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Towards a Silent Aircraft

A good friend of mine, Ed Hall had told me about a lecture at the University of Leicester this Tuesday (10th February 2015). This lecture was the 17th Annual Industry lecture organised by department of Engineering and was given by Professor Dame Ann Dowling. Professor Dowling is President of the Royal Academy of Engineering and a professor of engineering at the University of Cambridge where she ran the University Gas Turbine partnership with Rolls-Royce between 2001 to 2014. She is a world authority on combustion and aero acoustics and researches efficient, low emission aero turbines and low noise vehicles among other things.

The lecture hall was completely packed, I’m guessing over 200 people were in attendance. Professor Dowling started the lecture by showing how the noise from modern civilian airliners is significantly lower than that from airliners of the early jet age and thar they are also more fuel efficient.

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As you can see from the graph, this decrease in noise has plateaued (is this a word!?) in recent years. Perhaps a reason for this is that environmental concerns have led to an increase in attention paid to improving efficiency, she mentioned that increasing the size of the engines has increased fuel efficiency but increased the noise level (from the large fans).

More recently, during take-off the noise of the rear jet has been balanced by the noise of the fan at the front of the engine (on approach the noise of the airframe becomes significant too, I will be doing a write up of another talk which discussed this in more detail), and so, the noise of the rear facing jet is not the only thing to be controlled.

Below is a section diagram of the Rolls-Royce Trent 1000 (the picture is taken from their excellent infographic which is available here), I have included it here as I will refer to some of the key components.

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The large size of the fan – ~2.8m for the Trent 1000 – contributes to the large noise from the fan, so making this as light as possible can help, for this reason Rolls-Royce now use hollow fan blades with an internal girder like structure. Something, that to me is less obvious, mentioned by Professor Dowling is the noise generated by the wake from the fan impinging on the static stators that are downstream of the fan. This can apparently be improved by designing the stator so that the wake hits them at different points leading to the different frequency components of the noise cancelling out (this is how I understood it anyway – I am not an aero-acoustic engineer!) – this is very cool! Modern jet engines also often have chevrons on the trailing edge of the jet nozzles to reduce noise by smoothing the mixing of hot and cold air, since the source of the noise is the velocity fluctuations in the air coming out of the nozzle. However, these are generally designed empirically since running a full large eddy simulation of a jet flow through a chevron-ed nozzle takes around 4-6 months on a national supercomputer. Professor Dowling’s team developed a far quicker method that was validated experimentally.
She then presented the SAX-40 concept design for a silent aircraft:

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This image is taken from some slides available here. As you can see, this looks very different from todays conventional jets!
As the engines are buried inside the airframe, instead of in nacelles hung below the wings the airframe can be used to shield some of the noise, with more space being given over to the use of acoustic liners to contain the noise. The noise from the engines would also be reflected above the aircraft reducing the noise level when compared to a conventional aircraft even further. The wings of conventional planes reflect the noise below the aircraft and towards the ground. As mentioned earlier, the airframe noise is significant when a plane is coming into land; one factor contributing to this is the use of the flaps and slats on the wings which introduce large holes into the wing surface – the air flowing around these holes creates a lot of noise. For this reason, the concept features a deployable drooping edge.
The noise due to the undercarriage of an aircraft is also significant and Professor Dowling’s researchers looked into this too. To accurately measure the source of the noise from the undercarriage they installed 108 microphones in a section of floor from an aircraft which could then be used in a windtunnel. The undercarriage of modern civilian jets are pretty similar with most having multiple wheels that are exposed. The analysis discovered that having two wheels produced the largest reduction in noise, however this obviously limits the weight that can be born by the undercarriage. So, a design with the rear set of wheels staggered inside of the front, different shaped wheels at the front and rear and a fairing were found to reduce the noise from the undercarriage significantly.

This talk was very interesting and thought provoking. However, I’m not convinced we will ever see a plane like this due to how radical it is!!

It was also nice that there was a buffet reception after the talk where I had some interesting conversations with a retired engineer – you can’t beat free food! I had also been lucky enough to arrive in Leicester early and have time to talk to Ed about upwind fluxes, Runge-Kutta time stepping methods, discontinuous Galerkin methods and the Navier-Stokes equations – all exciting topics which I don’t have much of an opportunity to talk about anymore.

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Symbols are meaningless without understanding

NB: this post is more opinion led than most of mine!

