Welcome to Issue 48 of the Secondary Magazine. It will soon be the shortest day, closely followed by Christmas, so prepare yourself for the end of the autumn term by using our Christmas Mystery and enjoying some of the other interesting features in Issue 48.

Contents

From the editor – rhymes and mnemonics Hey Diddle Diddle, SOHCAHTOA, Along the corridor, apple pies R delicious…do you use rhymes to help pupils remember things in mathematics? How useful are they?

Up2d8 Maths – What would you do? The fortnightly Up2d8 Maths resources explore a range of mathematical themes in a topical context. The news has been full of the enormous £90 million Euromillions lottery win which was shared between a syndicate of IT workers in Liverpool and a couple from South Wales. ‘What could you buy with all that money?’ is a question that has been on many people's lips. This resource looks at one way that you might like to spend the money – by buying every CD, DVD and game on amazon.co.uk! Will you have enough?!

The Interview – Jane Jane’s work as a fitness instructor is underpinned by mathematics – find out how in this Issue’s Interview.

Focus on…interpretations of infinity Cantor and Hilbert both made a significant contribution to our understanding of infinity? Check out your interpretation in this Focus.

An idea for the classroom – Who killed Santa? Do you fancy yourself as an Inspector Clouseau or are you more of a Sherlock Holmes? Here’s a murder mystery to try in your classroom this Christmas.

5 things to do Check out the last dates for posting cards and presents – or find out about some mathematical events in this festive period.

Diary of a subject leader – Real issues in the life of a fictional Subject Leader So what has our subject leader’s ingrown toenail got to do with his school’s performance and his mathematics department? Do your students live up, or down, to your expectations? Join our subject leader as he considers how the school’s reputation affects the achievement of its students.

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

From the editor – rhymes and mnemonics

I suppose I am quite lucky – I’ve got a reasonable memory and things tend to stick in my mind if there is a reason for remembering them (that doesn’t always apply when I’m in the supermarket but I absolutely refuse to ring my significant other while staring at the breakfast cereal and shouting, “Have we got any oats left?” so that the rest of the store can hear me). I used to have a problem with ‘necessary’ until someone advised me ‘Never Eat Curry, Eat Salmon Sandwiches and Remain Young’. Our head of science was absolutely stunned when I started to tell him that ‘Henry Heinz Baked Beans Can Not Offer Full Nutrition So Munch Apples Since Prunes Sometimes Cause Awfully Painful Cramps’. He told me that he knows the properties of the elements of the periodic table and can visualise their positions/numbers because of their properties – there is the difference between a scientist and me! What has caused me to write about mnemonics in this issue? Well, I was in a colleague’s lesson last week and heard a pupil half singing, half saying:

Hey diddle diddle the Median's the middle. You add then divide for the Mean. The Mode is the one you see the most. And the Range is the difference between.

Feeling a bit like my head of science, I felt a bit uncomfortable to start with and started to question the pupil about mean, median, mode and found that he actually had a really good understanding of these measures of average but used the rhyme to remember which label to attach to each one. What could be wrong with that? It’s a bit like SOHCAHTOA isn’t it? Pupils can have a strong feeling for the ratios in similar right angle triangles and just need reminding that the one that involves opposite and hypotenuse is called Sine. That is completely different from pupils that tell me that what you do first is write down SOHCAHTOA, label the sides, cross out the one you don’t need etc… So, on my drives to work this week I have been thinking about the memory aids that we use in the classroom which include: Cherry pie’s delicious (c = πd) Apple pies are too (a = πr2). Twinkle, twinkle little star, circumf'rence is 2 π r I didn’t really know you cared, area is π r squared. A litre of water's a pint and three quarters. Two and a quarter pounds of jam weigh about a kilogram. But then I start to feel a bit uncomfortable again when I think about rhymes like: It’s not for you to reason why, Turn upside down and multiply!

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Now – that takes things into another league for me. I feel I have moved away from helping pupils to remember labels for objects or definitions, and into the realm of memorising algorithms. Of course I want pupils to reason why. Why on earth do we turn upside down and multiply? (If we think about the multiplicative relationships between 2 and 5, 2 x 5/2 = 5 so using inverse operations 5 ÷ 5/2 = 2 but we also know that 5 x 2/5 = 2 so ÷ 5/2 and x 2/5 are equivalent operations.) So I also feel uncomfortable with the Welsh couple ‘Taf and Di’ (the self inverse functions Take From and Divide Into) and ‘All you need to do to add fractions is to remember the upside down picnic table’. I’m sure you use similar aids in your classroom. Are they all ways of remembering labels or do they play another function. Why not tell us about them?

