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Math(s) Teachers at Play 83

Earlier this week I was honoured to hear that Stephen Cavadino (@srcav) had featured my post on conics with GeoGebra in Issue 83 of the Math(s) Teachers at Play blog carnival. 

The Teachers at Play blog carnival was started by Denise Gaskins (@letsplaymath) who regularly blogs herself. I can’t believe I didn’t know about the existence of this carnival until this week! There some great posts in this issue, and I have spent a lot of time this weekend reading through some posts from previous editions. 

Please do check out this issue here.

Thanks again for including me. 

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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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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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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?