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.

Sagetex Integrals page


I've added a new page for integral problems using sagetex. The Sagetex: Integrals page can be found on the sidebar. Two indefinite integral problems have been added.

There's been a lot of things that caught my eye this last week. Here are some which are more noteworthy:

  • Nevada, Montana and North Dakota may have problems getting money from the Department of Education because of testing problems with Common Core.  From the Tyler Morning Telegraph, "The state of Montana offered waivers Wednesday to the mandatory assessment for this year, which could put millions of dollars in federal funding at risk." The issue is getting enough students taking the tests and the important section: "This week's debacle was the second technical problem Measured Progress has had in recent weeks with the computerized English language arts and math tests for selected grades. In March, testing was delayed because of a coding issue. The company said its servers couldn't handle the number of students even though it increased capacity beyond what was indicated by the tests' creator, Smarter Balanced.Nevada likely overloaded the system because it has 210,000 of the 345,000 total students expected to take the test across the three states. The state was put on its own server to do limited testing Thursday, which will continue Friday. But problems appeared again as early as Thursday morning and led Nevada's largest school system to cancel plans for the day."We can't keep putting our kids in front of an error screen," said Leslie Arnold, an assistant superintendent with the Las Vegas-based Clark County School District."
  • Lock the doors and close the shades! Spiked Math Comics has "The SM Top 10 List of Dirty Mathematics". There's a lot of content at the site to distract you for hours.
  • I've mentioned Statistics isn't really math and cited AMSTAT news and David Moore, statistics educator and former president of the American Statistical Association to reinforce the point. I also linked to this page of W.M. Briggs which says, "Worse, we routinely reify the mathematics; for example, p-values positively wriggle with life: to most, they are mysterious magic numbers. Equations become a scapegoat: when what was supposed to have been true or likely because of statistical calculation turns out to be false and even ridiculous, the culprits who touted the falsity point the finger of blame at the math.". Now Scientific American reports, "In apparently the first such move ever for a scientific journal the editors of Basic and Applied Social Psychology announced in a February editorial that researchers who submit studies for publication would not be allowed to use a common suite of statistical methods, including a controversial measure called the p-value.....Unfortunately, p-values are also widely misunderstood, often believed to furnish more information than they do. Many researchers have labored under the misbelief that the p-value gives the probability that their study’s results are just pure random chance......the p-value is a statement about imaginary data in hypothetical study replications, not a statement about actual conclusions in any given study.....critics have complained that in practice the p-value in the context of significance testing has been bastardized into a sort of crude spam filter for scientific findings: If the p-value on a potentially interesting result is smaller than 0.05, the result is deemed “statistically significant” and passed on for publication, according to the recipe; anything with larger p-values is destined for the trash bin....Significance testing became enshrined in textbooks in the 1940s when scientists, in desperate search of data-analysis “recipes” that were easy for nonspecialists to follow, ended up mashing together two incompatible statistical systems—p-values and hypothesis testing—into one rote procedure. “P-values were never meant to be used the way we’re using them today,” says biostatistician Steven Goodman of Stanford University.". In other words, books and teachers have been teaching it incorrectly for decades and, as a result, researchers have been drawing false conclusion from using it incorrectly--the article estimates p-values have been used in 3 million papers. Statistics isn't really math.
  • A manuscript of Alan Turing went for one million dollars at a recent auction.
  • Atlanta Journal Constitution reports that 8 teachers have been sent to jail over the test cheating scandal. CNN has video here.
  • With respect to the story about Success Charter school from the last post, Education Next has a piece attempting to explain the school's success. According to the article: "What separates Success, in my opinion, is a laser focus on what is being taught, and how."
  • An inquiry has been launched into whether the LA Unified School District has "....properly used bond funds for a beleaguered $1.3 billion project to provide an iPad for every student.". According to the article, "Questions were raised after emails were disclosed showing Deasy had been in communication with vendors Apple and Pearson before the contracts were put to bid.".
  • Chess super tournament Shakmir 2015 has started and includes Carlsen, Anand, Caruana, Kramnik, and more. There is streaming commentary here.
  • Democracy Now! reports on parents engaging in test "mutiny". There is video and text. From the text, "In an act of mass civil disobedience, tens of thousands of parents in New York state had their children boycott the annual English Language Arts exam this week. At some Long Island and upstate school districts, abstention levels reached 80 percent. Protest organizers say at least 155,000 pupils opted out — and that is with only half of school districts tallied so far.".

