Brexit: because statistics isn't really math

A last minute change in my post due to Friday's historic day that wiped out 2 trillion off of the worlds equity markets. The cause? Virtually all the experts predictions were wrong on Brexit: the pollsters got it wrong, the betting line was wrong, and the markets got it wrong-- thank the statisticians for another job well done! CNN reports, "Ahead of the 2012 U.S. elections, Nate Silver, from the website FiveThirtyEight, correctly predicted who would win all 50 states, even as pundits were saying the race was "too close to call." In 2008, he had also correctly projected all but one state. As this year's British election results started trickling in, Silver tweeted that the world "may have a polling problem." "Polls were bad in U.S. midterms, Scottish referendum, Israeli election and now tonight in UK," Silver said....In a commentary on FiveThirtyEight, Silver suggested that forecasters had been overconfident. "Polls, in the UK and in other places around the world, appear to be getting worse as it becomes more challenging to contact a representative sample of voters. That means forecasters need to be accounting for a greater margin of error," he said....Prediction models for the U.S. elections had also become more reliable, Parakilas said, something he didn't believe had happened yet in the UK...." And while they tell you NOW that the models in the UK are less accurate, the Nate Silver that CNN trumpeted as the expert is the same Nate Silver I mentioned here, who performed so badly.

The great thing about statistics is you can explain why you were wrong after you learn that you were wrong. Here are some other explanations floating around as to why the results were so wrong:

  • there was a lot of rain in London and that could impact the turnout of city dwellers who tended to support Bremain.
  • Brexit voters were criticized as racists and not too bright, as all the "experts" came out for Bremain. As a result, Brexit voters were not honest when polled. I think this is a big factor, and a reason why applied statistics is more of an art than science.
  • the inability to predict how many people would turn out to vote
  • Bremain is what the establishment wanted to win so the dissenting voices were minimized. This seems unlikely given how genuinely surprised the establishment was.
  • From Reuters, "Predicting the outcome of Thursday's referendum was harder than that of a national election because there was virtually no historical data to draw on, said David Rothschild, an economist at Microsoft Research. He said pollsters also did not pay enough attention to working class and less educated voters....Rothschild, who also is a fellow at Columbia University’s Applied Statistics Center, said he expected forecasting to improve with a transition from polls using small, random representative samples to large Internet-based ones with rich demographic data. "If I have one million respondents with a large amount demographic data, I should be able to predict outcomes better, or I'm not a very good statistician," he said." OK, but right up to the end statisticians were confident of the result even though many polls were based on only a few thousand people. How representative were the samples they claimed were representative and why were so many experts wrong.
  • Statsblog.com gives their 5 reasons, including "Survey respondents not being a representative sample of potential voters (for whatever reason, Remain voters being more reachable or more likely to respond to the poll, compared to Leave voters)". Hard to argue with the samples not being representative of the whole AFTER you've been proven wrong. But aren't statisticians supposed to be guarding against this happening?

Lots of excuses; of course some or all arguments could be right BUT:

  • From NYTimes, "Britain’s decision to leave the European Union on Thursday was a big surprise. As late as 6 p.m. Eastern in the United States, less than five hours before the results became clear betting markets gave “Remain” an 88 percent chance to win the election, but it wound up losing by four percentage points....One could certainly argue that the polls were “wrong” in the sense that they tended to show a slight Remain advantage heading into the vote count. But it was clearly a distinct possibility that Brexit would win, based on the available survey data. So it’s hard to argue that this was a big polling failure, and it’s a bit strange that the financial markets appear to have been caught completely by surprise." The American Thinker responds to this stupidity,  "Sure it wasn't.  Once you are done rolling around on the floor in laughter at claims that this wasn't a massive polling failure, read on....Not a single one of the well known polling aggregators/predictors picked Brexit in their last-minute final projections...Thus, we had a systematic bias in the aggregated polling data that ranged from 4% to almost 11%.Individual polls leading up to the vote were publishing ridiculous results.  In the week prior to the vote, 9 of the 13 polls predicted a victory for Remain ranging from 1% up to 10%.  Just three polls had Leave in the lead, but just by 1% to 3%– i.e., still below the actual margin of victory – and one poll had a tie.  Not a single individual poll got the result correct, or overpredicted a Leave win....The overall bias in favor of Remain was effectively uniform, which is statistically impossible if the bias was random.  The bias was systematic."
  • In region after region the Brexit numbers were consistently underestimated by several percentage points.

