The above tweet [link] popped up in my twitter feed this morning, and it got me thinking.

Not about whether or not Dominic Raab’s claims* were valid.

No, I spent quite some time trying to figure out what that “graph” (info-graphic is probably a better term) was trying to say.

I just couldn’t figure it out.

Now, I’ll be the first to admit, I’m no economist and I’ve never formally studied the subject. But I would describe myself as reasonably numerate and (as I have written before) as a mathematician I am far more interested in the applied side of the subject to the pure; I am used to taking equations, data, charts and graphs and interpreting them. But on seeing the above, I just couldn’t understand it.

First schoolboy error was no axis labels (and no numbers on the y axis at all.)

The headline **in bold** mentioned house price increase from 1991 to 2016, suggesting a time series graph, where we are accustomed to seeing time flow from left to right. The title did imply that we were looking at a change over time, yet this makes no sense in the context of the graphic (I’ve given up callling it a graph because, although presented to try and look like a graph for (I presume) gravitas, it ain’t a graph).

I was now becoming increasingly confused.

Having twigged it was not a time series graph, my mind then picked up on a couple of key features of the graphic. The title said “**average house price**” and the top number on the x-axis was 275. I knew that the average UK house price is around £275K (I’ve since checked – its a little lower, but in that ball park) so perhaps the graphic was meant to represent the average house price in the UK? But that made the chart even more nonsensical.

By this stage I was genuinely perplexed. I genuinely had no idea what this tweet and graphic was trying to say.

I could have (and perhaps should have) left it there and got on with my day. But I couldn’t. It was bugging me, so I did a bit of digging to see if I could fathom what the Economics Editor of the FT was trying to convey. It seems I wasn’t alone in my confusion, finding this thread on Reddit

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I stumbled across the above the other day (source) and it made me chuckle, reminding me a little of of this post about cute angels and other mathematical bloopers.

The cartoon above was drawn by Dan Piraro and can be found on his Bizaro website – well worth a visit. By virtue of the fact that you are reading my blog, I’m guessing you are of a mathematical bent (but whether that bend forms an acute or obtuse angle, who knows!) and therefore may particularly enjoy this cartoon of his.

]]>Back in 2014, I wrote “What a turnip” as the then Chancellor of the Exchequer, George Osborne, refused to answer the simple times table: “what is seven times eight?”

Today it was the turn of the School’s Minister, Nick Gibb, who was on TV announcing his scheme for all eight and nine year old children to sit a compulsory times tables test.

Of course, the inevitable happened: he, himself, was asked a times tables question (what is 8 x 9?) and he, like Osborne before him, refused to answer.

I get why he (and other politicians) choose not to answer – what is in it for them? Nothing. Get it right and “meh, you should know that”, get it wrong and it’ll haunt you forever, could even spell the end of a glittering* political career. (* tongue firmly in cheek)

But the presenter, Kate Garraway, skewered him perfectly – in the context of the world of an eight year old, sitting a formal test is as high pressure as doing a TV interview is for a government minister.

To be fair, I don’t disagree with all that the Minister has said and done. The results for individuals, or individual schools, will not be published, the data will be used as a tool to measure performance in this skill across LEA and the country. And he spoke of the need for instant recall of times tables allowing the freeing up of working memory for other tasks, and I agree with this. As a maths teacher, I support anything that will promote numeracy and mental arithmetic – a solid foundation in these skills is not essential to future success in mathematics, but they certainly do help. A lot.

Anyway, you can watch the cringe-worthy ministerial squirming on the video below – it should start at the point the question was asked, but if you can spare six or seven minutes, its worth watching the whole interview from the beginning.

Anyone walking past my classroom of late might have come to the conclusion that I was fomenting a revolution, hearing me inciting my students to

Bring down the power!

The Head can sleep easy in his bed – I am not encouraging the students to rise up, burn their books and storm the staff room. I am merely teaching the rules of logarithms, and the power rule in particular. You will, of course, be familiar with the rule:

log x

^{n}= n log x

but students need to be taught this, so I constantly find myself telling them to “bring down the power.”

