Sheldon can't do probability

For the uninitiated, the people mentioned here are characters in the TV programme The Big Bang Theory.

In the episode The Russian Rocket Reaction Howard explains excitedly that some equipment his lab has been working on is going to the International Space Station and that someone needs to go there with it.  "Guess who that someone is?"

Sheldon's reply is "Mohammed Lee" (or other variants of the spelling).

When Howard asks "Who's Mohammed Lee?", Sheldon replies "Mohammed is the most common first name in the world, and Lee the most common surname. As I didn’t know the answer, I thought that gave me a mathematical edge."

This could be a way to introduce pupils to the idea of independent events.

First of all, why does Sheldon think this is a good answer?  If Mohammed is the most common first name in the world and Lee (or Li) is the most common surname then the most common full name is presumably the two of them combined.  Using independent probability, we can multiply p(firstname is Mohammed) by p(surname is Lee).

What is the problem with this reasoning?  The surname Lee/Li is one of the most common names in China and Korea.  The vast majority of people with this surname are likely to have family who originally came from China or Korea.

The boy's name Mohammed (which may be spelled in a number of different ways, including Muhammad and Mohamed), according to Wikipedia has its origins in the Arab World, and is particularly popular in the Middle East, north Africa, Pakistan, Bangladesh and India.

If names were distributed randomly then Sheldon would be right, but given the cultural, religious and geographic ties with names, the two events are not independent and it is likely that there are fewer people who have Mohammed as a forename and Lee as a surname than Sheldon expects.

Thought experiment:  What if we looked at the names of all the pupils who attend a school and found that the most common first name was "Jack" and the most common middle name was "Louise".  How many pupils would be named "Jack Louise"?

I suspect the answer is 'none', although Johnny Cash might know differently.

HS2 - Reams of Rubbish

The Daily Telegraph is carrying a story with this headline:

Two things immediately jump out here: we can work out how quickly we need to read each page, and we can work out the density of the paper involved.

The sub-heading says it “would weigh one tonne if it was (sic) printed out”.  This is because it was presented on a memory stick, which is significantly lighter, cheaper and easier than providing it in book form. 

Let's work out the density of the paper the campaigners intend to use.

One tonne = 1000kg = 1000000g

Divide by the number of pages to get 1000000g/50000pages = 20g per page.

Presumably it has been formatted to print on A4.  I know that A0 is 1m² of paper, so that means A1 is ½m²,  A2 is ¼m², A3 is one-eighth of a square metre and A4 is one-sixteenth of a square metre.

That gives the ‘weight’ of the paper that would be used to print this on as 20g x 16 = 320gsm.

The paper usually used in a photocopier is 80gsm, I think.  Card is 160gsm. 

320gsm seems rather heavy to me.

If we then revisit an earlier, unstated assumption, whereby I had assumed the document would be printed on a single side of the paper.  Surely some of the arguments against HS2 are environmental ones, so we would presumably want to print on both sides of the paper, meaning a 50,000-page document would need only (!) 25,000 pieces of paper. 

Let’s start a more sensible way around.  If the paper is 80gsm then a sheet of A4 (1/16 of a square metre) weighs 5g.  25,000 pieces of paper therefore weigh 125kg, which is sizeable but is a less impressive one-eighth of a tonne.

[The amount of reading is good.  Without any sleep I reckon you would need to read one page every 96 seconds to be able to get through it all.]

HS2 may well be A Bad Thing - I don’t know enough about it to be able to comment.  And giving people a 50,000-page document to read in under two months seems a little excessive (surely there is a summary?).   But overstating the size of the document by a factor of 8 doesn’t seem to help your cause.

Don’t do that.

The internet as soma

Over the next few days fall three 50th anniversaries.  Two of the events have become a major part of western culture and recent history, namely the assassination of President John F. Kennedy on 22nd November 1963 and the first episode of Doctor Who, broadcast a day later.  Here I focus on the third, the 50th anniversary of the death of Aldous Huxley, which also falls on 22nd November.

Huxley’s best-known work is Brave New World and for some reason this book is frequently compared to George Orwell’s Nineteen Eighty-Four.  While both books imagine a dystopian future and both predicted  a number of things that have come to pass, they are very different in their imagining of future society.

Nineteen Eighty-Four imagined a brutal world in the grip of perpetual war, where history is rewritten, the meaning and use of language is altered and surveillance is the norm.  There are many links here not only with the Soviet Union under Stalin but also with the level of surveillance carried out today, with CCTV cameras and the tapping of emails, text messages and phone calls being routine.

