## To CAST, or not to CAST

Today I was working with some Y12 students on solving equations involving trigonometry, the type where they have to find all the solutions within a given domain. *Incidentally, always refer to.it as a given domain, rather than a given range. I know that it is a “range” of values, but the **range** of that sine curve you’re using is -1 to 1, not 0 to 2pi. If you refer to this, as I’ve seen any resources including textbooks do, you’re setting the students up for trouble when they come to look at domains and ranges in more detail.*

So we were looking at finding values in a given domain, discussing the graphs and how they could help us etc and then once they’d got their head around that we discussed the CAST graph.

When I was an A level student I preferred the CAST graph, I found it easier to use than the full trigonometric graphs and found it speeded up my working. For this reason I encouraged it’s use when I started teaching, but now I have a different view on the matter.

Each year I discuss with students the different approaches they can take to tackle different problems, giving them the widest possible armoury to tackle problems in exams, and always discuss with them which way they prefer. Over the years my students have consistently told me that they prefer using the trig graphs, rather than the CAST graphs.

In the first few years I couldn’t understand why, students explained that they “just understood it better”, but that still didn’t quite get why. I was reflecting on it today, as we were discussing it, and I think I have much more of an understanding now. I think that the CAST graph is a procedural short cut, but to fully understand it you need to fully understand the trig graphs and other angle properties, and the relationship that exists between the graphs and the ratios and the angles you are looking at. I think when students fully understand the trig graphs they can use them easily, and the process is quick. To then learn the CAST graph, an additional piece of knowledge and set of rules, to speed up that process fractionally seems silly.

As I’ve grown more aware of this over the years my teaching of the subject has shifted, I still show them the CAST graph, but I spend more time concentrating on the trig graphs to ensure a fuller understanding. Sometimes I wonder if there is even a point to doing the CAST graph at all.

*Do you teach both methods? Which method do you prefer? What about your students? Why do you/they prefer that one? I’d love to hear your thoughts either in the comments or via twitter.*

## Numerical Methods

For a long time I’ve held negative views towards numerical methods, particularly “trial and improvement” and the trapezium rule, but I’ve been reconsidering those views. This has been quite a long process that began when Tom Bennison (@DrBennison) questioned negativity towards them, probably around a year ago. We had a brief discussion around them and some of the thoughts have been stewing since.

Tom reminded me that numerical methods are important as in the real world there are many things that cannot be done another way (yet!). The discussion left me thinking that rather than numerical methods themselves being bad, it’s could be more to do with the way they are framed.

I remember when I was studying towards my own A levels I was taught the trapezium rule for numerical integration. My teacher said it was what was used before calculus was invented and that it had no real use now but was still taught, it wasn’t until I got to university I discovered that actually there are many intergrands that cannot be integrated and that the trapezium rule is an excellent method for approximation. This was a fact I’d forgotten between university and entering the teaching profession, but a fact Tom reminded me of.

This seems to me to be a very good reason to keep the trapezium rule in the syllabus. I was teaching it last week and I was thinking about this, and I realised that the way we assess the trapezium rule at A level is silly. We always ask students to approximate an integral than integrate it using calculus, oven via substitution or parts. This can only add to the feeling among students that the trapezium rule is pointless, as they can instantly see a way to find a much more accurate value. I now make a concerted effort to examine it’s importance and to state why I feel it gets a bad run, this had a positive effect on my class this year and they were much more engaged with it than previous classes.

This is not the only numerical method that gets a poor deal on our exams, another that jumps to mind is trial and improvement, a simple iteration method that can be used to find a reasonably accurate solution to an equation, however at GCSE the equation is often a quadratic, which students can find an actual solution to relatively easily via the formula or by completing the square. Why not use an equation they can’t solve otherwise?!

*What are your views on numerical methods? Have you had similar thoughts? Is there anything you used to dislike teaching but have changed your mind on? If so, I’d love to hear in the comments. *

## Half term revision

Today I spent a few hours at school leading Y13 on a half term revision session. It’s something that I feel I would have shunned in my days as a student, but my students were keen and who am I to quash that enthusiasm.

We spent the time looking at problem topics from C3 and C4, mainly Trigonometric Identities and Integration. They are two of the topics that students struggle with most, but they are favourite two subjects too.

The students did some recalling of facts they needed to know, and i filled in aome gaps, they applied their knowledge to challenging exam questions, both individually and as a class and I modelled some answers on the questions they were really struggling with.

It was an enjoyable day, spending a few hours talking about some of my favourite areas of maths with some great young people who are eager to do.well in the subject. Let’s hope it has a positive effect on their scores.

## Flipping the classroom

So flipped classes are something I’ve read a lot about over the last few years, I’ve seen many people who do it claim fantastic results but I’ve always been a tad sceptical about the process. The reasons for this scepticism have been that often in advanced mathematics the topics are really hard, and that I work in a school where I often have to chase homework which could potentially derail the whole flipped class process.