The Telegraph has a lot more analysis with charts and graphs as to what wrong, "Professor Curtice was cautious throughout the campaign, saying that“some of the polls are definitely wrong” in “a cloud of uncertainty”. There were also clear distinctions between phone and online polls - phone polls invariably scored higher results for Remain compared to online." [Comment: Aren't you glad you know this now, after the fact?] "Interestingly, the Leave vote remained constant across both phone and online - it was the “don’t know” score that decreased for phone polls, and Remain seemed to be securing most of these. This was falsely encouraging for the final result. According to YouGov’s analysis, the reduction of don't knows for phone polls was because people were more likely to give an opinion when in conversation with someone, rather than admitting they didn’t know what they thought about such an important choice.Analysts dismissed the idea that different methods would reach different demographics.." So they got it wrong again. But at least their hindsight is 20/20.

Some lessons according to the Washington Post,

  • "First, we did not see this coming. For some weeks now, Stephen Fisher and Rosalind Shorrocks have been tracking referendum forecasts. They consider a wide range of sources, from forecasting models based on polls, to citizen forecasts, to betting markets. None of these methods saw a Leave outcome as the most likely outcome."
  • "Second, this was not a systematic polling failure of the same magnitude as last year’s U.K. general election, where opinion polls badly underestimated the Conservatives’ chance of victory." So the defense is: the failure isn't as bad as when they REALLY messed up last year. That should inspire confidence.
  • "Third, we learned something about campaign dynamics in referendums — and we went wrong by believing too firmly in a claim about how voters decide. Part of the disparity between relatively close polls and relatively confident betting markets was due to the belief in status quo reversion — the idea that undecided voters will be more likely to choose the status quo option (in this case, Remain) than the alternative."
  • "Fourth, given the types of areas that voted to Leave, and given the available polling evidence, it seems likely that a majority of Britons have traded economic benefits for restrictions on people from the European Union coming to live and work in Briton. The areas which voted Leave were older, whiter, and less likely to have a university education."

Another black eye for statistics and statisticians but you can't expect statistics to have the accuracy of mathematics--it isn't math any more than mathematical economics is.

Here are some stories that caught my eye last week:

