Strictly Stats

The Independent newspaper used to run a feature called “Questions to which the answer is no”.

The Radio Times has an article about Strictly Come Dancing that could fall into that category… maybe.

I am not a Strictly fan (someone sent me the link), but it’s a nice article to use with a class.

I am going to show this to a class and ask them to comment on whether this will be a good way to determine the eventual winner.  So, what did the Radio Times do with their trusty calculator?

They added up ten numbers and then divided the answer by 10 to work out the mean.

They then compare the ‘average score per dance’.  But wait: everyone has done ten dances, so everyone’s total is divided by 10.  So actually they could just have used the total.  That is a nice talking point.

Then there is the differing methodology.  The winner is the one who gets the most points in the final, not the dancer with the most points across the series, so there is no nailed-on reason as to why this must work.  It might be a good pointer to the eventual winner though.  Can we test that out?

Well if this is an accurate method then surely it will have worked last year?  The pupils could download the data themselves and find out.  It’s all available on Wiki (which is where my data below comes from).

Let’s have a look.  Here they are shown in the final order, so Caroline & Pasha were the eventual victors.

After 10 weeks last year the leaders (on average … and in total) were waiting to be eliminated after the next round, whereas the eventual winners were in third position.

I looked at two other years.  In 2013 the eventual winner was leading after 10 rounds.

In 2012 the eventual winner was third after 10 rounds.

This could all be discussed with the class and they could find out similar info for all the other editions of Strictly too.  (Maybe the years could be shared out so two or three members of the class analyse each year as a homework task?)

Of my sample of three years, in 2/3 of them the eventual winners weren’t in front after ten weeks.  I wonder what happens overall.

Hannah’s Sweets – a great time to be a maths teacher!

Lots of ink has been spilled about this question elsewhere.  I just want to respond to a couple of things that I have seen written, either on tweets or on blogs.

“A question shouldn’t have material from different areas of maths.”  I think this is absolutely wrong.  The idea of ‘using and applying mathematics’ was that you can, um, apply the mathematics you know.  There is no requirement that it must all come from one topic area. 

I think one of the big problems with modular mathematics specifications was the expectation that only the content of that module would be tested.  This meant that questions that involved ideas from two particular parts of the syllabus just couldn’t be set. 

“Kids haven’t been taught to use algebra with probabilities.”  I would think it reasonable for pupils to see unfamiliar contexts on an exam paper, given how many different contexts there could possibly be.  Let’s take a fairly typical sort of exam-style question:

Neo’s doctor tells him to take a blue pill every 10 hours and a red pill every 12 hours.   He takes the first red pill and the first blue pill at the same time.  After how many hours will he next take both at the same time?

The underlying mathematics here is LCM.  Do questions of this type say “use the lowest common multiple to work out …”?  No.  Is it an unfair question if pupils haven’t covered questions about pills?  What if Neo is waiting for Trinity next to the bus stop and notes that buses leave to go on route A every 10 minutes and route B every 12 minutes (always dead on time)?  He sees buses A and B leave at the same time.  How many minutes will he have to wait until he again sees buses A and B depart together?.  Is that unfair if pupils have covered questions about buses? 

“Grade boundaries should be adjusted downwards.”  Gosh – this one has cropped up a lot.  I assume that people saying this don’t realise that grade boundaries do change anyway.  Every time.  Here, from the Edexcel website, are the boundaries for linear maths from the last two exam sittings.  The grade boundaries are different.  (This raises a host of other issues and questions, but those are not relevant here).

“The question involved some irrelevant information.”  Again – it is perfectly reasonable to have to scythe through irrelevant stuff when solving problems. 

“The context for the question was rubbish.”  Well – yes it was, but it’s quite hard to create exam questions that build up to a genuine and sensible context in the space of a couple of marks.  The context of this question is no worse than those about people taking pills or waiting for buses.  I think you could attack almost any exam question on these grounds.

So why did this question make so many people so cross?

I think the big thing is that people who have learned their maths solely by being shown identikit questions will have found this difficult. 

There were several memes that cropped up after that paper.  One was a very positive response to the stem and leaf diagram question.  Why was that?  Because it was an easy question about an easy topic and was very similar to questions that have appeared in previous years. 

Revision nowadays seems to focus very largely on past papers, to a much greater degree than it used to.  (I wonder whether that is because everything is now so readily available on the internet.)  I helped out with some AS-level Psychology revision a few weeks ago.  I know no Psychology, so my role was to read out the question and then, later, to read out the markscheme.  After three or four papers I was familiar with the main texts and ideas and felt that _I_ could mount a reasonable answer to some of the questions.  If pupils get used to seeing very similar questions in past papers, then when an unusual one crops up perhaps it is not surprising they feel it is unfair. 

The most recent Edexcel probability-with-quadratics question I could find was from June 2004, so perhaps those pupils didn’t go back that far in their past paper practice?

Here is that question (and it happens to be question 19, the same as Hannah's):

This question is arguably harder than the Hannah question, and I was surprised to see it was worth 13 marks!  Was it a good job that twitter didn't exist when this question was published?  Perhaps. 

(Steve Wren (@Yorkshire_Steve) pointed out that the modal score for part c of this question was zero, so candidates certainly found it difficult!)

Finally, why is it a “great time to be a maths teacher”?  It has been lovely to see how many people have been interested in how to answer the question, and wanting to talk about maths.  But most of all, it has prompted a debate about the nature of maths and how it should be taught, and that has been potentially rather valuable.

Blog – Politics Venn Diagram

I hadn’t thought much about Venn Diagrams in the past.  I had used them and enjoyed the wonderful “Venn that tune”, but I hadn’t really thought about them.

A link on the front page of the Telegraph website brought me up short:

33?  Seemed rather unlikely.

Here’s a screenshot of the article.  What a beautiful Venn Diagram!

The diagram helps explain what the journalists have done.  They have used this as a nice way to show each of the possible combinations of 5 parties, from the ridiculous (SNP forming a government by itself – which clearly can’t happen because they only have candidates in Scotland), to the ridiculous (a 5-way coalition between Con-Lab-Lib-UKIP-SNP).

It feels as if this ought to be a Pascal’s Triangle scenario, with 1 way for no parties to be involved, 5 ways for one party to lead, 10 ways for two parties to govern, etc.

The diagram lends itself to being coloured to show this, with the 1-party parts shown in red, the 2-party sections in yellow, etc.  Including the outside of all of the sets (no parties govern) that gives us 1, 5, 10, 10, 5, 1.  This totals 32, so there are actually 31 different options for governing rather than the 33 advertised.  [This also ignores the possible contribution of other small parties.]

It is then rather neat to colour other Venn diagrams to display the relevant row of Pascal’s Triangle:

2 sets:  1, 2, 1 (with the initial ‘1’ being the white ‘outside’ section).

3 sets: 1, 3, 3, 1

4 sets: 1, 4, 6, 4, 1

So this is yet another place in which Pascal's Triangle crops up.  Beautiful!