Tuesday, 1 September 2015

8. Addicted to the laptop

Many parents provide their children with a range of technology (smart phone, tablet, desktop computer…) in the belief that they are investing in their education. In my view, this is largely self delusional. The reality is that much of the time that children spend on their gadgets is brain-numbing game-playing and using social media, neither of which offers them any real educational benefit.

I’ve always been a bit of a gadget person and love messing about with computers, tablets and calculators. As an educator, I have been excited by the opportunities that these magical, electronic boxes can bring to maths learning. In my work at the Open University I was usually at the front of the queue when it came to incorporating into students’ learning the use of mathematical apps, spreadsheets, calculators and the like. Indeed, many students have reported that they have enjoyed these parts of their courses and gained a good deal of benefit from them.

But these courses were written for adults. Is the same thing true for children’s learning? Some years ago, I would have said ‘Yes’, but over the past couple of years I have come around to the view that this is probably not the case. First, there is the obvious concern that staring at a screen for 4 or 5 hours a day will probably dull a child’s brain, It will also prevent them from taking exercise and will discourage them from talking to people face-to-face. But I’m also dubious about whether the much hyped learning benefits really happen in practice.

For many parents, helping their children to be computer-literate is an important aim. Courtesy of mum and dad’s generosity, a growing proportion of children have a laptop or tablet in their bedroom. The barcharts below really reveal the pace of change in a single year.

So here’s my problem with this explosion of technology inside children’s bedrooms. We shouldn’t fool ourselves into thinking that spending a lot of screen time with a computer is equivalent to ‘real learning’. If you put a tablet or smart phone into a child’s hands and ask yourself what they are more likely to do with it – practice their maths skills or play Angry Birds? Well, you know the answer to this. 
So what IS ‘real learning’? There are many elements to this but here are five features that I particularly value. Helping children to:
  • engage with ideas and concepts at a deep level;
  • bring two or more different ideas together to reach a new conclusion;
  • take an idea learned in one context and adapt it to apply to a new context;
  • develop problem-solving skills;
  • be intellectually curious.
Where technology becomes of real benefit to learners is when it is used to support these learning skills, rather than being a goal in itself.

One consoling thought with children spending so much screen time is that perhaps, along the way, they are developing useful keyboard skills and a confidence with the computer environment. Well, this may have been true once, but children pick these skills up very quickly. Also, as technology gets better, it becomes much easier to master, so skill and confidence development become non-issues.

The reality is that, left unsupervised, too much of the screen time that children spend on their computers involves game-playing or social media, whatever they may say to the contrary. For quite a number of children, this can become so addictive that they will strongly resist attempts to remove or limit their use.

These and many other reasons for doubting the usefulness of technology as the great panacea for education can be found in a new book by computer whizz, Kentaro Toyama, called Geek Heresy. Toyama suggests that:
’What’s not being taught through exposure to technology are things like the basic mathematics that you need to become a good engineer or the writing and communication skills you need to become a good journalist.’

Based on his experience of testing out technology initiatives in Indian schools, he argues that technology-based initiatives are often successful in pilot schools when the exciting new ideas are first introduced. Here, the teachers are well supported by the researchers and are excited to be part of a cutting-edge project. But too often it all tends to fall apart when the same approach is offered to ordinary schools, where the teachers are unsupported and where they may have a very different agenda about educational priorities.

Postscript
In this post I suggested that unsupervised technology offers few benefits for children's learning. However, I still strongly believe that, when carefully planned  and properly integrated into their learning, technology can have a very positive effect. I'll be looking at some examples of this in future posts, starting with an exciting and growing feature of children's education – a new topic on the curriculum called coding

If you are interested in reading more about these issues, have a look here in my website, www.MathinPictures.co.uk.


Tuesday, 23 June 2015

7. Logical Puzzles

I was chatting to my daughter, Ruth, on the phone yesterday and she mentioned that she was struggling to solve the latest of a series of regular puzzles posted by Alex Bellos in the Guardian newspaper.  The puzzle, which is called The Three Switches, is as follows:

The Three Switches
Downstairs in a house are three identical on-off switches. One of them controls the lamp in the attic. The puzzle is to work out which switch controls the lamp.
The rules are as follows. You are allowed to manipulate the switches all you like, and then you are allowed only one trip to the attic. How do you do it?

You might like to have a go at the problem yourself before reading on.

Ruth pointed out that the problem seemed to be impossible. If there had been just two switches the question would be trivial but with three, there wasn’t enough information to solve it. From my personal experience of tackling these sorts of puzzles, I work with the following assumptions.
1. Since the problem has been posed, it must be solvable (unless there has been a misprint, which is unlikely).
2. If I have reached the point of finding it impossible, then there must be an extra dimension or layer of information available to me that I have not yet considered.

