Math teachers at play carnival
This is a Math teachers at play carnival, issue #
which I am going to reveal shortly. See if you can make it from what is known as a single image stereogram. Try focusing your eyes behind the screen.
In this month issue:
What is the number of this issue?
There are three books in my library that are exclusively devoted to listing all kinds of information about all kinds of numbers – as is well known no number is uninteresting. The granddaddy of them all (at least in the recent times) is The Penguin Dictionary of Curious and Interesting Numbers by David Wells. There is The Kingdom of Infinite Number by Bryan Bunch and the latest Lure of the Integers by Joe Roberts. There are marvelous websites, the wikipedia and Number gossip that place at one’s fingertips a huge number of properties of a multitude of numbers.
Well, in the announcement of the current issue I offered a simple problem whose solution is this issue’s number. Subtracting this number of years from 2010 gives a number whose last two digits are this number written with the digits swapped, i.e., in the reversed order. The carnival is in its prime, but the number, being triangular, is not. Now, what’s number is the next issue? There are two triangular numbers that are good candidates for the main condition: 28 and 55, for
is a distinguished number rich in properties and associations. 28 is happy, perfect, hexagonal, triangular, and what not … Here are a few less known (i.e., to me until recently) facts.
Bryan Bunch wrote of the discovery of the main asteroid belt. In 1776 the astronomer J. E. Bode published the discovery made four years earlier by J. D. Titius, that the distance of each of the planets from the sun is determined by a simple sequence which is now known as Bode’s Law. Write 0, 3 and then double the last number on each step: 0, 3, 6, 12, … Then add 4 to every term: 4, 7, 10, 16, 28, 48, … If the distance to from Earth to the Sub is taken 10 units, then Mercury is at 4, Venus at 7, and Mars at 16 units from the Sun. Jupiter, the next planet, is at distance of 52 (think of it as close to 48), skipping the expected 28.
On the New Year’s Day, 1801, the Italian astronomer G. Piazzi found a previously unknown, but very small, “planet”. The great K. F. Gauss devoted many hours to calculating its orbit and found that it corresponded to the number 28 in the Bode’s sequence. The “planet” was named Ceres and in time was shown to be the largest (950 km or 590 mi in diameter) among the myriad of asteroids in the main asteroid belt.
Joe Roberts gives several uncommon properties of 28. The least positive integer having h divisors is denoted
There exist exactly 14 values of N for which the least common multiple of all integers from 1 through N is minimal. 28 is the largest integer with this property.
0.28 is commonly cited as an element of the 14-terms long solution to Steinhaus’ problem (Find an N-term sequence with terms in the closed interval [0, 1], such that the first two are in different halves of the interval, the first three are in different thirds, …, and all of them are in different Nths of the interval.)
Sad news
One of the topmost Russian mathematician Vladimir Arnold, after whom one of the asteroids – Vladarnolda – in the main asteroid belt has been named, passed away on June 3, 2010. Arnold was an outspoken critic of the trend in math education known in the US as the New Math, while curiously, his teacher, Andrey Kolmogorov (indisputably one of the greatest mathematicians of the 20th century) was the initiator of that trend in the USSR. Arnold considered mathematics in part as a natural science. In his view, mathematics is being developed by trial and error, modeling and experimentation, with proofs coming last as a method of hypothesis verification. Both aspects of mathematics ought to be reflected in mathematics education.
In June I learned of passing in January 31 of Steve Fisk, professor of mathematics at Bowdoin College. Fisk became a math legend having dreamed up an almost trivial proof of Chvatal’s Art Gallery theorem while on a bus trip somewhere in Afganistan. The incident puts another big question mark to the notion that mathematics is the product of the left side of the brain.
Interesting and relevant news
The controversial Russian mathematician Grigory Perelman has been awarded the Millennium Prize by the Clay Mathematics Institute in Cambridge, MA. The award honored his solving of the Poincaré conjecture, which characterized the three-dimensional (which is a four-dimensional object) sphere among other three-dimensional manifolds. As was not unexpected, Perelman turned down the $1 million prize; four years previously he declined to accept the Fields Medal. In a telephone interview, Perelman, 43, told Interfax: “I don’t like their decisions, I consider them unjust.” He judged his contribution not greater than that of Columbia University mathematician Richard Hamilton.
Jim Carlson, President of the Clay Institute, said institute officials will meet this fall to decide what to do with the prize money. “We have some ideas in mind,” he said. “We want to consider that carefully and make the best use possible of the money for the benefit of mathematics.”
