## Observations, Ofsted and the Trial of Alfred Wegener

Last week I had a morning conversation with a colleague from the science department that got me quite excited. I was about an hour before lessons were due to start and the colleague in question came into the workroom and started cutting up some cards for his lesson. I noticed one mentioned “the jury” so asked him what he had planned. He informed me that he was looking at continental drift and was running the lesson like a trial. It was to be set at the time when Wegener had first come up with his theory and pupils were role playing parts of defence and prosecution barristers, expert witnesses on both side. The lesson sounded awesome, I was gutted not to have a non-contact period when it was on so I could go and see he lesson!

While we were discussing this I reflected that often when cutting up resources in the workroom the question gets asked “are you being observed?” This is something that normally bothers me, I don’t understand why people would change their approach to a lesson because an observer is coming in. Obviously, there are things you wouldn’t do for you PM observation, I can’t imagine there being any point in observing a mock exam where the class are working in silence for instance.

Our discussion moved on, as my colleague suggested that he wouldn’t do the lesson if he was being observed, as there was potential for it to go wrong. This was the polar opposite to the usual expectation, and I wasn’t sure what to make of it.

This got me thinking about observations. I think that by altering the way you teach for an observation gives a false picture, and means there is absolutely no point in the observation taking place. But, if you are planning exciting lessons, but are using safe and steady lessons for observations, you are also giving a false picture.

It think the key word we all need to keep in mind, is appropriate. My colleague Mark Miller recently wrote this piece exploring the Ofsted annual report. The evidence he found within is that Ofsted are finally moving towards an approach that recognises that a single one-size-fit all prescribed lesson format is ridiculous. The context of each school is vastly different; the context of each class within a school is also vastly different. Even classes of similar age and ability will have a different context, and what works for a class with one teacher may not with another. It’s all about finding the appropriate lesson for any given class at any given time.

I think that, as professionals, we should be striving to give all our classes the best lesson for them. Making sure the lesson is planned appropriately. The right amount of stretch and challenge. The right sort of activities for the class, and the right seating plan to enable the class to all make the best progress over the course of the year. And that should be the same for all lessons, whether you are being observed by SLT, HOD, Ofsted or no one at all.

## (Tenuous) Christ-maths Links

Firstly, I should probably explain the photo: my year 13s (well, two of them) drew it for me because they said my room didn’t look Christmas-y enough!

They drew it after a special Christ-maths lesson. We did a C(hristmas) 3 past paper lesson. They told me it was a tenuous link (it was – and I’ve just realised I missed a major opportunity it could have been a C(hristmaths) 3 (Ghost of Christ-maths) past paper lesson!)

This wasn’t the first tenuous link to Christmas I’d used, and it won’t be the last! I thought I’d share a few here:

With year 7 this week I’ve been doing prime factor decomposition using “Factor Christmas Trees” – I did an example using 64 and split it symmetrically. I did it in green and circled the primes in red, the class were amazed at how much it looked like a Christmas tree and didn’t call me up on the tenuous link!

In a similar vein I’ve looked at some Probability Christmas Tree Diagrams.

Today I was discussing Eulerian walks and Konigsberg, they had to find some Eulerian walks round France with start and end points. “So Santa doesn’t have to retrace his steps!”

My favourite though, is tomorrows lesson: “Christ-maths Newton Raphson” – what’s Christmas-y you ask? Well Isaac Newton was born on Christmas Day! (*nb He was born on Christmas Day according to the Julian Calendar in use in England at the time. We have now adjusted and there are people who think his birthday should be adjusted accordingly, as he would have celebrated his birthday on Christmas I think it’s still fair to use it. Although he’s so great he deserves two birthdays, so I celebrate both on his behalf.*)

Other Christ-maths lessons I’ve done have had even less tenuous Christmas links. Christmas indices, Christmas Simultaneous Equations and Christmas Binomial Expansion were great lessons, but the pupils in question all called shenanigans on the fact that it was an entirely non-Christmas-y lesson where I had just added the word Christmas to the title!

If circles are coming up you could talk about mince pi s. For data you could do some mince pie charts. And don’t forget yule logs and exponentials!

Have you done any tenuous Christ-maths lessons?

## Why calculators should be banned.

“*Sir, there’s a fraction in it. Have I got it wrong?”*

*“Sir, what do I do with this fraction one?”*

*“Sir, I get confused when there’s a fraction.”*

All these phrases are far too common in my year twelve class at the moment. We’ve just finished c1 and are doing some past papers, and I’m fairly worried by the way some of them baulk at fractions. This isn’t a problem that is solely theirs though. Some of my year 13s sometimes have trouble with fractions too. It’s not an isolated problem either. I think it’s symptomatic of the “calculator culture” which we live in.