Plotting with Sagetex: Cantor Function


In the previous post, I mentioned an article about the Cantor function--and that got me thinking about plotting it. Recursive functions have that extra level of complexity that makes them more complicated than a typical plot--perfect for Sage and sagetex. I was able to get 9 iterations out of it in Sagemath Cloud as you can see above. You can download the file on the Plotting with Sagetex page.

Here are some issues that caught my eye recently:

  • Infowars gives a look inside middle school where a student was charged with a, "...felony cybercrime Wednesday after changing his teacher’s desktop background.". The child in question stated,"“I logged into a teacher’s computer who I didn’t like and tried putting inappropriate pictures onto his computer to annoy him,” Green said.Green also revealed that multiple students regularly logged into admin-level accounts in order to communicate with fellow classmates through computer webcams during school hours....Aside from the felony charge, Green also received a 10-day suspension following the incident.".
  • A crazy article, "Want Kids to Learn Math? Stop Teaching It" which spouts all sort of nonsense: "The U.S. has a math problem. Despite all the time, energy and money the country has thrown into finding better ways to teach the subject, American children keep scoring poorly and arriving at college woefully unprepared. Just as bad, if not worse, too many students think they hate math.I propose a solution: Stop requiring everyone to take math in school." followed later by, "Babies are born with an intuitive knowledge of numbers. It wouldn’t take much for schools to teach every child how to add, subtract, multiply and divide." and "People who invent new industries are rarely using math they learned in school, and often aren’t using any at all. Why drag all elementary school students through a compulsory curriculum that turns off as many as it prepares, on the off chance that a few might need it?" There's more but you get the basic idea. Now check out the information on the author (top right) who: " the Class of 1959 Director of Program in Teaching and a senior lecturer in psychology at Williams College. She is the author of six books...". With "experts" like this giving advice it's no wonder the education is such a mess.
  • Remember how Common Core would set higher standards for students  and that would lead to an improved education? The NY Post tells us, "On last year’s tests, The Post has revealed, state officials quietly lowered the number of right answers kids needed to pass six of the 12 exams. They also erased the results of three third-grade questions, including an essay, after 5 percent to 6 percent of students didn’t get to them, and the results of a seventh-grade multiple-choice question because the answer was unclear.". That's a basic problem for math teachers: the typical "good" math student isn't that strong. If you try to raise the standard you will (in most cases) have most of the class struggling and then the teacher has a problem as all the parents come in saying their child never had a problem with math before you. Decreeing higher standards, which students can't meet, will dropping the standards (as the the NY Post reports). So a more difficult test + lower score to pass =not much change.
  • The battle over ditching Common Core in Nevada is getting intense, "Assemblyman Brent Jones (R-Las Vegas) is sponsoring Assembly Bill 303, which would roll back the standards in Nevada."
  • Alabama is looking to repeal Common Core, too.
  • NYTimes has  an eye-opening piece on some extreme practices at a charter school in NY. "Success has stringent rules about behavior, down to how students are supposed to sit in the classroom: their backs straight, and their feet on the floor if they are in a chair or legs crossed if they are sitting on the floor. The rationale is that good posture and not fidgeting make it easier to pay attention." It's incredible to find out what students are subjected to in order to avoid having to go to the local public school.

