## 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.

## A conic extention

This week has been a strange one, I’ve been trying to shake an illness but it keeps on getting worse, and my lessons have been disrupted a little bit by mock examinations. This, however, has given me a chance to work.woth small groups in some classes and to see how some classes are getting on with their courses.

In one of further maths lessons a student who takes maths but not further maths asked if he could.sit in the classroom and revise for his maths, asking if he needed help, I said it was fine and started the lesson,which was a review of the conic sections topic and an assessment sheet to identify any areas of weakness that may be apparent. I gave the none further maths student a spare copy of the assessment sheet, told him it was basically C1 skills but applied in a much more algebra heavy context, and asked him to have a crack.

Not only did he have a good crack at it, he answered it near perfectly. I was extremely pleased with his resilience in working through a question that has way more algebra than anything he’s looked at before and was glad he could make the links to the C1. His thoughts on the question were interesting, and I think they allowed the rest of the class to see more clearly how the conic section of FP1 fits with, and build on, the coordinate geometry sections in C1 and 2.

I think that in future I will use these parabola and hyperbola questions with all my high attaining AS maths students.

*This was cross-posted to the “One Good Thing” blog here.*

## Unstructured problems

Currently I’m in the process of completing a dissertation based around problem solving in A level mathematics and how this can be improved. This is a focus as in our setting students have struggled with this in the past. It was timely, then, that the article picked for this week’s Maths Journal Club was around the same subject.

The article was by Sheila Evans and was entitled “Encouraging Students Formative Assessment skills when working with non routine problems”. Available here.

The article itself was interesting, it set out an approach to teaching based around these unstructured problems and designed student responses to get students talking and thinking about the way they are approaching the questions. The article seems to suggest that students with an instrumental or procedural understanding are less likely to succeed at this type of problem than those with a relational understanding, and that is something I’ve been thinking myself.

I think the approaches mentioned in the article sound interesting and I am going to tailor them to students and trial them in my own context to see if there is an effect.

It’s certainly an article that has got me thinking and has given me ideas for things to investigate in teaching, as well as signposting a raft of other pieces of literature that I want to investigate further too.

## 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.

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