Problems: discrete math

I had to change the topic of my post this week once I read the article "Math is my religion" on the website. The author, Brian Gentry, is a high school student and self described "math geek" who has been taking college level classes. He writes, "But my interest in math has allowed me to see the holes in our math curriculum. ... Rather than teaching kids integration, we should teach them the math that is most applicable to their life goals.". Two important courses that Mr Gentry thinks are particularly useful are geometry and discrete math. With respect to geometry, he writes "A fundamental piece of our geometry class is proofs, and the logic taught through proof-writing is used not only in math, but also in journalism, history and every other field that requires the construction of a logical argument.".  To which I say, "Amen"! With respect to discrete math, (singling out number theory) he writes "I was taught that I need to cite each theorem I use in my proofs and justify each application, just like a history major has to cite quotes and explain how each quote is relevant. I can say without a doubt that this class, if implemented in a high school curriculum, would be beneficial to everyone who took it. ". And that deserves a "Hallaleujah!". Brian goes on to quote a teacher, Barry Garelick, about the "...decreasing number of proofs in geometry textbooks over the decades. He contends that proofs are integral to geometry...In Garelick’s mind, proof-based courses teach students how to construct logical arguments, which I argue is not only central in mathematics but also in a variety of other fields.". To which I say, "Testify!".

As you can tell, I agree with Mr Gentry but, unfortunately, Brian knows more than the experts who are moving us in the wrong direction. It's not enough to have a good idea in the public school system, you've got to get some "experts" on board to change policy and then the devil is in the details of how they implement it mess it up. You see, Brian has the mathematical knowledge that so many " experts" are missing. He also has a sincere desire to improve the situation--but no real power to do anything. In our centrally planned model with high paid "experts" who don't seem to have either. It's no surprise education flounders decade after decade after decade. When you see well intentioned people get "Zuckered" out of 100 million dollars by experts and when you see in state after state that "experts" have set the bar for mathematical knowledge needed for a math teachers by a a multiple choice test which requires a calculator--at the same time they turn away people with STEM degrees-- you do tend to question motives. They build failure into the system and look for superficial ways to "improve" on some contrived school rating (such as paying for students to take AP exams).

The fact is I've never talked to anyone with a graduate degree in math who thinks ripping proofs out of the curriculum is a good idea but that's where the current mathematical curriculum has taken us. The proofs that "us old folks" associate with geometry are gone and/or watered down. Is that because proofs don't prepare students for higher level math?  Of course not---but if students can't master proofs and much of the proof content is replaced and what remains is watered down test scores might rise.

There is no reason, based on math, to remove proofs--proofs are the essence of mathematics--and getting students some foundation in proofs would help better prepare them for college. So today's curriculum prepares a student for math less than before in that key area. Clearly the educational central planners have no clue when it comes to math. But even if they did many teachers have no idea what that discrete math means and that presents a huge obstacle to implementing Mr Gentry's excellent idea: remember one third of the high school math teachers don't have a degree in mathematics so there's going to be far less who have taken and are qualified to teach discrete mathematics. And finding these people is at odds with the various Bull**** certification requirements that create an artificial teacher shortage in various states.  Take a look at the story below on California and Common Core to see the complete lack of planning and the resulting fiasco. Whose accountable for the mess? Nobody. Who pays the price? The kids. It's difficult for me to imagine anything more than centrally planned failure of implementation.

With that in mind I've posted an example of a discrete math problem which is more understandable/natural than "Two parallel lines are cut by a transversal...". The problem is posted on the Problems page. There are n \geq 2 people are at a party. Prove that there are two people who know the same number of people." Of course, there's a little explanation needed: Two people either both know each other or they don't. That is, it's impossible for A to know B but for B not to know A. Also assume that a people don't "know themselves". A very surprising result that can be proven using mathematical thinking/logic. It uses the Pigeonhole Principle and of course, you can relate it to graph theory, too.

Here are some stories that caught my eye this last week:

