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Mathematician or Educator?

A couple of weeks ago I was at an IMA organised Mathematics and Industry event at Rolls-Royce in Derby. One of the speakers posed the question “Are you a mathematician or educator?”. This has led me to think about where I stand on the related  “Mathematician or educator?” question.

Ultimately I would describe myself as a mathematician who teaches. Whilst I’m passionate about good teaching and teaching concepts in a way that everyone can understand I don’t feel that teaching defines my outlook on life. Whereas being a mathematician in some way seems to define how my brain works and how I look at the world. 

I think it is easy to forget the mathematician in the day-to-day world of teaching but I think I owe it to my students to let them see the mathematician side of me and not just the teacher side of me. 

I’m really interested to see what other maths teachers think, particularly those who didn’t do a maths degree. Please leave a comment or message on Twitter 🙂 

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A Level Software Teaching

Geogebra and Further Maths

This week I had an observed lesson, where the remit was that I do something risky that I perhaps wouldn’t have done otherwise.

My Year 12 Further Maths class had been selected as my observation lesson – I found thinknig of something risky to do with them harder than if it had been any other group. They are used to me doing odd things, that are a bit off the wall with them and they always respond well. In the end I decided to use ICT and try to get them to do some independent discovery work.

We have just started the coordinate geometry section of FP1 (Edexcel) and wer due to look at the parabola this week. I have used GeoGebra a bit in the past but I had never created my own geogebra worksheet before and this seemed like the perfect opportunity.

So, one evening I installed the latest version of GeoGebra on my MacBook, signed up for a GeoGebra Tube account and set about creating a worksheet where students would be able to explore the parametric equations of a parabola and the focus directrix property for themselves. I then uploaded the file to GeoGebra Tube, and got the content ID from the embed option. As I didn’t want to rely on GeoGebra being installed / working on all the computers in my teaching room I decided to upload a html version to my own webpage. The GeoGebra tean have made this really easy by poviding a javascript library that you can just source at the top of your html code and then a really simple API to embed a dynamic worksheet in your webpage. They have provided examples of how to do this.The web apps for the Parabola and Hyperbola that I created are here. On loading the page you should see a screen that looks like this:

I then wrote a sheet with some questions to guide the students’ explorations here, these questions should prompt them to derive the parametric equations of the parabola and notice the focus-directrix property. It is significantly harder to answer the questions concerning the hyperbola – I saw these as hard extension questions.

All of my class seemed to enjoy using these and engaged well with the work. Walking round the room I also saw some great responses to questions. 

To produce the worksheets, upload to GeoGebra Tube and then host on my website took in total about 2 hours which I don’t think is bad for a first time. 

My intention is to use GeoGebra moe across the keystages, any worksheets I create I will share through my website as well as GeoGebra Tube. The original GeoGebra files (in case you want to modify them) are here (parabola) and here (hyperbola). I will also be adding a worksheet for the Ellipse to the webpage later too. 

Some examples of students responses to the questions are shown below:





Update: Please be aware that for the web applets to load there has to be communication with the GeoGebra website, so make sure that is not on your schools block list. 

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Nottingham NQT Differentiation

On Thursday this week I went to the Nottingham University School of Education NQT session on differentiation. 

I was quite concerned at the beginning when someone presented slides titled “Differentiation by Gender” and “Differentiation by Learning Style” but the subject specific discussion was your interesting. In true ITT style we had to summarise our thoughts on a big piece of paper resulting in this:





The Venn diagram exercise I have used before with KS4+ but I’ve been inspired to try something similar with KS3 now. 

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Problems Teaching

My First @solvemymaths Problem

This morning I woke up to the problem shown below by the excellent @solvemymaths on my Twitter feed.