The society in Brave New World is less physically oppressive but rather focuses on the importance of consumerism, advances in medicine and on societal control.  The ideal sport is one that involves lots of equipment and to do your duty as a member of society you ought to spend your salary as you enjoy your leisure time.  Babies are not only created in a laboratory but are fully gestated there too, meaning that no-one has a parent and ‘mother’ has become a taboo word.  Children are conditioned, before birth with the use of chemicals and after birth using electric shocks and also by listening to audio while they sleep.  This conditioning not only prepares people physically for their future (with workers who will live in the tropics being given an immunity to malaria before they are born) and mentally (in terms of their capacity for learning) but also emotionally (a ‘Beta’ remarks that she is glad she isn’t an ‘Alpha’ because they have to work so hard but also glad she isn’t a member of one of the other classes because they wear horrible-coloured uniforms).

Leisure time revolves around the use of the drug soma, which is not only legal but the use of which is encouraged by the government.  The slogan: “a gramme is better than a damn”, refers to taking the drug.  The use of soma is linked with the state religion, ensures people are placid and that they spend their leisure time in a way that keeps them out of trouble.  If a character in the book displays any level of passion then another character is sure to suggest they take some soma.

In the same way that I earlier made links between Nineteen Eighty-Four and the present-day world, it is possible to draw comparisons between some of the themes of Brave New World and what we see around us, 80-odd years after it was written.  Examples include genetic modification and gene therapy, the acceptance and importance of consumerism and the way simple recreational pursuits now involve vast amounts of equipment (I used to put on a pair of shorts and a pair of plimsolls and go for a jog.  Nowadays there are shirts and shorts made of special fabrics, particular shoes to run in, hand-held computers, energy drinks, fluorescent tops, mobile phone apps, etc, all designed to help us run better.  All of them, coincidentally, cost rather a lot.)

What, then, is the 21st century equivalent of soma?  One could argue that it is a drug such as those used to combat ADHD, or a non-medical drug.  I want to make the case for it being the internet.

For me, born in the 1970s, TV was old hat.  It had existed before I was born and I grew up with it there.  I didn’t wonder how it worked but consumed what it produced and enjoyed it.  At that time there were three television channels and in the evening there would be ‘closedown’, when everything stopped (except perhaps on the evenings when the Open University would broadcast into the early hours).  For me “the internet” is still an amazing concept.  I watched the film War Games when it came out and marvelled at the idea that computers could be connected together.  I didn’t have an email address until after I had finished university and still have the habit of loading up lots of news webpages before beginning to read them from the days when my dial-up connection was so very slow.  I got excited when I learned how to use html and when I worked out how to create a basic website.

For children today the internet is old hat.  It has always been there and is only a means to access the content they want.  And ‘the internet’ is very different now from how it was only a few years ago.  Now almost everything children used to do is available on the internet, but with the only way of turning it off being the self-control of the user.  Instead of phoning up a friend and talking to them you can now interact with all of your friends using Facebook or Twitter.  Instead of watching TV you can see any TV programme or film by watching it on the internet.  Instead of playing physical games you can link up your games machine over the internet and play with others there.

Is this a bad thing?  Not in itself, and it is of course wonderful that we have so much choice, but it does bring to the fore the importance of self-control.  I used to be able to watch an episode of Doctor Who on the television on a Saturday night.  One episode.  If I missed it then I didn’t get to see it unless my friend who owned a video machine had recorded it.  Now it is possible to get hold of every episode with David Tennant as The Doctor online.  This might be through the BBC iPlayer, via Netflix, or on YouTube or an illegal file-sharing website.  When one episode finishes Netflix will immediately cue up the next one so you can continue to watch without distress!

It really does require a large amount of willpower to stop watching and would be very easy for a child to watch episode after episode.  The only problem with this comes for children who don’t have that willpower, or who aren’t aware that there are other things out there they could be doing instead. 

So how is the internet like soma?  Well, if you want to disengage from thought in Brave New World then you take soma.  Nowadays we can use the internet.  If you want to disengage from thought then there are scores (literally) of episodes of The Big Bang Theory you can watch, or an impossibly vast number of photos of cats looking cute, or videos of “the world’s biggest fails”, or of every goal Lionel Messi has scored this year, or … .

After spending several hours doing this you are as sedated as if you had taken a gramme of soma.  