Students of all ages have become far too reliant on the infernal contraptions! My year 13s think I’m obsessed with triangles (so do my year 10s, 11s and 12s. Perhaps I am?! They are amazing shapes with endless possibilities though.). The reason for my year 13s is that I try to encourage them to calculate trig functions using triangles, rather than using calculators.

*“Why? When you’re allowed a calculator in an exam?”*

Because it’s quicker, because there’s less chance of error, and because it will ultimately make you a better mathematician.

This isn’t a problem that is limited to the sixth form either. I was observing a year 9 lesson yesterday on pie charts. The class are quite bright, and the tasks involved dividing 360 (ugh, degrees) by some nice numbers like 90, 60 and 12. When one of the girls near me reached for her calculator to divide 360 by 60 I took it off her. She looked at me in shock and I simply asked “what’s 360 divided by 60?” she said “6” without even thinking. I then asked her why she had reached for the calculator and she said “because it was there.” I then circulated the room and all the pupils were at it.

Recently I wrote a post on multiplication methods which was inspired by a twitter chat on the subject and itself inspired a further chat. During one of them the someone inevitably suggested “just use a calculator”.

I don’t agree. I think calculators are responsible for a major decline in basic maths skills. I think they are responsible for creating lazy A-level mathematicians. And I’m sure they will have cost many gcse students many marks in exams.

A while ago the government announced a ban on the use of calculators in primary maths tests. Perhaps I should have written this then. I thought about. I’m in complete agreement on this one. I’d go further and at least encourage against them for most things across all key stages. I don’t allow my pupils to use them unless it’s necessary. I want them to be fluent in the maths, not good at following instructions to type stuff into a calculator.

## Supermarket Maths

A while ago Colin Beveridge. (@icecolbeveridge), or possibly his friend the mathematical ninja, wrote this piece which looked at ice cream containers and chastised them for the practice of measuring themselves in mass (ie grams) and giving portion sizes in capacity (ml), yet not giving a density or any other sort of help to convert between the two. Don’t worry (spoiler alert) Colin did, of course, find a fix, and although I thought the practice strange I thought little more about it.

Until today, that is, when I went into the supemarket to pick up a few bits and the first thing I picked up was this bottle if Encona Hot Sauce (Other sauces are available, but this is my favourite):

As you can see, the good people at Encona have quoted the mass **and **the capacity in their bottle! My second thought was “if only Colin’s Ice Cream company thought like this!” (my first had been, “I bet I can use this in a lesson”!)

As I wandered round the supermarket I couldn’t help but think “I don’t remember ever seeing that on Encona bottles before.” So when I got home I checked, and sure enough, just mass no capacity:

I wondered what prompted the change, then, while unpacking the shopping I noticed this:

Heinz are at it too! This is the back of their garlic sauce! Now my mind started to race, have the government passed legislation to enforce the labelling of packaging in both capacity and mass? If so, why? Did Colin, or one readers, start a petition? A campaign? Was there a March on Downing Street? Will the law be enforced by the maths police?

Who knows? But I certainly think there are some good lessons to be had from it. I also noticed this whilst unpacking:

6 bars, 147 grams, so 24.5 grams each? Is it just me, or does it seem a strange number? The weird e means “estimated” (or so I’m told), so why estimate to 1dp when an integer value would be so much simpler? Ah, the strange world of supermarket Maths. I also wonder whether STD is the best abbreviation to use when selling a product…

## Euler, Konigsberg and WW2

Recently I taught a D1 lesson introducing the terminology involved in graph theory. Looking at trees, networks and all the other associated things. Graph theory is something I love. I never studied the decision modules at A level, so when I arrived at Manchester University in the autumn of 2001 and saw the module title: “Trees and Networks – with Professor Nigel Ray” I was intrigued by the name, but entirely unsure as to what it would contain. I loved every minute of that course, and have loved graph theory ever since. I’m certain that Nige’s inspirational lecture style deserves some of the credit, but I’m also fairly sure that graph theory itself deserves some credit too.