  • ZeroHedge reports "The percentage of new doctorate recipients without jobs or plans for future study climbed to 39% in 2014, up from 31% in 2009according to a National Science Foundation survey. Those graduating with doctorates in the US climbed 28% in the decade ending in 2014 to an all-time high of 54,070, but the labor market - surprise surprise - has not been able to accommodate that growth. "The supply of PhD's has increased enormously and the demand in the labor market has increased but not nearly as fast. When you can import an international workforce or outsource research, you have a buyer's market" said Michael Teitelbaum, senior adviser to the Alfred P. Sloan Foundation."
  • Wow! RT notes "Around one in 10 of the students attending the largest four-year public university system in the US is homeless, while one in five cope with food insecurity, according to a new study by the California State University system."
  • I've got nothing against unions, my issue is against the terrible decisions/policies they (or anyone) support. Case in point. ChicageCBSlocal reports "He crossed the line – the CPS teachers’ one-day strike — out of his love for the classroom. Joseph Ocol stuck with his kids and brought home a chess championship. Tonight, he’s expelled from the union and wonders if he’ll even have a jobCBS 2’s Brad Edwards reports.The union’s decision came via certified mail, in a letter signed by Chicago Teachers Union President Karen Lewis.....the CTU said in a statement: “Mr. Ocol has been informed of his member privileges and is talking to us through media, which is unfortunate. All members are well aware of what happens to strike breakers and are informed by their own peers of the process for both suspension and reinstatement. CTU is a democratically led member-organization.”"
  • CounterPunch explains how Common Core helps bust the unions.
  • HeatStreet looks into how colleges are letting students censor speach, "For many students and professors, one of the great appeals of college life is being exposed to new and different ways of thinking. But that age-old process is now under threat at schools around the country. Take the University of Northern Colorado. After two of the school’s professors asked their students to discuss controversial topics and consider opposing viewpoints, they received visits from the school’s Bias Response Team to discuss their teaching style. The professors’ students had reported them, claiming the curriculum constituted bias. These incidents, both in the 2015-2016 academic year, reflect a growing trend in higher education. College students increasingly demand to be shielded from “offensive,” “triggering” or “harmful” language and topics, relying on Bias Response Teams to intervene on their behalf. Such teams are popping up at a growing number of universities....To date, more than 100 U.S. public colleges and universities have established Bias Response Teams."
  • HeatStreet again with "Kayla-Simone McKelvey will serve 90 days in jail, five years of probation and 100 hours of community service for her role in a racially charged hoax threat issued to Kean University students. McKelvey, who is black and the former president of the New Jersey college’s Pan-African Student Union, used a fake Twitter account to send a message threateningto kill a group of black students at an on-campus rally in NovemberThe Twitter account, @Keanuagainstblk, claimed that the anonymous user would “kill all male and female black students” at Kean and issued a bomb threat against the school. The account was quickly suspended from Twitter, but not before causing an uproar on social media. Supporters of #BlackLivesMatter across the country called on the university to take action to protect protesting students, and demanded that Kean President Dawood Farahi resign. They tried to use the threat to demonstrate that Farahi had not done enough to diffuse racial tension on campus....McKelvey told the court she was sorry she issued the threat, and that she still believes her actions helped expose racism on campus...But if McKelvey’s excuse sounds a bit strange, she’s not alone, even at Kean, in thinking that her clearly illegal actions “helped” fellow social justice warriors to bring Kean’s “systemic racism” to light. Some Kean students said that the threat’s author didn’t matter that the threat was still evidence of strong racial bias on campus."
  • LA Times reports that Pat Haden used an educational foundation to enrich himself and his family, "Under Haden's leadership as board chairman, however, the $25-million foundation became a lucrative source of income for him and two of his family members -- even as its scholarship spending plunged to a three-decade low and the size of its endowment stagnated, a Times investigation has found.Haden, his daughter and sister-in-law together collected about $2.4 million from the foundation for part-time roles involving as little as one hour of work per week, according to the foundation’s federal tax returns for 1999 to 2014, the most recent year available. Half of that, about $1.2 million, went to Haden. His annual board fees have been as high as $84,000; the foundation paid him $72,725 in 2014....Many foundations do not pay their board members, philanthropy experts say. The $1.5-billion California Community Foundation, for example, does not pay board members. Foundations that do compensate board members, those experts say, typically pay far less than the amounts received by Haden and his relatives. The $12.5-billion Ford Foundation paid its board chairman about $30,000 less than the Mayr foundation paid Haden in 2014. Mark Hager, an associate professor of philanthropic studies at Arizona State University, said in an email the Mayr payments to the board would be high “even for a foundation that was giving out more than $50 million in grants each year.”“I’ve never heard of fees that large,” said Adam Hirsch, a law professor at the University of San Diego who specializes in trusts."

Sage: Polynomial Interpolation

PolyI1

In an earlier post I mentioned that it's not that uncommon for math teachers and even education professionals making up academic resources to mess up badly on sequences. For example, the IXL site, which is, in general, an excellent resource for teachers and students makes a common error with sequences both here and here. Experiment with the problems that are randomly created at those 2 links. Try entering a "wrong" answer and you'll get an explanation of what the "right" answer is. But the "logic" they're using is that here's a formula that describes the sequence, therefore the next term is... The problem is that if you follow the same logic you can justify the missing terms of the sequence to be whatever you want. Therefore those problems have no correct answer and should not be given. Moreover, they shouldn't mark other answers as wrong.

Whenever teachers and math professionals are wrong on the math, you've got a teachable moment. This lesson would be for someone teaching about matrices who has gotten through reduced row echelon form. I'll suggest that you start a class by working through the material on the IXL website and have them figure out what the terms of the sequence are. Write the various sequences on the board to refer to later. When they've gotten comfortable ask them what will happen if you put in a term that doesn't seem to fit the pattern. They'll predict that you will be marked wrong. After showing them that you are indeed marked wrong by IXL figure out a formula to justify your answer and have students confirm that it works. You now have a formula that justifies a sequence that IXL was marking you wrong for. Get the class to discuss what it must mean for an answer to be correct (you can find a formula) and for what it means to be incorrect (it's impossible for anyone to find a formula).

Now it's time for math! Elicit that a sequence is a function from the positive integers into the real numbers (or integers, depending on how you teach it). Remind them that this means the sequence -5, 3, 11, 19, 27, ... corresponds to the function where f(1)=-5, f(2)=3, f(3)=11, f(4)=19, f(5)=27. Give them the mathematics known as polynomial interpolation or Lagrangian interpolation, and using one of the IXL sequences written on your board, set up a Vandermonde matrix. Work through the mathematics to create the polynomial. Get those calculators out to solve the matrix equation. And at the end you'll have a polynomial which they'll need to confirm works to generate the sequence. By the end of class your students will have learned about Vandermonde matrices, seen the math behind polynomial interpolation, used the calculator to power through some of the calculations and construct a polynomial that shows that even the "experts" get things wrong. That's a lesson the class will remember long after they've forgotten how to do the math.

