Posts
A useful feature of eWeek is the list of “Most Read” stories, which is currently:
- Microsoft's Big Windows 7 Week. Microsoft's 2006-2007 release of Windows Vista was reported as disappointing; causing many PC users to stick with Windows XP, so the release of Windows 7 has been widely anticipated.
- NASA Space Funding Reaches Critical Point. NASA is planning the future of space travel against a background of controversial decisions, such as: whether to use the Moon as a stepping stone to Mars, whether to use humans or robots to explore space, and whether space travel should be a commercial or government funded venture.
- Why the Droid Can Challenge the iPhone. Should mobile phones be locked to applications and services that Apple Inc. permits, or should you have a free choice to customise mobile applications and services, as with (An)Droid?
- Windows 7 Comes on USB Drives for
Netbooks. Software and documents have traditionally been stored on spinning magnetic discs that are prone to mechanical failure, but we are now in the decade where solid-state memory (with no moving parts), has become a cost effective and more reliable alternative.
- Tilera Talks 100-Core Processor. Some calculations, such as simulating the collision between two black holes, would be too dangerous to try as experiments (1) and too complex to solve on an office PC, so scientists use dozens of computers on a single silicon chip, to complete their calculations in days rather than years. This new technology makes supercomputers available for business applications.
The business-like format of weekly publications made sense when they were printed on paper, but the Internet has enabled the era of reader generated content. The popular http://slashdot.org/, by contrast, has become the classic site where technology news is posted and rated by its readers. It is worth wading through the sometimes inane comments to find an occasional gem of alternative opinion.
Disclaimer: I have no interest in any of the above mentioned companies or organisations, and write this blog post for general interest purposes only.
People who haven't had the right opportunity to learn math have
something in common with people who were born before math was invented.
Perhaps some of the techniques of Euclidean geometry could be revived in a format
that is more accessible today. Specifically, the ancient tools of straight-edge and compass, could be reinvented to suit today's teaching needs. Take this geometry problem: What is the largest area of the semi-circle that can be inscribed in a square of edge length 1 unit? That was the question posted by harpreet in the topic " maxima & minima" on a Mathematics forum on Orkut.com. Some mathematics enthusiasts took a team approach to this problem on the Mathematics24x7 social network, Christian drew a diagram of the solution and Steve calculated the radius of the semicircle. We calculated the radius as 2-sqrt(2) and the area as pi*(3-2*sqrt(2))
.(sqrt() represents the square root.) Danny proposed a less abstract expression of the problem, which would be more interesting to students. "I have been asked to help paint a mural on the outside wall of a
grocery store in my neighborhood. My task is to create the background
for the mural. The instructions are to create the largest possible
semicircle on the wall, with the semicircle touching all 4 sides. The
wall is square with 10 feet on each side. I need to find out how to
position the semicircle to satisfy the instructions. I also need to
know the radius and center of the semicircle. How can I figure this out
with the basic math that I know?" In the discussion Straight Edge and Compass Construction For Developmental Math, I described a method of drawing the mural using
a long plank, lengths of rope, a few pegs and some chalk. Use the rope to extend the base of the wall, to the right, by its
width, and mark the point with a peg. This defines an imaginary square
that is side by side with, and on the right of the square wall. Draw a
diagonal on the original square, because we know that the solution is
symmetrical about the diagonal. Mark a diagonal on the imaginary square
by stretching a rope from the peg to the top right corner of the square
wall. This is a way of calculating the square root of two. Using the
peg as the center point and the rope as a radius, follow an arc down to
the base of the wall, and mark the point where the arc intersects the
base. The distance of that point from the left wall is 2-sqrt(2), which
is the radius of the semicircle. Draw a perpendicular from that point,
and where it meets the diagonal, peg the center point of the semicircle.
Attaching a length of rope to the peg, stretch it to the furthest wall,
and with chalk held fixed on the rope, draw the semicircle. I also described the construction in more abstract terms, and used the GeoGebra geometry software to demonstrate.(See the diagram, where the biggest semicircle that fits in the square is positioned diagonally in the top left of the diagram.) Draw the square as a 4 sided regular polygon. Draw
an identical square to its right Draw the diagonals of the right hand
square and use them as radii of arcs that intersect the left hand
square. Create points where the arcs intersect the base and top side of
the left hand square. Join these two points with a vertical line. Its
intersection with the diagonal of the left hand square defines the
centre of the semicircle. Draw the semicircle through any one point on
the square and notice that it touches or intersects the square at 4 points. Draw the
base of the semicircle through the two intersection points. (I mistakenly drew the wall as 11 feet instead of 10, but that does not detract from the construction.) Constructive geometry involves no measurement of length, except use of
the compass as a tool to copy a length and duplicate it somewhere else
on the plane. It also involves no algebra. We multiply a length
by extending it with additional equal lengths. We draw the diagonal of a
square without realising that we are calculating the square root of
two. Constructive geometry is available to people who don't do algebra. They can solve ancient problems, rediscover the
history of mathematics and apply it to their own environment. Straight-edge and compass problems range in difficulty from simply drawing a hexagon to the more complex procedure for drawing a regular pentagon. Refer to comment below for public domain geometry files of this example.
