- The 2014 Sinquefield Cup has started. This year's event is actually the strongest tournament in the history of chess. It's a 10 round tournament consisting of World Champion Magnus Carlsen along with Hikaru Nakmura, Levon Aronian, Fabiano Caruana, Veselin Topalov, and Maxime Vachier-Lagrave. The first round featured some fighting chess with the position above from a wild encounter between Magnus Carlsen and Maxime Vachier-Lagrave which eventually ended in a draw when Carlsen went for the strong looking 25....Qd2 rather than the computer recommended 25....Qd3. There is live streaming of the event here and it includes analysis by Yasser Seirawan, Maurice Ashley, and [EDIT: Jen Shahade].
- I've added two pictures of a 3 ring and 4 ring dartboard (with scores in the rings) to the Graphics page. The scores are randomly created with Sagetex. They'll form the basis of a combinatorics question later on.
- As if students weren't coddled enough, now the American Academy of Pediatrics has issued a statement that “The empirical evidence [of] the negative repercussions of chronic sleep loss on health, safety and performance in adolescents … has been steadily mounting for over the past decade” and the article states the suggested solution is “In most districts, middle and high-schools should aim for a starting time of no earlier than 8:30 a.m. However, individual school districts also need to take average commuting times and other exigencies into account in setting a start time that allows for adequate sleep opportunity for students”. That simple solution is undercut by their own words, "Owens acknowledges that there is no single, simple answer to the problem of chronic sleep deprivation in adolescents. But “it's important to recognize that school start time delay is a necessary but not sufficient step to ensuring that teens get enough sleep,” she wrote in an email. “If school starts at 7:20 a.m., it's almost impossible for students to get enough sleep, but delaying start times is not a guarantee. This needs to be accompanied by healthy sleep habits (regular sleep-wake schedule, avoidance of caffeine, shutting down electronics before bedtime, etc.) as well as making sleep a health priority for the whole family.”". So good parenting is essential to this working.
- Remember when kids acting badly at school meant a fight during lunch? Infowars reports of 3 teens in Ohio (ages 14, 15, 17) have been been caught for performing the Knockout Game on unsuspecting victims and then stealing from them. Of course they filmed their exploits (see link). Now the eldest is going to be tried as an adult. He faces 16 years in prison(!)
Various points of interest.
- Quiz Time!: First question: What's the average teacher's salary (2012 data)? Second question: What's the amount of money spent on each student per year (2011 data)? You can get the answer to these questions and more at Education Next's recent article on Common Core. For those who can't wait, the answers are at the bottom. The article contains a lot of statistics and a major take away is that common core support has dropped noticeably amongst the public and has plunged with the educators who teach it.
- Some schools have already started the school year. And there's already been someone suspended when a Facebook post revealed the teacher allegedly instructed the students Tuesday to reenact a Ferguson, Mo. shooting known nationwide.
- There's a great video "Humans Need Not Apply" posted on You-tube. It's not directly math related but certainly related to the importance of education. But even good jobs aren't necessarily safe. Perhaps a good video for those early days when you still don't have class textbooks.
- The influence of the government in public college system is on display in Tennessee. In order to raise the graduation rates funding is now based on how many people graduate . Guess what? Graduation rates are now climbing resulting in most schools getting more money from the government. Problem solved! Now 25 states are following the new model.
- A (former) school administrator in Las Vegas was "...indicted last month in an alleged scheme to steal or misuse nearly $300,000 in public money". Her office was beautifully decorated: ""Persian rugs, vases, pottery, artifacts, Kachina dolls, waterfalls, a fire place. I was shocked.""...."Malich had only been Rocha's boss a few months when she saw the extravagance and immediately informed her supervisor. Grand jury testimony suggests the fine furnishings may have been there as long as three years."
- GM Nakamura battles Stockfish on Saturday, Aug 23rd. Chess.com has the details. It will be covered live.
