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Isosceles triangles and deeper understanding

March 30, 2016 3 comments

When marking paper 3 of the Edexcel foundation Sample Assessment Materials recently I came across this question that I found interesting:

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It’s a question my year tens struggled with, and I think it is a clear marker to show the difference between the current specification foundation teir and the new spec.

The current spec tends to test knowledge of isosceles triangles by giving a diagram showing one, giving an angle and asking students to calculate a missing angle. This question requires a bit of thinking.

To me, all three answers are obvious, but clearly not to my year 10s who do understand isosceles triangles. The majority of my class put 70, 70 and 40. Which shows they have understood what an isosceles is, even if they haven’t fully understood the question. They have clearly mentally constructed an isoceles triangle with 70 as one of the base angles and written all three angles out.

What they seem to have missed was that 70 could also be the single angle, which would, of course, lead to 55 being the other possible answer for B. One student did write 55 55 70, so showed a similar thought process to most but assumed a different position for the 70.

I already liked this question, and then I read part b:

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Now students are asked to explain why there can only be one other angle when A = 120. Thus they need to understand that this must be the biggest angle as you can’t have 2 angles both equal to 120 in a triangle (as 240 > 180), thus the others must be equal as it’s an isoceles triangle.

The whole question requires a higher level of thinking and understanding than the questions we currently see at foundation level.

In order to prepare our students for these new examinations, we need to be thinking about how we can increase their ability to think about problems like this. I think building in more thinking time to lessons, and more time for students to discuss their approaches and ideas when presented with questions like this. The new specification is going to require a deeper, relational, understanding rather than just a procedural surface understanding and we need to be building that from a young age. This is something I’ve already been trying to do, but it is now of paramount importance.

There is a challenge too for the exam boards, they need to be able to keep on presenting questions that require the relational understanding and require candidates to think. If they just repeat this question but with different numbers than it becomes instead a question testing recall ability – testing who remembers how they were told to solve it, and thus we return to the status quo of came playing and teaching for instrumental understanding, rather than teaching mathematics.

What do you think of these questions? Have you thought about the effects on your teaching that the new specification may have? Have you any tried and tested methods, or new ideas, as to how we can build this deeper understanding? I’d love to hear in the comments or social media if you do.

Further Reading:

Teaching to understand – for there thoughts in relational vs instrumental understanding

More thoughts on the Sample assessment materials available here and here.

Cross-posted to Betterqs here.

A request for help

October 6, 2015 1 comment

So I’m working on some research, looking at motivation and effective teaching in A level maths. As part if my preliminary research has I’ve put together a short questionaire to help me shape the direction of that research.

If you have completed a maths A level in the last 5 years I’d be eternally grateful if you could fill in this short survey. It shouldn’t take more than 4 minutes.

I’d also be eternally grateful to anyone who knows anyone that has sat a level maths in the last 5 years if they’d share the survey with them.

The survey can be found here. (https://www.surveymonkey.com/r/RPNHR78)

Thanks.

Categories: A Level, Exams Tags:

To Ebacc, or not to Ebacc

June 19, 2015 Leave a comment

This post was originally published on Labour Teachers, available here, on 25th June 2015.

My Geography teacher; the head of year 9; the head of year ten; the head of geography; two deputy heads; the head of maths and my form tutor. They’re the people I remember “having a conversation” with regarding my GCSE options, and more pertinently the fact I’d chosen wrong.

The school I attended had a two year key stage 4, as was the norm then, so it was the end of year 9 that we needed to pick our options in. The choice itself wasn’t massively wide. We had to do maths, English, double science, RE, A language (mine was Spanish) – and obviously we had to do core PE, although this wasn’t examined. This left room for three choices, one was technology- I’m led to believe technology was a legal requirement. I chose IT and electronics (2 short courses and I was told this was because IT didn’t count as a technology).

Then there were two option blocks. One had a limited number of subjects. History, Geography, IT and maybe a couple more. The other had these and all the other subjects one would expect. I chose history and music. The school encouraged all students (well the vast majority) to take either history or geography. Those deemed bright were supposed to take both. I was deemed bright.

I felt under a but of pressure from a few directions, and if I hadn’t had supportive parents and a supportive music teacher I may have folded. I’m glad I didn’t. I enjoyed my music GCSE, I studied it beyond GCSE and I found it as academically demanding as the others. I also set a precedent, I was the first male for years to take music but that increased quickly.

What’s this got to do with Ebacc?