In this particular case, the restriction in most people’s thinking (mine included, I have to admit) is that they only consider their sense of vision to test the lightbulb. But (most) lightbulbs also generate heat and this is the missing link for tackling this puzzle. If you were unable to solve it before, have another go with this hint in mind. And when you’re ready, check your solution against my 2-minute video below.


Interestingly, Bellos claims that he offered this puzzle to many people from a wide variety of backgrounds and only one was able to solve it directly. This was Jo Nesbø, a well-known Swedish crime writer. Perhaps this is not surprising, since crime writers make an art form out of misdirection, to ensure that the reader only finds out the answer right at the end. So perhaps Nesbø's mind was well prepared for the sort of lateral thinking that this puzzle might required.

If you like these sorts of puzzles, Bellos has put together a nice collection at:

More lightbulbs
And talking of lightbulbs …




Hannah’s Sweets
Over the past couple of weeks I’ve had a number of emails from friends drawing my attention to the recent hullabaloo about a GCSE exam question – referred to as ‘Hannah’s sweets’. The outcry from students has been that this question was too hard. Anyway, here it is. Have a look and see if you can answer it. 


If you'd like to see a solution, I’ve posted one here.  

Postscript
This post looked at the puzzle, The Three Switches. I also looked briefly at a GCSE question from this summer's mathematics exams that has upset a few people. 
If you are interested trying some more puzzles, have a look here in my website, www.MathinPictures.co.uk.



Tuesday, 19 May 2015

6. What are the chances?

We’ve recently returned from a few days holidaying in deepest Pembrokeshire – a beautiful unspoilt part of west Wales with rugged coastlines and wind blown trees.

Rugged (and wind blown)

Wind blown

One of the highlights was visiting the cathedral in St Davids, which has roots that go back to the 6th century. 

St Davids cathedral

On our last evening, we tried to find a table at a small fish restaurant in the tiny coastal town of Dale. The owner explained that he had a party of eight people about to arrive but thought he could squeeze us into a corner. 

Dale in Pembrokeshire

Just as we were about to tuck in to our fresh lobster, we heard a shout from behind us, ‘I don’t believe it!’. Who should troop in but eight of our near neighbours and good friends from back home in Warwickshire.

After a fair bit of hilarity, our conversation turned to unlikely events in general and then the national lottery in particular. We were drawn into that old chestnut about what on earth we would do with those awkward Lottery millions and how much of our fortune we would want to give away. I was asked if there was a best strategy for choosing the numbers, and it caused a bit of a surprise when I said that there was! 
The UK National Lottery
But before I reveal this hot lottery tip, there are a couple of issues that need to be sorted out.

Issue 1: Is any one combination of six lottery numbers more likely to crop up than any other?
My Answer: No! Provided that the lottery numbers are chosen randomly from the lottery ball machine, all numbers (and all combinations of numbers) are equally likely to occur.

Issue 2. People say that lightning doesn’t strike in the same place twice. So I’d expect that the six numbers that came up last week are less likely to come up this week?
My Answer: This is a fallacy. As a matter of fact, lightning does sometimes strike in the same place twice, especially if that place is a tall building or tower. Indeed, several people have been hit by lightning more than once. For example, Roy (‘Human Lightning Conductor’) Sullivan who was a United States park ranger in Shenandoah National Park in Virginia was hit by lightning on seven different occasions and survived all of them. 

But anyway, even if it were true that lightning didn’t strike in the same place twice, how would this work? Are we to believe that there is some lightning god from above saying, ‘No, hang on, we hit that particular spot last year – better move this bolt’s trajectory a smidgen so that it hits the next field.  The same principle is true of the selection of lottery numbers from the lottery ball machine – last week’s numbers are as likely to appear this week as any others. Just as was true of the lightning bolt, it’s hard to believe that there’s an invisible force subtly pushing last week’s six balls towards the bottom of the machine in order to give the other 43 balls a better chance, in order to even up the outcomes. 

Now you might be thinking, ‘But surely by the Law of Averages, all the numbers should come up roughly the same number of times, so that should mean that previously favoured numbers are less favoured next time around. Well, sorry but this is not how the Law of Averages works. If you want to know why, have a look at the follow-up to this post in the Blog page of my website, MathinPictures.co.uk.