As you may or may not know, graph theory (sometimes called topology) is an area of maths which was built by Leonhard Euler (Oi! Read that name again, its pronounced “Oi-ler” not “you-ler”… don’t ever let me catch you making that mistake again….). Euler was prompted to discover this area of maths by the “Seven bridges of Konigsberg” Problem. In brief: The city of Konigsberg in Prussia (Now called Kaliningrad and in Russia) was dissected by the Pregel so that there are 4 distinct parts of the city. There were seven bridges, as shown below, and the problem was posed: was it possible to take a walk round the city traversing all the bridges once and only one?

Euler solved the problem. Of course he did, he was a total maths legend. Although by solved, I don’t mean he worked out a way to do it (or that he built an extra bridge), but rather he managed to prove that it could not be done. Something many had thought, but had been unable to prove beyond doubt. It was his method of analysis and proof that led to Graph Theory. All this, and my general love of all things Eulerian, meant that I digressed in my D1 lesson away from terminology and spend a lot of time discussing Eulerian graphs, semi-Eulerian graphs, Konigsberg and Euler himself. Not exactly what I was supposed to cover, but time well spent none the less.

The following day, I read this amazingly funny piece by Colin Beveridge (@icecolbeveridge) on Flying Colours Maths, and the two related articles that it linked to (this and this). I was excited, I could see the bridges on street view (although only two remain from Euler’s Time) so finally got to see (some of) the bridges that this area of maths was devised for! I was surprised to discover that since WW2 there are only 5 bridges, leaving the network traversable! (It is now semi-eulerian, you must start on one island and end on the other. So if you were to do it properly you would need to get to your start and from the finish point on a boat or other none bridge means). And I was happy to hear that google themselves use graph theory to find the best routes for their streetview cars! (Incidentally, I’m on streetview, sat in a bus shelter in LS1 –or at least I used to be…)

These two events (teaching D1 and Colin’s post) have reawakened my love for graph theory, and I’m looking to widen my knowledge of this wonderful area. If you’ve read anything good on it do let me know! First stop: Nige’s Lecture Notes!

## “Manglish”, and a mastery curriculum

Today I attended #pedagoowonderland, it was a wonderful event with some superb sessions. One of the workshops I attended was by @lisajaneashes about “Manglish”, this is her philosophy on maths and English across the curriculum (you can read her blog or pre order her book if you are interested in learning more).

During the session Lisa said something that got me thinking about a whole host of things and I wanted to share these thoughts. She said during the session that it would be really effective to cover certain topics in other subjects at the same time as you are doing them in Maths.

There are a number of things happening at the moment, and this idea, to me, links them together.

Firstly, with the curriculum overhaul coming out of Whitehall, (see what I feel is missing here) we have the opportunity to develop a new and exciting curriculum for our school. Secondly, we are trying to look at whole school numeracy, and thirdly, we are hoping to increase the number of our pupils who go onto further study maths.

**Mastery Learning**

I’ve been reading a lot about New curricula recently, and something that strikes me as interesting is the idea of a mastery curriculum. (You can read Joe Kirby’s (@joe_Kirby) blog here. Michael Tidd’s (@michaelt1979) here, and check out this website). The basis of mastery learning seems to be to spend longer on each topic, covering fewer areas each year and ensuring that classes have mastered a minimum level of learning before moving on. This strikes me as exciting. A SoW with short units means you cover a topic for a fortnight, complete a unit assessment, and move on. This can work really well, especially for the high achievers, but it has its draw backs. If students have failed in year 7 to fully master how to solve one or two step equations then when equations next come up you have to revisit that. As they haven’t managed to learn it in two weeks the first time, they may not have retained much and they may fail to fully grasp the topic again. This can be come a cycle and can lead to pupils in year 11 becoming stuck on problems they should have solved at a younger ages. A mastery curriculum would enable deeper learning, and give pupils more time to learn these skills, offering those who master them quicker to mover further on in the curriculum. The theory being that the longer, deeper, covering of the topic would ensure retention rates were higher and when the class returned to the topic they could move on.

**KS3/4**

I’ve been involved in a few discussions recently on the need for separate keystages, do we need a specific KS3 and KS4 scheme of work or could we have a five year scheme of work? In the absence of levels, I’d imagine many secondary schools are looking at moving to GCSE grades as a way of reporting from yr7. If we are using these grades from the start why not a singles scheme of work?

**Feedback**

The shorter scheme of work system gives rise to a lot if summative feedback. (You can read more about our feedback here). This means that formative feedback happens in lessons, but written feedback tends to be summative, with pupils receiving written feedback on the topic they have completed, an extension question (or consolidation question) for them to try and then move on. A move to the mastery curriculum would mean that marking with the same frequency would give more chances for formative written feedback which could create a much better dialogue in the pupils books.