But it's easy to make mistake with calculations so I've constructed a Sage program to go through the steps to create an interpolating polynomial using Vandermode matrices. It's posted on the Python/Sage page. A little warning is necessary, however. Usually you just go to a SageCellServer (or SageSandbox on this site) and copy/paste the code and press "Evaluate" to get the code to run. For reasons I don't understand, sometimes you get some sort of I/O Error such as the one below.

PolyInterpError

That error can pop up at various stages in compilation. The code runs, sometime it's just a matter of pressing "Evaluate" several times.

In order to set the code for your sequence, you need to alter xvalues and yvalues in the code. For example, if your sequence is 3,2,_,0,-1,-2, _,... and someone says that the pattern is to subtract 1 from the previous terms so the missing numbers are 1 and -3 then you'll need to pick the values you want in your sequence. If you choose to complete the sequence as 3,2,5,0,-1,-2,11,... then you'll need to set xvalues = [1,2,3,4,5,6,7] because you have 7 terms in your sequence and yvalues = [3,2,5,0,-1,-2,11] because those are the terms of your sequence. Press "Evaluate" and you'll get a polynomial that goes through those points. So there is a formula for your sequence--it's just not obvious to most people.

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

  • The74million notes that Common Core lowers standards and that's a good thing. HUH? "Implementation of the Common Core has run headlong into high school exit exams, which many states require students to pass in order to graduate. But now states that have adopted the Common Core are grappling with whether raising academic standards should also mean making it harder to graduate.To supporters, tough graduation requirements are necessary to encourage student effort and ensure a diploma “means something.” Some have even pushed for requiring students to demonstrate “college and career readiness” in order to graduate.But decades of research now show that exit exams have not really raised standards, and have actually harmed disadvantaged students....In other words, the unintended consequence of the Common Core may have been to lower the bar to graduate — and research suggests that this was a good thing."
  • Decades ago, a high school degree was normal and few people had college degrees. Yet people could get jobs that allowed them to provide for a family. Now the quality of high school education has been watered down, graduation rates are higher and a college degree is "necessary" to get a job. And that means people are some $30,000 in debt after college (which is more likely to be 5 years now) and too many students have taken courses in topics that lack rigor and meaning. And while I've taught algebra to many young students outside the US, somehow in the US algebra is too complicated for 18-22 year old students to master--it's actually standing in the way of people graduating. How can you get horribly educated students through college with a pesky math course in the way? Simple--water down standards at the college level. Now Wayne State University leads the charge in dumbing down education, "Up until now, students had to take one of three different math classes before they could earn their degree. Now, depending on their major, students may be able to squeak through college without taking math. The university is leaving it up to the individual departments to decide whether math will be a required part of a degree's curriculum." So in the future, students can graduate high school and college and still not have the math skills of someone who only graduated from high school 50 years ago...and pick up a lot of student loan debt along the way. But at least more people are graduating from college! Progress?
  • Now compare the American drive to banish math with this clip from NextShark called "Watch Korean Students Take the American SAT Math Section For the First Time". You'll hear some people talk about Americans as "exceptional"--that probably shouldn't be taken as a compliment. Even the weak Korean students are feeling better about their math now....
  • The Columbus Dispatch reports "The State Board of Education is expected to lower minimum proficiency standards on two new high school math tests after results came in lower than expected.The move raises questions about whether benchmarks for new assessments will accurately gauge a student’s readiness for college or a career" Somehow I doubt they care about college readiness...
  • Wired.com with an article on "deep learning" that underpins AlphaGo and other computer programs.
  • He's gone viral! ABC News with a video on "8TH GRADER GRABS BELLY LAUGHS FOR CANDIDATE IMPERSONATIONS"
  • Mental Floss with "15 Observational Facts About Isaac Newton"
  • The bit-player blog has a fascinating post on the non randomness of the prime numbers. "These remarkably strong correlations in pairs of consecutive primes were discovered by Robert J. Lemke Oliver and Kannan Soundararajan of Stanford University, who discuss them in a preprint posted to the arXiv in March. What I find most surprising about the discovery is that no one noticed these patterns long ago. They are certainly conspicuous enough once you know how to look for them...For the past few weeks I’ve been noodling away at lots of code to analyze primes mod m. What follows is an account of my attempts to understand where the patterns come from. My methods are computational and visual more than mathematical; I can’t prove a thing. Lemke Oliver and Soundararajan take a more rigorous and analytical approach; I’ll say a little more about their results at the end of this article."
  • A May 18th, 2016 interview with graph theorist Maria Chudnovsky on anthonybonato.com. Looking for a female mathematician to inspire the girls in your classes.  Look no further; she's even been in 2 commercials.