On Tuesday evening I attended a webinar on the ARK Mathematics Mastery programme (@MathsMastery I found this a very interesting hour or so, but the one quote that I took away from it was the following

Symbols don’t mean anything unless you understand the concept.

They showed a picture similar to the one below, and commented that the symbol for ‘5’ means nothing unless you associate it with the concept of ‘five-ness’.

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I’m constantly thinking the same thing when teaching something like expansion of brackets. When faced with the following question “Expand \( 3(2x -4) \)” I have had a fair number of students ask how you know if the second term is a plus or a minus – this just shows that sometimes I end up teaching the mechanics of a concept of method without ensuring there is solid understanding to build on. This is most definitely not a good thing – there is no point, in my opinion, in being able to complete the square of a quadratic for example without understanding why it is useful, and what it tells you about the quadratic. In my experience there are a significant number of undergraduate mathematics students who don’t realise that completing the square gives them important information about the turning point of a quadratic.

Having a concrete understanding of number and the number system is incredibly important. I see the overall goal of education to be “to prepare children for the world”. If by the time they leave school they don’t have a solid understanding of how numbers work and how to use them but can mindlessly expand a linear bracket because we have been told that it is on the scheme of work, have we really prepared students to deal with real life?!

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Secondary Curriculum Development Meeting

Last Wednesday (21st January 2015) I attended the first East Midlands Secondary Curriculum Development meeting of this year. These are hosted at the University of Nottingham and organised by the East Midlands West maths hub (@EM_MathsHub).

This was a great, thought provoking session about problem solving led by Dominic Hudson, from Heanor Gate Science College, with some additional input from Malcolm Swan of The Centre for Research in Mathematics Education in the School of Education at the University of Nottingham (@UoNSoE)

Dominic kicked off the session by presenting us with 4 problems and asked us to have a think about how a student would approach them. I particularly like the Matchless activity from nRich, available here which I think could promote a nice discussion on how to work out if you have enough information to solve a given problem – and with older pupils it could provide a nice opener to the world of under-determined systems in Linear Algebra. The activity concerning plane turn around times from Bowland Maths was also great, and I’m ashamed to say i hadn’t seen this one before. The activity is here and I think it would be fascinating to see how different pupils approach this task, and how long it takes them to realise that some tasks are independent of others.

I was very interested to hear Dominic talk about the week long problem solving activities that they have been trialling at Heanor this year; this is very similar to the few weeks of problem solving that we tried with Year 8 last year. It seems to have gone well from what he was saying and it’ s nice that he has shared the activities for us to try out too!

Malcolm Swan presented the following Framework for Designing a Balanced Mathematics Curriculum

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I don’t think I was alone in feeling a bit sad that in my classroom I spend a lot more time focussing on the top section of the table than on the rest. This is a real shame as the true nature of mathematics is not just becoming proficient at calculations!!
Malcolm, also gave us a copy of a conference proceeding “Designing tasks and lessons that develop conceptual understanding, strategic competence and critical awareness” from a conference talk he gave in November 2014. Unfortunately I haven’t got an electronic copy to link to here.
A key message that I took from this paper is that a problem solving exercise requires that a solution method is not known; indeed a question where a “problem” is given to be solved, but at the same time requires students to implement a given approach is an “illustrative exercise” . I immediately thought of the “Problem Solving” questions at then end of each double page spread of the new MyMaths textbooks, where it is quite clear that the questions are normally no more than worded questions about a particular topic.
I will be trying out some of the real-life graph activities that Malcolm discusses in this paper this week with my Year 9 class (these are available from the Math Shell website here).

I really value these meetings and hope they continue for longer than just this academic year!

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FMSP Favourite Problems – Number 1

In a fairly shameless copy of the style of some of @srcav‘s posts where he solves a maths problem and briefly goes through his reasoning I have decided to do the same with some (easier) problems that I have recently discovered.

The Further Mathematics Support Programme (FMSP) have produced a series of 6 posters of problems that should be accessible to GCSE students. The first one of which is a fairly nice area problem – the poster is available here and shown below.

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As always, when solving any kind of geometric / visual problem I find drawing a diagram (even if there is a printed one in front of me) helpful, and I then mark on everything that I already know.

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Each distinct area I have labelled with a number (maybe lower case letters or greek letters would have been a better choice, but never mind!) and I now proceed to write down as much as I can quickly calculate. Here, that involves a bit of Pythagoras’ theorem and the formulae for working out the area of a right angled triangle:

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Now comes the slightly harder bit, where I actually had to think and work out how I could find the shaded areas that I was interested in using only what I had already worked out. It was actually fairly easy, as soon as I had written some things down!!

