’s tough to imagine Indian chess. Without his mother, it’s tough to imagine Anand”. • CinemaBlend posts on a Tobey Maguire starring as former World Chess Champion Bobby Fischer in "Pawn Sacrific". Check out the movie trailer. • Hundreds of SATs go missing. If they aren't found soon the teenagers will have to take them again. Let's hope nobody has their college plans derailed. • GreenBayPressGazette.com reports "Wisconsin may be the first state in the country to certify teachers who don't have bachelor's degrees under a provision put in the state budget....Under the change, anyone with relevant experience could be licensed to teach non-core academic subjects in grades six through 12. They would not need a bachelor's degree and they could even be a high school dropout.". This doesn't sound like a good idea. # Understanding "The Test" The post Understanding the Prediction explained the mathematics behind Richard Wiseman's brilliant magic trick that's posted on his Quirkology site. It's a great way to introduce graph theory to your class. Richard Wiseman's video "The Test" is a similar type of magic trick that can be explained using digraphs and, once again, is a great way to introduce a class to digraphs. I've added an explanation of "The Test" to the Other page. Here are some thngs that caught my eye last week: • Discover blog has an article, "The Purpose of Harvard is not to Educate People". From the article, "Don’t believe me? Here is the test: when was the last time Harvard made a senior tenure offer to someone because they were a world-class educator, rather than a world-class researcher? Not only is the answer “never,” the question itself is somewhat laughable.". • Patch.com reports on a "...former interim director of special services for the Brick Township School District, failed to reveal a 1990 conviction on heroin and cocaine charges on his job application with the district, the Ocean County Prosecutor’s Office has confirmed. Morgan, 65, was charged Thursday along with Brick Schools Superintendent Walter Uszenski and Uszenski’s daughter, Jacqueline Halsey, in a scheme that supplied Halsey with full-time day care for her preschool child paid for by the Brick Township schools, with official misconduct and theft by deception, said Al Della Fave, spokesman for the Ocean County Prosecutor’s Office.....According to a 1989 report in the New York Times, Morgan was arrested and charged with selling cocaine on five occasions, in amounts ranging from a half-ounce to more than 3 ounces, to undercover detectives assigned to the Brooklyn District Attorney’s investigation unit.Morgan -- who taught English to 9th and 10th graders in a special education program at Canarsie High School -- also was charged with possession of cocaine, as well as with conspiracy to sell heroin, in which he is said to have agreed to travel to Thailand to buy heroin for undercover agents posing as drug dealers. The heroin was supposed to be brought into this country concealed in disposable diapers, the authorities said.Morgan later was convicted of felony drug charges, though a follow-up article in the New York Times does not specify the exact counts. Morgan, who had worked for the school for 20 years, was fired but later won a civil case against the schools over the firing, despite his conviction.". So you've got a school official arrested and charged with selling drugs multiple instances (and the details are such public knowledge that it made the NY Times) and was teaching kids. So the question becomes "Was a background check ever done?" If it was, "How did it miss such well known information?". Also ask yourself what it says when Morgan was hired at the "request and recommendation" of someone still working in the educational system. • (Stephen) "Colbert Fune$800K in Grants for SC teachers".
• Huffington Post has an interview with 2010 Fields Medal winner Cedric Villani.

# Math Models and a Birthday Problem resource

Mathematical models is one of those ideas that students should know, but don't. Even after they've studied them. Ask your class the following: "A coin is tossed. What's the probability that it lands as heads?".

Most students have are quick to say 1/2 but that's wrong--the correct answer is we don't know the probability for any particular coin. We could use experimental probability to estimate it but even that's an approximate answer. The probability of heads that the students think is reality is actually a based on a mathematical model with a "fair coin". Mathematical models are approximations of reality. Unfortunately most students who have had some probability don't know coin flipping is based on a model and think the number of outcomes determines the probability (not realizing the equally likely assumption is an assumption which could be false). Some of these problems invariably trace back to the teachers who have taught them incorrectly.

Mathematician William Feller was a well known expert in probability who wrote a classic book An Introduction to Probability Theory and Its Applications in which you can find (by click on "Look Inside") the following quote on page 19: "As a matter of fact, whenever refined statistical measures have been used to check on actual coin tossing, the result has invariably been that head and tail are not equally likely. And yet we stick to our model of an "ideal" coin even no good coins exist. We preserve the model not merely for its logical simplicity, but essentially for its usefulness and its applicability.".