Thomas Jefferson HS and School Quality

Every now and then I read something which gets me thinking: "Diversity -- or Meritocracy", by Patrick J. Buchanan on the site is such a piece and it got me thinking about school quality. I know it's a fuzzy term but school quality is essentially "the ability to set and maintain standards"--the better the standards the better the school. Every school has to battle with school quality issues on some level, even the best. In a poor quality school the lack of standards slaps you in the face: in my case, trying to teach algebra 2 to students who struggled with arithmetic. There's very little you can do when students lack the basic foundation needed for the course. When students are in classes they don't belong in (especially because they "passed" the previous course) you know your in a poor quality school. Behind most of those students were parents who wouldn't return phone calls, never came to parent-teacher meetings, and would never call to find out why their child has an average below 20%. Schools that operate on this poor level don't care about the basics. That's why the problem is there and why bad schools tend to stay bad. From what I've seen, admin know about the problem but aren't ready to make the difficult choices to create change. They just want the kids to go through the school and then it's the problem of the colleges--if the students can get into one.

But problems with standards also occurs in better schools with "good" students. This shows up, for example, when students who don't belong in accelerated classes get in just because they want to. At the public schools I've worked at the teacher would determine which students shouldn't be in the accelerated track--only to be overruled by the guidance counselor. Maybe that's because school "quality" is often determined by formulas that give more points to schools that have many students taking AP classes. As a result it's common nowadays to have students in AP calculus even though they struggle mightily with fractions. Years ago these poor quality students would be kept out of the classes; now they're encouraged to take the classes and they confidently assert they're good at math. Parents can feel good about saying their spn or daughter is taking AP classes but the level has, in so many schools, been been dumbed down.  And in response AP seems to have dumbed down their exams to compensate. The article  "Universities Raise Standards for Earning Advanced Placement Credit" by Heather Zimar (2005) states, "According to the Web site for the College of Arts and Sciences, the minimum required score students needed to earn college credit increased from September 2003 to September 2004 for the following AP exams... In addition, in 2005, the required minimum score for AP exams will increase from four to five in the following subjects.......“I have to assume that they felt that they had been too generous for giving fours and fives,” Lunkenheimer said of the departmental faculty who establish the standards for accepting AP credit. She said she suspects that students were not doing as well in the courses that followed those for which they received AP credit......more and more students who are not ready for college-level work are taking the AP in order to boost college applications. These students do not do well in the courses that follow their AP courses, she said. “They take the test in order to write it down on their application,” she said. “This is a national problem, and it’s hard to see a way out of it.”".

Welcome to today's world where it's more about placating people trying to get an edge than setting and maintaining high standards. Which is one reason why I favor having less public education; it's so difficult to maintain a standard when the student has to be at the school. A private school can say no to the parents: if you don't like it, leave. Many parents I've talked to over the years have learned to tolerate issues at magnet schools because they realize the quality is so much worse if they were to take their child out of the magnet school and put them in the regular public school.

Which brings us to Thomas Jefferson high school. It's the gold standard of public education. Thomas Jefferson is not a typical school; it's a magnet school and it's success comes from borrowing the ideals of the private school model: in particular there are higher standards to get into the school. These higher standards keep the worst kids out. That policy has helped create a school that offers an education that puts many private schools to shame. Buchanan's piece makes it clear that even the public nature of the school means there is always pressure to weaken the quality.

Thomas Jefferson high school has decided entrance based on merit--and that's the problem in today's world because some 70 percent of the students are asian in the latest class. And, unfortunately, according to the Washington Post article: "The data also shows that for the fifth year in a row, 10 or fewer black students were admitted to TJ.". This prompted a lawsuit in 2012 when " activist group representing those students filed a complaint against Fairfax County with the U.S Education Department’s Office for Civil Rights, alleging that the admission process discriminates against black, Hispanic and poor students. That complaint is ongoing. Bob Frye, one of the longest serving black members of the Fairfax County School Board, who voted to establish TJ as a magnet school in 1985, said that the administration should take a closer look at the school’s admissions process.“I have no interest in lowering the standards at TJ,” said Frye, 78, who served as chairman in 1999 and 2000. “I believe even now with the proper amount of preparation and interest the numbers [of black students] could surely be higher than they are now.”...Jeremy Shughart, the TJ admissions director, said that a committee is reviewing the application process to improve diversity at the school.".

