The ultimate educational insight – from the most unlikely source.

Crash!

The chair flew across the room - followed by a stream of expletives.

“I hate this @$%@#$%$% school!”

Smash!

Books and papers were swept to the floor.

“I $%@%$#$ hate this place!”

The teacher was surprised. Such outbursts, once common, had become rare and had trailed away to almost never.  Until today.

Once such behaviours had been common – almost daily events.  “Matty” was known to have issues – but not a diagnosis.  His unwillingness to co-operate in the school setting had become legendary and his refusal to co-operate with the school psychologists and support staff had resulted in the lack of even a label to describe his behaviours. His physical issues were also obvious – but he would not co-operate with the optometrists either so the glasses he clearly needed could not be obtained.  He was on the case list of many departmental officials – but on the case load of none.

That changed when he met this year’s teacher.  She made contact – more than that – she established a relationship with his parents. She tactically ignored the “small stuff”.  She differentiated his experiences in the classroom – and ensured that other staff did too. When things went wrong – and they did, frequently – she was unblinking in her defence of him.  She disregarded rumours and insisted of facts. When he did wrong he got the same consequence as anyone else – but no worse. She insisted that departmental staff revisit and find other ways to help this child.  She did not want a diagnosis, or a label, she just wanted techniques that would help him. She even provided him with food when he was hungry.

 Slowly, step by pain staking step, progress was made.  “Matty” learned first the letters of his name and then how to write them – a massive step that had not been achieved in the previous years of his schooling. He learnt to recognise the basic numerals and then how they indicated numbers – which he began to learn to manipulate. He learnt to recognise money – and how to use it. He still did not have a meaningful score on any recognised test or assessment – but he was able to demonstrate that he was capable. In fact, he occasionally stunned staff with some of his efforts – such as when he made a cave in the classroom and angled a reflective surface precisely so that it provided natural light inside his cave. 

But thoughts of his progress were blown away by this latest unexpected outburst. It was nearly the end of the school year and he had made so much headway – and now this.  What had prompted it?

“I’m not going to $%$%@#$% grade 5!” he screamed.  “I’m not @$%#@$% going!”

And then the answer became clear.

“Why not mate?” his teacher quietly asked.  His answer brought a tear to her eye.

“No-one is going to love me there.”

I would have loved to have been present when he found out who was going to teach him again the following year. A most perceptive principal had recognised the relationship and the two – teacher and student – would again share a classroom the following year.  And I don’t know who was happier about it – “Matty” or his teacher.

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“Matty” may never learn as much as his peers. He may never even get a score on a formalised test. But it seems to me that he already knows what is truly important in life. I just hope we have the wisdom to learn from him.

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Post Script.

Some years later "Matty" met his teacher again in a supermarket.  Now in High School, Matty pushed her shopping trolley around for her while and had a chat about what had happened in his life.  There was more than one "Special" in the supermarket that day.

Credits:

Image via Google images - here.

The hoax of Hanoi

The Tower of Hanoi is one of the more popular puzzles amongst recreational mathematicians and in education.  Almost everyone is familiar with it and the story that usually accompanies it - often couched in terms something like this...

“There is a legend of a temple near Hanoi in Vietnam.  The temple has three golden poles. One one pole sits 64 golden discs of different sizes. The rings are arranged with the largest at the base progressing in order to the smallest disc which sits on top of the pole.  The monks who tend the temple have been charged with moving the rings from their current pole to the end pole - one disc at a time - but there is a special rule that they must follow. They must never place a larger disc on top of a smaller one. When they complete this task the world will end. How many moves will it take the monks to do this?”

We probably don’t need to work ourselves up into a sweat worrying about the monks finishing their task which will signify the end of the world.  It has been calculated that if the monks were able to move the one disk per second it would take them around 585 billion years to complete the task - 18,446,744,073,551,615 turns.    In classrooms the number of disks is usually drastically reduced to around five in the early stages - which is enough to prompt some interesting explorations.   In this age of the Internet there are several interactive versions - which have the advantage of being readily customised and also of making suggestions for optimal moves. Examples can be accessed here and here.

It is a great puzzle - a surefire hit in the classroom.  But what makes me love this puzzle so much is the simple fact that it is a hoax.  Traditionally there was no such legend in Vietnam - or in India either, which is another popular location for the site of the temple.  The whole thing is the invention of mathematician Edouard Lucas. When the puzzle was first published in 1883 it appeared complete with legend and was credited to  N. Claus de Siam (an anagram of Lucas d’Amiens - Amiens being the town of his birth).  

Lucas was a significant mathematician in his own right - so significant that he has his own number series named after him - the Lucas sequence (similar to the Fibonacci sequence but instead of starting 1,1,2,3,5,7...  Lucas’ starts at with 2,1,3,4,7,11,18...) and he developed a test for large prime numbers still  in use today.