The coin flipping model has two assumptions built into it:

1. There are two outcomes (heads and tails)
2. The two outcomes are equally likely.

Since its possible for coins to balance on their sides (nickels and quarters more easily than a dime), it's possible (though admittedly remote) for a coin to land on its side. And it seems like most people have had experiences of a dropped coin which lands on its side only to roll away. Heck, it's even happened during a football game. This paper estimates the odds of an American nickel landing on its edge as 1/6000. Tested.com says that a study (broken link) indicates "...the "randomness" of a toss is actually weighted ever so slightly towards the side of the coin that's facing upwards when a flip begins....The paper, written by statistics and math professors from Stanford and UC Santa Cruz, also points out that a perfect coin toss can reproduce the same result 100 percent of the time.".

So coin tossing is a simple example of a mathematical model that students should learn. Another model is the famous "Birthday Problem". As I mentioned in an earlier post:

Answering the Birthday Problem involves creating a mathematical model. The model rests on two assumptions that aren't true and should be discussed with the class:

• there are 365 days in a year (Feb 29th is ignored to simplify the model)
• birthdays are equally likely to be on any given day (Also false. This varies from country to country; in the US birthdays are more towards the middle of the year. Count back 9 months and you've got cold weather. Nothing random there.)

I'm revisiting this post because I ran across a chart showing the distribution of birthdays referred to in the second point above. The Gizmodo post "How Common is Your Birthday" says, "The visualization used data from 1973 to 1999 to chart popular birthdays and figured out when the most popular time to pop out babies were."  So now you have a source to back up my claim and a nice chart to use in the classroom.

If you teach the Birthday Problem then you should get a copy of the chart or bookmark the page above.

Here are some stories that caught my attention this week:

• In an earlier post I pointed out the case of a high school student accused of stealing a backpack who was in prison for almost 3 years because he had been accused of stealing a backpack and would not plead guilty. The charges were eventually dropped and, because he was able to specify the specific dates on two occasions when he was abused, video has surfaced of these events. Democracy Now! has  "Explosive video obtained by The New Yorker depicts extreme violence inside New York City’s Rikers Island jail complex." and interviews New Yorker staff writer Jennifer Gonnerman, who has reported on the issue.
• NY Post reports on, "A copy of the state’s English Language Arts test that students took last week was leaked online Wednesday in an apparent act of sabotage by anti-testing activists......“This is a political act and it will be interesting to see whether [test-creation company] Pearson or the state Department of Education understands it as that or goes after them for civil or criminal liability,” said Brooklyn College education Professor David Bloomfield, who called the post an act of “civil disobedience.”"
• The Washington Post covers the widespread resistance of New York to Common Core testing: ,"Newsday has translated raw numbers into percentages, estimating that over 40 percent of all Long Island 3-8 students refused to take last week’s ELA Common Core state tests. Numbers in some districts reached well over 70 percent, with at least one district exceeding 80 percent. It appears that no more thanseven of the 124 districts on the island will meet the testing threshold of 95 percent. And that is before this week’s math tests, when opt-out numbers are expected to climb, as they did last year...It seems clear that the final 2015 tally will well exceed 200,000 students. New York State will likely not make the minimum 95 percent federal requirement for testing.."
• Shamkir 2015 has ended in victory for Magnus Carlsen. Chessbase has the report here. Anand took second with Caruana and So tied for third. Anand's second place performance has put him at number 2 in the world with a 2803.7 Live Chess Rating. I never thought he'd get there again. At 45 with his peak years ago and he's still in the hunt: incredible!
• My sympathies go out to Nigel Short. Not for his 1.5 - 8.5 versus Kasparov (Chessbase only has part 1 out here) but for the savage "beating" the "PC-police" are inflicting on him. The brouhaha, discussed here, starts with Nigel's comment, "“Men and women's brains are hard-wired very differently, so why should they function in the same way? I don't have the slightest problem in acknowledging that my wife possesses a much higher degree of emotional intelligence than I do. Likewise, she doesn't feel embarrassed in asking me to manoeuvre the car out of our narrow garage. One is not better than the other, we just have different skills. It would be wonderful to see more girls playing chess, and at a higher level, but rather than fretting about inequality, perhaps we should just gracefully accept it as a fact.”" which became sensationalized with The Telegraph's article, " Nigel Short: 'Girls just don’t have the brains to play chess".  You can even see Nigel defending his common sense position on Sky News and being given the absurd argument that he's wrong because J Polgar has a plus record against him. Given that there are distinct differences in the hardwiring  of male versus female brains (not to mention the differences between males and between females) and the study Short points out, it's difficult to believe his comments have become so controversial. Sorry Nigel! You deserve better.
• Huffington Post has an article about Ramanujan whose birthday was April 26th. The brilliant mathematician who was most definitely wired differently than others: "The biggest question is how an untrained teenager, and later young adult who repeatedly flunked out of college in his native south India (generally the area of Madras, today's Chennai), was able to obtain--all on his own--mathematical expressions that later would take some of the world's leading mathematicians years and even decades to ascertain and prove.".
• I've deleted the CTAN Mail Archive link and replaced it with the updates gmane.org. They were notifying about many changes in LaTeX that never made it to the other link. The link is CTAN announcements, located on the sidebar.

