## Class sizes

Today the labour party released a list of 40 things they would change should they come to power in may, (you can see the list here). When I was reading it I was pleased to see “smaller class sizes” included at number 4, the first non-nhs point on the list.

In my experience smaller class sizes lead to better progress. The smaller the class the lower the amount of low level disruption and the higher the amount of time the teacher can devote to each learner. Both these facts mean student progress is enhanced.

**But, I’m sure I read that class size has no effect? **

I’m sure I have too, I think it’s in the Sutton Trust toolkit produced by the EFF. Which is itself a meta analysis. I’d like to read the studies individually, as it goes against my own experience and the experience of every teacher I’ve ever spoken to.

I haven’t read the studies, so can only hazard a guess as to why the effects don’t show what we see. It could be that the analysis is distorted by other variables. I know that in the four schools I’ve been involved with the top sets have been bigger, so analysis run on class size vs progress would be distorted by the fact the most able make the most progress. If you have experience of, or links to, these studies do let me know, I intend to investigate further.

**So how do you know your experience isn’t distorted by similar other variables?**

There are two cases that I’ve been involved in that make me sure class size does have an effect in progress, certainly within the environment of a lower middle ability class at a school within a deprived area.

When I was an NQT I had a year 7 class who were set 7 or 9. The class had 26 pupils in it and over the first three half terms had made minimal progress, this was across the board in all subjects. A teacher was employed to take half their class in each subject and we specialists were to plan the lessons. Each pupil in that class made significantly more progress after the split than before it.

Then last year I had a year ten class, lower middle ability, with nearly 30 pupils in it. The class were mostly progressing well, but around 8-10 of them weren’t progressing as well as we knew they could. Extra capacity became available within the team and we split the class, from that point on the whole class made much more progress than they had before.

*What are your views on class size? Have you any personal experiences? Have you seen a positive effect? Do you agree with the view of the Sutton Trust toolkit that it doesn’t have an effect? I’d live to hear your thoughts.*

## Where’s the Function?

This question was another question from the January 2013 C3 paper that I remember almost foxing some of my students of the time:

It’s a strange question in that it asks for ff(-3) but, unusually, doesn’t define the function algebraically. I remember the day of the exam three of them discussing the paper with me after sitting it saying “it was a well weird function question, it just showed you the graph.” That day they had worked it out, but I feel that the time taken had contributed to at least two of them running out of time later in the paper.

So on Friday when a couple of learners asked for my help and I saw they were working on this question I didn’t even need to ask what their issue was. I started by reminding them that the graph was the function and that they didn’t need it algebraically to get an answer as there was enough information on the graph to answer it. We discussed this and they realised that the y coordinate is what comes out of the function when you put the x value in.

It’s a nice question, it highlights a possible blind spot that I need to address and shows us one of the pitfalls of an over reliance on past papers in revision. Past papers are a key part of revision, don’t get me wrong. But they shouldn’t be the only part of it, as otherwise a curveball question like this, one that asks something in a slightly different way, can really throw you off.

## Alternative Issues

Today my year 13s started work on the masses of past papers I gave them as an easter present (the lucky people that they are). One of those papers was the Edexcel C3 paper from the January sitting. One of my students today asked for help on this question from said paper:

He had managed part a easy enough and had then attempted to solve part b, when he checked his answer he had a different one so he asked for assistance. In tge first instance he said “Sir, what’s the maximum of this function?” I read the question and answered “2”, which was of little help and elicited the response “that’s what it says in the markscheme too, but how did you get it?”

The direct response for this would have been, “when you put it into alternative form R is 10, as 6,8,10 is a Pythagorean triple, so the function is maximised when the bottom is 2, meaning the max value is 2.” But I didn’t want to explain it that way, as I knew he knew how to get there. I knew exactly what he had done wrong, because I was hit with a memory from January 2013. I remember three year 13s showing up at my classroom after the exam and discussing the test, I remember the discussion around this question and I remember the look on two of their faces when one said “I nearly sodded up the alternative form question, I nearly didn’t realise that because it’s in the denominator you need to minimise the function to get the maximum.”

I knew this morning that the same thing was happening, “Did you get 2/11?” I asked. Once I had an affirmative response I asked what gave a higher number, a bigger or a smaller denominator, after a short discussion the learner grasped why you needed to minimise the alternative form bit to maximise the function. I thought I’d addressed this better since the Jan 2013 scenario, but I guess I need to ensure I fully get this across in future.