Is this post a case of "when I were a lad ...", or "the youth of today ..." ?  Do I think that society has gone to hell in a handcart?  No, actually!  But it is important for those of us who remember a time Before the Internet to realise that children today can turn on, tune in and drop out in front of their computer.

So amongst the reviews of the life and times of JFK and the anniversary of Doctor Who, I would encourage everyone to read some Huxley and to enjoy the prescience of Brave New World in particular.

We’re going to need a bigger boat

Here are my initial reactions to the new GCSE subject content document published on 1 Nov 2013 by the DfE.

So far this document is all we have.  We don’t know what exams will look like, how the tiers will work, and how each statement will be interpreted (for example a “simple proof” will mean different things to different people), so much of what follows is necessarily speculative. 

At the moment it seems sensible to be aware of some of the implications (like the need to talk to SMT about an increase in mathematics teaching time) but not to do any detailed planning (like rewriting schemes of work) until we have more information.

New material

First of all, there are new topics.  Some of these are clear and obvious (such as finding the equation of a particular tangent to a circle centred on the origin), while others can be interpreted in a number of ways.  

An example of this is the way gradients of lines will be used.  The word “calculus” is not mentioned, but at the higher tier pupils are expected to “interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents)”. 

I hope this means that essentially we will need to teach the concepts behind calculus (that the gradient of the chord is an approximation to the gradient of the tangent and that as the length of the chord shrinks the approximation is likely to improve) rather than telling pupils the way to differentiate polynomials.  A potential exam question for this could give an unnamed graph and ask pupils to draw a tangent at a particular point and then find the gradient of this, with an appropriate level of variation of the position of the tangent line being permitted.

Other queries include whether function notation will be needed, whether estimating a square root is as straightforward as noting that sqrt(20) is between 4 and 5 (or whether an algorithm is required) and exactly what is meant by “find approximate solutions to equations numerically using iteration”.

So far this is the DfE document.  The next stage is for the awarding bodies to create specifications and these will presumably include guidance that exemplifies topics like these one.

Beyond the headline new topics we will also need to ensure pupils are happy with using the kinematics formulae (which will be provided in exams) and that they have learned the other formulae they will need (because these, including the quadratic formula and the non-right-angled trig formulae, will no longer be provided).

Foundation will include harder topics

Aside from this there are a number of topics that will now fall into the Foundation tier.  This includes solving quadratic equations algebraically by factorising, and using the trigonometrical ratios.

Assessment Objectives

Assessment Objective 1 (AO1) is headed “use and apply standard techniques” and will be weighted at 40% of the higher and 50% of the foundation papers.  Much of the content in the document will fall under this heading.

The remaining marks on the papers will be equally split between AO2 (“reason, interpret and communicate mathematically”) and AO3 (“solve problems within mathematics and in other contexts”).  In the notes about AO2 and about AO3 there is the statement: “Where problems require candidates to ‘use and apply standard techniques’ … a proportion of those marks should be attributed to the corresponding Assessment Objective.”

I assume this means that a problem that involves reasoning within the context of Pythagoras’ Theorem will count as part of the AO1 marks (for Pythag) as well as part of the AO2 marks (for reasoning). 

This means that pupils will need to be comfortable with the problem solving and reasoning sections of the specification to be able to get all of the content marks.

What will the exams look like?

One of the key differences may well turn out to be in the structure of the exams.

A press release from Ofqual said this: “Exams only in the summer, apart from English language and maths, where there will also be exams in November for students who were at least 16 on the preceding 31st August.”  Finally, this gets rid of early entry completely.  

It appears that the highest attaining pupils will be stretched within the GCSE rather than needing to consider an extension course.  We don’t yet know what each grade will look like, so it is not clear whether the old G grade will map onto the new grade 1, up to grade 8 being the old A*.  I think this is unlikely because here is the only opportunity to recalibrate the system, so there will not be a direct link between old grades and new ones.

“Maths: Tiered with an improved overlapping tiers model. A foundation tier will cover grades 1-5 and the higher tier will cover grades 4-9. Assessed by external exam only, as now.”

If you just look at the tiers there doesn’t seem to be anything different about it.  Currently we have grades C-G (the lowest five grades) on the Foundation tier and the new version will cover the lowest five numerical grades.  There is currently an official overlap of two grades with the higher tier (with a compensatory E grade available for those who just miss a D) and the new system will have a two grade overlap. 

