Problems: discrete math

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

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

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

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

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

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

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

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