I’m not sure, I know when reading Nicky Morgan’s comments today I felt a little annoyed, having been in the situation described above. However, I do feel that the Ebac ensures that all learners have access to a good broad grounding. I’m glad I did music, but I’m equally glad I did the others as well.

I worry that the focus shifting as it is will see subjects like music and art shoved a side and that would be a tragedy. I like the curriculum model Tom Sherrington (@headguruteacher) has shared recently, as it offers a good grounding which includes something creative.

What about Vocational Education?

For too long we’ve got vocational education wrong. The rise of GCSE equivalent qualifications meant that learners could in fact walk away with a bagful of “GCSE equivalents” but arrive in the post school world to discover they are anything but. The Ebac and other recent changes have been positive in that respect, but they seem write off Vocational Education completely. Which is a shame as the idea is sound, we’ve just got it wrong for a long time.

So, what are you saying?

I think the Ebacc is nice in theory, but there are potentially worrying side affects for creative subjects. I also think that all the policy at the moment is patching up holes, instead of sorting out the structural damage.

I like Tristram Hunt’s recent ideas regarding scrapping GCSEs and implementing a baccalaureate system that has two truly equivalent qualifications, one academic and one vocational, or technical. This is an idea that seems to be backed by John Cridland of CBI and could link in to changes on HE too, with technical degrees being introduced to increase the expertise in manufacturing.

All you need is sine

June 2, 2015 12 comments

Today I was going through an M1 question with a year 13 student and was surprised to see the method he had used. The question involved finding an angle in a right angled triangle given the opposite and adjacent sides. The learner had used Pythagoras’s Theorem to find the hypotenuse then used the sine ratio to find the angle.

Puzzled I questioned further, thinking he may have instinctively found the hypotenuse without fully reading the question then having all 3 sides so going with the first. This turned out not to be the case:

“I know sine equals opposite over adjacent innit sir, I have trouble remembering the other ones so I just always use sine.”

This was extra interesting as earlier I had come across a markscheme which suggested the way to resolve a force at an angle of 30 degrees was to use Fsin30 for the vertical and Fsin60 for the horizontal! Further checking showed this learner did that too.

I wasn’t too sure what to make of it. It’s mathematically correct,  so there’s no issue there. The learner has a grasp of the other ratios but is more confident with sine so I can see why he would default to that position, although I hope the extra time it takes isn’t an issue tomorrow. I can’t fathom, however,  why the markscheme would show it this way in the first instance. (Not the only time a markscheme has confused me recently!)

What do you think? Have you got any quirky methods like this? Have any if your students?  Do you have an idea why a markscheme would default to this position? I’d love to hear your response.

A missed opportunity?

April 8, 2015 6 comments

I’m sure you all have had a look at the new GCSEs by now and have either started teaching it or have at least started thinking about the boards and looked at what’s on offer. One of the things that pearson have produced is a baseline test aimed at year 9 to decide whether to teach them higher or foundation, today I was looking through it and a couple of questions jumped out:

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This is a lovely question, although there’s one thing that annoys me about it. Can you guess????? Yes, that’s it! It’s that stupid picture of a stupid calculator. We need our students to be confident working in terms of pi, if this was a non-calculator question it would be brilliant:

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We could even equate areas instead and thus include fractions:

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In reality, as it is, it’s plugging numbers into a calculator and rounding, giving a non exact answer, a grrrr moment and a missed opportunity!

The other question that jumps out is this:

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A lovely opportunity to play with numbers and fractions, a non-calculator question too. I love it:

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All that confusion and a brilliant simple answer.

Have you any favourite questions from these baseline tests? Any favourite resources for the new GCSEs? I’d love to hear them.

A look back at 2014, and forward to 2015

December 30, 2014 7 comments

This time last year I wrote this post reviewing 2013 and looking to 2014. In the summer I looked again at it here, and discussed my year so far. Now, as the year draws to a close and I am reading all these #Nurture1415 posts it feels like a good time to reflect again.

My 2014

At home

I’ve had a good 2014, I’ve spent some great time with my family, watched my daughter grow from a baby/toddler into a real little person and seen a lot of other family.

Studies and the blog

I’ve continued to work on my masters, and to write this blog. Both of which have helped me improve as,a teacher, and both of which have been enjoyable. There has been a higher proportion of maths puzzles finding there way onto the blog this year. This hasn’t been a conscious decision, but I have really enjoyed working on them.

Maths

I’ve managed to read a few more maths books, and I have managed to get deeper into topology, as I had hoped to this year. I’ve also delved deeper into group theory, another old favourite of mine, and I particularly enjoyed exploring the tests of divisibility.

Teaching

When I wrote the post last year, I thought I’d be in the job I was in for a long time to come. In reality I’d decided to move on and found another job by the end of February (I think). This was a massive change. I moved jobs, schools and authorities and there were some real challenges. I had a full set of new classes to build relationships with, and a full set of new colleagues to get to know. The new school is similar on many levels to the old one, but it is also infinitely different too. I feel I’ve joined a great team, and that I’ve already made some great friends amongst my colleagues. I feel that with most classes I’ve built up decent relationships and am making progress, and I feel I’m getting to grips with the new role.

CPD

I’ve been on some great CPD events this year. I’m on a Teaching Leaders course, I attended ResearchEd York, Northern Rocks, Maths Conference 2014 and teachmeets (I even presented at one!) The key messages for me is challenge everything, don’t just accept anything, ensure there’s something to back it up, and even then don’t just assumed it will work in all contexts.

Education in 2014

2014 will forever be remembered for that day in July when the news that Gove had gone shocked the nation. I wrote about my feelings at the time and you can read them here. I can’t say I’ve seen much difference in policy since he left, and I feel the move was made purely to detoxify the brand in the run up to the election.

It was also the year we got to see the draft maths A Level curriculum, which looks good, but not radically different, and the approved specifications for the new maths GCSE. I’m excited about the new GCSE as I think it addresses many if the short comings that the current one has, although I’d have liked to see calculus and Heron’s Formula make an appearance.

The Sutton Trust released a report in 2014 entitled ‘what makes great teaching”, it was my favourite type if report, one that backs up the things I thought with plenty of evidence. The crux of its finding being “great teaching is that which leads to great progress“. You can download the report in full here free of charge.

It also saw the first teaching of the new “Core Maths” suite of post 16 qualifications. We are a pilot school, and I’m quite excited by the prospect, although it’s not been without teething problems so far.

Hopes for 2015

Last year I hoped that the new curriculum would increase the rigour of the maths being taught and that it helps prepare learners for A Level. I still hope this, although I realise now it is a longer term hope. As is the hope that the new GCSE system will eliminate the threshold pass and the gaming we have seen with early entry and other such things. And I think it’s too early to tell if the new routes into teaching can bring down the high turnover we experience.

I still hope that the inherent inequalities present in the UK education system, and wider society, can be addressed.

I hope to find more time to spend with my family, to read and investigate further areas of maths this year.

I hope to continue to improve my practice and to get better at my job.

I hope to see an end to the ridiculous pseudo-context “real life” problems we often see in exams.

And I hope to make a real difference to the learners I’m in front of in 2015, to increase their maths knowledge and skill but also their respect for, and love of, mathematics. A number of my Y13 learners have applied for maths degree courses, and I hope they enjoy them.

I hope you have enjoyed reading this post, and have had a great year, and festive period, yourselves. Here’s hoping we all have a happy new year, and a fantastic 2015

“Next Level” question

December 15, 2014 3 comments

Back in November I wrote this post outlining the changes to the new GCSE maths curriculum and included some questions I had enjoyed from the SAMs and some that I had thought were less good.

One of my Y13 students had read the post and had had a go at some of the questions, he came to me in school and told me that he’d seen a “next level” question on my blog, that it had taken him ages to realise how to do it and that he couldn’t believe it was on the new foundation GCSE. I asked him which question it was and he said it was this one:

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I considered the question. A shows a sector of a circle with an angle of pi/2 radians. I should say 90 degrees I suppose, as it’s a GCSE question. B shows the same square but this time there are 4 sectors, each with the same angle, again pi/2 radians 90 degrees. There are no lengths marked on.

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This question is relatively straightforward once you pick up the key piece of information, that the squares are the same. You don’t even need to use the angles to work out the sectors as we are dealing with as they are quarter circles. You assign a value for side length and work through each one, showing the areas are the same.

I think it’s a great question, and I love the fact that these two shading arrangements give the same area. I did, however, wonder how future foundation students would cope with it given that a Y13, who scored a good A in maths AS and As in his other 3 ASs in Y12, referred to it as “Next Level”.

When discussing it with said student it turned out that the maths wasn’t the issue, it was that there was no lengths marked on, and he hadn’t realised straight away that you could assign a variable to work though.

I assume this is down to the nature of the current examinations, and their tendency to give scaffolding all the way through. I like the rigour of the new qualifications, and the fact they are designed to build mathematical thinking all the way through, but I fear that for some teachers ensuring this is built in will take some getting used to.

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