Will they seek Perelman’s advice? I doubt it.
Guillermo Bautista came up with the idea of a new Mathematics and Multimedia Blog Carnival. The first issue saw the light of day on July 12, 2010. Meanwhile, the Carnival of Mathematics saw its 67th issue.
Next Math teachers at play carnival will be hosted at The Number Warrior blog.
From the trenches
Tom DeRosa of the I Want to Teach Forever blog shared his experience of teaching operations with fractions using the play cards and a Simple Graphic Organizer to “make fraction a little less painful to student.”
Bill James of the Discrete Ideas blog has put together a table of the most common fractions and showed how “Learning the first 12 fractions can make it super-easy to do division in your head and produce answers down to the 10ths or even 1000ths quickly and easily.”
I once observed that there is a way of introducing addition of fractions without a formal definition. Nothing more difficult than dividing a few apples between a group of boys. I also found a couple of fractions related bloopers that illustrate the persisting difficulty the populace is having with fractions.
Caroline Mukisa who is said to be on a mission to help parents to support their children Math learning and is a big fan of sneaking math into children’s (and parent’s) diets, authored a guest post at Sol Lederman’s Wild About Math blog. Caroline shows how the many fascinating facts gathered in a Guinness World Records book naturally lead to simple math inquiries. Caroline’s post reminded me of Eric Charlesworth’s book 225 Fantastic Facts Math Word Problems (Scholastic, 2001). He, too, tries to enliven the study of mathematics by exploiting various world records, incredible animal facts, historical anomalies, and such.
Denise from the Let’s play math blog offers an advice on how to start a math teacher blog. With her blog high ranking, who could know better. A must read for all aspiring bloggers.
Cindy from the love2learn2day finds absorbing activities for little kids even in the insect world. Seeing and identifying patterns at an early age contributes greatly to the mathematical development of children.
Guillermo P. Bautista Jr. submitted one of his great tutorials. This one is a very lucid Introduction to Permutations.
Tracy Beach made a well argued pitch for the DreamBox, a project of virtual manipulatives to help teach early numeracy.
Maria Droujkova and Linda Fahlberg-Stojanovska have co-hosted an Elluminate Workshop on using and teaching with GeoGebra. Materials for this workshop are available at GeoGebra’s wiki site.
Math curiosities
Patrick Vennebush, the Online Projects Manager for NCTM, hosted a July 7 webinar, where he discussed Calculation Nation, an online world of math strategy games, which is part of the NCTM Illuminations project. In his personal blog, Math Jokes 4 Mathy Folks, Patrick ponders over an argument he came across recently on the web. It was said that “the Senate gives just 18% of the U.S. population the power to stop a bill from passing Congress. That is, if 50 Senators vote ‘no’ to a bill, then it fails, and the 25 least populous states represent just 18% of the population.” This piece of statistics served a basis for a claim that the US Senate was no longer necessary. Along the way, it was implied that the Senate might have been necessary when it was first created, to give a voice to smaller states. Patrick checked the statistics of the first Senate that had only 24 senators from 12 states. At the time, according to his calculations, a bill might have been stopped by the senators representing only 17% of the population. As he put it, “Please understand, I’m not arguing that the Senate should be retained or abolished. But by the numbers, it appears that the Senate might have been even less necessary in 1789 than it is today.”
Shecky Riemann from Math-Frolic shares his life-long fascination with the Galton Box (the device which is also known as a “quincunx.”) His post, A Museum Piece describes the appeal of the contraption held to his even in childhood: “… like a magic, unseen hand guiding the fate of those individual spheres — even though each one took a rather random, unpredictable journey, the end result was highly predictable and little-changing. Even as a youngster I sensed there was something profound in that.” The post contains a list of links to an explanation of the mathematics behind the workings of the device, a video of a demonstration and an interactive simulation.
James Pollack in his new blog takes a fresh look at the Birthday problem.
Mimi Yang does an investigation into the problem of the best viewing angle in an iMax theater. Regiomontanus would have loved her submission, complete with diagrams and graphs.
Real mathematics occurs on most elementary levels. In a well known story, J. Piaget tells of a child who grew excited after discovering that the result of counting is independent of the order in which items have been counted. Another example that never fails to surprise both children and grownups is the question of whether it is possible to draw the same curve on two very different surfaces, say, a sphere and a cube. (For a cube, stipulate that a vertex must be inside the curve.) I would like to solicit such examples of “minimal mathematics”. I have put together a post Most elementary aha! moments to which comments can be added.
