Recently Ed Southall shared this problem from 1976:
I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. Edit: it’s O level I saw the question and couldn’t help but have a try at it.
First, I drew the diagram – of course:
I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.
Next I found the equation, and hence the X intercept.
And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!
The word in brackets is duh…..
Now I had all three point.
It was a simple division to find the tangent ratio of the angle.
The next 2 parts were trivial:
And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).
Because I decided calculators were probably not widely available in 1976 I did it without one:
I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:
Which makes the simplifying in part d simpler:
I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.
Edit: The front cover of the paper:
Today I finally had time to sit and look through the government white paper “Educational Excellence Everywhere”. A catchy title I thought, and I was interested to read what it actually said. I didn’t get chance to read all 150 pages – I will – but I did get to read the first chapter, and I thought I’d frame some initial thoughts.
A fantastic aim
In the foreword Nicky Morgan states that ‘Access to a great education is not a luxury, but a right for everyone.” – Definitely a sentiment I agree with, and certainly ine James Kier Hardie would be proud to hear espoused by a conservative politician, but not one that has always been an obvious policy driver over the last six years.
Academisation and Local Authorities
The white paper continued in this way, setting out an idealistic vision, but in the early stages not much was said about how this would be achieved. There was a lot of talk on the forced academisation of all remaining local authority schools.
There were some qualifying statements about Local Authorities (LAs). The government are hoping to keep the current experience and envision those who run LAs to go and work for academy chains. This fits the Conservative ideology of small state, bigger private sector, and seems to hint that this ideology is the driving force.
They also claim that moving school control from LA control will give greater accountability, as those elected can’t are there to further the interests.pf their constituents and they apparently can’t do this when LAs control schools. This is a nonsensical argument and the reality is in fact the complete opposite. When schools are under local authority control they are run by officers of the local authority who are answerable to elected members. Thus they HAVE to respond swiftly and allay concerns. Academy chains have no such in built accountability to the elected members and hence the electorate.
LAs will focus their role on core functions. These will be – ensuring all have school places, acting as champions for children and families and ensuring the needs of the vulnerable are met. It’s the third one that worries me. Currently local authorities provide a great deal of support to vulnerable children through Ed Psychology, CAMHS, and a whole host of other services and agencies. In the new world of tiny LA budgets, how will they afford to keep up this level and meet this core function?
Teachers, Training and Research
The next section turned it’s attention to teachers. I was a little worried that this white paper seems to ignore the recruitment and retention crisis we are experiencing, and the idea of placing responsibility for accrediting teachers into heads hands worries me. I’m certain that for the vast majority this would be fine, but I have heard some terrible horror stories about bullying from heads, particularly in the primary sector, and to give more power to do this worries me.
There was an extremely positive line on ITT content though:
“We’ll ensure discredited ideas unsupported by firm evidence are not promoted to new teachers.
So no more VAK pushed on unsuspecting ITT students! This is part of a wider drive to get more teachers to engage with research and development a research based profession. This is an idea I am fully behind, but with the caveat that we need to include training on how to engage with research. Every class in every school is a different context. Just because research shows something works some places doesn’t mean it will work everywhere, there are no magic bullets, no snake oils. We can take ideas from research and try them, but we have to adapt them to our own contexts and be able to see when things are just not working.
There are my initial thoughts on the first chapter. Some positives, some worries, and some signs that we are in the process of full privatising our education system. What are your thoughts on the ideas mentioned here?
As part of George Osborne’s budget statement today he made some comments about mathematics education. He said that they would look into teaching mathematics to 18 for all pupils. This has caused a lot of discussion on twitter and the treasury have since clarified that by “looking into teaching mathematics to 18 for all” he actually meant “look to improve a level teaching” – why he didn’t just say that is beyond me….
The bigger debate that seems to have opened is whether mathematics should be taught to all. There seems to be people in both camps on this one, and it’s something I’ve thought about many times.
Some of the arguments for it that I read suggest that for non a level students this would be a great time to learn about the life skills. I would argue that that’s not actually mathematics, it’s more numeracy. And I’ve often thought that they should be taught as distinctly different subjects, with numeracy a core subject and mathematics one that is chosen as an option from KS4 onwards. I sometimes think this would be a great idea, strip back the core curriculum entirely to just numeracy, literacy and citizenship, leaving a wide range of options and a lot of time in the timetable to build truly bespoke schooling. Students could study academic or vocational qualifications and perhaps we could get both right. However I realise this would be a logistical nightmare, and I worry massively that 14 year olds would be picking things that defined the rest of their life, so the other part of me thinks actually we should be prescribing a broad curriculum giving everyone a fair grounding and allow them to choose at 18 what to specialise in.