During the half term just gone a colleague and I were discussing the merits and worries of flipped classes and decided to try it on the small KS5 classes we share. We provided them material to prepare for lessons. For Y13 the topic was Integration and for the Further maths class the topics covered were Traveling Salesman, Transportation problems and the hungarian algorithm.

The set up of the lesson was such, the class would arrive and complete a check in question based in the previous lesson, then we would discuss the preparatory material to draw out the understanding the class had manged to gain from it, answer as a group any questions that any had drawn put of it (with me or my colleague only inputting if no one could help) and then looking to apply these skills in an exam context.

My fears about the classes not doing the work have been unfounded, they all completed each bot of prep. Although these groups are small and all are very committed to doing well in maths, so I still have these concerns regarding this.

My fears about the difficulty were also unfounded. It’s true that, for the most part, the students would not have been able to go straight into answering questions on the skills the preparatory material covered, but they had gained enough of an understanding to discuss the topic and they had identified the areas they didn’t understand, allowing the lesson to focus on this, rather than cover everything. The only lesson that was met with entirely blank faces was the lesson on volumes of revolution, but through their misinterpreted ideas of it I was able to focus in on misconceptions that had arisen from prior knowledge and correct that as well as teaching them about solids of revolution.

At the end of the half term we checked the student voice and they were all positive about the process and wanted to continue in this manner for the rest of the year, we will be looking again at student voice at the end of the year and the results to determine whether we want to roll this out across the key stage, but so far the results look extremely positive.

## Travelling Salesman

This year we are doing the module “Decision 2”, D2 for short. And I’m really, really enjoying teaching it. This week’s topic has been the Travelling Salesman problem, which is a fantastic springboard into a whole host of other areas of maths. When we looked at route inspection during the last module I made mention of the traveling salesman problem and the fact there is no known way to solve it in a reasonable amount of time and I briefly mentioned P vs NP and the millenium prize.

When we started this week with a slide that had the title on the class were automatically hooked. They had been eager to reach the traveling salesman and had even looked up P vs NP and the aforementioned millenium prizes. This meant before we even started d the lesson we had an awesome conversation about these amazing unsolved mathematical problems, with the class telling me what they had read and what they thought they understood of it and me filling in the gaps around it a bit and linking to other areas of maths.

Towards the end of this discussion one asked “but what are we going to study? We all already know it can’t be solved quickly enough for an exam!” Which led me onto the discussion of lower and upper bounds and optimal regions, and how we can find a good solution (within 1% of an optimal solution) within a reasonably short time.

This left around enough time to discuss least differences and tackle the nearest neighbour algorithm for an upper bound. The following lesson we looked at using minimum connectors for upprbounds and how we could identify the best upper bound. Then we looked at lower bounds, and how to identify of we had found an optimal solution or an optimal region. I do hope TSP makes it into the optional content of further maths when the new specification starts.

*This post was cross-posted to One Good Thing here.*

## Stop with the negativity

*This post was first published here, on Labour Teachers, 9th February 2016.*

So, this went viral this week. The latest in a long line of post that surely impacts on the already crisis hit recruitment of new staff into the profession. These articles are seemingly written by people eager to combat the myth of lazy teachers working 9-3, but I don’t think that this myth exists anymore. Certainly no one I know actually believes it, and even if they did it wouldn’t matter. I know I don’t only work 6 hours a day, and so do those closest to me, who cares what others think.

I worry for the author of the article, if this truly is their day then I can see a burn out happening for them in the very near future. I will admit, a few of the things rang true, but if your day truly contains all of these elements everyday then you need to stop putting insurmountable pressure on yourself.

I work long hours, but I certainly don’t work from 7am to 11pm every day. I would never get time to see my family if I did, I would miss seeing my daughter growing up. That’s 15 hours a day. 75 hours a week. That’s an unsustainable life. If you have found it is actually your life you need to take stock of what your doing. You need to take a breathe and reflect. You need to work out how you can do what you’re doing more efficiently otherwise you’ll cripple yourself. And if the weight of this pressure is coming from external places, then you may need to look for a new school. If you intend to make a career in this you may be looking at 50 years til retirement. And no one can work 75 hour weeks for 50 years.

I doubt that anyone actually does encounter all of these issues in a single day, most of us will have encountered most of them, at some point in our teaching lives, but to frame them as a daily occurrence is a worryingly dangerous thing to do at a time where we cannot recruit enough teachers into our schools. How many fine young minds have read this viral article and switched away from thoughts of the profession? I know at least 1 of my Y11s and at least 1 of my Y13s who have been put off.

The negativity needs to stop. I love my job, I basically get to talk about the beauty of mathematics all day long, a lot of the time with people really keen on the subject. I am alway pleased when students go on to study it at higher education. It would be a shame if others missed out on such a great job because of articles like this, and the often negative secret teacher.