When I do, I do like to think of myself as some sort of latter day Lenin, inspiring my students to throw off the shackles of ignorance, to rise up and seize the power that a knowledge of mathematics will bring them!!!

]]>A year ago, I wrote this blog post, introducing Large Data Sets, a new feature to be taught on the “reformed” new AS and A level specification. Back then, it was a lot of guess work as to how best to use this new element on the syllabus, and how they will be examined in the exams.

One year on, and I must confess, I’m not much the wiser, but time waits for no man and, with much of the “pure” content having been taught the elephant in the room that are large data sets can no longer be ignored.

Helpfully, OCR have published some teaching activities for use with large data sets. They can be found by following this link. I have also uploaded the Word documents and Large Data Set and you can download the directly on the links at the bottom of this post.

They are described as “Starter Activities”, designed to familiarise your students with the large data sets and should take circa 10 minutes per activity. Not sure I agree with this. The activities/discussion points are good – for example, in Activity 5 you might get half your class to be “Team Bristol Mayor” and argue the case for how they have successfully got commuters out of the car and travelling to work by foot or bike, and ask the other half be “Team Paxman” taking down the Mayor’s argument and highlighting car use has increased over the last decade. The statistics can be used to support both arguments and I do think that it is right that we are teaching our students how data can be – and is – used in the real world.

But be warned: these are not trivial activities that you can print out 5 minutes before your lesson – you will want to spend some time looking at the activities yourself before presenting them to the students, even if it is only to understand what the various graphs show as, for example, OCR have not labelled the axis in many cases.

They are also liberal with the use of abbreviations – perhaps this deliberate, forcing the student to consult the Large Data Set to remind themselves with what they mean, but to help you, below are a few of the more common abbreviations, and what they mean:

UMLT: Underground, Metro, Light Rail, Tram

BMC: Bus, Minibus, Coach

MSM: Motorcycle, Scooter or Moped

LDS: Large Data Set

LA: Local Authority

Download the documents:

Investigating Bicycle Use (Sampling Activity)

]]>I could spend hours pontificating, explaining and lecturing and still not explain the difference between a Type I error and a Type II error as simply and as effectively as the image above. A picture really can be worth a thousand words.

Credit where credit is due: the image can be found in

The Essential Guide to Effect Sizes – Paul D. Ellis

This succinct and jargon-free introduction to effect sizes gives students and researchers the tools they need to interpret the practical significance of their results.

… a useful (and readable) book that aims to equip the reader to be able to distinguish between statisically significant and practically significant results.

]]>A little over a year ago I completed my collection of 15 works of fiction.

As a tutor to fifteen Upper Sixth students I was obliged to write a glowing UCAS reference for each one, to support their application for university. My line, above, is of course, a joke – every word that I penned was factually correct, 4,000 characters painting an accurate – if positive – picture of the hopeful applicant. I did joke with my students that what I had written may well be the best thing anyone ever writes about them – my aim was to make each and everyone one of them appear an attractive prospect for the universities to which they were applying.

It was no mean feat, if only in terms of time. If I spent only an hour on each, then it would have added an extra two working days to my load; a more realistic three hours per student – including time to discuss them with their subject teachers, draft my reference, proof read it, upload it etc – would add an extra working week to my “normal” teaching load.

And I couldn’t help but wonder how much emphasis the universities would place on my wordsmanship.

It seems no-one (other than the institutions themselves) know.

The Sutton Trust has today published a report looking at universtity application and admissions, and it makes interesting reading. The report looks at three areas:

- The UCAS form
- The Predicted Grades system
- Personal Statements

… so not teacher references, but relating to personal statements the report does state:

They have different approaches to the use of contextual admissions and apply different criteria when analysing personal statements.

so it is reasonable to assume that different criteria will have been applied to my references by different universities – some may have read them, some may not. I have no way of knowing if my efforts were of use, or simply vanished into the ether, never to be read by anyone but me.

The Sutton Trust’s report is quite damming of the admissions process and as a teacher/sixth form tutor and a parent (my Year 13 daughter (at a different school) has been applying to University this term) I agree with them.

