Angles or Angels?

When I marked my year 11 books the other day I noticed that quite a few had been working that morning on “Angels in triangles’. This peturbed me a little, surely by Year 11 they should know the difference and be able to spell each one.

To counteract this massive literacy issue I played a game of “Angles and Angels”. I spoke to them first about the difference, then about the spelling and then did a show me activity where I showed them various pictures and they had to show me on their whiteboards if it was an angle or an “Angel”. I was impressed that they even got the picture of Kurt Angle,  although none of them recognised David Boreanas…..

The activity led to a discussing with a couple of them as to why it was important to discuss these things in maths lessons. Stemming from the inevitable question “why we learning about this? It’s maths not English.”

I explained my opinion that we may be learning maths, but that literacy is important in all subjects. As a maths teacher I educate these students and literacy has to be a big party of that, as I hope numeracy is a big party of those subjects that deal with numbers but aren’t maths. I also expressed the importance of maths specific vocabulary, such as ‘angles’ and how it’s not necessarily going to be covered in English.

It is these sorts of things that we need to be thinking about, literacy wise, to ensure our students are in the best position when they leave.

A latitude and longitude question

A friend of mine, a geography teacher, tweeted me this question earlier:

A lovely set of questions that I thought I’d run through here. I hope you’ve had a go at them before you read on….

The first one appeared at first to be a very simple question, just looking at the proportion of the circumference you have travelled. I then noticed the sleight of hand with the units and realised it involved an extra step. Still relatively straightforward though. Convert the distance from miles to km and work out the proportion of the circumference you have travelled. Multiply that by 36o and you have your angle, in this case 50.57 degrees to 2dp.

The second question is more tricky. The first thing we need to work out is the radius of the circle we would get if we cut a cross section parallel to the equator 30 degrees north. In order to do this I drew a diagram  (always imperative! ).

This allowed me to find a right angled triangle with the hypotenuse being the radius of the earth. This allowed me to find the radius of the circle I was after.

Once I had this I could work out the distance travelled easy enough, as we have travelled 180 degrees so half the circumference.

Or 15349 km (to the nearest km.) Here I’m assuming we want the distance we have travelled around the earth, rather than the displacement which would obviously be the diameter of the cross section or 9772km.

More maths in the playground

My daughter is now a little over two, and as such I spend quite a lot of time at children’s play parks and the amount of maths you can find there is incredible.

I’ve written along these lines before. From silly things, like the fact our local park was the only place I’ve seen with a simple abacus that’s set up for a base ten, rather than base 11, world. To the more intricate mechanics behind swings and roundabouts.

Today, however, it was this that got me thinking:

“A slide?” I hear you thinking. “Surely there’s nothing too mathematical about that? Gravity means you go down it.” Well in some respects you would be exactly right. However, we don’t live in a model world, and there is the question of friction.

Today the park was rather damp, it had been raining over night, so we took a towel to dry the slide. Once the slide was dry my daughter went down it, but at an incredibly slow speed and stopped about halfway down. I have always noted that the speed of the slide varies incredibly, and had assumed it was due to the coefficient of friction varying between different materials. Her cotton trousers provide a much faster slide than her jeans, for instance. But today she had denim on and it doesn’t normally stop halfway down. I then thought about another recent trip where the slide had seemed slow, and that day it had been particularly warm.

It got me thinking, does the temperature affect the coefficient of friction between two materials? Or perhaps it’s humidity that has an affect? A little bit of research leads me to believe that both can be contributing factors. Definitely more ammo for investigation in a mechanics school trip!

After the slide, she wanted a go on this:

As you can see, the local see saw is fairly basic and works best if two people of a similar mass are on each end (obviously we aren’t). It’s still easy enough to work, but it made me think about another local-ish park that has a much more complex see saw where there are 3 seating positions on each end. That gives more variability to the user’s and would mean two children of different mass could select positions,that give similar, but opposing, moments and hence work the see saw as well as if they had equal mass. What a fantastic idea, and a fantastic use of maths!

I love the amount of maths that is present at local playparks, and one day I do hope to take a mechanics trip (perhaps a cross curricular one with physics) to investigate all these things. It would be awesome to make a TV show around it. Perhaps I’ll make my own Numberphile – esque video on it one day!

The maths of the playground

Since becoming a parent I have seen maths in many places I wouldn’t necessarily have thought I would have. The playground (or park) is one source that just keeps giving.

A week or two ago we were at a new playground and I found this abacus, and of course the maths links to an abacus are obvious, but what else is there?

