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False Variables and Simpson’s Paradox

February 5, 2014 2 comments

Last weekend I attended a day of lectures as part of my MA course. The focus of the day was on barriers for learning and it was quite intensive. Part of the day involved looking at the statistics involved in various things and seeing how they related to the development of children and the lecturer mentioned the idea that a false variable can skew ones ideas, and can make it look like something is having an effect, when in reality it is something else.

This idea of false variables is one that has been “following” me around recently. The first book I read this year was “The Simpsons and their Mathematical Secrets” by Simon Singh. In the book he discusses “Simpson’s Paradox”. The example he uses is in relation to the US government vote on the American civil rights act of 1964. In the north, 94% of democrats voted for the act compared to 85% of republicans. In the south 7% of democrat voted for, and 0% of republicans did. However, overall 80% of republicans voted for the act, compared to 61% of democrats. This example is great for showing Simpson’s Paradox and really emphasises the fact that stats can be deceptive. The worrying thing is that these stats can be manipulated to show that a higher proportion of democrats in the north and in the south supported the bill, or that a higher proportion of republicans supported the bill. Meaning both sides can legitimately lay these claims and hence really confuse the electorate. The fact of the matter is that the real variable that was feelings towards the bill differed largely due to attitudes in the north vs attitudes in the south, rather than a political allegiance.

Simpson’s paradox also appeared at school recently. A teach-firster in our department was planning a lesson on probability and asked me if I knew “that thing where you have a higher probability of picking one colour in each bag of balls, but if you put them all into the same bag you get a higher probability of the other.” This produced a rather interesting discussion, around Simpson’s Paradox, no one else in the department were familiar, and they all found it pretty interesting. We both then included it in our lessons. The question was around bins with coloured counters in them and showed that you had a higher probability of picking black counters from the blue bin in two cases, but if you combined the counters into the same bin, the higher probability came from the red bin.

The example of this false variable situation given in our lecture was that of breast feeding. The stats suggest that breast feeding equates to a better academic achievement for the pupil. But if you drill down into the stats you see that there is a far higher proportion of breast feeding mothers in the “middle class” as opposed to the “working class”, and that academic achievement may be more down to socio-economic status, rather than the breast feeding itself. This could be due to a plethora of reasons which may include: a higher level of education to the parents, enabling them to provide more support to learning at home; a higher income in the house which may enable private tuition if a child is falling behind or even that more working class families are reliant on shift work, longer days and multiple jobs, leaving them less time to spend with their children to aid their development. This is clearly a complex issue, and it highlights the fact when reading anything that includes statistics you have to ask yourself, “does the author have an agenda, and are they twisting the facts to suit it?”

A Book Review: Fermat’s Last Theorem – Simon Singh

January 12, 2014 3 comments

Recently I’ve notice that there are a lot of signposts online to great maths books that are available. Flying Colours Maths, Wrong but useful and the maths book club (Which i hope to get involved with soon) are but three examples of places where you can find mention of good maths books. As a maths teacher I have found an array of purposes for reading them. Firstly, I love maths so I find them interesting. Secondly, they give me a deeper working knowledge of the subject and improving your deeper subject knowledge should be a key priority for all teachers. And thirdly, I have found many lesson ideas in these books.

I figured that this blog would be a good place to review some maths books as and when I finish them. I did think about going back and writing reviews of all the ones I’ve read, but I don’t think my memory will allow it. Although I have decided to review one I finished at the end of 2013 which is still fresh in my mind.

Fermat’s last theorem – Simon Singh

When I started in the sixth form my pure maths teacher, Mr Armitage, had a poster up which intrigued me (as the problem had Wiles, decades before). It was based around Fermat’s last theorem. It contained the theorem, Fermat’s cryptic note about a proof but the margin being too small and a timeline of near misses up to Wiles’ achievement. I asked Mr Armitage if he could show me Wiles proof, as I was intrigued. He response was, “I’m sorry, I can’t. The proof runs to over 300 pages and some of the maths involved is beyond my own comprehension.” I was a little disappointed but understood, and decided that one day I would like to be in a position to understand the proof (Not there yet… I’m still struggling a bit with Modular forms…).

Anyway, since then I have always meant to read this book, but I hadn’t got round to it until 2013. And boy was it worth the wait.

The book is fantastically written and takes the reader on a whirlwind rollercoaster ride through mathematics history, from Diophantus to Wiles. There is a real suspense thriller feel to it in parts, and even though I knew the outcome I found myself entirely absorbed in the story and needing to find out what happened next, at points even questioning what I thought I knew.

Simon has kept the writing in the main text to a level where top GCSE level knowledge would be enough to follow, introducing new concepts in a way that is easily digestible. He also uses the appendices well to explore some of the deeper issues. This is a good way to appeal to those of us with a deeper thirst for maths without eliminating any of his potential readership.

There were a few occasions I thought I would like more information on certain things, but the further reading section has signposted me to places to look.

All in all, I think this was the best book (that’s any book, not just maths book) I’ve ever read. It isn’t just limited to Wiles’ battle with Fermat, but rather an amazing look at the world of maths, and mathematicians, through history that keeps you guessing and leaves you wanting more.

If you have even the slightest interest in maths, you MUST read this book.

You can see Simon on Numberphile discussing the maths involved in the book, and his more recent title “The Simpsons and there mathematical secrets” here

You can buy his books here.

You can follow Simon on twitter @slsingh

And as I’m currently reading “The Simpsons and there mathematical secrets”, you can watch this space for a further review.

Quadratics

November 27, 2013 3 comments

Currently my year 11 class are working on quadratic equations. I have always loved quadratic equations and as such i am quite excited to be teaching them. I have recently gone over them with my year 12 class so I had some good stuff on and I have built on that to create a good set of lessons around solving equations. I’ve shared them on TES here. Most of the stuff is mine, but a couple of the worksheets and the tarzia came from elsewhere, but i have included them for ease.

As part of my teaching with the group I have been trying to expand their matrhematical thinking. I want them to be able mathematicians who can think mathematically and solve a mathemnatical problem put before them, rather than pupils who are good at answering questions phrased a certain way. They all already have C’s and B’s, and to ensure they hit the A’s and A8’s they need they must be able to think mathematcially. For tomorrowa lesson i have come up with this starter (It is in one of the notebooks on TES):

Pythagquadstarter

I’m hoping that they will be able to make the jump and combine the ideas to come up with the right answer. Even if they don’t, I’m sure they will produce some good maths to help them on their way!

I also intend to show them this photo (also included in the lesson):

Peppa

It provides a good talking point as the discriminant is replaced by a delta in the picture and next to it the delta is than defined as b^2 – 4ac.  Why the quadratifc formula shows up in Peppa Pig is another question entirely. It seems stranbge how many cartoons involve higher level maths problems, and I may highlight Simon Singh’s new book “The Simpsons and their Mathematical Secrets”, as something they can potentially add to their christmas list! (Its certainly on mine)!

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