  • One of my favorite nontechnical math books is  "The Man Who Knew Infinity". A movie based on the book is finally out. IFLScience talks about Ramanujan and has a clip from the upcoming movie.
  • The TCEC Season 8 Superinal is about 60% done and Komodo leads Stockfish with 4 wins, 1 loss, and 65 draws. With so many games to play Stockfish has a mathematical chance to win but given the consistent nature of computer play; i.e., computers don't blunder or get tired/overconfident, this match is essentially over. Nevertheless, you can keep following the match here.
  • Kevin Knudson with an excellent article on Forbes: "I then casually mentioned that if you take the harmonic series and throw out the terms whose denominators contain a 9 then the resulting series converges...And, there’s nothing special about 9; you can toss out terms containing any particular digit. In fact, you can pick any finite string of digits, toss out the terms containing those, and the result converges. With that set-up, let’s talk about what all this means and how we can prove it..". Read the article to find out the details. If you teach AP Calculus, you really should take a look.
  • Student protests are happening at college campuses all over the country. The Chronicle of Higher Education mentions a bunch here. The Washington Post has an in depth piece on Yale and the student demands, "The students also are asking Salovey to remove Nicholas and Erika Christakis from their positions at the helm of Silliman College, one of Yale’s 12 undergraduate residential communities. The pair became the subject of students’ ire when Erika Christakis, the associate master and an early childhood educator, sent an e-mail to students encouraging them to view offensive Halloween costumes as a matter of free speech and free expression."
  • ZeroHedge looks into the demands from students at the Amherst College. There is a long list of demands but take a look at demand number five: " President Martin must issue a statement to the Amherst College community at large that states we do not tolerate the actions of student(s) who posted the “All Lives Matter” posters, and the “Free Speech” posters that stated that “in memoriam of the true victim of the Missouri Protests: Free Speech.” Also let the student body know that it was racially insensitive to the students of color on our college campus and beyond who are victim to racial harassment and death threats; alert them that Student Affairs may require them to go through the Disciplinary Process if a formal complaint is filed, and that they will be required to attend extensive training for racial and cultural competency." Did you get that? People who posted a flyer on how free speech is dead (picture posted) need to be disciplined and re-educated --- along with those who post "All Lives Matter"---if these intolerant zealots get their way.
  • NYDailyNews posts a disturbing and "chilling" video in the case of the student on trial for killing his math teacher.
  • has an all too typical  story of Common Core implementation problems. "During the five years since California adopted the Common Core State Standards for mathematics and English language arts, the search for high-quality textbooks and curriculum materials has been a sticking point, in some cases a major one, in effectively and speedily implementing the new standards....The root of the problem, argued Phil Daro, a principal author of the Common Core math standards, is that “districts tried to switch to the Common Core before there were any books aligned with them.”That, however, was not the fault of districts. The state adopted the Common Core in 2010, but the State Board of Education only  approved a recommended list of K-8 math textbooks and materials in January 2014 – and only did so two weeks ago for K-8 materials in English language arts. But focus on the fact that even though Common Core was known to be coming years in advance and that it is 5 years after it is adopted and they still don't have quality curriculum materials. How bad is the state DOE when teachers still don't have "the basics" under control after 5 years, especially when they had years of planning before Common Core was implemented? California's plight is going on in many states and it's a big reason why the educational system doesn't improve much. But with a new election around the corner don't be surprised if another educational model takes its place. Then more years to transition to implement another bad system. More money to spend designing tests,etc. Wash, rinse, repeat.
  • has a Walter E. Williams piece "Education Disaster" which looks at the latest Nation's Report Card, "When it comes to reading and math skills, just 34 percent and 33 percent, respectively, of U.S. eighth-grade students tested proficient or above — that is, performed at grade level or above. Recent test scores show poor achievement levels in other academic areas. Only 18 percent of eighth-graders are proficient in U.S. history. It’s 27 percent in geography and 23 percent in civics. The story is not much better when it comes to high schoolers. According to 2010 and 2013 NAEP test scores, only 38 percent of 12th-graders were proficient in reading. It was 26 percent in math, 12 percent in history, 20 percent in geography and 24 percent in civics." So failure is the norm. Moreover, Williams writes "Richard Vedder, emeritus professor of economics at Ohio University, argues that there has been a shocking decline in college academic standards. Grade inflation is rampant. A seminal study, “Academically Adrift,” by Richard Arum and Josipa Roksa, argues that very little improvement in critical reasoning skills occurs in college. Adult literacy is falling among college graduates. Large proportions of college graduates do not know simple facts, such as the half-century in which the Civil War occurred. Vedder says that at the college level, ideological conformity is increasingly valued over free expression and empirical inquiry."
  • EAGNews on the teacher arrested for running his own brothel, "McCrimmon was arrested when authorities shut down his Memphis nightclub, Walt’s Place, over the weekend. Undercover officers allegedly made eight separate prostitution transactions there, including deals organized by McCrimmon himself, before they raided the Parkway Village establishment Saturday, according to The Commercial Appeal. Police allege the club charged patrons a $20 “membership fee” for events that featured strippers and other activities, but did not have a compensated dance permit. Walt’s Place also served booze without a liquor license and provided a VIP room for $50 sex sessions, police said....McCrimmon has since resigned from his teaching position, My Fox Memphis reports.". Perhaps he'll be moving to another state? Be on the lookout...
  • I was shocked to see someone claimed to have solved the Riemann Zeta Hypothesis. Whose that? From where? What the? A quick search made it clear someone was full of BS. With no "reputable" site proclaiming the amazing story I had to wait to see how it played out. Now Quartz has an article explaining how he "fooled" the British media:  "Leading British media, including the BBC and the Daily Telegraph, ran the story of Enoch winning the award, but a little digging suggests they might have jumped the gun.". Very little digging, in fact. The article continues, "Enoch has an page where the “proof” of the solution to the Riemann Hypothesis has been uploaded—but that has also come in for criticism, as the proof is believed to have been plagiarized.". So bottom line is it doesn't take much to fool the press---and the Riemann Zeta Hypothesis is still open. The Aperiodical gets more in depth on the deception of what did and did not happen. Hey, at least he's not running a brothel.
  • EAGNews with reporting the lengthy and somewhat outrageous demands, including "A mandatory class for everyone, including staff and administrators, about the “historical racial violence of this University and town …”...Housing and bathrooms that are not segregated by gender.". LewRockwell hosts a smackdown piece by Fred Reed directed at bad universities like what we see at UNC: "In all likelihood you will waste these four years of your time and mine in this institution...during which you will take absurd courses of your own devising, courses having nothing to do with the purposes of education, of which you know nothing....When you graduate, a terrible shock awaits you. You will find that employers have no interest in your wearisome righteousness. They will not pay you for Victims’ Studies or  contemplation of grievances. They will not care about the high GPAs you got through grade inflation or sleeping with the professor. They will expect you to do your job, if there is a job for you to do."
  • With intolerance and free speech under assault at the universities, has a video clip from the documentary "Can We Take a Joke". I haven't seen the movie, but I'm guessing the answer is no."