SageTex: Complex Numbers

Complex1

It's been a very busy week as schools wind down for the year. I've added a new page Sagetex: Complex numbers to the sidebar. I will, of course be adding problems to the page in the coming weeks.

Here are some stories that caught my eye this week:

  • Mathematician Dr. Ken Ono has been written about in Quanta Magazine.
  • Dr. William Stein is leaving academia to start a company around Sagemath. His pdf slideshow can be found on this Reddit thread. He has a talk about this in a video posted here.
  • Ars Technica reports, "Google's DeepMind AI division will face off against humanity's number one Go player, Ke Jie, according to Chinese state news agency Xinhua. Ke Jie is currently the top-ranked Go player in the world."
  • DurhamLocalNews has a piece on how "Legislation recommended Wednesday by a Senate committee would require North Carolina schools to offer traditional high-school-level mathematics classes alongside newer “integrated” classes that arrived with Common Core...If the bill were to become law, districts would have to offer both math class sequences to students starting in the 2017-18 school year."
  • Click2Houston tells us a creepy story about a math teacher who enjoys being with kids a little too much: "The video shows Gregg Gustafson wrestling and tickling minors at a student's home when the parents were not around...One of the students told the police Gustafson gave him several wedgies that broke his underwear. The boy said he was wrestling with Gustafson and that he would not let him tap out.The student posted two videos of his broken underwear. Gustafson allegedly asked the boy to get rid of the evidence."
  • Washington Post on the new school report "Many people in education and the workplace don’t think some of the English Language Arts and math standards — which are being used in most states — are what students and workers need to be successful in college and career." One of the findings is that calculators are prevalent in math class. That's exactly what I'm seeing--but you'll see the teacher teaching how to work a problem and then they jump to using a calculator, so that the students quickly forget the skills they were supposed to learn. Whose "best practice" is that? More disturbingly, "There may be disagreement across K-12, college, and workforce about which mathematics topics are important to success in postsecondary STEM coursework and STEM careers. In K-12, there may also be disagreement about when these topics should be introduced in the mathematics curriculum." Shouldn't these issues have been worked out before forcing them upon everyone?
  • Ars Technica with an interesting article on "That ’70s myth—did climate science really call for a “coming ice age?”"
  • EAGNews on how "For a mere $6.5 million, New York City’s School Construction Authority renovated a former clothing store on Fifth Avenue into a “state-of-the-art” facility for 18 students to participate in its new, free pre-K program....The renovations cost the city about $362,222 per pupil, which was about $160,000 per seat more than the next most expensive renovation in Staten Island, for creating partitions, pouring concrete slabs, adding sprinkler systems and HVAC, and other upgrades." It's easy to spend public money.
  • ZeroHedge on the embarrassment that is the new generation. But it's not just America. "In the western world, political correctness is often taken to absolutely ridiculous extremes in attempt to keep people from being exposed to anything that could remotely be considered “offensive”.  For instance, just consider a couple of examples from the United KingdomThis hyper-sensitivity has prompted the University of East Anglia to outlaw sombreros in a Mexican restaurant and caused the National Union of Students to ban clapping as “as it might trigger trauma”, asking youngsters to use “jazz hands” instead. Could you imagine banning clapping? But this is actually happening.  Anything that might make someone feel the least bit “uncomfortable” is now being labeled as a “micro-aggression”, and at schools all over America “safe spaces” are being set up where young people can avoid anyone or anything that may make them “feel uncomfortable, unwelcome or challenged”.And this is not a fringe movement at all.  These “safe spaces” are being established at some of the most prestigious universities in the entire country, and in at least one case a “safe space” included “calming music, cookies, Play-Doh and a video of frolicking puppies”
    At Brown University – like Harvard, one of the eight elite Ivy League universities – the New York Times reported students set up a “safe space” that offered calming music, cookies, Play-Doh and a video of frolicking puppies to help students cope with a discussion on how colleges should handle sexual assault. A Harvard student described in the university newspaper attending a “safe space” complete with “massage circles” that was designed to help students have open conversations.Are you kidding me?...Now that we have defined “safe spaces” and “trigger warnings”, I am going to define a term that I used in the title of this article.