Lucas’ death is also worthy of interest - he is possibly the only mathematician in history to die as a result of a piece of crockery. A clumsy waiter dropped a plate which broke and shard from the plate cut Lucas on the cheek. The cut became infected and he later died of erysipelas - a serious streptococcus infection.

The myth and the puzzle are well known … but the puzzle has a rather curious property that  is less so.  The movements of the pieces can be represented in a graph as depicted here.
This bears an uncanny resemblance to  to the famous  Sierpinski triangle- which is a fractal.   The fact that the movements of a game can be linked to fractals is, to say the least, intriguing. (More on this at the source.)
 


The Tower of Hanoi has recently become a movie star. A disguised version of it appears in the move “Rise of the Planet of the Apes” - disguised, fittingly given the manner in which it as first published, as the Lucas Tower which was used to test the intelligence of the apes.

What can we learn from this wondrous hoax? When it comes to education sometimes a touch of showmanship and playfulness can improve a strictly mathematical experience. It is important that the mathematics remains central - but the “power of story” is a significant tool to generate engagement and interest  - which may contribute to the “holy grail” of teaching;  a self-directed learner.  We can help foster this when we show our students that there is more to mathematics than just the numbers.

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Credits: http://www.wooden-toy-store.com/images/Tower%20of%20Hanoi.jpg

Undirected graph of movements: http://en.wikipedia.org/wiki/File:Tower_of_Hanoi-3.svg
Sierpinski triangle: http://upload.wikimedia.org/wikipedia/en/thumb/8/88/Sierpinski_Triangle.svg/220px-Sierpinski_Triangle.svg.png

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.

Classroom teachers - the real educational experts

Our staff had a professional development day recently.  The group was asked some “simple” questions such as “How do children learn” and “What do you do to facilitate learning”.  The thoughts of the group were then collected and presented as a word cloud.  Obviously this is a relatively crude way of summarising the ideas of the group - but it does capture the essence of what was discussed.

It seems to me that this is a solid representation of the “wisdom of the group” approach. There were no “outside” educational leaders or gurus present at this workshop, no-one was flown in from interstate to facilitate, no-one earned a cent in presenters fees.  Yet the result is a more than fair representation of the things that effective teachers do. In fact, if each item was to be expanded and clarified we would have the basis of a fairly satisfactory professional text.

Obviously knowing what to do and being able to do it are two different things. If our staff can enact these practices in the classroom our students will be the beneficiaries.  However, it strikes me that the knowledge base in our staffroom is a fairly significant one. There is an expression that an expert is someone who lives more than 100 miles away. Upon reflection it is obvious that we all must live 100 miles away from someone - and thus can all be classified as experts.

Whilst it must be acknowledged that effective teachers need to be effective learners themselves and always strive to develop their skills and knowledge base, perhaps it is time to place more value on the “home grown” wisdom within our schools.

Effective classroom teachers may well be the real educational experts.

When is a school a school?

I was privileged to be given a tour of a new school recently.  It was, without exception, the most exciting school building I have ever been into.

The design of the buildings was thoughtful and, dare I say it, “modern”.  ICT was present everywhere (but not always obvious), major walls were sound “proof” but flexible in that they could be opened (in effect removed) to increase teaching space and allow for co-operative learning, or inter-class interactions.  The school had avoided the trap of creating computer labs - instead each area had access to a half dozen or so desk top machines which were supplemented by a generous supply of lap tops via a booking system.  Wireless technology with fast connection speeds was ubiquitous and covered every area in the facility (or so we were told). Teachers had pleasant individual office space.  Each section of the school, or “pods” as they were called on site, contained significant art installations.  Large windows dominated the wall spaces filling the  facility with light and enhancing the impression of spaciousness. Classrooms seemed to have adequate resources. One of my fellow visitors commented that the staff room had the feel of an upmarket hotel rather than a school staff room.

Outside it was just as impressive - attractive from all angles, functional yet not overly institutional.  Play areas were clearly well used and functional.  It was, quite simply,  a wonderful building - but it wasn’t a school.

Then the siren went.

The empty rooms quickly filled with chatter, children moved to desks, teachers started explaining tasks, students started asking questions, books were opened, computer screens burst into life, the sounds of productive work became the back-ground audio-track for the visit.  Parent helpers slid quietly into a classroom to be greeted with smiles and accepting nods of “hello” from the students. A class of children walked past in an orderly but not overly regimented group. Seeing their teacher smiles burst onto the faces of the first few students as they entered the room.

Then it was a school.

Credits:
Image: http://www.wes3rdgrade.com/uploads/7/8/3/3/7833021/8576233.jpg?252