Basics: Simplify Like Terms

SimplifyLikeTermsI've added another worksheet to The Basics page; this one is on "Simplfying Like Terms". In this process of trying to solve a problem in the last worksheet (where Sage turn multiplication into *) I did a search on how to fix the problem and found the answer on this website: latex() the expression. So with that problem solved, I've revised the previous Worksheet on evaluating expressions so that it prints without * for multiplication. If you had downloaded the earlier version you should get the updated version to replace it.

Here are some stories that caught my eye over the past week.

  • Okay, I HAVE to start with a real honest to goodness mathematical breakthrough. László Babai, a mathematician and computer scientist at the University of Chicago  appears to have proven a very important result. has the details, "In the "graph isomorphism problem," the challenge is to determine whether two graphs are really the same or not. Babai has found a new algorithm to solve that problem, as he announced today....For the previous best method, invented in 1983 by Eugene Luks, a mathematician and computer scientist at the University of Oregon in Eugene, the number of steps grows almost exponentially with the number of nodes in the networks to be compared. In Babai's algorithm, the number of steps grows only slightly faster than polynomial time.... If it holds up, the new algorithm simply proves that the tough cases that stymie the current algorithms can also be solved efficiently, "
  • A MUST READ article. In an earlier post I mentioned the problems Kentucky was having with Common Core. The Federalist looks into the details. Some key passages, "In connection with federal Race to the Top grant applications in 2010 and No Child Left Behind waivers in 2011, states had to demonstrate that their institutions of higher education (IHEs) would “exempt from remedial courses and place into credit-bearing college courses” students who attained a certain score on Common Core-aligned assessments.". So the "Common Core is just a bunch of standards" trope that has become the first line of defense is shown to be false again. Back to the article, "So what happens when those unprepared students matriculate at a college that has already agreed to place them in courses that count towards graduation, without remediation? Exactly what is now happening in Kentucky... students who formerly would have gone through remediation are now to be thrown into credit-bearing courses. But since such students obviously won’t be ready for real college work, the courses will be designated “co-requisite”—meaning lagging students will receive extra help of some sort so they can catch up...By and large, professors weren’t consulted before their colleges and universities signed onto the Common Core scheme. They are only now beginning to understand that Common Core will result in hordes of unprepared students showing up in their freshman classes, and that the professors will be expected to relax or suspend course quality to hide the problem...Common Core’s promise of “college readiness” means nothing if the definition is set not by colleges themselves but rather by the standards-writers.".
  • The TCEC Superfinal is nearing the halfway mark. It's 3 wins for Komodo versus 1 win for Stockfish. There are 36 drawn games. You can follow the matches here.
  • A connection between quantum physics and \pi isn't as irrational as it sounds. has the incredible details, "In 1655 the English mathematician John Wallis published a book in which he derived a formula for pi as the product of an infinite series of ratios. Now researchers from the University of Rochester, in a surprise discovery, have found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.". Continue with Science20: "Friedmann did not set out to look for \pi nor for the Wallis formula. The discovery began in a quantum mechanics course taught by Carl Hagen, a professor of physics at the University of Rochester and one of the six physicists who predicted the existence of the Higgs boson. While the quantum calculations developed by Danish physicist Niels Bohr in the early 20th century give accurate values for the energy states of hydrogen, Hagen wanted his students to use an alternate method--called the variational principle--to approximate the value for the ground state of the hydrogen atom...Addressing the centuries-long gap between the 17th century Wallis formula, the 20th century quantum theory, and the decades that passed from that time to now, Doug Ravenel, a professor of mathematics at the University of Rochester, points out that Friedmann and Hagen used long-established concepts of their fields to arrive at their result, so even mathematicians and physicists who lived many decades ago would have been able to appreciate it."This is a beautiful connection between pi and quantum mechanics that could have been found 80 years ago, but was not discovered until now," said Ravenel, congratulating the two authors."
  • Inside Sources has the article I want to quote in one place, "Two multistate testing groups — the Smarter Balanced Assessment Consortium (SBAC) and the Partnership for Assessment of Readiness for College and Careers (PARCC) — received $360 million in taxpayer funds to create Common Core-compliant tests. But there are growing concerns over the program, such as the cost and classroom time consumed by state tests.". It's difficult to imagine that a committee couldn't come up with appropriate test questions in the years that led up to Common Core being implemented without spending one tenth the amount. Open source the problems, cut the costs, and spend the money on the kids, not on the corporations.
  • That hysterical, profanity laden tirade by the Yale student in my last post is just one example of the nonsense going on in public and private education. ZeroHedge has a response by a UNC-Wilmington law professor that's gone viral. Here are some excerpts, "Let’s get something straight right now. You have no right to be unoffended. You have a right to be offended with regularity. It is the price you pay for living in a free society. If you don’t understand that you are confused and dangerously so. In part, I blame your high school teachers for failing to teach you basic civics before you got your diploma. Most of you went to the public high schools, which are a disaster. Don’t tell me that offended you. I went to a public high school....Unbelievably, a student once complained to the Department chairwoman that my mention of God and a Creator was a violation of Separation of Church and State. Let me be as clear as I possibly can: If any of you actually think that my decision to paraphrase the Declaration of Independence in the course syllabus is unconstitutional then you suffer from severe intellectual hernia. Indeed, it takes hard work to become stupid enough to think the Declaration of Independence is unconstitutional. If you agree with the student who made that complaint then you are probably just an anti-religious zealot...".