    “Wussification” is the act of turning someone into a “wussy”.  And urbandictionary.com defines “wussy” in the following manner… A person with no guts. A person who whines all day and sits around and cries like a little baby for years over nothing. Will blow anything out of proportion and create drama to forget about their sad miserable lives.If our young people need cookies, Play-Doh and videos of frolicking puppies to deal with the challenges in their lives right now, what in the world are they going to do when the things really start falling apart in America?The real world can be exceedingly cold and cruel, and our young people need to be equipped to handle whatever life will throw at them."

  • RT with the latest on "A breakthrough in the study of a mysterious Hellenistic clockwork device which was found at the bottom of the Aegean Sea more than a century ago has led researchers to conclude it was designed for philosophers to peer into the future.Dubbed the ‘Antikythera mechanism’, after sponge divers hauled the bronze mechanism from a shipwreck off the coast of the Greek island in 1900, the machine has been described as an ancient computer because of its advanced technological design."
  • SOTT has an interesting hour long interview "Stefan Molyneux and Dr. Duke Pesta on the dangers of Common Core"

1,1,2,3,5,....

Contrary to what you might be thinking, this post is not about the Fibonacci sequence; it's about an all too common mistake that teachers and test makers make. I've found this error to be very common at the high school level but what brought it back to mind was when I was exploring the IXL.com website recently. Let me start by saying I actually think the website is an excellent resource for teachers. Just find a skill that your students are deficient in and send them to the IXL site for practice on that very topic. If you get your class registered you can even get a lot of analytical data on how your students are doing. That said, the website has some problems, too. You might not think that an educational site would teach students incorrectly but you'd be wrong. A quick look around the website reveals some problems in probability, irrational numbers, and sequences. I'm going to focus on sequences. Sequences are covered in many grades so there are are many different links that are "teaching" students incorrectly, such as here or here. If you experiment (the problems get randomly generated) you'll have no trouble coming up with the type of wrong answer that I've already talked about in my post "...is Ambiguous". In that post I showed that given a formula for the first k terms of the sequence you can produce a formula to justify whatever real number you wanted to be next. But there are some variations you will see on the IXL site that have a slightly different format:

  • Find the next move of a recursive sequence (e.g. 1,1,2,3,5,...)
  • Find more than one missing term from another sequence (e.g. _, 3, 11, _, 27, 35)

Since neither of those is covered by the previous post (I'm ignoring the fact that recursive formulas can be rewritten as an algebraic formula--because we don't need that formula to create our function, unlike last post) I'm revisiting the issue. If these mistakes are so common then the subject should be the foundation of a lesson--to make sure students learn the correct theory and, hopefully, future teachers and educational sites won't teach students incorrectly. That, by the way, is one reason why I'm so adamant that you can't have teachers without a bachelors degree teaching math in high school. It doesn't mean that they all will know this (many still won't); it's just that there are many such pitfalls that a teacher with a poor math background will make. For example, I've tutored students recently that had a certified and tenured teacher "teach" that there are 6 ways to form a committee from 3 people. Turns out she had taught students about permutations for over a week using a word "committee" that should only be used in talking about combinations. Needless to say, I had to fix what she broke. But I digress.

The first type of problem is illustrated by the sequence 1,1,2,3,5,... which you might recognize as the Fibonacci sequence. Yes, that's AN answer, but it's not THE answer. In fact, THE answer doesn't exist; it's possible to produce a polynomial p(x) such that p(1)=1, p(2)=1, p(3)= 2, p(4)=3, p(5)=5 and p(6)= whatever we want. That is, ... is still ambiguous for reasons beyond what I mentioned in the earlier post.

Similarly, in the second case we can pick whatever missing numbers we want and produce a polynomial to go through them. Like a statistician, we can produce any answer we want. Unlike a statistics there is real math underlying the method. Don't believe me? For the sequence 1,1,2,3,5,.... I'm going to claim that the 6th term is -3 and give you a formula to prove it. Try this:

PolyInt1

Notice that I've got some help getting the answer. Sage is going through the calculations to make things quick and avoid mistakes. You can see p(1)=1, p(2)=1, p(3)= 2, p(4)=3, p(5)=5 and p(6)=-3. Also notice I said "the polynomial"; there are other polynomials that would work but in the context used above it's about THE polynomial that comes about by following  a particular sequence of steps. Suppose, instead of -3, I want p(6)=11. No problem:

PolyInt2

The same method works with the sequence:_, 3, 11, _, 27, 35. It looks like THE formula is 8x-13, so that the first term is -5 and the fourth term is 19. But it's not THE answer, it's just AN answer. Here's one such polynomial.