Up2d8 maths The fortnightly Up2d8 Maths resources explore a range of mathematical themes in a topical context. The resource is not intended to be a set of instructions but rather a framework which you can personalise to fit your classroom and your learners. The news has been full of the enormous £90 million Euromillions lottery win which was shared between a syndicate of IT workers in Liverpool and a couple from South Wales. ‘What could you buy with all that money?’ is a question that has been on many people’s lips. This resource looks at one way that you might like to spend the money – by buying every CD, DVD and game on amazon.co.uk! Will you have enough?! The activity poses a simple problem for students to solve and allows them the freedom to develop their own methods and work with different degrees of accuracy. Initially students are introduced to the lottery-winning couple and the amount that they have claimed (£45 570 835.50) and are then asked whether this would be enough to buy all of the games on amazon.co.uk, or all of the CDs? the DVDs? or more?? It is likely that students will decide to sample the prices of DVDs, CDs, games etc. to allow them to estimate the value of Amazon’s stock and consideration will need to be given to the accuracy of different sampling strategies. This resource is not year group specific and so will need to be read through and possibly adapted before use. The way in which you choose to use the resource will enable your learners to access some of the Key Processes from the Key Stage 3 Programme of Study. Download the Up2d8 Maths resource - in PowerPoint format.

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

The Interview Name: Jane Wrafter About you: I’m a fitness instructor and personal trainer, and I run my own business offering pay-as-you-go classes outside gyms.

The most recent use of mathematics in your job was... taking payment for a class and having to work out the change. Some mathematics that amazed you is... that pi really does go on for infinity…………

Why mathematics? Because it underpins much of the fitness work I do – either working out appropriate weights to use for exercise, trying to improve a client’s body mass index or advising clients on calorific intake. Your favourite/most significant mathematics-related anecdote is… One of the strongest muscles in the human body in ratio to its size, is the tongue.

A maths joke that makes you laugh is… Why was 6 afraid of 7? Because 7 8 9. Something else that makes you laugh is… Ricky Gervais.

Your favourite television programme is… The X Factor and Dragons’ Den. Your favourite ice-cream flavour is… toffee. Who inspired you? The amazing late Jane Tomlinson (cancer sufferer who raised loads of money for charity and overcame amazing physical challenges throughout her illness to do so). If you weren’t doing this job you would… be earning more money doing something else! But it would be something people-oriented (and nothing too mathematical by the way...).

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Focus on...interpretations of infinity

How have views of infinity changed through time? The Compact Dictionary of the Infinite gives a timeline – just go to the bottom left hand side of the screen and click through to infinity.

It is a common misconception that infinity is a number rather than a concept. Wolfram

MathsWorld says that infinity is “an unbounded quantity that is greater than every real number”. This is different from being a number and certainly different from being the ‘biggest number’. Infinity is not a number at all.

German mathematician Georg Cantor (1845 – 1918) explored and re-imagined the concept of

infinity. His theory depends on a simplified idea of counting – a one-to-one correspondence. Imagine a teacher on a school trip – as each student gets off the bus, she puts a mark on a piece of paper. After the visit, as each student gets back on the bus, she crosses out the mark. Without using numbers she has used the one-to-one correspondence of students and marks to ‘count’ the students. Cantor described any set of numbers which has this one-to-one correspondence with the natural numbers (1, 2, 3, …) as countably infinite. This leads to the counterintuitive result that there are ‘the same number’ of even numbers as there are counting numbers as there are fractions! Cantor defined this number, the cardinality of these sets, as 0א(aleph-null).

Cantor showed that the set of Real numbers (which includes the irrational numbers), are ‘more

infinite’ than the set of Natural numbers. In fact, he showed that the set of Real numbers between 0 and 1 is more infinite!

A set is countably infinite if ‘a prescription can be given for identifying its members one at a time’ put simply, they can be written in an ordered list.