To describe the new tiering system as an “improved overlapping tiers model” must mean that model is improved.  Does this mean that the exam papers will be structured differently?  The document has some items in ‘standard’ type (for all pupils to know fully), some underlined (for all pupils to use and higher pupils to be more confident with) and some in bold (for higher pupils).  An early version of GCSE mathematics (when I took it in the first year it was available) had a common paper for all pupils and a lower paper and a higher paper (were they called Foundation and Higher back then?).  It will be interesting to see whether we return to a model like this.

If this is the case then we will need to have an increased focus on the higher tier topics.  At present the probability that any given different topic will crop up in a GCSE paper is incredibly small.  Half of the higher tier papers currently have to be questions at grades C and D, so A* questions topics can only be tested on about a sixth of the paper.  The new system might well give us more A and A* questions (sorry, grade 7/8/9 questions) so the pupils will need to study this higher content more thoroughly.  (As an aside, at the moment some students start AS-level mathematics not having previously been taught ‘completing the square’, or ‘rationalising denominators’ because they are unlikely to crop up in a GCSE exam.  An increased focus on the more difficult material in exams would make it less likely that this would happen.)

Teacher support

This will be important on a number of levels.  For those fully conversant with the mathematical topics involved there will be the need to flesh out exactly what is meant by each statement (such as the gradients referred to earlier).  For those who have taught mathematics in school up to GCSE level there may need to be some refreshing of subject knowledge for the new topics.  Teachers who are not maths specialists but who have taught GCSE (either at both tiers or only at foundation tier) may need to learn some new material. 

All mathematics teachers are likely to need to consider some new pedagogical ideas, because even the topics that currently appear at A-level will have a different emphasis in the new GCSE specifications.  (For example, we do not currently have long at A-level to delve into the meaning of the gradient and the new GCSE topic will afford us this opportunity.)

Teaching time for mathematics

In a written answer to the House of Commons, Michael Gove said:

“The new mathematics GCSE will be more demanding and we anticipate that schools will want to increase the time spent teaching mathematics. On average secondary schools in England spend only 116 hours per year teaching mathematics, which international studies show is far less time than that spent on this vital subject by our competitors. Just one extra lesson each week would put England closer to countries like Australia or Singapore who teach 143 and 138 hours a year of mathematics respectively. We announced on 14 October that mathematics, alongside English, will be double weighted in secondary school performance measures from 2016. This will also provide a strong incentive for schools to ensure that they are strengthening their mathematics provision.”

It will be important for mathematics departments to have the teaching time they need.  This will be important for the teachers, so they have sufficient time to cover the material (including AO2 and AO3), important for the pupils so they are able to achieve well on this new, more difficult course (if they don’t achieve a good enough grade the pupils will need to continue to study mathematics after Year 11), and important for schools because the maths grade will be double-weighted in performance tables.  (Incidentally, this double-weighting will start in 2016, before the new exams come into play.)

Where will this lesson that Mr Gove suggests we need come from?  Which subjects will lose out?  I am afraid I don’t know, but what I do know is that it will be important for the mathematics teaching to be done well and the extra time will be necessary for this.

It is also worth bearing in mind that it appears this extra period per week should happen in every year group at secondary school and not just at KS4.

Implications

These changes are for first teaching Sept 2015, so the current Year 8 will be the first year group to take the new exams.  Presumably it would be ideal for next year’s Yr 7, 8 and 9 to have an extra lesson per week. 

Are there enough mathematics teachers for the country to be able to staff this?  Probably not at the moment, so schools will want to appoint new teachers fairly early next term.

Scheme of Work

Much of the language in the KS4 document echoes language in the new KS3 national curriculum.  We will want and need to update our scheme of work to ensure it will still be appropriate not only for Sept 2015, but also to get the current Yr 7 and 8 pupils to 2015 with the requisite background knowledge and skills. 

If feels sensible to have an updated SoW for KS3 in place for Sept 2014, but then to wait until the final GCSE specifications and associated document (exemplification, sample papers, etc) are published and approved by Ofqual before making major changes to the KS4 SoW. 

Summary

So, we are going to need to start thinking about some of the implications (such as timetabling for next year), and will need to consider our schemes of work, etc.  We know we will need a bigger mathematical boat, but will also need to be patient and wait for further information before we can do all of our planning for the new GCSE.

Finally, I haven't yet mentioned that I like many of the new features in the document.  For example, being able to explore the background to calculus will be important and interesting.  It is good that AO2 and AO3 will maintain the focus on reasoning and problem solving.  Having more time to teach mathematics well will be welcome, as will be the increased importance of the subject.