This is, however, the reason past paper practice can be an imperative part of the revision process, it gives an opportunity to diagnose these misconceptions and address them before its too late.

## Under pressure?

I seem to be reading a lot lately about “Mocksteds”, these mock inspections that seem to be all the rage, and other seems to me that they can cause more damage than good. I have a number of friends who have been subjected to these over the last few months, all felt heavily pressured, all felt they had had their already massive workloads increased and some felt undermined and have lost confidence in themselves. I can’t imagine any of these are positive outcomes for the schools.

**So you think it’s all nonsense?**

No, I can see there may be a need to review and evaluate a school, it’s the manner in which it is often done that I take issue with. Part if the philosophy in my school is that the “why” and the “how” are as important as the “what”. The reasoning behind the review should be shared, and when designing the how staff welfare should be taken into consideration. During my career so far I’ve been through Ofsted inspections, internal “mock inspections”, external “mock inspections”, internal “reviews” and external “reviews”. On the face of it, the latter four seem very similar but in reality there can be a major difference.

We had an external review in the winter term, it lasted a day, reasoning was shared beforehand, and it seemed like the senior leadership had gone out of their way to ensure that staff remained relaxed about the process, while still putting their best foot forward. The process went well and I didn’t notice any colleagues crying in the cupboards.

On the other side, I have a friend who teaches at a different school who has recently been through a “mocksted”, she was put under an immense amount of undue pressure and was left questioning her own practice.

I have friends at another school who have been through a series of events that would test the strongest of people. They are expecting Ofsted, so are in the midst of Ofsted fever, they are being hit with no notice “Marking” scrutinies, they had an external company come in to do a mock inspection to ensure they are ready, and had an internal review conducted by SLT from within their Academy chain to make sure they were ready for that. If ofsted come next week that’s three two day review processes in a term. One staff member received 5 observations in 6 days. As if teachers weren’t under enough pressure.

The worst though, is this harrowing account, where school staff were told that Ofsted were here, but actually it was an external review team conducting a mocksted. A fact not shared until after the inspectors had left.

**What are the outcomes**

If a review is conducted in the right manner it will find areas to develop for the school, it will build teachers confidence in themselves and their leadership and will ultimately improve outcomes for the learners in our care. If a review is conducted wrongly it can lead to increased stress, reduced confidence, reduced staff well being, increased workload and can lead to learners getting a raw deal.

Of course we need to know how are schools are performing, we need to know where staff strengths and weaknesses are in order to tailor our development plans, but it doesn’t need to be a painful process.

## Books

*It’s world book day today, and despite dressing up as Rudy Baylor from “The Rainmaker” (you know, suit, shirt, tie etc) my pupils didn’t believe I was dressed up! I have been working on a pupil facing site, and wanted to include a page on suggested further reading, so far I have:*

You all have textbooks to help with your course, and revision guides. If you want any further books/websites to specifically help with your course, try the revision page.

This page is a selection of books that are about maths, but not specifically related to the course. The maths in them should be accessible to A level maths students and they will help deepen your knowledge of maths.

For those of you considering further study they may be particularly helpful in shaping the direction you go in, and may provide excellent fodder for UCAS statements and university interview discussions.

These books are available from all good bookshops, we are looking at getting some into the library, and Mr Cavadino has a few of them which I’m sure he’ll lend you if you ask nicely.

**Fermat’s Last Theorem – Simon Singh**

This is Mr Cavadino’s favourite book. It is based around an enigma known as Fermat’s Last Theorem. Fermat was an amateur mathematician, but a brilliant one. He did Maths for the live of it and he came up with, proved and solved many great mathematical theorems and puzzles. When he died he left a number of theorems unproven. Slowly as the years progressed mathematicians proved them all, except one. His last theorem. One he posed with a note “I have a truly marvellous proof that this margin is too narrow to contain”. In the book, Simon Singh looks at the evolution of maths, and how this amazing theorem drove so many people to make so many amazing discoveries. If you only read one maths book in your life, make it this one.

**The Simpsons and Their Mathematical Secrets – Simon Singh **

Have you ever watched an episode of The Simpsons, or its sister show Futurama, and noticed a maths reference? Well so did the author. It turns out it’s not just a coincidence, but that the writing team are all mathematicians! For years they’ve been sneaking mathematics into the world’s most popular cartoon. The book looks at the maths they’ve included, why they’ve included it and how it relates to the episodes it’s in. If you like maths, and the Simpsons, then this book is for you.