  • with an article that says "According to statistics released by the U.S. Department of Education and published by NBC News, in the 2011-2012 school year, teachers called the cops on students a total of 31,961 times in the state of California alone, leading to 6,341 arrests. With 175 8-hour school days, that means a cop is called every 2.6 minutes. At one California school district, in particular, East Side Union High School District in San Jose, police were called on students 1,745 times during the 2011-2012 school year. This one school called the police on students more than 10 times a day."
  • The University of Kansas gives in to the PC extremists. Infowars notes "
    A governing body made up of students at the University of Kansas has voted to eliminate their use of gender specific pronouns, stating the terms pose “microaggressions” towards people who don’t fit traditional gender roles....The student government’s move to eliminate gender nouns comes after the National Science Foundation spent $125,000 at KU last year studying how adjectives could be perceived as racist or sexist."
  • The University of Missouri has had the spotlight turn on it. ZeroHedge has a piece that says, "For months, black student groups have complained of racial slurs and other slights on the overwhelmingly white (79% white and 8% black) flagship campus of the Missouri's four-college system. Today, amid a campus in open revolt and at least 30 black football players announcing that they would not play until the president was gone, AP reports that Mizzou President Tim Wolfe has resigned effective immediately urging students and faculty "to heal and start talking again to make the changes necessary." Protestors demanded that Wolfe "acknowledge his white male privilege," that he is immediately removed, and that the school adopt a mandatory racial-awareness program and hire more black faculty and staff.". So much is going on there that I have no idea about, but what has gotten my attention is the behavior of faculty in student demonstrations. The Daily Mail has captured a lot of the stupidity. "The civil rights protests at the University of Missouri took an unexpected turn on Monday, when a media teacher was caught on camera harassing journalists trying to cover the national story....Melissa Click, an assistant media professor at the university, is seen coming up to Mark Schierbecker, another photographer recording the exchange, to cover his camera and demand that he leave....She then comes back to the photographer and starts yelling in his face: 'You need to get out. You need to get out'. When he explains that 'this is public property' and he can stay there because 'it's owned by the university' Click takes on a mocking tone.'That's a really good one. I'm with the communications faculty and I really get that argument - but you need to go. You need to go!' Click says.'Who wants to help me get this reporter out of here?' she yells out. 'I need some muscle over here.'...Her aggression towards journalists is perhaps strange since just two days earlier she publicly reached out to the media on Facebook to cover the story." The website indicates she wasn't the only faculty involved: "Janna Basler, Mizzou's director of Greek life and leadership, tells Tai that he needs to "back off" and "go." She brushes against him, and Tai asks if she's employed by the Office of Greek Life. Basler responds, "My name is 1950." She also tells Tai that he is "impinging on what [its members] need right now, which is to be alone," that report states."As the students behind Basler begin to push forward, she makes physical contact with Tai, prompting him to object, to which she responds, "I don't have a choice." The students seem to decide that since he's not going to move, they're going to move forward as a human chain, physically pushing him back with their bodies. A student adds, "It's our right to walk forward."". But there is a consequence for adults who act like kids: TheManeater reports, "Multiple petitions have been created calling for the removal of two MU employees after a video surfaced documenting an incident on Monday, Nov. 9 in which they demanded that journalists leave the Concerned Student 1950 campsite. Assistant Director of Greek Life and Leadership Janna Basler and Assistant Professor of Mass Media Melissa Click can be seen in the video shouting at MU student and former Maneater staffer Tim Tai and other journalists."
  • New York Magazine looks at the Missouri situation from the PC angle: "The student protest at the University of Missouri began as a response to a serious problem — outbursts of vile racism on campus — and quickly devolved into an expression of a renewed left-wing hostility to freedom of expression...It is also undeniably true that outbursts of political correctness disproportionately take place in campus settings. In recent weeks, UCLA, Wesleyan, and Yale have seen left-wing student activism aimed at shutting down the expression of contrary viewpoints....As far as the students are concerned, they represent the cause of anti-racism, a fact that renders the need for debate irrelevant....People on the left need to stop evading the question of political correctness — by laughing it off as college goofs, or interrogating the motives of p.c. critics, or ignoring it — and make a decision on whether they agree with it.". Well said. Forbes has noted "Melissa Click has become of symbol of what many parents dread when they send their children off to college. From her bullying of students to her doctoral thesis on the whiteness of Martha Stewart and her classes in “visual literacy,” she crystallizes the view that tuition dollars are spent on nonsense, and sometimes worse....That an assistant professor of “mass media” in the department of communication was unaware of the instantaneous power of YouTube and social media is another reason for parents to wonder about the wisdom of spending their money on Click.". ZeroHedge posts when Melissa Click resigned.  Huffington Post looks at her apology, line by line, and calls it Bull****. Fox2Now reports that "...Janna Basler has been relieved from her duties as director of Greek Life & Leadership, pending a university investigation into her actions."
  • EAGNews on how "...the Buffalo school district spent $5,045,586 on the union’s insurance “cosmetic surgery rider” from July 2014 to June 2015....Just look at what the district has spent over the past four years on cosmetic procedures for teacher union members:* July 2014-June 2015: $5,045,586
    * July 2013-June 2014: $5,439,218
    * July 2012-June 2013: $5,221,293
    * July 2011-June 2012: $4,966,179That’s $20,672,276 that has been diverted out of the classroom for expenditures that have nothing to do with educating children."
  •  BoingBoing ask "What would happen if you mixed a math education tutoring site with a late night 900 number?". Click on the link do the reading and watch the video to find out. You'd be very naughty to watch the video at school; wait until you get home. But if you click on the link at the bottom you'll get to BostInno which says "Joking aside, Solve X 4 U is a legitimate homework help business. According to the startup’s website, they can help with problems pertaining to a wide range of difficult STEM subject areas, including statistics, accounting, economics and chemistry. Depending on how many customers they’re servicing, Solve X 4 U will come to your aid within about 24 hours, so plan your homework assignments accordingly."