Issue 3. So if each number is equally likely to crop up, how can I have a ‘best strategy’ for choosing my numbers?
My Answer: Unless you can find a clever way of cheating the system (which would be very hard to pull off), it is true that you cannot find a way of choosing a combination of six numbers that is more likely to win than any other combination. But there is another factor here. You need to remember that if you choose the same six winning numbers as someone else, you have to share the jackpot with them. And if nine other people choose your winning numbers, your share is reduced to one tenth of the jackpot, So if you want to maximise your winnings, try to be aware of which numbers other punters are likely to choose … and avoid them! 

Here then are my hot tips for maximising your lottery winnings.
1. DON’T CHOOSE numbers between 1 and 31: people are more likely to choose numbers linked to birthday dates, so numbers in this range tend to be more widely used.

2. DO CHOOSE consecutive numbers: many people mistakenly think that a consecutive string of six numbers is less likely than any other combination and so they avoid such combinations. However, don’t choose the particular consecutive combination 1, 2, 3, 4, 5, 6 as it seems that several hundred punters regularly choose these numbers each week.

The joke
Finally, no lightbulbs this time – just this:

Teacher: What does “coincidence" mean?

Pupil: That’s funny, I was going to ask you the same thing.

Postscript
This post looked at the idea of chance in the context of the UK National Lottery… If you are interested in finding out a bit more about the mathematics of lotteries, have a look here in my website, MathinPictures.co.uk.

Monday, 27 April 2015

5. The Golden Rectangle

‘Quick, Alan. We’ve got to go NOW or we’ll lose them. They’re only a tenner each.’

I didn’t question Sheila further. We jumped into the car and sped off in the direction of our local second-hand furniture shop. What she had spotted was a collection of ammunition boxes – leftovers from World War II that probably originated from the US army. But of course, what Sheila was after was wooden boards for her paintings. 

Normally Sheila paints on commercial canvases. She was delighted to have the opportunity to paint on wood but also to be free to make this wooden ‘canvas’ to any size and shape she fancied. Now, as I am a big fan of a particular rectangular shape – the so-called ‘golden’ rectangle’ – and also because, I was the one wielding the saw, I suggested that we should cut up the box lid to create a board of this ‘golden’ shape.

A week later, this little ‘Apple Blossom’ beauty was hanging on our kitchen wall.

Apple Blossom

This isn’t the only art object that Sheila has produced that was inspired by this particular 'golden' proportion. Here is her Golden Window, painted from a wrought-iron window frame in a crumbling fromagerie in Normandy.

Golden Window

So what exactly is a golden rectangle and why is it considered to be special? Have a look at this 3-minute video to get a quick overview.


Golden Rectangle

Maori drawings
A few years ago, we spent a wonderful two months travelling around New Zealand. I had arranged a deal with a number of NZ mathematics advisors that I would provide a series of ‘inspiring’ talks/workshops in return for a night’s board and lodging in each location. That way we had a great time meeting and chatting to many teachers and their families in their own homes. But there was also plenty of time for sight-seeing. We particularly enjoyed having the opportunity to explore Maori culture and art. 

On our return to England, we decided to mark the trip by commissioning, from my friend Roger the glass artist, a pair of small stained glass windows that celebrated the fusion of mathematics (upper) and Maori art (lower). 

The Golden Spiral


A Symbol from Maori Art

The mathematical one shows a logarithmic spiral (sometimes referred to as the ‘golden’ spiral) inscribed inside a golden rectangle. Here's a detail to clarify the mathematical part of the picture.


A Detail of the Golden Spiral

Starting out from the bottom left corner, the spiral takes a path between opposite corners of each square in turn. In order to construct it, Roger used compasses to form an arc inside each square which he drew as a quarter circle, centred at one of the other corners of its square. 

Golden rectangles have been known about and celebrated since the time of Euclid. He didn't use the term 'golden' but referred to this special ratio as the "extreme and mean ratio". It's sometimes said that shapes formed from 'golden' proportions have a particular aesthetic appeal. I'm not totally convinced by this claim but it would be lovely if it were true!. 
Anyway, this 'special' ratio is all very well, but it's not getting these lightbulbs sorted out ... .


Postscript

This post looked at the idea of a golden rectangle, which has proportions of approximately 1.618 to 1. The number, 1.618… is often referred to with the Greek letter Φ (phi). If you're interested in finding out where this number comes from and indeed other matters 'golden', have a look in my website, MathinPictures.co.uk

Thursday, 9 April 2015

4. The Simpsons and Mathematics

I recently celebrated my n+1th birthday (yes, a year older…) and received from Sheila, ‘The Simpsons and their Mathematical Secrets’ by Simon Singh. Having already read several of Simon Singh’s books – Fermat’s Last Theorem, The Code Book: The Secret History of Codes and the fabulous Trick or Treatment?: Alternative Medicine on Trial – I was confident that this one would prove to be an entertaining page-turner. And I’m pleased to say that it is indeed a little cracker so I thought I’d share an extract from it in this post.