The goal of the Sutton Trust is to improve social mobility and, in its report, has focused on the challenges faced by poorer and disadvantaged. My students certainly don’t fall into that category so, by implication, they benefit from the current system, but I think that they, too, would benefit from a leveling of the playing fields.

At the heart of the fallibility of the current system are predicted grades. Applications are made, and offers given, well before students sit their A level exams and so predicted grades – not actual grades – form the central pillar of the process. The Sutton Trust correctly argues that those students from disadvantaged backgrounds are less likely to have pushy parents picking up the phone to demand that their child’s predicted grade be raised. If the student doesn’t have high enough predicted grades, they won’t be made an offer.

The predicted grades that I give are aspirational – a grade to aim for, but one that *may* ultimately prove to be beyond the reach of the student. I am not alone in this, the report states:

Evidence shows that the majority of grades are over-predicted, which could encourage students to make more aspirational choices.

Other than in exceptional circumstances, a student shouldn’t do better than their (university) predicted grade, otherwise we’ve done them a disservice and, possibly, shut an educational door on them that should have remained open.

However the report tells us:

However, high attaining disadvantaged students are more likely to have their grades under-predicted than their richer counterparts. This could result in them applying to universities which are less selective than their credentials would permit.

It goes on to tell us that circa 1,000 students each year have their grades under-predicted.

There is, of course, a way, a simple way, to overcome this problem:

Don’t allow pupils to apply to university until they have their grades.

Once you know you have three As, or two Bs and a C, you can then, with confidence, apply to the university and course for which you have made the grade. No more opaque applications, no more unfair predicted grades, no more clearing, just a simple, straight forward and transparent admissions system.

The educational calendar my have to be “tweaked” to make this work, to give universities and students time to make to make their choices – or perhaps not, thousands of students get their results in mid-August, go through clearing and then pack their bags ready for a September start. Or, my big idea, start the university year in January: students finish their exams in July, giving them 5 months to work/travel/grow up before starting university, with a brief punctuation in August to make their application.

Perhaps I can leave the last word to Sir Peter Lampl, founder and chairman of the Sutton Trust:

“Access to leading universities has improved and they are working hard to attract a wider applicant pool. However, the brightest disadvantaged students, given their grades, are under-represented at leading universities. The admission process itself may be responsible for this.

“Accordingly, the Sutton Trust is recommending we move to a post-qualification applications system. This is where students apply only after they have received their A-level results. This does away with predicted grades. Having actual grades on application empowers the student. They can pick the right course at the right university with a high degree of certainty they are making the right choice.

“Additionally, we are recommending using contextual data in admissions. Also, the format of the personal statement should be reviewed to see how it could be improved and how it could become more transparent.”

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Over on Twitter, @missradders has sparked a really interesting discussion on subtracting fractions by offering the above method to subtract two fractions. (see the Twitter thread here)

I must admit, it is a method new to me and, from reading the replies to her tweet, it was unfamiliar to many other teachers, to.

But is it a valid method?

Again, reading through the comments on the thread, its taking a bit of a kicking. Before we come to a conclusion, perhaps its worth asking why, in the age of the scientific calculator, do we even bother to add or subtract fractions?

Beside me on my desk, I have the new(ish) Casio Classwiz fx-991 EX calculator, that will integrate, solve cubic inequalities, handle matrices – adding and subtracting fractions are well within its capability: what is the point of asking recalcitrant pupils to manually perform an algorithmic task that can be done electronically in moments?

The answer is algebra.

Being able to add or subtract – mentally, or on paper – straightforward fractions such as, say ^{3}/_{4 } – ^{1}/_{2 } has it’s merits, but I would be reaching for my aforementioned fancy calculator to do the subtraction at the top of the page: ^{8}/_{9 } – ^{3}/_{5 } but I need to know how to do it with numbers as it is the stepping stone to being able to work algebraically.

High end GCSE – and **all **A level – mathematicians need to be able to work with algebraic fractions and being secure with a method of adding and subtracting numeric fractions is crucial for this skill.

So the test of @missredders method is to see how well it works with algebraic fractions. So I tried it to subtract the fractions: ^{a}/_{x } – ^{b}/_{y }, my working is below.