The Basics: Evaluating Expressions


I've added a worksheet to The Basics page. This is programmed using \LaTeX and the sagetex package so that every time you run \LaTeX on the file it creates another randomized worksheet. You should get a free Sagemath Cloud account to run it (that link is on the sidebar as well). I used a "quick and dirty" approach by generating the same type of problem over and over using a for loop. The problems were built into the Sagetex Test Template created some time back and posted on the Handouts page and you'll need to change the teacher name to avoid any questions about who "Ima Putz" is. The extra wrinkle in the worksheet is the creation of answers. It's done on the fly as the problem is being created. The string outputP holds the typesetting of the Problems and the string outputA holds the typesetting of the Answers. Each time a problem is created, outputP is modified and then outputA has the answer appended to it. In this way you've got an answer key which should be correct if I didn't make any mistakes.

Here are some stories which caught my eye over the past week.

  • Who hasn't seen the video of the school officer "choking and slamming" a student who wouldn't hand in her cell phone after she was caught using it in class? Sputnik News has the original report here, followed by a report here on how the officer has a history of problems. The officer was fired, according to the latest report.
  • An article you have to read to believe. What happens if a private school does a terrible job educating students? They have to fix the problem or go out of business. After all, who will pay to get a miserable education? It has to be better or students will go to the "free" public school. The public school system is different--it lacks that accountability. The teachers aren't going to lose their jobs for poor performance and the students are stuck in the local public school. A typical public school will focus on getting all students to meet minimum standards. In an attempt to get some of the poorer students more proficient (it helps the school's score) material gets dumbed down. But in many cases the worst students are not interested in making an effort to improve and performance at all levels suffers. The lower bar means students need to do less to pass and so many do less--a negative cycle that's hard to break. If administration is poor then you've got the next story. Daily Caller reports, "Officials in at public school district about 50 miles north of San Francisco have found a new and exciting way to coddle kids. Under a new policy, students will be able to earn passing grades with scores of just 20 percent...Also, a school district-wide rule forces teachers to give every student a score of 50 percent even if they don’t complete a scintilla of homework or make an effort to mark down an answer on a test or quiz." While the article quotes some teachers who hate the system, there are education professionals some people who disagree, "Excited school district officials have also praised the generous grading system. The officials say students who are academically terrible won’t become demoralized as long as teachers are forced to hand out passing grades. And students who somehow manage a 19 percent or lower will remain encouraged.". From my own experience, students who "manage" a 19 percent do it effortlessly. That is, they put no effort into the course and their parents aren't there to keep them on track. They aren't demoralized; they just don't care. I think vocational educational is the only possible way of getting those students on track. Lowering the bar is a recipe for failure. Why do education professionals some people try to implement such an obviously stupid idea? It seems more about making the admin look good than about improving student performance. Yet in a public school this type of incompetence doesn't get fixed.
  • US News and World Report has a report on all the testing. "The drop in proficiency, which is a first-time occurrence in math since the test was first administered in 1990, comes after a series of years in which the country experienced gains in math and reading on NAEP. "
  • with a piece on Dr. Wendy Bradshaw. Although she " far from retirement age and she has PhD in education, she's leaving her profession because of standardized testing, which she explained in a Facebook post which has been shared more than 44,000 times since it was posted on Oct. 23.". The letter is an indictment of the current system and although she never mentions Common Core by name, it sounds like she doesn't like it. You decide, "Like many other teachers across the nation, I have become more and more disturbed by the misguided reforms taking place which are robbing my students of a developmentally appropriate education. Developmentally appropriate practice is the bedrock upon which early childhood education best practices are based, and has decades of empirical support behind it. However, the new reforms not only disregard this research, they are actively forcing teachers to engage in practices which are not only ineffective but actively harmful to child development and the learning process. I am absolutely willing to back up these statements with literature from the research base, but I doubt it will be asked for."