This is a story of the promotion of mathematics and science through social networks, digital repositories and other Web 2.0 technologies. It began in August 2008 when I was inspired by the 1999 Cluetrain Manifesto, and wrote a discussion paper "Use the Cloud to Get a Clue", which I published as a PDF file on Scribd.com and later as a slide show "Openness and Social Networking" on Slideshare.net. To demonstrate the Creative Commons licenses, I threw together a quick presentation "The Cool Physics of Heat", and was surprised that it notched up almost 1000 views on Slideshare.net and 1400 views on Scribd.com, much more than any other document that I have released. I then focussed on my interest in mathematics (not my main subject), in particular blogging about it here, at http://cmcallister.vox.com/ on the free blogging service, Vox.com. I discovered an active niche social network, Mathematics24x7, on the Ning.com platform, where teachers and other academics were expressing their enthusiasm for maths. I participated in a weekly #mathchat Twitter conference (2am GMT Thursdays). That exchange inspired me to create a public wiki "Online Mathematics Access", about math markup as a tool for discussing mathematics online. There were failures too; my "Mirimatics" forum on Friendster.com stimulated absolutely no discussion. I added a Math Problem Solving group to Mathematics24x7, which is more successful, and attracted a dozen participants in as many days. My online activity is neither scientific research nor publishing, in the formal sense. However, it is still worthwhile, and involves the exchange of academic ideas with a network of new online acquaintances. It is promising that this discourse has grown, without being published in an academic journal, or having any research focus. The discussion has drifted across such diverse topics as the solution of geometric problems and the significance of colour in cognition, which someone "Liked" on Facebook. I'm simply writing about a subject that I enjoy, not striving to maximise page hits. The only metrics are the view count on my uploaded files, and a few red dots on the visitor map on my blog. I uploaded snapshots, of both the math markup wiki and this Vox.com blog, to Scribd.com, to make them more available. The online discourse is dynamic, refreshing, and involves a broad cross section of maths enthusiasts. We exercised teamwork in solving maths problems and discussed some new ideas. The discussions are linked to other networks too, including the Math, Math Education, Math Culture group on LinkedIn.com and a Mathematics community on Orkut.com.
This story is dedicated to the many individuals who are persecuted for publishing their ideas, a few of whom I mention on my blog at http://cmcallister.blog.friendster.com/.
Links:
Social network: http://mathematics24x7.ning.com/ created by Rashmi Kathuria
#mathchat, hosted weekly by Maria Droujkova at http://twitter.com/mariadroujkova
Math Markup wiki: http://onlinemathematicsaccess.wikispaces.com/
Math, Math Education, Math Culture, managed by Opher Liba, at http://www.linkedin.com/groups?gid=33207
Math 2.0 Interest Group, managed by by Maria Droujkova, at http://mathfuture.wikispaces.com/
Yesterday, during a discussion about cognition, a colleague informed me that Physicist Richard Feynman perceived colours when he saw equations. "When I see equations," he once said "I see the letters in colors - I don't know why." (Ref 1). This ability is a form of synesthesia by which there is cross-over from one sense to another. Daniel Tammet, an amazing savant, also has this gift. He "sees numbers as shapes, colors, and textures, and performs extraordinary calculations in his head", as he describes in his book, Born on a Blue Day. This suggested to me that colourised equations would be a good aid to learning for some people. (This is just an idea. I have no evidence that it would be effective, and it could put a person who is colour blind at a disadvantage. Ref 2.)
I took the well known roots of the quadratic equation:
and used blue to identify irrational parts and red to identify negative terms. The mapping is not clear cut, because one term is plus-or-minus, and it is only the result of the square root that is irrational.
\frac{\uc{red}{-b}\pm{\uc{blue}{\sqrt[]{(b^2 \uc{red}{- 4*a*c})}}}}{2a}
into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/. The LaTeX command \uc{red}{-b} defines that the term -b is to be coloured red.
The benefits of colouring may not be apparent for a single equation. I suggest that you try it the next time you need to present a long and tedious algebraic evaluation. The low-tech approach would be to use coloured highlighting pens to mark up a printed copy of the equations. This is not a new idea, accountants have been using red ink to represent negative numbers for generations.
For the benefit anyone who is unable to view the images, the equation shown by the above images is:
(-b +/- sqrt(b^2 - 4*a*c))/2a
The negative terms -b and -4*a*c are coloured red and the sqrt() function, which may be irrational is coloured blue. The square root may also be an imaginary number, if it is the square root of a negative number.
Another application of colouring is to present each digit in its standard resistor colour code. The pattern of repeating decimals for certain fractions becomes much more memorable. For example 264/999:
This was generated from the LaTeX code:
\pagecolor{gray}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}/\uc{white}9\uc{white}9\uc{white}9 = 0.\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}...
using into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/.
Ref 1: The book Sparks of Genius, By Robert Scott Root-Bernstein. The page containing Feynman's description may be viewed in Google Books.
Ref 2: In "Colourised Equations As An Aid To Mathematics" on the Mathematics24x7 network, Bob Mathews commented that we need to create presentations that are "color-blind friendly".
The open discussion about support for mathematics on Web 2.0 and social networking sites, is now taking place on the public wiki, http://onlinemathematicsaccess.wikispaces.com/.
You are invited to add your comments to the following Discussions:
- How can we share math online?
- What sites support math conversation?
- Is it OK to use plain text math?
- Considerations for impaired vision or hearing?
There is a related discussion on the Mathematics24x7 social network, Editing Equations on Web 2.0 Discussion Forums. This discussion is also on LinkedIn.com, in the group Math, Math Education, Math Culture, with the same title Editing Equations on Web 2.0 Discussion Forums.
I wish to thank Maria Droujkova for organising the weekly math events, including 22nd July online meeting at Mathfuture on this topic, and for asking the above questions. I created the wiki in lieu of my participation in her scheduled weekly discussions.
Twitter hashtag: #mathmarkup, for LaTeX, MathML and related topics about mathematics markup.
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.
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.
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.
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.
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.
