Friday, 1 May 2020

Effective questions when using dynamic graphs

Most graphing tools have a feature to add a variable constant in a way that can be changed dynamically. In Desmos and GeoGebra this is by using sliders; in Autograph it is via the constant controller. When the parameter is changed you can view its effect on the graph and this can be discussed with students. You can even animate the change so you don't need to do it manually!

This feature of being able to dynamically change a variable constant is a very powerful tool to help students build their understanding; however, to make the best use of it, it is useful to have some questions to ask to direct students' thinking.

Effective questions when using dynamic graphs

  • When I vary … how does it move and why?
  • For what value of … will …?
  • If I vary … how will it move?

Examples using the intersection of a line and a parabola

To exemplify this I'm going to use the intersection of the parabola y=x2 with the line y=mx+c. To do this I've plotted:

  • y=x2 
  • y=mx+c
  • (0,c) - this point isn't strictly necessary but it helps students "see" the motion of the line as vertical when  is varied.

You can play with a dynamic version of this in GeoGebra ( or Desmos (
In all of the examples below I want the students to focus on the number of points of intersection of the line and the parabola.

When I vary … how does it move and why?

I've set m=2. When I vary c how does the number of points of intersection change and why?

For the responses to this I would expect students to be able to observe that the number of points of intersection can be 0, 1 or 2. I would hope that some would be able to link this to solving the simultaneously and using the discriminant of a quadratic equation. This would lead to directing them to thinking about the related equation x2mx−c=0.

For what value of … will …?

I've set m=4. For what value of c will the line and parabola have a single (or repeated) point of intersection?
This is a less open question but when used as a follow-on from the previous type of question it allows for students to apply their understanding and obtain an answer. The answer can then be verified dynamically.

If I vary … how will it move?

I've set c=2. If I vary m how many points of intersection will there be between the line and the parabola? Will it be 0, 1 or 2 or will it be always 2?

This more demanding style of question requires the students to be familiar with dynamic graphs and should be used after they've experienced answering questions of the first two types a few times. Being able to predict how varying the constant will affect the graph is a very useful stage in supporting students with generalising.

Monday, 24 February 2020

NCETM article on the use of calculators in A level Mathematics

My article for NCETM: "Ten calculations that A level students should be doing on a calculator: can yours?" can be read at:

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…