**Maths Across the Curriculum**

To start with, I think we should call it maths, rather than numeracy. I don’t think it should be just about numeracy. There are many other areas that can link in, rather than just simple number tasks. Similarly, I think we should talk about English across the curriculum, as it shouldn’t just be kept to “key words”.

I also think that Maths across the curriculum needs to be a culture embedded in a school. Lisa spoke today about how she wasn’t good at maths at school and how she didn’t care about it. She told us how this was compounded by her English and Art teachers telling her they were rubbish at it and that as long as she got the c it didn’t matter and she could just forget about it. This is a problem which is still rife today. Last year one of my year tens informed me that one of his teachers had told him she could never do algebra and it hadn’t had a negative effect on her. This infuriates me. A lot of pupils tell me they hear things like that at home, which is bad enough! The whole grade C culture is detrimental too, as my sixthformers are finding out when unis want Bs. (You can read more on this here).

Once the culture is embedded, maths links can be made with other subjects. This sort of link could be strengthened, as Lisa suggested, by covering these at the same time. Logistically, this would be a nightmare to embed with the 2 week unit scheme of work, but I think it would be more doable given a mastery curriculum which covered topics in more detail for a longer time. The whole school would know that in this half term year seven were looking at representation of data, and they could build that into their lessons accordingly. If in geography pupils were collecting some data, they could analyse that in their maths lesson. If the scheme of work was written in such a way that pupils in each year group were covering the same strand of maths, this could provide exciting whole school opportunities. Assemblies could tie into the topics. Cross curricular projects could be in abundance. Pupils would be seeing the links, seeing the importance, seeing the context and having the learning consolidated and embedded.

**Drawbacks**

There are drawbacks to this idea. There is the worry that pupils may get restless and lose interest if the same topic was covered over and over again, although I think this is avoidable with planning. Set changes would be much harder to implement as different classes would have reached different points in each area. It may be harder for pupils to catch up if they moved from another school partway through the course.

**So is it the answer?**

In short, I don’t know. I think there are many plus sides to moving to a mastery based curriculum and I am currently swaying towards thinking it would be a great way to go. But to be sute I need to read more on it and discus it more.

**What do you think?**

Have you implemented this sort of curriculum? Did it work? Are you thinking of it? Does it sound good to you? Or do you think it’s daft? Whatever your opinion, I’d love to know.

## Fractal Christ-maths Trees!

Yesterday was the 1^{st} of December, and as such it was deemed time I our house to put up our Christmas tree. The tree itself is an artificial tree which my parents bought when I was a child sometime in the early 1990s and have since given to me. As such, I have been putting it up for more than 20 years, but this year I noticed something interesting that I hadn’t noticed before.

The tree comes in many parts, and fits together quite easily. The majority of the parts look like this:

They are packed away flat, then you spread them out and attach to a pole (the trunk, as it were) and fit together to form a tree that looks like this:

As we were putting it up his year my partner said, “These branches all look like mini trees themselves, don’t they?” This caused a real revelation. The designer of this tree had harnessed the power of fractals to create the most real looking artificial tree I’ve ever known. And this would make sense, given that real Christmas trees do also have a fractal quality about them. Every year I’d been putting up this tree and never realised the mathematical significance it held!

Here is the (almost) finished product:

And of course, no post about fractals wold be worth it’s salt without the following joke: “What does the B Stand for in Beniot B Mandlebrot?” “Benoit B Mandlebrot, of course!”

## Edublog Awards 2013

I read about these awards the other day, and in the first instance it struck me as rather an odd thing, and I wasn’t going to nominate anyone. However, having since read plenty of other nominations and discovered some excellent blogs I figured that, actually, it is a great idea and that if this post enables just one person to find a new blog they get something out of then the nominations will be firsthand worthwhile.

So here are my nominations for the Edublog Awards ( http://edublogawards.com):

Best individual blog: Reflective Maths – Dave Gale (@reflectivemaths)- http://reflectivemaths.wordpress.com/ this is one of the first blogs I discovered and helped me really get into the world if blogging. There is a lot about maths teaching, but also plenty about maths in general. A great blog we’ll worth a read.

Best Group Blog: The Echo Chamber (@theechochamber2)- Old Andrew leads a group of dedicated educationalists out to ensure that as many education blogs as possible reach a wide audience. http://educationechochamber.wordpress.com

Best resource sharing blog: Mr Collins Maths (@mrprcollins) http://mrcollinsmaths.blogspot.co.uk/?m=1 This is the blog I have gotten the most resources from. They are always superb and it is interesting and informatively written.