But what about in our current situation?
Given the situation we have at pre 16, I started to think about the idea of compulsory maths to 18. Clearly making A level maths compulsory won’t work. I’m told that around 50 % of those who attempt it with a grade B fail in Y12, that’s a massive amount of students we would be setting up to fail, and that’s not counting the A grade students who can’t handle the step up or the C grade students who wouldn’t have a strong enough grounding in algebra to succeed.
What about core maths?
I’ve been teaching this as part of the early adopters programme and I am quite impressed by the qualification. We do the AQA version and I’ve found the specification has enough stuff that fits the “life skills” heading to cover that aim of it while also having some more mathematical elements. The optional papers give the option of creating a course that fits the needs of each student best, and I’m looking forward to continuing teaching it and seeing it develop.
But should it be compulsory?
Again, I’m torn on this, I can see that the life skills bits would be good for anyone to learn. On top of that the other bits offer help with a vast range of other subjects and future job roles and help build logical thought, all of which I feel would be a good argument for making it compulsory. But it eats into the time they could be spending working on the things that are really important to them and the qualifications that they directly need to move to the next stage of their lives plan.
One thing I find ill thought out about the qualification is the 2.5 hours a week for 2 years suggestion. The idea was that it was to ease the burden and to spread it out, however I found that students were disengaged around exam time as it was the only subject they weren’t examined in. We also lost a lot of candidates after year 1 as they secured apprenticeships and basically had a years working without any sort of credit. We think going forward that it is better suited as a 1 year 5 hours a week course, perhaps students could do core maths in Y12 followed by EPQ in Y13? This would mean, however that the objective of keeping students in maths education to 18 was no longer being met.
I certainly agree with the compulsory resitting of GCSEs up to 18, although the previous comments around Maths and numeracy are certainly highlighted in this issue too.
As you can probably tell, I have conflicting views on a lot of this, and I’m still trying to.make sense of them. I’d love to hear your views on this. Do you thing all students should have to do maths to 18? Do you think they even need to do it to 16 or should we split maths and numeracy? What are your views on the idea of a stripped bare curriculum where students build their own? Would you have the same 3 core subjects as me, or different ones? Or would you prefer my other idea of a broader curriculum where students are a bit older by the time they need to make those massive decisions? Please let me know in the comments, via social media or email.
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.
This post was originally published here, by Labour Teachers on 29th December 2015
Curriculum, it’s an issue that plays on my mind a fair bit, and I think the reason for that is that I don’t really know what I think is best. I don’t mean the maths curriculum here, I mean the overall curriculum.
I’ve been asked before by students “why do we need to study maths”, and this is really the sort of thing I mean. I can see that some skills are necessary for all. We all need to know how to read, to write, to understand the laws of the land and our democratic model. But the rest of it? I’m not sure I could argue that knowing Hamlet is more or less important than knowing how tectonic plates work, or that knowing how to use trigonometry is more or less important than understanding the difference between a bass clef and a treble clef.
These thoughts lead me to understand an argument for a really slimmed down core curriculum and plenty of option choices to allow students to choose a really unique and bespoke curriculum. The slimmed down core would include numeracy, literacy, digital literacy and citizenship. And everything else would be optional, allowing students to choose their own truly bespoke path.
In many ways I love this idea, but in many ways I hate it too. At what age would we teach a wider core curriculum and at what age would we introduce this wide choice? Would we start it at KS3? KS4? KS2? The current system sees a choice given at KS4 -typically started by 13 or 14 year olds (depending on whether the school counts Y9 as KS3 or 4). Do 13 year olds have enough knowledge of themselves, the world, the subjects and the future careers they feed to be able to make an informed choice? Do 16 year olds for that matter?
These reasons lead me to see the alternative argument too, perhaps there should be no real choice for education. Perhaps a broad base which covers all subjects is the best option? That way all students would have a fair crack of the whip. They would be able to make a much more informed choice on their future, when the time came, given their broader knowledge base. But who chooses that broader base? Who decides what bits of the curriculum are the most important? We certainly wouldn’t have time to fit all of all the subjects in if there were no opt outs. At what age would we then allow choice? Would it come at A level? Or would we wait until undergraduate study? Would this cover both academic and vocational subjects? Answers either way to that could see some students put at a disadvantage.
