Wednesday, 17 July 2019

Slice of advice 2019: What I have learned this year

I've been asked again this year to contribute to Craig Barton's end of year podcast: Slice of advice - what have I learned this year:

The main thing that I've learned this year is how few teachers are familiar with Richard Skemp's ideas on instrumental versus relational understanding. I meet a lot of teachers in my job and I will often have only one or two teachers in the room who have read it. These ideas have been fundamental to my development as a mathematics educator and I see it as the key concept that is most illuminating when considering ideas about teaching and learning in maths. It is also one of the main reasons why I am so passionate about the use of technology in maths.

Instrumental versus relational understanding

Skemp published his ideas in the 1970s but they are still available via the ATM website at:

The paper contrasts two meanings to the word "understanding" that are used in relation to maths:
  • Instrumental is just teaching the "how" of mathematical methods. Most commonly this will be when a method is presented to students, usually by working through an example, and then their activity is to be able to repeat this "recipe" for a number of different examples.
  • Relational understanding emphasises the "why" of mathematics. Teaching for relational understanding focusses on the students being able to explain why mathematical methods work, especially by making connections, or relations, with other mathematical ideas.
For example, an instrumental approach to learning completing the square would be based on showing the students the mechanics of the method on particular examples and having them repeat this method themselves on some further examples. A relational approach would focus on relationship with the transformed graphs of quadratic equations and, in particular, the special cases of quadratic equations with repeated roots.

Instrumental understanding is shallower whereas relational understanding is longer lasting and more easily applied in solving novel problems.

Using technology to support relational understanding

There are a number of reasons why I believe technology, especially dynamic graphing software such as GeoGebra or Desmos, is so powerful in supporting relational understanding. The main two of these are:
  • it constantly reinforces the link between algebraic, graphical and numerical representations of mathematical objects;
  • it gives us the power to create many instances of a situation very quickly so that we can observe and generalise patterns.

An example: straight lines that intersect a parabola at one point

As an example I am going to think about the following problem:

What are the possible values of m and c so that the line y=mx+c has
a single repeated point of intersection with the curve y=x².

If you've not met this problem before then you might like to explore it in a graph-plotter yourself.

By putting this into a graph plotter it is fairly quick to find a number of different cases where the line cuts the curve at a single point (i.e. a tangent). These cases can be listed and observed algebraically, graphically and numerically. 

One way of solving this is based on using completing the square (I'm assuming that this is a task being explored by students before they've met calculus). Being able to observe that completing the square is applicable here is difficult for a student that has been taught the method instrumentally compared to a student who has a strong understanding of the relationship between completing the square and quadratic equations with repeated roots:

Skemp is still relevant

Although these ideas were first published over 40 years ago I think they are just as relevant today. There are increased demands for reasoning and problem solving at both GCSE and A level and teaching for relational understanding is key to supporting students with these. If you've not read Skemp before I'd suggest doing so and if you have I'd suggest a re-read. 

Wednesday, 15 May 2019

Masters assignment: Using Dynamic Geometry/Graphing software to set students open-ended mathematics investigations

In 2011 I completed an MA in ICT and Education through the University of Leeds. My critical study was on "Using Dynamic Geometry/Graphing software to set students open-ended mathematics investigations". It can be downloaded from:

Friday, 8 March 2019

Neverware CloudReady: turning old laptops into Chromebooks

Over the past week I've been exploring CloudReady from Neverware. The idea behind it is that it takes an old laptop and effectively converts it into a Chromebook.

I took an old Windows laptop (I'm not sure of the exact age but I think it was about 10 years old). It was running so slowly that it was practically unusable. I replaced it with their operating system, which is based on the Chromium OS, to turn it into a Chromebook. The process was pretty straightforward - it required me to make an installer on a USB stick and then I ran the installation. Overall it took less than an hour but it would be quicker if I were to repeat it. It was a full replacement of the OS, but given that the laptop was destined to be disposed of, this wasn't a problem.

I've now got a fully working Chromebook and I'm really impressed with it. It runs really quickly - the browser is at least as fast as my current Windows laptop and I can watch/stream videos on it clearly. You can see a picture of it below running the GeoGebra app that's available for Chromebooks.

The only issue I had was that it initially wouldn't connect to WiFi. This wasn't a problem with CloudReady - the WiFi was temperamental on the laptop when it was running Windows. I ended up buying a USB WiFi adapter and it now works fine. I found it difficult to find any information online as to which adapters were compatible with Chromebooks - a couple of forums suggested that I needed 802.11n so I bought this one: and it worked immediately.