The Simpsons
Unless you’ve been hiding out in another planet for the past 20 years, you’ll know already that The Simpsons has been one of the longest running and most successful TV series in history. Singh's book explains how it came about in the first place.

Back in 1985, the British entertainer Tracey Ullman was having a successful run of shows in the US. Her sketches were performed before a live audience and many of them required extraordinary makeup and prosthetics. Scriptwriter James L. Brooks thought that it would give Ullman more time between sketches if they could include some cartoon animations. He met with cartoonist Matt Groening and minutes prior to this meeting Groening thought up the characters of the Simpson family. Later, some of the short animations featuring the Simpsons were pieced together into a longer show. To everyone’s surprise, they went down very well. Their first full-length show was ‘Simpsons Roasting on an Open Fire’ which was aired as a Christmas special on 17 December 1989. 

'Simpsons Roasting on an Open Fire'

Where’s the mathematics?
If you are a regular follower of The Simpsons, you’ll be aware of the many amusing mathematical references it has included over the years. This is down to the strongly mathematical backgrounds of many of the writers, some of whom gave up teaching and research posts in mathematics to join the writing team. Singh includes many examples of mathematical references but don’t be put off if your own mathematical background is a bit thin – the emphasis is just as much on the humour as on the mathematics.

Various mathematical topics are touched upon in the book but I’ll mention just one – the number known as pi. An episode, first aired in 2001 and called “Bye, Bye, Nerdie” is about the bullying of nerds, a question that Lisa decides to investigate. She discovers and isolates a pheromone emitted by ”every geek, nerd and four-eyes” that could be responsible for attracting bullies. She decides to deliver a paper on this theme to the ‘12th Annual Big Science Thing’. The conference is hosted by John Nerdlebaum Frink Jr, Springfield’s favourite absent-minded professor. Frink attempts to introduce Lisa but the audience is so excitable that he is unable to bring them to order. In desperation, he eventually calls out:
“Scientists… Scientists, please!  
I’m looking for some order.  
Some order, please, with the eyes forward…  
And the hands neatly folded…  
And the paying of attention…  
Pi is exactly 3!
Suddenly the noise stops. Professor Frink has correctly judged his audience. Stating that there was an exact value for pi has stunned this audience of geeks into silence!

If you have only a hand-waving grasp of this number, pi, about where it comes from, or indeed, why this story is funny, you might like to look at the 3-minute video below where I have provided a cheap and cheerful explanation of pi and how it is linked to a fundamental question about what makes a circle a circle.

The Circle and Pi

Of course, this story just scratches the surface of the many gems contained in Singh’s book – definitely 5 stars from me.

Now on to weightier matters. How are those lightbulb operatives doing, I wonder.





Postscript
If you are interested in reading more stories about pi and this delightful book about the Simpsons, have a look here in my website, MathinPictures.co.uk.

Tuesday, 17 March 2015

3. A Troublesome Tweet

I come from a very sporty family and anything that involves chasing after a ball generally catches my interest. I couldn’t avoid reading about the following tweet from cricketer Stuart Broad (the England fast bowler) that made a stir in the news recently.

Stuart Broad, England Fast Bowler

"I've heard if you earn minimum wage in England you're in the top 10% earners in the World. #Keep Humble”

The backlash caused by this tweet resulted in Broad having to issue a rather grovelling apology. 

Now setting aside for the moment the tactfulness of how he worded his tweet, was his claim actually true? There are two parts to this – the earnings of someone on the minimum wage and the pattern of all earners, globally. 

Minimum Wage Earnings: A quick scan in Google revealed that the national minimum hourly rate for UK adults, at the time of writing, is £6.50. Working, say, a 40 hour week, this gives a weekly wage £260 or roughly £13,500 annually. (Assume a 37 hour week and this figure drops to £12,500 annually.)

Global Earnings: A harder task was to get a picture of world earnings. Again, with the help of Google, I was able to find a Daily Mail article which provided some useful tidbits of information – for example:
* You need to be earning just £22,000 per annum to be in the world’s top 1% of earners.
* Half of the world’s richest 1% live in the USA.

Now this is all interesting stuff, but it didn’t quite resolve whether Stuart Broad’s claim was true. Some further googling turned up what I was looking for – an on-line calculator called the Global Rich list, posted by the charity, Care International. You simply enter a figure for your annual earnings and it tells you where you fall within the global rich list. 