Last year I was pushing my daughter on a swing and I couldn’t help but see the swing as a pendulum and wonder whether we could take a physics and maths cross curricular trip to the playground and investigate the simple (or more likely damped) harmonic motion on display from the children’s swings. I think the mechanics involved would be interesting and different swings could be tested to see which ones are more efficient. Given that some swings are extremely noisy, I would assume that these are the least efficient and would love to test this hypothesis.

Today I had another thought about playground maths. My daughter was playing on the roundabout, and after a while she got off and decided to start picking up the bark chips that cover the floor and plonking them onto the roundabout. We told her to stop and to spin the roundabout as that should make the chips fall off. She did this but the chips didn’t move. This was due to the fact her spin didn’t have enough speed to generate a centripetal force big enough to cause a reaction (or whispers centrifugal force shhh) larger than the maximum frictional force acting in the bark. This made me wonder how large the coefficient of friction would be between the painted metal roundabout and the bark chips and what the minimum speed required would be to move them. Again, I figured a cross curricular trip would be great to investigate this too. I didn’t have my phone handy, so couldn’t photograph the chips on the roundabout, or film the sight of them flying off quickly when I gave the roundabout a spin!

There you have it, children’s playgrounds, the perfect school trip for A Level maths and physics!

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.

“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.

Observing, and cross-curricular ideas

This week has been a good one. It’s been the last real teaching week before summer, as next week is ICE week (Immersion Curriculum Enrichment week) which means we are off timetable doing a week of activities around a theme. The theme this time us the environment, and the activities look to be fun. One of my favourite bits about ICE week is the chance to do something different and hone my teaching skills in an unfamiliar situation, but more on that later.

Back to this week. We have had most of our new staff in this week, including one of the new NQTs who I am mentoring. I am looking forward to the role of mentor, and have now informally observed him twice. Already there has been a marked improvement and a response to advice, which is pleasing. I was a bit worried about observing in this capacity but the mock observations project I completed with my PE colleague earlier this year and the ITT student observations I have done definitely helped. I also gained a good insight into my own teaching by observing and there are always bits to pick up.

Also this week I completed a joint observation with my HoD on a colleague who was kind enough to volunteer. Again this was unofficial and the purpose was so she could check my ability to grade and give feedback. This was great, my confidence as an observer was boosted as she picked up on exactly the same positive points and areas for development that I did and we agreed on the grade. I also learned a lot about giving feedback and she gave me some great pointers in that respect.

There were also tons of things from the lesson that I picked up, my favourite being this: During a traffic light show me activity he put three wrong answers up. The majority of the kids chose the nearest one and some assumed they must be wrong. This was a superb discussion point and there was some real good contributions from the class. I was unsure if he had done this on purpose, so asked him afterwards. He said that he had in this instance and often does this because he did it once by accident and the results were great. I think this is a superb idea that I will use myself.

Also in Thursday I observed a new y9 science lesson. A couple of my form were in the class and have been in trouble a bit in science so I went to see how they were and offer assistance if required. The lesson was on penguins (always a winner) and cooling rates and included an experiment where the pupils were simulating the huddles emperor penguins stand in to keep warm. It was good fun and the difficulties some members of the class had with graphing made me think that as a maths department we need to embed this skill better in KS3. It also got my brain flying about cross curricular lessons with science on graphing and I hope to implement those next year.

This is one of three cross curricular projects I have in the pipeline, all of which excited me. The second is with an English colleague (@goldfishbowlMM) and involves looking at “The maths of Shakespeare”, and is very exciting. The third is once that a music colleague has suggested to me and involves trying to help improve the times tables of our pupils using the medium if hip-hop!

With these projects and mentoring an NQT, next year is looking incredibly exciting already!

Probability and Sex Ed

This week we had our third CT day of the year. (CT Days, or citizenship themed days, are collapsed timetable days where pupils do a range of topics linked to a theme.) I was with my coaching group and we had a great day on the topic of “personal wellbeing”.

The new year 11 (we move up year groups at spring bank) had a day on sexual education. Currently in maths they are learning about probability and one of my colleagues and I decided this was a perfect opportunity to merge the two.

We gathered some data on the probabilities if contracting STIs from an unprotected sexual encounter and they looked at the probabilities involved in contracting things after multiple encounters (here).

We also looked at expected values, and given the effectiveness of different types of contraception, (from here) how many pregnancies a year would you expect if a couple who were always safe made love twice a week. The answer shocked the whole class. They were also amazed by the difference when I asked them to complete tree diagrams and work out the expected value if the couple used condoms and the pill.