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The result is lovely, in that the area of the shaded regions is the same area as the triangle

Overall, I think this is a nice, accessible problem that would get students to think.
As, an aside this problem demonstrates (though it isn’t immediately clear from the picture) the fact that Pythagoras’ theorem can be extended for other shapes attached to the sides of a right angled triangle – in this case semi-circles.

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Nottingham Lakeside’s George Green Exhibition

This is a post that I should have written months ago when this exhibition opened. Unfortunately it is closing on Sunday, it’s open 12pm-4pm tomorrow and Sunday, the details are available here. If you are local to Nottingham and have some spare time it is well worth a visit.

George Green is probably one of Nottingham’s most famous residents. The science library of The University of Nottingham is named after him. He was a miller based in the Sneinton area of Nottingham, he had little formal education but published a paper (by subscription) in 1828 where he presented the following theorem, given below in modern notation.

Green’s Theorem
Let \( R \) be a simply connected plane domain whose boundary is a simple, closed, piecewise smooth curve \(C\) oriented counter-clockwise. If \( f(x,y) \) and \( g(x,y) \) are continuous and have continuous first partial derivatives on some open set containing \( R \), then


\( \int_C f(x,y) \mathrm{d}x + g(x,y) \mathrm{d}y = \int \int_R \left ( \frac{\partial g}{\partial x} – \frac{\partial f}{\partial y} \right) \mathrm{d} A
\)

This theorem has had wide ranging influences, but at the time received limited exposure due to the local nature of the publication and it not being published in a scientific journal.

An interesting fact about this paper is that Green used Leibniz’s notation for the calculus as opposed to Newton’s notation which was in common usage in England at the time.

If you want to learn more about George Green the following two videos are a good place to start:
Maths history trail of Nottingham – George Green
Sixty Symbols – George Green

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Verbalising mathematics

One thing that I think is very important to develop in students is their ability to describe mathematics, both verbally and in written form. I think that the ability to describe their thinking in a mathematically correct and concise way is a skill that can significantly improve the transition from GCSE to A Level in addition to the transition from A Level to University.
I recently stumbled on this activity on the Mathsbox website.

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I used this activity as a starter for a lesson with my Set 1 Year 8s. I gave them 10 minutes to work in pairs and write as much as they could, using the terminology we had been learning, to describe the figure in front of them. We then discussed as a class the various things they had come up with. I was really impressed with the language that my class were using, and it also allowed me to tackle some misconceptions with labelling angles, angles on a straight line etc. The key words at the bottom of the page were great for differentiation as some of the students didn’t even notice they were there and managed fine, but I could point them out to the pupils who were struggling.

I can’t recommend Mathsbox enough. I persuaded my school to buy their very reasonable subscription this year and they have some great activities for use as starters. Follow them on Twitter here. They also seem to have an open house for half an hour tomorrow (Sunday 30th November 2014) where you can try out their resources and activities!

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Exams and tests

I’ve just been marking some Year 7 end of term tests and I wanted to get people’s views on tests split up in to tiers.

I did two tests from different tiers with my class, and inevitably the levels reported by the tests don’t vary wildly.

To me, the whole purpose of summative testing is to get robust data. Why then do we not use as a matter of course one test that covers all possible levels, make it a bit longer and have everyone in a cohort sit the same exam paper. This approach would at the very least get rid of the “which paper will my students be able to get a C on more easily” debate at GCSE level. I also think that doing this throughout will lead to more robust data which is actually useful.

I’ve heard some arguments for having different tiers, such as students having lots of questions they can’t do being bad for self esteem.

What does everyone think? Do you have a preference?

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TexWriter app for iPad

I am aiming to transition away from using a laptop at work, ideally going to using an iPad for almost all work whilst I am in school.

There have been a few sticking points that have prevented this, mainly

  • Lack of a decent LaTeX editor on iOS.
  • Inability to compile computer code natively on an iPad
  • Lack of equation support in Microsoft Office applications for iOS

Finally, I have found an app that can do most of the LaTeX compilation I want. In the past I have used TexTouch with dropbox syncing to compile remotely on my Mac, however the delays of sending files, waiting for compilation, copying back the pdf made this a fairly unwieldy approach, suitable only for quick and minor edits.