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I’ve always been very lazy and not properly solved any of his puzzles before, partly because I have a strange irrational fear of geometry. Wanting to conquer this fear I thought I’d have a quick go at this.
It turned out to be quite straight forward really (although I did make a silly mistake and got an original answer that seemed odd). My workings with my mistake are shown below:

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Books

Classic Maths Books 2 – Numerical Solution of Partial Differential Equations by the Finite Element Method

This book isn’t actually that old, it was first published in 1987, the edition I have is a reprint by Dover Publications.

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For my PhD I studied the application of a particular class of finite element methods for the solution of the neutron transport equation. Finite element methods are a way to numerically solve a partial differential equation – maybe I’ll write more about this in the future. It works by discretising the domain into regions known as elements and then producing an approximate solution on each element. Importantly, these approximate solutions converge to the true solution as the mesh is refined and the elements become smaller.
In the field of finite elements, it is widely regarded as a classic, and is often the first thing that someone new to the area reads, indeed it was the first published book on finite elements that I read. This is due to the relaxed and easy to read style of the book.
There are the following 13 chapters:

  1. Introduction
  2. Introduction to FEM for elliptic problems
  3. Abstract Formulation of the finite element method for elliptic problems
  4. Some finite element spaces
  5. Approximation theory for FEM. Error estimates for elliptic problems
  6. Some applications to elliptic problems
  7. Direct methods for solving linear systems of equations
  8. Minimization algorithms. Iterative methods
  9. FEM for parabolic problems
  10. Hyperbolic problems
  11. Boundary element methods
  12. Mixed finite element methods
  13. Curved elements and numerical integration
  14. Some non-linear problems

A really nice thing about this book is the mix of theoretical analysis and implementation details it provides. After reading this, you would be able to go away and code up a simple two dimensional finite element method. The book is well illustrated which helps with the understanding of concepts such as ‘basis functions’

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As well as discussing finite elements the direct methods for solving linear systems is a fantastic introduction to this important area. The efficient solution of linear systems of equations is vital for any numerical solution of a real life problem that has been modelled by partial differential equations. This chapter serves as a great introduction to this topic (which will be considered more in future posts, and in a future classic books post). The chapters also contain a few stimulating questions, such as these from the linear systems chapter.

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I think it is really nice that this book is written from a computational viewpoint (in addition to being mathematically rigorous) as after reading it, you can go away and code up the discretisation and a solver to try the methods out.

I would recommend this book to anyone who wants to learn about finite element method. The author did write an expanded book after this with some other authors, but it doesn’t read as nicely, and hasn’t generated the wave of good feeling in the finite element community as the original. In fact the sequel it is colloquially referred to as “The Black Death” (supposedly a term coined by one of the authors) as it is so bad!!

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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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FMSP Favourite Problems Teaching

FMSP Favourite Problems – Number 2

Here is the second post in my ‘FMSP Favourite Problems’ series looking at the problems selected for the Further Maths Support Programme’s favourite problem series of posters.
This second problem I’m not very keen on as it just seems a bit dull.

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As usual I started with a small sketch, annotated with what I knew – that fact that at every bounce the ball rebounds to a height 75% of the previous bounce.

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As this is repeated for every subsequent bounce you can easily obtain a relation between the height of the initial bounce and the height of the \(nth\) bounce:

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Clearly this is easily solved using logarithms

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The answer above isn’t the full answer, since you don’t have decimals of a bounce, the ball must be on the 9th bounce before it rebounds to less than 10% of the original height.

Unfortunately, the typical GCSE student will not have come across logarithms – though I sort of feel that logarithms should be in the GCSE syllabus – and so couldn’t solve it this way. Unless I have missed something obvious, the only way that they could solve it would be to either calculate the powers of 0.75 until they got the correct value or to plot a graph. Not wanting to plot a graph by hand, I quickly loaded Matlab on my iPad and plotted the following:

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In conclusion, as a problem I don’t find this particularly satisfying, however I think it could be used as an opening to an interesting lesson discussing exponential decay, mathematical modelling, ensuring answers you report are meaningful and interpreting graphs.

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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.