Would some sort of middle option be better? Or would that make it worse? I really don’t know, and I’m conflicted massively on this issue. I’d love to hear others thoughts on the topic.
This morning I read this interesting little post from Andy Lyons (@mrlyonsmaths) which looked at teaching the exact Trigonometric Ratios for certain given angles (namely 0, 30, 60, 90 and 180 degrees). The post gave a nice little info graphic linked to the unit circle to show what was going on and then focused on methods yo remember the ratios.
While reading it I thought about how I introduce these exact Trigonometric Ratios. I first like to know that my students have a thorough and in depth understanding of right angled triangles and the trigonometry involved with them (including Pythagoras’s Theorem). I feel this is imperative to learning mathematics, the Triangle is an extremely important shape in mathematics and to fully understand triangles you must first fully understand the right angled triangle. The rest follows from that.
Once these are understood then you can move on to the trigonometric graphs, showing how these can be generated from right angled triangles within the unit circle, as shown in the info graphic on Andy’s post. Once the graphs are understood then the coordinates f the x and y intercepts and the turning points give us nice exact values for angles of 0, 90 and 180 degrees. This leaves us with 30, 60 and 45 to worry about.
At this point I introduced 2 special right angled triangles. First up is the right angled isosceles triangle with unit lengths of the short sides. This obviously gives us a right angled triangle that has two 45 degree angles (as the angle sum of a triangle is 180) and a hypotenuse of rt2 (via Pythagoras’s Theorem).
Using our definitions of trigonometric ratios (ie sin x =opp/hyp, cos x = adj/hyp and tan x = opp/adj) we can clearly see that tan 45 = 1 and that sin 45 = cos 45 = 1/rt2. This aids the understanding more than just giving the values and allows students a method of working these values out easily if stuck.
The second triangle is an equilateral triangle of side length 2 cut in half. This gives us a right angled triangle with hypotenuse 2, short side lengths 1 and rt3 (again obtained through Pythagoras’s Theorem) and angles 30, 60 and 90.
Again we can use our definitions of trigonometric ratios to conclude that sin 30 = cos 60 = 1/2, sin 60 = cos 30 = rt3/2, tan 30 = 1/rt3 and tan 60 = rt3.
This is again good for deeper understanding and for seeing why sin x = cos 90 – x, and cos x = sin 90 -x. This can lead to a nice discussion around complementary angles and that the word cosine means “sine of the complementary angle”. This triangle is also a good demonstration that tan x = cot 90 – x, when you come to higher level trig.
This post was first published here on Labour Teachers on 19th April 2015
Monday is the last day you can register to vote, which means I have spent a large portion of the week checking in with my year 13 class and my year 13 form group that they have registered. This has led to various discussions around politics, how they can decide who to vote for etc. It seems a lot of young people take on the political views of their parents. I was pleased to hear a number of them had actually downloaded the manifestos of the major parties and we’re using them to help them decide. I was shocked, however, at how many of them knew very little about any of it.
One asked if she could vote for Barack Obama. One asked if we were choosing a new queen. One thought Ed Miliband was the prime minister at the moment. There were many others like this and they echoed conversations I’ve had in previous years too. When one young lady asked who I thought our next president would be another year 13 overheard and joined the conversation. He was very knowledgeable about the political spectrum and the democratic process and it was quite refreshing to hear him explain the ins and outs of it rather than having to explain it myself.
As the conversation drew he said: “the thing is sir, there are far too many people who know nothing about it. It’s arguably the most important part of life. Why isn’t more taught about it in schools?”
I don’t have an answer. I don’t know much about citizenship, but I thought that this was something that fell under that banner. The current provision clearly isn’t helping engage young people with the process and it clearly isn’t helping to enhance their understanding of politics. I am a firm believer that we need to be giving our learners the knowledge they need to make these decisions, and that means we need to increase this aspect of their education.
Maybe the citizenship curriculum is too wide and/or the time allocated is too small meaning teachers cannot give each topic the time it deserves? Maybe it’s not seen as important by schools because it doesn’t fall into the same category of core subjects as others? Perhaps if it were made a requirement that all students study it to GCSE and it were given double weighting on progress 8 similarly to English and maths then it would be seen as more important by schools? Perhaps it needs a revised curriculum and a rebranding? Perhaps it suits the political classes to keep the majority uneducated on politics and it’s importance? One thing is for sure, there are far too many people leaving school without a sound knowledge of these matters.