Monday, January 30, 2012

A track less travelled

Like most people I am a creature of habit.  I am also a keen bush walker - partly for exercise, partly for photography, partly for personal therapy.  My most favourite site just happens to be a World Heritage area less than two hours from my home - Cradle Mountain.

I tend to stick to well  trodden paths when I visit. With scenery like this waiting why not?

When you know you will be seeing this why would you go elsewhere?

Yet that is just what I did recently - due, in part, to weather conditions.  Seeing as though my favourite tracks were shrouded in morning mist I went on some of the less well known tracks... and “discovered” this...
and this...

I even met some of the locals...

Not only did I add some gems to my walking catalogue but I also discovered tracks to two other destinations that I had been wanting to visit.  In short, by breaking my habits I discovered some “unknown” locations - but also found some exciting places to explore in the future.

So how does this relate to education?  It struck me that my habits prevented me from exploring other options and opportunities.  This happens in classrooms too.  How many teachers cover the same material every year?  How many people use the same task design year after year? No doubt these are successful teachers - if it didn’t work they wouldn’t do the same things repeatedly.  Yet this approach reduces innovation and exploration of alternate methods - especially in the nature of what the students do.  The curriculum may dictate what students are to be taught - but usually teachers have freedom to select how they  present and get students to explore material.  This is easy to say but possibly a little harder to do - but there are some simple ways for teachers to expand their instructional horizons.  

One might be to use this resource - the Kelly Tenkely's blooming peacock.  This resource identifies areas of Blooms taxonomy and then identifies software that addresses that mode of thinking.  Another llist, by Kathy Schrock ,does the same thing but limits the software to that produced by Google - meaning free and web based. Andrew Churches' Blooms Digital Taxonomy  worksheets also provide advice on how to address Bloom via technology. One way for teachers wishing to expand their skills and move beyond their current habits into 21st Century skills might be to choose one new piece of software per unit and become an accomplished user of that site or program. Once done, select another site that suits your purpose from the next level up in Bloom’s taxonomy. Over a relatively short period of time teachers will have learnt not only a range of useful skills and programs  but shifted their teaching towards the higher end of Bloom’s spectrum.

Taking a fresh look at how we teach and trying something different is likely to be very rewarding - for the teacher as well as the students.

If you like the images displayed in this post you may like to view this video I made of walks completed last year. 

Tuesday, January 24, 2012

Real life is not virtual

It has been said that a picture is worth a thousand words.  Some images are worth far, far more.

One glance a the image reveals that it is not recent - the drawing of the TV reveals the rampaging advances of technology since the image first appeared.  But the trend of viewing life via a screen rather than engaging with it directly is depicted clearly. Since this cartoon was first drawn technology has continued to evolve - we now not only have TVs, but computers, laptops, tablets, and smart phones, all of which allow us to access information from around the world - often in “real time”, as it happens.

In fact we have added a new word to the lexicon - “Screenager”. The term,  first used in 1997 by Douglas Rushkoff in Playing the Future, refers to the tech-savvy youngsters who seemingly have their retina’s permanently glued to some form of screen (or, more likely, scanning various screens almost simultaneously) and paying little attention, if any, to others around them.  

As educators we  need to remember that, despite the numerous strengths of seeing the world through a screen it is still a vicarious experience. It doesn’t matter how many manuals you read or videos you watch of other people riding bikes, you only really learn how to do it when you get on a bike and have a turn yourself.  Watching other people’s lives is not the same as living your own.

Real life is not virtual. 

The image above, drawn many many years ago by Michael Leunig is typical of his cartoons. Some links to his work can be found here.

Cartoon by Leunig.
Image source.

Monday, January 16, 2012

A slice of Biblical pi...

Author Alex Bellos has been rightly praised for his fascinating book Alex's Adventures in Numberland. It is a book full of mathematical gems presented in an accessible style not normally associated with mathematics books - which possibly explains it's success.  One of the chapters is devoted to the study of pi and the people who have pushed back the boundaries of this number - a dry topic transformed into a engrossing read.  

