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."