One of my many criticisms of the public school system is that their funding is coming from the government which, due in part to lax oversight in a huge system, becomes a big piggy bank. Trying misappropriating $300,000 from a private high school and I doubt you'd have an easy time because most organizations will miss that much money. Notice in 4 that the government funding is encouraging weaker standards so more money can be drained from the system.
Quiz answers: 1) $57,000 2. $12,400
I've added a PDF version of the picture above to the Graphics page and if you're like most people, it needs an explanation. The curriculum I'm covering, along with the supporting book, talks only about the greatest integer function which is denoted by the book as . I find that annoying for several reasons. First, it isn't that clear from the name what the function does. Wikipedia mention that a similar looking symbol was used by Gauss and was the standard until the floor and ceiling functions were introduced in 1962. Secondly, the notation doesn't give any indication what the function does. So when you follow the book all sorts of needless confusion follows because the notation and naming aren't really intuitive. Compare that with the floor and ceiling functions with notation and . The little "feet" at the bottom of the floor function are suggestive of rounding the values down and the "feet" at the top of the ceiling function suggest rounding the values up. The floor function is the one that occurs most naturally in the curriculum, such as in determining the number of integers less than 1,000,000 that are divisible by either 5 or 2 (using the inclusion/exclusion) but learning about just the greatest integer function causes confusion because the students aren't sure (poor naming) about how to plot it combined with where is the open circle and where is the closed circle. Students should learn about the floor and ceiling functions at the same time.
The diagram helps to make sense of the floor function and ceiling function and the relationship between them and the line . All three functions have been plotted together and from that diagram you can see:
- if and only if is an integer.
- if and only if is an integer and if and only if is an integer.
- when n is an integer then the intervals have the ceiling function above the floor function; in fact, .
- By visualizing the graph (or using as a guide) the student can construct floor and ceiling function more easily and will remember which sides gets the open and closed circles.
The drawback in changing the notation and introducing the more standard notation are students complaining that the book doesn't cover the material so they shouldn't be responsible for it. Teaching to US students has been quite an experience. It would be great if the makers of math textbooks would teach floor functions and ceiling functions rather than the greatest integer function. Half a century behind--time to change! It will make the material easier to learn and take away student excuses.
- I've posted 2 more problem on the Sagetex: Limits page.
- Quanta Magazine (video and article) and Stanford News (article) have the best coverage of Maryan Mirzakhani, the recent winner of the Fields Medal. She is also the first woman to win this prestigious prize.
- New Scientist reports that a conjecture of Kepler on stacking fruit as a pyramid is, in fact, the most efficient way to do so. The proof "...used two formal proof software assistants called Isabelle and HOL Light" to verify much of the mathematics.
- The World Chess Olympiads have ended and the winner is: China! It's their first victory, 2 points ahead of the competition. Hungary was second and India was third. For the women's section the winners were Russia, China, and Ukraine. The chess world has changed so much! You can read about the results at Chessbase or Chess.com.
Several additions for today:
- I've added a new page Sagetex: Limits along with 2 limit problems (plus solution). You can find the link on the sidebar as well.
- Sage learners: he Sagemath Twitter feed reports that Sage tutorial videos have been posted. You can find them at YouTube's matsciencechannel.
- A professor emeritus at UCal Berkeley claims Common Core will move the US,
- Python learners: the e-books directory (see sidebar) has posted "Python Scripting for Computational Science". This is a must have book!
- I've added another problem to the Sagetex: Combinatorics/Probability page. The problem has students determine all the divisors of number. The problem has also been added to the Problems page.
- The NY Times has a good article: Why Do Americans Stink at Math?" and it's focus is not what you might expect from the title. One of Japan's most famous math teachers who, "once attracting 1,000 observers to a public lesson" goes to the United States to learn more about the best methods for teaching math only to find that, "The Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them."
- A judge dismisses a lawsuit filed by a teacher who was fired for critical comments she made on her blog about her math class. The judge, "...ruled that the defendants were within their rights to conclude that the teacher's posts 'would erode the necessary trust and respect between Munroe and her students.' ".
Several months ago I was reminded of how old I was. In a discussion involving numerous teachers I casually stated that "...statistics isn't really math" and the result was confusion. I quickly clarified to my position to say that although theoretical statistics is math (just analysis) a lot of the application (confidence intervals, regression, data analysis) isn't. That distinction didn't clear up the confusion. With time to think about the conversation I think it's a generational issue. The fact is I never had statistics in high school while in college the course wasn't required to major in mathematics (so I didn't take it). It wasn't until I was in graduate school that I had to take statistics and the closely related EDA (exploratory data analysis). Nowadays, however, statistics is a required part of the mathematical curriculum at the high school level so it's not so surprising that my younger colleagues have identified statistics as mathematics--they've been forced to study it in math class.
But just because statistics has a lot of mathematical calculations doesn't mean it's math. Engineering, physics, mathematical economics, and many other courses rely on mathematics yet they aren't called mathematics. You can see some of that distinction is recognized f you look at the departments of many universities. Bigger universities (such as Berkeley or Texas A&M) have a statistics department which is separate from the math department (much like engineering would be a separate department). Other schools (such as here or here or here) have a department of mathematics AND statistics; they specifically differentiate between the two subjects.
Even many people well versed in statistics/EDA recognize the difference; my EDA professor was quite vocal in telling us that EDA was more of an art than a science. He emphasized that no set of statistical measures (mean/median/mode/std/...) could do a better job at determining whether data was normally distributed than he could do by judging normal probability plots with his eyes. The behavior "in the tails" was particularly important.
There are other statistical experts who feel the same. In addition to W.M. Briggs (see his explanation of confidence intervals, regression, p-values) look at the work of statisticians George W. Cobb and David S. Moore who published among, among other articles, "Statistics and Mathematics: Tension and Cooperation". The American Mathematical Monthly 107 (7): 615–630 and "Mathematics, Statistics, and Teaching"
George W. Cobb; David S. Moore, The American Mathematical Monthly, Vol. 104, No. 9. (Nov., 1997), pp. 801-823. These articles and others can give you a more in depth, nuanced view (with plenty of examples) on why statisticians think statistics isn't math.
AMSTAT NEWS gives a quick summary:
Statistics, however, is not a subfield of mathematics. Like economics and physics, statistics uses mathematics in essential ways, “but has origins, subject matter, foundational questions, and standards that are distinct from those of mathematics” (Moore, 1988, p. 3). David Moore, statistics educator and former president of the American Statistical Association, gives the following four compelling reasons why statistics is a separate discipline from mathematics:
- Statistics does not originate within mathematics
- The aims and foundational controversies of statistics are unrelated to those of mathematics
- The standards of excellence in statistics differ from those of mathematics
- Statistics does not participate in the inter-relationships among subfields that characterize contemporary mathematics
Besides my simplistic observations that math gives exact answers while statistics will give you confidence intervals for the answer or that EDA can have multiple regression models for the data set (there isn't 1 correct model) you'll also find that people practicing statistics often get the wrong answer. The older crowd might remember "A Random Walk Down Wall Street"; the financial industry used the normal distribution for decades to model risk in the market. With decades of data showing the black swan events occur much more frequently than the normal distribution would predict, its been abandoned for "fat tailed" distributions. That doesn't sound like mathematics, does it?
Statistics is important, but it isn't really math and the spread of statistics into the math curriculum is deluding people that it is. The thinking process, as explained in the links above, is much different than the mathematical thinking process, so pushing statistics into math class is at odds with students learn the mathematical thinking process. As WM Briggs says, "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.....Much nonsense in the last century has been promulgated because of sloppy thinking in statistics. It is time to stop thinking about the mathematics and more on the meaning.". If we want students to get better at math we should stop the spread of statistics into math classes and replace it with math. Discrete math would be the natural candidate as it has applications to computer science.
I've added 2 problems to the Sagetex: Combinatorics/Statistics page.
Problem Type 10: Suppose a system has 6 independent components, each of which is equally likely to work. Now suppose the system works if and only 5 components work. What is the probability the system works?
In problem type 10 the number of independent components is random. Problem Type 11 has students calculate C(n,k) and P(n,k) for random n, k.
Several odds and ends:
- I've updated the Sage Cheat sheet for version 6.2; correcting errors, typos, and any deprecated commands. You can find it on the Python/Sage page (at the top).
- The OP/ED page of the NY Times has letters related to the common core article mentioned in a previous post. The letters are here.
- Wisconsin is gearing up for a war on common core. "If you think there was a fight over Common Core educational standards in the Capitol earlier this year, think again. Interviews with dozens of candidates show that it was just a warm-up for the war coming next year over those standards."...."WisconsinEye interviews with more than 70 candidates in Aug. 12 primary elections show that scrapping Common Core standards, or reasserting the authority of local school boards to set their own academic goals, is a top priority of Assembly Republicans. Most Democratic candidates defend Common Core."
Summer's here and I've had the luxury to just play around. I try to put extra math into the curriculum through either warm up problems, extra credit problems, or examples in the curriculum that I follow. Discrete math figures prominently in the extra math. In my opinion, discrete math deserves a prominent place in the high school curriculum; it certainly deserves a bigger share of the curriculum. But in a top down system you can't add extra material to the curriculum that the average math teacher has never seen and expect it to be implemented properly.
Graph theory is ideal because it requires little mathematical background to understand and solve problems yet it also provides a way to introduce theorems (many of a basic nature that high school students can understand) along with counterexamples thereby providing a basic primer into the sort of considerations that mathematicians are looking at. In the past, geometry had been the topic for introducing proofs but it's dry as dust and now common core has ripped out much of the proof content so that your typical student can go through math and with very limited exposure to theorems/proofs/counterexamples.
Graph theory would be a way to get that back in a way that is much more interesting than the dry "2 parallel lines are cut by a transversal" proofs that have bored generation after generation. You can even connect to matrices through the adjacency matrix of a graph. Topics like recurrence relations and generating functions complement the curriculum of calculus (power series) and probability. Unfortunately, common core has de-emphasized proofs and matrices and upped the exposure to statistics, so I think it's my duty to undue the damage of the curriculum.
Which brings us back to the current post. Bipartite graphs which have the same number of vertices in each partite set provide a way of determining whether a suitable job can be found for each person so that everyone has a job (see the "Matching" section for the link above). One partite set with vertices, say, A,B,C,D have edges to the other partite set of vertices (a,b,c,d) representing jobs based on whether the person is qualified to do the job. The question becomes, Is there an assignment of jobs so that everyone has a job they're qualified to do? If yes, then there is a (perfect) matching for the graph. If no then there needs to be an explanation as to why no such assignment is possible (Hall's Marriage Theorem).
As a teacher the problem I usually face is finding lots of examples that can be used in the classroom or on a test/quiz. Sage/sagetex is the tool that works best for me. The code generates bipartite graphs so that the number of vertices in each partite set is equal and the probability of an edge being selected for the picture is 1/2. The code is posted on the Graphics page along with sample output (the PDF shown above). The number of vertices is a random number set between 4 and 8 here: N = Integer(randint(4,8)). The probability of the edge being chosen is set here: if Integer(randint(0,1))==1:
It would be relatively simple to choose a larger number than 1, such as 9, and draw the edge if the number chosen is between 0 and 7 but don't draw the edge if the number chosen is 8 or 9. That would make the probability of the edge appearing to be 80%.
Through setting the parameters to your liking and repeatedly compiling the code you can generate many examples that you can use. If you're particularly ambitious you can add extra code spit out example that have (or fail to have) a perfect matching. Of course, Sage can help with that.