PolyInt3

And there is no reason that the terms have to be integers; it just makes the polynomial nicer. In this one pi is used as a missing term in the sequence. You can click on the image below to make it bigger.

PolyInt4

The bottom line is that an nth degree polynomial can be created to go through any sequence of points [where the x values are different]. As a result, ...is ambiguous and it's a shame that many teachers and even educational websites keep making the same mistake. Going forward, I'll look at how Sage can help us compute these polynomials.

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

  • Truth in American Education reports "Even though the Bill & Melinda Gates Foundation acknowledged that the Common Core State Standards and related assessments have not accomplished what they had hoped they are bringing the standards, assessments and accountability movement to higher education.....Of course we have to have standardized tests to go with the common standards

    “It may be difficult to list everything students should know and be able to do,” the book said, “but when faculty are asked to focus on essential elements they are quite ready, willing and able to define priorities for student learning in their disciplines.”

    One of the project’s goals is for the white papers to be used for the creation of tests, or assessments, that colleges can use in a standardized way. However, those possible assessments must be voluntary, the book said, and based on multiple measures rather than a simple box-checking, multiple choice test.

    College faculty, the article noted, may be forced to adopt measures like these." I get the feeling that success isn't measured by academic performance but by the amount of money spent on standardized testing.

  • The Valley Patriot explains "The Ugly Truth About Common Core Education" with such observations as "Private testing companies and textbook publishers are pushing for spending millions of dollars pushing Common Core because these corporations see the possibilities of hundreds of millions of dollars in profits generated from the new tests, computers, preparation guides, and Common Core-aligned textbooks.... The new ELA texts contain graphic pornographic depictions, which should cause deep concern for every parent. One especially graphic text in the curriculum is Toni Morrison’s The Bluest Eye, which graphically depicts rape, incest, and pedophilia. Is this what we want our students to read and “analyze” in English class? Dr. Sandra Stotsky, who wrote a large part of the ELA Framework for Massachusetts, which made Massachusetts the highest achieving state in the nation, resigned in disgust from the Common Core Advisory Team and is now actively opposing the implementation of Common Core."
  • Graduation is approaching and you might think it's time to reward academically superior students. Not in this Texas school. Kens5 has the story: "National Honor Society (NHS) stoles are frequent sight at high school graduation ceremonies around the country, but one Plano Senior High School student is frustrated that he won't be allowed to wear one when he puts on his cap and gown next month. According to school practices, students are not allowed to wear NHS regalia....KellyAnn Frederick says a National Honor Society sponsor claimed school administrators want everyone to feel included in graduation and not single students out." Since said teenager wants to be singled out for his hard work the issue is apparently that the students who didn't excel will feel badly.
  • The Dayton Daily News reports "Ohio students performed dramatically worse than expected on two state exams this spring according to preliminary results, and state officials want to soften the scoring system so more students pass....An Ohio Department of Education memo leaked by Wagner shows that only 24 percent of students scored proficient or better on the Geometry test, and only 21 percent did so on the other sophomore math option — Integrated Math II. Students had to get roughly half of the questions correct to be proficient....So Jim Wright, director of ODE’s office of curriculum and assessment, on Wednesday proposed softening the scoring system so that 52 percent of students would pass the Geometry exam and 35 percent would pass Integrated Math II." The rhetoric of higher standards sounds great until your numbers look bad. The result? "Wagner calls the system “class warfare,” pointing out that for years, test scores have closely tracked income levels, with poorer students scoring worse." Luckily some educators are actually interested in standards, "“We’re increasing the rigor as we should. We should make sure that every child has the ability to perform at something after high school,” Gunlock said. “Hiding behind the fact that they all got a diploma even if they weren’t qualified doesn’t make any sense to me. … The idea that this is holding back poor kids is baloney.”". The point was to raise the academic bar because student achievement was poor. The fact that the DOE wants to pass them anyway shows you how academic performance is not a priority.
  • A surprising result in Shakmir 2016. Tournament favorite Caruana lost in the penultimate round and drew the final round to tie for first with Mamedyarov--Mamedyarov, having beaten Giri in the last round then beat Caruana in the playoff to win the title. Chessbase has the report here "Tied for first, both Fabiano Caruana and Shakhriyar Mamedyarov faced each other in a Playoff shortly after the end of Mamedyarov's game against Giri. In the first mini-match of games at 10 minutes plus 3 seconds, Fabiano Caruana achieved a decisive advantage in both games, but each time failed to deliver the killing blow and the Azeri managed to escape. With two draws and a score of 1-1, the momentum had to seem to swing to Mamedyarov's side, if only for psycholgical reasons: escaping 'certain defeat' twice in a row had to be a confidence booster, feeling the opponent is just not firing all cylinders at this point. Furthermore, Caruana is not known to be the best blitz player (his FIDE Blitz rating is not a fluke), an area where Shakh might now feel he had he edge.Whatever the truth of it, Mamedyarov won the first game in a very tense battle that had everyone on the edge of their seats. Vishy Anand, watching online, was no less riveted and commented about their great fighting spirit, "We are not worthy!". Game two saw Caruana in a must win situation with black, but being blitz, anything goes. After reaching a clear advantage in a rook and opposite-colored bishop ending, it was the opinion of grandmaster pundits that Fabiano was going to win it and level the score, but everything went south for the American and after a wild scramble he was suddenly worse with no chance of winning whatsoever, and the players shook hands." Hou Yifan had a disappointing last place finish. Playing with the men is quite different when every game is a battle. Judith Polgar's place as a (former) top 10 player in the world still puts her as the best woman ever.
  • Gen Whiny is on the move in Yale University yet again. According to Reason, "Some Yale University students are demanding changes to the English Department curriculum: specifically, they don't think it should feature so many English poets who were straight, white, wealthy, and male. "It is your responsibility as educators to listen to student voices," the students wrote in a petition to the faculty. "We have spoken. We are speaking. Pay attention....Students should feel comfortable challenging the notion that a Shakespeare or a Milton deserves his place in the canon: in fact, that sounds like an excellent subject for a classroom discussion facilitated by a professor. But professors need to actually teach students about these pivotal figures before those discussions can be had....In a brilliant piece for Slate, Katy Waldman eviscerates the idea that non-white students have nothing to learn from dead white poets: ...

