Teaching Uncategorized

MathsHUBS and Steel Cables

I’ve been pretty slow writing about this but on Wednesday 11th March Mathshubs East Midlands West held this half term’s Secondary Curriculum Development Meeting. 

Aswell as being a chance to have a few cakes (I particularly like the granola bars topped with strawberry jam and seeds) Matilde Warden (@jordanvorderman), the maths lead for the East Midlands West Maths Hubs discussed improving reasoning in lessons.

She started by highlighting aspecs of the new national curriculum concerning mathematical reasoning and problem solving before we looked at a couple of problems from the nRich website, specifically the Stage 2 problem Fitted and the Stage 4 problem Steel Cables. The nice thing about these is that the nRich website (along with all the others in this collection) is that they show a few approaches to the problem that some students have tried. I had seen the Steel Cables problem before, but had never really considered different approaches before – the default to me was to find the quadratic nth term rule to predict all the terms. 

We also briefly looked at an article by Malcolm Swan, also available on the nRich website here. In the past, when doing problem solving lessons I confess that I normally spent only one lesson on them, and valued the thinking during the lesson and the verbalising of mathematics that happened with this. However I probably didn’t build on this terribly well to develop pupil’s problem solving approaches and build resilience. 

In the article, Malcolm suggested a two lesson approach where for the first lesson pupils work individually on a problem without help. At the end of the lesson you collect in the work and look through them (but don’t formally mark them!) so that you have an idea of how to move their thinking forward. Then, in the second lesson allow pupils to work in pairs to share their thinking, prompting them if necessary to move thier thinking forward. 

I decided to try this approach with one of my Year 8 classes and chose to look at the Steel cables problem – we had done quadratic sequences before Christmas so was interested to see if anyone would go along those lines. A couple of pictures of pupils work are below (the first is at the end of the second lesson, and the second piece isfrom a different student at the end of the first lesson)



Lots of different approaches were used, though drawing them out and counting methodically was a dominant approach. However the second picture shows the first person to answer the “how many strands in a size 10 cable” question, and she noticed that by counting in rings outward each new ring contained 6 more strands than the previous ring, and used this to work out the number of strands for a size 10 cable, before spotting that the pattern was related to the 6 times table. I also saw pupils spot the vertical symmetry of the cable and split the cable up into large and smaller triangles. Strangely though I didn’t see anyone split it up into quadrilaterals like one of the sample pieces of work on the nRich website. 

Malcolm also suggests showing sample work to pupils and getting them to critique it, and develop it further. Unfortunatey I didn’t have time to do this, though this is something I intend to do in the future.

He ends his article by suggesting that teachers let pupils see their reasoning and tackle a problem unseen oon the board. This is something I try to do every lesson with my sixth formers. I enjoy tackling a question that I haven’t though about before, I think it is important for pupils to see me struggle with arithmetic sometimes and try iincorrect approaches and double back on myself. After all, making mistakes is really what maths is about.

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. 

A Level Software Teaching

Matlab in the Classroom

One of the things I have missed since teaching in a school is the fact that I don’t have access to Matlab….

Matlab is a product from The Mathworks, designed originally for numerical computation (indeed the name derives from MATrix LABoratory as the software was originally focussed towards operations on matrices) but which now is capable of much more. More information about Matlab and The Mathworks is available here.

I had got used to using Matlab in teaching an demonstrating, not least because of the ease with which you can use it to check matrix computations, plot graphs of functions and demonstrate numerical methods.

Last term I set my Year 12 Further Mathematicians some homework which involved them researching the Jacobi Method and performing 2 iterations of this method on a \(2 \times 2\) matrix system. I wanted to be able to demonstrate this straightforward numerical method to them, without having to work out how to use the Python IDE on the Windows computers at school, so I decided to give Matlab Mobile a try.

Matlab Mobile was originally released in 2010 (see here for an early blog post discussing Matlab Mobile) but I hadn’t had much reason to use it before. Originally the app needed to connect to your computer with a running copy of Matlab, but since it’s release it has become a lot better. Th app is currently on version 4.1.1 which is a universal app suitable for both the iPad and the iPhone. With this version (as long as you have a valid Matlab Licence for desktop) you can connect to a copy of Matlab running in the cloud to execute commands. You can also upload your files to the cloud so that they can be run from the app.

On opening the app you are presented with a screen that looks like this:


The left hand portion of the screen displays commands that you have typed and on the right hand side you can toggle between Figures and viewing your command history.

Using a keyboard (I recommend using a bluetooth keyboard here as the on screen keyboard takes up so much space) it is easy to type short commands that accomplish complex things. For example, in the screen shot below, I create a matrix and then find its eigenvalues using the eigs command. As you can see this command can also return the corresponding eigenvectors of a matrix simply by explicitly giving two output arguments to the function.


One of the great uses of the Matlab desktop product is to prototype functions before coding in another language. Unfortunately, the mobile app does not allow you to edit or create .m script files or function files. This is a real shame, especially as it extends to the inability to make small edits to files you have already stored in the cloud. Luckily though, you can at least view the test of functions that you have written by using the command type in the command window, as shown below.

The code shown above is a naive implementation of the iterative Jacobi scheme for solving systems of linear equations – I stress it is not the most efficient way to code this algorithm, but it is clear for teaching purposes. Using Matlab Mobile I was able to discuss this short function with my class, and relate it to the algorithm that they had to research and then express in pseudocode.
If you are going to use Matlab Mobile for a similar purpose I would recommend avoiding writing script files and instead use function files with enough arguments to allow you to vary everything you will want to demonstrate the affect of. I would also say it is probably useful to print more of the internal workings of the function to the command line than you would normally.