Overall I think this has the potential to be really useful for schools - if they have old laptops that run really slowly then they can convert these to fast Chromebooks. It's free for home users and £10 per device, annually, for schools. You can see more details at:   

Monday, 16 July 2018

Slice of Advice: What have I learned this year?

I recently submitted a short clip for Craig Barton’s podcast on the theme of Slice of Advice: What have I learned this year?  In this I suggested two things that had had an impact on me this year: the use of graphing apps on smartphones and diagrams representing functions as mappings between number lines.

Graphing apps on smartphones

I’ve been using GeoGebra for over ten years now and it has had a huge impact on how I think about mathematics and teaching mathematics. I’ve also been massively impressed with Desmos though I don’t know it inside-out in the way I do with GeoGebra. My use of these graphing tools, and Autograph before it, was always based on using them on a computer and this means I have a preference for the computer software over the phone apps: I find them much more usable on a large screen with a proper keyboard and mouse.  This works fine for me but, as I’ve commented on here before, I think the greatest impact of technology in the mathematics classroom is as a result of students using it. Consequently I’ve long been an advocate of the use of graphing apps on phones by students, especially in A level classrooms. However, even though I’d promoted the use of phones I had always thought that this was an inferior device for students to be using the software on and that the ideal would be to have them using the software on laptops if it were possible.

What has changed for me this year is that I no longer consider phones an inferior device for using the software for students. Today’s generation of students have grown up with small touch-screen devices and find it completely natural to operate software in such an environment. For many students their phone is the device they are happiest using and so the opportunity to situate such fantastic bits of software as GeoGebra or Desmos in this environment is one that we should be capitalising on. The quality of modern phones is such that the apps work beautifully on them. In particular I think the GeoGebra 3D app on Android phones renders the image better, and is more responsive, than the equivalent computer-based version.

This year we recorded some more PD videos for the Further Mathematics Support Programme – short 5 minute videos that address a particular bit of pedagogy. In one of these we looked at using graphing apps in the Further Maths classroom. In this video you’ll see students using them and hear the teacher comment on how surprised he was that the students didn’t find the small screens an impediment to them using the app effectively. This video, along with questions for reflection, can be seen at 

At MEI we’ve developed a series of classroom tasks that are designed to be compatible with the phone apps for both GeoGebra and Desmos. These can be found at and

Representing functions as mappings between number lines

At the BCME conference at Warwick in April I attended a talk by Martin Flashman, a visiting professor from the US, about the use of mappings between number lines for graphical representations of functions. This is in contrast to the usual convention of plotting them on a pair of cartesian x-y axes.

This was an idea I’d seen before but I hadn’t really thought about it in detail. It was a revelation to see how a change of representation could give whole new insights into mathematical ideas I thought I’d understood. In particular I think the representation of composite functions and their inverses is much clearer in this way. I was also amazed at how this could be related to the cartesian system: for example a “linear” function is the set of points that lie on a line; if this is transferred to a mapping between number lines this becomes a set of lines that pass through a single point. What is more, these two representations are the duals of each other!

Martin went on to give a lot more detail to how this system could be extended to represent ideas such as calculus and functions of complex variables represented as mappings between two planes in 3D space. There is a lot more information, including presentations from his BCME conference sessions, at:

Overall this has had a wider impact on me than just thinking about representation of functions: it has made me consider how the way an idea is represented impacts on my understanding of it. For example I think setting out long multiplication in a grid helps me focus on the two-dimensional nature of a product and I’m now searching for more such examples…

Thursday, 31 May 2018

Simultaneous equations: an insight from playing with technology

I've noticed a couple of tweets recently about the importance of being playful when doing mathematics and this has brought to mind an insight I had about simultaneous linear equations that occurred when I was being playful with them in technology. As a result of this I now have a different method for solving that I  prefer to the standard textbook approaches.

Playing with simultaneous equations

In trying to construct a pair of linear simultaneous equations where the solution went through a given point. I can't remember the point but I'll use (3,2) for the example. I did this by creating a point A at (3,2) then two more points B and C and finding the lines that went through these and A. This gave me the simultaneous equations:
3x + y = 11
x + 5y = 13

I then moved the points B and C around to find some different equations that would have the same point of intersection. This is probably best displayed here by showing all the different lines I got in a different colour.
The equations for these lines are:
4xy = 10
3x + y = 11
2x + 3y = 12
x + 5y = 13
y = 2

By playing around with these I noticed some interesting features:
  • There are infinitely many lines through the point A.
  • The coefficients of these lines are related linearly.
  • One of these lines will be horizontal and one will be vertical (and hence have a simple equation just in terms of x or y).
NB rewriting the final equation as 0x + 7y = 14 makes the linear relationship more obvious.