First, I tried entering the earnings of a UK pensioner receiving the full state pension, which is currently £113.10 per week – i.e. roughly £5880 per year. Using the online calculator, this puts a British pensioner into the top 18% of earners, globally.

Next, I entered the earnings of someone on a minimum wage – £13 500 annually. 

The on-line calculator put this earner into the top 5.41% of earners, globally – a figure even smaller than the 10% claimed by Stuart Broad. 

So these calculations do seem to bear out Stuart Broad’s claim with something to spare. I wondered afterwards why making what appears to be a factually correct statement could cause such a furore. A likely explanation is how Broad presented it – you may have noticed that he had tacked onto the end of his tweet the words: #Keep Humble. He claimed afterwards that this hashtag was aimed at himself. But for someone who reputedly earns upwards of £1million per year, he is open to the accusation of lecturing at those earning a tiny fraction of what he earns by telling them how lucky they are!

Finally, here's what the lightbulb team are getting up to.

Postscript

I wondered whether these figures from the Global Earnings calculator were to be believed? When it comes to statistical claims, it’s always sensible to look for loopholes and alternative explanations. I consider some of these here in my website, MathinPictures.co.uk, along with the sources I used.

Thursday, 5 March 2015

2. The rat, the pallet and the logs


The rat: The recent cold snap has encouraged Sheila and myself to be more enthusiastic about putting seeds and nuts on the bird table. But in recent weeks, an uninvited guest has joined the party. Yes, Mr. Rat has been busy helping himself as well. Last week, I had my camera at the ready and here’s the moment when he realised he was rumbled. 

Rumbled
In a flash he was off, making a rapid leap for freedom, disappearing around the side of the house.

The leap for freedom
‘I bet your life he’s living under that tarpaulin where we store the logs,’ Sheila said straightaway and I rather felt that she was right. We removed the tarpaulin and carefully started reorganising the logs. But all we uncovered was a very sleepy, plump frog. 

We could see that the logs at the bottom of the pile had started to rot and gather moss. What we needed was a large pallet to go underneath, but where would we get one of those?

The pallet: Two days later, I was driving into town to pick Sheila up from a little light shopping. The footpaths were jammed with pedestrians and the roads were busy with commercial vehicles trying to unload goods. As I approached our usual pick-up point, I could see that there was a bit of a commotion. It seemed that the footpath was blocked – mothers with prams and pushchairs were having to step into the road to get past. It was then that I spotted the source of the blockage, waving at me frantically. There she was, making her way slowly along, staggering under the weight of the largest pallet I have ever seen – it looked like it weighed a ton. It turned out that she had found it at the back of a local TV store (what was she doing there?).

With difficulty we managed to get it into the car and home. It now forms the base of our new log store. So I just need to send Sheila out to liberate a few more pallets of a similar size and shape and our log store will be complete.

Logs: While we were working on the wood pile, I asked Sheila what she remembered of logarithms at school.  She laughed. “I remember a book with big numbers in. I don’t know what we did with them though.”

The book of mystery from schooldays
So, what does the average person understand by the word logarithm – a term I’ve worked with all my life? I emailed a few of my friends and they all graced me with interesting and entertaining responses, from which here's a selection.

* Umm, log tables, looking things up that didn't make sense, then I had an exotic looking calculator and I still have no idea. (age 44)

* I know that a logarithm is something to do with sums, and that there are log tables, but I can’t help you any further. I have to say that I would have expected you to know what a logarithm is without having to ask. (age 65)

* Absolutely no idea whatsoever. (age 42)

* Something from school days -– slim book with columns of very long numbers in. Some kind of crazy multiplication? (age 67)

* I cannot think of any other bit of schooling where the books on the memory shelves have snapped so firmly shut. (age 62)

* Without looking it up I wouldn’t know the word logarithm, other than thinking it is a mathematical equation of some sort. (age 31)

One reply was a perfect textbook definition and it came from my neighbour Andy: 

* It's the power to which the base number is raised to get the number in question.

Have a gold star Andy and go to the top of the class! 

So, overall, the little book and its columns of numbers was well remembered by most of the over sixties age group, but very little else of any substance!

What IS a logarithm?: Although Andy's definition is correct, it may be too dense for most people to grasp at a single reading. Learning about logarithms needs a little patience. If you're interested in knowing what a logarithm is, I've made a short YouTube video which you can view here.




But a trickier question is, what’s the point of logarithms and why do we need them. Relax – I won’t explain that now but I hope to return to it in a future post. Instead, here's another 'lightbulb' joke.




Postscript
If you are interested in finding out more about logarithms and their properties, have a look here in my website, MathinPictures.co.uk.