No proof of unfairness in the admissions process, just some people believing that there "could surely be" more students of color and the willingness to bring a lawsuit. Indeed, the "evidence" of unfairness seems to be a result that favors asian students, and for that people are essentially advocating discrimination against students who have earned their right to study at Thomas Jefferson in order to get the right distribution. While a private organization such as the NBA doesn't have to face discrimination lawsuits over the demographics of their players, Thomas Jefferson high school is public and now acceptance based on merit is under pressure. As Buchanan writes, "And while Fairfax County generously supports its school, it does not spend what D.C. does. And how are D.C. schools doing? The Post reported yesterday: “Only 58 percent of D.C. students graduate high school within four years, and only about half of students are proficient in reading and math."".

Pat Buchanan's article raises all sorts of questions. Obviously, Thomas Jefferson administration wants the lawsuit to go away, but what will it take for that to happen? Will letting in 3 more students of color do that or is 30 the right amount? Who decides the "proper" amount and what is the criteria to do it, if not by merit? Would a quota be used even if the students of the "desired demographic" is too much too weak? And what about the qualified applicant who is now discriminated against because the merit system shows one group outperforms the others? Perhaps the magnet schools that hold lotteries to determine admission will be next if the luck of the draw results in the wrong racial distribution.  The admissions process to get into Thomas Jefferson was set up to be fair but now merit takes a back seat to other factors. This is the sort of nonsense that promotes racism and undermines school quality.  It's a shame, in the public arena quality standards aren't valued properly.

Here are some other stories that caught my attention:

  • the untamed scribe has a piece on "Cell Phones in the Public Schools: a Tool or a Distraction?". Seems like he read my mind but obviously is a much better writer.
  • Business Insider has a great article and video on "A college math professor brilliantly pranked his students and won the internet". A first class April Fool's stunt.
  • The Atlantic Journal Constitution has coverage on 11 former Atlanta educators who were convicted in the cheating scandal.
  • Scientific American has a piece on "The Cantor Function: Angel or Devil?".
  • The Telegraph has an article on Pietro Boselli, his students, and, "That moment when you realise your maths lecturer is a top designer model."

SageTeX: Derivatives problems


I've added two more problems to the SageTeX: Derivatives page. One problem for finding the derivative of a logarithm (with Chain Rule) and one problem for the derivative of an exponential (with Chain Rule). Taking the derivative of logarithms with Sage is a bit problematic: the output of the derivative of a natural logarithm prints out as a common logarithm, so I've limited the scope of the problems to make sure the output is typeset correctly.

A link has been added to the sidebar for the CTAN mail archives. This is where you can see the new packages being added to LaTeX as well as package updates. Take a look and you'll find sesamanuel, a fresh addition to CTAN which, "....could be used to compose a stu­dent's class­room book...". This looks like a great addition but unfortunately the documentation is only in French.

Here are some issues which caught my eye:

  • HuffPost tells us that mathematician John Urschel, "...The 6-foot-3, 308-pound offensive guard [who] absolutely loves math..." has come out with a paper "A Cascadic Multigrid Algorithm for Computing the Fielder Vector of Graph Laplacians.”. You can find his explanation at Forbes along with a link to the paper.
  • DailyHerald is out front on the latest emerging issue: "A New Jersey student's tweet about a question on new Common Core tests was deleted after it was flagged by a testing company, spurring a national debate about how to balance children's privacy with test security in the age of social media.". Monitoring of social media is "widespread" and "commonplace" on a variety of tests. "It used to be that all students across a state took a paper-and-pencil test on the same day or days. Now children are taking the same Common Core tests in multiple states during a period lasting longer than a month. That has created a more intense test-security problem for K-12 standardized exams: If a student from New Jersey reveals questions via Twitter -- or even just the name of a poem that appears on the exam to test reading comprehension -- it could give an advantage to students in New Mexico who are still preparing to take the test."
  • PRWatch looks at the corporate profits in testing, "The expanded testing has fueled a testing boom worth nearly $2 billion annually, giving the main corporations getting the testing contracts a huge return on investment for their lobbying while generating a growing backlash from parents across the country."
  • The Arizona senate is getting ready to debate ending Common Core.