**The Code Book – Simon Singh**

Another classic from the pen of Simon Singh. This one looks at the evolution of cryptography and cryptanalysis over the millennia and includes some fascinating accounts of where codes and encryption have been used throughout history. If you feel you may be interested at pursuing this as a career, or just have a passing interest in it, then make sure you read the book!

**From Here to Infinity – Ian Stewart**

Ian Stewart is a Professor of Mathematics and has written many fantastic maths books. This one is a particularly good one for learners who are interested in picking Maths at university. The book tracks the evolution of mathematics and gives a great introduction to many of the mathematical topics that will be covered on the course. The maths in this one does get quite heavy, and there may be a coupe of points where you can’t follow it. This shouldn’t matter as you should all be able to follow the majority of it and if you do read it and want to discuss any of it then mention it to your teacher. The variation of topics included in the books gives a good start point to future mathematicians who are unsure which areas of maths they would like to study.

**Music of the Primes – Marcus Du Sautoy**

Marcus Du Sautoy is a another Professor of Mathematics and he has also written many great maths books. This one is based around unsolved problems in Maths. He says the reason he wrote it was that when Fermat’s Last Theorem was getting a lot of press the non-maths world seemed to be of the feeling that when it was solved that was it, we had “done” maths. This is obviously not the case, as maths is infinite, and he uses this book to explore some of the big unsolved problems of the subject.

*I am also going to include more books by Marcus Du Sautoy, some Rob Eastaway ones and Tony Crilly’s “How big is infinity”, then add books as I read them, but I’d love feedback on it and I’d love suggestions on what other books to include.*

## The Code Book – A Book Review

It may not surprise you to discover that Simon Singh (@SLSingh) is one of my favourite authors. I have previously reviewed “The Simpsons and Their Mathematical Secrets“, and “Fermat’s Last Theorem“, the latter of which is still my favourite ever book.

“The Code Book” came out quite a while ago, but I’ve only just read it, and as with Simon’s other books I was hooked pretty much straight away. The narrative Simon weaves throughout the ages is amazing. Seamlessly switching between the hardcore maths of the subject and the historic events that drove the discoveries. What did Mary Queen of Scots use codes for? What about Julius Caeser? How brilliant was Alan Turing?

I was lost I’m a world of espionage, war strategies and amazement at how cryptographers (code makers) and cryptanalysists (code breakers) managed to keep in out doing each other, whether the driver was military power or purely academic.

The book covers some heavy maths, but it is broken down into terms that anyone with a high school education should be able to follow. At times I felt it was broken down a bit too much, but I realise I gave quite a strong mathematical background, and the subject of codes may appeal to people who don’t.

*This book is a great read, a must read for anyone with the slightest interest in codes, which is probably a growing number in tge wake of the imitation game! I think if I’d read it as a teenager I may have ended up in cryptanalysis! There is also a version of the book aimed at **young adults**.*

## Math(s) Teachers at Play – 83rd Edition

Hello, and welcome to the 83rd Edition of the monthly blog carnival “Math(s) Teachers at Play”.* For those of you unaware, a blog carnival is a periodic post that travels from blog to blog. They take the form of a compilation post and contain links to current and recent posts on a similar topic. This is one of two English language blog carnivals around mathematics. The other is “The carnival of mathematics“, the current edition of which can be found here.*

It is traditional to start with some number facts around the edition number, 83 is pretty cool, as it happens. Its prime, which sets it apart from all those lesser compound numbers. Not only that, its a safe prime, a Chen prime and even a Sophie Germain prime, you can’t get much cool than that can you? Well yes, yes you can, because 83 is also an Eisenstein prime!!!! Those of you who work in base 36 will know it for its famous appearance in Shakespeare’s Hamlet: “83, or not 83, that is the question…..”

Firstly, to whet you appetite, here is a little puzzle, courtesy of Chris Smith (@aap03102):

The solution can be found here. *And there are tons more puzzles here.*

So, what delights do we have for you within the carnival?

**Maths Fun**

Firstly, we had a few submissions that were based around having fun learning maths.

Firstly, Mike Lawler (@mikeandallie) submitted this on a 242 sided Zonohedron: *This project plus the follow up project, were projects out of Zome Geometry that we did in the open space in our new house (i.e. we don’t have much furniture yet!) Really fun project for kids. Lots to learn about geometry, symmetry, and especially perseverance! Really shows how amazing the Zometool sets are as learning aides, too.*

Our Carnival Founder, Denise Gaskins (@letsplaymath), has submitted this on “Fun with the impossible Penrose triangle.”