Imagine one of your students, in an effort to prove Cantor wrong, tried to write every number between 0 and 1 as a decimal in an ordered list. If they’re not able to do this, then Cantor is correct! They bring you their list to be marked (!) - if you can find a number that’s not on the list, then Cantor is right. If their list is a set of decimals

0.a1a2a3a4a5…

0.b1b2b3b4b5… 0.c1c2c3c4c5… and so on

To find a number that’s not on the list, ask the student where the number r is where r = 0.r1r2r3r4r5… made in such a way that r1 is different to a1, r2 is different to b2, r3 is different to c3 and so on. Your number, r, is different to every number on the student’s list in at least one decimal place and, therefore, Cantor is correct! This set of numbers is uncountably infinite.

German mathematician David Hilbert (1862-1943) devised the idea of the Hilbert Hotel to

illustrate different types of infinity. Hilbert’s idea has been turned into a short story, a short film, and even inspired the lyrics to a song!

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

The ∞ symbol for infinity is sometimes known as the lemniscate (from lemniscus, Latin for ribbon). It is thought to have been introduced as the symbol for infinity by English mathematician John Wallis in his 1655 work De sectionibus conicis.

“This video is either going to be my greatest accomplishment or my stupidest video ever. Today,

I’m going to count to infinity.” Can you guess what happens?

An idea for the classroom – Who killed Santa? “I‘d love to do more activities in lessons but we just haven’t got time – the syllabus is so full.” Have you heard that said? Have you heard that said in your school? So often the end of term is a time when little useful learning takes place so why not use the end of this autumn term to do a ‘fun’ activity which also gives pupils access to some of the process skills in the curriculum and contributes towards their developing functionality.

A Christmas murder mystery always keeps people alert after the Christmas pudding, so at the risk of being a bit early – here is a murder mystery for the classroom which is perfect for the end of term. Pupils are given a set of clue cards which define the task – of finding out who killed Santa – and give them all the necessary information to solve the mystery. Pupils have a map of ‘Santa Town’ where distances are measured in ‘reindeer miles’; the characters are simply called Red, Orange, Yellow etc.

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

How would I use this in the classroom? Ideally I would organise pupils to work in pairs, give each pair a set of the

clue cards and a big piece of paper to work on, then stand back as pupils ‘sort out’ what they need to do, observing some of their useful strategies to feed back to pupils later in the lesson. If pupils are not used to working in this way it may be necessary to ‘oil the wheels’ of the activity by scaffolding the problem. It is tempting to dive in and tell them what to do, but by doing this you have diluted some of the rich learning experience. So some useful teacher behaviours might be:

you could stop the class after five to ten minutes to ask them to clarify what the task is (ie. find out who killed Santa)

ask them to tell you some strategies they are using which might be: o copying out the table o copying out the map to jot down the positions of the colours o putting different possibilities on the map o putting distances on the map o putting cards to one side after the information has been used

ask pupils to suggest which is a useful card to start with ask pupils to suggest which may be the final card they will use.

Towards the end of the lesson, it would be useful to stop and ask pupils to talk through how they arrived at their solution but also to suggest some of the skills they have been using – it may be worth pointing out instances where pupils have justified their answers or worked logically. Finally, it may be useful to ask pupils if they feel they have done any mathematics. Do you think they have?

5 things to do this fortnight

How to ensure progress for all in maths is a series of three events being hosted by the SSAT. The first of these took place in November but there are two more, one in Birmingham on 10 December and the final event in London on 26 January. The day will comprise a mixture of showcases from teachers in leading schools and a chance to discuss progression in maths via round-table rondevals.

If you’re near the University of Surrey on 11 December then why not head along to the STEM

event GMES – Using science from space to safeguard the future on Earth, a masterclass with Anu Ojha. Anu Ojha is Director of Education and Space Communication at the National Space Centre, Leicester, and GMES is Global Monitoring for Environment and Security. The National Space Centre and University of Surrey have developed a range of new activities and innovative teaching approaches that use GMES topics as contexts for post-16 science and mathematics curriculum areas. Current GMES scientists have worked with Advanced Skills Teachers and other outstanding educators to produce a series of intensive, all-day teacher masterclasses and extra resources to help bring GMES into the classroom. Although these sessions are primarily aimed at science and mathematics teachers in further education and sixth form colleges, they are also open to post-16 science and mathematics teachers in other institutions. The event is free and a bursary is available to cover supply and other costs.