Recently I have discovered the TeXWriter app (http://www.texwriterapp.com/) that can compile LaTeX natively in the app without the need for an internet connection. First impressions are good; on loading the app you are presented with the following window:

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To compile your code it is a simple case of pressing the play button in the top right hand corner. Once compiled the output is viewable, as is the full LaTeX error report.

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I have tried compiling some fairly involved documents, drawing in many different libraries and TexWriter seems to handle it well. The only downside with this app is the time it takes to compile. A set of lecture slides that take less than 5 seconds to build from scratch on my MacBook Air takes over 3 minutes on my iPad. However this is an iPad 3 with only an A5X SoC, I’m looking forward to examining the performance boost when I get an iPad that is running with an A8 SoC.

I am also investigating the writeLaTeX service that runs in a web browser, though this does need a working Internet connection to compile. I’ll make that a subject of a further blog post in the future I think.

I always thought that Apple’s App Store terms and conditions prohibited code compilation, I’m curious to know how TeXWriter have got round this?! Unless it is because you aren’t generating an executable program…

At the moment TexWriter is definitely the best solution I have found for compiling LaTeX on the go on the iPad, however I need to see significant improvements in compile time before it replaces compilation on my laptop.

If anyone else has used TexWriter please let me know what you think.

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Explaining what maths is in a tweet…

During this weeks #mathschat @LearningMaths posted something like “it comes down to what we think maths is”.

This gave me the idea of trying to crowd source a definition based on peoples tweets. I follow a wide selection of mathematicians, maths teachers, computer programmers so I think I could get some interesting responses.

Please tweet to me (@DrBennison) your definition of what mathematics is. The only rule is that it must be no longer than a tweet, i.e. 140 characters.

My attempt is the following

A creative discipline with problem solving as a focus; to expose truth and beauty in the world around us in a rigorous, reproducible way.

Sounds a bit ….. I’m sure you can all do a lot better than me! I look forward to seeing your tweets.

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Everything We Know Is Wrong

On Tuesday 26th August 2014 BBC Radio 4 presented a programme looking at the validity of scientific research. This programme is Everything We Know Is Wrong, available on BBC iPlayer here.

Jolyon Jenkins presents, among others, the results from some research done by Dr. John Ioannidis of Stanford University. His paper “Why Most Research Findings are False for the journal PloS Medicine surveyed 50 of the most cited (more than 1000 times) papers in medical research. For each paper, he tried to find if there was a subsequent study that was larger to back up these cited papers. 5 out of 6 of the papers whose results had been based on non-randomised data had been proved to be wrong, yet these papers were still being cited. In addition, about 25% of the claims based on randomised data have been shown to be wrong or widely exaggerated.

This issue of reproducibility is important to me, and something I feel is a major problem for educational research. As a numerical analyst, generally when results are presented in a paper it is possible to go away, write some computer code and reproduce the computations / results, checking their validity. This does not seem possible with any of the educational research papers that I have read. For a start data is often anonymised (even if it is from the public domain) and so cannot be verified. Not only this, there are so many factors at play in a classroom it is hard to strip everything out apart from the factor the researcher is interested in. Another problem is mentioned in this programme: many papers seem to use very small samples, to me this throws into question the power of the test. Has the conclusion presented in a paper really been found, or is it just by chance.

Of course, their are ethical considerations (leading to the use of anonymity) and cost considerations at play here, the larger a sample, the larger the cost of the study and os the less likely it is to be funded, especially if the outcome is unclear.

John Ioannidis also discusses the influence of the culture of scientific research in what is published. Having spent a few years of researching a topic, it is not ideal to have nothing to publish, and so, if the original outcome hasn’t been realised researchers may start looking for something in the data to publish, even if the study was not designed to investigate this. This I am sure happens in the field of education.

Academic researchers, in all fields, build their career on their publications. This leads to a pressure to publish results (potentially limiting the time spent verifying a theory) and also means that previously published results are very rarely checked – You are not going to get a blockbuster publication by re-doing the work of others and verifying it. Even if some work has been checked and shown to be wrong, this fact may not be published. Suppose that a new researcher, R, has shown that a result due to a big name in the field doesn’t hold. How would publicly criticising an acknowledged expert in the field affect this researcher’s career? The big name may even be on the refereeing panel of R’s paper – is this going to be fair?

All of these issues should be considered when reading a piece of educational research. It is important to not take research at face value without first passing a critical eye over it.