# 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 Portlandtribune.com 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.
• Edsource.org 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.