## On-the-fly

*The best laid schemes o’ mice an’ men, Gang aft agley – R. Burns*

There are many reasons that a lesson goes awry, and being able to deal with that is key. During teacher training a lot of weight is put on planning, pikes of lesson plans are produced by student teachers and often they are very helpful and certainly aid development and allow us to consider the subject we are teaching, consider the questions we need to ask and what exactly we want our students to take away. But the heavy emphasis on planning can make some teachers too reluctant to deviate from said plans.

I remember during my NQT year being told about hinge questions and how I should include them in every lesson, a deputy head said I could come to his y11 class and see how he used them, but I was left underwhelmed as the result wasn’t any different to how the lesson would have been without it. He just had a “hinge question” in the middle and carried on regardless. Hinge questions are useful, but only if you then have 2 separate paths for the students to take.

Similarly starters that check prior knowledge are good, they’re useful for filling in gaps and they can aid a lesson, but you need to be ready to change your plans on the fly if needs be.

This week I had a lesson planned on cones and spheres, some of the questions towards the end of the lesson included cylinders and prisms as well as spheres, cones, pyramids and frustums, so I set a couple of cylinder and prisms questions in the starter. I was met with blank faces. Totally blank. I hadn’t taught them this before, but I had assumed they had met them in previous years, but they hadn’t (or at least if they had they’d lost their memory of it). At this point I jettisoned my plan and started over.

I talked through some examples, explaining how they had got it and then set them off on some tasks I had saved on my hard drive while circulating to check the understanding. It was an enjoyable lesson and the students now have a good grasp of cylinders and prisms, plus I have the added bonus of one less lesson to plan next week now.

It can be terrifying when this becomes necessary. During my NQT year we lost all the power from sockets in the school – the lights were still on but the smart boards were unuseable. When they went off I was 5 minutes in to a year 10 lesson on constructions with a class who had a reputation as the worst in the school. It was my first time being without the presentation I’d planned and my first time teaching construction. I did my best to demo on the boards, then set them doing simple constructions while circulating and teaching the more complex ones. It was a success, but it was a terrifying ordeal.

*Being able to adapt on the fly is key, and it’s something we need to prepare new and trainee teachers for. I’ve had thoughts about how to do this, but nothing concrete. One idea is to have them “wing it” occasionally- ie show up to a lesson every so often unprepared. Do you have any ideas on how we can help prepare for the times when we need to act on the fly.*

## Core, Decision and the new GCSE

Well January went by in a flash, and as we enter February 2016 seems rather light on posts so far. It’s always the way at the start of the new year, mock exams create piles of marking and it’s all coming at such break neck speed it’s hard to find time to write anything. So here are a few thoughts:

**Decision maths**

It’s a real shame there’s no content from these modules on the new A level. I’ve been thoroughly enjoying certain aspects this year, as usual. I won’t shed any tears about flowcharts, bubble sort and binary search (etc) but I will very much miss the graphs and networks section.

**Core Maths**

I’ve enjoyed teaching this subject this year and I hope to carry on with it. It has, however, been massively frustrating at times, especially when trying to assess the students and try to gather evidence to make a prediction on what they will score.

**The new GCSE**

The frustrations with core maths are all applicable to the new GCSE which I’m teaching to year ten, we still have no way to grade them on it, and in a culture where grades need to be entered regularly this can be contentious. I’d love to hear any bright ideas you have for grading CORE maths or the new GCSE spec.

The bonus of this time of the year is it tend to be when those exam classes (and all in bar y10 are) start to short it up a gear. I’ve been impressed by the change in attitude from some of the most challenging pupils and I’m hoping for more of the same.

## The SLT effect

The SLT effect is something that happens when a member of SLT enters your classroom. I remember in my NQT year having a Y10 class I struggled to control and a deputy head walking in to my classroom and the class all of a sudden becoming the very model of good behaviour. I’ve seen this effect time and time again, in my lessons and in the lessons of others. I’ve only ever known 1 class be immune to it.

The effect can be welcome, if your being observed for performance management or Ofsted, and it can be unwelcome – like it was I that year 10 class in my NQT year, as I’d asked the deputy to observe to give me ideas of how I could improve the behaviour.

A couple of weeks ago I welcomed a member of SLT into Y13 class, and I experienced the effect in a really different way. The class are normally a really bubbly class who are constantly discussing the maths and asking questions to further their knowledge and help them make sense of the new content, and to aid their application of that content. But this lesson they froze, they responded to questioning timidly and with short answers and didn’t ask any questions themselves. It was like a totally different class.

It got me thinking about the class and why this happened. I can only imagine it must stem from a lack of confidence, perhaps they were worried that our visitors would think they were no food at maths (they are in fact very good). I feel I now need to look at building their confidence to discuss their learning in front of others, and to build their confidence in the learning itself.

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