  • Science Daily with some educational research from Sweden, ""Most digital learning tools used in schools are unsatisfactory and only test the knowledge the pupils already have"..."However, digital learning tools can provide great educational benefits, as long as they do not become books on a screen, but use their digital advantages. This involves providing good feedback, showing that there are different ways of thinking to reach a goal, and presenting consequences that that cannot be demonstrated in a book," says Björn Sjödén."
  • A nice article in Quartz on Singapore teaching "productive failure" in math: "Students who are presented with unfamiliar concepts, asked to work through them, and then taught the solution significantly outperform those who are taught through formal instruction and problem-solving. The approach is both utterly intuitive—we learn from mistakes—and completely counter-intuitive: letting kids flail around with unfamiliar math concepts seems both inefficient and potentially damaging to their confidence..On procedural knowledge, or applying the formula, there was no difference between productive failure and direct instruction. But on conceptual understanding—understanding what it means and possessing the ability to adapt the information—the productive failure students dramatically outperform their direct instruction peers.."
  • The Baltimore Sun on Maryland's PARCC performance, "The first results of testing on the Partnership for Assessment of Readiness for College and Careers tests — introduced as part of sweeping educational changes begun several years ago — showed only 31 percent of students met the standard for Algebra I and 40 percent of students met the standard for 10th-grade English....Only a quarter of African-Americans, 7 percent of special education students and 23 percent of students who qualify for subsidized lunches met the benchmark in English. The worst performance was by students learning to speak English, many of them immigrants. Only 2.3 percent of those students were proficient."
  • The Atlantic with a nice article on the new changes to the SAT: "The college-admissions test is being restructured as an extension of the controversial public-school reading and writing standards.". With respect to the math section "The math test will consist of nearly 60 questions split between two sections, one that allows a calculator and one that doesn’t." but this passage sounds bad "“The current SAT asks questions where the material is remarkably simple, but students have to figure out what exactly they are asking for,”". What's wrong with harder math and more straightforward question?
  • NJ PARCC results were mentioned last post. But Newsworks reports there is a twist: "The proficiency rate for PARCC’s geometry test was only 24 percent. In Algebra II, the proficiency rate was 23 percent.....Under state law, students must pass the state’s exit exam to graduate. But under an improvised system put together by the administration for the first three years of the new testing, 12th-graders will have other options....When asked whether the state even has the capacity to handle an appeals process that may include tens of thousands of students, state Education Commissioner David Hespe said the necessary resources will be found: “We’re a big department; we’ll deploy whatever we need.”". Sounds like a big mess.
  • The Obama administration has changed its stance on testing in schools. Vox has this story "After seven years of trying to hold schools and teachers to higher standards — and testing to make sure they meet them — Obama said he's taken it too far. "When I look back on the great teachers who shaped my life, what I remember isn't the way they prepared me to take a standardized test," Obama said, saying he's concerned about "too much testing, and from teachers who feel so much pressure to teach to a test that it takes the joy out of teaching and learning."Beginning immediately, the Education Department is going to start directing states and districts to spend less time testing and to give fewer tests.". More details here, "They've promised to give grants to states to review the tests they're giving and determine which ones to cut. They plan to provide specific instructions on how states can use other federal money to study and cut tests..Most importantly, they're also backing down, at least slightly, on linking test scores to teachers' evaluations. ."