Best podcast: Wrong, But Useful (by @reflectivemaths and @icecolbeveridge) http://www.flyingcoloursmaths.co.uk/category/podcasts/ a superb maths podcast I’ve written about before here.

Best administrator blog: the always wonderful Chris Waugh (@edutronic_net) for the #blogsync initiative http://blogsync.edutronic.net

Most inspirational blogpost: this from Chris Hildrew (@chrishildrew). Chris’s Blog is always worth a read http://chrishildrew.wordpress.com

Best Twitter hashtag #MTBoS

Lifetime achievement : David Didau (@learningspy) http://www.learningspy.co.uk There is nothing he hasn’t blogged about!

You can find out about other blogs and pages I like here.

## A look back at 2013, and forward to the future

I guess inevitably, given the time of year, I have been in a rather reflective mood. I suppose the season builds to NYE, Christmas passes, you catch up with family and friends you haven’t seen for ages and all this brings the year to the forefront of your mind.

On top of this, I’ve been reading many #nurture1314 blogs and these have made me think more about my year. (I would highly recommend reading this offering from @andyphilipday).

My 2013On the whole, I’ve had a great 2013, I’ve seen my daughters first birthday, her first steps and I’ve seen her grow from a babbling baby to a chatterbox toddler. My partner has embarked on a return to education, and I myself have started working on an MEd (student discounts all round!)

Professionally, I have received a promotion. I’m now the KS5 coordinator for maths, which is something I enjoy immensely and something that gives me experience in the direction I want my career to go.

I’ve worked on my own practice, and my CPD highlight was the #pedagoowonderland event at Joseph Swan academy in December. I attended a number of other teachmeets and thoroughly enjoyed and learned form them all.

I had this blogpost picked up by the TES and it was published in full on their website.

Real maths teaching based highlights for me this year were: when a leaving year 13 told me I had restored his faith in maths; when a current year 13 told me he wanted to forge a career as a mathematician; when a year 12 pupil told me she had “gone off maths” in year 11, but I had “made it awesome again”; when a year 11 told me, I used to hate maths but now I can’t wait to do it at a level! And when a year ten came to find me to tell me he had just “worked out how long and at what angle to cut a roof joint in construction using that pythagoras thing and the tan ratio trigonometry thingy, sir was well impressed I knew how to do it!” (He meant his other teacher, but I was too.)

I finally managed to read Simon Singh’s (@slsingh) book “Fermats last theorem”, which is the best book (any book, not just maths book) I’ve ever read.

This, along with the awesome numberphile (@numberphile) and wrong but useful (@reflectivemaths and @icecolbeveridge), have meant 2013 has seen my already great love of maths grow exponentially.

2013 in education2013 has seen many changes in the wider educational landscape, and I think some of them are for the better. The new maths curriculum looks good, but it could have been better. This news from ofsted (via @andrewold) is brilliant, and shows that they feel, as do I, that there is no perfect pedagogical style, but that teaching is about teaching the appropriate style for you and your class. (you can read more on this here). On the down side, there have been some terrible pieces of legislation passed, the worst being PRP (you can read more on that here).

My hopes for 2014I hope to continue to watch my daughter develop into a thoroughly amazing person. I hope that my partner and I both thoroughly enjoy the education we are undertaking, and I hope to continue to improve my classroom practice.

I plan to attend more teachmeets and other events (including June’s #northernrocks event). I might even present at one.

I plan to continue writing this blog and reading the amazing other ones that are out there. (A list of my favourites can be found here)

I hope to read more! Maths books, learning books and others. Currently I’m reading Here to Eternity by Ian Stewart, Philosophy if the big bang theory, and Hattie’s Visible Learning for Teachers. My list of maths and education books to read us massive, so I hope to make a dent in it.

I intend to investigate more areas of maths I am not too familiar with, starting with topology, as I know the basics and have a good starting point.

I hope 2014 sees a move towards ending the inequality we see currently in the British education system (see this post for more details). I also hope that the newly redesigned ITT pathway can see an end to the high turnover of teachers (see this post).

I hope the changes to the GCSE sees the education system alter the way it reports, and thus removes the damaging idea of a threshold pass.

I look forward to the new a level curriculum with relish, and hope we can get plenty of exciting new content involved.

So, finally I hope you enjoy tonight and have a happy new year.

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