There's still no closure to the Shouryya Ray story I posted on several days ago, especially if you're looking at the news. The best summary was here, and it refers you to conversations on various websites. Missing from this account, though, is a different thread from Reddit. The basic conclusion: "User pelli at /r/math has worked out what Shouryya Ray has determined, and jsantos17 in one of the replies claims that this solution was implicitly known from a 1977 paper. Also, I agree with jsantos17's other remark, and wish to state so myself, that what Ray has worked out, almost certainly independently, would definitely put him in the 99th percentile amongst his peers and maybe even more. This effort is certainly appreciable and his result commendable."
That is, it appears that Shourrya Ray wasn't the first one to solve the problem. This conclusion matches up with talk on physicsforum.com:
When the paper was almost done, I discovered a 2007 paper that claims to have found an analytic solution. Here it is the reference:
Yabushita, K., Yamashita, M. & Tsuboi, K. (2007). “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method”. Journal of Physics A: Mathematical and Theoretical, Vol. 40, pp. 8403–8416.
Two different papers with the same result? Maybe this story is not as incredible as it seemed initially but I'll keep an open mind and keep looking for a more authoritative conclusion. If you hear any news, let me know!
If we're really interested in improving the math abilities of high school students it would make sense to have some idea about what mathematicians actually do. We're at an interesting time where all sorts of changes are taking place; some thoughtful changes might go a long way towards improving student performance. One of the more misguided changes is the pervasive use of (expensive) technology. Properly used, technology can be very powerful but when it's used improperly, it can do some damage. Often times, for example, it seems that students are pushed into using a calculator before they've the solid math background needed to use them properly. As a result, the numbers sense and algebraic skills of high school students are shockingly poor. From my experience most mathematicians use calculators sparingly, if at all, so why the pressure to use them?
The mathematician Alfréd Rényi has said, “A mathematician is a machine for turning coffee into theorems”. Plain and simple, mathematicians prove things but it's a little more complicated. Here's a list of things that mathematicians do.
- Communicate with other mathematicians about interesting problems and results.
- Find interesting ideas, concepts, and questions to study by reading mathematical papers, attending math conferences, and interacting with other mathematicians.
- Gather evidence on the problem they’ve chosen to study.
- Form conjectures based on their investigation of a problem. This is inductive reasoning.
- Prove their conjecture or come up with a counterexample to show it is false.
- Write a formal explanation and constantly review it for mistakes as well as clarity of explanation.
- Look for ways to apply or extend the result.
- Submit their proofs to mathematical journals for review and publication.
- Present their findings at mathematical conferences.
The important news for the week has to be about a 16 year old German high school student, Shouryya Ray, solving a problem related "... to the movement of projectiles through the air, in the 17th century. Mathematicians had only been able to offer partial solutions until now." as well as a second problem (mentioned in the link). Some other accounts provide additional details:
- here: "Many professors, mathematicians and physicists had previously stated the problems Shouryya Ray solved were “uncrackable.” In fact, only partial solutions had been discovered up to this point with the help of a supercomputer."
- here: "On a visit to the Technical University in Dresden pupils received raw data to evaluate a direct numerical simulation – which can be used to describe the trajectory of a ball when it is thrown."
The original article, if you're interested seems to be this; it's in German but my Chromium web browser provides a translation (which is quite clumsy in places). The Telegraph (in India) seems to do a better job translating it but at one point says, "Some physicists are still unclear about Ray’s achievement. 'The flight of a ball is classical Newtonian dynamics —all we need (to describe the motion) is the angle at which the ball is projected and the coefficient of elasticity which describes the interaction of the ball with the wall,” said K. Subbaramaiah, an executive member of the Indian Association of Physics Teachers.' ". I'm certain there will be more details in the weeks to come. This video from a news channel is about Shouryya Ray.
UPDATED: This story addresses some of the concerns mentioned above:
- "A research paper that claims to fill in a gap in Isaac Newton's formulas for the physics of falling objects has drawn worldwide attention to a 16-year-old student in Germany, but physicists are reserving judgment until they've seen the proof."
- "Ray's paper was a school project submitted for a contest, and thus not subject to the publication process and peer review that professional work typically goes through. For that reason, the experts are reluctant to weigh in."
- " 'This story seems rather suspicious," Richard Fitzpatrick, a physicist at the University of Texas in Austin, told me in an email. "None of the news reports give any details of the calculation. None of the people who hailed Shouryya Ray as a genius are scientists, and none of them give the impression that they have seen the calculation in question. It is impossible to gauge the scientific merit of the calculation until it is made public.' "
- "His paper putting forth an "analytical solution to two fundamental unsolved problems" may not be the breakthrough that some of the reports have made it out to be, but that doesn't take anything away from the teenager's achievement."
- "University of Bristol physicist Michael Berry: 'Without seeing the details of what Ray has claimed, it's impossible to comment intelligently. It depends crucially on how he has modeled the air resistance. But a falling body with air resistance (however modeled) is hardly a 'fundamental unsolved problem,' as he seems to think. There's a powerful aroma of hype.' "
The link also shows Shouryya Ray holding up a piece of paper with his equation on it. Perhaps that will help the experts decide exactly what Shouryya Ray has accomplished?
There's an interesting site you should keep your eye on: TED-Ed. As one of the videos explains, they're looking for educators to submit their best lessons, after which they'll work with the teacher to get the lesson down to under 10 minutes. With the help of some animators, they'll try to make that lesson come to life. These videos are posted along with a lesson; they have quizzes, open ended questions, and even resources for those who want to investigate the topic further.
The site also allows teachers to flip the lesson so they can adapt the video to better suit their class (changing the title, adding further instructions, changing the quiz questions, and adding their own follow up resources). The resulting flipped lesson is then accessed at a different URL and you can track the results of the people who go there (did they pass the quiz?). You can even flip lessons from YouTube as well, giving you access to a lot of videos that can be used for your class.
The site has only recently opened so there aren't that many lessons available right now, but take a look at the video How Folding Paper Can Get You to the Moon. The video is an introduction to exponential growth which has been beautifully explained and animated. Note the "Quick Quiz", "Think", and "Dig Deeper" links on the side.
This site looks promising, so I've added it to my list of links.
I've already posted 2 different proofs of Thales' Theorem; the second version used coordinate geometry to show that slopes of 2 sides were negative reciprocals of each other. The proof today also uses coordinate geometry along with the converse of the Pythagorean Theorem. As with Thales' Theorem (2), regardless of the radius of the circle, it can placed at the center of the coordinate plane where an individual unit is one radius. That is, we can represent any circle as a circle centered at the origin with a radius of one. The proof is on the Handouts page. The graphics are coded into the .tex file. If you want the graphics inserted as PDF files, the figures can be found on the Graphics page (Thales and Thales2).
It's been almost 10 years since I had to teach matrices in high school. At the time, I structured the course around using matrices to encode and decode secret messages. It went over well; the tedious process of matrix multiplication is a lot easier to bear when there is a secret message waiting for you. I didn't have Sage to help me back then, so I thought I'd try putting together a bit of Sage code to help illustrate the process for 2x2 matrices. The use of a table to show the encoding scheme helps make the output look a little nicer.
The code is posted on the Python/Sage page and the output (above) is posted on the Sage Output page. The code runs in Sage 4.8 and 5.0.
As you're learning the basics of chess tactics, make sure devote a lot of time to skewers because they happen so frequently. Skewer are an attack on 1 piece (directly) and indirectly an attack on another piece behind it. The most effective type of skewer is one that involves check because it limits the opponent's possible replies, making it more difficult for the opponent to escape the skewer safely. I've add a piece on skewers to the Chess page, both the .tex file and the PDF.
There were 2 fascinating interviews on the Chessbase website this week: One was the interview with Steve Giddens on May 16th, and the other with former World Chess Champion Garry Kasparov on May 18th (video, partial transcript given). The Giddens interview was centered around the current world chess championship and the influence of computers on the game. He gives his thoughts on how computers are killing the game. I'd go further and say the game is dead. Computer scientists solved checkers some years ago and although the game hasn't been solved (so that we know whether the game is a win/loss/draw with best play) parts of it have been (endgames with 6 or fewer pieces and perhaps the King's Gambit). Computer scientists focused their attention on the game of go some years ago.
Giddens points out that the leveling effect of computers make it difficult because "...everybody is analysing the same opening lines, using the same powerful computers and programs. As a result, everybody is coming to the board, with much the same opening preparation, with the result that nobody can get a serious opening advantage any more.". When 2 top players (such as in a world championship match) are playing against one another they also rely on the same computer programs and, "The result is a whole series of effectively contentless games, where the players are just checking each other's computer-aided preparation. Once in a while, they will hit on a gap, and get some advantage, but most of the time, there will just be what we have already seen in Moscow – 15-20 moves of preparation, 4-5 more accurate moves, a dead position, and a draw.". Giddens, in my opinion, is right on target and his well spoken interview will give you plenty to think about.
The second interview (with Kasparov) was rather long but I found 2 salient points: the decline of Anand and Kasparov's opinions on the changes that chess has gone through (around 20:45 on the video). Kasparov talks about how his match with Anand was a milestone in using the computer in a world championship match. Kasparov used a computer to check the analysis of an opening line he used in Game 10. He says that the trend of using computers has gone "...one way..." with computers playing an increasingly important role in every strong players preparation. They even change the way players evaluate the positions. Kasparov goes on to talk about the evolution of chess (26 minute mark on the video) about the shortened time controls, cancellation of adjourned games, various regulations (Sophia rule, wins being awarded more than one point), and the various calls by some for Fischer random chess to compensate for the computer killing the opening.
If you're interested in chess, I think you'll find both interviews worthy of your time.
I've added some 3D solids to the Graphics page: a rectangular prism, cube, sphere, cylinder, pyramid, and cone.
The newest version of Sage, version 5.0, was released a couple of days ago. I wasn't in a hurry to download the latest version because my previous version is working fine and updating has been a hassle. Was a hassle; it looks like things have steadily improved for the Linux crowd. Searching around on the internet I found this, which explains that Ubuntu 12.04 users can quickly get Sage onto their computers with a couple of terminal commands:
apt-get install sagemath-upstream-binary
Encouraged, I decided to take the plunge and after about 15 minutes I had Sage 5.0 ready to go on Ubuntu 12.04. After the download the Sage icon was in the applications menu (because Unity sucks) under "Other", so I just dragged it onto the Cairo dock.
Now Sage is just a click away. Great job!