PEMDAS: notes


As I mentioned last post, I've been interacting with some middle school students. PEMDAS has been an important topic. I've put together some brief notes, shown above, that could form the basis of a lesson: warm up problems (which could then be the basis of discussion), teaching points, and then some more difficult problems involving PEMDAS, The solutions are at the end. The level of difficulty is more towards high school school students or an "honors" course at the middle school level. I've created a new  page called The Basics where you can download the PDF. I will post other introductory material there as needed. The Basics link is located on the sidebar, too.

Here are some stories that caught my eye this week:

  • Zero Hedge has commentary on a provocative piece in the The Economist magazine: "MICHAEL WANG, a young Californian, came second in his class of 1,002 students; his ACT score was 36, the maximum possible; he sang at Barack Obama’s inauguration; he got third place in a national piano contest; he was in the top 150 of a national maths competition; he was in several national debating-competition finals. But when it came to his university application he faced a serious disappointment for the first time in his glittering career. He was rejected by six of the seven Ivy League colleges to which he applied.". The article has the nice quote from, "...Supreme Court Chief Justice Roberts on this subject: “The way to stop discrimination on the basis of race is to stop discrimination on the basis of race.”". So true; but as private institutions they can do what they want. They are, however, undermining the prestige of their school. The loss of confidence in quality, combined with the cost of a US college education doesn't bode well for the future of US college education.
  • Payson Roundup wants to know why math scores of Arizona students have "plunged". According to the article, "Arizona students are worse in math than students in 33 other states according to data provided by the Nations Report Card. Arizona is not the only state that is struggling with proficiency in mathematics; the entire United States has poor mathematic proficiency scores. The National Assessment of Educational Process found that 7 out of 10 students in the U.S. scored at or above basic level in mathematics in 2013."
  • Huffington Post's Peter Greene delivers a verbal smack-down to a nonsensical Politico article. Nice job Mr. Greene!
  • The Atlantic has a piece on "The Anti-Free Speech Movement at UCLA". From the beginning of the article, "A half-century ago, student activists at the University of California clashed with administrators during the Berkeley Free Speech Movement, a series of events that would greatly expand free-speech rights of people at public colleges and universities.Today, activists at UCLA are demanding that administrators punish some of their fellow students for expressive behavior that is clearly protected by the First Amendment.". The article dishes out some blame, too: " The San Francisco Chronicle put it this way: “Regent Dick Blum said his wife, U.S. Sen. Dianne Feinstein, D-Calif., ‘is prepared to be critical of this university’ unless UC not only tackles anti-Jewish bigotry but also makes clear that perpetrators will be punished.” The lawyer Ken White wrote that “Blum threatened that his wife … would interfere and make trouble if the Regents didn’t commit to punish people for prohibited speech.”"
  • Listverse has "10 Things You Probably Didn't Know about Albert Einstein". What do you know about the Einstein refrigerator?
  • California has banned schools from using "Redskins" as a team name or mascot beginning Jan 1, 2017. "The new law will affect four California high schools in Merced, Calaveras, Tulare and Madera counties.". Does this mean they condone the "fighting Irish"? Perhaps that's the next lawsuit. Curiously enough, Governor Jerry Brown "...vetoed a separate measure that would have barred public properties from being named after individuals associated with the Confederacy."
  • Edsource has an article on poor California test scores, "In fact, only one-third of California students in grades 3-8 and grade 11 met the math standard – compared to 44 percent of students who met the standard in English language arts.". Did you get that? Failure is "the norm". Always remember that the test scores would be lower if it weren't for the many students who get a tutor to help them learn. And take a look at all the it's-no-big-deal talk and Common Core challenges talk. With respect to Common Core: "“Los Angeles started implementing Common Core three years ago,” Dorado said. “It takes time and a tremendous amount of work. In LAUSD, we’re talking 500 elementary schools alone.”Another challenge has been the shortage and quality of curriculum materials aligned with the standards. “Many teachers are in the implementation phase,” said the California Mathematics Council’s Vierra. “Many districts are still getting around to buying the curriculum (materials they need).”“A lot of teachers are cobbling together old materials with lessons they find online and material the district is providing,” she said. “Much of the math curriculum is still very fragmented.”". That's similar to my experience--despite knowing years in advance what was coming the school systems are still behind the curve years after Common Core was "implemented".