Another great piece of functionality in Matlab is the ability to produce professional looking graphs with very little effort. You can still generate figures such as the one below (a visualisation of the Barnsley fern after 10000 iterations – I may write a blog post about this in the future) in the Cloud, but a lot of the data exploration tools in the figure window of the desktop product are not usable. You can however, save the figure to your camera roll or email it to yourself.

As a final example of the graphical abilities of Matlab Mobile the 3D surface plot below was created using 4 lines of code.


One important thing to note is that I couldn’t connect to the Matlab Cloud through my schools wifi (I suspect because of the firewall) and so had to resort to using my phone as a 4G hotspot.

All in all, if you have a Matlab licence it is well worth familiarising yourself with the Mobile app, and I managed to do all that I wanted to do in the classroom with it. I just needed to do a bit more forward planning than I would have done if I had had access to the fully fledged Matlab product! Hopefully as the app develops they will remove some of the limitations that are currently present.

A Level Books Teaching

Classic Maths Books 1 – Bostock and Chandler textbooks

This week I posted the following picture on Twitter:

A few people retweeted, favourited and commented how good they were for questions to stretch A Level students so I thought I would write a bit more here.
They were first published in 1982 and are still available from Amazon here and here.
I think I got my copies when I was doing my A Levels and asked my teacher if there were any other books he would recommend that weren’t the Edexcel ones.
I think they are still fantastic books, vastly superior to the Pearson published textbooks, though I think some students may find them a bit dry. They certainly assume more knowledge than the current books and the later questions are generally more involved (or it is less clear on how to start them). There are some really nice examples in the book, such as:

  • Find the complex roots of the equation \(2x^2 + 3x + 5 = 0 \). If these roots are \( \alpha \) and \( \beta \), confirm the relationships \( \alpha + \beta = – \frac{b}{a} \) and \( \alpha \beta = \frac{c}{a} \)

All these examples are well explained, and it is easy to follow them line by line.
One other aspect of them that I appreciate are the multiple choice questions at the end of each chapter; these are great for a quick test of understanding and I am using some of them in some QuickKey quizzes that I am trialling this half term.
I may make a blog post about a classic mathematics book a fairly regular thing… gives me an excuse to get some of the older books off my shelf again 🙂



A lot of people seem to be doing a #Nurture1415 post – Sue Cowley (@Sue_Cowley on Twitter) has compiled a long list here.

So here are my, hopefully short, 5 positives and 5 negatives.

5 Positives from 2014

  1. I saw the final completion of my PhD in applied computational mathematics by graduating from The University of Nottingham
  2. I joined Twitter, properly. I had a while a go signed up to Twitter, but never really used it. This year I decided to set up a professional Twitter account (@DrBennison). I was a bit dubious at first, but I now use it all the time. Everyone on Twitter is very welcoming, happy to get involved in discussions and I have learnt a lot from everyone I have interacted with. It was particularly good to meet people I had spoken to on Twitter at Mathsconf2014.
  3. I started working at a great school, with a great department that is very supportive of each other.
  4. I also started a blog in the summer, which I have enjoyed updating!
  5. Doing maths by hand. It has been nice to practice integrations etc by hand again. I had become very reliant on Mathematica over the last few years.
  6. First wedding anniversary. It has been a great first year married to my beautiful wife and I feel very lucky to always have her support.

5 Wishes for 2015

  1. Develop my website properly. At the moment my website just hosts my blog, a few files and a Javascript game – I need to spend time designing a proper homepage.
  2. Do some coding. After 5 years of programming everyday I have been missing doing some programming. I have two projects I would like to work on: 1) an iFEM finite element iOS app; 2) a useful online computer based assessment and grading system. Hopefully if I am disciplined in school holidays I can make a start on these.
  3. Blog more frequently. I’ve learnt loads in the last six months reading other people’s blogs and updating mine, however my posts have been infrequent. My goal for 2015 is to post at least one decent post every week – fingers crossed I can manage this.
  4. Continue learning from colleagues at work, people on twitter and in any other way so that I can be the best teacher that I can be.
  5. Manage a good work-life balance

You got one bonus positive.. sorry for not sticking to the rules.


Why teach?!

So, I thought as a good way to introduce myself to people who may not know me that way it would be good to describe why I have gone into teaching…

During my School Direct year I had to write a short assignment on “My Starting Point as a New Teacher”, I gave this a quick re-read before writing this post. Whilst finding it, on a re-read, all a bit woolly it did convey the main reason why I feel the teaching of mathematics is so important.

To me, mathematics is the truly universal language that can be used to explain the world and establish truth amongst conflicting media reports. I am passionate about equipping students with the skills necessary to use mathematics productively in their every-day life as well as seeing the inherent beauty of mathematics.

I studied for a PhD in Applied Computational Mathematics at the University of Nottingham and during this time I found the time I spent teaching and running outreach events with local school students far more rewarding than my actual research (I was looking at numerical methods for the Neutron Transport Equation – I may do a post about it one day!). Because of this I decided to apply for a school direct post at a school in the East Midlands, where I have been lucky enough to obtain a post from this year too.

From my experience of tutoring undergraduate students and lecturing postgraduate students (across all disciplines) in quantitative methods one problem that you notice is that British students are not as prepared for a mathematics degree as their international peers. This is of course a sweeping generalisation, however, as a teacher I want to work towards developing the independent study skills required for students to successfully engage with a mathematics degree course if that is where their studies lead them. I believe, very strongly, that mastery of the basic mathematical skills is very important for all students, I aim to develop this mastery in my lessons while still maintaining interest.

Having said all the above, I guess the main reason why I have decided to go into teaching is that I love it! I hope this blog will reflect my love of both teaching and mathematics. I also hope that it will prove useful to other people as well as being a development tool for me.