# Sagetex: Derivative 5/6 without the Chain Rule

I've added two more problems (Problems 5 and 6) to the Sagetex: Derivatives page. Problem 5 requires the student to calculate the derivative of a product of polynomials using the Product Rule. Problem 6 looks like a Quotient Rule but the derivative is more easily calculated by carrying out division before taking the derivative.

Here are some stories which caught my eye:

• L.A. Times reports 9 students at Venice High School were arrested and 14 accused, "...in connection with a series of sex crimes that began more than a year ago and involved at least two female classmates. All but one of the arrests were on campus; authorities were attempting to locate five other students." No doubt this story will continue to evolve.
• Mathematically challenged: it could cost you your marriage. The New York Post reports "Bride Walks out on groom after he botches simple math problem". In a country of arranged marriages, it was a sign to the bride she had been "misled" about his education. "“The groom’s family kept us in the dark about his poor education,” said Mohar Singh, the bride’s father. “Even a first-grader can answer this.”"
• LaTeX Community has a nice piece on "Why I like the TikZ Math library".
• Welcome to school today: a student intentionally runs into a school security officer "..in what has been described as an attempt to ‘chest bump’ the officer...". This knocks the officer to the ground. The officer gets up, grabs the boy and according to Infowars, which has the video posted: "The video shows Hardin lifting the 13-year-old child into the air with a chokehold. The child kicks and flails around before becoming limp and lifeless. Hardin then drops the student to the ground, resulting in what a doctor has described as “an injury to the brain” due to loss of blood flow.". Be aware of the graphic nature before you watch this.
• It's obvious to "older" people that school quality has dropped dramatically over the years (think Calculus students who struggle with fractions) but at least the students of today are tech-savy, right? According to this Fortune article, no. "American Millennials are among the world's least skilled" reports that a new study shows, "...Millennials in the U.S. fall short when it comes to the skills employers want most: literacy (including the ability to follow simple instructions), practical math, and — hold on to your hat — a category called “problem-solving in technology-rich environments.” Not only do Gen Y Americans lag far behind their overseas peers by every measure, but they even score lower than other age groups of Americans."

# 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.
• CBS MoneyWatch has an article on "Teacher Pensions: the Math Adds Up to a Crisis". From the article, "Teacher pension plans across the country are staggering from a half-trillion dollars in debt. Put in perspective, that's more than \$10,000 worth of debt for every student in the nation's primary and secondary schools.".
• The Houston Chronicle tells us: "Teachers would be able to use deadly force against students, and would be safe from prosecution, under legislation filed last week in the state House. by Rep. Dan Flynn, R-Van, would allow educators to use force or deadly force  if they feel they need to protect themselves against a student or anyone else on school grounds.". The article references, "The bill was filed just days before a video of a New Jersey physics teacher being body slammed by his 9th grade student went viral. In the video, it’s clear the teacher is avoiding fighting back or touching the student." Check out the video of the teacher being slammed; he's remarkably calm during the attack. But is giving a teacher the right to use deadly force a solution?