Sagetex: Indefinite Integrals 6/6b

Integral6I've added two more indefinite integrals to the Sagetex: Integrals page. The integrals are of the form \int e^{-\alpha x}\cos(\beta x)\,dx and \int e^{-\alpha x}\sin(\beta x)\,dx where \alpha, \beta are random (positive) integers.

Here are some issues that caught my eye over the last week:

  • has an interview with "...filmmaker Ted Balaker who is currently finishing up his latest documentary, "Can We Take A Joke?." The film, which features comedians Gilbert Gottfried, Jim Norton, Lisa Lampanelli, Adam Carolla, Karith Foster, and Penn Jillette, examines the role of comedy in our culture of constant outrage. "Comedians don't even have the freedom of conscience to just be neutral on something," Balaker told Reason TV's Nick Gillespie. "[They] have to affirm what the cool kids believe."". Finally someone taking on the PC zealots.
  • Inofwars has the local news on a Wisconsin school to randomly drug  test the students: "Given that it is actually unconstitutional to randomly drug test students, the school district is using a loophole to do so. Students taking part in extracurricular activities or students who park vehicles on school property will be subject to the random testing.“Participating in extracurriculars, um in public high schools is a privilege and it’s not a right, as well as parking on our school parking lot,” Dorschner explained.Tests will be conducted by randomly picking student identification numbers via computer every fortnight.Should a student test positive, or refuse to be tested, they will be barred from athletic involvement, mandated to attend counseling, and their parents will be alerted. The school says it will not expel any students or involve police."
  • on presidential candidate Carly Fiorina's criticism of Common Core. The beginning of its expected prominent place in election topics? She said, "Common Core may have started out as a set of standards, but what it’s turned into is a program that honestly is being overly influenced by companies that have something to gain, testing companies and textbook companies, and it’s becoming a set of standards, not on what a kid has to learn but instead on how a teacher has to teach and how a student should learn, and that kind of standardization is always going to drive achievement down, not up." I couldn't agree more.
  • A new pentagonal tiling has been discovered. See RedOrbit or check out what NPR says, "In other words: It's possible that that there are dozens — hundreds, thousands even — of these convex pentagon shapes waiting to be discovered. Up until last month, only 14 had been found, and for all anyone knew, that list could have been final. But last month, a cluster of computers that Von Derau was using to run though different shapes spit out an intriguing possibility...The three mathematicians had discovered the first new convex pentagon able to tile the plane in some 30 years. The scientists had become a part of a legendary history that dates to 1918, when the German mathematician Karl Reinhardt described the first five types of pentagons to be able to tile the plane."
  • again with a story about, "A Minnesota student who had to transfer high schools to avoid an expulsion for an incredibly short, wholly inoffensive Tweet can sue the district for violating his First and Fourteenth Amendment rights, a federal judge ruled. The student, Reid Sagehorn, first landed himself in trouble with Elk River School District administrators in January of 2014, according to Education Week. He was asked on an internet message board whether he had made out with a certain 28-year-old teacher at Rogers High School; he tweeted his two-word answer: “actually, yeah.” Sagehorn later claimed that he was joking."
  • with a good piece on Brazil's Artur Avilla, winner of a Field's Medal in mathematics.
  • Raven's guard, football player Dr. John Urschel decides to test his mathematical skills after receiving a concussion: "No word as to how Urschel performed on the math questions, but we're willing to bet pretty well, concussion or not."
  • The Sinquefield Cup, hailed as the highest rated chess tournament in history last year and won by Caruana in historic fashion, has the first round today. It will continue until September 3. Spotskeeda gives a preview. The rounds can be followed live here.


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: texts, audio, moving images, 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 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."
  • 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. 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 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: 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.
  • 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.".
  • 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. 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 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.