It works. In fact, I think it helps to see the method in algebraic form as it explains why it works.

I don’t think that it is a method that I will be teaching (but happy to keep it in my armoury, just in case) but I don’t think that it is the dog’s dinner that some are claiming on Twitter.

As a method it works, and there is a clear route from using the method numerically to using the method algebraically, and that is important. I suspect that if a student is taught the “grid method” shown at the top of the page, masters it and goes on to use it algebraically for higher level maths, they will probably develop their own shortcuts for using the method (i.e. going straight to writing down the common denominator) and that – find short cuts – is what maths, and being mathematical, is all about.

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It was shortly after my standing tree pose had inadvertently become a fallen tree, and I lay on my mat, a crumpled collection of arms and legs listening to the calm, soothing words of my Yogi that it happened, that I had my epiphany, my moment of clarity, when I suddenly understood.

A mid-week evening in late September, the summer holidays a fast receding memory with only a damp autumn followed by a long, dark, bleak winter to look forward to. Back in the classroom for three weeks, and tiredness was already gnawing away at my very being. The obvious thing to do, the easy thing to do, would have been to flop on the sofa, fire up Netflix and nod-off in front of another American import. But no, it was Wednesday night, and Wednesday night means Yoga night, one of the highlights of my week (the other being my Friday night spinning class.)

Many, including myself, have asked me why I enjoy my yoga and spinning classes so much. Not too many people look forward to an hour of punishing pain pushing the pedals on a stationary bike every Friday night, but I do. I’ve tried to explain why I enjoy it so much – the virtues of physical exercise, the camaraderie of the class, the relief when its all over. When all you can think about is the burning pain in your thighs the problems of an unmarked pile of books and that difficult phone call earlier in the day shrink into insignificance, and that certainly cleanses the mind. But it still didn’t get to the nub of why I enjoyed it so much, why I would look forward to it all day long.

But then it came to me, on my yoga mat. Enlightenment.

As a teacher, in your classroom, you are the boss. It’s your domain, you are – you have to be – numero uno, top dog, the big cheese. You spend your day making decisions, giving instructions, controlling your environment. And when you’ve been doing the gig as long as I have (too long to count) you inevitability end up with various positions of responsibility which means more problem solving, telling others what to do, taking charge, and I love it. My siblings would describe me as bossy, I prefer organised and efficient. Regardless, I’ve found my niche and it’ll take a lot to prise me out.

I wasn’t looking for enlightenment, I only started doing yoga as I wasn’t very bendy and getting older wasn’t helping my suppleness. Spiritual awareness? Nah, thanks, but no thanks, not for me.

But my epiphany had been stalking me, and then: Wham! It stepped out of the shadows, taking me by surprise, and hit me in a moment of realisation. I suddenly understood why I do it, why I look forward so much to my yoga and spinning classes.

For an hour or so, twice a week, I’m no longer in charge. I’m on the other side of the fence, I’m the one listening to and following instructions and its hugely liberating. Letting someone else take control, doing what they say, surrendering to their direction frees the mind of all its cares and troubles. For a short while, a couple of times a week, you have no responsibilities other than to yourself.

Now, don’t get me wrong, this isn’t the birth of a new me – most of the time I definitely want to be in charge. For a megalomaniac like me, teaching is the perfect job: every hour a new group of subjects file into your domain to be inspired, nurtured, led on a pathway to discovery – a captive audience. But for a couple of hours each week to be the student, and not the teacher, to let someone else take the lead is a wonderful experience.

Maybe you should try it.

]]>**Possibly the best Venn Diagram, ever.**

Two words – absolute genius.

Witty, visually appealing but, most importantly, mathematically correct.

Sets and Venn Diagrams have been on the IGCSE syllabus for some time, and made it onto the new GCSE syllabus, so we’ve all got to teach them. Project this image onto your whiteboard, sit back and put your feet up – job done. You could spend hours telling your classes about intersections and unions, or you could just show them this and they’ll grasp it in a moment.

The image is just one of many great diagrams, charts and infographics to be found in Stephen Wildish’s fantastic book: Chartography – worth a tenner of anyone’s money.

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