For those who need a reminder - pi is the ratio of a circle’s circumference to its diameter.  As Bellos puts it ”...if you take the diameter of a circle and curve it around the circumference, you will find that it fits just over three times.”  It is that “just over” bit that makes pi interesting.  The “just over” bit is actually an irrational fraction - meaning it never ends  - it simply goes on forever.  The accuracy of pi has been calculated to a ridiculous degree - billions of digits. According to Bellos, manufacturers of precision instruments only need an accuracy of four decimal places so the quest for a more and more accurate figure for pi is no longer driven by any practical reasons.  However, early methods of calculation were ingenious and the chapter in Bellos’ book provides an entertaining overview of many of them.

The chapter contains this discussion of pi in, of all places, the Bible.
"...A line in the Bible reveals a situation in which pi is taken a 3: 'Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about' (I Kings 7:23)".

Thus the Bible is on shaky ground from a mathematical perspective.  After a very brief discussion on some squeamish “explanations” from devote believers on what the bible may have meant when giving that description Bellos throws in an absolute gem.
"A mystical explanation is much more enticing: due to the peculiarities of Hebrew pronunciation and spelling, the word 'line', or qwh, is pronounced qw. Totting up the numerological values of the letters gives 111 for qwh and 106 for qw. Multiplying three by 111/106 gives 3.1415, which is pi correct to five significant figures.

Today we would call this  numerology - but the practice has a long history. Traditionally the practice of ascribing numerical values to words was known as gematria and was wide-spread amongst the ancient Greeks.
I’m not sure what I find hardest to believe - that someone in Biblical times not only knew pi to five significant figures and could hide it via gematria in a passage to be decoded by the enlightened...or that purely random chance and an accident of linguistics has delivered pi to five significant figures - in a passage describing a circle.

What has this mathematical musing have to do with education? Two things. As teachers we don’t always have the answers - and we should admit this to our students - not necessarily in relation to the concepts discussed here but in general terms. The second is that the simple fascination of the unknown and the delight of exploration that is associated with mathematics should be shared with our students. Again, not necessarily these concepts, but with mathematics in general.  We should present mathematics as a subject full of fascination to be explored rather than allow it to be reduced to a collection of rote exercises, algorithms and meaningless formulae.

The worth of a word:
Gematria is an ancient belief. For most of us it is as valid as the notion of a flat earth or that the earth was a disk sitting on the back of a turtle floating through space.  However, if you want to calculate the numeric value of certain words or names then clicking here will take you to  a site that may be useful as well as amusing.  The site also provides words with the same numerical value as the text you input.

Have fun.

Pi graphic:

Tuesday, January 10, 2012

Sierpinski surprises

 One of the wonderful things about mathematics is that it is frequently surprising.  Often relatively simple things reveal unexpected hidden depths - and do so with minimal probing.  One such aspect of mathematics is the Sierpinski triangle (aka Sierpinski gasket).  
The Sierpinski triangle was named after a Polish mathematician who explored the concept around 1915,  although it is not true that he “discovered” it as the basic shape apparently appears in art work dating from some centuries before.

The Sierpinski triangle is a self similar set - a pattern that can be made larger or smaller indefinitely while maintaining the same pattern,  in other words, it is a fractal.  Fractals are not just found on the pages of maths books - which is one of the reasons that they are so fascinating.  Fractals are found in many places in nature - from snowflakes to certain leaves, from ferns to forked lightning, even, most unexpectedly, in broccoli. Making Sierpinski triangles is something that even relatively young students can do. Start with an equilateral triangle. Inscribe that triangle with an inverted copy of itself (or, more simply, make an upside down triangle inside the first one). And keep repeating. (One set of instructions for use with young mathematicians can be found here.)  The pattern created, as depicted above, is surprisingly pleasing for what is essentially a mathematical process.

Creative types have added colour, and combined them to make them even more so as shown here.  So, the Sierpinski triangle is interesting and visually attractive in its own right...but there are some fascinating attributes that do not immediately meet the eye.  (Source)

The other famous triangle in mathematics is Pascal’s triangle. It has become customary to credit the discovery of this triangle to Blaise Pascal although the concept was known well before
him.  In Pascal’s triangle the two numbers above a cell are added to create the  number in the next row and the process repeats.  Now, this is where things get interesting.  If we shade only the odd numbers in Pascals triangle we get … something VERY similar to Sierpinski’s triangle - and the similarity strengthens as the larger the triangle continues.   

Why might this be?

