Colin McAllister's blog

New Mathematics Problem Solving Group

You are welcome to join the new "Math Problem Solving" group at Mathematics24x7.ning.com. Rashmi Kathuria, creator of the Mathematics24x7 social network, has kindly permitted me to add the discussion group. The social network has 145 members including maths teachers and maths enthusiasts. I'm sure that some of them will be happy to join you in discussing and solving maths problems. Pose your maths problems there, or add comments that lead towards a solution or understanding. Embed equations in your text as images, if they improve the presentation of your discussion. Mathematics24x7 has diverse blog posts and topics about teaching mathematics at all stages.

Although I am not a mathematician, I have added three problems to get the discussion started:

Algebra and Graph Theory
Combine Resistors to Achieve Minimum Current
and save the planet!
Based on Ohm's Law, Kirchhoff's Rules and 10% tolerance resistors.

Geometry and Algebra of Irrational Numbers
Largest Semicircle Inscribable in a Unit Square
or Corn Circles 101.
From the topic "maxima & minima" on a Mathematics forum on Orkut.com

Algebra and Differential Calculus.
Problem of Monkey Climbing a Chain
Can you solve it faster than the monkey?
From the Friendster group "ELITE MATH CIRCLE".

The Combine Resistors problem is my own design. The other two problems are borrowed from maths groups on Friendster.com and Orkut.com, and have hyperlinks to those groups for reference.

Problems may range from Kindergarten to Postgraduate level and may cover any subject involving mathematics. Discussions on Ning.com social networks support images embedded within the text, which is useful for algebra. You may render equations as images, using an on line equation editor, and upload them into your discussion. 

Editing Equations on Web 2.0 Sites

Quadratic_Solutions

Support for equation editing is a weakness of all Web 2.0 sites that I have encountered. Students and lecturers of science and mathematics need to use equations and formulae to develop their work. The blackboard or whiteboard is convenient in the classroom, but academics also need to share their work online. Social networks and Web 2.0 sites are the ideal forum for sharing, but they are not geared for academic use. Programmers of FORTRAN, ALGOL and C already have a solution; they have been expressing equations in plain text for half a century. It is common to see x-squared written as x^2, and the square root of x as x^.5, (x to the power of one half), but many people would not be familiar with that shorthand. Some Web 2.0 sites support HTML markup when editing posts and comments, and provide format controls like Bold and Italics on the toolbar. Academic discussions would be considerably improved if Web 2.0 sites would add just three buttons: Superscript, Subscript and Greek letters to the editing toolbar. Some social networks, such as LinkedIn.com and Friendster.com do not support any HTML editing. The Hi5.com social network supports HTML in discussions, but you need to code it by hand, or with HTML editing software such as Kompozer, and paste it into the edit box. Most blogs let you attach an image to your post, so a solution is to use an online equation editor, as at CodeCogs or Thornahawk, to generate an image of your equation. For example, the expression "\frac{-b\pm{\sqrt[]{(b^2 - 4*a*c)}}}{2a}" renders the formula for solutions to the quadratic equation a*x^2+b*x+c=0. I uploaded the rendered image "Quadratics_Solutions.gif" to this blog post. One popular Web 2.0 service, the online collaboration tool Zoho Writer has a LaTeX Equation Editor. According to the Zoho Blog: "As you may know, a significant % of our users are students. We got a lot of requests from this user segment to build an Equation Editor into Zoho Writer. And Zoho Writer has it now." The tool that Zoho added is LaTeX, a document markup language developed in the 1980's. If you host a blog or web page, you can use equation rendering plug-ins or cgi programs such as Yourequations.com or MathTex to embed equations into your online content.
This topic is also being discussed at Mathematics24x7.ning.com and on the Math, Math Education, Math Culture group at LinkedIn.com.

Representing Scale in Mathematics

Smallest_Transistor_1999

The conception of scale and its representation and exploration are critical mathematical skills. Analogies are commonly used. A transistor is two thousand times smaller than a human hair. Antarctic ice is as deep as a dozen football fields. Analogies are not always helpful, and it should not be necessary to repeat them every time that science or technology is discussed. Exponential or Scientific Notation or powers of ten provide a convenient representation of distance, time, money and other measurements. We use words like "trillion" in Finance and "nanometer" in Physics, as symbols for powers of ten, to help us reason about the very large and the very small. Exponential notation is used on a scientific calculator. Sometimes powers of two are used, as in defining a Kilobyte of memory, or repeatedly folding a sheet of paper in half. An understanding of scale helps us to understand science. For example, travelling to Mars would be much harder than travelling to the International Space Station. Both are in space, but Mars is about one million times further away. Zooming, as used in photography, is a transformation that helps us to appreciate different scales. It is the dynamic version of the relationship with the real world that we infer when looking at a model car or the map of a country. Scaling relates to the concept of similar triangles in geometry. M.C. Escher used one scaling transformation, now known as the Droste Effect, in his mathematical prints. Zooming is an essential tool in navigating computer media and makes graphical content more accessible by letting us adjust the scale to match our visual acuity.The space simulation software Celestia provides an "exponential zoom feature that lets you explore space across a huge range of scales". Fractals show self-similarity at many scales, and graphical presentations let you zoom in to theoretically limitless depth, as shown on this video, Baroque Mandelbrot Zoom on Youtube. In summary, representations of scale include: exponential notation or powers of ten, maps and models, zooming out to explore simulations of space, or zooming in to explore the structure of fractals. This post was my response to a discussion Multiple Representations, with math teachers on mathematics24x7.ning.com.
Transistor image is from the linked 1999 BBC article.