Odds and Ends: March 27,2015



Lots of odds and ends today:

  1. I've added 2 figures to the Graphics page: the one above showing how the behavior of the \sqrt{x} is linearized by the tangent line. This sort of "trick" (technique) allows the physics teachers to impress their classes by calculating square roots in their head. The other figure shows how line segments can help estimate the length of an arc.
  2. I'm calling it already: The Atlantic has one of the most important education articles you'll read this year. In "The Deconstruction of the K-12 Teacher" the future of education is examined, "there will be a local teacher-facilitator (called a "tech") to make sure that the equipment works and the students behave.... and that "tech" will earn about $15 an hour to facilitate a class of what could include over 50 students...When I did some research to see if it was just me sensing this transformation taking place, I was overwhelmed by the number of articles all confirming what I had suspected: The relatively recent emergence of the Internet, and the ever-increasing ease of access to web, has unmistakably usurped the teacher from the former role as dictator of subject content. These days, teachers are expected to concentrate on the "facilitation" of factual knowledge that is suddenly widely accessible.". I think this vision is further off than predicted but when knowledge of content isn't an important part of teaching it does make sense. While it's sold as a way of improving education, please note that elite schools haven't been going this route. They don't care about certification, hire teachers with masters and PhDs, keep class sizes small, make sure students are in the appropriate class, etc.. And because they provide such a superior level of education and avoid so many of the negatives (large class size, poor quality students, etc) people with the financial means willingly pay enormous sums of money to have education done right.
  3. Nancie Atwell, a public school teacher in Maine, has won $1,000,000 for the Global Teacher Prize (considered the "Nobel prize of teaching"). Her advice for the future generation is to avoid public schools: ""Honestly, right now, I encourage them to look in the private sector," she said during an appearance on "New Day." "Public school teachers are so constrained right now by the Common Core Standards and the tests that are developed to monitor what teachers are doing with them. It's a movement that's turned teachers into technicians, not reflective practitioners." "If you're a creative, smart young person," she continued, "I don't think this is the time to go into teaching."".
  4. Science Daily reports, "A 60-year old maths problem first put forward by Nobel laureate Enrico Fermi has finally been solved."
  5. Computer glitches in Wisconsin delay Common Core exams, "On Thursday, officials from the Wisconsin Department of Public Instruction said they would delay the testing window for the new Badger Exam because the online delivery platform is not yet reliable. That means about 375,000 children originally scheduled to take the online tests in English and mathematics between Monday and May 22 will now take the tests between April 13 and May 22."
  6. Yahoo News reports, "Two Maryland high school students have been accused of using Twitter to cheat on Common Core state standardized tests, officials said on Monday."
  7. Yet again, The Atlantic, has another good piece. "Does Students Motivation Even Matter?" which is "...pell-mell of counterintuitive findings that call into question many of the widely held tropes about what works in boosting student learning...".

LaTeX: Plot template with spy

PlotTemplateSpyOver the last week I found myself working on graphics for calculus (arc length and linearization) which required learning a bit about the spy capabilities of tikz. Spy allows you to magnify a part of your picture which is useful when you're dealing with small changes as is typical in calculus. After creating said graphics it seemed only appropriate that I create a plot template which uses spy. I've worked the capability into the previous plot template for consistent plotting graphics; you can see the output above. That's for illustrative purposes only; in my opinion there's too much happening in the picture above: so many functions, the grid and the spy area, and the vertical line. Recall that changing the "grid" option from "both" to "none" will turn off the graph paper. The new template is posted on the Handouts page, under the first template.