**Pedagogy and Reflections**

There’s a few post around the pedagogy of teaching mathematics, including reflections on what’s been tried in the classroom.

Cody Meirick submitted this on “Maths Investigations”: *The developers of this series argue that math should not be viewed as a history lesson, teaching formulas and concepts that mathematicians “invented” centuries ago. Instead, math time should be an active and even creative process, allowing students to learn through experimentation and exploration*.

Benjamin Leis asks “can we get there from here?”: *I’ve been blogging about my experiences running a math club for the first time. This one was a planning exercise to figure out how to make a particular problem accessible to the kids.*

Rodi Steinig has submitted this nice little post around tessellation.* In the 5th of our 6th Math Circle session about Escher and Symmetry, middle-school students make some discoveries about assumptions, and also discuss the pros and cons of inventing your own math.*

The superb Ed Southall (@solvemymaths) has produced more posts in his excellent complements series, aimed at helping to further subject knowledge within the profession. The latest instalment is on Highest Common Factors (or Greatest Common Divisors, to those of you across the pond) and Lowest Common Multiples.

Manan Shah (@shahlock) wrote this excellent piece entitled “100 days of school“. He looks at the maths behind animation.

Emma Bell (@el_timbre) looks at maths hubs and some of the issues involved with them.

Kris Boulton (@krisboulton) explores the question: “If we cannot see the learning win a lesson hat are exit tickets for?”

Colin Beveridge (@icecolbeveridge) has written this post on the three “tricks” he wishes his GCSE students would use. *(SPOILER: They aren’t really tricks!)*

Bodil Isaksen (@bodiluk) has written this fine piece on embedding maths into all aspects of school life.

**Resources **

Also this month many people have shared great resources, here are some brilliant posts on that.

Jo Morgan (@mathsjem) has produced another of her maths gems series. The series looks at great ideas and resources Jo has discovered recently, and it is another of my favourite series. This is issue 25.

Mel Muldowney (@just_maths) has put together this lovely post entitled “Two is the magic number” sharing some resources aimed at wiping out misconceptions and checking answers.

Sam Shah (@samjshah) has shared his thoughts, experiences and resources from a recent lesson on angle bisectors in triangles. It is well worth a read.

Paul Collins (@mrprcollins) has shared these fantastic mathematical plenary sticks.

Bruno Reddy (@MrReddyMaths) shared this great post which includes hints, tips and resources to aid peer tutoring.

Tom Bennison (@Drbennison) shared this superb post on using geogebra to teach conic sections.

Martin Noon (@letsgetmathing) shared this excellent trigonometry calculator, check it out!

**Teachers at play**

February saw maths teachers from across the UK gather at “A celebration of maths“, which was a superb event put on by the White Rose Maths Hub (@WRMathsHub.

Dave Gale (@reflectivemaths) and Colin Beveridge (@icecolbeveridge) have produced another sterling episode of “Wrong, but useful” the nations favourite maths podcast (well, mine at least).

*Well that rounds up edition 83, I hope you have enjoyed it. If you want to submit a post to the next carnival you can do so here. If you’d like to host contact Denise (@letsplaymath). And make sure you catch next moths carnival which will be hosted by John Golden (@mathhombre) over at mathhombre.*

## An interesting discussion

Yesterday I was teaching perimeter of compound shapes with my year 9s. After they had been solving problems based on shapes made up solely from squares and rectangles one if the pupils asked “Sir, will the perimeter always be even?” I thought this was a great point for discussion so I opened it up for the class.

They decided that for the type of shape we were looking at the perimeter would indeed always be even, as you had to cover every distance twice, meaning 2 was always a factor of the perimeter. I was impressed with their reasoning, but a little disappointed that one had to prompt non-integer side lengths. When I did they quickly dismissed this with “of course we knew that sir, but be were only considering whole numbers!”

I then asked them to consider is this applied to all shapes. They quickly concluded that triangles could have odd perimeters with integer side lengths, and circles, then they extrapolated to any shape with an odd number if sides. They concluded quickly that regular polygons with an even number of sides couldn’t have an odd perimeter if it had integer side lengths. Irregular polygons provided a more difficult challenge, and I left them to ponder it over the weekend!

## Share this via:

## Like this: