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An Interesting KS3/KS4 Problem from @m4thsdotcom

 In January I noticed that Steve Blades (@m4thsdotcom) was posting some interesting problems, many designed to stretch and promote thinking in students studying for their GCSEs.

I was particularly drawn to this question in the middle of January 

 

Initially I thought this was harder than it was and was intending to use the information about the interior and exterior angles to form two simultaneous equations in \(x\) and \(n\) where \(n\) is the number of sides of the underlying polygon. However, there is of course a much simpler way if you use the fact that the interior and exterior angles of the polygon must add up to \(180^{\circ}\). 

Using this fact you can obtain \(7x+1+28x+4 = 180\) and so \(35x = 175\) giving \(x = 5^{\circ}\). Hence the exterior angle of the polygon is \(7 \times 5 + 1 = 36^{\circ}\) meaning that the polygon has \(10\) sides and is a decagon.

I would be fascinated to see how students go about solving this question.

Steve has also collected a paper worth of challenging problems in a “Grade 9” paper available here. I particularly like this question about functions: 

 

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A Level Software

An Undiscovered MATLAB Function

Regular readers here will know that I love using MATLAB for some things but earlier this week I discovered a function that I hadn’t used before and is surprisingly useful.

I wanted to produce some diagrams showing portions of a normal distribution shaded and had tried to do it in Geogebra but wasn’t quite happy with the results. I then decided to try R but for some reason (probably to do with X-Windows or Quartz) the graphics output on the R installation on my Mac wasn’t working properly so I fired up MATLAB. My original intention was to plot the normal distribution function I was interested in, then plot the vertical lines at the limits and then shade the region. However, I discovered that in the “Statistics and Machine Learning Toolbox” there was a function called \(\texttt{normspec}\).

The syntax for this function is described in detail in the documentation but essentially this function can be used to shade a region under a given normal distribution. The syntax is \(\texttt{p = normspec(specs,mu,sigma)}\) and this plots the normal distribution with mean \(\texttt{mu}\) and standard deviation \(\texttt{sigma}\). It then shades the portion inside the specification limits given by the two element vector \(\texttt{specs}\) Рyou can use \(\texttt{-Inf}\) or \(\texttt{Inf}\) if there is no lower or upper limit respectively.

For example, the command \(\texttt{p = normspec([-Inf,65],65,5)}\) produces the plot shown below:


I’d encourage you to go and try it out if you have a MATLAB licence.

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Maths Teachers – How do we See Ourselves!?

For a while I’ve been thinking about how maths teachers see themselves, in fact I wrote about it in March last year in my post “Mathematician or Educator”.

Last week I decided to re-open the debate with one of the relatively new twitter polls asking whether maths teachers saw themselves as mathematicians, educators or both. The final results are shown below:

twitter_poll

I’m very pleased that 50% of the respondents saw themselves as both a mathematician and an educator it does concern me that 45% of people said only an educator.. I’m very interested in to why some maths teachers don’t see themselves as mathematicians. mrvman (@vahorai) suggested a possibility:

Screenshot 2016-02-03 21.46.39

I think more generally a big reason is the common belief that to be a mathematician you must be researching mathematics or using mathematics professionally was opposed to teaching “elementary” mathematics to school children. I think this is a sad way to think –  I’m sure many teachers will write things such as “.. is an exceptionally gifted young mathematician” so why wouldn’t a teacher see themselves as a mathematician?

Personally it is important to me to feel that I am a mathematician and a teacher of mathematics if I am to truly show a love of mathematics.

I’m not sure how to change this, I guess in a way we perhaps need to fight against some intellectual snobbery!

I’d be really interested to hear your views on this topic…

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The Difficulty of Predicting A-Level Achievement

Last year my predictions of grades for my A-Level were unfortunately a little off. Since then I have been thinking about what I did wrong or whether it was purely a consequence of the exams changing slightly and exam pressure….

I do think the exams were slightly harder the last time round, but I have also decided that I should make any end of chapter tests (I also do full past paper mocks) a bit more rigorous. As an example of my slightly improved tests I have uploaded the one I used for Taylor Series approximations here. I particularly like Question 9,

Screenshot 2016-02-02 21.02.21

The easy way is to apply Leibniz’s formula for derivatives, but of course A-Level students won’t have come across this explicitly and so it relies on them spotting a pattern in the derivatives to answer the question successfully. I’d love to hear your views on this assessment!

Predicting A-Level achievement seems, to me, harder than predicting KS4 achievement. I know that over a country wide cohort predictions based on previous attainment prove to be accurate but anecdotally the spread away from predictions is significantly higher at KS5. I think that part of this is due to the increased importance of students actually working at A-Level. In maths, for instance good students can get an A* at GCSE without doing any work outside of the classroom and then can sometimes fail to recognise the importance of hard work to achieve similarly at A-Level. I often tell my students that the step up from GCSE to A-Level is harder than the step up from A-Level to degree level and I firmly believe this to be true: We expect a lot more independence, tenacity and perseverance from an A-Level student than we tend to during previous Key Stages and if they can crack this then the transition to university study shouldn’t be too hard as they will have already developed the skills required for success.

How do you predict A-Level grades (particularly in Maths and Further Maths)?

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A plea for a calm debate….

So my fellow¬†#summerblogchallenge writer¬†@missnorledge¬†alerted me to the¬†#29daysofwriting¬†challenge organised by staffrm and I thought I might as well have a go. I will be cross posting from my own blog at¬†www.blog.ifem.co.uk¬†(if you haven’t already check it out, there is a wide mix of stuff on there!) I think these posts will be different to my usual posts because I don’t have the flexibility with the staffrm editor that I do with my own hosted wordpress site – for instance I don’t believe I can insert LaTeX into my posts easily on this.

I was struggling to choose a topic to write about for this first day until a conversation on Twitter earlier this evening reminded me about an article that I really enjoyed reading last week. The article was from¬†Naveen Rivzi¬†and published on the TES entitled “Why new teachers should not have to plan lessons. They should just get on with teaching.” I found this article very thought provoking, and whilst I may be still unconvinced by all of it – I think I would miss planning my own lessons – there are some points definitely worth thinking about. Subject knowledge is important and it’s natural that pedagogic knowledge develops with time so anything that supports new teachers or teachers with weaker subject knowledge in certain areas (we all have those areas!) gets my vote.

I am all for debate about maths education, and this is actually one of the things I love about Twitter – many differing views to digest and think about; however this article (for some reason) seemed to generate a huge amount of negative publicity. Of course there is nothing wrong with disagreeing with people’s views; it would be a very boring world if we all agreed with each other! In this case, however, I feel that the response took a more personal line, with some tweets I saw appearing to call into question the professionally of teachers who weren’t planning their lessons. To me this is un-called for, an open honest debate about the points put across in an article is one thing ¬†but to dismiss it out of hand and ridicule the ideas in it isn’t constructive. I don’t think many teachers would allow that kind of response to a suggestion made by a child in the classroom, I know I wouldn’t!

I’m nearing the end of 29 minutes and the word count….

To sum up my views: I feel their is a danger of the maths ed debate becoming a polarised “them versus us” discussion – I don’t think this is helpful, and I feel that over my time on twitter I have learnt lots of things from people with very differing views. To close yourself off to viewpoints that you perhaps hadn’t considered before is, in my mind, fool hardy at best, and dangerous at worst.