Here are some issues that caught my eye recently:

  • A NY school district is apologizing after letting a student recite the Pledge of Allegiance in Arabic during National Foreign Language Week. Fury follows and the article mentions a tweet: "State Education Department regulations specifically say the Pledge of Allegiance should be read in English.".
  • Illinois Watchdog reports "After working one day as a substitute teacher in Illinois, David Piccioli could be entitled to an annual pension of more than $30,000." This story is attracting attention and is well worth reading.
  • Chessbase hosts John Stewart's piece on U.S. buying (chess playing) nerds. No offense Fabiano!
  • RT reports on over 1,000 arrested in a mass cheating scandal. Watch the video of them scaling walls to position themselves by windows to help out. The article notes,"More than 1,000 people were detained, half of them were parents and teachers while the other half consisted of friends and relatives," Gupteshwar Pandey, a police official in Bihar state, told AFP on Sunday. "Fifty percent have been released but I believe that the others are still probably in jail." followed by "According to Pandey, at least two policemen were also arrested and 10 more were dismissed after being implicated in the scam." and "Numerous photos and a video presented earlier this week showed the students using smuggled crib sheets, which friends and relatives had handed to students after climbing the walls of examination centers." You have to see it to believe it.

March 20, 2015: Odds and Ends



  • I've added code for plotting the upper Riemann and upper Riemann sums using sagetex. You can find them on the Plotting with Sagetex page, in the Riemann sums section.
  • The problem: "Find a function whose domain is the positive real numbers and whose range is the integers." has been added to the Problems page.
  • Infowars hosts the CBS news video on a student who was tased, arrested, and "...charged with assault on an officer, and two misdemeanors."
  • In the last post there was an article on the poor performance of American Millennials compared with the rest of the world. ZeroHedge hosts  a more in-depth piece by Michael Snyder: "It's Official: Americans R Stoopid". From the article: "In Japan, Finland and the Netherlands, young adults with only a high school degree scored on par with American Millennials holding four-year college degrees, the report said." and reference stats from a USA today article on college students, showing,"-“Students also spent 50% less time studying compared with students a few decades ago”-“35% of students report spending five or fewer hours per week studying alone.” “50% said they never took a class in a typical semester where they wrote more than 20 pages”, “32% never took a course in a typical semester where they read more than 40 pages per week.”. Sobering.
  • TheConversation reports on the roll of Twitter in the Common Core debates and studies the content of a large selection of Tweets: "There are upwards of 40,000 tweets using #commoncore each month right up to the present."

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, " 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 " 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."

Python/Sage: Finding roots with Newton's Method


An earlier post dealt with a fractal that resulted from Newton's Method. Today I've posted some Python code for finding the root of a function using Newton's Method on the Python/Sage page. Copy and paste the code into a Sage Cell Server, change the function to what you want and set the number, N, of iterations. It's quicker than using the calculator, less prone to mistakes, and shows the estimates of roots along the way (which is especially useful in cases where Newton's Method fails to find a root).

Here are some stories which caught my eye:

  • Over 1,000 people turned out in Long Island to rally against Common Core. Long Island Press has the coverage.
  • edSurge explains "Why the Smarter Balanced Common Core Math Test is Fatally Flawed".
  • South Carolina has abandoned Common Core.
  • In the last post I mentioned Saturday is Super Pi day because 3/14/15 matches up with 3.1415. But as this article points out, "in 2015, Pi Day really is significant as the mathematical moment of Pi–3.141592653–will only come around once in a lifetime, on March 14th, 2015 at 9:26 a.m. and 53 seconds....A moment like this won’t be back for another hundred years, March 14th, 2115.". Get your geek on!
  • The Guardian has an article "Confessions of a mathematical Olympian: an insider view of film X+Y", which opens this weekend.
  • The Daily Signal has a good piece on the growing concern over the Common Core standards. "A Pioneer Institute report coauthored by Milgram detailed that, by seventh grade, Common Core mathematics standards leave American students two grade levels behind their peers internationally and do not prepare them for admission into highly selective four-year universities and STEM (Science, Technology, Engineering, and Mathematics) programs....And recently, reports surfaced that the Common Core architects left what some consider holes in the standards. Richard A. Askey, a professor emeritus at the University of Wisconsin-Madison and former member of the math standards’ feedback group, later noticed an omission of a geometry standard in Common Core. In fact, according to Education Week, Askey said “the process toward the end was so hurried that an entire high school standard was left out of the final draft.”"