Sierpinski and Chaos
There is a fascinating game invented by Michael Barnsley called “The Chaos Game” - which, ironically, proves that order can come where chaos is expected.

To “play” this game grab a sheet of paper and mark three points of an equilateral triangle.  Label the points A, B and C. Make a mark at any random point on the paper - call it X. Use a die - numbers 1 & 2 relate to point A, 3 & 4 to point B and 5 & 6 to point C. In our example imagine you roll a 1 (which therefore indicates point A).  Measure half the distance from P to A and make a new mark. This is the next point. Roll again and mark the point half way between the relevant point indicated by the die and the last point obtained. Repeat and repeat...and repeat.  The longer you “play” the game the more strongly a pattern emerges - and that pattern is the Sierpinski triangle!  
When I first read about this years ago I wasted hours “playing” the game in an attempt to test it. It proved to be true. These days actually playing the game is not strictly necessary as there are any number of interactive versions of the game available on the Internet.  One can be found here - I recommend that you try it for yourself.  Concentrate on your sense of wonderment as the patten unfolds. This is the feeling that we want our students to have when they explore mathematics.   Incidentally, this “game” has been called the “Creationist’s worse nightmare” - for those who like their maths flavoured with mysticism - and the site that uses the term explains why the pattern works well.

Counter-intuitive results such as that produced by the Chaos game can intrigue students - which leads to engagement...which leads to learning.

Tower of Sierpinski?
For me personally this is perhaps the most perplexing of the unexpected appearances of the Sierpinski triangle.

The Towers of Hanoi is a popular game / puzzle where the player is required to shift a number of disks from one of three “posts” to another of the three available and reassemble them in order with the largest disk at the bottom and the rest of the disk sitting on the bottom disks in order. It is a surprisingly simple yet engrossing game. An interesting thing occurs when the moves leading to a solution of the Towers of Hanoi are graphed.

(source)The resemblance of this graph to the Sierpinski triangle is startling. It is worth pausing a moment to think about this. The Tower of Hanoi is a mental experiment, a “game”  devised by a human.  It is not a “naturally occurring”  phenomena such as a fractal snowflake or a symmetrical fern leaf - it is totally the product of human imagination.  Yet the solution to this totally invented  game, when graphed, has a strong resemblance to the Sierpinski triangle - which is a fractal.  It could be argued that Sierpinski triangle is also the product of the human mind - yet this does not diminish the sense of surprise when the link between the two concepts is established.  Why should this link exist? The sense that there is some intriguing connection between two such different things is tantalising.  In mathematics the enjoyment is in the exploration and the discovery.  Sometimes no answer is much more satisfying than a clinical definition that puts everything in its place.
Sierpinski triangle image link: 
Sierpinski hexagon image link:  

Wednesday, January 4, 2012

Maths that makes you go “Hmmm”

Mathematics often puzzles me.  Sometimes it also makes me wonder.  We get accustomed to the Fibonacci sequence and the Golden Ratio appearing in nature so often that we cease to register just how remarkable this actually is and sometimes fail to ask ourselves why this should be.  

Sometimes it takes something new to bring us to a fresh realisation that the presence of some forms of mathematics in nature is not only unexpected but  possibly unexplainable.  Some unexpected occurrences of  mathematics are truly remarkable.  The following clip provides a worthy example.

It is taken from the BBC series “Why Maths doesn’t add up” and features
features Marcus du Sautoy,  Professor of Mathematics at Oxford and Simonyi Chair for the Public Understanding of Science with actor, comedian and maths fall guy  Alan Davies. Early in the clip we see a graph first devised by Bernhard Riemann showing the random distribution of prime numbers. In this excerpt  the pair see what happens to the vibrations of a sphere of quartz when the quartz is struck with a ball bearing.    Quite unexpectedly the two graphs are startlingly similar.

As du Sautoy says in the clip, the similarity of the graphs is so striking that “it can not be a co-incidence”.

So if it isn’t a co-incidence...what is it?

Perhaps in pondering such “oddities” we will also remember that the “common” occurrence of the Fibonaci sequence and the Golden Ratio in nature are also worth of contemplation.  

I believe it is this sense of wonderment and intrigue that we need to impart to our students as much as computational skills and procedural understanding.

If you enjoyed this post you may enjoy my other maths related posts which are compiled on my maths page - accessible via the home page or by clicking here.