Heartening Success of Nesin Mathematics Village

The Nesin Mathematics Village, or Nesin Matematik Koyu, in Turkey arose from controversial beginnings. I first read the story "Mathematics under arrest" of its controversial beginnings in 2007, when Professor Ali Nesin was unfairly charged as a criminal for operating an unauthorised summer school. Closure of the school by authorities in 2007 was condemned by mathematicians around the world, including Alexandre Borovik, of Manchester, England. I recalled that story while contemplating what it means to be a mathematician and the waste that occurs when authorities deny mathematicians that role, as in the case of Chen Jinrun, when criticised by rebels during the Communist Revolution in China. A quick search of the web shows that Professor Ali Nesin's small school has grown into a heartening success, an educational village dedicated to giving young people the opportunity to focus on mathematics. "Nesin Mathematics Village, random faces" by Alexandre Borovik shows that the village is thriving, and includes a photo of the founder, Ali Nesin. A maths undergraduate in England tells more of the story, in "Nesin Mathematics Village in Turkey", and shows interest in beginning something similar.

Mathematicians or Mathematics Enthusiasts?

Wikipedia defines that "A mathematician is a person whose primary area of study and/or research is the field of mathematics.” Although I use some mathematics in my work, and the subjects that I teach have some mathematical content, I do not call myself a Mathematician. As academics, we must be careful not to misrepresent our expertise. In my blog post The Agile Mathematician Chen Jingrun, I began with the disclaimer "I am not familiar with his work" lest I misrepresent myself as being a mathematician. Conversely, we should not be hasty in denying that title to someone who claims it, like the rebel leaders who misjudged Chen Jingrun. What are the circumstances in which a teacher, technologist or amateur may use that title? The Wikipedia definition may be too limited, and it is not definitive, as Wikipedia may be edited at will. Everyone is a mathematician (with a small "m") at least some of the time. But the title Mathematician is not like the title Motorist for someone who drives a car. It implies that the holder is advancing the field or at least striving to do so. Old mathematicians die and young ones take their place, so teachers are advancing mathematics, even if they do not publish papers. I won’t press this matter any further, but will focus on using mathematics, rather than discussing the meaning of words. For now I will say that I am a Mathematics Enthusiast. This post is also on mathematics24x7, with comments.

Thinking Upstream About Ideas and Human Rights

WithoutArticle19

My other blog Thinking Upstream, about ideas and human rights, is based on online news of some of the bloggers and journalists who remain unjustly imprisoned in several countries. As one blogger among many, I hope to draw attention to the people mentioned, urging that we each seek peaceful ways to secure their freedom. They have been hurt by authoritarianism and its cruel repression, and by those who execute that repression or gain from it. I have no experience of the countries mentioned, but am convinced of the accuracy of the news, by the variety of reputable publications, to which I refer in the hyper-links of my posts. I blog about human rights because: all freedoms and rights are fragile, and at risk of being lost, if we are even a little careless. Thinking Upstream is now a PDF file on Scribd.com for easy offline reading, even in places where Friendster is blocked. I used Google Translate to machine-translate it into various languages, also in PDF format on Scribd.com.Thinking Upstream is at: http://cmcallister.blog.friendster.com/. Your comments are welcome. The image refers to United Nations' Article 19, and contains photos of victims of repression.

Boosting Motivation For Learning Mathematics

Mandelbrot_Set

In the discussion, On teaching math successfully, on the LinkedIn.com forum, Math, Math Education, Math Culture; Dan, a math teacher at a community college, asks "... how to motivate students to learn math?" I replied: Dan, there are two branches of your question on this forum, but I think that we can handle that. David's reply on the other branch made me laugh, as he joked about bad teaching practices that we can see in others but not in ourselves. He was joking about their value as teaching methods? Can we appeal to heroes, like Chen Jingrun or Grigory Perelman? Or are they in a different league from most of us who merely use mathematics, rather than creating it? One student rebuked me, in the group All Science Professionals on Hi5.com for suggesting some famous scientists as role models, saying that he would find his own way as a scientist, and I think he will be right. Mathematics is crucial to many subjects, and I have been attempting to promote it on my Vox blog and in unlikely places like the Hi5 (Mathematics group) and Friendster (Mirimatics group) social networks. Mathematics is created, and we cannot predict what it will be like until it happens, so there is a sense of adventure. The visible originality of the Mandelbrot set (1978) is a good example of a recent mathematical discovery. The unconventional role models whom I mentioned, discovery of the unknown (even as a spectator), the mere usefulness of mathematics; these certainly motivate me. Feedback is important. How will you know if you succeed in boosting your students' motivation? If you have a website, that is a tool that you could use to show off your interest in mathematics. (The above image of the Mandelbrot set is the first of a fascinating sequence of images at Wikimedia.org.)