Sam Alexander from xamuel.com submitted an article on a variant of König’s Lemma that deals with trees in a graph theoretical sense. A tree is a connected graph without loops, meaning there is always one and only one way to get from any one node to any other along the edges of the graph. One of the nodes is usually chosen to be a root. The length of the path from a node to the root defines the level of that node. König’s Lemma says that an infinite tree all of whose levels are finite must contain an infinite branch. Xamuel applies the lemma to family trees under an apparent assumption that the big crunch (see, for example, S. Hawking’s A Brief History of Time) is never coming, and to games on an infinite board.
Vi Hart co-authored a paper with Erik and Martin Demaine, Computational Balloon Twisting: The Theory of Balloon Polyhedra. One motivation for balloon twisting is education. Balloon twisting is fun: the activity can both entertain and engage children of all ages. Thus balloon twisting can be a vehicle for teaching mathematical concepts inherent in balloons.
Vi Hart offers step by step instructions for creating ballon polyhedrons. She also wrote a music piece for a music box on a Möbius strip.
This kind of musical composition is known as a canon. In a canon, copies of a single theme are played by the various participating voices. The copies can be shifted relative to the main theme, played backwards, or be inverted. The difficulty is of course to have the various voices coalesce in a harmonious whole. Douglas Hofstadter’s Gödel, Escher, Bach gives a literary expression to this idea while exploiting the manner in which it permeates the mathematics of K. Gödel, the art of M. C. Escher and the music of J. S. Bach.
I am grateful to Victor Gutenmacher for pointing me to the following youTube video
Humor
Lemma: when communists put you to death, you lose on average half of your expected lifetime.
On a lighter note, my wife came across a couple of jokes in a Beginner’s Spanish Reader book. Both are probably folk tales.
A farmer went to market and bought 4 donkeys. He rode one and drove the rest home. Upon reaching his house he decided to check on the acquisition. He counted: 1, 2, 3 in bewilderment. Agitated, he called his wife. “Look here! I know I bought 4 donkeys but there are only three of them now: 1, 2, 3.” The wife looked and replied, “How strange! You see only three and I see 5 donkeys: 1, 2, 3, 4, 5.”
A student of mathematics was visiting his parents on a Christmas break. At breakfast, his mother put on the table a plate with two hard-boiled eggs. The son decided to play a joke on his father. After all, wasn’t he a very advanced math student?! Having removed and hidden one egg, he asked the father how many eggs were there. Naturally, the father counted 1. The student put the hidden egg back on the plate and asked the same question. “2″, replied the father. “So, in all, there are 3 eggs, right?”, said the student. “We counted one egg before, and two eggs later, and together this makes three eggs.” At this point, his mother intervened. “You are absolutely right, there are three eggs indeed. But let’s eat already. I’ll take the egg nearest to me and the brown one will go to your father. You’ll have the third one.”
P.S.
With sincerest and humble apologies to all the carnival participants – reader and authors – I am adding submissions that were received before the deadline but after the carnival issue went out due to an unfortunate accident. What I was thinking when hitting the Publish button two days earlier? I am certain I was not thinking about pressing that button.
John Golden, a.k.a. Math Hombre, celebrates a full circle day (6/28) with an article on Similarity and π.
John Cook, in his The Endeavor blog, writes about the Lincoln Index which is commonly applied to population estimates. As John shows, the index reliably estimates the number of errors that cropped up in a computer program – the place the bugs live in and thrive.
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July 13th, 2010 at 12:31 pmReally get carnival, so many interesting ideas. A great range of posts!
July 13th, 2010 at 5:30 pmThank you. I was quite worried doing this the first time.
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July 14th, 2010 at 12:51 pmNice collections. Thanks for including mine. You are also invited to host the Carnival of Mathematics and Multimedia.
July 15th, 2010 at 1:23 am[...] July 15, 2010 Guillermo Bautista Leave a comment Go to comments The Let’s Play Math Math Teachers at Play #28 Blog Carnival is now available at CTK insights. My article on Introduction to Permutations was one [...]
July 15th, 2010 at 4:15 amThank you. I may consider doing that in the future, but certainly not right away. I still need to make up on a few plans I had to postpone because of the carnival.
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July 15th, 2010 at 7:42 am[...] to send in your submission for the Math Teachers at Play carnival, coming this Friday now posted at CTK Insights. THE NUMBER 1 Welcome to the Mathematics and Multimedia Blog Carnival Number 1. Before beginning, [...]
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August 23rd, 2010 at 7:29 pm