A simple method for solving linear simultaneous equations 

Putting all this together I realised this could be used to form a simple method for solving simultaneous linear equations. They can be solved by moving linearly along the coefficients until one of the coefficients of either x or y is 0. For example:

3x + y = 11
x + 5y = 13

The gap from 3x to x is –2x, so if you move half this gap again (–x) from x you'll have 0x.

Moving an equivalent amount for the other terms means you need to move half the gap from y to 5y, i.e. +2y, on from 5y to get 7y and half the gap from 11 to 13, i.e. +1, to get 14. 

This gives 0x + 7y = 14 and hence y = 2. Then using the first equation 3x = 9 gives x = 3.

I've tried this method with lots of linear simultaneous equations and I think it's quicker and makes more sense to me. It's the method I know use if I need to solve them.

The importance of being playful

This method only occurred to me because I was being playful with the mathematics in the technology. The dynamic aspect of technology offers a fantastic opportunity to be playful with mathematics as it allows the users to change aspects of a situation and observe the impact on representations (here it was graphical and algebraic but it applies to others such as geometrical or numerical too). The playfulness here is not in the sense of 'lets have fun messing around' but being open to explore a situation and then try to explain what is happening and why.

Friday, 23 March 2018

EEF report: Calculator use has a positive effect on students’ calculation skills

EEF (Education Endowment Fund) have published a report today on "Improving Mathematics in Key Stages Two and Three". The report is a meta-analysis of research into teaching and learning strategies and can be access in full at:

The report has been widely publicised, mainly for what it has to say about calculator use. Although Key Stages Two and Three are outside my area of expertise I think there are useful reflections that can be made with reference to the use of technology, including calculators as well as other tools, in GCSE and A level Maths.

Calculator use has a positive effect on students’ calculation skills 

The conclusion in calculator use states: "When calculators are used as an integral part of testing and teaching, their use appears to have a positive effect on students’ calculation skills. ... When integrated into the teaching of mental and other calculation approaches, calculators can be very effective for developing non-calculator computation skills; students become better at arithmetic in general and are likely to self-regulate their use of calculators, consequently making less (but better) use of them." 

This emphasises the importance of calculator use being integrated into the teaching and learning (and assessment). As with any technology simply adding in a technology without changing the teaching or activities is unlikely to have positive impact on students' understanding. However, if they are used purposefully, with appropriately designed tasks, this is suggesting that students' calculation skills will improve. The challenge is to design appropriate tasks that take advantage of this opportunity. This point is further highlighted by a suggestion that such tasks can enhance students' problem solving skills.

I think much of this will apply at GCSE and A level too. Some classroom activities that can be tried are:
  • attempting the same problem with and without a calculator and comparing;
  • using a calculator to investigate a function numerically, such as sin(x) or ln(x).

Technology: technological tools and computer-assisted instruction

A separate section of the report discusses technology tools other than calculators. These are split into two categories: technological tools and computer-assisted instruction. I am pleased that this split has been made. I think technology tools for learning mathematics have a much greater potential than computer-assisted instruction.

The analysis looked at three types of tools: dynamic geometry software, exploratory computer environments and educational games. Here the report suggests that dynamic geometry software has huge potential but stresses the importance of how these tools are used if they are to have a positive impact and that there is a need for professional development for teachers to keep pace with this change.

I think there are parallels with the use of technology in GCSE and A level Maths; however, the technology tool that is most widely used (apart from calculators) is graphing software. This is probably due to the nature of the GCSE/A level content: there is a much greater emphasis on the behaviour of functions and understanding them through a combination of graphical and algebraic techniques. For graphing tools it is still essential to use them carefully, and in a structured way, if they are going to have the most impact. This is similar to the use of calculators in that care should be taken to design and use tasks that take advantage of the technology and not just replicate what is done with pen and paper in the software. Detailed consideration is required about how the features of the software, such as sliders, can be used to illuminate mathematical concepts and this then needs to expressed through appropriate tasks (and professional development for teachers).

I would be interested to see a similar analysis of graphing software at GCSE and A level. I have definitely found it more difficult to design classroom tasks for dynamic geometry software than I have for graphing software, though this possibly warrants a different blog post.

It's all about the maths!

Reading through the technology sections of this report I'm left with a strong impression that calculators and other technology tools are useful for doing and learning maths and so should be integrated into students' experiences of the subject. This can then have a positive impact on their understanding and skills. Calculators and technology tools are just one of many strategies that can be employed and they should be judged in the same way as any other strategy - can they help students understand maths better?