This was a much easier concept for them to relate to than picking sweets out of a bag as they could see that this was something that would affect everyone at some point in their lives. It also got across some messages that are important, especially as our school is located in an area with quite a lot if young parents.

Vocabulary and maths are not mutually exclusive

My favourite area of maths is the pure maths side. Following it down to KS3 and 4 that means I like number and algebra best (and bits of shape). Leaving data handling as my least favourite. (Although teaching some S2 this half term has kindle a desire to teach more A level stats, and I do love some probability and the associated game theories, and other areas of data, but that’s for another day.)

The reason I mention this here is by way of explaining why I have taught only a small amount of the data topics over the last year. (I don’t mean I skip over them, just that when a class is split and the other teacher asks “do you wanna do data or algebra?” I always choose algebra, which normally pleases them too.)

Next week, however, I am embarking on a module in probability with my new year 11 class (they are “new year 11”, as in we move up at spring bank, not a new class to me.) They are a high set of intelligent pupils, and as such I will be teaching topics such as tree diagrams. Today I was planning the lesson and I noticed something strange: there is no mention of the terms “mutually exclusive” or “independent events” anywhere, not on the examboard spec, the SoW nor the resources I was looking through. It struck me as strange.

I then had a look at the DoEs draft spec for 2014, the term independent IS there (hurrah!), but the term “mutually exclusive” is missing. I feel vocabulary is important in general, but subject specific vocabulary is imperative if we are to ensure the next generation of mathematicians and game theorists are to make any ground!

Another thing happened today too, one of my sixth formers referred to brackets as “Parentheses”, this is a term I am familiar with, and one the girl in question grew up with, but the rest of the class had never heard. Again, it was an example of vocabulary disappearing in Britain, and I think this is something we should be aware of. This coupled with the fact a student teacher we had needed to explain the words “tedious” and “petulant” to a top set year ten a few months ago makes me think this vocabulary gap needs to be addressed. (Mark Miller has written a series of posts on vocabulary here )

We all go out of our way to include literacy in our lessons, which is great. Words like “linear”, “quadratic”, “expression” etc are revisited again and again, even the lowest ability classed I teach can use these well, and correctly, in the context of mathematical discussions. However, I think we need to go further, we need to use vocabulary such as “mutually exclusive”, and “iteration”, and other such things that don’t specifically appear on our syllabus (“iteration” is on the new draft!). We also need to go further, we need to use a wider vocabulary in our discussions with pupils to increase their own vocabulary.

Addition: I’ve been thinking, this could be an issue specific to the type of pupils we get at our inner city school, I’d be interested to hear how wide vocabularies of teenagers are at other schools with different pupil compositions.

The skills needed for the future…

Last week I was lucky enough to see two more of my colleagues deliver lessons, and from each one I learned a few things.

The first I saw was an A-level physics lesson, this was delivered by an experienced colleague of mine who has a background in FE colleges and as such is very experienced in post 16 teaching. He welcomed me into a year 12 lesson on Hooke’s law, and I am pleased to report that I enjoyed it immensely, The lesson atmosphere was very similar to my own post sixteen lessons and his approach to the topic was similar to the one I would have taken if I as teaching it, so I could take away the knowledge that I’m not completely wrong! I was a little worried as to the length of time it took the pupils to draw and plot two graphs though. This made me think about my own teaching of graphs. From year seven upwards I normally give out graph paper with axes already drawn on to pupils when touching on graphs, as the drawing of axes can take a while and they are always given axes in the exam paper. I realise now this is a rather insular way to look at it. My question usually is: “when will they ever need to draw a set of axes?” this question hasn’t changed, but the answer has. Before, my answer would have been “never!”, but now I know the answer is “for A-Level physics”. They need this skill for their A-Level in physics and I would assume Biology and Chemistry also. It is something they need to do as part of their assessed practical, and it is a skill I can start to hone in my pupils way before they get there, meaning their A-Level science might be a tad easier. I am going to insist that the higher ability pupils I teach learn these skills early, so that in the future our A-Level science students are already equipped with them. There is, of course, the further question: “is it necessary for them to draw them by hand in the digital age we live in?”, but I feel that’s a question for another blog post on another day.

The other lesson I saw was taught by an NQT within the department, we share a class and I went to see her teach the class we share. It was great to see the class in a different light, and to see how they respond to another teacher. There was also an IT failure at the start of the lesson, and I was amazed by the calmness the teacher had and the way she adapted her plans to go without it. I also gained a superb starter task out of it (see below)!