The Advisory Committee on Mathematics Education (ACME) recently launched a paper designed

to provoke discussion on the future of post-16 mathematics. An online community has been created which aims to stimulate discussion on the paper and to collect your views on the proposals. ACME then intends to digest those views with the intention of producing a position statement in spring 2010 on the issue of Level 3 mathematics. Get involved and have your say.

What maths has surprised you recently? In this video David Acheson (who will be President of the

Mathematical Association in 2010-11) talks about some of the mathematics that has surprised and inspired him. Remember, we looked at some tricks that you might use to surprise your students in Issue 41 of the Secondary Magazine.

Have a great last couple of weeks of term but remember to get your cards and presents in the

post! The last posting dates are 18 December (second class) and 21 December (first class).

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Diary of a subject leader Real issues in the life of a fictional Subject Leader A theme current in our political debate, and perhaps not always thought of as an issue to debate, is the expectation that service users should have increasing control over the service they use. I think I have heard it called the ‘consumerisation of state services’. I can’t quote a source, probably because my brain is already full of school stuff and to-do lists, but also because I find the whole thing boring and annoying. I don’t particularly want to have to spend ages investigating which hospital and which surgeon is the best to operate on my slightly ingrown toenail (apologies to any ingrown toenail sufferers. I once played hockey with a goalkeeper who had an ingrown toenail, and the lengths he would go to in order to avoid kicking the ball were indicative of the pain he was in) Similarly, the release of Schools Adjudicator Ian Craig’s report, in which he estimated 3 500 parents lied on school application forms each year, indicates the lengths parents go to in order to get into ‘good’ schools. I want my local hospital to be able to operate on my toe expertly, and hopefully at a time that is in some way convenient to me. I’m not too worried about how well it would be done in Aberdeen. I also want my school to be the best it can be, regardless of the quaintly-titled, ‘Historic-Roman-name-of-city Academy for Learning’ that is ten minutes down the road. Generally speaking, when I buy something I want to pay a fair price for a good quality product. It would be much easier if I didn’t have to look up on the Which? website before I bought. I think I can understand the idea that competition can raise performance. And that is fine, so long as there is an understanding that alongside the winners, there are inevitably losers. And even more importantly, that the difference between first and last place may be very small. The problem I have is that those in the losing institution can too easily expect to be a poorer version of the winning, when in fact this may not be the case. Previously, I taught in a school that was just a few percentage points above the local competition. I would often hear senior staff admonishing behaviour and ‘recalcitrant’ attitudes. The parting shot would so often be: “You do as you wish, because if we decide we don’t want you we can easily arrange for you to go to ‘Just-2%-less, shockingly-awful-normal-school-down-the-road’ – we will even buy you the new uniform so there is no hassle for you.” It had an effect way beyond that it deserved or should have had. We could not have arranged the move. We could have bought the uniform and to be honest, for many of the cases, a whip-round the staff room would have easily covered the cost. I am embarrassed to have to admit to using the line a few times myself, but I knew colleagues at ‘Just-2%-less, shockingly-awful-normal-school-down-the-road’ and their teaching was at least equal to mine. I felt a bit uneasy about it. I now teach down the road from a thriving, soon-to-expand-with-sixth-form, historic Roman name of city Academy for Learning. I am at ‘Sizeable-15%-less, shockingly-awful-normal-school-down-the-road’. The hardest bit of my job is keeping heads up. Our students expect to do worse. In some subjects, more of our students met their FFT targets than students at ‘Historic-Roman-name-of-city Academy for Learning’. However, the perception among students and their parents is that we do not do a good job. The only grounds they have for this is that ‘Historic-Roman-name-of-city Academy for Learning’ gets better results. They do not have time to properly investigate the difference; they use the simple headline figure. Yes, they want a good education for their offspring, but the fact we are lower than ‘them down the road’ means we must be worse, and thereby not good enough. From this stems so many issues in our school. Our parents constantly look for the smallest error; openly chastise teachers in front of their children, and then expect us to earn back the respect. But worst of all, is

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

.www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

that students genuinely believe that the reason they will not do well is because it is a bad school, so there is absolutely no point in them trying to better themselves. Having to work within this context is very draining and I regularly watch able students underachieve just because they have an excuse. They are sucked into the self-perpetuating myth that we are a bad school. As teachers, we argue that they are able to succeed in our school. But... we would say that wouldn’t we. Well, that’s what the kids say....?