    But even if you disagree, there’s no getting around the facts. Although you’ve written that the English department “actively contributes to the erasure of history,” what it really does is accurately reflect the tainted history we have—one in which straight white cis-men dominated art-making for centuries—rather than the woke history we want and fantasize about. There are few (arguably no) female poets writing in Chaucer’s time who rival Chaucer in wit, transgressiveness, texture, or psychological insight. The lack of equal opportunity was a tremendous injustice stemming from oppressive social norms, but we can’t reverse it by willing brilliant female wordsmiths into the past.

    In tangentially related news, Nicholas and Erika Christakis—who declined to protect students from offensive Halloween costumes—have finally resigned their positions at Yale." The adults in the room are now gone. 

  • The Federalist explains how the Gen Whiny came to be, "Forty-five years later: By this time, we had instituted a wildly successful pogrom in higher education, eliminating most of the conservative faculty and driving the rest underground. We ran amok, instituting freethinking, progressive reforms at every level of the system....Gradually, new codes of conduct were instituted to ensure everyone was comfortable at all times. ...The enlightened former freaks who now inhabit these campuses have become increasingly hypersensitive and nasty, spitting tacks at people for all manner of imaginary crimes such as “cultural appropriation.” Recently, a white guy got hammered for wearing dreadlocks. This is deeply ironic because, as I recall, we hippies were masters of cultural appropriation. Hookahs, Nehru jackets, bead curtains, reggae, Eastern religions, sitar music, Tibetan prayer flags, chakras, ethnic food, dashikis, Rasta shoulder bags, ironically worn military apparel, mandalas, henna tattoos, muumuus, hand-woven Guatemalan tunics, pyramid power, Maori tattoos, macramé... Meanwhile, at Emory University, some fascist from outside the perimeter jumped the razor wire and wrote “Trump 2016” on the sidewalk in chalk —a situation that could easily be remediated with a bucket of water. Instead, the student government allocated emergency funding for counseling sessions, and the college administration issued the usual limp apologies and assurances. Students were afraid to attend classes because they might sit near someone with a different worldview.....Today’s outraged, privileged, fragile snowflakes conjure up utterly trivial nonsense to consider as an affront: microaggression. This can include using the wrong one of more than 50 gender pronouns, sideways glances, snort-chuckling, eye rolling, resigned sighing, and even merely existing in proximity to a person with raw sensitivity. "