The Agile Mathematician Chen Jingrun

Chen_Jingrun

I am not familiar with his work, but I found the story of Chen Jingrun (1933-1996) fascinating. He graduated from Xiamen University in 1953 and became a researcher at the Chinese Academy of Sciences. His work led to progress in analytic number theory. His Chinese home page is at the Chinese Institute of Mathematics. A short biography "A great mathematician Chen Jingrun" is on http://city.chinaassistor.com/. His 1966 paper was on ""On the representation of a large even integer as the sum of a prime and the product of at most two primes". The cultural revolution put a halt to research, so Chen's 1966 Theorem was not made public until 1973. His work continues to be relevant to recent research in number theory. Mathematician Jason Dyer reports on "Carnival of Mathematics #43" in the blog "The Number Warrior". He explains why 47 is a Chen prime and 43 isn't. He also introduces Roth’s theorem, linking to a paper "Restriction theory of the Selberg sieve, with applications" by Ben Green and Terence Tao, which is available as a free PDF file or PostScript file from arXiv.org. There is a telling anecdote about Chen Jingrun at "Extraordinary Chinese Sayings, 1840-1999 - Part 1", from the book Extraordinary Sayings (非常道) by Yu Shicun (余世存). During the Cultural Revolution, the people criticizing Chen Jingrun (陈景润) said: "Let the Goldbach conjecture go to hell! What is so big deal about 1+2? Isn't 1+2 equal to 3? You eat the food grown by peasants, you live in a house built by workers, you are protected by the People's Liberation Army and your wages are paid by the nation so that you can study 1+2. What is this? This is fake science!" So Chen jumped on the table, went through the open window and leapt downwards. But when he jumped out of the third-floor window, he was nicked by an awning and therefore only suffered some scratches on his leg when he landed on the ground. A rebel leader looked at Chen and said: "It is no wonder that you are a famous mathematician. You even know how to select the angle when you jump out of the window!" As reported in How Chinese's idols have changed: On February 17 1978, People's Daily and Guangming Daily introduced Xu Chi's article "Goldbach Conjecture" from People's literature. This was the day when hundreds of millions of Chinese learned of Chen Jingrun, who was honored as China's "jewel in the mathematical crown".

Century Old Topology Problem Solved in 2006

Grigori_Perelman

In case you missed the solution of the century old Poincaré conjecture in 2006, I'll repost a link to the New York Times article "Elusive Proof, Elusive Prover: A New Mathematical Mystery". The story of the brilliant Russian mathematician Grigory Perelman (photograph) who identified the solution in 2003 is as mysterious as Poincaré's problem. Wikipedia describes the "Solution of the Poincaré conjecture", for non-mathematician, explaining the problem with doughnuts, balls, a cigar and even mozzarella cheese that is stretched to represent Ricci flow, the method of solution. For further reading, there is a repository of original material here "Notes and commentary on Perelman's Ricci flow papers". The New Yorker covers the story of the Poincaré conjecture in the article Manifold Destiny, August, 2006. Photo of Grigory Perelman Courtesy International Congress of Mathematicians Press Office.

A New Game to Help Children Learn Mathematics

Prototype_Mathematics_Game

I reviewed the prototype of a web based graphical game that aims to help children learn mathematics. It was developed by Mr Veljko Sekelj, a freelance Internet consultant (LinkedIn profile). He invited me, via the "Math, Math Education, Math Culture" discussion group on LinkedIn.com, to try out the game and I was quite impressed. At first I couldn't understand its intuitive interface, because I am an adult accustomed to pull down menus and tabbed panes. The game aims to help children associate fractions with percentages. A cartoon Santa Clause introduces a snow scene with a group of identical parcels that contain either a fraction or a percentage. It is operated using the mouse, and the goal is to pair up matches, for example 1/3 and 33%, which pop out of the boxes. There are no instructions, but the game would be quite intuitive to children, though perhaps not to less inquisitive adults. It needs to be made more international, with alternative themes, as the snow scene would not be familiar to children in the tropics. Use of red or green colouring to indicate wrong and right selections might be a problem for someone who is colour blind. I was delighted to get all the correct answers, but I was greeted by a cartoon character in tears of distress, with a sign telling me in German that I was “Zu Langsam!” or “Too Slow!”. The game is at the prototype stage, so I hope that the feedback is made more constructive. The game is online at http://www.lernmathe.com/, where you may try it out for yourself or with your children. The author would be glad to hear your feedback. He is also seeking funding for developing the game. His development project "Innovating the way how children learn math" and "Contact Me" link, is at Kickstarter.com. Kickstarter.com is a new way for people in the USA to seek funding for ideas and endeavours. Read the Kickstarter FAQ to find out more about how it works.